Mathematical Explanation in Science

by

Alexander Koo

A thesis submitted in conformity with the requirements for the degree of Doctorate of Philosophy

Institute for the History and Philosophy of Science and Technology University of Toronto

© Copyright by Alexander Koo 2015

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Mathematical Explanation in Science

Alexander Koo

Doctorate of Philosophy

Institute for the History and Philosophy of Science and Technology

University of Toronto

2015

Abstract

Inspired by indispensability arguments originating from Quine, mathematical realists such as

Alan Baker argue that since mathematics plays a key explanatory role in our best scientific

theories, then the very same reasons which convince us to be scientific realists should lead us to

be mathematical realists as well. Baker’s enhanced indispensability arguments (EIA) makes use

of inference to the best explanation (IBE) to deliver the realist conclusion. Mathematical

nominalists have resisted this argument by asserting that there is no such thing as a genuine

mathematical explanation (GME) where the mathematics is playing an indispensable explanatory

role. In this dissertation I will argue that the nominalist is incorrect and that GMEs do, in fact,

exist. My methodology will be to develop a set of criteria that clearly defines a GME which the

nominalist would gladly accept. From there, a new example of a GME will be advanced that

satisfies all of the criteria. To solidify this result, I will show that Strevens’ kairetic account of

scientific explanation clearly points to mathematics playing an indispensable explanatory role in

our supposed examples of GME. While this all bodes well for mathematical realism, I will

further argue that the EIA still does not lead to mathematical realism. Baker assumes that using

IBE to infer the existence of mathematical objects is unproblematic. I will challenge this

assumption, and without IBE the EIA does not deliver. By no means does this result block the

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realist project. Ultimately, I believe that freeing ourselves from the yoke of traditional

indispensability arguments and focussing on how it is that mathematics can explain physical

facts explain will advance the realist position even further.

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Acknowledgments

I would like to thank my supervisor, Professor James R. Brown, for all the emotional and

intellectual support that he has given me over the years. Jim was the first to introduce me to the

philosophy of mathematics, and I have found inspiration in both his work and his kindness ever

since. I truly could not have completed this project without his guidance. I would also like to

thank my committee members, Professors Joseph Berkovitz and Christopher Pincock, for their

incredibly helpful comments and the many fruitful conversations that we had. All three have

contributed greatly to my development as a philosopher, and I am a better person for it.

Special thanks to Professors Alan Baker and Mark Colyvan whose works inspired my own

research. I am fortunate to have received helpful comments and criticisms from both, as well as

the encouragement to continue my research. Thanks also to Professor Sorin Bangu for originally

introducing me to indispensability arguments, and for being instrumental in my decision to apply

to graduate school.

This dissertation would not have been finished without the love and support of my friends and

family. Thanks to my study partners, Michael, Agnes, Rebecca, and Anita, who were always a

reliable source for motivation, decompression, and for a lunch buddy. To Jimmy for keeping

things in perspective and helping me unwind before the next work day. To Don for being my

biggest fan. A big thanks to my mom, Sunda, for all her love and support, and for all that she has

done and provided for me. Words cannot express my gratitude. Finally, to my wife Rosie, thank

you for putting up with me over all these years. Your support, encouragement, and love got me

through this journey, and continues to get me through each and every day.

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Table of Contents

Abstract ...........................................................................................................................................ii

Acknowledgments.......................................................................................................................... iv

Chapter 1 The Quinean Picture ....................................................................................................... 1

1 Mathematical Realism ......................................................................................................... 4

2 W. V. O. Quine .................................................................................................................... 7

3 The Quinean Indispensability Argument ........................................................................... 13

4 Three Strengths .................................................................................................................. 16

Chapter 2 Enhancing Quine .......................................................................................................... 20

1 Weakness One: The Quinean Picture ................................................................................ 20

2 Weakness Two: Disrespect towards Mathematics ............................................................. 23

3 Weakness Three: Vagueness .............................................................................................. 28

4 Internal and External Mathematical Explanations ............................................................. 42

5 The Enhanced Indispensability Argument ........................................................................ 45

6 What About Naturalism? ................................................................................................... 49

Chapter 3 Genuine Mathematical Explanation ............................................................................. 52

1 Supposed Genuine Mathematical Explanations................................................................ 53

1.1 Geometric Explanations ............................................................................................. 53

1.2 Contrived Explanations .............................................................................................. 55

1.3 Optimization Explanations ......................................................................................... 56

2 The Indexing Argument Revisited ..................................................................................... 59

3 The Indexing Criteria for Genuine Mathematical Explanations ........................................ 71

4 Genuine Mathematical Explanation: Electron Spin .......................................................... 77

5 Blocking the Nominalist’s Response ................................................................................ 84

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5.1 The Unknown Explanation ......................................................................................... 84

5.2 The Ad Hoc Physical Explanation .............................................................................. 87

5.3 No Explanation ........................................................................................................... 95

6 The Honest Conclusion ...................................................................................................... 97

Chapter 4 Scientific Mathematical Explanation ........................................................................... 99

1 Accounts of Scientific Explanation ................................................................................. 101

1.1 The Deductive-Nomological and Pragmatic Accounts ............................................ 102

1.2 The Unification Account .......................................................................................... 105

1.3 The Statistical-Relevance and Counterfactual Accounts .......................................... 107

2 The Kairetic Account ....................................................................................................... 110

2.1 The Kairetic Criterion ............................................................................................... 111

2.2 Adapting the Kairetic Criterion ................................................................................ 113

3 Applying the Kairetic Criterion ....................................................................................... 120

4 Difference-Making Revisited........................................................................................... 126

4.1 Internal Mathematical Explanations ......................................................................... 126

4.2 The Roles of Mathematics in Genuine Mathematical Explanations ........................ 131

5 Takeaway ......................................................................................................................... 139

Chapter 5 Inference to the Best Mathematical Explanation ....................................................... 140

1 The Inference to the Best Explanation ............................................................................. 142

1.1 Problems with Inference to the Best Explanation .................................................... 143

1.2 Types of Inference to the Best Explanation .............................................................. 147

2 Fictionalism: A Better Explanation? ................................................................................ 150

3 Unjustified Inference to the Best Mathematical Explanation .......................................... 157

4 The Not-So-Enhanced Indispensability Argument .......................................................... 166

Chapter 6 Conclusion .................................................................................................................. 172

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1 Moving Forward .............................................................................................................. 174

2 Final Thoughts ................................................................................................................. 186

Bibliography ............................................................................................................................... 187

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Chapter 1 The Quinean Picture

An indispensability argument for mathematical realism aims to convince the typical scientific

realist that the very same reasons that motivate their commitments to the unobservable entities

postulated in our best physical theories, such as positrons and quarks, should inevitably lead

them to be realists regarding the mathematical entities utilized in science, such as numbers and

sets. The key consideration behind the realist position is the indispensable nature of unobservable

entities. For the time being it is sufficient to characterize a scientific realist as believing in the

objective existence of the entities postulated by science. Scientific realists need not commit to

every entity utilized in science, but rather realists believe in the existence of those unobservable

entities that are indispensable for generating the results, predictions, and explanations in our best

scientific theories. It is this indispensable nature that provides the basis for our belief in the

objective existence of said entities. Indispensability arguments make a parallel argument for

mathematical entities. At first glance it surely seems inconceivable to do science without the use

of mathematical entities. These entities are indispensable, so this points to the objective existence

of mathematical entities in the very same way as it does for physical unobservable entities.

Originally popularized by Quine and Putnam, many different forms of indispensability

arguments have since emerged. The most influential of these indispensability arguments is

championed by Mark Colyvan and is deeply inspired by Quine’s work. Recently, Alan Baker

(2005) has put forward a new enhanced indispensability argument (EIA) that claims to have

significant advantages over its predecessors. Baker’s EIA focuses on how mathematics is

indispensable in scientific explanations in order to lead to the realist conclusion. The move to

mathematical realism is achieved by an inference to the best explanation (IBE), which is a

standard inference for the scientific realist. The EIA hinges on two much discussed topics in the

philosophy of science: scientific explanation and IBE. What makes Baker’s argument interesting

is that these two topics are transplanted from a purely scientific domain, which deals with

concrete physical objects and phenomena, into the abstract realm of mathematics. An upshot of

this move is to build up an argument for mathematical realism using concepts that the scientific

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realist is already intimately familiar and comfortable with. No additional concepts or reasoning is

needed to extend one’s ontology over mathematical objects.

The benefits of the EIA are quite strong in that it aligns itself well with the present discourse in

the philosophy of science, but such a move comes at a cost. At present, there are many

unanswered questions which prevent the EIA from being a convincing argument. Is it the case

that using IBE to infer the existence of abstract objects is the same as the standard usage in

which we infer the existence of physical objects? If not, how can we justify such an inference?

Given the contentious state of our understanding of scientific explanations, can we base an

argument on the even less understood idea of mathematical scientific explanations? Are

mathematical scientific explanations well-defined, and do such things even exist? Only when

these questions are addressed will we be able to properly asses the EIA.

The principle aim of this dissertation is to evaluate whether the EIA represents a significant

advancement over other indispensability arguments or not. In order to critically assess the EIA,

the first task will be to understand the development of indispensability arguments for

mathematical realism. Chapter 1 will be dedicated to a brief history of the influence of Quine,

Putnam, and scientific naturalism. Quine’s naturalism is the driving force behind the

development of indispensability arguments that culminates in Colyvan’s presentation of the

Quinean argument. Chapter 2 will present Baker’s enhanced indispensability argument and

demonstrate how it differentiates itself from previous indispensability arguments by improving

on several critical weaknesses that Quine’s argument faces.

Unfortunately, many of the concepts utilized in the debate surrounding the EIA are unclear or

underdeveloped. As mentioned above, the two most critical are those of mathematical

explanation of scientific facts, and the use of IBE for mathematical realism. In order to properly

assess the worth of the EIA it will be important to make precise these key aspects of the

argument. Largely due to the EIA, the topics of mathematical explanation and IBE have been

typically treated together. The most common approach in the literature has been to assume that

the use of IBE is unproblematic, and then focus on determining the explanatory power of

mathematics. Broadly speaking, this has divided philosophers into two predictable camps.

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Realists support the existence of mathematical explanations of physical facts, whereas anti-

realists deny it. I feel that this is unfortunate as it is the attraction or fear of ontological

commitment that is pointing the direction of analysis of mathematical explanations in science.

My methodological approach will be to not assume anything about IBE and to analyze both

mathematical explanations in science and inferring mathematical realism separately and

independently. Understanding the explanatory role of mathematics in science is a worthwhile

endeavour entirely independent of potential ontological commitment. In addition, such an

understanding will facilitate a much better grasp of how the EIA works, and whether or not it is a

good argument for mathematical realism. The first step will be to consider what is meant by

mathematical explanation while remaining agnostic about any sort of inference to mathematical

realism. Chapter 3 will start down this path by developing criteria for exactly what we mean by a

mathematical explanation of a scientific fact. Once these criteria have been laid out we will

analyze many of the standard examples, and ultimately present a new example of a mathematical

explanation of a scientific fact. Chapter 4 will take this process one step further by making use of

theories of scientific explanation. Specifically, we will make use of Michael Strevens’ kairetic

account of scientific explanation to analyze our examples of mathematical explanations. The

goal here is to reinforce our claim that mathematical entities can play a genuine explanatory role

beyond simply presenting examples and criteria that speak to our intuitions. If the existence of

mathematical explanations of physical facts can be corroborated independently by theories of

scientific explanation, then this will be a large step forward in accepting the explanatory power

of mathematics.

The final task will be to consider if IBE can be employed to infer mathematical realism. This

discussion will be performed with the knowledge that mathematical explanations of physical

facts do exist as argued for in chapters 3 and 4. Yet if this is the case, how can such an analysis

remain independent? Anti-realists would certainly need to deny that IBE is a legitimate tool for

inferring the existence of abstract mathematical entities, whereas realists would argue that there

is no problem with using IBE. My position in chapter 5 will be that the problem with these

attacks and defenses of IBE is that they are ad hoc or question begging in nature. What needs to

be analyzed and understood is how IBE actually works, and such an analysis should be as

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independent as possible from potential ontological outcomes. Ultimately, while I do support the

existence of mathematical explanations of physical facts, I will argue that an independent

analysis does not point to IBE leading to mathematical realism.

My ultimate position will be that the EIA does not represent a significant improvement over

previous indispensability arguments as IBE cannot deliver mathematical realism. However, this

is not to say that studying the EIA is not worthwhile. The EIA has thrust the topic of

mathematical explanation to the forefront. Attempting to understand how mathematics can or

cannot explain physical facts adds critically to our understanding of scientific explanation in

general, and also to our picture of the precise roles that mathematics plays in our best scientific

theories. Similarly, examining whether or not the use of IBE can be extended over abstract

mathematical entities clarifies the boundaries and the spirit of arguably the most important

inference for scientific realists.

1 Mathematical Realism

The goal of indispensability arguments is to convince us to be mathematical realists, but what

this means exactly is a bit unclear. Mathematical realism is often described in a variety of

different and conflicting ways. This is because realism is habitually conflated with 'ontological'

platonism and 'epistemological' platonism. This conflation has as much to do with the word

‘Platonism’ itself indicating its historical roots in the works of Plato, as it does with the actual

differences in beliefs. At the bare minimum, realism is the commitment to at least one of the

following three theses:

(1) Existence: Mathematical objects exist,

(2) Abstractness: Mathematical objects are abstract. They are non-

spatio-temporal,

and,

(3) Independence: Mathematical objects and their properties are

mind and language independent.

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It may be difficult to consistently believe a particular combination of the above three, such as

accepting existence and abstractness but denying independence. Regardless, belief in at least one

is sufficient for being a mathematical realist.

Ontological platonism is the belief in all three of the above theses. This is clearly a strong brand

of realism, and it is often the type of realism that people refer to today. For example, Michael

Dummet says:

Platonism is the doctrine that mathematical theories relate to systems of

abstract objects, existing independently of us, and that the statements of

those theories are determinately true or false independently of our

knowledge. (Dummett, 1991, p. 301)

Hartry Field similarly claims that:

A mathematical realist, or platonist, (as I will use these terms) is a person

who (a) believes in the existence of mathematical entities (numbers,

functions, sets and so forth), and (b) believes them to be mind-independent

and language-independent. (Field, 1989, p. 1)

Notice that Field does not insist that mathematical entities are abstract. Most often this is an

implicit assumption based on the affirmation of the existence and independence theses. However,

to be clear, it could be that someone may still deny the abstractness thesis in which case, by my

classification, they would not be an ontological platonist, but simply a mathematical realist of

some other form.

Epistemological platonism maintains the truth of the above three theses, but at the same time

adds to it some sort of epistemological thesis. This thesis states that we somehow come to know

mathematical truths and properties of objects that are abstract and mind-independent. Exactly

how we do this tends to vary, but generally speaking an appeal is made to some sort of intuition

or mental perception that is unlike our regular senses. Kurt Gödel, perhaps the most famous of

modern epistemological platonists, says that:

[Platonism is] the view that mathematics describes a non-sensual reality,

which exists independently both of the acts and [of] the dispositions of the

human mind and is only perceived, and probably perceived very

incompletely, by the human mind. (Gödel, 1995, p. 323)

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Similarly, James R. Brown explicitly admits an epistemological thesis in his description of

platonism above and beyond those required for ontological platonism.

Mathematical entities can be ‘seen’ or ‘grasped’ with ‘the mind’s eye’.

These terms are, of course, metaphors, but I’m not sure we can do better.

The main idea is that we have a kind of access to the mathematical realm

that is something like our perceptual access to the physical realm. (Brown,

1999, p. 13)

If this talk seems vague, that is because, as Brown admits, it is. There simply is nothing better

available to clarify the matter. This sort of talk is what is often pointed to in a condescending

way for people to argue against mathematical realism; however, for our purposes it is enough to

point out that the weakness of the epistemological thesis does not actually hurt mathematical

realism or ontological platonism as neither position subscribes to this extra-sensory connection.

For the remainder of this dissertation whenever I speak of platonism I am referring to ontological

platonism unless otherwise stated. I aim to distinguish between the use of platonism and

mathematical realism in the same way as laid out above. Of course there are other forms of

realism that I have not discussed here, such as plenitudious platonism or structuralism, but as we

shall see, indispensability arguments do not say anything at all about these respective and distinct

brands of realism.

One last thing remains to be clearly defined. As I have laid out the differences between

mathematical realism and platonism, what then is an anti-realist towards mathematics? The

answer to this is somewhat trivial. An anti-platonist is simply someone who denies at least one of

the three realist theses, whereas an anti-realist is someone who denies them all. However, as was

the case for the realist positions, the ‘anti’ positions are often similarly conflated. For our

purposes we will be mainly interested in two anti-platonist positions: nominalism and

fictionalism.

Nominalism and fictionalism are not actually two distinct versions of anti-platonism. Rather,

fictionalism is just a particular brand of nominalism. Nominalism in its traditional philosophical

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usage refers to the view that universals do not exist.1 In the context of mathematics, nominalism

denies the existence and abstractness theses, but remains silent regarding independence.

Certainly once the nominalist position is adopted it would seem natural to also deny the

independence thesis; however, one could still remain a mathematical realist by not doing so,

although the type of realism remaining may be nothing more than lip-service.

Fictionalism is a nominalist position but it adds to it an account of how we can understand the

practice and application of mathematics given that mathematical entities do not exist and are not

abstract in nature. Mathematical entities are merely fictions in the story of mathematics, just as

Bilbo Baggins is a fiction in the story of the Hobbit. Statements such as ‘the number 3 is less

than 5’, are strictly speaking false, just as saying that ‘Bilbo Baggins has hairy feet’ is also false

as 3, 5, and Bilbo do not really exist. We can admit that statements about 3, 5, and Bilbo are true

in their respective stories, but they will never be true in reality. Mathematical fictions are

certainly more useful than other fictions in that they have proven to be incredibly useful in our

best scientific theories, but that is not to say that they are special in any way that necessitates

existence or abstractness. Given the fictionalist account, it is easy to see how it can be extended

to deny the independence thesis and thus turn into a full anti-realist position. These extensions

will vary depending on whose variety of fictionalism one subscribes to.

2 W. V. O. Quine

In the latter half of the 20th century, Quine put forward what is often regarded as the strongest

argument for mathematical realism.2 The Quinean indispensability argument (QIA) is built upon

Quine’s naturalism and his theory of confirmational holism.3 Essentially, Quine believes in

rejecting any attempt to practice ‘first philosophy’. Science, which is nothing more than an

1 See Burgess and Rosen (1997) for an excellent discussion on nominalism.

2 Quine was not the first to make use of some sort of indispensability argument. Gottlob Frege, for example, said

that “it is applicability alone which elevates arithmetic from a game to the rank of science.” (Frege, 1970, p. 187)

3 Quine is also famous for many other theories, such as his views on translation, underdetermination, and semantic

holism. The QIA need not depend on these other positions.

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extension of our natural common sense, is the sole arbiter for our beliefs. More importantly,

science itself requires no external justification. Naturalism is “the recognition that it is within

science itself, and not in some prior philosophy, that reality is to be identified and described,”

(Quine, 1981c, p. 21) and that science is “not answerable to any supra-scientific tribunal, and not

in need of any justification beyond observation and the hypotheticodeductive method.” (Quine,

1981a, p. 72) Any attempt to impose conditions on the practice of science from outside of

science is unnaturalistic.

This does not mean that science is infallible or beyond critique. Our scientific methods and

beliefs are constantly being revised and improved. The salient point is that criticism and

improvement comes from within the practice of science.

As scientists we accept provisionally our heritage from the dim past, with

intermediate revisions by our more recent forebears; and then we continue

to warp and revise. As Neurath has said, we are in the position of a mariner

who must rebuild his ship plank by plank while continuing to stay afloat

on the open sea. (Quine, 1966, p. 210)

One may ask what role the philosopher has to play given the prized position of science in

Quine’s naturalism. Philosophy is still important, but the practice of philosophy is in line with

that of science; “[t]he philosopher and the scientist are in the same boat.” (Quine, 1960, p. 3)

Quine’s naturalism stems from his attempt to rescue empiricism from two large errors

perpetuated by the logical positivists in the early 20th century. In his famous paper Two Dogmas

of Empiricism, Quine rejects the classic distinction between analytic and synthetic statements. He

argues that there is no such thing as analytic statements which are “true by virtue of meanings

and independently of fact.” (Quine, 1961b, p. 21) Quine bases his rejection of analytic statements

on the claim that any attempt to clearly define analytic relies on other ill-defined notions that

leads to circular reasoning. Quine considers the seemingly analytic statement ‘no unmarried man

is married’. A similar statement, ‘no bachelor is married’, is also seemingly analytic as the word

‘bachelor’ can be replaced with ‘unmarried man’ because they are synonymous. In this sense,

analytic statements depend on the idea of synonymous replacement. Although such a move

appears innocuous, Quine claims that the challenge is to clearly explicate the concept of

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synonymy. Appealing to definitions will not work as definitions are dependent on synonymy in

the first place. Instead, Quine looks towards the idea of interchangeability. This too runs into

problems as understanding interchangeability depends on some concept of necessity or

analyticity – the very concept that we were originally hoping to comprehend in the first place.

Thus the notion of analytic statements is circular and ought to be rejected. Without any analytic

truths, Quine is a true empiricist at heart. All of our knowledge is learned through our

experiences. Moreover, the best way to organize and make sense of our knowledge is through

natural science. The traditional concept that philosophy, independent of any experience, can

deliver us truths about the world is simply incorrect. This leads to Quine’s abandonment of ‘first

philosophy’ and his prizing of science as the main arbiter for our beliefs about the world.

The second dogma that Quine repudiates is the dogma of reductionism which states that “each

statement, taken in isolation from its fellows, can admit of confirmation or infirmation at all.”

(Quine, 1961b, p. 41) Quine believes that no hypothesis can ever be tested as a single unit.

Rather, it is our entire body of knowledge and beliefs that are put to the test every time. “[O]ur

statements about the external world face the tribunal of sense experience not individually but

only as a corporate body.” (Quine, 1961b, p. 41) Our knowledge, which is all acquired through

experience, forms a complex network which Quine calls our ‘web of belief’. Science helps us

organize this web such that at the center are beliefs which are central to our understanding of the

world, such as laws, theories, or mathematics and logic. At the edges of our web lie beliefs that

we may be less certain of, or of which less of our other knowledge depends upon. Everything in

the web is interconnected.

One implication of Quine’s ‘web of belief’ is that it is impossible to test or confirm specific

statements in isolation. Instead, Quine argues that experience confirms or denies in a holistic

sense. Confirmational holism asserts that if a prediction from a theory is experimentally verified

then not only has that prediction and theory received confirmation, but all of the auxiliary

scientific and mathematical beliefs that the theory depends on have received it as well. Consider

a hypothesis X that is predicted by theory T. An experiment is run that is meant to test for X and

the result turns out positive. Such an experiment depends on many other scientific factors other

than T, such as the laboratory equipment being used, the methods of analysis, etc. Each factor is

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supported by other theories, T1, ... , Tn, such as theories of optics or electromagnetism, etc. In

addition, there are many auxiliary assumptions maintained and utilized by the scientists, A1, ... ,

An, such as the constant nature of laws, and of course mathematics. The positive test result for X

is actually a product of not just the theory T that predicted X in the first place, but actually of the

collection of theories and auxiliary assumption {T, T1, ... , Tn, A1, ... , An}. What Quine points out

is that the positive result does not single out and confirm T individually as there is no way to

isolate T from the other theories and auxiliary assumptions. At best, all we know is that at least

one of {T, T1, ... , Tn, A1, ... , An} is supported by our positive experimental result. The only

logical conclusion is that such a positive result confirms everything that was required to generate

it; {T, T1, ... , Tn, A1, ... , An} is confirmed holistically. This directly implies that positive results

empirically confirm the mathematics that are employed in our scientific theories as it is one of

the auxiliary assumptions utilized in almost all scientific experiments.

In the case of a theory facing a recalcitrant experience, the theory and all the other theories and

auxiliary assumption may be suspect. If X was not discovered then at best all we know is that at

least one of {T, T1, ... , Tn, A1, ... , An} is false. It may not be the case that T, the theory we were

hoping to test, is actually the culprit. Logically speaking, it could be any combination of our

other theories or auxiliary assumption that were part of our experimental setup. It could be that

some other theory, such as the theory of optics, or some auxiliary assumption, such as

mathematics, is mistaken and is causing our false prediction of X. We may bristle at this notion

by pointing to the historical success of the theory of optics, or even more so of mathematics and

claim that such assumptions surely cannot be false. But Quine famously maintains that nothing is

impervious to revision: “Any statement can be held true come what may, if we make drastic

enough adjustments elsewhere in the system... Conversely, by the same token, no statement is

immune to revision.” (Quine, 1961b, p. 43) Ultimately what guides our decision as to which

statements we elect to revise is purely practical. Quine cites several factors in his writings which

all boils down to “considerations of simplicity plus a pragmatic guess as to how the overall

system will continue to work in connection with experience.” (Quine, 1966, p. 210) Although

there is nothing stopping us from rejecting and revising something central in our ‘web of belief’

like mathematics, such a change comes at a great cost. Mathematics is central in our web as so

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much of our other knowledge depends on it. If we were to alter our mathematics in the face of a

recalcitrant experience, this would then require a massive reorganization of our web in order for

our knowledge to remain consistent, assuming that this is even possible. Conversely, if we were

to reject a belief on the periphery of the web, such as our theory T, this would be much simpler

as far less knowledge depends on it. Such a move is clearly the more pragmatic choice. If,

however, we were continually getting recalcitrant experiences, there is nothing that is stopping

us from looking to alter more central theories and assumptions.

Quine’s naturalism and confirmational holism have important consequences for the objects and

entities that saturate our natural sciences. It seems obvious that our knowledge of medium-sized

physical objects, such as tables or chairs, is different than our knowledge of entities that we

cannot see, such as electrons or protons. On a trivial level they are different in that our

knowledge of things like a chair seems more certain than that of things like a proton. From this

distinction there is a temptation to conclude that this difference is actually a difference in kind.

Knowledge of medium-sized physical objects is a different sort of knowledge than that of

unobservable entities on the basis that the former are accessible by our senses, whereas the latter

are not. Quine argues that this division in kind is a mistake. The physical objects that are utilized

in science, from macroscopic to microscopic, are all, epistemologically speaking, on par. There

is no actual difference in this sense between a chair, a proton, and even the gods of Homer. What

we perceive to be a difference in kind is merely a difference in degree:

[I]n point of epistemological footing the physical objects and the gods

differ only in degree and not in kind. Both sorts of entities enter our

conception only as cultural posits. The myth of physical objects is

epistemologically superior to most in that it has proved more efficacious

than other myths as a device for working a manageable structure into the

flux of experience. (Quine, 1961b, p. 44)

What we see as a difference between our knowledge of medium-sized physical objects and

unobservable entities is solely based on the fact that macroscopic physical objects are more

central and critical in our ‘web of belief’ to understanding the world as we perceive it than

microscopic objects. But what of mathematics, and things like numbers and sets whose use is

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ubiquitous in the sciences? Quine maintains the same position: “[e]pistemoligcally [abstract

entities] are myths on the same footing with physical objects and gods, neither better nor worse

except for differences in the degree to which they expedite our dealings with sense experiences.”

(Quine, 1961b, p. 45)

Quine’s position that all the entities posited in our scientific theories are epistemologically equal

spills over into his views on ontology with some important clarifications. It is easy to admit into

our ontology objects that we have direct sensory experience of, like a table or chair. The

evidence that unobservable physical objects exists is different in that it is indirect. Again, it

seems natural to point to this difference in evidence as justification for inferring that medium-

sized objects and unobservable objects are different in kind. However, the difference here, just as

above, is solely “a matter of degree.” (Quine, 1969, p. 97) Quine extends this perspective over

abstract objects such as numbers and sets as well. What matters for Quine is what role these

objects play in our scientific theories, be them medium-sized, unobservable, or abstract in nature.

All these entities play the same role in our scientific theories; they make sense of and organize

our experiences and knowledge, and this is why they are all on par with one another. How do we

know that they all play the same role? Quine puts forward the criterion of being a bound variable

under an existential quantifier, and associates that with being indispensable for the truth of the

theory. “[A] theory is committed to those and only those entities to which the bound variables of

the theory must be capable of referring in order that the affirmations made in the theory be true.”

(Quine, 1961a, pp. 13–14) Quantification and indispensability leads to accepting an entity into

our ontology, no matter the type of entity in question, and this is “the only way we can involve

ourselves in ontological commitments.” (Quine, 1961a, p. 12)

Quinean naturalism tells us that science is the sole arbiter for making sense of our experiences

and beliefs. The two key principles of his view is his commitment to empiricism – no knowledge

can be gained independently or prior to science – and to confirmational holism – all our

knowledge is the same in kind and faces confirmation and revision as a whole. Everything is a

part of our ‘web of belief’, and all the objects that our scientific theories depend on are

epistemologically and ontologically equal. With this framework in place, Quine naturally

extends his ontology to include the mathematical objects that are utilized in science. He simply

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has to show that mathematics is indispensable to the truth of our scientific theories, and that

entities such as numbers and sets are existentially quantified over, just like tables, chairs, and

electrons are. Quine mostly assumes this to be the case without much justification. In terms of

the indispensable nature of abstract mathematical objects, Quine is “persuaded that one cannot

thus make a clean sweep of all abstract objects without sacrificing much of science.” (Quine,

1981a, p. 69). Quine also states that mathematical entities, such as numbers, functions and sets,

“figure as values of the [quantified] variables in our overall system of the world. The numbers

and functions contribute just as genuinely to physical theory as do hypothetical particles.”

(Quine, 1981b, p. 150) Given that mathematical entities are indispensable to our best scientific

theories, and that they are existentially quantified, then Quine’s conclusion is that mathematical

objects must exist.

3 The Quinean Indispensability Argument

The QIA has proven to be incredibly influential. One of its most famous supporters was Hilary

Putnam who summarized Quine’s argument as follows:

[Q]uantification over mathematical entities is indispensable for science, both

formal and physical; therefore we should accept such quantification; but this

commits us to accepting the existence of the mathematical entities in question.

This type of argument stems, of course, from Quine, who has stressed both the

indispensability of quantification over mathematical entities and the intellectual

dishonesty of denying the existence of what one daily presupposes. (Putnam,

1979a, p. 338)

In perhaps the strongest defense of the QIA, Mark Colyvan (2001a) presents the argument in a

more formal manner which I have altered slightly here.4

(P1) We ought to have ontological commitment to only those entities

that are indispensable to our best scientific theories.

(P2) We ought to have ontological commitment to all those entities

that are indispensable to our best scientific theories.

4 Colyvan treats the first two premises as one, but the content of the argument is identical to his presentation.

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(P3) Mathematical entities are indispensable to our best scientific

theories.

Therefore:

(C) We ought to have ontological commitment to mathematical

entities.

(P1) is Quine’s commitment to science as the sole arbiter of our beliefs, (P2) reflects

confirmational holism, and the conclusion of mathematical realism follows from (P3). Laid out

in this way the QIA seems to be a valid argument for realism. One difference between Colyvan’s

formulation and Quine’s original version is the absence of requiring quantification over entities

for ontological commitment. Colyvan believes that (P1) and (P2) together capture the

quantification requirement. Implicit in (P2) is that the only entities that are indispensable to

science are those that are quantified over. (P1) restricts ontological commitment to things that are

indispensable to science. Thus, only entities that are quantified in our best scientific theories are

under consideration.

The QIA is meant to target scientific realists who at the same time maintain a nominalist position

towards mathematical entities. If you are not a scientific realist to begin with, the QIA, and any

other indispensability argument for that matter, possesses no force as you would certainly reject

both (P1) and (P2), and for that matter any other realist premise that need not depend on Quine’s

views. Given this, for the remainder of this dissertation we are only interested in the

aforementioned nominalist who does count physical unobservable objects posited by our best

scientific theories amongst their ontology. The QIA means to show that the very same reasoning

that the nominalists’ scientific realism depends on inexorably leads to the conclusion that they

should also be mathematical realists. The power of the QIA is that nominalists of this sort are

trapped. If they wish to withhold commitment to the existence of mathematical entities, then they

are guilty of being ‘intellectually dishonest’ as they accept the argument for realism in one case,

but reject it in another.

One way around the QIA is to say that existence means something different when we speak of

mathematical objects. These objects exist, but not in the same sorts of ways that physical objects

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exist. This would allow us to maintain that mathematical objects exist in some trivial way, such

as mental constructions or simply marks on a page. Quine says that such a move is to employ

“double-talk,” (Quine, 1969, p. 99) Existence should mean the same thing no matter what type of

object we are referring to. To have two different meanings would be to maintain a “double-

standard” (Quine, 1961b, p. 45) that is not supported by any of our best scientific theories. There

is no principle or law in science that tells us to treat observable objects differently than

unobservable objects. To maintain that we should have different standards comes from concerns

stemming from a philosophical nature. Nominalists may have reasons to abhor unobservable or

abstract objects and to prefer their ontology to resemble a ‘desert landscape’, but any such reason

is motivated by supra-scientific concerns which runs directly against Quinean naturalism. Issues

of existence should be governed solely by science, and in science there is no explicit cleft

between observable and unobservable objects and their type of existence.

Fictionalism fares no better against the QIA. Putnam states that to believe that mathematical

entities are indispensable but to maintain that they are merely useful fictions is to be rejected as,

it is silly to agree that a reason for believing that p warrants accepting p in

all scientific circumstances, and then to add ‘but even so it is not good

enough’ [for mathematics]. Such a judgment could only be made if one

accepted a trans-scientific method as superior to the scientific method; but

this philosopher, at least, has no interest in doing that.” (Putnam, 1979a, p.

356)

Like Quine, Putnam wants to uphold a naturalist position, and treating mathematical entities as

fictions is unnaturalistic. Putnam does not stop there. He belittles fictionalism for simply being

an untenable position in the first place, regardless of it being unnaturalistic.

It is like trying to maintain that God does not exist and angels do not exist

while maintaining at the very same time that it is an objective fact that God

has put an angel in charge of each star and the angels in charge of each of

a pair of binary stars were always created at the same time! (Putnam,

1979b, p. 74)

It would seem that the QIA is successful in showing that nominalism and fictionalism are not

tenable alternatives to mathematical realism.

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4 Three Strengths

Beyond simply being a powerful tool against the nominalist, the QIA has three important

strengths that have made it one of the strongest arguments for mathematical realism. The first

strength is that the QIA does not presuppose any truth claims of mathematics or any sort of

abstract objects. Traditional arguments for platonism and mathematical realism often make these

very assumptions. From this starting point an argument is made to add mathematical objects into

our ontology as well. Gödel, for example, famously asserted that the axioms of set theory “force

themselves upon us as being true.” (Gödel, 1983b, p. 270) Once we accept the truth of the basic

axioms we can then argue that there exist some mathematical objects.

It seems to me that the assumption of such objects is quite as legitimate as

the assumption of physical bodies and there is quite as much reason to

believe in their existence. They are in the same sense necessary to obtain

a satisfactory system of mathematics as physical bodies are necessary for

a satisfactory theory of our sense perceptions and in both cases it is

impossible to interpret the propositions one wants to assert about these

entities as propositions about the “data,” i.e., in the latter case the actually

occurring sense perceptions. (Gödel, 1983a, p. 220)

Gödel’s argument only makes sense given that he already takes for granted that the axioms of set

theory are true. The issue here is that for a nominalist who rejects the truth of mathematics, these

arguments for mathematical realism never get off the ground. For the nominalist, any argument

that presupposes the truth of mathematics or the existence of some abstract object to argue for

the existence of abstract mathematical objects is simply begging the question.

In stark contrast are indispensability arguments. No assumptions about the truth of mathematical

entities needs to be made at all in order to deliver the realist conclusion. All that is required is an

analysis of science and scientific practice. The QIA is an argument that can engage nominalists

on their own grounds. Its premises are prima facie acceptable by a nominalist who is also a

scientific realist. Field, an ardent nominalist, echoes this view by stating that “[t]he only non-

question-begging arguments I have ever heard for the view that mathematics is a body of truths

all rest ultimately on the applicability of mathematics to the physical world.” (Field, 1980, p.

viii)

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The second and third strengths of the QIA are related to the works of Paul Benacerraf. In two

famous papers, Benacerraf (1965, 1973) raises significant challenges for the platonist position.

First, if mathematical objects such as numbers do actually exist, a basic question is: what are the

properties of these numbers? Assuming that numbers are sets, the difficulty lies in the fact that

numbers can be represented by sets in an infinite number of unique ways. One way, for example,

is with the progression {Ø}, { Ø,{ Ø}}, { Ø,{ Ø},{ Ø,{ Ø}}}, ... , and another is to use {Ø}, {{

Ø}}, {{{ Ø}}}, ..., to represent the natural numbers. A potential question about the properties of

numbers is: “is the number 3 a member of the number 5?” In the first progression the answer is

yes, but according to the second progression the answer is no. Many more questions about the

properties of the natural numbers generate this same contrary scenario. What this all boils down

to is whether, say, the number 3 = Ø,{ Ø},{ Ø,{ Ø}}}, or 3 = {{{ Ø}}}, or some other

representation. One option is to bite the bullet and say that 3 actually equals all of these

representation, but Benacerraf says that such a conclusion is “absurd.” (Benacerraf, 1965, p. 56)

Benacerraf’s belief is that it must be the case that one of these representations is the right one –

only one representation correctly picks out the natural numbers. However, the problem is that

there is no way to tell which unique representation of numbers is the correct one as they are all

functionally equal. The platonist is left in the embarrassing position that either they do not know

which representation is the true one, which means that they do not know the true nature of

numbers, or they pick one representation to be the one but they cannot justify their choice as all

representations produce the same mathematical consequences.5 Neither horn of the dilemma is

attractive, thus the conclusion is that platonism cannot be true.

While this first challenge seems to pose a problem for platonism, it is neatly sidestepped by the

QIA. The QIA is often confused as an argument for platonism, but it is in fact not that strong.

The QIA does not conclude with the acceptance of all three theses that comprise platonism. In

fact, the conclusion is somewhat vague. Surely there is some sort of mathematical realist

conclusion, but it does not pick out a specific realist position. The QIA could, for example, be

5 In this latter case, one potential response that Benacerraf does not consider is that this is simply a case of

underdetermination. The platonist need not be bothered by the fact that our body of mathematics cannot pick out the

true nature of numbers. Thanks to James R. Brown for pointing this out.

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used as a foundation to justify ones belief in mathematical structuralism advanced by Stewart

Shapiro (1997) and Michael Resnik (1997). One feature of structuralism is that it says that our

mathematical theories are not about mathematical objects at all. Rather, mathematics is about

structures and relations, and not the objects that might instantiate the structure. The structuralist

can then answer Benacerraf’s challenge by stating that the number 3 does not actually equal any

of the set theoretic representations at all. 3 is just a place in the structure of the natural numbers

and has no inherent properties other than its relation to other places. Or, the QIA could be used to

conclude a form of semantic realism wherein we need not commit to the existence of actual

mathematical objects. Rather, we only commit to the independence thesis and withhold talk of

existence altogether. I will have more to say about the multiple ways that we can interpret the

conclusion of the QIA below, but for now it is enough to observe that the QIA is not married to

platonism. Hence, even if we grant that Benacerraf’s first challenge is embarrassing for the

platonist, this does little to harm the effectiveness of the QIA.

Benacerraf’s second challenge is the epistemological problem of access. Again, if platonism is

true, mathematical objects exist and have properties. But how can we know these properties, or

the objects at all for that matter, given that mathematical objects are by their very nature abstract,

non-spatio-temporal objects? Benacerraf subscribes to a causal theory of knowledge which has

since fallen on hard times, but the force of his second challenge remains even without the causal

theory. In the absence of direct contact with the abstract realm, the platonist cannot justify their

knowledge without appealing to some sort of extra-sensory or intuitive connection. As we saw,

some realists, such as Gödel and Brown, embrace this move with open arms. This leads to their

brands of epistemological platonism. Yet Benacerraf and many others find such an option

anathema. The appeal to some extra-sensory connection to the abstract realm operates as a

reductio ad absurdum, and Benacerraf argues that this challenge points to the conclusion that

mathematical objects do not exist and that platonism is fundamentally flawed.

The QIA has an elegant response to Benacerraf’s problem of access. Advocates of the QIA also

shun the move by epistemological platonists in accepting an epistemological thesis. They accept

that mathematical realists who believe in abstract mathematical objects need to be able to say

something about how we learn about these objects. The QIA offers an alternative response that

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Benacerraf did not address. We learn about mathematical objects through their applications in

our best scientific theories. Our knowledge is empirical. This is a natural repercussion from

Quine’s critique of analytic statements and his naturalistic attitude. A committed anti-realist

could press the question further and demand an account of how exactly we come to this

empirical knowledge of things such as numbers. There is an easy response for the Quinean. We

acquire such knowledge in the exact same way that we acquire knowledge of unobservable

physical objects as well. The anti-realist is making a mistake in assuming that knowledge of

abstract objects is different in kind than knowledge of physical objects. They are mistaken as

their difference is a matter of degree only. Others, such as Penelope Maddy, have attempted to

answer this further question more directly. Maddy (1990) argues that we can actually see sets in

the physical world, and that our knowledge of sets, which she considers to be the foundations of

mathematics, is empirical.6 Regardless, the QIA is committed to the belief that knowledge of

mathematical objects does not come through some extra-sensory connection to the abstract

realm, but rather solely through our use of mathematics in our best scientific theories.

Benacerraf’s second challenge is a non-issue.

Benacerraf’s challenges are arguably the most influential and powerful critiques of platonism in

the 20th century. The fact that the QIA easily circumvents them is no trivial matter. Add to that

the fact that the QIA is elegant, straightforward, non-question-begging, and that it aggressively

blocks the nominalist position, then what we have is a potent argument for mathematical realism.

This much has been recognized by mathematical realists and nominalists alike.

6 Maddy has since abandoned this view.

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Chapter 2 Enhancing Quine

The fact that the Quinean indispensability argument (QIA) does not beg the question against the

nominalist and successfully deals with Benacerraf’s two challenges makes it one of the strongest,

some say the strongest argument for mathematical realism. This does not mean that the QIA is

without its flaws. Opponents of the QIA have attacked the argument in a wide variety of ways. In

general, the QIA has three weaknesses where the criticisms have been the fiercest. First, the QIA

depends entirely on Quinean naturalism, and in particular on confirmational holism. Second, the

QIA disrespects mathematical practice. Third, the argument is inherently vague. Once we have

explored these weaknesses we can introduce Baker’s enhanced indispensability argument and

see why it is considered to be an improvement over the QIA.

1 Weakness One: The Quinean Picture

The QIA is intimately connected with Quine’s naturalism. In some ways this is a strength. A

believer of Quinean naturalism is well-equipped to fend off certain attacks against adding

mathematical entities into our ontology. The difficulty for the QIA is that it is not obvious why

we should adhere to Quine’s naturalism in the first place. For our purposes, the primary point of

contention surrounds the thesis of confirmational holism. Maddy (1997) launches several

convincing arguments against confirmational holism that stem from the actual practice of

science. Maddy grants that confirmational holism makes sense on a logical level. It may be

perfectly reasonable that experimental success should be taken to confirm the theory as well as

all auxiliary assumptions such as mathematics, but the sticking point for Maddy is that this is

simply not what happens in practice. Instead of accepting confirmational holism based on some

abstract argument, Maddy urges us to look towards the practice of science directly in order to

draw our conclusions. If we look at the practice of science we see that scientists do test

hypotheses in isolation and that they do not treat confirmation in a holistic manner.

Maddy’s main example to refute confirmational holism is the discovery of the atom and its

adoption into chemical and physical theories around the turn of the 20th century. Even though

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atomic theory was in wide use and its power and usefulness had been well established, scientists

still desired a direct verification, or some crucial experiment that would resolve the issue of

existence. Einstein’s 1905 paper on Brownian motion provided the experimental framework,

and Perrin shortly after confirmed the result.7 Only after this crucial experiment had been

performed was the atom confirmed and the issue of existence finally settled. Maddy’s key point

is that prior to 1905, scientists were withholding confirmation of the atom even though it had

proven to be so useful, perhaps even indispensable to many scientific theories. According to

Quine’s criteria, the atom should have been confirmed and added to our ontology by this point.

Yet confirmation only came when Einstein and Perrin devised a way to test the atomic

hypothesis in isolation. Maddy argues that historical cases such as this shows that the actual

behaviour of scientists does not square away with the Quinean picture of confirmational holism.

Scientists do not confirm in a holistic manner, and indispensability is not a sufficient condition

for ontological status. Instead, scientists often look for crucial experiments in order to test

hypotheses in isolation, and only once these are successful does an entity or theory truly gain

acceptance beyond merely instrumental status.

Maddy further alleges that confirmational holism fails to account for the many idealizations and

strictly false assumptions that scientists use every day. Idealizations such as frictionless planes,

or an infinite number of molecules within a fixed volume are seemingly indispensable to our best

scientific theories, but no one would claim that these mathematical idealizations somehow

receive confirmation when our theories are successful. Without a way to account for the use of

such false idealizations within the holistic framework, there is reason to doubt that scientists

confirm in a holistic manner. If we doubt the veracity of confirmational holism, then we must

reject the premise that confirmational holism is true (P2) from the QIA. Without (P2) there is no

way that we can extend any sort of ontological commitment over mathematical identities even

though their use may be indispensable to our best scientific theories.

7 Maddy utilizes this example in many works, but the most detailed account can be found in Maddy (1997) part II,

chapter 6.

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Maddy’s criticisms are convincing. What is worse for the Quinean is that her attacks stem not

from some ‘supra-scientific’ level, but rather from a look at the actual practice of scientists. This

is the practice that Quine takes so seriously and argues that we should accept at face value. In

essence, Maddy is arguing that the thesis of confirmational holism itself is somehow

unnaturalistic according to Quine’s own standards. Colyvan attempts to defend the QIA from

Maddy’s attacks. With regards to the use of false idealizations, Colyvan argues that consistency

trumps all. Any entity that renders a theory inconsistent, no matter how useful, is “unlikely to be

indispensable to that theory... because there exists a better theory (i.e., a consistent theory) that

does not quantify over the entity in question.” (Colyvan, 2001a, p. 99) Colyvan's argument

depends on the idea that any idealization can be de-idealized into a faithful description of the

physical system. In this way, false idealizations are not indispensable. However, this stance is

controversial, and some argue that certain idealizations are not de-idealizable at all.8 Colyvan

also takes issue with Maddy’s atom example. Maddy argues that prior to Perrin’s experiments in

1905, scientists were withholding ontological commitment to the atom, whereas Quine’s

naturalism says that they should have committed already based on its previous success and

indispensable nature. Maddy’s conclusion is that Quine is wrong. Colyvan agrees with Maddy’s

analysis, but instead concludes the opposite. He believes that the scientists were wrong, and that

the philosopher is in a position to make such judgements. There seems to be something fishy

with this move. It is unclear if such a position runs against Quinean naturalism even though

Colyvan claims it does not. Maddy’s example suggests that the scientists had good reason to

withhold commitment from the atom until 1905. They were waiting for a more direct

confirmation of the atom which is perfectly in line with scientific methodology and practice. If

we say that the scientists were wrong, are we also saying that the methodologies and practice of

the scientific community were also misguided? If so, are we applying an external standard to

judge the methods of science? Arguing that the scientists were mistaken is suspiciously

unnaturalistic, and seems to be a classic example of first philosophy in practice. In fact, Colyvan

himself admits that he “appreciate[s] that many would not share my intuitions here.”(Colyvan,

8 Batterman (2008) argues against de-idealizable idealizations. He claims that certain idealizations lose their

explanatory force when they are de-idealized.

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2001a, p. 100) Colyvan does offer some other possible explanations for this historical example,

such as invoking degrees of belief, but these remain as nothing more than undeveloped

suggestions for possible ways around Maddy’s critique. Ultimately, Colyvan’s defense of the

QIA falls short as it either depends on taking an unnaturalistic stance that is in conflict with the

very position that he wishes to defend, or it rests on other underdeveloped and complicated

concepts such as de-idealizable idealizations and degrees of belief.

2 Weakness Two: Disrespect towards Mathematics

The QIA extends our ontology over those mathematical entities that are utilized in our best

scientific theories. But what of the many mathematical entities that have not yet found a way to

be applied in our sciences? According to Quinean naturalism, science is the sole arbiter of

existence. If certain mathematical entities are not indispensable to science, then we cannot grant

them any ontological rights. Putnam makes this clear in discussing the use of set theory in

physics.

When we come to the higher reaches of set theory, however – sets of sets

of sets of sets – we come to conceptions which are today not needed

outside of pure mathematics itself. The case for ‘realism’ being developed

in the present section is thus a qualified one: at least sets of things, real

numbers, and functions from various kinds of things to real numbers

should be accepted as part of the presently indispensable or nearly

indispensable framework of... physical science... and as part of that

existence we are presently committed to. But sets of very high type or very

high cardinality (higher than the continuum, for example), should today be

investigated in an ‘if-then’ spirit. One day they may be as indispensable to

the very statement of physical laws as, say, rational numbers are today;

then doubt of their ‘existence’ will be as futile as extreme [nominalism]

now is. But for the present we should regard them as what they are –

speculative and daring extensions of the basic mathematical apparatus of

science. (Putnam, 1979a, pp. 346–347)

Quine’s position on unapplied mathematics is similar.

So much of mathematics as is wanted for use in empirical science is for

me on a par with the rest of science. Transfinite ramifications are on the

same footing insofar as they come of a simplificatory rounding out, but

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anything further is on a par rather with uninterpreted systems. (Quine,

1984, p. 788)

This ‘rounding out’ adds a bit more set theory than what is needed for the practice of science, but

it is still clear that the QIA creates a cleft within mathematics between the applied and unapplied.

The difficulty with this division is that it seems absurd to many philosophers of mathematics

who hold strong realist positions, such as platonists, and it also may seem ridiculous to practicing

mathematicians regardless of their philosophical leanings. Higher order set theory is based on the

same axioms and principles as the portions of set theory that have found successful application in

science. To the mathematician, there is no obvious dividing line in the discipline of set theory

like the one that the QIA supporter insists on. Quine eventually softens his position by

considering the higher reaches of set theory to be “meaningful because they are couched in the

same grammar and vocabulary that generate the applied parts of mathematics. We are just

sparing ourselves the unnatural gerrymandering of grammar that would be needed to exclude

them.” (Quine, 1990, p. 94) Although this is generous of Quine, it is a hollow victory at best.

While some portions of mathematics are fortunate enough to luck into having ontological status

simply because it is too inconvenient to explicitly rule them out, thus does not change the fact

that there is still a divide within mathematics.

Quine’s position leads one to wonder what practicing mathematicians are actually doing. The

QIA divides mathematical practice into two basic camps. Mathematicians can be working in

areas and on mathematical entities that are already accepted and exist as they have found

indispensable application to our best scientific theories, or mathematicians are working in areas

and on mathematical entities that do not exist as they have not yet found application. For this

latter camp it is unclear how we can make sense of their pursuit. Is it just some formal game? Is

it some mental exercise in logic? This is even harder to understand when we consider historical

examples where mathematical entities were developed and only later found application in

science.

Consider complex numbers. When Cardano and Bombelli first introduced the concept it was met

by staunch resistance throughout the 18th century by most mathematicians. (Kline, 1972, pp.

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592–596) Understanding of complex numbers continually improved until Gauss provided a

geometric interpretation in 1831. The association of a complex number with a point in a plane

led Gauss to believe that the “intuitive meaning of complex numbers [is] completely established

and more is not needed to admit these quantities into the domain of arithmetic.” (as quoted in

Kline, 1972) The mathematical community was confident that complex numbers should be

admitted to their body of knowledge. Later on, complex numbers were found to have vast

application to physics such as in dynamics, electromagnetism, and quantum mechanics to name a

few. According to the QIA, complex numbers did not exist until this later indispensable

application to science was uncovered. From a mathematician’s perspective this is simply

incorrect. What Cardano and Bombelli were discussing was interesting but ultimately

undeveloped and hence unacceptable in their own time. The important event was when Gauss

provided a clear understanding of complex numbers that satisfied the mathematical community.

The scientific application that came later did nothing to change the minds of mathematicians

regarding the nature of complex numbers as it had already been settled according to

mathematical standards.

This example illustrates two major problems for the Quinean. First, the QIA gives no account as

to how it is that recreational mathematics develops without any notion of application in mind, yet

can one day end up one day being indispensable to scientific practice. Surely we do not want to

admit that complex numbers, which was simply a ‘speculative and daring’ extension or

exploration just coincidentally happened to be useful and true.9 Secondly, on Quine's philosophy

the practice of science imposes certain restrictions on how we can interpret and understand

mathematical practice that conflict with how mathematicians wish to interpret and understand

their own discipline.

9 Of course we could actually admit just this - it is a cosmic coincidence. It is perfectly reasonable that our intuitions

do not find such an admission embarrassing. However, accepting coincidences such as these seem to run against the

fundamental intuitions of the scientific realist, and those are the very group which the QIA is meant to target.

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Maddy highlights this second problem through an analysis of the continuum hypothesis and the

search for a new independent axiom for set theory.10 In order to resolve the status of the

continuum hypothesis, a new axiom must be added to the standard ZFC axioms of set theory.11

The axiom under consideration is the axiom of constructibility, often written as V = L.12 Gödel

proved that V = L implies the truth of the continuum hypothesis, but whether or not we should

adopt V = L as a new axiom is still an open question. Quine utilizes a naturalistic analysis to try

to decide on the new axiom.

Further sentences such as the continuum hypothesis and the axiom of

choice, which are independent of those [set theory] axioms, can still be

submitted to the considerations of simplicity, economy, and naturalness

that contribute to the molding of scientific theories generally. Such

considerations support Gödel’s axiom of constructibility, V = L. It

inactivates the more gratuitous flights of higher set theory, and incidentally

it implies the axiom of choice and the continuum hypothesis. (Quine, 1990,

p. 95)

Quine believes that adopting V = L, which implies the truth of the continuum hypothesis, as an

axiom of set theory would be the most advantageous for the practice of science, thus we should

do exactly that. However, Maddy points out that this is the exact opposite conclusion that the set

theoretic community have proposed. V = L restricts the possibility of many potential sets from

our universe. This ‘minimizing’ property has been met with resistance from set theorists who

prefer to ‘maximize’ the universe of sets. Hence, set theorists wish to reject V = L as doing so

would maximize the potential of set theory as a foundational tool, and thus invalidate the

10 The continuum hypothesis was advanced by Georg Cantor in 1878. It states that there is no infinite set that has

cardinality greater than the set of the natural numbers, and less than the set of the real numbers. The continuum

hypothesis is independent of the ZFC axioms of set theory.

11 There are infinitely many candidates that would resolve the status of the continuum hypothesis. Consider the

axiom which simply states that ‘the continuum hypothesis is true’, for example. However, the only axioms with any

serious potential for being adopted would have to satisfy the mathematical community of having certain properties

that qualify it for being an axiom, such as being ‘self-evident’ or ‘obvious’. Of course the picture is much more

complicated than that in practice. See Maddy (1988) for an excellent discussion on the axioms of set theory.

12 This axiom states that the universe of all sets is exactly equal to the universe of all constructible sets.

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continuum hypothesis. Notice that these arguments have nothing to do with scientific practice or

application, but are solely based within the methodology of mathematical practice.

If a mathematician is asked to defend a mathematical claim, she will most

likely appeal first to a proof, then to intuitions, plausibility arguments, and

intra-mathematical pragmatic considerations in support of the assumptions

that underlie it. From the view of the indispensability theorist, what

actually does the justifying is the role of the claim, or of the assumptions

that underlie its proof, in well-confirmed physical theory. In other words,

the justifications given in mathematical practice differ from those offered

in the course of the indispensability defence of realism. (Maddy, 1997, p.

106)

In the case of the continuum hypothesis, not only are the justifications different, but even worse,

so are the conclusions are drawn by the mathematical and scientific communities. Is it reasonable

for us to reject what actual mathematicians believe to be the case in order to satisfy concerns

regarding indispensability? This would certainly be a troubling conclusion for mathematical

practice, and moreover one that mathematicians would certainly ignore.

Maddy finds this conflict between mathematical and scientific practice introduced by the QIA so

problematic that she rejects the QIA and Quinean naturalism altogether. Instead, she advocates

for her own brand of mathematical naturalism which we will discuss in chapter 6. Although she

is correct in pointing out this inconsistency between indispensability concerns and actual

mathematical practice, I think she has missed the mark in her conclusion. Maddy is too quick to

take Quine’s word as the correct application of indispensability concerns when evaluating the

continuum hypothesis. It could certainly be that Quine was incorrect in his attempt to weigh in

on the practical application of V = L, and that he erred in accepting it rather than denying. Of

course, such a claim appears to have an ad hoc flavour to it meant solely to save Quine from the

embarrassing predicament that Maddy has cornered him in.

Regardless of Quine’s reputation, what is really at stake here is the independence of

mathematical practice from that of science. In the face of a conflict between the interest of

scientific and mathematical concerns regarding mathematics, which discipline should win the

day? Siding with scientific practice seems to be somewhat ludicrous. Mathematicians would pay

no heed, and there is no historical evidence that such an approach has ever been fruitful, while at

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the same time there is ample evidence that mathematicians developing mathematics by their own

standards has been incredibly successful indeed. Siding with the mathematicians thus seems to

be the more reasonable choice, but this is troubling for the Quinean naturalist. If mathematicians

govern mathematics with no concerns about applicability towards science, and subsequently

science finds indispensable application of said mathematics, does this mean that science does

answer to some ‘supra-scientific’ tribunal – that of mathematics? We could conceivably get

around this problem by noting that in general mathematical and scientific practice tend to be in

accordance with regards to mathematics. Quine was simply wrong in the V = L case. But now we

are left with having to explain how scientific and mathematical practice are always in line while

at the same time the QIA tells us that only applied mathematics is meaningful and real.

Explaining this could pose to be even more difficult than the original problem.

My point here is that it is not enough to simply condemn the QIA because Quine believes

something that set theorists do not. The bigger issue is how Quinean naturalism and

mathematical practice relate, if at all. Sadly, this question has been all but ignored. There is a

general disrespect towards mathematical practice within this naturalistic framework even though

it acknowledges the indispensable nature of mathematics in our best scientific theories.

Resolving this should be a primary concern for any naturalist.

3 Weakness Three: Vagueness

At the very core of the QIA lie the notions of indispensability and that of existence. An inherent

problem with the QIA is that these notions are somewhat vague. What do we really mean by

indispensable? Is something indispensable to science if we cannot do science without it? Or does

it require some stronger condition that would narrow the field? What does existence or

ontological commitment really imply? Do abstract objects like numbers and sets exist in the

same way that tables and chairs exist? Or do they exist in some other sense, perhaps as a mental

or non-spatio-temporal object? These terms mean different things for different people and

without a clear and common understanding of their usage in the QIA the argument cannot be

followed.

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Quine talks about existence in a very ambiguous way. Even though they are outside space-time,

abstract objects exist in the same sort of way as unobservable physical objects. This is because

abstract objects are posited and utilized in science in the very same way and for the exact same

reasons that physical objects are posited and utilized. Our theories also commit to them in the

same way via quantification. This may be the case, but surely Quine does not mean that objects

like numbers and sets are just like protons and quarks. By most standard accounts of

mathematics, if mathematical objects are to exist at all they would be non-spatio-temporal in

nature. That is certainly different than the way physical objects exist. Quine’s formulation opens

the door to many different mathematical realist positions depending on how one caches out their

understanding of existence.

Putnam’s version of mathematical realism is not the standard platonistic account. Putnam wants

to be,

realist with respect to mathematical discourse without committing oneself

to the existence of ‘mathematical objects’. The question of realism, as

Kreisel long ago put it, is the question of the objectivity of mathematics

and not the question of the existence of mathematical objects. (Putnam,

1979b, pp. 69–70)

This type of 'semantic' realism is inspired by Michael Dummett (1978). Semantic realism

towards a theory maintains only that the sentences of the theory are true, and that their truth is

independent of us – there is no commitment to existence. Putnam only endorses the

independence thesis of mathematical realism. Such a conclusion is certainly compatible with

Quine’s general assertion that mathematical objects exist just like physical objects. For scientific

realists, a true statement about physical objects is true independent of our minds.

Colyvan sees no inherent problem with Putnam’s semantic realism, but for him (and for me), the

most important and interesting question is: “Do mathematical objects exist?” (Colyvan, 2001a, p.

3) His interest is entirely metaphysical, and his formulation of the QIA reflects that. The

conclusion shifts from simple talk of existence to that of ontological commitment. Furthermore,

Colyvan endorses all three of the realist criteria of existence, abstractness, and independence.

There are other mathematical realists who actually deny abstractness while still maintaining

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existence and independence, but these positions are not the norm. Following Colyvan’s lead, I

will specify existence to mean ontological commitment and take the standard interpretation that

mathematical objects exist, are abstract, and are independent.13

The meaning of indispensability is much more difficult to understand. The first to critically

consider the claim that mathematics is indispensable to scientific practice was Field in his book

Science Without Numbers (1980). Recall that Quine associates existential quantification with

indispensability, and he basically assumes the indispensable nature of mathematics to be true.

The problem with this is that it is always possible to restate a theory so that it does not quantify

over any mathematics. We can simply just take our new theory to be the set of all consequences

that do not contain any reference to mathematical entities, or we can perform a Craigian

reaxiomatizaion to get a recursively axiomatized theory. The problem with this trivial approach

is that these new ‘theories’ are hardly of any significant interest as they are entirely parasitic on

the original mathematized theory in question. It does point out though that existential

quantification is not a suitable criterion for indispensability.

Putnam also believes in the importance of quantification and reference, but he adds to it the idea

of actually being able to ‘do’ science.

Now then, the point of the example is that Newton’s law has a content

which, although in one sense is perfectly clear… quite transcends what can

be expressed in nominalistic language. Even if the world were simpler

than it is, so that gravitation were the only force, and Newton’s law held

exactly, still it would be impossible to ‘do’ physics in nominalistic

language. (Putnam, 1979a, p. 338)

If we cannot ‘do’ Newtonian gravitational theory in this simplified world without mathematics,

then surely we cannot ‘do’ physics without mathematics in the actual world where things are

13 Colyvan also maintains that mathematical knowledge is not a priori as is commonly believed, but is actually a

posteriori. This is because the existence of mathematical objects is only established by empirical means through

their indispensable use in science. This leads to the belief that mathematics is contingent – that is, there exist

mathematical objects, but it isn’t necessarily so. These positions are not always associated with indispensability

arguments, and Colyvan admits that many often point to these conclusions to undermine the QIA. For the most part I

will ignore these issues since indispensability arguments do not explicitly commit themselves to mathematics that is

a posteriori or contingent. See Colyvan (2001a) chapter 6 for details.

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significantly messier. Field takes Putnam’s assertion as a challenge. Field advances his brand of

fictionalism by trying to show that it is indeed possible to do and understand Newtonian

gravitational theory without the use, reference, or quantification over any mathematical entity or

property. His hope is that in showing that this portion of physics can be understood and

performed in a purely nominalistic language, it is then reasonable to believe that all of physics

can be nominalized in a similar way. Thus, mathematics is actually dispensable to scientific

practice. This result is important to Field as he accepts the QIA as a valid argument, but at the

same time he wishes to maintain his nominalist beliefs without maintaining a double-standard.

The only way to do so is to show that the QIA is not sound. If he can demonstrate that

mathematics is not indispensable then there is no need to believe in the existence of

mathematical entities.

Field’s position is that strictly speaking mathematics is false and that mathematical entities are

merely useful fictions. When we say statements like ‘the number 2 is the only even prime

number’, this is like saying ‘unicorns have horns’. Both statements are false as neither the

number 2 nor unicorns actually exist. An important question is how is it that this false language

is so useful and important to the practice of science? This reflects a general problem for any

nominalist account. How can we explain what physicist Eugene Wigner (1960) famously calls

the ‘unreasonable effectiveness of mathematics’? Mathematical realists can utilize the claim that

mathematics is true and that mathematical statements genuinely refer to real objects.

Mathematics is so effective at producing true results in science just because mathematics itself is

true and mathematical objects exist. If Field is to deny the existence of mathematical objects,

then there is no recourse to the notion of truth in explaining why and how mathematics is so

effectively employed in our scientific theories.14

14 Colyvan (2001b) argues that the ‘unreasonable effectiveness of mathematics’ is as much of a problem for the

mathematical realist as it is for the nominalist. What is required by any position is an account of how mathematics is

actually applied, regardless of its ontological status. Mathematical realism merely assumes that it has a superior

solution to the problem of applicability without providing any actual account or explanation of how mathematics is

applied. Colyvan argues that this assumption is false.

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Field’s solution is to argue that mathematics is conservative. In order to have a believable

fictionalist position mathematical theories need not be true; they need only be conservative to be

useful. Mathematics is conservative if when it is added to a nominalist scientific theory then

every nominalist consequence from this expanded theory would have followed from the

nominalist theory alone. Field argues that conservativeness follows from the consistency of

mathematics. A consistent theory only allows us to deduce consequences that are logically true.

Hence if a consequence follows from a mathematized theory, then it should also logically follow

from the original nominalist theory as well. Of course it could turn out that mathematics is not

consistent, but this would be a shocking result and one that we would try to overturn in a variety

of ways. If mathematics is indeed conservative, then Field can explain how it is that mathematics

can be false yet still be extremely useful. The fact that it is false is irrelevant as it is consistency

and consistency alone that ensures we deduce true facts about the world. Furthermore, all these

true facts about the world would have followed from a purely nominalistic theory in the first

place. Mathematics, then, is solely a useful deductive tool that is not, in principle, necessary.

Field does not believe that this gives the nominalist license to utilize and quantify over

mathematical objects freely. The nominalist still needs to show that it is actually possible to have

a nominalist theory in the first place that could do the work on its own. Conservativeness

guarantees that “once such a nominalistic axiom system is available, the nominalist is free to use

any mathematics he likes for deducing consequences.” (Field, 1980, p. 14) This sets the stage for

Field’s next move.

Field’s second task is to show that reference to any existential claims regarding mathematical

objects is eliminable from Newtonian gravitational theory. Field makes use of representation

theorems from measurement theory to give a nominalistic version of space-time. From there he

extends his treatment to laws, differentiation, and Newtonian gravitational theory in general. The

details of this approach are not important to our present discussion. What is worth noting is that

Field believes that this nominalist formulation of Newtonian gravitational theory is ‘attractive’. It

is certainly more attractive than the trivial methods presented above, but more importantly Field

claims that this version is attractive as it does not appeal to any arbitrarily selected objects to

serve as particular units of length or basis for a coordinate system, and that it reveals that all

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physical explanations are ‘intrinsic’ explanations. This demonstrates what Field actually believes

to be at the root of indispensability. Something is dispensable not just because it is eliminable. It

must be eliminable, and the resultant theory must be ‘attractive’.

Many commentators remained unconvinced that Field succeeded in his ambitious project.

Objections stem from the use of second-order logic, the utilization of some notions in set theory,

problems with determinism, projected difficulties in nominalizing general relativity, the

treatment of phase-spaces, and more.15 These technical objections are not important to us here,

but it is safe to say that in the face of all these difficulties the general consensus is that Field’s

project of nominalizing physics did not, and is not likely to succeed. What is important to us is

how Field understands the notion of indispensable. Clearly the notion of ‘attractiveness’ of a

nominalized theory is crucial to this understanding, but Field does little to expand on it. He even

admits that actual usefulness is not important. “I do not of course claim that the nominalistic

concepts are anywhere near as convenient to work in solving problems or performing

computations: for these purpose, the usual numerical apparatus is a practical necessity.” (Field,

1980, p. 91) But how can we possibly claim that a nominalized theory is ‘attractive’ yet at the

same time be practically useless compared to its mathematized counterpart? What we need is to

look at other theoretical virtues, such as simplicity, explanatory power, fruitfulness, etc., in order

to facilitate an informed judgment.

In a critique of Field’s argument, Colyvan takes this very route. His understanding of

indispensability is that an entity is dispensable if there exists an alternate theory with the exact

same observational consequences where all mention of the entity is eliminated, and that this

theory must be preferable to the first. (Colyvan, 2001a, p. 77) Not surprisingly Colyvan asserts

that the regular mathematized theory is preferable to a nominalized counterpart mainly due to the

simplicity and explanatory power that the mathematics brings to the table. Colyvan introduces

the practice of theory choice in order to define what it means to be indispensable. His reasoning

is that mathematics is indispensable to science because any alternate theory that eliminates the

15 For a good survey of many criticisms of Field's project see Burgess and Rosen (1997).

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use of mathematics would always be less preferable. The biggest advantage for mathematics is

that it has the virtue of making a theory significantly simpler.

Joseph Melia also agrees that theoretical virtues are important for theory choice, and that

mathematics makes a theory simple, however, he argues that though “such principles may justify

postulating quarks and space-time, it is a mistake to think that these principles justify our

postulation of mathematical objects.” (Melia, 2000, p. 472) He asserts that it is still possible to

‘weasel’ away from commitment to mathematics even if mathematics is indispensable to science.

Melia looks to the actual practice of scientists for his inspiration. He claims that even though

scientists use and quantify over mathematical entities, they still do not believe that mathematical

objects exist. Melia uses a naturalistic argument and states that we should take the practice of

science at face value and assume that scientists are not unreasonable in their beliefs. His goal is

to account for scientists being realists towards physical objects indispensable to science, while at

the same time maintaining a nominalist attitude towards mathematics. What makes Melia’s

position interesting is that he happily accepts both premises of the QIA; most importantly he

agrees that mathematics is indispensable to our best scientific theories according to Colyvan’s

criteria. The crux of Melia’s argument is that mathematics is not indispensable in the right way

such that it leads to ontological commitment.

An unsatisfactory element to Melia’s position is that he offers no actual evidence for his claim

that most scientists have this split attitude towards physical and mathematical entities employed

in science. Although he cites this as a motivating force, it definitely remains to be seen if the

majority of scientists truly hold such a position. Regardless, Melia could easily construct his

argument as a defense of those that do hold this position independent of whether or not they

comprise the majority. It would lose the naturalistic foundation which adds to the credibility of

his position, but would still be sufficient to undermine the validity of the QIA. Another issue

with his approach is the role of philosophy within the naturalistic framework. As we saw earlier,

the practice of philosophy is supposed to be in line with that of science. However, this does not

mean that it is outside of the philosopher’s role to criticize the beliefs and practices of scientists.

Philosophers are not there simply to provide idle justification for these beliefs and practices

simply because the majority of the scientific community holds them. Rather, philosophers aim to

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make sure that the beliefs and practices that scientists maintain truly hold up to the standards set

out by our scientific methodologies. It is one thing to say that beliefs and practice should be

informed and follow from good and accepted methodologies and principles. It is an entirely

different thing to assert that they actually do follow in the appropriate way. Melia seems to

assume that the goals and the actual practice of scientists are one and the same. Thus, the

philosopher’s role is only to justify, not critique. However, this is not necessarily what

naturalism implies. There is certainly a role for the philosopher to contribute meaningfully to the

practice of science by ensuring that scientists live up to the high standards and methodologies of

science, whatever those may be. So even if we grant Melia’s unjustified claim that the majority

of scientists are realists towards physical objects but nominalist towards mathematical entities,

this does not mean that this is the right attitude to have. We as philosophers should not be in the

business of justifying such a stance if it is somehow unnaturalistic.16 This, of course, is the very

position that Quine maintained when he devised the QIA. Melia needs to go further and argue

that scientific realism paired with mathematical nominalism is the appropriate way to divide our

ontology.

All this talk about the majority of scientists is merely posturing meant to enhance the importance

and justification of Melia’s argument. Even if it is not as well motivated as Melia would have us

believe, this does not affect the importance or legitimacy of his attack against the QIA. Melia’s

strategy is two-fold. First he presents the ‘weaseling argument’ which is a method so that we can

make use of mathematics in the practice of science but at the end of the day still maintain a

16 In the previous discussion regarding Maddy’s atom example I was skeptical of Colyvan’s stance that the naturalist

could claim that scientists were wrong for denying the existence of the atom prior to Perrin’s crucial experiments.

Here I endorse the idea that the philosopher can criticize the practice of scientists within the Quinean naturalistic

framework. These positions are not contradictory. In the atom example, what is fishy about claiming that the

scientists were wrong is that they were discussing a physical entity that was posited by their own physical theories as

a useful tool or explanatory device. It was their very own scientific methodologies and principles which caused them

to withhold ontological rights from the atom until it could be more clearly experimentally verified. All of this seems

perfectly within the desired practice of science. In the present example, scientists who deny the existence of

mathematical entities are doing so not for any reasons dictated by scientific methodology or principles. No further

experiment or criteria can ever be satisfied that would change their minds. The reason being is that this prejudice

against mathematical objects originates from ‘supra-scientific’ beliefs. It is this behaviour that the philosopher of

science should question and be critical of.

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nominalist position towards mathematics. Secondly, Melia argues that there is indeed good

reason to employ the weaseling method.

Melia first points out that Field’s attempt to show that mathematics is dispensable to science

fails. His objection is not about the technical work of nominalizing Newtonian gravitational

theory. Melia (2000) argues that it is Field’s claim that mathematics is conservative that is false.

He considers two theories of mereology, T and T*, where T is a nominalist theory and T* is an

extension of T which utilizes and quantifies over functions and sets. Melia shows that T* does

not entail any new sentences in the nominalist theory, T; however, T* does entail the existence of

a special infinite region that T does not guarantee to exist. Such a region, although possible

according to T, is not directly implied by T alone. In essence, T* is a conservative extension of T,

but at the same time it has implications about the physical world that T does not. By rejecting

conservativeness, Melia is admitting that mathematics can help us express certain physical facts

about the world that we could not do with a nominalistic theory alone – mathematics is

indispensable to scientific practice. So, for any nominalist theory T and mathematical theory T*

as above, we may have to use T* to get the facts about the world correct. Regardless, Melia

insists that scientists and philosophers make use of mathematical theories all the time yet still

maintain a nominalist stance towards mathematics. Surely this is inconsistent. As Melia himself

asks, “[h]ow can anyone coherently assert P, know that P entails Q, yet deny that Q is the case?

How could it ever be rational to assert that P whilst denying a logical consequence of P?” (Melia,

2000, p. 466) The answer is that we can weasel ourselves out of such a seemingly incoherent

predicament.

The crux of the weaseling method is that it is perfectly legitimate to say “T* - but there are no

such things as functions or sets.” By doing this we can ‘subtract’ or ‘prune away’ the abstract

mathematical entities that are implied by T*. The idea here is that simply because we assert T*

does not mean we have to believe that every aspect of T* is actually true. In particular, the

mathematics employed in T* could be false. This may seem strange at first, but Melia contends

that we do this regularly in both everyday and scientific language. Consider the statement:

(1) All F’s are G’s, except for Rosie.

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There are two ways in which we can interpret (1). The first is to claim that (1) is contradictory.

What (1) is really saying is that “all F’s are G’s, and Rosie is an F that is not a G.” This is surely

not what is meant by statement (1), and is an uncharitable interpretation. Instead, we can

understand (1) as saying:

(2) All F’s, except for Rosie, are G’s.

There is no paraconsistency in (2) and it is perfectly clear in meaning. Statements such as (1) are

a manner of speaking wherein we can legitimately ‘take back’ certain claims that we made about

the world in a clear and coherent way. Melia also presents a scientific example where we

introduce a three-dimensional sphere in order to define a two-dimensional world by only taking

the surface of the sphere and pruning away everything else. His claim is that these examples in

common-language and in science make it plausible that we can make statements like

(3) There exists a function from space-time to the real numbers – but there

are no such things as functions or numbers,

without being incoherent. We should understand (3) to be a (2)-like statement. In this way we

can weasel away from the mathematical realist conclusion of the QIA.

A disanology with statements like (3) and statements like (1) is that for (1) there is a clear

reformulation in (2), whereas (3) does not have a clear reformulation. An obvious question for

people who utter statements like (1) is why would we ever do this when we could have just said

(2)? (1) is inherently more misleading and unhelpful. Melia says that this may be the case, but

there is no fault in someone simply being ‘longwinded’ so long as their actual meaning is clear.

In addition, sometimes we simply have to say things such as (1) as the clear reformulation is not

available to us. This is exactly the case for statements like (3). Mathematics allows us to say

things about the physical world that we simply cannot without. “Mathematics is the necessary

scaffolding upon which the bridge must be built. But once the bridge has been built, the

scaffolding can be removed.” (Melia, 2000, p. 469) The fact that a clear reformulation of

statements like (3) is unavailable does not matter. We can still weasel away from commitment to

mathematical objects.

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This last assertion has come under attack from Colyvan (2010). Colyvan argues that (1) is clear

and coherent because we have access to (2). Since there is no reformulation of statements like

(3), which Melia admits, then statements like (3) are neither clear nor coherent. Colyvan is

certainly correct that the availability of (2) is a sufficient condition for us understanding (1), but

is it necessary? Melia alleges that it is not necessary, and Colyvan asserts that in some cases it is

a necessary condition. Neither provides a concrete argument, but instead both use analogies to

make their case. As we saw, Melia says that understanding (3) is just like understanding (1),

irrespective of the presence of a nominalized reformulation. This analogy is weak at best. The

very way that Melia convinces us that (1) is okay is by using (2). He provides no example of

understanding a statement in the absence of a clear reformulation, so how can understanding

statements like (3) be just like understanding (1)? The very route that Melia takes to clarify (1) is

unavailable. That is what makes statements like (3) interesting and challenging to understand in

the first place. Nominalists such as Melia may find this analogy convincing as they are naturally

inclined to a nominalistic attitude, but for someone who has no prejudice towards the existence

of mathematical objects, or even worse for someone who leans towards the realist position, the

analogy that Melia presents is unsatisfactory.

Meanwhile, Colyvan accepts that in certain cases where we only weasel away trivial details the

weaseling method may work, but there also are situations where we clearly try to prune away too

much. In these cases, without a clear reformulation we are unable to weasel in a coherent way.

Colyvan’s analogy, like Melia’s, leaves much to the imagination.

We can change the story we are narrating by adding or subtracting minor

details, but we can hardly be thought to be telling a consistent story (or in

some cases, any story at all!) if we take back too much. In short, there are

limits to how much weaseling can be tolerated. J.R.R. Tolkien could not,

for example, late in The Lord of the Rings trilogy, take back all mention

of hobbits; they are just too central to the story. If Tolkien did retract all

mention of hobbits, we would be right to be puzzled about how much of

the story prior to the retraction remains, and we would also be right to

demand an abridged story – a paraphrase of the hobbitless story thus far.

(Colyvan, 2010, p. 10)

Taking back our commitment to mathematical objects in science is supposed to be another

example of subtracting or pruning away too much.

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Colyvan’s analogy is also problematic. If Tolkien had written near the end of The Lord of the

Rings trilogy that ‘oh, and there are no such things as Hobbits in Middle-earth’17, it is certainly

possible that we could make sense of the story. Middle-earth is full of magic, and Tolkien’s

writing is rich with metaphor. I see no reason why such an ending could not in principle be

believable. Colyvan merely states that it would be right to distrust such a bizarre ending.

However, for people who are predisposed to disbelieving in hobbits even prior to reading The

Lord of the Rings, then they would not only understand and accept this retraction of hobbits, they

would probably feel that it was entirely the right thing to do. This is the case for people who are

predisposed to disbelieving in the existence of abstract objects such as numbers or sets. They

find nothing strange at all about the scientist who weasels away from commitment to the

existence of mathematical objects, and moreover, they feel it is the only sensible thing to do. I

grant that there are some who could not understand such an unpredictable and bizarre ending to

The Lord of the Rings, but it is not clear why these people would have the right to demand, in

their opinion, a better tale.

Another problem with Colyvan’s example is the role that hobbits play in The Lord of the Rings.

Arguably, the trilogy is about hobbits. Hobbits are the central focus of the tale and the entire

trilogy is meant to tell their story. Yet no scientist or mathematician would contend that our best

scientific theories are actually about the mathematical entities that permeate them. Taking back

mathematical entities from our ontology after using them in our theories is not as drastic as

Colyvan’s example suggests, as mathematical objects are not the focus of those theories. Rather,

they are secondary to the true object in question: the physical world. One could easily grant that

Colyvan is correct in pointing out that it would be nonsensical to retract the existence of hobbits

in The Lord of the Rings, yet still not acquiesce to Colvan’s main point that weaseling away from

mathematical objects in science is nonsensical as well. The two are simply disanalogous.

17 Middle-earth is the fictional land in Tolkien’s books. I stipulate that Hobbits don’t exist in Middle-earth to remove

the obvious point that Hobbits don’t exist at all as they are fictional characters in a fictional land. Allowing truths of

fictional entities, then it is the case that Hobbits exist in Middle-earth according to Tolkien.

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As it stands, neither Melia nor Colyvan have done enough to show the legitimacy or illegitimacy

of the weaseling argument. Melia has definitely revealed an interesting avenue for the nominalist

to pursue in avoiding the force of the QIA, but it is unclear what the limits the weaseling method

is. As I have suggested, whether or not you believe that the weaseling method can successfully

apply amounts to how predisposed you are to nominalism or mathematical realism in the first

place. This is a problematic conclusion as then the weaseling argument does little to convince

either side of anything. Regardless, the weaseling argument offers a potential solution for the

nominalist who wishes to make use of mathematized scientific theories without committing to

mathematical objects, and without having to pursue a Field like project of nominalizing all our

scientific theories.

The weaseling argument represents a new tool for the nominalist. What Melia does next is

provide reasons for why we should use it. His claim is that although mathematics is

indispensable to science, it is not indispensable in the sort of way that would normally lead to

ontological commitment. The main attraction of using mathematical objects in our scientific

theories is that it makes our theories simpler, albeit at the cost of a complex ontology. Simplicity

is one of several theoretical virtues that are generally used in assessing the merits of scientific

theories. However, Melia claims that mathematics does not make the world a simpler place, but

rather it only makes our theories simpler. Consider any unobservable entity postulated by our

scientific theories. These unobservables are put forward to account for a wide range of other

different kinds of objects or observable phenomena. They work in simplifying our account of the

world. But mathematics does not behave in this way. The world is not the way it is in virtue of

the fact that mathematics exists. In the mereology example, the infinite region that T* guarantees

to exist does not exist because of the mathematical objects employed in T*. Melia believes that

mathematics only serves as an aid in descriptions and in indexing physical facts in our theories.

Mathematics allows for ways of expressing concrete possibilities that we may not be able to

express without it, yet ultimately it has no real role in simplifying the world as we know it.

Claiming that indispensability leads to ontological commitment is a naïve view of

indispensability. Committing to mathematical objects does not make the world a simpler place,

and without this virtue accepting them only comes at a cost to the complexity of our ontology.

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Melia’s argument is that mathematics only serves to index or represent physical facts, and thus is

not indispensable in the right way to lead to ontological commitment. This indexing argument

provides the justification for the nominalist to make use of weaseling to restrict mathematical

entities from his ontology. We will take a critical look at the indexing argument in chapter 3.

Interestingly, although Melia argues that mathematics does not add to the simplicity of our

scientific theories due to its role of indexing physical facts, he does admit that there could be

other theoretical virtues that mathematics do actually enhance.

Of course, there may be applications of mathematics that do result in a

genuinely more attractive picture of the world – but defenders of this

version of the indispensability argument have yet to show this. And

certainly, the defenders need to do more than point to the fact that adding

mathematics can make a theory more attractive: they have to show that

their theories are more attractive in the right kind of way. That a theory

can recursively generate a wide range of predicates, that a theory has a

particularly elegant proof procedure, that a theory is capable of making a

large number of fine distinctions are all ways in which mathematics can

add to a theory’s attractiveness. But none of these ways results in any kind

of increase in simplicity, elegance or economy to our picture of the world.

Until examples of applied mathematics are found that result in this kind of

an increase in attractiveness, we realists about unobservable physical

objects have been given no reason to believe in the existence of numbers,

sets or functions. (Melia, 2000, pp. 474–475)

Colyvan (2002) and Baker (2005, 2009) were quick to take up the challenge by providing

examples of how mathematics does genuinely enhance our explanatory power. For his part,

Melia accepts that this is the best way to refute his own position.

In my view, Colyvan’s strategy is the best way for those who want to defend the

indispensability argument. Were there clear examples where the postulation of

mathematical objects results in an increase in the same kind of utility as that

provided by the postulation of theoretical entities, then it would seem that the

same kind of considerations that support the existence of atoms, electrons and

space-time equally supports the existence of numbers, functions and sets. (Melia,

2002, pp. 75–76)

This has opened the door for mathematical realists so that if it can be shown that mathematics is

indispensable to our best scientific theories in the right way – namely that mathematics is

explanatorily indispensable – then we should commit to the existence of mathematical objects.

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What indispensability actually means is quite controversial and complex. Quine’s view was that

quantification is the key to indispensability. It turns out to be trivial to remove quantification via

Craigian reaxiomatization, so this is not a sufficient way to characterize indispensability. Field

introduces the notion of conservativeness and a non-trivial reformulation that is ‘attractive’ in

some respects. Colyvan argues that this is not enough as any such nominalized theory needs to

be at least as attractive as the one with mathematics. Any mathematized theory has an

overwhelming advantage in the simplicity that it provides. Finally, Melia points out that

simplicity is not enough. Making a theory simpler is not the same as making our understanding

of the world simpler. Mathematics needs to be indispensable in the right way by impacting our

views and understanding of the world, such as by being indispensable in scientific explanation.

Although there are still many points of contention, mathematical realists and nominalists

presently generally agree that the best way in which we should understand the notion of

indispensability in the context of the QIA is that mathematics is indispensable to our best

scientific theories if it is indispensable to scientific explanations.

Since the exchange between Melia and Colyvan, the discussion has almost exclusively focused

on the explanatory role of mathematics in our best scientific theories. Is mathematics

indispensable in scientific explanation? In a trivial sense the answer is most likely yes. As noted

above, the general consensus today is that Field was unsuccessful in showing that mathematics is

dispensable to Newtonian gravitational theory, and thus also unsuccessful in showing that

mathematics is dispensable in our best scientific theories in general. Seeing as how explanations

are such an integral part of science, it seems perfectly reasonable to assume that mathematics

will be indispensable to scientific explanations. However, as per Melia’s line of reasoning,

mathematics may certainly be indispensable to scientific explanation, but the important question

is it indispensable in the right way? Is it explanatorily indispensable? Unfortunately, the present

discussion by the main actors is unsatisfactory as no one has developed an account of what it

would take for mathematics to be considered explanatorily indispensable.

4 Internal and External Mathematical Explanations

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At this point we need to develop a better understanding of what is meant by mathematical

explanation. There are at least two major types of mathematical explanation. We can have a

mathematical explanation of a mathematical fact, called an internal mathematical explanation, or

a mathematical explanation of a physical fact, called an external mathematical explanation.18

Internal explanations somehow explain particular facts in the realm of mathematics, such as why

a particular theorem is true. A proof of a theorem certainly demonstrates that it holds, but the

belief is that there is an important difference between a proof that merely demonstrates the result,

and a proof that genuinely explains it. Internal explanations need not even be a proof at all. It

could simply be a demonstration of how otherwise seemingly disparate mathematical facts are

actually related. We will return to internal explanations in chapter 4.

External explanations are explanations such that the mathematics explains a physical fact about

the world. Although it seems perfectly plausible that internal explanations exist, it is unclear if

external explanations do at all. No one doubts the immense utility and the ubiquitous use of

mathematics in scientific explanations. What we mean here, then, is to distinguish between

standard mathematical scientific explanations, where the mathematics is utilized for its

simplicity or deductive power, and genuine mathematical scientific explanations. Genuine

mathematical scientific explanations are explanations where the mathematics involved actually

confers an indispensable explanatory role to the overall explanation.

The goal is to differentiate between the use of mathematics as simply a representational tool and

the use of mathematics as an explanatory device. For example, suppose I want to explain the

migratory patterns of birds by modeling certain facts about the Earth. I could construct a

mathematical model that incorporates factors such as temperatures, daylight, food availability,

wind patterns, past migration information, etc. From this mathematical model I could deduce,

and thus explain the migration patterns in question. Mathematics is employed extensively, but at

the end of the day no one would say that the mathematics actually explains the behaviour of the

birds. What is really at the core of the explanation are ecological and biological facts – the very

18 See Hafner and Mancosu (2005) for a nice discussion on internal and external mathematical explanation.

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facts that we represented using mathematics. Mathematics was an important, and perhaps

indispensable19 tool for making sense of all of the information and deducing the relevant

conclusion, but it was not in itself explanatory. The original mathematical explanation is not

genuine, but rather is an example of a standard mathematical explanation in science.

So what is required for mathematics to be explanatory? There are two reasonable ways to

understand how mathematics could be explanatorily indispensable. The first is to claim that if we

have an external mathematical explanation, and at the same time we cannot provide an

alternative explanation in purely physical, nominalistic language, then the mathematics in the

explanation is explanatorily indispensable. This approach is unsatisfactory. It is simply parasitic

on the more general claim that mathematics is indispensable to science at large; i.e. there is no

reformulation of science in purely nominalistic terms. Melia and other nominalists would agree

with this claim, but they would still reject the notion that mathematics is genuinely explanatory.

The second way is to assert that there is something about the mathematics itself that explains the

physical fact in question. If we were no longer privy to the mathematics employed then we

would actually lose the explanation entirely. In essence, the mathematics is contributing to the

explanation in an explanatory way. If such an explanation exists, then all parties would concede

that mathematics genuinely explains.

Consider the bird migration patterns example. If we were to somehow not know or forget our

mathematics, would we no longer be able to explain migration patterns? We certainly would not

be able to explain it in the same way as a sophisticated mathematical model could, but we could

definitely cite the same underlying factors that the mathematics represented in the first place.

Our explanation would appeal to the very same ecological and biological factors as before.

Without a doubt we would lose substantial predictive power and other virtues, but we could still

explain the phenomenon at hand. Contrast that with what would happen if we lost all knowledge

of certain ecological or biological factors, such as temperature or sunlight. In this case, even

19 When I say indispensable here I mean that mathematics may be indispensable in expressing certain physical facts

or as a deductive tool. This notion of indispensable is perfectly acceptable to nominalists such as Melia and need not

confer any ontological commitment.

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though we have our mathematical tools available to us we would fail in explaining the migration

patterns. The intuition here is that without knowledge of temperature or sunlight we would lose

the explanation entirely. Hence it is these factors that are genuinely explanatory. The same

cannot be said about the mathematics.

It may be hard to imagine what sort of explanation would depend on mathematics for its

explanatory force. We will look at many supposed examples in chapter 3 when we analyze

mathematical explanations closely. Hence forth, when I speak of standard mathematical

explanations I mean external mathematical explanations in which the mathematics is not

indispensably explanatory. By genuine mathematical explanation (GME) what I am referring to

are external mathematical explanations where the mathematics is explanatorily indispensable; the

mathematics is not playing a representational role, but actually confers explanatory power to the

overall explanation. Finally, any general reference to mathematical explanation is to the external

sort, and not internal.

5 The Enhanced Indispensability Argument

With the focus placed squarely on mathematical explanation, Baker (2005, 2009) puts forward a

new indispensability argument for mathematical realism. His Enhanced Indispensability

Argument (EIA) is meant to be an improved version of the QIA.

(EP1) We ought to rationally believe in the existence of any entity that

plays an indispensable explanatory role in our best scientific

theories.

(EP2) Mathematical objects play an indispensable explanatory role in

science.

(EC) Hence, we ought rationally to believe in the existence of

mathematical objects.

(Alan Baker, 2009, p. 613)

Recall that (P1) and (P2) of the QIA are the naturalist and the holistic premises respectively.

Here they are replaced by a single premise, (EP1), which is implicitly appealing to inference to

the best explanation (IBE). Like its predecessors, the EIA is meant to target scientific realists

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who have not yet extended their realist sympathies over mathematics. The appeal to IBE in (EP1)

reflects this as IBE is generally considered to be the inferential tool that characterizes scientific

realism, and is often rejected or restricted by those who are anti-realist towards science. We will

perform a critical examination of IBE in chapter 5, but for now we will assume that scientific

realists have no problem with IBE as a tool to infer the existence of objects that genuinely

explain physical facts. Given that IBE is the key inference for the EIA, we can now see why

internal mathematical explanations are not considered in the argument. The standard usage of

IBE requires that we treat the explanandum in question as true, and then we infer that the best

possible explanans is also true. In internal explanations the explanandum would be some

mathematical fact. But if we are to assume this to be true, then this immediately implies that

mathematical objects must exist. Seeing as this is the very conclusion that we are trying to

establish in the first place with the EIA, such an assumption entirely begs the question. Thus,

only external mathematical explanations can be considered as this way we need not assume

anything about the nature of mathematical entities.

(EP2) is the indispensability claim. Unlike its Quinean counterpart, here the emphasis is that

mathematics is indispensable in scientific explanations. This move is clearly motivated by the

exchange between Melia and Colyvan. The claim is that GMEs exist, and hence that

mathematics is indispensable to science in the right way. From the first two premises it follows

that we should believe in the existence of mathematical objects.

In many ways the EIA is just a simple extension of the QIA. It leads to the same realist

conclusion, and it possesses the same three strengths that made the QIA so attractive in the first

place: it can face both of Benacerraf’s challenges, and it does not beg the question against the

nominalist. What makes Baker’s argument enhanced is that while it retains the same strengths as

the QIA, it supposedly does not suffer from the same three weaknesses discussed above.

The first two weaknesses of the QIA are directly related to its reliance on Quinean naturalism.

The first weakness is tied to the use of confirmational holism as the key tool for delivering

mathematical realism. As we saw, there are many good reasons to be suspicious of this holistic

thesis, but perhaps the most troubling is simply the fact that it is perfectly reasonable to be a

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scientific realist without also being a supporter of confirmational holism. If this is the case, then

any argument relying on confirmational holism is critically limited. The EIA is able to avoid this

weakness by making IBE the inferential tool. All the criticisms of confirmational holism are

entirely irrelevant to the enhanced argument. Even better, by tying its success to IBE, the EIA

more closely aligns itself with the standard views of scientific realists. The assumption here is

that any scientific realist necessarily endorses IBE. This assumption is unjustified, but it is a far

more believable than claiming that every scientific realist endorses confirmational holism. By

freeing itself from confirmational holism, the EIA is immune to the first weakness of the QIA.

The second weakness of the QIA is that it disrespects the actual practice of mathematics. The

reason for this is based off of its first premise which states that:

(P1) We ought to have ontological commitment to only those entities

that are indispensable to our best scientific theories.

A core principle of Quine’s naturalism is that science is the sole arbiter of our beliefs. This

allows us to reject epistemological or ontological claims from non-scientific disciplines such as

religion or voodoo. However, this also results in the unfortunate situation where mathematical

practice has no say regarding the epistemological or ontological status of mathematical entities.

In particularly embarrassing cases, Quine’s naturalistic views seem to put the claims and beliefs

of the scientific community directly at odds with those of the mathematical community. The EIA

appears to fare better in this regard. (EP1) is significantly weaker in that while it necessarily

asserts that we must commit to the existence of any entity that is explanatorily indispensable to

our scientific practice, it does not at the same time explicitly state that this is the only way in

which we can add to our ontology. This opens the door for mathematics to contribute to

ontological questions regarding its own subject matter.

Although it does appear that the EIA improves on the second weakness in that it respects

mathematical practice more than its Quinean counterpart, it is actually not clear that this is the

case. While it is certainly true that the EIA does not explicitly state that mathematical practice

has no say with regards to the status of its subject matter, this does not mean that the argument

endorses the input from mathematical practice either. Whether or not this is so depends entirely

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on what naturalistic views the EIA is committed to. Consider the case put forward by Maddy

where Quine believes that we should adopt the mathematical axiom V = L as it would benefit the

scientific community, but the set-theoretic community disagrees. Would Baker side with the

practice of science, or the practice of mathematics? Similarly, was the pivotal moment in the

acceptance of complex numbers Gauss’ geometric interpretation, or was it the indispensable use

of complex numbers in disciplines such as quantum mechanics? As it stands the EIA does not

give us guidance on these questions. The best we can say is that while the EIA may actually not

fare any better than the QIA with regards to the disrespect of mathematical practice, it certainly

can fare no worse.

The final weakness of the QIA is its vagueness with regards to the meanings of existence and

indispensability. The conclusion of the EIA is essentially identical to that of the QIA. For us, the

interesting interpretation of existence is that mathematical objects exist as abstract and

independent entities. It is in how the EIA commits to indispensability meaning explanatorily

indispensable that separates itself from the QIA and eliminates the vagueness surrounding the

Quinean argument. Although Baker codified this explicitly in his argument, he was not the first

to suggest that explanatory concerns be at the forefront. Colyvan wrote, “we could easily

construct an argument that relied on quantification over mathematical entities being

indispensable for explanations.” (Colyvan, 2001a, p. 13) Notice that Colyvan is still committed

to Quine’s view that quantification is the sign of indispensability. The EIA frees itself from this

and concentrates instead on the exact role played by the mathematics within a scientific

explanation. The EIA is precise in stating that mathematics plays an indispensable explanatory

role and hence is to be granted ontological rights. In doing say it faces Melia’s arguments head

on by stating that mathematics is explanatorily indispensable in the right way.

It is easy to see why Baker considers his argument to be an enhanced indispensability argument.

Yet for all the apparent improvements there are two significant obstacles to overcome before the

EIA can be considered a truly powerful argument for mathematical realism. The first is that it

must be established that there exist GMEs in science – it must be shown that (EP2) is true.

Almost all of the literature surrounding the EIA is focused on this issue and no consensus has yet

been reached. The second obstacle is showing that IBE is the right tool for the job. It is simply

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assumed that inferring the existence of abstract mathematical objects is the same in kind as

inferring the existence of unobservable physical objects. The justification of this assumption has

not been undertaken, but it is critical to the validity of the overall argument. The remainder of

this dissertation will be dedicated to overcoming these obstacles. Once this is complete, we will

be able to judge whether Baker’s argument truly enhances Quine’s argument or not.

6 What About Naturalism?

Quinean naturalism plays such an important role in the QIA. It helps to motivate the QIA, it is

essential for delivering the mathematical realist conclusion, and it provides a front line of

defense against many would be criticisms. Yet, as demonstrated above, Quine’s specific brand of

naturalism also leads to two critical weaknesses of the QIA. In order for the EIA to improve in

these areas it must be the case that the naturalistic views that the EIA depends on are different

from the Quinean picture. The difficulty here is that it is not an easy task picking out exactly

what type of naturalism the EIA subscribes to. Complicating this is that my present usage of the

term ‘naturalism’ is poor. Generally speaking, Quinean naturalism was defined in 1.2 as the

belief that:

(i) we should take the practice of science at face value,

(ii) science is not answerable to some supra-scientific tribunal,

(iii) science confirms in a holistic manner,

and,

(iv) science is the sole arbiter of our beliefs.

But when we refer to naturalism in general, what does this mean? At a bare minimum, naturalism

would surely include the basic idea of taking the practice of science at face value and thus would

endorse some form of scientific realism. But beyond that it is hard to say what else is necessary.

In his survey of naturalism, David Papineau begins with the statement, “the term ‘naturalism’ has

no very precise meaning in contemporary philosophy… It would be fruitless to try to adjudicate

some official way of understanding the term.” (Papineau, 2009) Maddy remarks that “these days,

it seems there are at least as many strains of naturalism as there are self-professed naturalistic

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philosophers.” (Maddy, 2001, p. 37) What, then, are we actually committed to if we endorse the

EIA?

(EP1) is the scientific realist premise of the argument, and thus we are meeting the bare

minimum requirement of naturalism and endorsing (i) from the Quinean view. In an earlier

discussion on indispensability, Baker gives us more clues as to what naturalistic position is

implied in his EIA.

There is one theme that will surface repeatedly in the subsequent

discussion and that I want to stress at the outset. It derives from the insight

that – given the naturalistic basis of the Indispensability Argument, which

rejects the idea of philosophy as a higher court of appeal for scientific

judgments, – the only sensible way of judging alternatives to current

science is on scientific grounds. If such alternatives are to be adequate,

they must preserve those features of our current scientific theories that are

of value to scientists. Many of these features may also be deemed valuable

from some broader philosophical perspective. But if there is conflict

between the verdicts of the scientist and the philosopher then it is those of

the former that must take precedence. (Alan Baker, 2001, p. 87)

It certainly sounds here as if Baker is endorsing a position similar to (ii). There is to be no supra-

scientific tribunal which judges our best scientific theories. Philosophers should work in line

with the current methodology and practices of the scientific community.

Even though this all sounds much like Quine, the fact that there is no mention of (iii) is what

makes Baker’s naturalism distinct. Confirmational holism is at the root of Quine’s beliefs on

science and ontology. Baker is clear that his position does not depend on confirmational holism

at all. The reason why we grant mathematics ontological rights is not because we are holists

regarding our best scientific theories, but rather because mathematical entities play the exact

same roles as do unobservable physical entities in our scientific theories: they are both

explanatorily indispensable. The very same reasons that lead to scientific realists believing in the

existence of protons and positrons are the very same reasons why we should believe in functions

and sets. More importantly, the very same inference used for adding unobservable physical

entities to our ontology is to be used to add mathematical entities as well. No extra baggage or

views on science are needed for the EIA to lead us to mathematical realism. If there was not a

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distinct departure from confirmational holism, then the EIA would not actually be an enhanced

argument at all. It would simply collapse back to the QIA in an essential way.

Laslty is whether or not the EIA assents to (iv). Maintaining that science is the sole arbiter of our

beliefs leads to the disrespect of mathematical practice. As shown above, it is possible for the

EIA to adopt a more respectful attitude towards mathematical practice based on whether or not

we commit to (iv) or not. At the present moment I am happy to leave this an open issue. The

reason being is that for the majority of the participants in the debate surrounding indispensability

arguments, respect towards mathematical practice is not a make or break issue. There are

certainly those who disagree with this and argue that disrespect towards mathematical practice is

a fatal flaw. I will return to this viewpoint in chapter 6. Regardless, the main claim that I wish to

advance is that even if the premises of the EIA are found to be true, the critical factor in gauging

whether the EIA is truly an enhanced argument over the QIA or not is if the EIA successfully

frees itself from any reliance on the thesis of confirmational holism. Even if the EIA is found to

improve significantly on the second and third weaknesses of the QIA, respect of mathematical

practice and vagueness respectively, dependence on confirmational holism would essentially

reduce the argument to a slightly reworded Quinean argument. This is not enough to consider it

enhanced as its success would still entirely depend on the acceptance of Quine’s holistic views.

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Chapter 3 Genuine Mathematical Explanation

The second premise of the Enhanced Indispensability Argument (EIA) asserts that mathematics

plays an explanatorily indispensable role in our best scientific theories. The veracity of this

statement has been subject to much debate. In this chapter we will take the first steps towards

determining if (EP2) is true or false. The determining factor will be whether or not a genuine

mathematical explanation (GME) exists. Recall that a GME is a mathematical explanation of a

physical fact such that the mathematics is explanatorily indispensable. Supporters of GMEs have

advanced many supposed examples, whereas detractors have been quick to argue that these

examples are not genuine at all, but merely are standard mathematical explanations where the

mathematics is not explanatory. My diagnosis of this difference in opinion is straightforward.

Neither side has put forward a clear and acceptable set of criteria for what makes a mathematical

explanation genuine. Without this there can be no consensus on what it means to be a GME, and

thus it is no surprise that there is no agreement on whether or not supposed examples are genuine

or not.

My methodology for making progress in this debate is threefold. First, the task will be to put

together a set of criteria which all parties can agree on such that, if satisfied, a mathematical

explanation would be considered genuine. These criteria will be developed from a close look at

the present stock of examples put forward by supporters of GMEs, and also by examining the

reasons for their rejection by nominalists. The most powerful argument against the supposed

examples comes in the form of the indexing argument inspired by Melia which we will dissect

carefully. With the criteria in hand I will then present a new example which I claim meets all the

requirements, and hence should be considered a GME. Although it may be tempting at this point

to claim that GMEs exist, I will withhold this conclusion until the third task is completed. What

still needs to be accomplished is corroborating this GME via an analysis using accounts of

scientific explanation. Chapter 4 will be dedicated to this final undertaking. Only once all three

of these tasks are complete will the status of GMEs and (EP2) be decided.

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1 Supposed Genuine Mathematical Explanations

Do examples of GMEs exist? Supporters of GME have put forward many examples in an attempt

to show that mathematics can genuinely explain. I present here a sampling of these examples as

well as some criticisms found in the literature.

1.1 Geometric Explanations

One of the most famous examples of mathematical explanation comes from Graham Nerlich

(1979) long before any of the present discussion on mathematical explanation got off the ground.

Consider a particle in motion along the surface of a sphere. Classical mechanics tells us that the

particle, free from any other forces, would move at a uniform speed along a geodesic.20 Now

consider a cloud of particles moving with the same direction and speed. If such a cloud was

moving along a flat surface of approximate Euclidean space, the size of the particle cloud would

remain the same. But if the same cloud is moving along the surface of a sphere, the size of the

cloud would change. Why is this so? The explanation is that geodesics, which are the paths that

the particles would follow on the surface of the sphere, are not parallel and thus the distances

between any two given geodesics constantly change. Thus the size of the particle cloud also

constantly changes.

This explanation seems straightforward, but what exactly is doing the explaining? Nerlich notes

that this is a “non-causal style of explanation.” (Nerlich, 1979, p. 74) It is the curvature of the

space that explains the changes in size. This explanation is clearly a mathematical one. The full

explanation invokes things such as geodesics, vectors, curvature, etc., and it is certainly

explaining a physical fact. But is it a GME? Is it the mathematics that is genuinely explaining?

The problem with considering it genuine is that the explanation seems to hinge on facts about

geometry. This is problematic as it is unclear if these geometric properties are actually

mathematical properties, or properties of space-time. If space-time naturally has geometric

20 A geodesic is the shortest path between two points on a curved space.

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properties, as many believe it does, then these properties are of a physical nature, and not

mathematical. It is these physical properties that are doing the explaining, and thus this example

is not genuinely mathematically explanatory at all.

Many mathematical explanations share this difficulty in that geometric properties can be seen as

physical and not mathematical in nature. Peter Lipton gives another example of a non-causal

explanation:

Suppose that a bunch of sticks are thrown into the air with a lot of spin so

that they twirl and tumble as they fall. We freeze the scene as the sticks are

in free fall and find that appreciably more of them are near the horizontal

than near the vertical orientation. Why is this? The reason is that there are

more ways for a stick to be near the horizontal than near the vertical.

(Lipton, 2004b, p. 9)

This explanation also relies on geometric properties as well as implicitly making use of some

basic probability, and so is certainly mathematical; yet we cannot consider it genuine in that

these geometric properties are typically considered to be physical rather than mathematical. It is

the physical nature of space that is conferring the explanatory power.

Due to this, such examples are not strong candidates for establishing the existence of GME. It is

difficult to determine if the geometric properties invoked are physical or mathematical. The

natural response of nominalists would be to side with the physical interpretation, so this is even

worse for the prospects of finding a GME of a geometric sort within the context of trying to

convince the nominalist that mathematics can genuinely explain. In criticizing some of

Colyvan’s purported examples of GME, Melia writes that the “explanation is a geometric

explanation... not a mathematical one.” (Melia, 2002, p. 76) Baker also admits that geometric

explanations are problematic. “[N]ominalists often object that geometrical explanations are not

genuinely mathematical... [I]t suggests that we should look elsewhere than geometry for a

convincing case of [genuine] mathematical explanation in science.” (Baker, 2005, p. 228) The

key here is Baker’s admission that it is the responsibility of supporters of GME to present an

example that will convince the nominalist. If the EIA is to have any force against the nominalist

we will have to come up with a GME that is not geometric in nature as these are too easily

dismissed by the nominalist.

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1.2 Contrived Explanations

Colyvan puts forward another example as a candidate for a GME. His antipodal weather patterns

example is as follows:

We discover that at some time t0 there are two antipodal points p1 and p2

on the earth’s surface with exactly the same temperature and barometric

pressure. What is the explanation for this coincidence? Notice that there

are really two coincidences to be explained here: (1) Why are there any

such antipodal points? and (2) Why p1 and p2 in particular? (Colyvan,

2001a, p. 49)

Colyvan notes that the second question can be explained in causal terms by looking at the causal

history of weather patterns to account for the behaviour of p1 and p2 at time t0. What this causal

explanation cannot explain is the first question. For this, the explanation follows from a corollary

of the Borsuk-Ulam theorem, a theorem of algebraic topology. This theorem proves that there

will always be two such antipodal points at any given time, thus explaining question (1). This

explanation makes direct use of things such as continuous functions, and hence is mathematical

in nature.

The problem with this example is that it is entirely contrived. Baker (2005) notes that no such

antipodal points have ever been empirically reported, nor are they likely to be discovered unless

we actually set out to look for them. But it seems that we would only set out to look for them if

we already knew the result from mathematics. It appears, then, that what Colyvan treats as an

explanandum, namely why two antipodal points have the same temperature and barometric

pressure at the same time, is actually not an explanandum at all but rather is a prediction. There

was never any phenomenon that scientists wanted explaining in the first place. If this is so, then

certainly the antipodal weather patterns example is not a GME as it is not even an explanation.

The underlying point here is that we still want to maintain our naturalistic sympathies. If we wish

to tie GME to the practice of science, we must actually look towards science for examples. It is

not naturalistic to artificially manufacture an example that fits the mold of a GME but is not

actually one utilized in scientific practice. We must seek examples that actual scientists are both

interested in and accept as good scientific explanations of physical facts that truly warrant

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explaining. In short, an example of a GME must be endorsed by the scientific community as a

good explanation. To claim otherwise would be equivalent to imposing external standards onto

the practice of science.

1.3 Optimization Explanations

Baker (2005, 2009) presents his widely discussed cicada example which avoids the previous two

pitfalls; it is not a geometrical explanation nor is it contrived. Certain species of North American

cicadas, large fly-like insects, have been discovered to share a peculiar property. These cicadas

exhibit a periodic life-cycle where they lie dormant in the soil for 13 or 17 years, depending on

their region, until they all emerge together as adults. What biologists have found interesting is

that the periodic length of the life-cycle is prime. What needs explaining, then, is the prime-

numbered-year cicada life-cycle lengths. Baker’s explanation of the 17 year life-cycle is as

follows:

(1) Having a life-cycle period which minimizes intersection with

other (nearby/lower) periods is evolutionarily advantageous.

(biological law)

(2) Prime periods minimize intersection (compared to non-prime

periods). (number theoretic theorem)

(3) Hence organisms with periodic life cycles are likely to evolve

periods that are prime. (‘mixed’ biological/mathematical law)

(4) Cicadas in ecosystem-type E are limited by biological

constraints to periods from 14 to 18 years. (ecological

constraint)

(5) Hence cicadas in ecosystem-type E are likely to evolve 17-year

periods.

(Baker, 2009, p. 614)

(1) is motivated by biologists who posit that minimizing intersection would be advantageous for

one of two reasons. First, it could be that at one point in time there were predators of the cicada

that also had a periodic nature. (Goles, Schulz, & Markus, 2001) Minimizing intersection would

minimize exposure to these predators, and this is certainly advantageous. Secondly, it could also

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be that minimizing intersection is advantageous to avoid hybridization with similar subspecies.

(Cox & Carlton, 1988; Yoshimura, 1997) (2) is a straightforward mathematical proof in number

theory that makes use of lowest common denominators of pairs of natural numbers. Finally, (4)

is established by looking at factors such as average sunlight, soil temperatures, etc., of the given

region. In a different region of North America a similar ecological constraint would appear but

with different values, such as between 12 and 16 years, that could be used to explain the life-

cycle duration of 13 years. (1) and (2) entail (3), and from (3) together with (4) we can deduce

(5).

The cicada example is the most discussed example in the current literature on mathematical

explanation. No one doubts that this is an example of a mathematical explanation, but the

important question is whether or not it is actually genuine. This question, as Baker recognizes, is

not easily answered.

What needs to be checked in the cicada example, therefore, is that the

mathematical component of the explanation is explanatory in its own right,

rather than functioning as a descriptive or calculational framework for the

overall explanation. This is difficult to do without having in hand some

substantive general account of explanation. (Alan Baker, 2005, p. 234)

Baker briefly canvases some accounts of scientific explanation to try to show that the cicada

example can be considered genuine, but it leaves much to be desired and is ultimately more

gesturing than it is argument. I will revisit this criticism in chapter 4.

Another explanation similar in form to the cicada example comes from Lyon & Colyvan (2008).

Honeybees make honeycombs to tile an area for the storage of eggs, pollen, and honey. All

honeybees make their honeycombs in the shape of a hexagon. Why is this the case? Why do

honeybees make honeycombs that are always made up of hexagonal cells and not some other

shape, or combination of shapes? I will recast Lyon and Colyvan’s explanation to follow the

form of the cicada example.

(i) Minimizing the amount of materials used to tile an area is

evolutionarily advantageous. (biological law)

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(ii) The hexagon is the most efficient way to tile an area using the

least amount of material. (mathematical theorem (Hales, 2001))

(iii) Hence, organisms that tile an area are likely to evolve to create

hexagonal structures. (‘mixed’ biological/mathematical law)

(iv) Honeybees build honeycombs to tile an area where biological

constraints limit the size and maximum number of possible

sides of their honeycombs to include the hexagon. (ecological

constraint)

(v) Hence honeybees make hexagonal honeycombs.

Honeycombs are made from beeswax which is produced in the glands of worker bees. Bees must

consume honey in order to produce the beeswax. Hence, minimizing the amount of beeswax

required to make honeycombs is advantageous as it minimizes the amount of food and energy

spent, thus implying (i). (ii) was proved mathematically by Thomas Hales in 1999. (iv) is simply

the observation that the hexagon is within the realm of possibility for the honeybee to construct.

The honeycomb example follows the same basic deductive pattern as the cicada example: (i) and

(ii) imply (iii), and (iii) together with (iv) imply (v). Interestingly, for reasons we will see below,

although Lyon and Colyvan present this as an example of mathematical explanation, they do not

think that it is genuine.

The cicada and the honeycomb examples exhibit the same basic structure. I call any such

explanation that follows this pattern an optimization explanation. Optimization explanations

must include the following three requirements.

(a) A scientific law that stipulates the requirement of being efficient

or optimal.

(b) A mathematical theorem that demonstrates the optimality of a

particular state of affairs.

(c) Real world constraints that show the optimal state of affairs is

obtainable.

These three requirements together can derive the specific conclusion that something, from living

things to elementary particles, obtains an optimized state. Optimization examples, or course,

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depend on much more than simply (a), (b) and (c). (a) is dependent on a body of science that we

accept and is able to generate the required law.21 Likewise, (b) requires a mathematical proof to

establish it as a theorem. (c) is most likely a contingent empirical fact that could be learned in a

variety of different ways.

Optimization explanations represent some of the most popular candidates for a GME. There are

definitely many other types of supposed GMEs that have been put forward22, but for now these

examples should suffice in giving us a basic understanding of the type of explanations that we

are considering.

2 The Indexing Argument Revisited

So far we have looked at three different ways in which nominalists have argued against the

existence of GMEs. First, supposed examples of GMEs can be rejected if they are found to be

geometrical or contrived in nature. Secondly, without an account of scientific explanation that

can make sense of mathematical explanations of physical facts it can be very difficult to make a

convincing case that mathematics genuinely explains. This was recognized by Baker as a

significant challenge for supporters of GMEs which we will attempt to address in chapter 4.

Finally, we saw that nominalists such as Melia believe that the sole role of mathematics is to

index physical facts. If this is the case, then even if mathematics is indispensable in scientific

explanations, it is not indispensable in the right sort of way that leads to ontological

commitment. This indexing argument draws a distinction between being mathematics being

explanatorily indispensable, and indispensable to explanations. Nominalists can grant the latter

but reject the former, and by making use of the weaseling argument they can freely utilize

21 I have not mentioned what I take ‘law’ to mean. Certainly I do not mean a strict law of nature, as (i) from the

honeycomb example would be disputed as being a fundamental law of nature. In this discussion I take a very loose

interpretation of ‘law’. If the reader prefers, ‘law’ can be replaced with something less problematic, such as

‘acceptable scientific generalization’.

22 See Pincock (2007, 2011) for his bridges of Königsberg example, Bangu (2012) for a probabilistic example from

economics, and Batterman (2002, 2008, 2010) for many examples of asymptotic mathematical explanation.

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mathematics without having to be mathematical realists. To better understand GMEs we first

have to make sense of how the indexing argument actually works.

The indexing argument has proven to be the most powerful weapon against the existence of

GMEs. Unfortunately, the argument itself has not been clearly defined. There is no explication

for what it means that the sole role of mathematics is to index physical facts, and due to this there

is no clear way to show that the indexing claim is not the case (or, for that matter, to show that it

is the case). My understanding of the indexing argument is that it is actually just an expression of

a core nominalistic intuition. This intuition is that mathematics cannot explain as the role of

mathematics in all mathematical explanations is solely to index or represent physical facts or

properties. In trying to explicate this intuition two notions of indexing have emerged. First,

mathematics solely indexes physical facts as the mathematics utilized in scientific explanation is

always arbitrary. Second, even if the mathematics is not arbitrary, it is still reasonable to believe

that purely physical facts underlie all mathematical explanations. The indexing argument argues

that since the sole role of mathematics is to index physical facts, then it is never the case that

mathematics is genuinely explanatory. Rather, it is the physical facts that the mathematics is

indexing which are truly conferring the explanatory force.

Let us first consider how mathematics indexes when its application is deemed to be arbitrary.

Suppose we have an explanation of some phenomenon that relies on the fact that the length of a

particular rod is 7

11 of a meter. Such an explanation would make use of the number

7

11, but this

number is entirely arbitrary. If we had chosen a different unit of measurement, then the length of

the rod would be a different number. The explanation is not explanatory in virtue of the number

7

11, but rather it is explanatory in virtue of the physical length of the rod. This example is meant

to illustrate Melia’s (2002) suggestion that if the use of mathematics is found to be entirely

arbitrary, then it is not explaining anything. Rather, the mathematics is merely indexing a

physical fact – in this case the actual physical length.

Daly and Langford (2009) borrow Melia’s suggestion and argue that Baker’s cicada example

suffers from this exact form of arbitrariness. Recall that what biologists wish to explain is why

the periodic-life-cycles of the cicada are prime. However, the life-cycles are prime only if we

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take years to be our fundamental unit. If we take months, days, or seasons, then the life-cycle

will be some other composite number instead. Baker (2009) defends the choice of years as being

unarbitrary in a variety of ways, such as appealing to the fact that biologists consider years to be

the proper unit of measurement, and also by pointing out that years do reflect a significant and

real regularity. My goal is not to take sides on this issue. It should be clear that the threat of

arbitrariness does not rear its head in all supposed examples of GME, nor do Daly and Langford

claim this either. In the honeycomb example, for instance, one would be hard pressed to assert

that the number of sides of the honeycomb cell is anything other than 6! The takeaway from this

discussion should be that in the eyes of the nominalist, lots of the usage of mathematics in

science seems to be arbitrary in the sense that Melia describes. It is this observation that leads us

to the intuition that the sole purpose of mathematics is to index physical facts. The supporter of

GME needs to produce an example that will fly in the face of this intuition by having an

explanation where the mathematics is not arbitrary.

The true force of Melia’s argument is that even in cases where the mathematics is not arbitrary it

can still be argued that mathematics is only indexing or representing purely physical facts. This

is the belief reflected in the second sense of indexing. Consider our two leading examples of

GME which are both optimization explanations. A key feature of the optimization explanation is:

(b) A mathematical theorem that demonstrates the optimality of a

particular state of affairs.

Supporters of GME claim that (b) is necessary for an optimization explanation; the explanation

would simply be lost without it. Thus, mathematics is playing an explanatory role. Opponents of

mathematical explanation argue that (b) is actually not necessary at all, or at least not necessary

in an explanatory sense. Instead, the (b) statement is merely a statement that indexes or

represents a purely physical fact. All the nominalist needs to do is show what these purely

physical facts actually are, and then the mathematical explanation can no longer be considered

genuine. For example, in the cicada example, the relevant physical fact would be that units of

time have the property such that 17 years is the most optimal number of years between 14 and 19

such that it would minimize intersection with other years. It is this fact about time that is

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conferring the explanatory power to the optimization explanation, and not the mathematical

theorem itself.

The realist could object that mathematics is still necessary in order to express some of these

physical facts. However, nominalists can simply agree with this and claim that this

fundamentally does not change the fact that in the end the explanatory force is not being supplied

by the mathematical theorem, but rather by the underlying physical fact. Another option for the

realist is to claim that we only know the physical fact to be true via the mathematical theorem

due to its mathematical proof. Juha Saatsi (2010) argues that the representational notion can be

extended to the mathematical proofs as well. In the cicada example, he suggests that we can

deduce that 17 should be the optimal number of years for the cicada life-cycle by using sticks to

represent annual periods of time.

[O]ne could represent periods of time with sticks as follows. Take a bunch

of sticks of 14, 15, 16, 17, and 18 cm. You’ll need fewer than 20 sticks of

each kind. Lay down sticks of each kind one after another to find the least

common multiple (LCM) for each pair... You’ll soon find out that the least

common multiple is almost always clearly longer for the pairs one member

of which comprises 17 cm sticks. (Saatsi, 2010, p. 8)

The point of this example is that the mathematics functions like the sticks. It merely represents

the concept of time and from this purely physical concept we can deduce the relevant physical

fact that is the genuine explanatory factor. To conclude that mathematics is genuinely

explanatory is as ludicrous as it is to conclude that the sticks are genuinely explanatory as well.

The honeycomb example is similar to the cicada example in that the nominalist can argue that it

is not the mathematical theorem that explains the hexagonal shape of the honeycomb, but rather

it is a purely physical fact about approximately Euclidean space. What makes this example

different and slightly more difficult to attack is that the mathematics employed is not as easily

recreated in a purely nominalistic fashion. A quick glance at Hales’ mathematical proof of the

honeycomb theorem reveals that it involves limits, disjoint measureable sets, and more.

Recreating the proof with physical objects like sticks is most likely impossible. Even so, Colyvan

sees a weakness in the honeycomb example.

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[T]he mathematical part of the explanation for the hexagonal structure of

the hive-bee honeycomb comes from the proof of the honeycomb

conjecture – a result in geometry and topology. But Field has shown how

we can speak nominalistically about the geometry of (Newtonian) space-

time, and so it seems likely that a similar result could be proven in his

system. If this is possible, then any explanation involving the nominalist

form of the honeycomb conjecture would arguably be at least as good as

the original form of explanation presented earlier. (Lyon & Colyvan, 2008,

p. 240)

If we take Field’s project seriously then it may be possible to create a nominalistically acceptable

proof that the hexagon is the most efficient shape to tile an approximately flat area. If this is so,

then this fact is no longer solely a mathematical fact, and thus the nominalist could assert that the

mathematics is merely indexing or representing the physical properties of space-time.23 So long

as the nominalist has the possibility of claiming that physical facts are the only explanatory

factors, then the indexing argument appears to be good enough to reject the existence of GMEs.

Colyvan admits this when he says “for an example of mathematical explanation to be of the

ontologically committing type, there must be no matching nominalist explanation.” (Lyon &

Colyvan, 2008, p. 240) In this light, the honeycomb example, and perhaps the cicada as well, are

not GMEs as there exists nominalistically acceptable counterpart explanations.

Colyvan (2010) presents another supposed example of a GME which he believes avoids the

problem of having a nominalistic counterpart explanation, and hence avoids the force of the

indexing argument.

The Kirkwood gaps are localized regions in the main asteroid belt between

Mars and Jupiter where there are relatively few asteroids. The explanation

for the existence and location of these gaps is mathematical and involves

the eigenvalues of the local region of the solar system (including Jupiter).

The basic idea is that the system has certain resonances and as a

consequence some orbits are unstable. Any object initially heading into

such an orbit, as a result of regular close encounters with other bodies

(most notably Jupiter), will be dragged off to an orbit on either side of its

initial orbit. An eigenanalysis delivers a mathematical explanation of both

the existence and location of these unstable orbits (Murray & Dermott,

23 It could also be pointed out that the honeycomb example fails as it is another instance of a geometric explanation.

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2000)... The explanation of this important astronomical fact is provided by

the mathematics of eigenvalues (that is, basic functional analysis).

(Colyvan, 2010, p. 303)

The mathematics of eigenvalues and functional analysis has not been successfully nominalized

by Field. It is the eigenvalue analysis that explains the Kirkwood gaps, and since there is, at

present, no nominalistic counterpart to the mathematical explanation, Colyvan claims that the

mathematics is genuinely explaining the physical phenomenon. Although Colyvan may be right

in his assertion that there is no nominalistic counterpart to the mathematical analysis in the

Kirkwood gaps explanation, his conclusion shows that he has missed the true force of the

indexing argument. The indexing argument does not tie its success to the complexity of a

mathematical proof, or to the explicit availability of a nominalistic counterpart. All it depends on

is the intuition that what is truly doing the explaining are purely physical facts. The nominalist

can easily assert that what explains the Kirkwood gaps are properties of space-time, gravitation,

planetary mass, etc. This is no different than their belief that the cicada and honeycomb

explanations are fundamentally physical as well.24 What makes the Kirkwood gaps different

from the honeycomb and cicada example is that all parties can agree that the mathematics is

indispensable. The mathematical portions of the cicada and honeycomb explanations can be

replaced with nominalistically acceptable counterparts, whereas in the Kirkwood gaps example

this cannot be done. However, all this indicates to the nominalist is that the mathematics is

indispensable in a deductive or representational way. This is not controversial as recall that

nominalists like Melia consent to this fact already. Colyvan has tied genuinely explanatory

mathematics directly to the unavailability of a nominalistic counterpart. Although I agree that

this is a necessary condition to convince the nominalist that GMEs exist, it is not sufficient. So

long as the nominalist is able to assert that there is something purely physical explaining the

phenomenon, they will be able to resist the conclusion that mathematics is genuinely

explanatory.

24 In a recent paper, Bueno (2012) draws similar conclusions.

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In this presentation of the indexing argument I have invoked the idea of ‘purely’ physical facts or

properties. The argument depends on the idea that mathematics indexes these purely physical

facts, and it is these facts that are genuinely explanatory. Naively speaking, by purely physical

fact I mean a fact that can be entirely understood without the use of mathematics – in essence,

they are facts that are solely physical in nature. Properties such as hardness or shape are

supposed to uncontroversially fall under this label. However, other physical properties such as

instantaneous velocity seem to resist our naïve definition. Is it possible to understand

instantaneous velocity without any mathematics?25 Nominalists take the existence of such purely

physical facts for granted, and their intuitive belief is that things such as time and space are

purely physical as well. Realists in this debate also seem to assent to the belief that we can

discuss purely physical properties in an unproblematic way, and I will assume this as well.

However, one important question that is seldom asked is: are there any physical facts or

properties that are not purely physical? If so, what are these non-purely-physical physical facts?

There are three potential answers to these questions. The first is to maintain that there is no

difference between purely physical facts and physical facts; the word ‘purely’ is just a redundant

descriptor. The second is to admit that it is possible for certain physical facts to be not purely

physical. There could be physical-mathematical facts, and what this precisely means would be up

to much debate. Lastly, one could maintain that this is all entirely mistaken, and that there are no

purely physical facts or properties at all as physical facts do not exist. We live in a purely

mathematical world and hence all facts and properties are actually mathematical in nature.26

Regardless of its independent merits or deficiencies, there are reasons pertinent to our present

discussion for rejecting this third position. If we are committed to the belief that there are only

mathematical facts and properties in the universe, then it is a trivial conclusion that mathematics

would genuinely explain the world around us. This entirely begs the question against the

25 See Berkovitz (forthcoming) for an argument against a purely physical understanding of certain physical

properties.

26 See Max Tegmark (2008) for a spirited defence of this position.

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nominalist. By the same token, ought we to reject the first view that claims all physical facts to

be purely physical in nature? It seems that this could be used in the exact same way to reject

even the possibility of a GME. Although I believe this to be a core belief held by nominalists, it

alone is not enough to a priori reject the existence of GME. It is still entirely reasonable to

maintain that mathematical facts or properties can explain purely physical facts. One way is to

assert that mathematical objects have causal power. Another is to take the more traditional view

that mathematical objects are acausal, but so long as we reject an entirely causal account of

explanation there is no inherent conflict in believing that mathematics can explain. So, asserting

that all physical facts are purely physical is compatible with the existence of genuine

mathematical explanation. Lastly, the second position that allows for mixed physical-

mathematical facts is also open to GMEs. The prima facie problem with this position is agreeing

on what we mean by a mixed fact or property. While all three views are open to the possibility of

GME, we will henceforth only consider the first two.

Although it does seem obvious that nominalists would happily assent to the first of these

metaphysical views, it is entirely unclear what supporters of GMEs believe. This lack of

commitment represents a weakness in the realists’ position, and is something that the indexing

argument successfully exploits. If you believe that all physical properties are purely physical,

then even though this alone does not rule out mathematics genuinely explaining physical

phenomena, it is extremely difficult to conceive of how it could possibly work. In my opinion,

this intuitive belief is what fuels the indexing argument. The crux of the problem is that there are

two separate questions being fused together. The first is ‘does mathematics genuinely explain?’,

and the second is ‘how could mathematics genuinely explain?’. Realists have yet to offer an

answer to the latter question, and without one it is proving difficult to answer the former question

in the affirmative.

We will not delve into these issues much further in this chapter. I will return to the much ignored

‘how’ question in chapter 6. Until then, I will help myself to the naïve understanding of purely

physical properties, and simply note that the indexing argument does not hinge on the nature of

all physical properties in general. If we can put forward an example that resists the intuition that

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there is something purely physical doing the explaining, then we will succeed in avoiding the

challenge presented by the indexing argument.

The indexing argument has proven to be the most challenging argument against GMEs.

Somewhat surprisingly, supporters of GMEs have not aggressively challenged the validity of the

argument, and instead have chosen to search for examples that are impervious to the indexing

critique. This will also be the strategy that I will employ for reasons I will detail below.

However, before we proceed it is worth mentioning some of the shortcomings of the indexing

argument.

Despite its name, the indexing argument is not much of an argument at all. It is actually a loose

application of a deep intuition that nominalists possess regarding the role of mathematics in

science. The intuition is that mathematics simply indexes, and this is why mathematics does not

genuinely explain physical facts. A serious problem is that this application of the indexing

intuition is difficult, if not impossible to defeat. The indexing argument essentially expresses the

intuition that mathematics does not genuinely explain because mathematics cannot genuinely

explain due to its sole role as an indexer of physical facts. This is dangerously close to question

begging. Whether or not mathematics can explain, and hence what precise role(s) mathematics

plays in scientific applications, is exactly what is in question. Obviously, if we assume that

mathematics cannot play an explanatory role then we are also assuming that mathematics does

not play an explanatory role, and in this light the indexing ‘argument’ is impossible to refute.

Consider the following from Daly and Langford. They feel that the indexing argument is strong

enough to rule out the possibility of any GME.27

We suggest that the nominalist’s view should be that there could not be

mathematical explanations. In this paper we have tried to show the

resilience of Melia’s indexing strategy. If his strategy works against some

cases of putative mathematical explanations, it works against all possible

27 In a more recent paper Daly and Langford (2011) are less enamoured with Melia’s indexing argument.

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putative mathematical explanations. (Daly & Langford, 2009, pp. 655–

656)

Even if we grant that the indexing argument is powerful enough to resist the force of the cicada,

honeycomb, and Kirkwood gaps example, Daly and Langford’s conclusion is entirely too strong

and unjustified. They perform an inductive generalization based off the conclusion that the

mathematics in the above examples was merely indexing and hence not genuinely explanatory.

But what justifies the generalization that this will be the case for all future possible mathematical

explanations? The only justification is the intuition that the sole role of mathematics is to index,

and nothing more. Daly and Langford are assuming the very issue that is under investigation in

order to motivate their general conclusion. A more charitable interpretation that could justify the

universal conclusion is simply that it is an instance of enumerative induction. So far, all instances

of supposed GMEs have been found to be not genuine based on the discovery that the

mathematics was solely indexing physical facts. It then stands to reason that all supposed

examples of GMEs will suffer the same conclusion, and hence we should believe that

mathematics does not genuinely explain. My phrasing here is weaker than above, and, like all

instances of enumerative induction, is ultimately fallible. Nominalists may be justified in holding

their position until an example is put forward where mathematics is not solely indexing physical

facts. In this light, the nominalist remains open to the possibility that mathematics could be

found to genuinely explain, and that the indexing argument can be defeated.28

The non-question begging nominalist is the one who we are interested in engaging with. This is

the position that most closely matches Melia’s who presented the indexing argument in the first

place. Recall that Melia says,

Of course, there may be applications of mathematics that do result in a

genuinely more attractive picture of the world – but defenders of this

28 This charitable interpretation of Daly and Langford is most likely not what the authors intended. The extended

quotation from above is: “Our topic is mathematical explanation. Baker thinks that there are mathematical

explanations. Is the opposing nominalist’s view simply that there are no mathematical explanations, but that there

could be, and that it is just an unfortunate coincidence that to date no mathematical explanations have been

produced? We suggest that the nominalist’s view should be that there could not be mathematical explanations.”

(Daly & Langford, 2009, p. 655)

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version of the indispensability argument have yet to show this... Until

examples of applied mathematics are found that result in this kind of an

increase in attractiveness, we realists about unobservable physical objects

have been given no reason to believe in the existence of numbers, sets or

functions. (Melia, 2000, pp. 474–475)

His version of the indexing argument is not undefeatable. It is open to the possibility that

mathematics could genuinely explain, and hence is not question begging. However, it still has

the problem that it is not so much an argument as it is an expression of an intuition. One reason

for this is that the concept of indexing is not elaborated on or explained in any detail. A standard

understanding of mathematics indexing physical things is how we can use, say, the natural

numbers to index physical objects or properties. A simple example of this is how the natural

number 2014 is indexing the current year, which is actually just the number of periodic cycles

completed by the Earth revolving around the Sun starting from an arbitrary starting point. We

can perhaps extend this notion of indexing unproblematically to other mathematical systems,

such as the real numbers indexing positions in space-time, assuming that space-time is

continuous in the right sort of way. But once you get into much more complicated or abstract

mathematics such as complex differential equations, eigenvectors, etc., it is entirely unclear how

these mathematic entities actually index physical facts at all. What is needed here is a robust

account of mathematical representation that details how mathematical concepts index physical

things, and how through this indexing mathematics is able to bestow all its benefits and utility to

the practice of science.

Nominalists have not presented any such account of mathematical representation; they simply

assume that the indexing argument is a coherent and defendable position. This assumption is

rooted in their intuitions regarding the sole role of mathematics, and this is why I do not feel that

the indexing argument is much of an argument at all. It remains to be seen if an anti-realist view

of mathematics such that mathematics only represents physical facts can retain all the epistemic

benefits that mathematics brings to the table.29

29 See Pincock (2011) as well as Bangu (2012) for a discussion on the epistemic benefits of mathematical

representation in science.

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One final issue with the concept of indexing is that nominalists such as Melia, Leng, and Daly

and Langford accept that mathematics is an indispensable tool for the indexing of physical facts

in scientific practice. This poses two interesting problems for the nominalist. The first is that

there is no account as to why mathematics is indispensable for indexing. The second is that due

to its indispensable nature, the language of science seems to directly imply the existence of

mathematics. The first problem remains somewhat of a mystery. Perhaps it is a brute fact,

perhaps there is no account at all, or perhaps our demand for an account is unwarranted.

Regardless, something should be said about this issue to either explain the indispensability of

mathematics, or explain why no explanation is needed. The second issue of the language of

science implying the existence of mathematical entities is supposedly solved by Melia’s

weaseling argument. Recall that the weaseling argument allows us to have our scientific

language imply the existence of mathematical entities, but at the same time we are able to weasel

away from any ontological commitment. In section 2.3.2, concerns were raised regarding the

viability of the weaseling argument. It is not clear that the weaseling method is actually strong

enough to apply to the indispensable use of mathematics when indexing physical facts. The

indexing argument critically depends on weaseling out of the apparent commitment to

mathematical objects in scientific language, and insofar as the weaseling argument is found to be

lacking, so too will the indexing argument be found unsatisfactory as well.

As previously mentioned, one potential strategy to argue against the nominalist is to show that

the indexing argument is not coherent due to any of the above objections. This is not the method

I will pursue. The reason for this stems from my critique that the indexing argument functions

less like an argument and more as a statement of the nominalists’ intuitions. Showing that the

present state of the indexing argument is grossly underdeveloped and lacking in rigour does

nothing to demonstrate that the core intuition that mathematics solely indexes physical facts is

false. The ardent nominalist can still, albeit stubbornly, hold on to their beliefs and assert that a

truly comprehensive account of the indexing argument could potentially be developed if needed.

The catch is that there is, at present, no need for this comprehensive account as no purported

example of a GME comes close to pressuring the intuition that mathematics only indexes

physical facts. Until that time comes, the onus is on the supporter of GMEs to come up with a

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superior example, and not on the nominalist to flesh out their intuitions into a full-fledged

position. My strategy of choice is to meet the nominalists’ intuition head on. I will treat the

indexing argument as a legitimate argument against GMEs, and accept the burden of proof by

presenting an example of a GME that is impervious to the intuitions that lie at the crux of the

argument. The first step will be to develop the indexing argument into a bona fide argument.

Only then will we be able to see precisely what the nominalist intuition that mathematics solely

indexes physical facts consists of, and subsequently how to avoid it.

3 The Indexing Criteria for Genuine Mathematical

Explanations

Contra Daly and Langford, the indexing argument that we will consider does not rule out the

possibility of a GME. Our target is the honest nominalist, such as Melia, who does not beg the

question against GME. Their indexing argument moves from the observation that all present

supposed examples of GMEs fall short as the mathematics is found to index physical facts and

hence does not genuinely explain, to the general conclusion that there are no GMEs. What needs

to be made clear are the criteria that are used to determine that mathematics is playing an

indexing role in these explanations. Since the indexing argument does not mention any such

criteria, our task will be to extract them from the various ways in which nominalists have

deployed the indexing argument to reject past examples.

One way that supposed examples of GMEs were rejected was due to their seeming contrived or

not being accepted by the scientific community as a proper mathematical explanation of a

physical fact. This is not actually a part of the indexing argument proper, but rather it is more of

a prerequisite criterion that an external mathematical explanation must meet before we even

consider it. The first actual employment of the indexing argument was in rejecting any

mathematical explanation where the mathematics is geometric in nature. Nominalists asserted

that geometric properties such as geodesics are actually physical properties. If this is the case,

then the mathematics that we use to explain is actually indexing these physical-geometric

properties.

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Next, the indexing argument was used to reject mathematical explanations where the

mathematics was found to be arbitrary. The claim is that if mathematics is arbitrary in a scientific

explanation, then there is nothing special about the particular mathematical entities being

employed as it could have been some other mathematical entity. Seen in this light, the nominalist

can claim that there is nothing explanatory about the mathematical entities or their properties at

all, as they could have been different. Rather, what is explanatory are the physical facts which

are not arbitrary at all. The arbitrary nature of the mathematics points to the conclusion that the

mathematics holds no explanatory significance to the overall explanation, and merely functions

as a tool to index the explanatory physical facts.

Closely related to arbitrariness is the rejection of examples where the mathematical explanation

was found to be not unique. If there exists another nominalistic explanation of the same

explanandum with no reference to mathematical objects or properties, then there is no reason to

believe that mathematics is explanatory. This line of reasoning was utilized even without direct

access to the nominalistic counterpart explanation, such as Lyon and Colyvan’s rejection of their

own honeycomb explanation as a candidate for a legitimate GME. Even though there is no

known nominalistic proof of Hale’s honeycomb theorem that they can point to, their admission

of merely the possibility of a mathematics free explanation of the hexagonal nature of the

honeycomb was enough to lead them to reject the mathematics as genuinely explanatory. Thus,

even if it is unattractive in other ways compared to a mathematical explanation, it seems that if

there exists the possibility of a purely nominalistic explanation of a phenomenon, then the

conclusion must be that the mathematics is not genuinely explanatory. The nominalist is able to

assert that the physical facts which appear, or would appear, in the nominalistic version of the

explanation are conferring the explanatory force.

Up till now, the applications of the indexing argument seems somewhat clear and plausible. The

final use of the indexing argument, however, is more complicated and controversial. Recall that

the Kirkwood gaps example seems to avoid all the above objections. It is not contrived, nor is it

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necessarily geometric.30 The mathematics is not arbitrary, and presently there is no purely

nominalistic explanation of the Kirkwood gaps, nor is there any reason to believe that such a

nominalistic explanation is even possible. However, even with all these points in its favour, I still

argued that Colyvan was incorrect in asserting that the Kirkwood gaps example is a convincing

example of a GME. The reason why is because it does nothing to address the underlying

intuitions of the indexing argument. Nominalists can still point to (or perhaps only vaguely

gesture at) purely physical facts or properties that they claim are the true explanatory factors. In

the case of the Kirkwood gaps, physical facts about space-time, gravity, planets, mass, etc., could

be cited as the true explanatory factors. If this is the case, then once again the indexing argument

can conclude that the mathematics is simply indexing these explanatory physical factors, and

hence the mathematics is not genuinely explaining anything at all.

This final use of the indexing argument is the most powerful as it seems to be strong enough to

apply in every imaginable situation. Even without a nominalistic counterpart proof in hand that

identifies the key physical explanatory factors, the nominalist is still able to assert that the

mathematics is indexing at least one member from a set of possible purely physical explanatory

facts or properties, and it is from these members that the true explanatory factors are found. My

line of attack against the nominalist is to treat the indexing argument as a legitimate argument

and then aim to defeat even the strongest version of it. However, if we allow for this final usage

of the indexing argument, are we granting the nominalist too much? Consider a mathematical

explanation of a physical fact that meets all the above criteria: it is not contrived, not geometric,

not arbitrary, and there exists no actual or possible purely nominalistic counterpart explanation.

Now let P* be a set consisting of all possible purely physical facts or properties in the universe.

The indexing argument essentially says that there is some subset of P*, called P, that represents

the true explanatory factors of the explanandum in question, and that the mathematics indexes

these physical factors. Notice that the nominalist does not (or perhaps cannot) even need to

identify the correct subset of physical factors! In fact, the nominalist need not even narrow P

30 This depends on whether or not we view space-time as purely geometric, and whether or not such a geometric

model is even the appropriate way to understand space. I will avoid this line of analysis and instead will follow the

lead of Colyvan and others by treating the Kirkwood gaps explanation as non-geometric in nature.

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down in any way for the argument to run. I take this usage of the indexing argument to be the

most accurate instantiation of the nominalists’ intuition that mathematics solely indexes physical

facts. It is also this usage which, I believe, motivates nominalists such as Daly and Langford to

claim that GMEs cannot in principle exist. It is easy to see why, as if this is a viable employment

of the indexing argument then how could we ever find an example of a mathematical explanation

of a physical fact that can avoid it?

I am willing to accept this usage of the indexing argument with two subtle but important

constraints. These constraints should be perfectly amenable to the nominalist, and are both

limitations on what the subset, P, made from all possible purely physical facts or properties in

the world can actually consist of. The first constraint is that P does not include all possible

physical facts or properties, but rather it can only consist of all known physical facts or

properties. This distinction is not motivated from any particular metaphysical view of natural

kinds or ontology. Rather, this restriction limits the nominalists to selecting from physical facts

or properties that our present best scientific theories acknowledge as genuine facts or properties

about the world. If we allow all possible physical facts or properties into P, then there is nothing

stopping us from asserting that mathematics is indexing the physical fact of, say, Koo-ness,

which is some physical fact that has yet to be discovered. At first glance this limitation seems to

be somewhat useless. It does not rule out any of the previous employments of the indexing

argument that we have looked at. Moreover, it seems that this restriction on P is implicitly

assented to by any reasonable nominalist. Surely their aim is to point to actual known physical

facts or properties. To cite future or possible physical facts is not scientific or naturalistic at all.

Rather, it would seem somewhat ludicrous and patently question begging against supporters of

GME. Regardless, making such a restriction explicit will prove to be important.

The second constraint on P is that the physical facts or properties in P need to have independent,

physical motivation or justification for their existence. This restriction is meant to rule out the ad

hoc creation of ‘physical’ properties meant to mimic mathematical properties. Consider again the

Kirkwood gaps explanation. Colyvan believes that it is the eigenvalue analysis that is conferring

the explanatory force. If, instead, I claim that it is not eigenvalues, but rather it is ‘p-eigenvalues’

that are genuinely explaining, then I could run the indexing argument successfully. A p-

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eigenvalue is, of course, a purely physical property. They are actually properties of p-

eigenvectors, which are also purely physical. Asserting the existence of p-eigenvalues arguably

passes the first constraint that the physical property must be known. We know that p-eigenvalues

exist because of the seemingly explanatory nature of the mathematical eigenvalues which

function solely as indexing p-eigenvalues. The problem here is obvious. The existence of p-

eigenvalues is entirely ad hoc and parasitic on the existence of a mathematical explanation and

mathematical formalism. There is no independent or purely physical motivation or justification

for believing in the existence of p-eigenvalues whatsoever, nor is there any independent physical

confirmation of the existence of things such as p-eigenvalues. This is entirely different from how

naturalists and scientific realists typically infer that physical facts or properties exist. As above,

the constraint that P only consist of independently or purely physically motivated or justified

physical facts seems trivially obvious. All previous instances of the indexing argument

considered do not violate this constraint, and I take it that any honest nominalist would assent to

this limitation on P. The reason for this is simply that to not accept such a limitation would make

the indexing argument seem ad hoc and question begging against GMEs. Although I formulate

these restrictions on P, the net effect of the two constraints is to force the nominalist to cite

known, physically independently motivated physical facts as those being indexed by

mathematics when employing the indexing argument. This is exactly what has been happening in

every use of the indexing argument so far, and I take it as uncontroversial that the nominalist

would accept these constraints.

Putting our analysis of the indexing argument together, I present here four criteria that a

mathematical explanation of a physical fact must meet in order to be impervious to the indexing

argument and the core nominalist intuition that underlies it. A mathematical explanation of a

physical explanandum can be considered to be a candidate for genuine mathematical explanation

if:

(A) the explanation is not contrived and is accepted by the scientific

community as a good scientific explanation,

(B) the mathematics employed is not arbitrary,

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(C) there are no purely nominalistic explanations of the

explanandum,

and,

(D) there are no known and physically independently motivated

physical facts that could potentially explain the explanandum

such that the mathematics is simply indexing these facts.

If a mathematical explanation of a physical fact meets all four criteria, then the indexing

argument cannot be used against it to show that the explanation is not genuine. Without the

indexing argument, the nominalist must concede that the mathematical explanation in question

opposes their core intuition that the sole function of mathematics is to index physical facts.

Hence, in the eyes of the nominalist, meeting these four criteria is a necessary condition for a

mathematical explanation to be genuine.

Even though the indexing argument is underdeveloped and quite vague in its presentation and

application, in taking it seriously with the most charitable interpretation possible the true worth

of the argument has emerged. The critical contribution of the indexing argument is in its

expression of the core beliefs and intuitions of the nominalist. Once these beliefs were clearly

laid out, we were finally able to arrive at a set of criteria that a mathematical explanation must

satisfy for the nominalist to take it seriously as a candidate for being a GME. This is not a trivial

achievement. So many of the problems surrounding the debate on mathematical explanations

have come from the fact that neither side has ever been clear on what exactly makes an example

genuine or not. Even advocates of GMEs do not agree on what makes an explanation genuine.

Their approach has been to present examples that they feel are genuine. However, without any

criteria to compare them to, all these feelings amount to are more intuitions. It is no surprise

then, that nominalists remain unconvinced as certainly examples that speak to the intuitions of

the realist would not have the same effect on the intuitions of the nominalist. Seeing as how the

target of all indispensability arguments is the nominalist, the inability to precisely define and

identify what criteria a GME needs to satisfy has been crippling to the success of the Quinean

indispensability argument and the enhanced indispensability argument.

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I have presented the four criteria as necessary conditions for a GME. The reason for this is it

reflects how nominalists have been rejected supposed examples. They show that examples are

lacking at least one necessary criterion, and thus cannot be genuine. What would be ideal is if the

four necessary conditions of a GME could be shown to be sufficient as well. Sadly, I do not think

that this is possible. Nominalists can still point to the fact that even if we had an explanation that

satisfied all four criteria, and thus successfully resists the intuition that the mathematics is

indexing physical facts, there is still no account of scientific explanation that corroborates the

claim that the mathematical explanation is a GME. One more criterion needs to be added to our

list:

(E) An acceptable account of scientific explanation must

corroborate the claim that the mathematics is explanatory.

The remainder of this chapter will be dedicated to presenting an explanation that satisfies (A) –

(D). Chapter 4 will take a detailed look at accounts of scientific explanation and attempt to show

that our explanation also satisfies criterion (E). Once this is complete the nominalist will be

forced to conclude that GMEs exist.

4 Genuine Mathematical Explanation: Electron Spin

Based on our understanding of the indexing argument, the strategy for finding a GME is

straightforward: we must find a mathematical explanation of a physical fact that satisfies criteria

(A) through (D). I advance here an example that I believe does the job, and is one that physics

students are intimately familiar with. The Stern-Gerlach experiment is a standard experiment

taught in every introductory quantum mechanics class. In 1922, Otto Stern and Walther Gerlach

conducted an experiment where they measured the deflection of silver atoms.31 Their goal was to

test that the direction of angular momentum is quantized, and hence act as a form of

confirmation for quantization and as a rejection of classical theory. Stern and Gerlach were

31 See Weinert (1995), and Friedrich and Herschbach (2003) for history and analysis of the Stern-Gerlach

experiment.

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famously successful in this, and in addition they were also the first to observe the quantization of

magnetic moment associated with electron spin. It was not until 1925 when Goudsmit and

Uhlenbeck proposed the concept of electron spin.32 This concept was subsequently formalized by

Pauli in 1927 and Dirac in 1928. Spin is now considered a fundamental quantum property that

explains the outcome of not only the Stern-Gerlach experiment, but it also accounts for the

Zeeman effect, it provides the basis for the periodic table of chemical elements, and much more.

The Stern-Gerlach experiment involves a beam of particles passing through an inhomogenous

magnetic field. If the magnetic field is homogenous then the net forces of the magnets will

cancel each other out and the path of the beam will remain unchanged.

An inhomogenous field results in a net force that will deflect the trajectory of the particles. At

the other end of the Stern-Gerlach are detectors to measure the deflection of the particles.

Consider a beam of electrons passing through a Stern-Gerlach oriented in the z-direction. It is

observed that the beam splits into two distinct beams that are the equidistant from the z-axis

when viewed from the side. What this shows is that the electrons have a form of intrinsic angular

momentum. It also serves as a counterexample to classical theory as according to classical

32 Goudsmit and Uhlenbeck are often credited with the discovery but they were not the first to conceive of electron

spin. In 1921, Compton proposed that the electron is “spinning like a tiny gyroscope” (Compton, 1921), and Kronig

also had an unpublished paper at least six months before Goudsmit and Uhlenbeck that explored electron spin.

South Magnet

z-up direction

Electron Beam

z-direction North Magnet

z-down direction

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physics the electrons should be continuously and randomly dispersed. Instead we see only two

trajectories that are deflected up or down by a discrete amount. The explanandum we are

interested in is: why does a beam of electrons oriented in the z-direction split into two distinct

beams that are the same distance from the z-axis when passed through a Stern-Gerlach

apparatus? A standard explanation of this is as follows.

In quantum mechanics, the state of physical systems is represented by the vector ψ defined in a

Hilbert space. ψ contains all the information of specific properties such as position, momentum,

and spin. Each property is represented by operator functions also defined on the Hilbert space.

The operator functions for spin, S, in the x-, y-, and z-directions are written Sx, Sy, and Sz

respectively and are defined as:

Sx = ½ħσx, Sy = ½ħσy, and Sz = ½ħσz.

According to the Pauli spin matrices, σx, σy, and σz for electrons are defined as:

σx = (0 11 0

), σy = (0 −𝑖𝑖 0

), and σz = (1 00 −1

).

Any square matrix has both eigenvectors and eigenvalues associated with it. An eigenvector, v,

of a square matrix, A, is a non-zero matrix that when multiplied by A yields a multiple of A by a

given number, λ. λ is called the eigenvalue of A. In quantum mechanics, the eigenvalues

represent the only possible magnitudes that the property represented by its linear operator can

possess. It is not difficult to deduce that the Pauli spin matrices have two eigenvectors associated

with each matrix. They are:

λx = 1

√2(

11

), 1

√2(

1−1

), λy = 1

√2(

1𝑖

), 1

√2(

1−𝑖

), and λz = (10

), (01

).

Similarly, it is straightforward to calculate that the eigenvalues for each of the Pauli spin

matrices are ½ħ and -½ħ, which for convenience we call spin up and spin down respectively.

Thus, the only possible values for the spin of an electron are spin up and spin down. This, then,

explains why the beam of electrons deflects into exactly two separate beams in the Stern-Gerlach

apparatus. According to the property of spin, there are only two possible values of the electron

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spin, and hence only two possible ways the electron can be deflected by the magnetic field.

Moreover, |½ħ| = |-½ħ|, and this explains why the spin up beam and the spin down beam are both

equidistant from the original plane or orientation.

This presentation of spin is standard in introductory quantum mechanics textbooks. From the

Pauli spin matrices we can make many predictions of what would happen to particles that pass

through multiple joined Stern-Gerlach apparatuses which are popular exercise questions for

physics students. To briefly summarize, the Stern-Gerlach apparatus leads to the observed

deflection of an electron beam in the z-direction into two equidistant discrete beams when

viewed from the side – one above the original beam and one below. The explanation of this

deflection is that electrons have the property of spin, which is defined by the Pauli spin matrices,

σx, σy, and σz. From the Pauli spin matrices we can deduce that there are only two possible spin

values of the electron, spin up and spin down, and that they have the same magnitude. This

explains why the original beam splits into exactly two equidistant beams.

I claim that this spin explanation of the splitting of the electron beam in a Stern-Gerlach

apparatus is an example of a GME. This is certainly a mathematical explanation of a physical

fact that is well accepted by the scientific community. It is not geometric, contrived, or otherwise

problematic, and hence the spin explanation satisfies criterion (A). The mathematics employed is

also not arbitrary in the way that Melia describes. No trivial shift in units would alter the result

that the eigenvalues for the property of electron spin are ½ħ and -½ħ, so (B) is also satisfied. (C)

requires there to be no purely nominalistic explanation of the splitting of the electron beam. This

would require a nominalistic version of quantum mechanics. Save for an interesting attempt from

Mark Balaguer, the standard assumption is that no such nominalized theory is possible. I will

adopt this assumption for now, but will address Balaguer’s view below. With the first three

criteria met, all that remains is to see if the spin explanation also satisfies (D) – there are no

known and physically independently motivated physical facts that could potentially explain the

explanandum such that the mathematics is simply indexing these facts. (D) reflects the core

intuition behind the indexing argument and is without a doubt the most difficult criterion to

satisfy. I will argue that the spin explanation does indeed pass this criterion.

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In my presentation of the spin example it may appear that I cheated. I only provided the

mathematical formalism of spin without ever saying what spin actually is. I did mention briefly

that spin is somehow related to angular momentum. In classical mechanics, angular momentum

can be thought of in two ways. An object’s center of mass can be in motion, called orbital

angular momentum, and there can also be motion about the center of mass, called spin. A classic

example is the Earth revolving around the sun. The actual revolution of the Earth about the sun is

its orbital angular momentum, and we also know that the Earth spins on an axis. This distinction,

though intuitive, is somewhat misleading. The spinning of the Earth about its axis, denoted spinc

to identify it as classical, is actually just a form of orbital angular momentum. So when we say

angular momentum in the classical sense, we actually just mean the orbital angular momentum,

and this includes spinc. Angular momentum in quantum mechanics is significantly different from

the classical case. In quantum mechanics, angular momentum genuinely refers to two separate

quantities. The first is orbital angular momentum. Here the classical definition of angular

momentum can be carried forward. However, unlike in the classical case, spin is its own unique

type of angular momentum. It is not simply a part of the orbital angular momentum like spinc is

for classical mechanics. This indicates that spin is unlike anything we know in classical physics.

This is troubling as in many ways the behaviour of the quantum world is notoriously difficult to

understand. One way to understand quantum mechanical properties is to relate them to their

classical analogue. We may not have a perfect understanding of position in quantum mechanics,

but we have a good enough understanding of position in classical physics to help us get by. This

strategy, although far from perfect, has proven to be quite successful. But if spin is critically

unlike spinc, then how can we understand this property? Of course there is one sense that we do

genuinely understand spin. We understand it in the way presented above – we understand spin as

Pauli spin matrices.

Perhaps I have not motivated the claim that spin is not understandable in terms of spinc enough.

What I have illustrated is that spin and spinc are not exactly the same, as spinc is essentially

reducible to orbital angular momentum, and spin is not reducible to anything at all. This is only

the tip of the iceberg. Essentially what this means is that a particle in quantum mechanics will

always have spin no matter what frame of reference you observe it from. This is entirely unlike

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how we conceive of spinc. Imagine that you are spinning around and holding a video camera at

arm’s length pointed towards your face. When you later watch the video, it does not actually

appear that you are the one spinning, but rather it is the background that is spinning around you.

Of course, to your friends who were sitting there watching you spinc around while filming

yourself, you were certainly spinning in the classical sense. This demonstrates that whether or

not you have spinc depends on your frame of reference. But what does it mean for an electron to

always have spin no matter how we look at it? For this reason, spin is often referred to as

intrinsic angular momentum, and any attempt to understand it in terms of spinc is nonsensical.

Another reason why spin and spinc are incommensurable is that particles in quantum mechanics

are thought of as point particles. If this is the case, then the notion of a point spinning around its

axis is fundamentally incoherent. Pauli makes the distinction between spin and spinc clear when

he said,

Bohr was able to show on the basis of wave mechanics that the electron

spin cannot be measured by classically describable experiments (as, for

instance, deflection of molecular beams in external electromagnetic fields)

and must therefore be considered as an essentially quantum mechanical

property of the electron. (Pauli, 1994, p. 164)

Modern physics textbooks also remark on the distinction. Griffiths states that “[i]t doesn’t pay to

press this analogy [between spin and spinc] too far.” (Griffiths, 2005, p. 171) Similarly,

Townsend writes:

We will see as we go along that such a simple classical picture of intrinsic

spin is entirely untenable and that the intrinsic spin angular momentum we

are discussing is a very different beast indeed. In fact, it appears that even

a point particle in quantum mechanics may have intrinsic spin angular

momentum... [T]here are no classical arguments that we can give to justify

[this]. (Townsend, 2012, p. 2)

In a final appeal to authority, I quote Richard Feynman. With regards to the mathematical laws

of quantum mechanics, he says:

One might still like to ask: “How does it work? What is the machinery

behind the law?” No one has found any machinery behind the law. No one

can ‘explain’ any more than we have just ‘explained’. No one will give

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you any deeper representation of the situation. We have no ideas about a

more basic mechanism from which these results can be deduced.

(Feynman, Leighton, & Sands, 1965, p. 1–10)

Later on, Feynman considers the possibility of trying to account for spin in a purely physical

way.

Until now, it appears that where our logic is the most abstract it always

gives correct results – it agrees with experiment. Only when we try to make

specific models of the internal machinery of the fundamental particles and

their interactions are we unable to find a theory that agrees with

experiment. (Feynman et al., 1965, p. 6–2)

The only way we can explain observed physical facts is through the mathematical formalism of

spin.

Now we can return to the question above: does the spin example satisfy (D)? I believe that it

does. The reason why is clear. If the nominalist asserts that spin is actually some purely physical

property that the Pauli spin matrices simply represents in an indispensable way, the challenge is

that although this may sound coherent, this assertion actually has no content to it. The reason

why such a move worked for the cicada, honeycomb, and Kirkwood gaps examples is because

the nominalist was able to identify, or gesture at, these purely physical properties that the

mathematics is supposedly indexing. Properties such as time, approximately Euclidean space,

gravitation, etc., can be appealed to as the true explanatory factors, and these factors are purely

physical. But what can the nominalist say when we ask what purely physical fact the

mathematics of spin is indexing? The standard route of pointing to some physical fact or

property will not work because, as we have seen, there is no physical property that we can

compare spin to. Without the ability to identify the physical fact(s) in question, this example of

electron spin represents a significant improvement as a candidate for being a genuine

mathematical explanation. There are three possible ways in which the nominalist can still resist

the conclusion that the mathematical explanation of the splitting of the beam of electrons in a

Stern-Gerlach apparatus is a GME. I will show that all three methods are not viable, and that the

nominalist must conclude that the electron spin example meets the necessary requirements of a

GME.

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5 Blocking the Nominalist’s Response

In order to preserve the nominalistic intuition that the mathematical explanation of the Stern-

Gerlach experiment is not genuine, the nominalist can argue either that there is some unknown

physical property that is actually doing the explaining, that there is some known purely physical

property that actually is electron spin, or that there is simply no explanation of the phenomenon

whatsoever. Each of these three possibilities leads to undesirable outcomes, and I argue that to

stubbornly commit to any of them would be ad hoc and ultimately untenable.

5.1 The Unknown Explanation

Consider the first option that there is some physical fact that explains the splitting of the electron

beam in the Stern-Gerlach apparatus which is indexed by the Pauli spin matrices, but we just do

not know what this physical fact is. It could be that we have not yet discovered the true physical

nature of spin, or we could even admit that it is beyond our ability to ever know, but this does not

change the fact that there is some physical fact or facts which are the true explanatory factors. If

this is true, then certainly the mathematical formalism of spin is simply indexing the true

physical nature of spin, whatever that may be. Hence, even though the mathematics may be

indispensable in this representation, it would not be considered explanatory due to the indexing

argument.

The problem with the assertion that there is some unknown physical fact that explains is that it

does not meet the standards set out in criterion (D). We specifically constrained the physical

facts that we can appeal to from possible physical facts to only known physical facts. This was

argued for on the basis that appealing to unknown physical facts is entirely unnaturalistic. There

is no scientific reason to believe in the existence of unknown physical facts.

At the time when this constraint to known physical facts was placed, there was no reason for the

nominalist to object. This constraint did not hurt any of the previous employments of the

indexing argument. In all other cases a known physical fact was cited as the true explanatory

factor that the mathematics was indexing. The nominalist could argue though that now they do

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have legitimate reason to object, and that the prior justification is not sufficient. The claim would

be that we have been guilty of some equivocation. There are actually two senses of explanation.

The nominalist could grant that the Pauli spin matrices explains the splitting of the electron beam

in the Stern-Gerlach apparatus, but it does not really explain it.33 In the first sense of explaining,

this type of explanation is commonly employed when we use things that we do understand to

explain things that we do not understand. I could, for example, explain the behaviour of electrons

by treating them as bullets or billiard balls. Call this type of explaining epistemological

explanation. In the second sense, we explain something that we understand well in terms of

things that we do not actually understand. Call this ontological explanation. So what type of

explanation do we have of spin? The nominalist can assert that the Pauli spin matrices are an

epistemological explanation of the splitting of the electron beam. It may be indispensable to our

understanding, but regardless it is not an ontological explanation. Only ontological explanations

can count as genuine. What is the ontological explanation of spin? We do not know, and in fact

we may never know. But that still does not change the fact that the mathematics explains only in

the epistemological sense and solely represents. The true ontological explanation, even though it

remains hidden to us, is ultimately purely physical.

There are two problems with this reply. First off, it patently begs the question. What the

nominalist is doing here is basically saying that even though mathematics can explain, it cannot

explain in the right way to be considered genuine. They are ruling out the very possibility of

GME to begin with. Of course this would result in their rejection of any supposed example of

GME as they assume that this is impossible right off the bat. The only way to get around this

obvious question begging would be to provide independent reasons for believing that

mathematics cannot explain physical facts in the ontological sense. Any such argument would

necessarily involve metaphysical views on the relationship between mathematics and the

physical world. But ultimately this relationship is the very thing that we are presently

investigating. Brown (2013), for example, maintains the view that mathematics is only

33 I am indebted to James R. Brown for pointing out this objection. See Brown (2013) for his take on quantum

mechanical spin.

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epistemologically explanatory and not ontologically explanatory, but he admits that this stems

directly from his prior belief that mathematics is solely representational. We saw that Daly and

Langford also attempted to suggest that this is the case. They tried to provide independent

reasons that mathematics is solely representational, but their argument was entirely unconvincing

and was more wishful thinking than anything else. It hinged on the generalization that since the

indexing argument is successful against certain supposed examples of GME, that it will be

successful against all possible supposed examples of GME. But this too was entirely question

begging. The target of interest is the honest nominalist who has not a priori decided that

mathematics cannot explain. This nominalist may believe that mathematics solely indexes

physical facts, but this belief comes from their experience of our best scientific theories. If an

acceptable explanation emerges such that the mathematics does not seem to index anything at all,

then the honest response is that it simply is not indexing. If the nominalist chooses to invoke the

distinction between epistemological and ontological explanations and assert that any

mathematical explanation is epistemological, then they are not being honest. For them, the role

of mathematics has been decided a priori, and this begs the question against the possibility of

GME.

The second problem is that all independent arguments that attempt to justify the claim that the

Paul spin matrices are an epistemic explanation fall short. Consider how we know that certain

explanations are epistemological and not ontological. Say we want to explain why we chose a

particular route to take on our road trip. The explanation we give is that when we looked at a

map, the route we chose was the shortest distance on that map from our point of origin to our

destination. This explanation is perfectly acceptable, but it is not ontological. Clearly the route

being the shortest is due not to the distance on the map, but the actual distance in the world. The

map is simply a representation of the physical region. In this case, the reason why we know that

the map explanation is not ontological is in virtue of the fact that we know precisely what the

map is representing. Now consider a scenario where we account for certain behaviour of the

electron by casting an explanation where we treat the electron as a billiard ball. This makes good

sense to do as we understand the nature of billiard balls well, and can thus use this knowledge to

help us understand electron behaviour. This, too, is not an ontological explanation. We know this

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because we know that in some respects, the electron is critically unlike a billiard ball. It is an

empirical fact that electrons at times do not behave like small hard balls, and that any accurate

ontological picture of the electron would have to reflect this.

The point of these two examples is that in each case we have good reason to believe that our

explanation is not ontological. In the first it was due to the fact that we already knew what the

proper ontological explanation was. In the second it was because we knew our explanation was

critically unlike what the actual ontological explanation should be. I claim that in the example of

electron spin, neither the first nor the second reasons present themselves. We are unable to point

to the physical facts that the Pauli spin matrices allegedly represent, nor do we have any reason

to believe that the mathematics somehow conflicts with anything that we know about the

properties of the electron. In the absence of either of these reasons, are we at all justified in

asserting that the mathematical explanation of the Stern-Gerlach experiment is not ontological?

If we are looking for reasons that come from within the practice of science, then clearly the

answer is that we are not justified in maintaining this conclusion. There simply is no scientific

reason to believe that the mathematical explanation is not ontological.

It is now clear that there is no non-question-begging way to maintain that the Pauli spin matrices

are representing some unknown purely physical explanation.

5.2 The Ad Hoc Physical Explanation

A second way around the spin explanation is for the nominalist to claim that the Pauli spin

matrices are actually representing some known physical fact after all. This is the route that

Balaguer (1998) takes. In his defence of fictionalism34, Balaguer argues that contrary to popular

opinion, Field’s project of nominalizing Newtonian gravitational theory was actually a success.

34 Balaguer defends both fictionalism and full-blooded or plenitudinous platonism in his book as being perfectly

viable metaphysical positions. From this he draws the following two conclusions. First, that there will never be an

argument that will settle the dispute over mathematical objects. Secondly, that this is not a limitation on our

philosophical ability, but rather that there is actually no fact of the matter whether or not abstract objects actually

exist or not.

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Even if we suppose this, a problem is that it is unclear if Field’s approach can be extended over

quantum mechanics. David Malament (1982) was the first to raise this objection. Balaguer

recognizes this as a serious concern, and in a similar fashion to Field, he sketches out an

approach to nominalize quantum mechanics.

Balaguer’s scheme for presenting a nominalistically acceptable version of quantum mechanics is

complicated. We will skip over many of the technical details here but will present enough to get

the flavour of Balaguer’s approach. Quantum events are represented by closed subspaces of

Hilbert spaces. It turns out that this representation relationship is quite strong such that the set of

closed subspaces of a Hilbert space is isomorphic to the set of quantum events for a given set of

mutually incompatible observables, such as position and momentum, or spin up and spin down

for spin-1/2 observables. From these two sets it is possible to construct orthomodular lattices.

Call L(H) the orthomodular lattice generated from the set of closed subspaces of a Hilbert space,

and L(E) the orthomodular lattice generated from the set of quantum events for a given set of

mutually incompatible observables. Like their parent sets, L(H) and L(E) are isomorphic.

Malament’s objection is as follows. L(H) is the mathematics used to represent the quantum

events, L(E). What is important to note though is that L(E) is by its very nature abstract; its

members are all possible quantum events, and thus include things that have not occurred which

make them abstract objects. So, even if we are successful in providing a nominalistically

acceptable version of the mathematics in L(H), the very thing that we are representing is still

abstract in nature. As Balaguer puts it, “to replace L(H) with L(E) is just to replace one

platonistic structure with another.” (Balaguer, 1998, p. 120)

Balaguer’s plan to nominalize quantum mechanics has three parts. First, he has to produce a

nominalistic structure that is embedded in L(H). This way the mathematical subspaces of the

Hilbert space just represent this nominalistic structure. Next, he has to get around Malament’s

worry that L(E) is abstract in nature. Balaguer must show that there exists some other ‘set’ which

is both nominalistically acceptable and isomorphic to L(E). Call this new nominalistically

acceptable orthomodular lattice L(P). Finally he has to prove representation theorems between

the mathematics and the physical properties picked out in L(P).

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What I need to do, then, is find a way of taking the closed subspaces of

Hilbert spaces as representing physical phenomena of some sort or other;

if I can do this, I should be able to construct nominalistic structures out of

these physically real things and then prove representations theorems that

enable me to replace the mathematical structures in question – that is, the

orthomodular lattices built up out of closed subspaces of Hilbert spaces –

with these new nominalistic structures. (Balaguer, 1998, p. 120)

We will focus on is Balaguer’s second task: that there are real physical properties of quantum

systems that the mathematics represents. Balaguer is inspired by Field’s approach where Field

argues that for properties such as temperature and length, the mathematics is simply representing

the actual physical properties of temperature and length possessed by the system in question. It is

in the technical details of this approach where most believe that Field failed. Putting aside the

technical details, we see that the fundamental motivation is that properties like temperature and

length are physical in nature and are possessed by the physical object or physical system being

considered. Mathematics is simply an indexing or representational tool for these physical

properties. The problem for Balaguer in using the same approach for looking at quantum

properties is that quantum mechanics is inherently probabilistic. For Field, it is not so

controversial to believe that a physical rod has a purely physical property of some determinate

length that we represent through mathematics. But for, say, quantum mechanical spin, what does

it mean to say that a probability is a physical property?

As we saw in the electron spin example, quantum mechanics tells us that the probability of the

electron having spin-up in the z-axis is 0.5. In order for the indexing argument to run, this

probability would have to be indexing a real physical property. Balaguer’s claim is that the

mathematics of quantum mechanics “represent propensity properties, for example..., the 0.5-

strengthed propensity of a z+ electron to be measured spin-up in the x direction.” (Balaguer,

1998, p. 120) It is propensities that are the physical properties which the probabilities represent.

If this is true, then the Pauli spin matrices are not genuinely explanatory of the behaviour of a

beam of electrons passing through a Stern-Gerlach apparatus. They merely index the true

explanatory facts: the electron possesses the 0.5-strengthed propensity for spin up in the z-axis,

and the 0.5-strengthed propensity for spin down in the z-axis.

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Balaguer’s nominalized quantum mechanics entirely depends on accepting propensities as purely

physical properties. If we do not believe that propensities are nominalistically ‘kosher’, then

Balaguer will have “merely replaced one platonistic structure with another.”(Balaguer, 1998, p.

126) Balaguer does not argue comprehensively for the claim that propensities are

nominalistically acceptable, but he does suggest two ways that he could endeavour to prove it.

The first is “to take quantum propensities... as the basic entities of our nominalistic structures and

simply argue that these things are nominalistically kosher.” (Balaguer, 1998, p. 126) Balaguer

has simply been assuming that this is the case, but he admits that such a view is controversial.

Still, he maintains that we should believe that propensities are physical properties. The

distinction he makes is between physical properties that are properties of particular physical

objects, exist in space-time, and are causally efficacious, and properties-in-abstraction, which

are abstract objects. This distinction is tenuous at best, and Balaguer accepts that he would have

to argue that physical properties can be clearly separated from abstract properties. The most

important step is that even if we grant the demarcation between physical and abstract properties,

Balaguer has to then convince us that propensities fall on the physical properties side, and not the

abstract. His strategy for this is to argue that propensities exist in space-time and are causally

efficacious. Balaguer does not make this argument, but instead suggests that it would run

similarly to how Field argues.

For instance, if we consider a particular particle b, it seems that b’s charge

causes b to move about in certain ways in a magnetic field; but given this,

it seems obvious that b’s charge exists in b (although it might not have any

exact location in b) and it seems almost crazy to say that it exists outside

of spacetime. What would it be doing there? And how could it have causal

influence from there? (Balaguer, 1998, p. 127)

The suggestion here is that propensities are just like charge in this example. They are both

properties of a particular object, exist in space-time, and are causally efficacious. Hence we

should believe that propensities, like charge, are physical properties and not abstract.

There are many problems with Balaguer’s analogy. The critical move is that we are supposed to

believe that abstract properties of particular objects are ‘crazy’ in that existing outside of space-

time leads to two unanswerable questions: where is the property, and how can it have causal

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influence from where it is? In an excellent paper, Baker (2003) convincingly demonstrates that

both these questions are fundamentally flawed. The assumption that Balaguer and most

platonists make is that mathematical objects are abstract which entails that they are non-spatio-

temporal, and are acausal in nature. However, if abstract entities do not exist in space-time, what

does it mean for Balaguer to even ask what they would be doing ‘there’? ‘There’ signifies a

spatiotemporal location, but we are all already assuming that abstract objects are not

spatiotemporal to begin with. As Baker correctly notes,

This sort of looseness is symptomatic of a general tendency to view

abstract objects as akin to ultra-remote, ultra-inert concrete objects... It is

all too easy, for example, to slip from talking of mathematical objects as

being non-spatiotemporal to talking of them as existing outside of space-

time. But ‘outside’ is of course a spatial notion, hence it cannot

legitimately be applied to abstract mathematical objects. (A Baker, 2003,

p. 249)

A similar looseness in reasoning can be seen in Balaguer’s second question. Balaguer is moving

from the assumption that abstract objects are acausal to the objection that they cannot make any

causal difference to the physical world. There is nothing wrong with this move. The problem lies

in Balaguer’s further inference that for abstract objects to make no causal difference means that

abstract objects make no difference to the world at all. This is a leap of reasoning that is not

justified.35 Balaguer is simply assuming that the only way to make a difference to the physical

world is through causation. This assumption is controversial, and also clearly begs the question

against genuine mathematical explanation. Abstract objects may have no causal influence to the

physical world, but that does not mean that they do not make a difference tout court to the

physical world either.

What Baker has shown is that Balaguer’s two unanswerable questions are no good. Without this,

it is not ‘crazy’ to believe that propensities are abstract, and even worse for Balaguer, there is no

reason to believe that propensities are purely ‘physical’ properties which can serve as the

35 Baker (2003) argues that this leap in reasoning critically depends on the argument that mathematical objects exist

‘outside’ or space-time. As we saw, such an argument is fundamentally flawed.

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nominalistic basis for quantum mechanics. Balaguer is aware that we may not be convinced that

propensities in themselves are nominalistically kosher, so he offers a second method in which we

can get around this issue. He asserts that we can nominalize away any commitment to

propensities. Recall that Balaguer believes that Field’s method of nominalizing Newtonian

gravitational theory was successful. Field believes that properties like length are physical

properties possessed by the physical object in question. This belief motivates him to develop a

system that eliminates reference to numbers when referring to the lengths of objects. Balaguer

aims to do the same thing for propensities. Propensities as they are presently formulated are in

the form of an ‘r-strengthed propensity’, such as the electron’s 0.5-strengthed propensity for spin

up in the z-axis. The trick will be to remove reference to the ‘r-strengthed propensity’ and

replace it with something nominalistically acceptable. Balaguer does not go over the technical

details, but he commits himself to employing the exact same method as Field did. A prima facie

problem with this approach is that the general consensus is that Field’s project failed in the

technical details. Balaguer is a notable exception to this consensus, but tying himself to this

approach is unlikely to convince anyone else.

Even if we accept Balaguer’s optimistic outlook that he “does not forsee any real problems”

(Balaguer, 1998, p. 126) eliminating reference of ‘r-strengthed propensities’, there is still a major

flaw in his approach. A strength of Field’s project is that the belief that properties like length and

temperature are purely physical seems perfectly reasonable. On the face of it, it is not difficult to

conceive of temperature and length as existing in space-time, being causally efficacious, being

independent of the mathematical units that we use to express them, and whatever other

conditions we may impose on a property being physical. The reason for this is because we have

independent purely physical experience of these properties. We can see and touch objects that

have length and temperature, and these properties have, or at least we believe they have, a purely

physical basis to them, such as the motion of molecules for temperature. Thus, we have

independent reasons to believe that length and temperature are purely physical properties.

Balaguer takes this belief and tries to appropriate it to the propensity property. At the very start

of his sketch of how to nominalize away reference of ‘r-strengthed properties’, he simply asserts

that “[p]ropensities are just physical properties, like temperature and lengths, and so we can get

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rid of them in the manner of [Field].” (Balaguer, 1998, p. 126) But why should we believe that

propensities are just physical properties? The argument meant to support this claim was shown to

be lacking above. Moreover, Balaguer even admitted that this core belief is controversial to

begin with.

My main criticism with Balaguer ultimately lies in the claim that propensities are just physical

properties like lengths and temperature. Specifically, the objection is that there is a critical

difference between propensities, and lengths or temperatures; the former has no physically

independent motivation for believing it to be purely physical, whereas the latter do. The belief

that propensities are physical properties aligns Balaguer with the propensity interpretation of

probability.36 Roughly speaking, this interpretation is that probabilities are really propensities or

dispositions of physical objects or situations to yield particular outcomes. There are many

challenges that face the propensity interpretation of probability, such as the connection between

propensity and long run relative frequency, or even the precise definition or meaning or

propensity itself. However, the other leading interpretations of probability also face their own

unique challenges. I will not take any sides on the issue here, and will be happy to treat the

propensity interpretation as perfectly viable. The challenge that I will present is the observation

that the propensity interpretation is an interpretation of the Kolmogorov axioms of probability,

and is thus mathematically motivated.37

36 Interestingly, Balaguer states that he is not committed to the propensity interpretation of probability and of

quantum mechanics. He claims that the broad claim that he is committed to is that “quantum systems are irreducibly

probabilistic, or indeterministic.” (Balaguer, 1998, p. 120) This claim is compatible with many different

interpretations of quantum mechanics, excluding hidden variables interpretations. In fact, Balaguer goes on to say

that he need not even commit himself to this claim if some other way to understand quantum mechanics in a

nominalistically acceptable manner surfaces. I find these assertions quite unbelievable. Balaguer seems to

necessarily have to commit to the belief that probabilities index propensities, and that propensities are physical

properties, as simply believing the broad claim does not imply a nominalistically acceptable quantum mechanics.

This is obvious as one could easily be a mathematical realist or believe in that the example of electron spin is a

genuine mathematical explanation of a quantum mechanical physical fact while still believing Balaguer’s broad

claim. I will not argue for this further, but will merely present a version of Balaguer that is committed to the

propensity interpretation of probability and quantum mechanics.

37 The Kolmogorov axioms of probability are not the only way to axiomatize probability calculus, but they have

earned the status of the orthodox interpretation amongst mathematicians. It is also worth noting that some

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The propensity interpretation can be traced back to Peirce (1910), and was advanced

significantly by Popper (1957, 1959). It seems that Balaguer is most influenced by the particular

brand of propensity interpretation put forward by Giere (1973). Popper and Giere were

concerned about how to interpret single case probabilities that present themselves in quantum

mechanics, such as the probability that a particular radioactive atom decaying being equal to 0.5.

Giere believes that statements such as these indicate that the quantum realm is fundamentally

indeterminate. Giere’s solution is to claim that probabilities represent propensities which are

physical properties of particular physical objects, and it is these propensities that lie at the

foundation of quantum mechanics.

At the moment there are no rigorous, widely accepted formalizations of

quantum theory, but it is a good bet that such a formalization will contain

a term with the basic formal characteristics of a probability. The physical

realizations of these quantities will be propensities. (Giere, 1973, p. 476)

In this way, Giere likens propensities in quantum mechanics to how properties such as charge

work in classical physics – they are both physical and not reducible to less theoretical concepts.

This certainly sounds extremely similar to the point of view that Balaguer is espousing.

Christopher Hitchcock presents an interesting challenge for Giere’s interpretation. If

probabilities represent propensities, then propensities necessarily have a mathematical structure.

Giere admits this when he argues that his single case propensity interpretation “provides a

natural interpretation for the whole mathematical theory of probability and statistics since

Kolmogorov.” (Giere, 1973, p. 477) But where does this structure come from? More specifically,

why should we believe that propensities should satisfy the basic laws of probability and statistics

at all? Hitchcock notes that Giere’s analogy of propensities to electric charge is flawed. All our

knowledge of charge is empirical in nature, but out knowledge of propensities is not.

It is surely an empirical matter, e.g., that charge comes in discrete

quantities (one-third the charge of the electron) and in both positive and

negative magnitudes. This cannot be determined a priori. But it is hard to

interpretations of probability do not satisfy the Kolmogorov axioms, however, we will not consider those

interpretations here.

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imagine discovering empirically, e.g., that chances could be greater than 1

or less than 0, or that they are not additive... it is hard to understand what

it could even be to discover empirically that chances can be greater than 1

or less than 0. (Hitchcock, n.d., pp. 10–11)

Hitchcock goes on to illustrate that there is simply no purely physical or empirical motivation for

having propensities model the axioms of probability. However, this assumption is needed in

order for propensities to be utilized as a building block of quantum mechanics.

My challenge to the claim that mathematics indexes propensities is that propensities themselves

are not purely independently physically motivated. The nominalist is helping themselves to

mathematical structure that is motivated solely from the axioms of probability. If this is true,

then applying the indexing argument against the example of electron spin will fail as criteria (D)

states that what the mathematics is allegedly indexing must be physically and independently

supported. Given that propensities as physical properties critically depends on the axioms of

probability, then running the indexing argument will only replace one platonic structure with

another. One way around this is to simply state that propensities follow all the axioms of

probability, but that this is simply a brute fact of nature. In this way it is not influenced by

mathematics, and is arguably physically motivated. However, the problem with this move is that

it is not independently motivated and is entirely ad hoc. This is the very reason why we ruled out

appealing to any facts that are not physically and independently motivated in the first place. Such

an ad hoc explanation is parasitic on the mathematical explanation, and poses no threat to the

genuine nature of the mathematical explanation in question.

5.3 No Explanation

The final option for the nominalist to resist the conclusion that the Pauli spin matrices are a

genuine mathematical explanation for the behaviour of a beam of electrons passing through a

Stern-Gerlach apparatus is to claim that there is simply no explanation at all of this phenomenon.

What we have stumbled upon is an excellent predictive tool in the mathematical formalism, but

this is not an explanation as nothing can explain the nature of the electron. What could it possibly

mean to have a phenomenon that we can accurately predict, but has no explanation? Notice this

is not the same as believing that we just do not yet know the actual explanation, which we

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addressed in 3.5.1. This is the much more controversial position that the explanandum is

fundamentally unexplainable – no possible explanation exists at all.

One way to attack this is to point out that the conclusion is so absurd that it functions as a

reductio to argue in favour of accepting that the mathematical formalism of electron spin is

genuinely explanatory. The conclusion that something exists that is perfectly accounted for and

predictable but is ultimately without explanation is so extreme that it ought to be rejected

immediately. The stubborn nominalist who holds this view would be cutting off their nose to

spite their face. In a last ditch effort to save themselves from admitting the existence of a GME,

the nominalist would be willing to abandon any hope of explaining the behaviour of our most

fundamental entities. This alone is enough to show that such a move is untenable.

While the above rebuttal is perfectly reasonable, I admit that it is not completely airtight. I am

assuming that the sole reason for maintaining that there is no possible explanation of the

behaviour of electrons is because the nominalist is unwilling to admit the mathematical

explanation as being genuinely explanatory. However, there could be entirely independent

reasons for believing that phenomena in the quantum mechanical realm are fundamentally

unexplainable, and if this is the case then such a belief is not as ludicrous as it seems. Our most

basic notion of what a scientific explanation actually is comes from the classical world of

medium-sized objects. Explanations of physical facts almost always cite other physical factors in

a causal relation, or they appeal to deterministic laws of nature. The problem is that the quantum

world is famously unlike the classical picture. On some interpretations of quantum mechanics

causation and deterministic laws are entirely out the window. Given that this is the case, then

perhaps the traditional links that we have become accustomed to between things such as

predictability and understanding to that of explanation do not hold in quantum physics.

This line of reasoning leads to the conclusion that there are two types of explanation: classical

explanation and quantum explanation. Classical explanation we know lots about, but for

quantum explanation we know next to nothing. So when we say that there is no explanation of

the behaviour of electrons, this is due in part to the fact that we do not even know what

explanation means in the quantum realm. Although we have some elements of a classical

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explanation in that the Pauli spin matrices are an excellent predictive tool, we should not confuse

that as having anything to do with quantum explanation. Until we can cache out what quantum

explanations really are, we are justified in saying that there is no explanation of the Stern-

Gerlach experiment.

I have little to say against this objection. If correct, it certainly weakens the force of the electron

spin example that I have advanced as a GME. The problem is that even beginning to explore the

differences between classical and quantum explanation would be an extremely large and difficult

task and is beyond the scope of this dissertation. All I can say is I feel that supporters of such a

radical view carry the burden of proof. If there is such a significant difference between classical

and quantum explanation such that classical factors like prediction and understanding are not

related to explanation in the quantum realm, then it is up to them to establish that this is the case

as such a conclusion is a drastic departure from standard beliefs. In the mean time I am happy to

simply assume that this is not so, and that quantum mechanical phenomena, just like other

classical physical phenomena, have explanations.

6 The Honest Conclusion

Overwhelmingly so, the largest problem in the present debate is that there does not exist any

clear understanding or consensus of what a GME even means. The first half of this chapter was

dedicated to remedying this problem. The strategy was to extract four key criteria from the

various employments of the indexing argument used to block past examples of mathematical

explanation. The remainder of this chapter was aimed to showing that the example of electron

spin does satisfy the four criteria. Satisfying (A) – (C) was not so difficult, but the crux of the

nominalist intuition lies in criterion (D). In order to be considered as a genuine explanation, it

had to be shown that the mathematics of the Pauli spin matrices was not indexing some known,

physically and independently motivated physical fact. This was demonstrated in two ways. First

we demonstrated a fundamental disconnect between quantum mechanical spin and any

classically understood physical property. The standard way of pointing to some other physical

property as being the true explanatory factor is simply unavailable to the nominalist. We also

showed that three other possible responses from the nominalist camp are untenable.

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What has been demonstrated, then, is that the electron spin example satisfies all four criteria laid

out by the indexing argument. The only conclusion for the honest nominalist is that the example

of electron spin is a legitimate candidate for being a GME. This is certainly weaker than the

desired conclusion that the example of electron spin is a GME. In order to move to this stronger

conclusion we must turn our focus towards accounts of scientific explanation.

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Chapter 4 Scientific Mathematical Explanation

Last chapter we developed the following set of five criteria for a genuine mathematical

explanation (GME). A mathematical explanation is a GME if:

(A) the explanation is not contrived and is accepted by the scientific

community as a good scientific explanation,

(B) the mathematics employed is not arbitrary,

(C) there are no purely nominalistic explanations of the

explanandum,

(D) there are no known and physically independently motivated

physical facts that could potentially explain the explanandum

such that the mathematics is simply indexing these facts,

and,

(E) an acceptable account of scientific explanation must

corroborate the claim that the mathematics is explanatory.

The example of electron spin which mathematically explains the behaviour of a beam of

electrons passing through a Stern-Gerlach apparatus was shown to satisfy the first four criteria.

In doing so we are able to avoid the indexing argument, which has been the most potent weapon

for the nominalist. The end result is that the nominalist has no reason to reject the electron spin

example as a candidate for a GME.

The aim of this chapter is to establish that the electron spin example also satisfies criterion (E).

So far, realists have made no attempt at verifying supposed examples of GMEs via accounts of

scientific explanations. The hope has been that simply showing that the nominalist cannot

intuitively resist a mathematical explanation is enough. However, this hope has proven

unrealistic and can be exploited by committed nominalists. Even if an example resists the

nominalists’ intuition that mathematics is solely indexing physical facts, it could still somehow

be the case that the mathematics is not genuinely explanatory. The only way around this is for

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the realist to make the case that according to acceptable accounts of scientific explanation, the

mathematics in our examples of GMEs is indeed genuinely explanatory.

Applying accounts of scientific explanation to mathematical explanations is no easy task. A

prima facie problem is that almost all of the standard accounts of scientific explanation were

never intended to analyze mathematical explanations in the first place. Add to that the

complication that there exists no single accepted account of scientific explanation within the

literature, and it is no wonder why realists have no desire to go down this road. In the last chapter

I made the demand that the nominalist be of an honest variety – one who is open to the

possibility of mathematical explanation. I feel that it is only fair that the realist be honest as well.

The realist needs to have something substantial to say when asked why we should believe that

the mathematics in our examples is explanatory.

Our first task will be to canvas many accounts of scientific explanation. Ultimately, we will

utilize Michael Strevens’ (2008) kairetic account as it is the most likely to be suitable for our

analysis. Next, the kairetic account will have to be adapted such that it will be able to apply to

non-causal, mathematical explanation. Finally, we will use this adapted kairetic account to

analyze some of our examples of GMEs, including the electron spin example. The conclusion

will be that the example of electron spin is corroborated by the kairetic account, and that the

mathematics is found to play the explanatory role of ‘difference-makers’ in our explanation.

Thus, the example of electron spin satisfies all five criteria and will be firmly established as an

example of a GME.

The framework that we will operate under for the bulk of this chapter will be of convincing the

nominalist that the electron spin example is a legitimate GME. Essentially, we will be focussed

on the question of whether or not mathematics can genuinely explain scientific facts. Once this

has been established, we can free ourselves from this restricted perspective and examine what it

is about mathematics in these examples that facilitates a good scientific explanation. We want to

be able to say something to the overlooked question identified in chapter 3: how is it that

mathematics can explain? The hope is to make some inroads into giving a more substantial

answer to what we mean when we say that mathematics is genuinely explaining a physical fact.

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1 Accounts of Scientific Explanation

The literature on explanation is vast and extensive. My aim here is not to assess which accounts

are better than others, but rather to select an account that can help us decide whether or not

mathematics is genuinely explanatory. Three difficulties present themselves in this selection

process. The first difficulty is that there is no widely accepted account of scientific explanation.

All of the classic accounts of scientific explanation have been shown to be fraught with

difficulties that make them each unappealing. In their stead are a collection of newer theories that

differ greatly in their approach and understanding of what an explanation actually is. Given this

landscape, choosing an appropriate account is not a simple or uncontroversial task.

The second difficulty is that many accounts of scientific explanation are exclusively causal

accounts. These theories say that some set of facts, A, explains some physical phenomenon, E,

only if A causes E.38 A causal approach is not surprising as plenty of scientific explanations are

of the causal variety; however, these accounts are not useful for our purposes. Mathematical

explanations by their very nature are noncausal, as by all standard accounts mathematical entities

and their properties are acausal in nature. This greatly restricts the types of accounts of scientific

explanation that we can even consider.

The final difficulty in our selection process is that we have a very particular goal. We do not

want an account of scientific explanation to simply verify that the example of electron spin is a

good scientific explanation. This much is already agreed upon by both realists and nominalists

alike. What is needed is an account that can isolate and identify the contribution of the

mathematics within the explanation. We need an account that can clearly decide whether the

mathematics is playing a genuinely explanatory role or not. In addition, our account must be

sufficiently discriminating in order to satisfy the nominalist. If our account of scientific

explanation trivially decides that all our examples, including the cicada and honeycomb

explanations, have genuinely explanatory mathematics, the nominalist is unlikely to agree that

38 See Salmon (1984) for a classic account of causal explanation.

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this account is the right one to use. Our ideal account of scientific explanation needs to be

sophisticated enough to rule out examples such as the cicada and honeycomb, but at the same

time be able to point to the Pauli spin matrices as conferring an explanatory role.

1.1 The Deductive-Nomological and Pragmatic Accounts

Any exploration into scientific explanation typically begins with Carl Hempel’s (1965)

deductive-nomological (DN) account. The DN account states that the explanans explains the

explanandum if and only if all statements in the explanans are true, there is at least one law of

nature in the explanans, and there is a deduction from the explanans to the explanandum.

Consider the cicada and honeycomb examples. These optimization explanations certainly satisfy

the conditions of truth and deduction, but what of the law of nature? If mathematical theorems

are laws of nature, then we have a good mathematical scientific explanation. If they are not laws

of nature, then we do not have a good explanation. However, there is no way to decide if

mathematical facts are to be laws of nature without entirely begging the question. Understanding

what we mean by laws of nature is complicated, but at the bare minimum laws of nature

fundamentally explain the world around us. If we assume that mathematical theorems are laws of

nature, then this amounts to assuming that they explain physical facts, which is the very thing we

are looking to establish in the first place. Likewise, if we do not grant mathematics law-like

status, then these optimization explanations are not satisfactory on the DN account and hence do

not explain. This too begs the question as not granting mathematics law-like status is equivalent

to claiming that they do not explain the physical world. The problem here, and moreover a

problem in general for the DN model, is that too much hinges on what we consider to be a law of

nature.

The pragmatic account of explanation is most famously championed by Bas van Frassen (1980).

Unlike the DN model, the pragmatic account does not require that an explanation be a formal

deduction. Instead, a good explanation is simply an answer to a why-question. Not any response

to a why-question counts as a good explanation. A why-question must be well-formed in order

for a satisfactory answer to be given. A good why-question has three parts and is fundamentally

contextually based. First, the question must have a topic: the phenomenon that needs explaining.

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There must be a contrast-class which is a set that includes the topic and many alternative

propositions. This contrast-class helps identify the context of the question, and thus the proper

type of answer as well. Lastly, a proper why-question has a relevance-relation which determines

what counts as a possible explanation or explanatory factor. An answer to a good why-question

gives a reason as to why the topic is true as opposed to the other alternatives in the contrast-class

in a way that is relevant to the context of the question.

The pragmatic account is much less formal than the DN account. In applying it to our

optimization explanations it seems obvious that they are good explanations. Why is the

honeycomb hexagonal? The contrast-class implied in the question is that we are asking why,

specifically, is the honeycomb a hexagon and not some other polygon, or even some other

irregular n-sided figure. Also, the type of answer that we are looking for is not simply that they

are hexagons because the honeybees built them that way. We are looking for a reason why all

honeybees build hexagonal honeycombs. The answer is due to Hale’s honeycomb theorem which

states that the hexagon is the optimal number of sides to minimize surface area when tiling an

area. This explanation certainly answers the question in showing why the honeycomb is not

some other n-sided shape.39 Similarly, the electron spin example is an answer to the following

why-question: why does the electron beam split into two distinct beams equidistant from the

center plane? The Pauli spin matrices adequately answers why there are two beams instead of

three or some other number, why they are distinct and not continuous or random, and why they

are equidistant from the center plane as opposed to some other distance.

The present literature goes no further than the DN or pragmatic account in analyzing the

examples of mathematical explanation. Baker (2005, p. 235) seems to think that this is enough to

show that GME’s are confirmed by our accounts of scientific explanation. I find this wholly

unsatisfactory. Even in the best possible scenario where we grant that examples such as the

cicada or honeycomb are indeed good scientific explanations on the DN or pragmatic account,

39 This is just a partial explanation as the full explanation would invoke the other required aspects of an optimization

explanation: appealing to evolutionary biology and ecological constraints.

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the major issue here is that neither account can actually tell us what aspects of the explanation

are genuinely explaining. It could very well be that the mathematics is still not genuinely

explanatory at all even though the overall explanation is mathematical and a good scientific

explanation. This is, in essence, the line of reasoning in the indexing argument.

Recall Nerlich’s example of the cloud of particles that changes in size when moving along a

curved surface. This explanation made use of geodesics, vectors, curvature, etc. If we grant that

Nerlich’s explanation is a good scientific explanation by the DN and pragmatic accounts, what

can we conclude about the mathematics employed in the explanation? Nothing! The reason is

that the role of the mathematics is still unclear even though we grant that the explanation in

general is a good one. Is it the mathematics that is explaining, or is it, as the nominalists contend,

the actual physical properties of curved space-time that explains the behaviour of the particle

cloud? Neither the DN nor the pragmatic account can help us answer this question as neither

account are in the business of actually picking out what the key explanatory contributors are

within an explanation. They are unable to identify the difference-makers to the explanandum.

When trying to establish if GMEs exist or not, what we really need is an account of explanation

that can pick out the proper difference-makers; we need an account that can identify what is

genuinely carrying the explanatory force. It is no surprise that our example explanations seem to

be compatible with the DN and pragmatic account as they are all deductions and answers to why-

questions. But what we are really looking for – the exact role that the mathematics plays – is

beyond the ability of these accounts of explanation to determine.

All that we can take away from this is that, at best, the DN and pragmatic account are open to the

possibility of GMEs, and nothing more. Baker (2005, p. 236) suggests that if we need more

evidence we should look towards our intuitions and those of the biologists who support the

cicada example for an extra nudge in the right direction, but this too is unsatisfactory. Certainly

there are those whose intuitions would lean in the exact opposite direction. For example, one

could have the intuition that mathematics is solely a representational tool; the true explanatory

factors are ultimately physical in nature. This is, of course, the very same intuition that the

nominalist holds. In any case, realists surely want to do better than leaving the status of GMEs

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and accounting for the explanatory role of mathematics to raw intuition, and thus neither the DN

nor the pragmatic account will be our account of choice.

1.2 The Unification Account

The unification account of scientific explanation was advanced by Michael Friedman (1974) and

improved upon by Philip Kitcher (1989). The unification account claims that what makes an

explanation good, or what enhances our understanding of the world, is that it unifies otherwise

seemingly independent different phenomena. Thus, a good scientific explanation leaves us with

less total phenomena that we accept and treat independently than we had before the explanation.

Many examples of such unification exist in the history of science. The kinetic theory of gases

entails the truth of many other laws, such as Boyle’s law, Charles’s law, Avogadro’s law,

Graham’s law of diffusion, and more. Newton’s laws of motion unifies the behaviour of

terrestrial and celestial bodies. This unification is what makes the kinetic theory and Newton’s

laws explanatory. No longer do we have many different, seemingly independent brute facts about

gasses or bodies. Instead we have a unified system where we can derive the exact same results,

and thus explain them.

It does seem that there are many examples in science where mathematics unifies seemingly

disparate phenomena. Colyvan (2002) gives an example of the use of complex numbers in

physics. Consider two differential equations:

(1) 𝑦 − 𝑦′′ = 0,

and

(2) 𝑦 + 𝑦′′ = 0,

where y is a real-valued function of a single variable. (1) describes some physical system

exhibiting growth, and (2) describes certain periodic behaviour. Equation (1) can be solved using

standard real algebra, but (2) requires complex methods to solve. Since complex algebra is a

generalization of real algebra, we can in fact employ the exact same method of solving (2) as we

can for (1). Complex algebra, then, unifies the mathematical theory of differential equations as

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well as the physical theories that make use of differential equations. Without complex numbers

we would have to treat (1) and (2) as disparate and independent phenomena.

Melia (2002) rejects this example. He points out that there is a major difference between

providing a unified account to solving methods and providing a unified account of apparently

independent physical phenomena. One can grant that complex algebra is a genuine example of

unification within mathematics, but this does not mean, contra Colyvan, that complex algebra

also unifies physical phenomena. Melia’s point is that the unification of algebra in mathematics

does lead to a sort of unification in solving methods in physics, thus allowing for a smaller tool

chest for the practicing physicists to work with, but this does not imply anything about the actual

nature of the physical world. The kinetic theory of gas unifies because it explains Boyle’s law,

Charles’s law, and many more laws in virtue of the fact that gas is made up of a large number of

molecules in constant, random motion. These laws hold as all gases are nothing more than the

molecules invoked by the kinetic theory. However, no one would claim that simply because (1)

and (2) have a unified solving method that this implies that they necessarily have a shared

underlying reality.

A more promising example is how the mathematics in the cicada example can be said to unify in

the sense that we use the exact same proof to explain the periodic life-cycle of the 13-year cicada

as for the 17-year cicada. The only difference between the two is that the ecological constraints

are different as they inhabit different regions of North America that possess different weather

patterns. Critically, there is also no appeal made to any sort of unification on the mathematical

end of solving methods in these explanations. It is the exact same mathematical theorem – prime

numbers minimize intersection – that unifies the physical phenomena of the 13 and 17 year

cicada. This avoids the challenge that there is a unification of solving methods on the

mathematical end only, and does seem to more clearly point to the conclusion that there is a

shared underlying nature to the cicada life-cycles, and that this nature is revealed by

mathematics. But herein lies a new but familiar problem. What is doing the unifying – the

mathematics or something else? Is the underlying unifying nature mathematical or purely

physical? Is it even possible for mathematics to unify the physical world?

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What we see is that when we invoke unification we end up with another version of the indexing

argument. Instead of claiming that the mathematics does not confer an explanatory role in

scientific explanations, the nominalist now asserts that mathematics does not confer a unificatory

role. All we have done is shift our attention from explanation to unification – we have solved one

problem by replacing it with another of equal difficulty. A major reason why unification does not

help us is that it is difficult to understand the exact relationship between unification and

explanation. Morrison (2000) makes a strong case that there are examples of unification in

science that are not explanatory. She argues that the unification which is achieved by using

mathematical structures and representations often results in a loss of explanatory power. Pincock

(2011) claims that there are also examples of potentially genuine mathematical explanation in

which their explanatory nature is not due in any part to unification.

All of this points to the fact that we should be reluctant to turn to the unification account in order

to determine the status of GMEs. Even without the major challenges the account faces in light of

Morrison and Pincock, there is still the overarching issue that the unification account cannot

precisely determine the role that mathematics is playing in even the explanations that seem to fit

the unification model best. Just as we saw in the DN and pragmatic accounts, the unanswered

question is whether or not the sole purpose of mathematics in these explanations is to represent

underlying physical facts, and as it stands the best we can do is point to our intuitions in order to

resolve this question.

1.3 The Statistical-Relevance and Counterfactual Accounts

The statistical-relevance account from Salmon (1971) and the counterfactual account popularized

by David Lewis (1973) are both causal accounts of explanation. As stated above, given the

standard assumption that mathematical entities are acausal, any causal account of scientific

explanation will not do the job of analyzing supposed examples of GME as none of these would

be considered a good explanation in the first place. Notwithstanding, there is still an important

benefit in looking at these approaches. These accounts are unlike the DN, pragmatic, and

unification accounts in that they are in the business of identifying the key (causal) factors that

explain an explanandum. The aim of statistical relevance and counterfactuals is to pick out the

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proper difference makers from the many other factors within an explanation. If it is possible to

adapt the statistical-relevance or counterfactual approaches to look at non-causal explanations,

then this could help us in determining if mathematics is genuinely a difference-maker.

The statistical-relevance account states than any factor which is statistically relevant to the

occurrence of the explanandum is explanatory. A good explanation reveals some (or all) of these

factors. To identify if a factor, a, is statistically relevant to our explanandum, E, we turn to

probability. If the probability of E occurring is different from the probability of E occurring

given that we know that a is true or has obtained, then we say that a is statistically relevant to E.

In conditional probability notation this is to say: 𝑃(𝐸|𝑎) ≠ 𝑃(𝐸). But how do we arrive at these

probabilities? For our mathematical explanations, let m be the mathematical theorem that our

explanation depends on. We need to know whether or not 𝑃(𝐸|𝑚) ≠ 𝑃(𝐸) to ascertain if m is an

explanatory factor for E. Even if we are able to assign a value to 𝑃(𝐸|𝑚) in some unproblematic

way, the difficulty is in trying to asses 𝑃(𝐸) alone. That is, we have to determine the probability

of E occurring as if m is false. But m is not like some standard physical fact or state of affairs; it

is a mathematical theorem, and thus on all standard accounts is necessarily true. Is assigning a

value to 𝑃(𝐸) alone even possible?

Closely related is the counterfactual account of explanation. The counterfactual account is rooted

in an account of causation where we can identify if a factor, b, is a cause of an event, E. We do

so by considering counterfactual conditionals. b is a cause of E if and only if the following two

counterfactuals are true: “if b occurs, then E occurs”, and, “if b had not occurred, then E would

not occur.” In the case of explanation, if we are at all considering b to be an explanatory factor

for E then we already know that b and E have occurred or are true and thus the first

counterfactual is trivially true. What needs to be checked is the second counterfactual. If the

second counterfactual is also true, then b is a cause of E and we can conclude that b (in part)

explains E. The greatest challenge for the counterfactual account is determining if the second

counterfactual, “if b had not occurred, then E would not occur”, is true or false. In order to

evaluate this truth value, Lewis uses his theory of possible worlds. Look to the closest possible

world in which b is false, or equivalently where ~b is true, and then determine whether or not E

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occurs. To evaluate closeness we use a relation of comparative similarity between worlds. The

more similar a world is to our actual world, the closer it is to our world.

As above, consider a mathematical explanation where m is the mathematical fact that supposedly

explains E. Running the counterfactual analysis on m to see if it makes a difference to our

explanandum, E, requires us to look towards the closest possible world where ~m is true. But

what would such a world be? In our world, m is logically implied by the axioms of mathematics

which we believe to be consistent.40 In a world where ~m is true, we have one of two options.

The first is that the axioms of mathematics are the same, but somehow we are able to deduce

both m and ~m. This world would be inconsistent. The other option is to have a consistent world

where the mathematical axioms are different than what we have in our world. However, under

the standard interpretation of mathematics this is an impossibility. Even supposing that it is

possible, there would be no way to decide which new set of axioms would represent those in a

closest possible world. In both of these options, the worlds we are looking at are either

inconsistent or are so far removed from our actual world, if they can even exist at all, that

evaluating the status of E in these worlds is impossible.

More sophisticated counterfactual approaches have been developed (cf. Lewis 2000, Woodward

2003); however, all these approaches fail in analyzing mathematical explanations for the same

reasons. Although they may be satisfactory accounts in causal cases, the fundamental problem in

using them to analyze mathematical explanation is that all these accounts depend on asking what

would be the case if a mathematical theorem is somehow false. This works just fine for physical,

actual events, but is essentially incomprehensible when discussing mathematics. If we are to

adapt an account of difference-makers in order to resolve whether or not mathematics is truly

making a difference in scientific explanation we must look beyond any account that requires us

to consider what would happen if a particular mathematical theorem is in fact false.

40 I speak of axioms of mathematics to maintain generality. If m was a theorem of arithmetic, then it would only

need to depend on the axioms of arithmetic, and not, say, of Euclidean geometry. No matter what axioms m actually

depends on, the following argument can still be run.

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2 The Kairetic Account

Strevens (2008) advances a new account of scientific explanation in the tradition of identifying

difference-makers in causal explanations. What makes Strevens’ kairetic account different is that

it frees itself from the use of counterfactuals. Because of this, Strevens suggests that the kairetic

account can be adopted to identify difference-makers in non-causal explanations, and perhaps

even mathematical explanations as well. Strevens uses an approach similar to

Mackie’s (1974) INUS condition whereby the statement ‘c is a cause of E’ is true just in case c is

an insufficient but non-redundant part of an unnecessary but sufficient condition for the

occurrence of E. The key for both the INUS condition and the kairetic account is that a non-

redundant part of a sufficient condition for E is one that cannot be removed from the explanation

without invalidating the entailment of the occurrence of E. The removal process here is different

than in the counterfactual. There is no need to consider ~c, but rather we only remove c from our

picture; that is to say, after our removal of c there is simply no mention of it, negation or

otherwise. This bodes well for mathematical explanation as we no longer need to consider what

happens if a mathematical theorem is false.

Even though the kairetic account avoids counterfactuals and possible worlds, it is still presented

as an account for causal explanations. Given that, how can we apply it to noncausal

mathematical explanations? The answer lies in Strevens’ belief that all theories of explanation

have two fundamental parts.

I will tentatively propose that the complete philosophical theory of

explanation is modular: it consists of two components, a criterion for

explanatory relevance that is the same in every kind of explanation, and a

domain-specific dependence relation. The relevance criterion selects from

the given domain of dependence relations those that must be appreciated

in order to understand the phenomenon to be explained. When the domain

of dependence is causal, the result is a causal explanation. When it is

mathematical, the result is a mathematical explanation. And so on.

(Strevens, 2008, p. 5)

The kairetic account is comprised of two parts. First is the kairetic criterion which is the criterion

for explanatory relevance. This criterion identifies those aspects of an explanation that “make a

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difference to whether or not the phenomenon occurs.” (Strevens, 2008, p. 5) Second is the

domain of explanation. The kairetic account that Strevens presents in his book is a combination

of the kairetic criterion coupled with a causal domain of explanation. But given this modular

style, it seems that if we are able to isolate the kairetic criterion and apply it with the domain of

mathematical explanation, then we should be able to determine if the mathematics is genuinely

making a difference to the physical explanandum.41

2.1 The Kairetic Criterion

The basic idea behind the kairetic criterion is straightforward. Start with a model, M, that entails

the explanandum, E. All the statements within M must be true. In other words, M must be

veridical. Now optimize M through a process of abstraction performed on the individual

statements within M. At each stage of optimization we must verify that M still entails E. Once we

can no longer abstract anything in M without violating the entailment of E, we stop. This

optimized model is called a kernel, K, of E. The claim is that every statement within K is a

difference-maker for E. What difference-makers we come up with may depend on what our

starting model looks like. This is not a problem for the kairetic criterion, but rather a feature. To

not be a difference-maker is to be removed from all models via the optimization procedure, or

equivalently, to not be featured in any kernel that entails E.

The process of abstraction is essential to the kairetic criterion. By abstracting we can remove the

mention of unnecessary and unexplanatory details, and replace them with more general and

explanatorily relevant counterparts. Technically, a model A is an abstraction of a model M just in

case all influences described by A are also described by M, and every proposition in M is implied

by the propositions in A. Practically speaking, we can take specific statements and make them

41 Although Strevens in the quotation above makes reference to mathematical explanation, it is unclear what type of

mathematical explanation he means. In a subsequent section, Strevens returns to mathematical explanation and

seems to mean mathematical explanations of physical facts. However, he makes it clear that in these explanations

the mathematics is simply representing something physical. He also at times seems to refer to mathematical

explanation as mathematical explanation of mathematical facts. At no time does Strevens seem to mean what I call a

genuine mathematical explanation of physical facts where the mathematics is not representing but is instead actually

conferring explanatory power to the overall explanation.

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more general. Consider a 10kg cannonball thrown at a window where the window shatters. The

fact that the ball weighs exactly 10kg should not make a difference to the window breaking. For

any model M that contains the precise weight of the cannonball, we could generate a more

abstract model by replacing the statement ‘the cannonball weighs 10kg’ with a statement that

says, say, ‘the ball’s mass is greater than 1kg’. In doing so we can arrive at an optimized kernel

that yields proper difference-makers. Notice thought that if we try to remove the weight of the

cannonball entirely, or set the weight incredibly low such as greater than 0.1 grams, then the

model would no longer entail the breaking of the window. Hence such an abstraction is not

allowed.

Strevens notes two potential technical problems with the abstraction method. The first is path

dependence. Depending on which statement you optimize first you may arrive at a different

kernel. This is problematic as although many kernels may exist, each particular starting model

should always arrive at the same kernel via the abstraction process. To solve this problem,

Strevens states that there should be a unique end point to the abstraction operation for a

particular starting model. This end point is the maximal abstraction of M while still entailing E,

and we should call this maximally abstract model the kernel, K. It could be that certain starting

models do not have a definitive maximally abstract model, but in these cases Strevens can appeal

to other notions to help him decide on a well-defined set of difference-makers.42

The second issue facing the abstraction process is that of cohesion. The cohesion problem arises

as a disjunction of two models results in a newer, more abstract model. Consider models M and

N such that only one of these models is veridical and entails the event E. Now create a new

model, D, that is the disjunction of M and N. D is a veridical model that entails E and is more

abstract than either M or N on their own. But this implies that the disjunction is a difference-

maker for E, and not anything from simply M or N alone. Strevens writes that “this is not a

42 Strevens suggests that we could look at the facts that all the most abstract models agree on as our difference-

makers. Or, we could use some other possible criteria for comparison, such as notions of generality, in order to

adjudicate between rival kernels generated from the same model. Either way, Strevens states that “[n]othing crucial,

I think, turns on the choice.” (Strevens, 2008, p. 101)

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tolerable conclusion. Even if the disjunction can be said in some extenuated sense to be a

difference-maker, the disjunct... ought to be a difference-maker too.” (Strevens, 2008, p. 102)

Strevens states that models such as D lack cohesion. Although D is, in a sense, a more abstract

model than M alone, its lack of cohesion makes it a bad explanation of E. Strevens defines

cohesion via causal contiguity. “A model is cohesive, I propose, if its realizers constitute a

contiguous set in causal similarity space, or, as I will say, when its realizers are causally

contiguous.” (Strevens, 2008, p. 104) There is a trade-off between a cohesive model and an

abstract model. Strevens makes cohesion and abstraction, then, not strict requirements of the

abstraction process, but rather recommended desiderata. The goal is not to strictly satisfy one or

the other, but rather to maximize combined cohesion and abstractness.

The optimization procedure can be summarized as follows:

the explanatory kernel corresponding to a veridical deterministic causal

model M with target E is the causal model K for E that satisfies the

following conditions:

1. K is an abstraction of M,

2. K causally entails E,

and that, within these constraints, maximizes the following desiderata:

3. K is as abstract as can be (generality), and

4. The fundamental-level realizers of K form a causally contiguous set

(cohesion). (Strevens, 2008, p. 110)

The kairetic criterion states that all statements within a kernel, K, that is arrived at via the

optimization procedure are difference-makers for E.

2.2 Adapting the Kairetic Criterion

Before we are in a position to use the kairetic criterion on mathematical explanations we must

resolve an apparent inconsistency. The reason why Strevens’ kairetic account is appealing in the

first place is due to its two-level approach. The kairetic criterion is supposed to be a method for

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finding difference-makers in all domains of explanation. However, when we take a look at the

definition of the optimization procedure, which is the essential part of the kairetic criterion, we

see that this is not the case. Steps 2 and 4 make reference to causation within the definition of the

procedure. It appears that the kairetic criterion is not independent of causation, and there can be

no two-level approach.

Strevens gives us reason to think that this may be a fixable mistake. He writes,

the difference-making criterion takes as its raw material any dependence

relation of the ‘making it so’ variety, including but not limited to causal

influence. Given a relation by which a state of affairs depends on some

other entities, the kairetic criterion will tell you what facts about those

entities are essential to the dependence relation’s making it so. (Strevens,

2008, p. 179)

An attempt to recast the optimization procedure results in the following:

1. K is an abstraction of M,

2ʹ. K entails E in a ‘making it so’ type of entailment,

and that, within these constraints, maximizes the following desiderata:

3. K is as abstract as can be (generality), and

4ʹ. K is cohesive,

and the ‘making it so’ type of entailment and how we define cohesion

are dependent on the domain of explanation.

The original version of the optimization procedure presented above had already selected causal

explanation for its domain, and thus step 2 defined the ‘making it so’ type of entailment as causal

entailment, and step 4 defined cohesion as causally contiguous. What is now needed is to use

noncausal mathematical explanation as our domain and define steps 2ʹ and 4ʹ accordingly.

In considering how mathematics ‘makes it so’, one place we could start is the deductive power

that mathematics brings to the table. Certainly a key benefit to mathematics is that it functions as

an excellent tool for deductions. Although this is pretty much universally agreed upon, Strevens

does not believe that deductive power is the key contribution of mathematics when it comes to

explanations.

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The derivational conception of mathematics entirely fails to capture the

illumination that mathematical facts bring in the explanations of...

[physical] phenomena. It is not enough to be told that these phenomena

follow mathematically from the relevant laws and boundary conditions.

Somehow, in grasping the way that they follow – in understanding the

mathematics as well as the physics of the scientific treatment of the

explanandum – you come to understand the phenomena better. It is almost

as though, by looking into the mathematical structure of the derivation,

you can see the forces at work in nature itself. (Strevens, 2008, p. 303)

There must be something beyond raw deductive power that mathematics contributes in a GME.

Strevens suggests two ways in which mathematics is more than a deductive tool. First, it is more

than just knowing that the explanandum logically follows from the explanans, it is the way in

which it follows. Second, in understanding the mathematics in itself we can gain insight into our

overall explanation. Beyond merely suggesting these two ways in which mathematics contributes

to scientific explanation, Strevens does nothing to expand or explain what he means. I take his

second claim to refer to mathematical explanations of mathematical facts and how this relates to

mathematical explanations of physical facts. I will analyze this relationship in 4.4.1, but for now

will leave it aside.

What can be said about Strevens’ first suggestion that we must grasp the way in which the

explanandum follows from the mathematics? Unfortunately, not much. Perhaps what we need is

a better understanding of the relationship between mathematics and our best scientific theories.

The problem with this is that any full account of this relationship is sure to have certain

metaphysical assumptions. Among them would be the very things that we are investigating here:

can mathematics genuinely explain, and is mathematical realism true? If we are to assume any

metaphysical stance on these positions it would render the rest of our argument utterly useless.

Fortunately, Strevens believes that it is unnecessary to make any metaphysical assumptions.

The prospect for making sense of mathematics’ explanatory role as

something more than derivational, then, might seem to turn on

foundational questions about the nature of mathematics and its relation to

the world. But it is possible to see how the derivational view of

mathematics’ role in explanation falls short without indulging in any of

this excitement... Mathematical reasoning in explanation is supplying

something more than deductive glue, but you need no metaphysical

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assumptions to spell out its additional contribution to explanatory

goodness. (Strevens, 2008, p. 304)

This sentiment is what makes the kairetic account so appealing to those interested in

mathematical explanation in the first place. We can allegedly use the account independently of

our metaphysical beliefs to analyze and understand how mathematics contributes to scientific

explanations, if at all.

Identifying the exact conditions for ‘making it so’ is a difficult task. The goal is to have

something not so weak as to imply that the bulk of our mathematical explanations are actually

GMEs; the condition needs to be strong enough to rule out explanations that we do not consider

truly genuine. At the same time, the conditions for ‘making it so’ cannot be so strong as to

trivially rule against the existence of GMEs. I propose that we borrow the criteria from the

indexing argument to help us set a condition for ‘making it so’. This makes sense in the present

context as it is the nominalist who we are trying to convince. If we pick a condition that is too

weak by their standards then they will have reason to reject our conclusions. Consider then:

2M. K entails E such that:

(a) there is no purely nominalistic counterpart that also entails

E,

and,

(b) there is no purely physical object or property that the

mathematics is representing.

2M is perfectly in line with the nominalists’ concerns raised within the indexing argument. The

electron spin example was shown to resist the indexing argument in chapter 3. If we tailor 2M in

the above fashion, then we can corroborate the claims from chapter 3 and draw an even stronger

conclusion: not only does the electron spin example resist the indexing argument, but by the

nominalists’ own standards, we can show that the mathematics makes the physical explanandum

so. I am not proposing that 2M is the correct description of how mathematics makes it so in a

GME. In fact, I feel that 2M is actually too strong, but this is exactly what we need to convince

the honest nominalist that GMEs exist. Once it can be established that GMEs exist, then we can

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revisit the issue of trying to pin down more reasonable criteria for how mathematics makes it so

when explaining physical phenomena without having to worry about satisfying the nominalists’

demands.

Now we must define what cohesion means for mathematical explanations. Unfortunately,

making sense of cohesion in noncausal explanations is much more problematic than

understanding the ‘making it so’ type of entailment. The problem of cohesion is strictly a logical

problem. Recall that the cohesion desideratum is meant to block the move of making more and

more abstract models via disjunctions which obfuscate the true difference-makers. Strevens’

solution was to define cohesion via causal contiguity. Clearly this will not work when our

explanations are not purely causal in the first place. A noncausal explanation is not without its

own relevant causal factors. In the electron spin example, the causal factors include the magnetic

field, the detectors, etc. So for the causal components of our explanation we can certainly make

use of causal contiguity as a requirement for cohesion. Unfortunately, this does not save us from

disjoining mathematical sets together that obscure the truly relevant mathematical factors in our

explanation.

In addition to the original presentation of cohesion, I present a new form of the problem

particular to mathematical explanation that has nothing to do with disjoining sets of statements.

Suppose we have a mathematical theorem, T, that our mathematical explanation, M, of some

explanandum, E, seems to depend on. Ideally we would like to conclude that T is the difference-

maker for E. In generating a kernel via optimization we arrive at the following difficulty. M

would necessarily have to include T and also the proof of T from a set of mathematical axioms,

A. Now let us make a more general model, N, from M by removing T and its proof and replacing

them with just the collection of axioms, A. Obviously T is deductively entailed by A, so our new

model N still entails E just as well as our original model. N is more general than M as A implies

many other theorems that T alone may not. We thus arrive at the conclusion that it is the set of

axioms A which are the difference-makers for E, and not T in particular. There appears to be

something wrong with this conclusion, and following Strevens, I claim that N lacks cohesion.

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To summarize, there are two forms of the cohesion problem that present themselves in

mathematical explanations. The first is in creating more general sets via disjunction, and the

second is making a more general set by only including the axioms of particular mathematical

disciplines instead of the specific mathematical theorems in play. Both forms of the problem

result in obscuring what we want to conclude as the true difference-makers in mathematical

explanations. I propose the following definition for cohesion in a mathematical explanation that

will block both of the above problems.

4M. K is cohesive such that,

(a) The fundamental-level realizers of the causal factors in K

form a causally contiguous set,

(b) The entailment of E from K requires no additional

mathematical derivation,

and,

(c) The mathematical statements in K form a maximally small

set.

(a) takes care of the causal factors in the same manner as Strevens. (b) successfully blocks the

new form of the cohesion problem. As above, consider a mathematical theorem, T, that our

mathematical explanation, M, of some explanandum, E, seemingly depends on. In showing that

M entails E, we would not have to do any further mathematical deductions as E depends only on

the mathematical result T which is already in our model. Let N be the more general model of M

that removes T and its proof, and replaces them with the axioms that imply T. In order to show

that N entails E, we would necessarily have to deduce T, but this time the deduction would be

done outside of our kernel. Condition (b) stops us from doing this. T and its proof must remain

within any kernel, and thus we block this form of the cohesion problem.

Finally, condition (c) together with (b) blocks the problem of making mathematical sets via

disjunctions that are more general. Again, let M be the model that is a mathematical explanation

of E. M includes mathematical theorem T and its proof, and thus M satisfies the cohesion

requirement (b). We have two options for generating a new model, D, using disjunction. First,

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we could disjoin T with some other mathematical statement. Call this resultant disjunction S. D

in this case is more general, but we have the unappealing conclusion that S is a difference-maker

for E. Now consider the sets created by taking only the mathematical statements in M and only

the mathematical statements in D. The latter set generated from D has at least one more element

in it than the set generated from M; in particular it includes the additional statement S. So N

violates the condition (c) and is not cohesive.

The other option is quite different. Now let D be a new model that includes any set of axioms, a

true mathematical statement, and its proof. For example, D could contain the axioms of

arithmetic and the statement ‘2+2=4’ along with a simple derivation. In order for D to entail E,

all we need to do is make a disjunction of this true mathematical statement with T. Call this

disjunction S. Now we can see that D entails E as S entails E, and N is more general than M. The

danger with this type of disjunction is that it is possible for the set of mathematical statements in

D to have a smaller cardinality than the set of mathematical statements in M. If this is so then

condition (c) is satisfied. However, what will result from this type of disjunctive explanation is

that condition (b) will be violated. While it is certainly true that S entails E, in order to show that

this is the case we need to derive T as a singular statement. But this mathematical derivation

would be performed outside of our kernel, which is not allowed by the cohesion requirement.

Thus, (b) and (c) together block this second type of disjunctive explanation.

The above discussion on cohesion is technical. All it boils down to are some restrictions on how

we should generalize the mathematics when creating a kernel such that we do not bury the true

difference-makers in strange statements generated by disjunction or by over-generalizing. This

problem will not present itself in my application of the kairetic criterion below. The main

takeaway from all this is simply to motivate the idea that the kairetic criterion can be

appropriated for analyzing noncausal mathematical explanations.

Putting everything together, we now have a kairetic criterion that can operate on mathematical

explanations. Starting with a veridical model, M, that is a mathematical explanation for an

explanandum, E, the corresponding explanatory kernel, K, for E is obtained by the following

optimization procedure:

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1. K is an abstraction of M,

2M. K entails E such that:

(a) there is no purely nominalistic counterpart that also entails

E,

and,

(b) there is no purely physical object or property that the

mathematics is representing,

and that, within these constraints, maximizes the following desiderata:

3. K is as abstract as can be (generality), and

4M. K is cohesive such that,

(a) The fundamental-level realizers of the causal factors in K

form a causally contiguous set,

(b) The entailment of E from K requires no additional

mathematical derivation,

and,

(c) The mathematical statements in K form a maximally small

set.

The kairetic criterion states that all statements found in the explanatory kernel obtained via the

optimization procedure are difference-makers for E. Thus, if mathematical statements appear in

the kernel, then we can conclude that mathematics is a difference-maker for a physical

phenomenon, and hence GMEs exist.

3 Applying the Kairetic Criterion

With a working kairetic criterion in hand, we can now apply it to see if the mathematics in our

many mathematical explanations counts as difference-makers or not. Recall Baker’s

mathematical explanation of the prime numbered life-cycle of the cicada.

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(1) Having a life-cycle period which minimizes intersection with

other (nearby/lower) periods is evolutionarily advantageous.

(biological law)

(2) Prime periods minimize intersection (compared to non-prime

periods). (number theoretic theorem)

(3) Hence organisms with periodic life cycles are likely to evolve

periods that are prime. (‘mixed’ biological/mathematical law)

(4) Cicadas in ecosystem-type E are limited by biological

constraints to periods from 14 to 18 years. (ecological

constraint)

(5) Hence cicadas in ecosystem-type E are likely to evolve 17-year

periods.

This explanation is incomplete as missing from it are the justifications for statements (1), (2),

and (4). The justification for (1) would necessarily include the fundamental laws or assumptions

in evolutionary biology. It needs to be shown that (1) follows from these laws. (2) requires a

mathematical proof in number theory to demonstrate its status as a mathematical theorem.

Statement (4) depends on many areas such as biology, ecology, weather patterns, and so on. A

complete explanation would include all these extra parts.

Let such a complete explanation be a model, M1, which entails the explanandum, E. Attempt to

generate a kernel, K1, from M1 via the optimization procedure for mathematical explanations.

The key to the optimization procedure is to abstract away or remove all explanatorily irrelevant

details while still ensuring that our model logically entails E. What sorts of things would be

abstracted away? It could be that our justifications of (1) and (4) contain superfluous details that

we could remove without violating the entailment. We may abstract away some of the biological

laws and replace them with laws of physics, if such a reduction is possible. Although this is

somewhat vague, what should be clear is that we cannot abstract away any of (1) through (5).

Imagine trying to remove (1) from the explanation. Without (1) there is no reason as to why

minimizing intersection is desirable, and thus the fact that cicada have a prime life-cycle is no

longer entailed by our model. The same problem obtains if we try to remove (2), the

mathematical theorem, from our model. This theorem is essential for entailing the conclusion

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that cicadas should have prime periods. Without (2) we would be unable to entail the

explanandum, and hence (2) cannot be removed from our model. K1, must include all of

statements (1) through (5), as well as the justifications and proofs of statements (1), (2), and (4)

that have been made abstract and general.

Now consider Saatsi’s explanation where we use sticks to explain why cicadas have a 17 year

life-cycle. Many features of this explanation would be the same as in Baker’s mathematical

explanation, such as statements (1) and (4) along with their respective justifications. What is

different is that there is no mention of primeness, and no mathematical theorem. Instead we have

facts about sticks and units. Call this second explanation that has no mathematical theorem the

model, M2, which entails E. Attempt to generate a kernel, K2, by the same process as above. An

interesting part of the process occurs when we run the abstraction procedure on the statements

that involve sticks. Certainly the existence of sticks is not necessary to the overall explanation.

All that is actually needed to perform Saatsi’s nominalistic ‘proof’ are many physical objects that

all have roughly the same length. This shows us that sticks themselves are not difference-makers

for cicada life-cycles, which is of course what we expect. However, what cannot be removed is

the ultimate conclusion of the stick ‘proof’ that 13 and 17 roughly equal length units minimize

intersection with other surrounding equal length units. Without this conclusion we would not be

able to entail the prime life-cycle of the cicada.

Finally, recall the third explanation of the prime-life-cycle of the cicada which does not invoke

mathematical theorems or sticks. Instead, this explanation invokes the physical fact that 13 and

17 units of time minimize intersection, and this is the critical explanatory fact. Call this

explanation the model M3, and attempt to generate a kernel, K3, in the same way as above. It

should be clear that the physical facts of units of time cannot be removed from our explanation

as without it we cannot entail the explanandum.

K1, K2, and K3 all satisfy criterion 1 of the optimization procedure – they are abstractions of their

respective starting models. Although I have not detailed it here, I will simply assume that K1, K2,

and K3 were generated in a way to maximize the desiderata of generality and cohesion. This

assumption is warranted as we are not looking to generate bizarre kernels that involve logical

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tricks, such as disjunction. The only thing that needs to be checked is whether or not K1 satisfies

the making-it-so criterion, 2M. If it does, then K1 is a true kernel for E, and thus all statements in

K1, including the mathematical theorem, are difference-makers. However, it is easy to see that K1

fails to satisfy criterion 2M. The existence of K2, which is a purely nominalistically acceptable

explanation of E, means that our mathematical model, K1, fails to meet 2M(a). K1 also fails to

meet 2M(b); the mathematics can be argued to be simply representing physical units of time

which is evident by the existence of K3. Hence, K1 is not a kernel for E, and the kairetic criterion

does not point to any mathematical difference-makers for the explanation of the cicada’s prime-

life-cycle.

The exact same conclusions can be drawn when analyzing the explanation of the hexagonal

nature of honeycombs. In the honeycomb explanation, the mathematical explanation will fail to

satisfy criteria 2M(a) as there exists a purely nominalistic explanation using a Field type

nominalization of Newtonian space-time. It also fails to meet 2M(b) as the nominalist can claim

that the mathematical honeycomb theorem is representing the physical fact that approximately

Euclidean space is most efficiently tiled by hexagons. We could construct an explanation that

utilizes this physical fact instead of the mathematical theorem that would still entail the

explanandum. Thus the mathematical theorem will not appear in a kernel for the explanandum,

and is not a difference-maker.

In the case of the Kirkwood gaps example, this explanation makes use of an eigenvalue analysis

to explain the existence of gaps in the asteroid belt. In this case, there is no nominalistic

counterpart explanation to the mathematics like there was for the cicada or honeycomb example,

and thus the Kirkwood gaps examples satisfies criterion 2M(a). Yet again, we are unable to

escape the nominalist belief that there is a physical fact that the mathematics is representing. As

discussed in chapter 3, the nominalist could claim that it is in virtue of physical facts such as

gravitation, mass of neighboring planets, space-time, etc., that actually explains the existence of

the Kirkwood gaps. The mathematics is representing these physical facts, and thus we do not

satisfy 2M(b). This blocks the mathematical explanation from being a kernel for the

explanandum, and like the cicada and honeycomb example, the mathematics is not a difference-

maker.

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These negative conclusions should come as no surprise. The making-it-so criterion, 2M, was

designed to reflect the nominalist intuitions raised in the indexing argument which were used

against the cicada, honeycomb, and Kirkwood gaps examples from being a GME. The fact that

the kairetic account corroborates this intuition should add to the credibility of the account in the

eyes of the nominalist. I want to restate that I am not endorsing the results of this section, but

rather I am trying to show that the kairetic criterion motivated by the indexing argument gives us

an account of mathematical explanation that agrees with the way nominalists such as Melia see

the world. Whether or not this is actually the right way to construct the kairetic criterion is not

being considered here. What has been shown is that when the kairetic account is constructed in

this nominalistic friendly way, we find that the mathematics featured in all these explanations are

not difference-makers for their respective physical explananda. This means that they are not

conferring the explanatory power in the explanation, and it follows that these examples are not

GMEs. Even better for the nominalist is that the kairetic account highlights the actual role and

contribution that mathematics provides in these explanations. Mathematics helps represent or

deduce the relevant physical facts, and it is these physical facts that are found to be the true

difference-makers. Up till now, the conclusions of the kairetic account as I have adopted it match

the intuitions of the nominalist in every way.

Now we will analyze the example of electron spin. Electron spin explains the splitting of a beam

of electrons that pass through a Stern-Gerlach apparatus where it splits into two equidistant and

distinct beams. What explains the splitting are the Pauli spin matrices – a mathematical entity.

Let MS be the model that entails the explanandum, E. MS will have to include lots of other

information in addition to what has been mentioned here, such as the theory of quantum

mechanics, magnetism, detectors, etc. Now attempt to generate a kernel, KS, from MS following

the kairetic criterion for mathematical explanations. While lots of information may be removed

during the abstraction process – perhaps we can reduce everything to quantum mechanics, or we

can generalize other facts such as the specific size of the magnets, the exact distance the beam

travels, etc. – one thing that cannot be removed are the Pauli spin matrices. If we try to remove

them from our kernel there would be no possible way for our model to entail the splitting into

two equidistant and distinct beams. Put simply, the Pauli spin matrices are exactly what allows

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our explanation to entail the explanandum. Although I have not precisely said what else is in KS,

I can confidently assert that the Pauli spin matrices will be a member. KS satisfies the

abstractness criterion, and again we assume that that nothing strange has been performed that

would upset the maximizing of abstractness and cohesion. It should also be clear from our

analysis in chapter 3 that KS satisfies 2M(a); there is no nominalistic counterpart to the Pauli spin

matrices that can explain the splitting of the beam.

I take the above analysis of the electron spin example to be perfectly amenable to the nominalist.

All that remains to be answered is: does KS also satisfy 2M(b)? This is where all the other

mathematical explanations critically failed. In each of those cases the nominalist was able to

assert that the mathematics was actually representing some real physical objects or properties

which were the genuine explanatory factors. With spin, however, we have a different story. As

shown in 3.4, there is no purely physical object, property, fact, or anything at all that the Pauli

spin matrices are representing. The nominalist cannot identify anything beyond the mathematics

that constitutes spin. There is no classical counterpart to electron spin that can even begin to

inform some physical understanding of the concept. All that we have is the mathematics, and

nothing more. It is clear that the spin explanation satisfies 2M(b), and hence 2M as well. This

means that KS satisfies all the criteria for being a kernel for E. The kairetic account states that

everything in a kernel is a difference-maker. Thus, the Pauli spin matrices, which are

mathematical objects, are difference-makers for a physical fact.

We have successfully shown that the electron spin example satisfies criterion (E) for GMEs: an

acceptable account of scientific explanation corroborates the claim that the mathematics is

explanatory. This result, together with those from chapter 3, means that the electron spin

example satisfies all five criteria for a GME, and therefore the mathematical explanation of the

splitting of an electron beam into two distinct and equidistant beams when passing through a

Stern-Gerlach apparatus is a genuine mathematical explanation.

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4 Difference-Making Revisited

Up till now we have been exclusively utilizing the standards of the nominalist in order to

examine mathematical explanation. This was done to meet the nominalist on their own ground,

and to show them that according to their very own standards there does exist a GME. Now that

we have demonstrated this, we can free ourselves from this restriction of satisfying the

nominalist and take a closer look at what it means for mathematics to genuinely explain physical

phenomena. We can now attempt to answer the question: how does mathematics explain? What

does it mean for mathematics to be a difference-maker for a physical phenomenon? Borrowing

Strevens’ phrase, we can now analyze how mathematics makes-it-so. I will consider two

approaches to answering these questions here: first by looking at the mathematics internally, and

second will be by closely examining what mathematics brings to our GMEs.

4.1 Internal Mathematical Explanations

Strevens suggested that one way in which mathematics could make a difference in a physical

explanation is through our understanding of the mathematics itself. Strevens is appealing to a

mathematical explanation of mathematical facts, or internal explanations. The idea is that there

is a patent difference between merely demonstrating a mathematical fact and explaining why that

fact holds. Internal explanations can reveal insight into mathematical theorems that otherwise

would have been lacking. Mark Steiner (1978a, 1978b) was one of the first to advance an

account of internal explanations. He claimed that internal explanations had to be in the form of a

mathematical proof that relies on the concepts of ‘characterizing properties’ and ‘deformation’.

A characterizing property is a property unique to a given mathematical object or structure that is

relative to a ‘family’ or domain of similar objects or structures. For example, the number 18 can

be characterized by it being the successor of 17. This characterization is relative to the family of

natural numbers defined with the successor relation. Or, 18 can be characterized by its prime

power expansion, 2 x 32, which is relative to the family of prime factorization of the natural

numbers. Deformation is to change the object or structure in question while holding the

characterizing property steady. Deforming 18 to 15 would change the characterization to be the

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successor of 14 or having the prime power expansion of 3 x 5. For Steiner, an explanatory proof

of a mathematical theorem is one such that the characterizing property is mentioned in the proof,

and that it is evident that the resulting theorem depends on the characterizing property which is

made obvious through deformation.

Steiner advanced an incredibly strong connection between an internal explanatory proof and a

GME. We only have a GME if we have an internal explanatory proof of the key mathematical

theorems being employed. As Steiner puts it, “when we remove the physics, we remain with a

mathematical explanation – of a mathematical truth!... In standard scientific explanations, after

deleting the physics nothing remains.” (Steiner, 1978b, p. 19) Call Steiner’s view the strong

relationship.

What Strevens suggests for how mathematics makes a difference is not nearly as strong as

Steiner. Strevens talks of how “understanding the mathematics as well as the physics of the

scientific treatment of the explanandum... you come to understand the phenomena better.”

(Strevens, 2008, p. 303) In a similar vein, Mancosu speculates that “it is conceivable that

whatever account we will end up giving of mathematical explanations of scientific phenomena, it

won’t be completely independent of mathematical explanation of mathematical facts.”

(Mancosu, 2008, pp. 140–141) The suggestions from Strevens and Mancosu have two major

differences from Steiner’s take. First, Strevens and Mancosu allow for any variety of internal

explanations rather than strictly an explanatory proof. There is good reason to believe that not all

internal explanations need to be mathematical proofs. Diagrams, pictures, or even

axiomatizations could all be explanatory in their own way. Secondly, Strevens and Mancosu do

not claim that for a mathematical explanation of a physical fact to be genuine it must have an

internal explanation within it. Rather, they simply assert that having an internal explanation may

increase the explanatory understanding gained from the external explanation. Having an internal

explanation is neither necessary nor sufficient for how mathematics makes-it-so in an external

explanation, but rather it is just one possible way in which mathematics can make-it-so. Call this

the weak relationship between internal and external explanation.

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There are several reasons why we should be skeptical of Steiner’s strong relationship. A prima

facie problem with this suggestion is that most proofs in mathematics are not explanatory by

Steiner’s own criteria. This results in an incredibly high standard when we are trying to use

mathematics to genuinely explain physical facts as we are limited to the small amount of

theorems that already possess an internal explanation. Even worse is that many GMEs could

depend on several mathematical theorems, all of which would need its own internal proof. This

criticism suggests that the strong relationship sets too high of a standard for GMEs, but there are

further reasons to believe that there is a more fundamental error in linking the existence of a

GME to an internal explanation in the first place. Steiner’s concept of what makes an external

explanation genuine seems to fly in the face of what supporters of GMEs actually believe. All of

the examples that we have looked at pay no heed as to whether or not the mathematical theorems

utilized possess internal explanations. The beliefs of Baker, Colyvan, and myself that our

respective examples are genuine have nothing to with the nature of the proofs behind the

mathematics. At no point did we invoke an internal mathematical explanation as evidence of the

explanatory force of the mathematics. Moreover, nominalists who reject examples such as the

cicada and honeycomb explanations do not do so due to their lack of an internal explanation.

Neither realist nor nominalist seems to attribute any importance to the existence of an

explanatory proof whatsoever. The reason why is because what is truly playing the explanatory

role in a GME is solely the mathematical theorem, and not its proof. By any account of scientific

explanation we would still have the exact same explanatory understanding of the explanandum

regardless of the style of mathematical proof employed. We would have the exact same amount

of predictive power, unificatory power, the same mathematical difference-makers, the same

entailment, and the same answers to why-questions. Steiner’s claim that a GME depends on the

existence of an internal explanation is unsupported and does not reflect how mathematics makes-

it-so in GMEs.

A final criticism of the strong relationship is that it relies on some objective conception of

internal explanation. Michael Resnik and David Kushner (1987) offer a detailed critique of

Steiner’s conception of internal explanatory proofs. They argue that the concepts of

characterizing property and deformation which are necessary for an internal explanation are not

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well-defined. Even worse, the very examples that Steiner uses as models of a good internal

explanation do not seem to satisfy his own criteria. This spells trouble for the strong relationship.

Resnik and Kushner propose their own account of internal explanation that is heavily influenced

by van Frassen’s pragmatic account of scientific explanation, and is drastically different from

Steiner’s. The context of why-questions determines the contrast class and relevance relation, and

thus what counts as an internal explanation is fundamentally relative. Consider the intermediate

value theorem which states that if a real valued function, 𝑓, is continuous on the closed real

interval [𝑎, 𝑏], and if 𝑓(𝑎) < 𝑐 < 𝑓(𝑏), then there is an 𝑥 in [𝑎, 𝑏] such that 𝑓(𝑥) = 𝑐. Someone

could ask, “why is the intermediate value theorem true?” To answer this question, any proof

would suffice. However, in a different context the asker could pose “why is it true for a real

valued function?”, or “why is it true for a continuous function?”, or “why is it true on a closed

interval?” The answers to these questions, and thus the explanations, would be entirely different.

It could be that some answers will be proofs of other facts about things like continuous functions

or the real number line, or it could be that the answer will simply be a counter-example in some

open interval. However, a generic proof of the intermediate value theorem which was

explanatory in the first place would not be explanatory for these more specific questions. What

counts as explanatory depends on the context of the question.

Resnik and Kushner conclude that there is no such thing as an internal mathematical explanation

simpliciter, but we do have explanations relative to the context of the why-question. If this is the

case, then certainly Steiner’s criteria for explanatory proofs are flawed as they do not take into

account the context of the question. Rather, Steiner searches for objective criteria within the

proof that makes it explanatory in its own right. But how does this relative theory mesh with the

intuitive feeling that some proofs are more explanatory than others? Resnik and Kushner write:

We have this intuition, we submit, because we have observed that many

proofs are perfectly satisfactory as proofs but present so little information

concerning the underlying structure treated by the theorem that they leave

many of our why-questions unanswered. In reflecting on this, we tend to

conflate these unanswered why-questions under the one form of words

‘why is this true?’ and thus derive the mistaken idea that there is an

objective distinction between explanatory and non-explanatory proofs. (M.

D. Resnik & Kushner, 1987, p. 154)

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Some proofs simply have the ability to answer more why-questions than others. This does not,

however, make one proof ‘explanatory’ and the other ‘non-explanatory’, as given the proper

why-question it could be that both turn out explanatory, both turn out non-explanatory, or one is

explanatory and the other is not. It all depends on the context of the question.

If we adopt Resnik and Kushner’s account of internal explanation, then the strong relationship

has no content. If we are scientific realists, then we hold an objective notion of scientific

explanation. How could it be, then, that a subjective explanation of the mathematics is somehow

a difference-maker within an objective explanation of a physical fact? To claim such a

relationship is inconsistent. It is certainly the case that if we are like van Frassen and do not

believe in objective scientific explanations, then the strong relationship is trivially true. Either

way, the strong relationship is problematic. On the one hand it is inconsistent, and on the other it

is trivial. Although I am sympathetic to Resnik and Kushner’s approach, I leave it open as to

which conception of internal explanation one should adopt. However, no matter your views on

internal explanations, maintaining that the internal mathematical explanation is a difference-

maker for explanations of physical facts is problematic.

The weak relationship shows more promise, but unfortunately in a trivial way. Advocates of the

weak relationship say that any sort of explanation of the mathematics, not necessarily a Steiner

style explanatory proof, may reveal some insight into the physical explanandum. No sort of

motivation or argument for this relationship is provided beyond the mere suggestion of it. My

issue with this position is that while Steiner’s claim was too strong, this claim is so weak that it is

certainly trivially true. Of course it is possible that an internal explanation would provide us with

some sort of added insight into a physical explanation. This is because supporters of the weak

relationship have not specified what an internal explanation actually is. Steiner was bold enough

to advance a comprehensive account of internal explanations in the form of his deformable

proofs. But for the weak relationship, it seems that anything goes. In this light, the mere

possibility that we gain insight into the physical world from some sort of unspecified type of

internal explanation is enough to make the weak relationship true. Yet this is such a feeble

conclusion that philosophically speaking it is not even worth maintaining. The weak relationship

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is nothing more than idle speculation that can neither be proved nor disproved. It contributes

nothing towards revealing how mathematics makes-it-so in external explanations.

The suggestion that having internal explanations of mathematics is in some way relevant to our

mathematical explanations of the physical world is either misguided or trivially true. If we are to

better understand how mathematics is a difference-maker in our GMEs we have to look

elsewhere. Focusing solely on the mathematics is not the correct approach. Instead we should

focus on what roles the mathematics plays within the explanation itself.

4.2 The Roles of Mathematics in Genuine Mathematical

Explanations

The way we cached out making-it-so above was through a comparative analysis. Criteria 2M of

the kairetic criterion states that mathematics entails a physical explanandum, E, if,

(a) there is no purely nominalistic counterpart that also entails E,

and,

(b) there is no purely physical object or property that the mathematics is

representing.

This idea is that if we cannot cite any purely nominalistic way that makes a physical

explanandum so, then it is reasonable that the mathematics is making-it-so. Although this

satisfies the nominalist and does accurately reflect how they view GMEs, it does very little in

telling us how mathematics actually makes a difference to the physical explanandum. All we can

say from the above nominalist criteria is that mathematical difference-makers make a difference

to physical facts only because we cannot find any non-mathematical difference-makers that can

explain the explanandum. This is not a satisfactory way of understanding how mathematics

makes a difference. We will briefly consider three ways in which we can improve on our

understanding of difference-making in GMEs that focus on what mathematics contributes in

genuine explanations.

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In a recent paper, Marc Lange (2012) correctly recognizes that most philosophers engaged in the

debate surrounding mathematical explanation do not focus on what makes an explanation

genuine, or as he calls it, what makes an explanation ‘distinctively mathematical’. For Lange, in

addition to being non-causal,

[genuine] mathematical explanations in science work by appealing to facts

(including, but not always limited to, mathematical facts) that are modally

stronger than ordinary causal laws—together with contingent conditions

that are contextually understood to be constitutive of the arrangement or

task at issue in the why question. (Lange, 2012, p. 7)

The two key aspects of a GME are that first the mathematics entails the conclusion in a much

stronger way than just the physical facts. The relationship between the explananda and the

explanandum “holds not by virtue of an ordinary contingent law of nature, but typically by

mathematical necessity.” (Lange, 2012, pp. 12–13) The second aspect of a GME is contextual.

The necessary entailment that mathematics brings is not enough to distinguish a regular non-

causal mathematical explanation from a genuine one. What is needed as well is an emphasis on

the role of the mathematical fact(s) being utilized. “[I]t is a matter of degree and of context.

Insofar as mathematical facts alone are emphasized as doing the explaining, the explanation is

properly characterized as distinctively mathematical.” (Lange, 2012, p. 23) Due to this

contextual nature, Lange argues that depending on how you frame explanations such as the

cicada and honeycomb examples, you could in one case conclude that these explanations are not

genuine, and in another case conclude that they in fact are.

Although I feel that Lange is making steps in the right direction in his approach, I find his

dependence on contextual factors largely unsatisfactory. Lange is clearly influenced by van

Frassen’s pragmatic account of scientific explanation. As we saw above, a potential issue with

van Frassen’s approach is that it somewhat trivially implies that GMEs exist as they are certainly

legitimate answers to why-question. This was unappealing as it did not do enough to identify

what role the mathematics was playing, and also because van Frassen’s pragmatic account does

not mesh nicely with scientific realism. Lange supplements the pragmatic approach by adding

the observation that GMEs are modally stronger than regular scientific explanations. My

criticism here is that this observation does not appear to actually add anything of substance at all

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over and above van Frassen’s original approach. Mathematical facts are always going to be

modally stronger than physical facts as on all standard accounts, mathematical facts are true in

every possible world. Once we realize this, Lange’s position essentially collapses into van

Frassen’s pragmatic account where only the contextual factors determine if a mathematical

explanation is genuine or not, as all mathematical explanations will necessarily trivially satisfy

the modally stronger requirement.

Lange’s analysis of the cicada example makes his reliance on contextual factors clear. Baker

phrases his explanandum as “that cicada life-cycle periods are prime” (Alan Baker, 2009, p. 624)

rather than composite in number of years. Baker’s explanation of this involves the idea that

prime periods have been ‘selected for’ for some evolutionarily advantageous reason. Lange, in

line with Sober (1984), argues that explanations that involve ‘selection for’ are classic examples

of causal explanations.

This explanation is also just an ordinary causal explanation. It uses a bit of

mathematics in describing the explanandum’s causal history, but it derives

its explanatory power in the same way as any other selectionist

explanation. Taken as a whole, then, it is not a distinctively mathematical

explanation. (Lange, 2012, p. 15)

Even though there is mathematics in the explanation, and that the mathematics is important and

perhaps even indispensable to the explanation, it is not, according to Lange, a GME. However, if

we rephrase the explanandum we obtain a different result.

But suppose we narrow the explanandum to the fact that in connection with

predators having periodic life-cycles, cicadas with prime periods tend to

suffer less from predation than cicadas with composite periods do. This

fact has a distinctively mathematical explanation. (Lange, 2012, p. 15)

Somehow, this rephrased explanandum is placing the emphasis squarely on the shoulders of the

mathematics, and not some causal selection process.

In a footnote meant to clarify this claim, Lange mentions that “[p]resumably, we would be

prompted to ask for an explanation of this fact only as a result of having used this fact to help

explain why cicada life-cycle periods are prime, an explanation that (I have just suggested) is

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causal.” (Lange, 2012, p. 15) Herein lies the problem with his position. Lange’s rephrasing is

simply not a rephrasing at all. The narrow explanandum is specifically looking to explain why a

prime period would minimize intersection. True, it is couched in language about predators and

cicadas, but ultimately this is asking to explain something about prime periods which is clearly a

mathematical fact. Notice that this is a remarkably different why-question than asking why

cicadas have a prime-life-cycle which is a fact about the physical world. Lange has simply

changed the explanandum to yield an internal mathematical explanation.

Even if we accept this as a legitimate move, a perfectly reasonable question is why would we

ever be prompted to ask for an explanation of Lange’s narrow explanandum? Baker’s

explanation already involves a mathematical proof that primes minimize intersection, which

would be the exact same explanation of the narrow explanandum that supposedly emphasizes the

mathematical factors. Typically, changing the context of a why-question also changes the

explanation that we supply. For example, if I ask ‘why is the traffic in Toronto is so bad?’, an

explanation could involve the poor infrastructure of the city. However, if I ask ‘why is the traffic

in Toronto so bad at 5:00?’, this implies a different sort of explanation entirely. The answer

would have to address my contrast-class of 5:00 rather than, say, 3:00 or noon. Such an

explanation would cite factors such as rush hour, when most office jobs end, location of most

jobs compared to living spaces, etc. The basic idea is that different why-questions are different

because they need different explanations to satisfactorily answer them. In Lange’s presentation

of the cicada example, this is not the case. The explanation of his narrow explanandum is just a

subset, specifically the mathematical subset, of the original explanation. How, then, does this add

anything to our understanding, or explain anything that we did not already know from Baker’s

original explanation? It simply does not. If we want to claim that there is a genuine mathematical

explanation somewhere within the cicada example, rephrasing the explanandum does not seem to

reveal it in any significant way.

Irrespective of my criticisms, I do feel that Lange’s observation of stronger necessity in GMEs is

worth pursuing if we free ourselves from van Frassen’s pragmatism. One way that mathematics

could make a difference in GMEs is that it entails the explanandum with more necessity than a

physical explanation, regardless of context. Pincock (2004, 2011) presents an example of a

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mathematical explanation that demonstrates this necessity nicely. His example seeks to explain

why it is impossible to traverse all the bridges in Königsberg and return to your original starting

point while crossing each bridge only once. We can make a mathematical representation of the

system using a non-Eulerian graph. The vertices of the graph represent the islands or land

masses, and the edges represent the bridges. A key property of non-Eulerian graphs is that there

is no path that starts at any single vertex such that you return to that same vertex while traversing

each edge exactly once. This mathematical property explains the impossibility of the physical

path in Königsberg. Bangu (2012) calls these ‘impossibility results’, and argues that this is an

important benefit that mathematics brings to explanations. Impossibility results are, or course,

just the flip side of mathematical truths being necessary. This necessity tells us that all other

possibilities are necessarily impossible.

It is easy to see that if we analyze the bridges of Königsberg example using the kairetic criterion

above, it will fail to meet the requirements of a GME. This is because it is clear that the non-

Eulerian graph is representing or indexing the actual physical layout of the land masses and

bridges, and it is this physical system that has the relevant explanatory property. This reveals that

the nominalist would most likely not find Pincock’s example compelling. Consider, then, what

happens if we change the making-it-so criteria from the kairetic criterion for mathematical

explanation to reflect impossibility results. Replace criterion 2M with:

2MI. The mathematical explanation, K, entails a physical

explanandum, E, if K shows that all other possibilities are

impossible.

Now we can see that the bridges of Königsberg example will meet the kairetic criterion and

should be considered a GME. This is because the way in which mathematics is making a

difference to the physical explanation is by showing that that it is necessarily impossible for a

return path to actually exist. Note that unlike Lange and the cicada example, I have not changed

the question or the explanandum in order to change the status of the Königsberg example from a

standard mathematical explanation to a GME. What happened is that now that we are free from

solely worrying about indexing or representation, we can focus more closely on what

mathematics actually brings to the table in a GME. By identifying some ways in which

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mathematics can make a difference, we can arrive at a clearer consensus on what counts as a

GME or not.

Applying 2MI to our other examples yields interesting results. One feature of the electron spin

explanation is that it not only explains why the electron beam splits into two distinct beams, in

virtue of the Pauli spin matrices we also know that it is impossible to have anything other than

the two resultant and equidistant beams. There are necessarily only two eigenvalues for the z-

direction Pauli spin matrix, and hence it is impossible for there to be any number of beams other

than two. The same explanation also shows why it is impossible for the beams to be anything

other than equidistant from the original plane. The mathematics is a difference-maker as it

demonstrates that all other possibilities are necessarily impossible, and hence by the standard of

impossibility results, the electron spin example is a GME.

The Kirkwood gaps example has a similar breakdown. Recall that an eigenvalue analysis shows

that it is impossible for the asteroids to settle in certain regions, and these regions represent the

observed gaps. Colyvan notes that it is perfectly possible to give a purely physical and causal

explanation for each individual asteroid that explains why that particular asteroid settled where it

did. However, this is not the same as explaining why no asteroid could ever be in the Kirkwood

gaps. The mathematical explanation delivers the impossibility result we are looking for. So, even

though I was critical of the Kirkwood gaps example in the sections above, that was due to its

arguable representational nature. Now that we are no longer concerned about indexing, we see

that the Kirkwood gaps can count as a GME as the mathematics is a difference-maker in the

context of impossibility results.

It is worth pointing out that there is an important difference between the electron spin and

Kirkwood gaps impossibility results, and the bridges of Königsberg impossibility result. In the

Königsberg example, we could in theory arrive at the very same impossibility result without

utilizing a non-Eulerian graph, or any other mathematics. Imagine someone painstakingly

charting out every possible path and attempting to walk each and every one. Eventually, that

person would realize that it is impossible to return to any starting point whilst crossing each

bridge only once, and this impossibility would be an empirical result. In the Kirkwood gaps and

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electron spin example, an empirical impossibility result is unimaginable. It is surely the case that

we would empirically discover that electrons only deflect into two beams, and they are always

equidistant, but this is not the same as knowing that the alternatives are necessarily impossible.

The former is potentially fallible and the latter is not. The situation is the exact same for the

Kirkwood gaps. Whether or not this difference is significant enough to lead us to rule against the

bridges of Königsberg being a GME I will not address here, but is an interesting open problem

that is worth further examination.

Unfortunately, the cicada and honeycomb examples do not fare as well. In the cicada example,

the key mathematical fact is that primes minimize intersection. In the honeycomb example the

mathematical fact is that a hexagon is the most resource-efficient way to tile an area. The theory

of evolution and ecological constraints provide both explanations with the added biological facts

that it is advantageous to minimize life-cycle intersection and to be resource-efficient

respectively. There is nothing wrong with either of these explanations, but note that neither of

them provides any impossibility results. It is not impossible for cicada to have a composite

number life-cycle. If it were the case that they had composite life-cycles, we could easily account

for this by asserting that cicadas are not maximally evolved. Similarly, it is not impossible for

honeycombs to be some other sided polygon. If so, honeybees would not be maximally resource-

efficient. The mathematics in these explanations is not a difference-maker by impossibility result

standards, and hence are not GMEs. What role are the mathematical theorems playing? In both

cases the mathematics plays the role of informing us what the most optimal conditions actually

are, which is why I call them optimization explanations. If we wish to include optimization

explanations as GMEs, then we need to argue that delivering optimal conditions is a difference-

maker for these explanations of physical facts. I will not make the case here, but will suggest that

although this move is possible, I feel that it will not be successful in the end.

Another way that mathematics could be considered a difference-maker is demonstrated in many

examples from Batterman (2002, 2008, 2010). Batterman’s examples are extremely technical and

I will only consider their features in broad strokes. Batterman is interested in explaining

regularities. He notes that “most (though not all) explanations in physics and applied

mathematics are explanations of patterns or regularities... Nevertheless, despite the fact that

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various details are completely distinct, we witness the same pattern. We would like an

explanation of why.” (Batterman, 2010, p. 20) The fact that the individual details are distinct do

not seem to matter to the overall pattern, which Batterman calls a universal. In fact, often these

individual causal details function as “noise” (Batterman, 2002, p. 4) which hinders our

explanatory efforts. Batterman’s solution is to employ ‘nontraditional’ mathematical

idealizations that are ways in which we can “remove details that distract from [our] focus.”

(Batterman, 2008, p. 430) These idealizations take the form of mathematical limits that allow us

to focus on the relevant explanatory details, and to eliminate the particular causal details of our

system that are irrelevant to the universal regularity. In this way, mathematical limits are a key

difference-maker for these types of GMEs. Pincock also suggests that how mathematics allows

us to focus on relevant properties is a key feature of acausal mathematical representations.

Mathematics can represent novel and explanatorily important features of a system that a faithful

causal representation cannot. Mathematics makes a difference as it “not only captures the feature

of interest but also has as part of its content that various aspects of the system are also irrelevant

to this feature.” (Pincock, 2011, p. 54) Pincock widely agrees with Batterman’s analysis except

that where Batterman is focused exclusively on mathematical limit operations, Pincock is open to

using other mathematical tools in representation. Putting Batterman and Pincock’s suggestions

together, mathematics could be considered a difference-maker in its ability to identify novel non-

causal explanatory features of a physical system, and also in identifying which causal features

are irrelevant.

We have briefly considered three ways in which mathematics can be said to make a difference in

a physical explanation. Mathematics can demonstrate impossibility results, it can demonstrate

optimal conditions, or it can identify both novel non-causal features and irrelevant causal

features. In no way do I see these as exhaustive of all the ways of how mathematics can explain

physical phenomena; nor are these ways exclusive of one another. The way forward in

understanding mathematical explanations should be to precisely cache out these and other roles

that mathematics plays in GMEs.

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5 Takeaway

The present state of affairs has been far too fixated on the problem of indexing and

representation as advanced by the nominalist. Sadly, this has halted progress on more interesting

and challenging issues. Instead of focusing on demonstrating that mathematics does not play

certain roles, we should be looking instead to discover and understand what roles it does play.

The true takeaway of chapters 3 and 4 is not that the electron spin example succeeds at being a

GME where all other examples have failed. This is far from the truth, as we saw above that the

status of a supposed GME rises and falls depending on what criteria we select to characterize

mathematical difference-makers. The electron spin example is special only because it was

carefully selected with a singular purpose in mind: to defeat the intuition that mathematics solely

represents such that we can make progress in actually understanding what it means for

mathematics to explain. My hope is to show that this takeaway is amenable to both the realist

and the nominalist. The fear for the nominalist is that by invoking inference to the best

explanation (IBE) they will have to become realists, and it is this fear that stops them from

seriously considering GMEs and all the benefits that come along with them. This fear is only

warranted if we assume first that IBE can legitimately infer mathematical realism, which I have

explicitly not assumed. With the assumption abandoned, there should be nothing to fear from

analyzing the explanatory contribution of mathematics in science. Clearly though this is nothing

more than a bait and switch if it turns out in the end that IBE is a legitimate inference that should

lead us to be realists. If this is possible, then the nominalist is justified in their aversion to GMEs.

In chapter 5 I will turn to IBE and show that, contra the enhanced indispensability argument, the

nominalist has nothing to fear.

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Chapter 5 Inference to the Best Mathematical Explanation

We have demonstrated that a genuine mathematical explanation (GME) exists. The question now

is whether or not we ought to be mathematical realists. Indispensability arguments are meant to

convince the scientific realists that the very same reasons which lead to their scientific realism

ought to lead them to mathematical realism. This line of reasoning requires two things to be

established. The first is that mathematical entities play the same role in our best scientific

theories as unobservable physical entities, namely that mathematical entities genuinely explain

physical facts just as unobservable physical entities do. The second is that indispensability

arguments must explain the method of inference that scientific realists utilize when they assert

that unobservable objects exist.

Recall Colyvan’s version of the Quinean indispensability argument (QIA):

(P1) We ought to have ontological commitment to only those entities

that are indispensable to our best scientific theories.

(P2) We ought to have ontological commitment to all those entities

that are indispensable to our best scientific theories.

(P3) Mathematical entities are indispensable to our best scientific

theories.

Therefore:

(C) We ought to have ontological commitment to mathematical

entities.

The method of inference in the QIA is the Quinean thesis of confirmational holism which was

found lacking in chapter 1. Arguments boiled down to the claim that actual practicing scientists

simply do not confirm in a holistic manner, and thus there is no reason to believe that

confirmational holism is true. In addition, it seems entirely plausible to be a scientific realist

without endorsing confirmational holism. Unless we can put forward a staunch defence of

confirmational holism and its necessity to scientific realism, the QIA cannot deliver its realist

conclusion.

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Baker’s enhanced indispensability argument (EIA) is meant to solve this problem by avoiding

confirmational holism altogether.

(EP1) We ought to rationally believe in the existence of any entity that

plays an indispensable explanatory role in our best scientific

theories.

(EP2) Mathematical objects play an indispensable explanatory role in

science.

Therefore:

(EC) We ought rationally to believe in the existence of mathematical

objects.

(EP1) ties the argument to the inference to the best explanation (IBE), which is widely

considered to be the standard inference of scientific realists. What makes the EIA compelling is

that at first glance, any scientific realist would gladly assent to (EP1). Now that we have

established that (EP2) is true, then the conclusion of mathematical realism follows more directly

than the route taken by the QIA. We need not justify any additional theses in order to extend our

realism over mathematical objects.

My aim in this chapter is to show that even though we, as scientific realists, endorse IBE,

utilizing this inference to infer mathematical realism is unjustified. There are two main

objections that have been raised in the literature that echo this belief. Some have argued that IBE

is in principle unable to infer the existence of mathematical, or other noncausal entities. I will not

address this objection here as it has already been satisfactorily refuted,43 and I will assume that

there are no fundamental limitations of IBE for inferring the existence of mathematical entities.

43 Steiner (1978b) and Bangu (2008) have argued that any use of IBE to infer the existence of mathematical objects

would necessarily beg the question. Baker (2009) has defended his cicada example, but also correctly points out that

many other examples would not be at risk of begging the question. Bangu (2012) has subsequently abandoned this

criticism. Others such as David Armstrong (1978), Brian Ellis (1990), and Hartry Field (1989) have advanced the

Eleatic Principle which states that IBE should be restricted to only those entities that possess causal powers, and thus

would be in principle unable to lead to mathematical realism. Colyvan (2001a), Lipton (2004a) and Psillos (2012)

have convincingly argued that the Eleatic principle is unjustified and patently begs the question against

mathematical realism.

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A more interesting objection comes from Leng who argues that treating mathematical entities as

fictions is actually the best explanation for GMEs, and hence IBE does not lead to the realist

conclusion. I will argue that Leng’s critique of the use of IBE is deficient, and hence the two

most prominent objections against using IBE for mathematical realism are not compelling.

The objection that I will advance is different from the two above in that I accept both that IBE is

applicable in principle to mathematical entities, and that there exist best explanations of physical

facts that indispensably appeal to mathematical entities which fictionalism cannot account for.

The real problem is that (EP1) is not an accurate presentation of IBE. In particular, (EP1) is too

weak, and does not faithfully reflect the discerning ways in which IBE is employed by the

typical scientific realist. When made precise, it is clear that IBE does not yet warrant inferring

the existence of mathematical objects. I will suggest that there is a way for the EIA to avoid this

new challenge, but it comes at quite a cost. The EIA can make use of IBE only if it at the same

time embraces confirmational holism. However, if this is the case, then the EIA collapses to the

QIA and can hardly be said to be an enhanced argument at all.

1 The Inference to the Best Explanation

The basic pattern of IBE is straightforward. If an explanation is considered to be the best

explanation of the phenomenon in question, then we should infer that the explanation is true.

Psillos (1999) advances the following example. You come down to the kitchen one morning and

hear some strange noises in the walls. You see some mouse-droppings on the counter, and the

bits of cheese that were left out the night before have gone missing. One possible explanation for

this is that your roommate is playing a strange prank on you and is hiding in the walls. Another is

that your kitchen is haunted by a cheese-loving ghost. However, it seems to you that the best

explanation is that there is a mouse in the walls. As this is the best explanation, you infer that this

explanation is true and that you have a mouse problem.

Of course we are not interested in cheese stealing mice. IBE is an extremely important inference

for the scientific realist. Belief in unobservable entities, such as positrons or neutrinos, is not

generated in the same way as our belief in things like tables and chairs; we have no direct

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sensory access to unobservables. We must instead infer their existence by other means, and this

means is IBE.

Suppose, now, that a scientist observes that in the standard account of β-

decay, the principle of the conservation of energy is violated. The energy

of the decaying neutron is not commensurate with the energy of the

emerging proton and electron. What needs to be explained here is not

mouse-droppings, but sure enough something needs to be explained.

Pauli’s positing of the neutrino (a particle with no charge and mass, but

with spin) is just another instance of [IBE]. (Psillos, 1999, p. 212)

There may be other potential explanations for the discrepancy in energy, but scientists feel that

the neutrino, if it exists, explains the observations best. They are justified in believing in the

existence of the neutrino in the very same way that we were justified in believing that we have a

mouse in the kitchen walls. In this way, IBE is the primary method which allows the scientific

realist to add unobservable entities to their ontology.

1.1 Problems with Inference to the Best Explanation

Famously, IBE faces two serious problems. The first is the problem of justification. No one

doubts that we use IBE all the time in our regular lives, but is IBE sufficiently justified in order

to license philosophical conclusions? The second problem is that IBE has been notoriously

difficult to precisely define. In his detailed analysis of IBE, Lipton states that the theory of IBE is

“more a slogan than an articulated philosophical theory.” (Lipton, 2004a, p. 2) This problem of

explication is important to our understanding of IBE, but is also critical for addressing the

problem of justification. The literature surrounding these two issues is vast, but the details are

not much of a concern to us here. Our task is to examine and understand the ways in which

typical scientific realists actually use IBE, which implicitly assumes that the inference is both

justified and reasonably well-defined. Given this assumption, I will only briefly examine

particularly important issues in the problems of justification and explication.

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Although van Fraassen presents several challenges to IBE, one of the most influential is known

as the “best of a bad lot” (van Fraassen, 1989, p. 143) objection.44 IBE allows us to move from

an explanation being the best, to that explanation being true. Van Fraassen notes that this

inference critically requires us to assume that the true explanation is actually within the set of

explanations that we are considering. He argues that this assumption is entirely unjustified.

Without this assumption, our best explanation could simply be the best of a bad lot of

explanations where none of them are true. Hence, IBE cannot reliably lead us to the truth. Van

Fraassen anticipates two potential responses by the committed realist. First, they could argue that

we are in the position to assert that the true explanation is within our set of possible

explanations. The reason we can do this is due to a belief in the “privilege of our genius.” (van

Fraassen, 1989, p. 143) A second potential defense is to soften IBE such that the best explanation

is not true simpliciter. Instead, IBE infers something weaker than truth, such as increasing our

personal probabilities, or potential truth.

Van Fraassen considers the claim that scientific realists have a privileged position that allows

them to claim that the true explanation lies within the available lot to be somewhat absurd. This

notion of privilege seems to be at odds with naturalism, rationalism, and empiricism. However,

Psillos (1996) takes exactly this route when defending against the best of a bad lot argument.

One should observe that the argument from the bad lot works only on the

following assumption: scientists have somehow come up with a set of

hypotheses that entail the evidence - their only relevant information being

that these hypotheses just entail the evidence - and then they want to know

which if any of the hypotheses is true… However… it is at least doubtful

and at most absurd to hold that theory-choice operates in such a

knowledge-vacuum. Rather, theory-choice operates within and is guided

by a network of background knowledge. (Psillos, 1996, p. 38)

Psillos calls this the background knowledge privilege. This privilege furnishes the scientific

realists with two important advantages. First, background knowledge can significantly restrict

44 Van Fraassen (1989) also argues that IBE conflicts with Bayesianism and thus users would be susceptible to a

Dutch book. Other attacks on the justification of IBE come from Lauden (1981) and Fine (1984) who argue that any

justification of IBE is circular, and hence illegitimate.

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the set of possible hypotheses that can potentially explain a given phenomenon. Second,

background knowledge also allows us to look towards explanatory considerations to select the

best explanation among our restricted lot. The key for Psillos is that there is something to be said

about potential explanations that are compatible with already confirmed theories. This

compatibility affects not only the types of explanations that we look at, but also how we

determine the best one. The best explanation is selected by appealing to the same explanatory

virtues that are already entrenched within our background scientific theories. Psillos concludes

that both these aspects of the background knowledge privilege do provide scientists with

significant support for the belief that the best explanation is true. Moreover, scientists are entirely

justified in maintaining the privilege of background knowledge, and thus the best of the bad lot

argument can be circumvented.45 This justification is strengthened by the confirmation that our

background theories accrue over time.46

Whereas Psillos bites the bullet and agrees that scientific realists are in a position of privilege,

others such as Musgrave (1988), and Lipton (1993) have modified IBE by adding other

necessary conditions that an explanation must possess in addition to being the best. Lipton states

that “Inference to the Best Explanation might be more accurately if less memorably called

'Inference to the Best Explanation if the Best is Sufficiently Good.” (Lipton, 1993, p. 92) Later

on, Lipton changes his description of the inference.

So the version of Inference to the Best Explanation we should consider is

Inference to the Loveliest Potential Explanation… This version claims that

the explanation that would, if true, provide the deepest understanding is

the explanation that is likeliest to be true. (Lipton, 2004, p. 63)

The basic idea here is that by adding extra conditions beyond just being the best explanation, we

can avoid the best of the bad lot problem as the extra conditions justify our belief that the true

explanation is within the lot we are considering. Maneuvering in this way leads us to the second

45 For a critical response, see Ladyman, Douven, and van Fraassen (1997).

46 Psillos does acknowledge that this does require us to assume that our background knowledge is approximately

true. While he does not defend against a full-blown skepticism, he does argue that even a constructive empiricist

would agree with accepting the important role of background knowledge.

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main problem with IBE: that of explication. One of the first expressions of IBE comes from

Gilbert Harman.

In making [an inference to the best explanation] one infers, from the fact

that a certain hypothesis would explain the evidence, to the truth of that

hypothesis. In general, there will be several hypotheses which might

explain the evidence, so one must be able to reject all such alternative

hypotheses before one is warranted in making the inference. Thus one

infers, from the premise that a given hypothesis would provide a “better”

explanation for the evidence than would any other hypothesis, to the

conclusion that the given hypothesis is true. (Harman, 1965, p. 89)

Psillos presents the following template for the IBE:

D is a collection of data (facts, observations).

H explains D. (H would, if true, explain D).

No other hypothesis can explain D as well as H can.

Therefore, H is (probably) true. (Psillos, 2007, pp. 442–443)

Although all these versions of detailing IBE seem very closely related, there are important

differences. In order to make sense of any of these expressions of IBE we have to have an

understanding of explanation, truth – be it probable or approximate – as well as a means of

identifying which explanation is the best amongst all candidates. For Lipton, we also need to

make precise the concepts of sufficiently good or loveliness, and its relationship to likeliness. All

these complications leads Psillos to admit that revealing ‘the fine structure of IBE’ may be

impossible. Despite this, scientific realists maintain that they have a good enough understanding

of IBE to legitimize its use, and we will adopt this belief here.

By no means is this section meant to be a comprehensive summary of the main objections and

responses regarding IBE. I have purposefully cherry-picked certain defenses of IBE from two of

its most vocal defenders of the inference, Psillos and Lipton. The reason for this is to motivate

the idea that IBE is not some blanket inference that applies to any explanation that is found to be

the best. Even if van Fraassen’s best of the bad lot argument fails at entirely undermining IBE, it

has at the very least forced the scientific realist to make more precise how the inference is

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actually employed. I will make use of this precision below when I argue that we are not justified

in inferring mathematical realism even though our genuine mathematical explanation is the best

explanation that we presently have.

1.2 Types of Inference to the Best Explanation

We need to distinguish between two typical ways in which IBE is employed: meta-IBE and

local-IBE. In both cases we are inferring the truth of the best explanation. The difference is that

in the meta-IBE case, the best explanation is a thesis, broadly conceived as a collection of

beliefs, which is inferred to be true. In the local-IBE case, a particular explanation is inferred to

be true, but ultimately what is required for this explanation to be true is the actual existence of

some entity or another. Local-IBE is in the business of granting ontological rights onto specific

entities, whereas meta-IBE asserts the truth of a particular theory.47

Within local-IBE lies an important distinction. One type of local-IBE is when a potentially

observable object is postulated in the best explanation of our observations. In the early 19th

century, irregularities in the orbit of Uranus had been well documented. Astronomers Urbain

Leverrier and John Couch Adams both independently began working on an explanation for these

observed irregularities. Leverrier and Adams suggested that an up till then undiscovered eighth

planet would explain the orbit of Uranus due to its gravitational pull. Within a year of their

suggestions the existence of Neptune was confirmed.48 This well-known episode in the history of

science is often used as an exemplar of local-IBE. The best explanation of the observed

irregularities of Uranus’ orbit was that an eighth planet in our solar system existed. This

explanation was treated as the true explanation, and so by local-IBE, astronomers believed that

an eight planet existed. This sparked the search for direct empirical confirmation of the new

planet which ultimately proved successful when Neptune was observed in the sky.

47 This distinction is often not so sharp in practice, but this does not concern us here.

48 See Grosser (1962) for a full account of the discovery of Neptune.

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The second important type of local-IBE is when the entity in question is not potentially

observable, or unobservable. This style of local-IBE was demonstrated above by Psillos’ account

of the discovery of the neutrino. The neutrino is not directly observable the way that Neptune or

a mouse in the wall is. But why does this difference matter? IBE says that the best explanation

should be inferred as true, and in both the Neptune and the neutrino case, these were the best

explanation of the phenomena. One clear difference is that in potentially observable local-IBE

there is a level of confirmation attainable that is not available in the unobservable case. Once

there was a direct confirmation of the existence of an eighth planet in the sky, scientists were

able to confirm the explanation that they previous believed to be true. Moreover, after the

confirmation, scientists no longer need to appeal to any form of IBE as to why they believe

Neptune exists. The reason we now believe that Neptune exists is due to our observation of it.

IBE did play a crucial role in the discovery of the planet, but it is no longer the evidence for how

we know that Neptune exists. This is the case in any instance of potentially observable local-IBE

once the entity in question has been observed. Local-IBE is important to the postulation and

discovery of the entity, but there is another higher level of direct confirmation which scientists

can then seek to attain. When the entity is unobservable, this higher level of direct confirmation

is in principle not available. We cannot go out and acquire any sort of direct observation of

things like neutrinos. For unobservables, there is no higher level of justification beyond local-

IBE.49

Returning to meta-IBE, the most famous employment of the inference is the no-miracles

argument for scientific realism. Putnam states that the “positive argument for realism is that it is

the only philosophy that doesn’t make the success of science a miracle.” (Putnam, 1975, p. 73)

The no-miracles argument claims that if we are not realists about science, then the only way to

account for the immense success of our scientific theories is to consider it a miracle, or a “cosmic

coincidence.” (Smart, 1963, p. 39) A miracle or cosmic coincidence is, of course, an unappealing

49 Some scientific realists would argue that there are other ways to confirm unobservables. Hacking, for example

makes the case that being able to manipulate unobservables is critical to our belief in them. I am not too concerned

with these particular brands of scientific realism mainly because they tend to limit their extra conditions to causal

factors which are not applicable to mathematics.

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explanation compared to the better explanation of scientific realism. Thus, by meta-IBE, we

should be scientific realists. The no-miracles argument is meant to convince anti-realists that

they should be scientific realists. If this does not succeed, at the very least the argument provides

justification for the realists’ set of beliefs.

The no-miracles argument has been criticized in a variety of ways from the antirealist camp.

What is interesting about this meta-IBE is that it has also received criticism from scientific

realists as well. A survey of the literature indicates that scientific realists as a whole, do not all

accept meta-IBE as a legitimate inference. Psillos accepts it and defends the no-miracles

argument, but take, for example, Lipton. Lipton recognizes that there is a difference in the use of

the IBE in the local and meta sense. Although he has aggressively defended the local application

of the inference, he questions whether or not the meta use is justified

Even if the miracles argument is not hopelessly circular, it is still a weak

argument, and weak on its own terms. This is so because the argument is

supposed to be an inference to the best explanation, but the truth of a theory

is not the loveliest available explanation of its predictive success; indeed

it may not be an explanation at all. (Lipton, 2004a, p. 193)

While Lipton rejects the no-miracles argument, he still has hope that there is some form of meta-

IBE that is justified. Other scientific realists such as Musgrave (1988) and Ben-Menahem (1990)

are less optimistic. Their position is that using IBE in the local sense is justified, but using it in

the meta sense, be it the no-miracles argument or some other form, is not.

The EIA aligns itself with IBE as the key inference to motivate mathematical realism. Since

mathematical entities are strictly unobservable, then the closest inference that scientific realists

employ is local-IBE for unobservable entities. It does no good to appeal to a local-IBE for

potentially observable entities, as mathematics can never be directly observed. Moreover, it is

important to note that the EIA is not making a meta-IBE either. The conclusion of the EIA is that

we ought to have ontological commitment to mathematical entities. It is these entities that, if

true, would best explain physical phenomena. The inference is being made to entities, and not to

some general thesis of mathematical realism such as platonism. Another reason why the EIA is

not making use of meta-IBE is that as we just saw, meta-IBE is not a widely accepted inference

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for scientific realists. Tying itself to meta-IBE would make the EIA susceptible to the exact same

arguments and criticisms leveled against meta-IBE, and would thus significantly weaken its

strength. Henceforth, any mention of IBE will refer to unobservable local-IBE unless otherwise

stated.

2 Fictionalism: A Better Explanation?

Leng advances an interesting challenge for using IBE to infer mathematical realism. Leng

accepts that there are mathematical explanations of physical facts in science, and that the use of

mathematics is indispensable in some of these explanations. She identifies herself as a committed

naturalist and scientific realist50, but at the same time is a nominalist with regards to

mathematical objects. Leng realizes that this seems contradictory as being a scientific realist she

gladly endorses the use of IBE. Her route to nominalism is to claim that mathematical realism is

actually not the best explanation for the existence of mathematical explanations in science.

Instead, fictionalism provides just as good an explanation, if not better. If Leng is right, then she

can use the EIA as an argument against mathematical realism as it is fictionalism that is the

superior explanation.

Fictionalism is the belief that mathematical objects do not exist, and that strictly speaking all

mathematical claims are false. They may be ‘true in the fiction of mathematics’ but they are not

true simpliciter. To this end, Leng accepts the burden of proof to provide an explanation for how

mathematics can be explanatory but at the same time be false. Leng’s goal is to show that

fictionalism is a better explanation than realism, but if she merely shows that fictionalism is just

as good, then “Ockham’s razor would counsel adopting the fictionalist alternative.” (Leng, 2010,

p. 218)

In a typical scientific explanation the explanandum is assumed to be true, and then we search for

explanans that explains the explanandum in question. IBE infers that the best explanans is true.

50 In her (2010), Leng identifies scientific realism with also being a mathematical realist. I do not use her definition

of scientific realism here.

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Leng has no problem with this general application of IBE. What she does note is that there are

many good scientific explanations where we actually know that not all the explanans are true. In

fact, some of the explanans are specifically designed to be false, such as false idealizations in

scientific explanations like frictionless planes or oceans having an infinite depth. The fact that

these idealizations are false has no impact on the value of the explanations that utilize them, and

also does not hinder them from being considered the best explanation of their explanandums.

Leng reasons that,

[i]f one is willing to accept that literally false theories can still serve to

provide accurate representations of physical systems, and that these

theories get their value because of conditions they impose on the behaviour

of these systems, then it is plausible that one should hold a similar attitude

to theoretical explanations. Couldn’t mathematical explanation get its

value as an explanation due to the conditions it imposes on concrete, non-

mathematical systems? And couldn’t these conditions be imposed equally

well by a fictional theory as they would be by a literally true one? (Leng,

2005, p. 180)

Leng’s main criticism of the realist is that they have been solely focused on demonstrating

scientific explanations where mathematics is indispensable, but at the same time realists “simply

assume that all good explanations must have true explanans, so that if we drop the assumption

that the mathematical objects posited by our explanations exist, then these ‘explanations’ cease

to explain at all.” (Leng, 2005, pp. 179–180) This assumption is what Leng challenges, and if she

is successful, we cannot use IBE to infer mathematical realism as realism is not the best

explanation.

One way to show that mathematics being false does not affect any of our scientific explanations

would be to take the Field route and produce a nominalized version of our best scientific

theories, but Leng does not believe this path will be successful. We could limit our

nominalization project to just scientific explanations and show that for every mathematical

explanation there exists a non-mathematical explanation of the same phenomenon. However,

Leng is like Melia in that they both grant that mathematics is indispensable to scientific

explanation, so this too will not work. Instead, Leng’s approach will be to argue that we can

retain all of our mathematical explanations even while maintaining that mathematics is false

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since the truth or falseness of mathematics makes no difference to the overall explanation. Again

like Melia, the reason why Leng believes this is because she claims that the sole role of

mathematics is to index or represent physical facts.

Since… mathematical theories are introduced in empirical science in order

to provide models which allow us to represent physical systems as having

particular physical properties, the question of existence of the

mathematical objects posited by these models makes no difference to the

utility of our mathematical theorizing… An understanding of the model as

a theoretical fiction will account for its role in theorizing equally well, and

as such no explanatory power is lost if we suppose that the mathematical

systems made use of in the course of providing mathematical explanations

of physical phenomena are mere fictions.” (Leng, 2005, pp. 167–168)

Leng assumes the indexing argument is correct, and that mathematics is only an indexing or

representational tool, albeit an indispensable tool. From this assumption she claims that all we

need to retain the explanatory force of mathematical explanations is the pretence that

mathematical objects relate to physical objects, and not that they actually do relate in any real

sense. This pretence is adequately supplied under the fictionalist interpretation. She writes,

could it be that the reason that a given mathematical explanation of a non-

mathematical phenomenon is a good one is not that the mathematical

utterances that make up its explanans are true, but rather that they are

fictional in our make-believe of set theory with non-mathematical objects

as urelements? … I will claim, the explanatory value of appeals to

mathematical objects is plausibly not a result of the existence of such

objects, but rather a result of the aptness of the pretence that such objects

are related to non-mathematical objects in the ways our ‘explanations’

suppose. (Leng, 2010, p. 244)

Leng does not advance a general argument for how the pretence of mathematical objects existing

is all that we need for every instance of mathematical explanation. Instead, her strategy is to

show that particular supposed examples of GMEs, such as the antipodal weather and the cicada

examples, can be treated fictionally without any loss of explanatory power. I take no issue with

Leng’s fictionalist interpretation based on the indexing argument here. In fact, it very closely

mirrors my own interpretation of the indexing argument in chapter 3. Leng ultimately concludes

that because the sole role of mathematics is to index or represent, then mathematical

explanations will be explanatory regardless if mathematical objects exist or not. So,

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even though we posit mathematical objects in the context of our best

explanations of empirical phenomena, the role that mathematical posits

play in these explanations is just the same as the role they play in any

theoretical representations of empirical phenomena. That is, what makes

the mathematical explanations good explanations is not that their

mathematical hypotheses are true of a realm of really existing

mathematical objects, but rather, that they allow for good representations

of the non-mathematical objects they model. (Leng, 2010, p. 245)

Of course, this alone does not show that fictionalism is correct, but Leng’s point is that the

fictionalist account of mathematical explanation is just as good as taking mathematical

explanations at face-value which would imply realism. Moreover, since fictionalism has a much

sparser ontology than realism, then by Ockham’s razor, the fictionalist explanation is better.

Thus, IBE does not license an inference to mathematical realism, but we should infer

fictionalism instead.

Leng does advance some cursory arguments on how the pretence of mathematics can still retain

all the explanatory and epistemic benefits that are utilized in scientific practice, but much is left

to be desired. Leng also does little to motivate her appeal to Ockham’s razor. This principle is

notoriously difficult to apply as the notion of simplicity is not clear. Even if we admit that the

fictionalist’s ontology is simpler than the realist, this does not mean that their theory as a whole

is simpler. Realists have the advantage that they can treat science at face value without any

reconstruction or interpretation, and they arguably have a simpler understanding of the world

without having to appeal to relations to fictional objects. In addition, the realist need not worry

about how to explain why fictional objects are so helpful and indispensable to science. Taking

the theories as a whole, it is unclear which one Ockham’s razor would cut. Regardless, the most

egregious error that Leng makes in her argument is that she patently begs the question against the

realist. Even if we assume that the fictionalist position can account for all applications of

mathematics in science adequately, and that the use of Ockham’s razor to rule out realism is

acceptable, Leng’s entire argument for fictionalism stems from her critical assumption that the

sole role of mathematics is to index or represent physical facts. But as we saw in chapter 3,

assuming this is equivalent to assuming that genuine mathematical explanations do not, and

cannot, exist.

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Leng admits that mathematical explanations where the mathematics is indispensable do exist, but

the mathematics is still limited to a representational role. This is what motivates her claim that a

fictional interpretation can account for such an explanation. Baker correctly notes that “her

argument misses the main point of the Enhanced Indispensability Argument, which is precisely

to draw a sharp line between representational and explanatory use of mathematics.” (Alan Baker,

2009, p. 626) Leng is assuming that GMEs do not exist, and from this assumption she argues for

fictionalism. However, as we have gone to great lengths to show, GMEs do exist. In such

explanations, the mathematics is not playing a representational role, and is indeed indispensably

explanatory. Given this, Leng’s argument for fictionalism is a nonstarter as it depends on a false

and unjustified assumption on the role of mathematics in science.

Baker also points out another serious problem with Leng’s argument. Leng, like Baker, is trying

to make use of IBE to motivate her position. The difference between the two is that Leng is

making a meta-IBE. What she is inferring is that the thesis of fictionalism, if true, provides the

best explanation for the success of mathematics in scientific explanations. Baker’s complaint

about this is that this type of argument is very different from claiming that that fictions are

actually doing the explaining.

Note, first, that what Leng is offering here is a kind of ‘second-order’

explanation. She is explaining how a mathematical explanation is possible,

given fictionalism. But she is not using fictionalism about mathematics to

explain a physical explanandum. In this situation it is crucial to distinguish

between acknowledging the possible falsity of the explanans being offered

and actively disbelieving in an explanans while simultaneously putting it

forward as an explanation. (Alan Baker, 2009, p. 627)

Another closely related objection to Leng’s use of a meta-IBE is that, as we saw above, meta-

IBE is not a widely accepted inference for the scientific realist. Many believe that meta-IBE does

not reflect the inference pattern that scientific realists utilize to generate their realism. Given this,

Leng’s argument does not carry much force, and is not one that would compel scientific realists

to adopt the fictionalist position.

The way to rescue Leng’s argument from the above criticisms is to rework it so that it is no

longer a meta-IBE. Doing so would require overcoming two key obstacles. First, we would need

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to revise all mathematical explanations in science such that they make explicit that mathematical

entities are false – that they are fictions. This revisionist project is potentially problematic as it

no longer takes the practice of science at face value. One would also wonder what the motivation

to perform such a reinterpretation of science is as it provides us with no greater understanding of

the physical world. Regardless, I am content with pretending that this obstacle can be overcome.

The second, and larger hurdle, is that the fictionalist would also need to argue that fictions can

genuinely explain physical facts. By genuinely explain, I mean that their explanatory

contribution is not restricted to mere representation or indexing.

The unstated assumption is that fictions can only play a part in an explanation when they are

playing a representational role. This assumption is certainly widely held within the community of

scientific realists, and it seems that Leng herself holds this view. If we believe that mathematics

are fictions, then GMEs where the mathematics is not representing anything run counter to this

assumption, and hence this creates tension between the existence of GMEs and fictionalism. The

committed fictionalist can dissolve this tension by abandoning this assumption and insist that

fictions can genuinely explain even when they are not representing anything.

Recently, Alisa Bokulich (2009, 2011, 2012) has taken up something akin to this position.

Bokulich develops and advances her own account of ‘model explanations’ which she argues can

make sense of fictions being genuinely explanatory. The fictions that she has in mind are

fictional models such as Bohr’s model of the hydrogen atom. Bokulich argues that based on her

model explanation account, Bohr’s fictional model gives us a genuine explanation of many

physical facts, such as the nature of spectral lines. Bokulich admits that “[p]urported

explanations, such as these, that appeal to fictional structures, are not easily accommodated into

any of the canonical philosophical accounts of scientific explanation,” (Bokulich, 2012) which is

what necessitates the development of her new model explanation account. If Bokulich is correct,

then the fictionalist could argue that mathematics explains just like how the Bohr model

explains, and that both are still ultimately fictions.

Without directly criticizing Bokulich’s account of model explanations, there are two important

obstacles in adopting her position to support fictionalism. The first is that believing fictions can

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genuinely explain may be at odds with scientific realism. One challenge for the Bohr model

explanation is to assert that the actual, genuine explanation is found in quantum mechanics; any

‘explanation’ that the Bohr model seems to provide is illusory. Bokulich’s responds by stating:

I think it is a mistake to believe that there can only be a single legitimate

scientific explanation for a given phenomenon. A closer examination of

scientific practice reveals that, not only can there be more than one genuine

scientific explanation for a given phenomenon (what might be called the

explanatory pluralism thesis), but that some of these explanations may turn

out to be deeper than others. To say that one scientific explanation is not

as deep as another is not the same thing as saying that it is no explanation

at all. (Bokulich, 2011, p. 44)

Granting that the quantum mechanical explanation is deeper than the one provided by the Bohr

model, Bokulich is correct in asserting that this does not imply that the Bohr model is itself not a

genuine explanation. However, is this explanatory pluralism something that a scientific realist

would assent to? Realists are interested in finding explanations that they can utilize IBE on; they

are interested in finding explanations that are most likely to be true. It is hard to see how any

pluralist thesis can be reconciled with a standard scientific realist position. The scientific realist

could surely be pluralist in that he could grant that the Bohr model is a nice explanation in the

epistemological sense, but it is not a genuine explanation. This is common practice as often

epistemological explanations are easier to use. Scientists still make use of Newtonian mechanics

when analyzing many simple dynamic problems, while at the same time recognizing that the

Newtonian picture is not genuinely correct. However, a key difference is that for the scientific

realist, genuine explanations are those which we would be willing to run an IBE on – they are

ontological explanations. Bokulich’s fictions are not like these.

Bokulich expands on her idea of explanatory pluralism when tackling an objection raised by

Belot and Jansson (2010). The worry is that if we accept explanatory pluralism then we end up

admitting far too many fictions as genuinely explanatory; the bar for a genuine explanation is set

too low. Bokulich’s response is to put forward a way to distinguish explanatory from non-

explanatory fictions.

My answer begins with the observation that some fictions are

representations of real entities, processes, or structures in the world, while

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other fictions represent nothing at all. We can even recognize that some

fictions do a better job of representing certain features of the world than

other fictions. What I want to say in answer to the challenge, then, is that

only those fictions that are an adequate representation of the relevant

features of the world are admitted into the scientist’s explanatory store.

(Bokulich, 2012)

The key to being a genuinely explanatory fiction is whether or not the fictions are representing

something real and physical. This leads to the second problem of asserting that fictions can

genuinely explain. As we have shown, the mathematics in GMEs specifically does not represent

anything at all. Their explanatory force is not due to any representational nature at all. According

to Bokulich’s account of genuinely explanatory fictions, then the mathematics in GMEs cannot

be fictions at all as they do not represent. We cannot coherently appropriate her suggestion that

fictions can genuinely explain to support mathematical fictionalism accounting for GMEs.

I have argued that there is no present way to account for the existence of GMEs from a

fictionalist perspective. If this is right, then Leng is incorrect in claiming that fictionalism is at

least as good as realism in explaining the applications of mathematics to science, and hence we

cannot use IBE to infer fictionalism. This by no means shows that there is no way in principle to

reconcile fictionalism and the existence of GMEs. One possible move would be to accept, contra

Bokulich, that fictions that do not represent anything can genuinely explain. This belief makes

fictionalism and GMEs perfectly consistent; however, the obvious problem is caching out this

belief in a coherent and appealing way without running into the problem above where the bar for

scientific explanations is set trivially low. Moreover, I suggest that any such system of

understanding scientific practice would be so complex that the appeal to Okham’s razor to

support fictionalism no longer seems realistic. Thus, granting that GMEs exist, fictionalism is not

the best explanation for the mathematical explanations in science.

3 Unjustified Inference to the Best Mathematical

Explanation

As argued in chapters 3 and 4, I believe that there are GMEs, such as the electron spin example,

which are the best explanation that we have. I also believe in IBE as a legitimate inference for

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the scientific realist. Lastly, I believe that there is no in principle objection to using IBE to infer

the existence of mathematical entities, nor do rival positions such as fictionalism pose a threat to

this inference either. Yet, at the same time I do not believe that we are presently justified in using

IBE to infer mathematical realism. I will argue that this is not a contradictory set of beliefs, and

is in fact supported by the actual way in which IBE is employed by scientific realists, and

advanced by its most ardent supporters.

The reason why believing in both GME and IBE, while at the same time rejecting the inference

to mathematical realism, may appear contradictory is due to an imprecise and naïve

understanding of IBE. As we saw above in 5.1, the best of a bad lot argument poses a serious

problem for any naïve presentation of IBE. This criticism from van Fraassen has led to more

precise and restrictive understanding of the inference. I will argue that once we adopt this more

sophisticated understanding of IBE that is motivated from the best of a bad lot criticism, it is

clear that inferring mathematical realism is not licensed by IBE.

The best of a bad lot argument states that there is no way for us to know that the true explanation

is among those which we are considering unless we assume that we are in some sort of

privileged position from which to make this judgment. Psillos assumes exactly this when he

asserts that scientific realists possess a background knowledge privilege that circumvents the bad

lot argument. The background knowledge from our best and strongly confirmed scientific

theories both restricts the potential set of explanations that we could consider to be true, and also

provides techniques that help us select the best explanation. The key for this process is that the

explanations must be compatible with our scientific theories. IBE does not work in a total

vacuum; rather we make use of our background knowledge as a critical guide for the inference.

While Lipton does not explicitly commit to a privileged position to avoid the best of a bad lot

argument, he does place additional criteria on the best explanation in order to ensure that the true

explanation is likely to be within the set we are considering. For IBE to be warranted, the

explanation needs to be the loveliest potential explanation available. By potential, Lipton just

means that we cannot already assume that we have the actual explanation as this would beg the

question. Still, defining what qualifies as a potential explanation is difficult. “We have to

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produce a pool of potential explanations, from which we infer the best one.” (Lipton, 2004a, p.

58) If our criteria for potential explanations are too broad, such as empirical adequacy, then our

pool would be far too large and would include explanations that we would not even consider

taking seriously. Lipton suggests that we restrict our pool to “only the ‘live options’: the serious

candidates for an actual explanation.” (Lipton, 2004a, p. 59) Doing so would restrict the pool

nicely, but this leads directly to van Fraassen’s complaint that we require a privileged position in

order to determine what counts as a ‘live option’, which he alleges is unjustifiable. Lipton

recognizes that this suggestion assumes some sort of epistemic filter exists, and that this filter

can both restrict our pool of potential explanations and serve to guide us in selecting the best of

them. This sounds very much like Psillos’ background knowledge privilege.

Lipton also discusses the difference between likely and lovely explanations. Likely explanations

are the most probable to be true, whereas lovely explanations are the most explanatory, or

provide the most understanding. It is certainly the case that sometimes the likeliest and loveliest

explanations are one and the same, but in other situations they will point to different explanations

entirely. What makes an explanation likely to be true is relative to our current body of evidence.

This evidence comes from our scientific theories, and how well the explanation fits with these

theories. So, when selecting the best explanation, should we choose the likeliest or the loveliest

explanation from our potential pool? Here, Lipton seems to differ from Psillos. Psillos looks to

the fit with background knowledge as his guide, which seems to point to the likeliest

explanation; however, Lipton argues that the loveliest explanations are those we should consider

to be the best. Lipton is quick to point out that loveliness and likeliness cannot be perfectly

separated, and in reality any defensible version of IBE would need to incorporate elements of

both in the selection process. (Lipton, 2004a, p. 61) In this light, Lipton is not as far from Psillos

as it first appeared. Still, Lipton emphasizes that explanations which are lovely, which provide

the greatest understanding, should be considered the best.

My strategy for showing that using IBE to infer mathematical realism is unwarranted is

straightforward. I will argue that all of our examples of mathematical explanation do not satisfy

the requirements for IBE advanced by Psillos and Lipton. I will not take sides on whose account

of IBE is better. If I can demonstrate that by the standards of the two most vocal defenders of

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scientific realism and IBE that inferring mathematical realism is unjustified, then that should be

sufficient to undermine the EIA.

Consider our optimization explanations: the cicada and the honeycomb examples. In both cases,

a mathematical explanation is advanced to explain the optimal behaviour of cicadas and

honeybees. However, in both cases there is a competing explanation that does not cite any

mathematical facts. Instead, these make use of physical facts such as sunlight, soil temperature,

and units of time in the cicada example, and facts of approximately Euclidean space-time in the

honeycomb example. Now, taking up Psillos’ version of IBE, do the optimization explanations

which depend on mathematical facts fit with our background knowledge privilege? I will not

make a knock down case here, as this would require a detailed look at what our background

knowledge genuinely consists of, but I will suggest that the optimizations explanations do not fit

well. The reason being is that in biology, and in typical biological explanations, it is the

biological, ecological, and physical facts that build up the core of our knowledge. The cicada and

honeycomb explanations, while interesting, are critically unlike almost all other biological

explanations in that they use mathematics for its explanatory force. This is not to take anything

away from the explanation in terms of its contribution to our scientific knowledge, but the

important point is that it does not satisfy the basic condition that would allow us to consider it the

best to warrant an IBE. This is made even clearer when considering the nominalist explanations

of cicada life-cycles and honeycomb shapes. Even if you reject the claim that optimization

explanations do not fit with our background knowledge, it is hard to deny that the nominalist

explanations do fit, and that they fit better than the mathematical explanations. So, even if both

the mathematical and nominalist explanations are in our pool, the nominalist versions are the

more likely to be true based on their compatibility with our background knowledge. Given this, it

is clear that using IBE based off of the mathematical explanations of the cicada or honeycomb is

not entirely justified. These mathematical explanations either do not meet the requirements of

fitting with our background knowledge, or if we are generous and grant them that fit, then they

do not fit as well and hence are not as likely as their nominalistic counterparts.

The situation is similar when we consider Lipton’s system. Are the mathematical explanations to

be considered ‘live’ options for us? It seems that when comparing them to the nominalist

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explanations that the mathematical explanations are not good candidates for being live by the

exact same reasons as above. This does depend on exactly what epistemic filter that Lipton has

in mind, so I will not base too much on the above conclusion. Still, there may be some hope as

Lipton’s focus is on the loveliest explanations. It can be argued that the main virtue of the cicada

explanation is that it genuinely enhances our understanding in a way that the nominalist version

does not, and thus while it may be true that the mathematical explanations are not more likely

than the nominalistic explanations, they are the more lovely and hence should be inferred as true

by IBE. While this sounds reasonable, it is, I feel, incorrect. While the mathematical explanation

invoking prime numbers does seem to answer some important questions about the life-cycle of

the cicada, it comes with an even larger question that remains unanswered. The question that is

blocking any actual understanding is how properties of abstract, noncausal, mathematical entities

influence, matter to, or make a difference to actual cicadas and honeybees. The nominalist

explanations, on the other hand, are lovely in that they explain the behaviour of cicada and

honeybees without introducing new unanswered questions. It is physical properties and physical

entities that relate to cicadas and honeycombs and are influencing or making a difference to their

behaviour. When comparing the two types of explanations, it is the nominalistic explanations

that lead to a greater understanding as there is no additional mystery as to how they work. Thus,

by Lipton’s criteria as well, it would be unwarranted to select the mathematical cicada or

honeycomb explanation to be the best explanation for an IBE as they are not the loveliest

potential explanation.

The Kirkwood gaps example does not fare much better when applying a more sophisticated

version of IBE. On the surface, it seems that the mathematical explanation does fit better with

our background knowledge as astrophysics is much more mathematical than biology. However,

even though mathematics may be ubiquitous in astrophysics, this is different than saying that

explanatory mathematical factors are standard in astrophysical explanations. When considering a

competing explanation that uses physical facts such as gravitation and mass as the key

explanatory factors, such an explanation certainly fits better. The same challenges that the

optimization explanations faced with fitting our background knowledge, or the live options for

Lipton, present themselves here. Moreover, while an eigenvalue analysis may explain the

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Kirkwood gaps in the sense of prediction, it is hard to say how eigenvalues contribute to our

understanding of the presence of the gaps. As above, the issue is in establishing some sort of link

between mathematical properties such as eigenvalues, and physical entities such as asteroids.

This issue is made even clearer when considering the nominalistic explanations that simply cite

how physical factors such as gravitation, planetary mass, etc., are the ones that influence the

asteroids. Without some sort of account as to how the mathematics is making a difference to the

asteroids, the mathematical explanation of the Kirkwood gaps explanation is not lovelier than its

nominalistic counterpart, and hence IBE would be unwarranted.

The criticisms of the cicada, honeycomb, and Kirkwood gaps explanations here mirror those

raised in chapter 3. One key factor that is making IBE unwarranted for these explanations is that

they may not be genuine mathematical explanations in virtue of the fact that there exists

nominalistic explanations of the same phenomena, which is precisely what was argued before as

a major downfall of these examples. Instead, we should consider the GME of electron spin. This

explanation is different as the mathematics is genuinely explanatory, and there is no nominalistic

explanation available to us. If IBE is able to infer mathematical realism at all, it would be most

likely to succeed when considering our best and uncontroversial example of a GME.

When considering how well the electron spin example fits with our background knowledge, it is

immediately apparent that the fit is much better as quantum mechanics is almost entirely

mathematical. Recall that the same claim was made for the fit between the Kirkwood gaps

explanation and astrophysics, but the problem was that explanations in astrophysics still almost

exclusively cite physical difference-makers. This problem does not present itself in quantum

mechanics. Many explanations in quantum mechanics point to mathematical difference-makers,

and lots of the mathematical formalism that is indispensable to doing any quantum mechanics

does not have any sort of physical interpretation at all. This certainly bodes well for using IBE to

infer mathematical realism; however, there is still a critical way in which a GME does not fit

well at all with our background knowledge. Some explanations in quantum mechanics have led

to our commitment to novel entities, such as quarks, through the use of IBE. But there is no case

yet where an explanation has led us to commit to an abstract, noncausal entity. Simply put,

mathematical entities just do not fit with the background knowledge of our best scientific

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theories no matter how inherently mathematical our theories are. This is because our best

scientific theories have only utilized IBE to infer physical entities. I do not mean this to be an in

principle objection to using IBE to infer mathematical entities. There is nothing about the

inference model that fundamentally restricts the possibility of such an inference. The issues

comes when we try to justify such a move using only the background knowledge from science.

Our background knowledge from our best scientific theories does not fit nicely with GMEs. It is

our background knowledge that is restricting our ability to infer mathematical realism.

Turning towards Lipton’s criterion of loveliness, the electron spin explanation fares no better.

The property of spin itself does increase our understanding and is a lovely explanation. In

addition, no one doubts the predictive power of the mathematical formalism of spin, but this is a

far cry from claiming that we are any closer to understanding what spin is. We saw above that

the primary reason for our lack of understanding is due to the absence of an account as to how

mathematical entities and properties influence physical entities and properties. This same

problem presents itself here in an even more bizarre way. Quantum mechanics as a whole is

famous for being incredibly accurate in its predictions, and notoriously difficult, if not

impossible, to understand in its present form. As the ever quotable Feynman remarked, “I think I

can safely say that nobody understands quantum mechanics.” (Feynman, 1965, p. 129) This cleft

between predictive power and understanding has been made famous by the Einstein-Podolsky-

Rosen and Schrödinger’s cat thought experiments. The core idea is that while mathematical

explanations in quantum mechanics, such as the mathematical formulation of spin and psi

function of superposition, are incredibly accurate predictors, they do not enhance our

understanding of the physical phenomena; in short, they are not lovely explanations. Even if

GMEs from quantum mechanics are among our pool of potential explanations, they are not

lovely enough to infer their (approximate) truth.

I admit that the above arguments have not been rigorously developed. I do hope that I have done

enough to motivate the claim that simply being a GME is not enough to satisfy the conditions of

IBE. The crux of my position is that there is a difference between being the best explanation that

we have, and being the best explanation with respect to running an IBE. GMEs satisfy the

former, but at present they do not satisfy the latter. While I have used the formulation of IBE

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advanced by two of its most ardent defenders, a brief look at the history of science also supports

the idea that there are times when an explanation is the best that we have, is incredibly useful in

terms of predictive power, but at the same time the scientific community has rejected any IBE

that would lead to new ontological commitment. In chapter 2 we saw such an example with

Maddy’s story of the discovery of the atom. Explanations that made use of the atom were

arguably indispensable, but still scientists rejected the existence of the atom. It was not until

Perrin’s 1905 experiments on Brownian motion that ultimately led to an almost wholesale

acceptance of the atom by the scientific community.

Another example of this surrounds J. J. Thomson’s discovery of the electron. Based on his

experiments on cathode rays, Thomson advanced the hypothesis that tiny ‘corpuscles’ with

negative charge are constituents of the atom.51 These electrons nicely explained experimental

results from Thomson, Lenard, and others in 1897; however, the scientific community still did

not accept the electron as a real entity. It took years of further experimenting to increase the

understanding of the electron before being accepted as a real entity. (Baigrie, 2007, p. 127) This

situation was summed up by Thomson almost 40 years later. “At first there were very few who

believed in the existence of these bodies smaller than atoms. I was even told long afterwards by a

distinguished physicist who had been present at my lecture at the Royal Institution that he

thought I had been 'pulling their legs.'” (Thomson, 1936, p. 36) The acceptance of the electron is

similar to the atom example, but a key difference is that in the case of the atom a crucial

experiment was critical in convincing the scientific community that atoms exist. For the electron,

increased understanding came more gradually. In both cases though, there was a time where the

best explanation of physical phenomena was still not good enough to be considered the best

explanation for an IBE.

My belief is that we are in the exact same position with GMEs as the scientific community was

with respect to the atom and electron prior to their ultimate acceptance. Some GMEs may be the

51 Thompson also claimed that these negatively charged corpuscles are the only constituents of the atom, which was

proven false.

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best explanation we have, and they may be excellent predictive tools that explain physical

phenomena, but we do not have neither a crucial experiment nor the developed understanding

that makes using IBE legitimate. What makes my position different from other critics of IBE for

mathematical realism is that my rejection of using IBE in this way is not an in principle rejection.

I do not believe that we can never use IBE to infer mathematical realism, but rather that we are

presently unjustified. This is supported by those who advance sophisticated versions of IBE, and

also by the historical record.

The only remaining question is how can we move forward such that an inference to

mathematical realism will be justified? This is difficult to answer, and I will merely suggest two

possible ways forward. One route that scientific realists typically take is to look for additional

confirmation. This was the case for the atom. If the atom actually existed, then Einstein predicted

we would be able to witness Brownian motion. Psillos also recognizes this with his example of

the neutrino.

When the neutrino’s energy is taken into account, there is no need to abandon

the principle of conservation of energy during β-decay. This degree of

confidence in the existence of the neutrino will depend on many factors. But,

sure enough accepting its existence will guide looking for further experimental

and theoretical confirmation. Pretty much like the mouse-case, the presence of

neutrino in a β-decay implies certain further predictions of neutrino-related

phenomena. (Psillos 1999, p.212)

So, if we take GMEs seriously, then we should be looking for further predictions of

mathematical-related phenomena. Herein lies the problem with this route of justification. It is

unclear what, if any, predictions are entailed by the existence of mathematical entities. The main

reason for this stems from their abstract, noncausal nature. The only hope for this way forward is

to gain an increased understanding of how noncausal explanations work in general. This is

important for not just mathematical explanations in science, but also any other type of noncausal

explanation such as geometrical or structural. While somewhat farfetched, it is possible that an

increased understanding of how these explanations work could yield interesting ways to

manufacture novel predictions and to test for them.

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A more promising route towards justification would be to show how mathematical objects truly

enhance our understanding of the physical world. For example, an increased understanding of

how the mathematical formulation of quantum mechanics is such a successful predictor, and how

it relates to the physical world, could lead to a critical mass of mathematical explanations and

understanding that would warrant an IBE. While this suggestion is highly speculative and

somewhat vague, the basic idea is that it more closely follows the path of the acceptance of the

electron where acceptance came gradually through increased understanding. Some inroads were

made in section 4.4 when we analyzed some of the possible contributions and ways that

mathematics can be a difference-maker in scientific explanations. What we need to be focusing

more on is how mathematics makes a difference to the physical world. The danger with this route

is that it may be impossible to gain this understanding without first presupposing some sort of

relationship between mathematics and the physical world. Undoubtedly, such a presupposition

would be charged as totally unsupported by our best scientific theories by the nominalist. This

would clearly make an inference to mathematical realism question begging as mathematical

objects or truth would already be taken for granted.

I have argued that when we consider a sophisticated version of IBE, inferring mathematical

realism based on GMEs is presently unjustified. I then presented two potential ways forward that

would allow for IBE to work on GMEs, but at the same time it should be clear that both these

ways are challenging and come with their own set of problems. By no means is this meant to be a

condemnation of either IBE or mathematical realism. Rather, the point of this analysis is just

meant to illustrate that if we take IBE on its own as our sole inferential tool for ontological

matters, then mathematical realism is not yet attainable. This leads to one final way in which we

can make use of IBE to infer mathematical realism legitimately: we can supplement IBE so that

it does not stand on its own. I will briefly consider two ways in which we can add to our

inferential tools, one below, and one in chapter 6.

4 The Not-So-Enhanced Indispensability Argument

We are now in a position to identify precisely what the problem is with the first premise, (EP1),

of the Enhanced Indispensability Argument (EIA). (EP1) states that “we ought to rationally

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believe in the existence of any entity that plays an indispensable explanatory role in our best

scientific theories.” This is meant to be an appeal to IBE. However, what is clear now is that this

conception of IBE is naïve. It does not reflect the understanding of IBE advanced in the

literature, nor does it accurately portray the actual way in which scientific realists have

historically utilized the inference. The version of IBE advanced in the EIA is not accurate, and

thus the argument is unsound.

There are two possible ways forward to rescue the EIA. The first is to redefine the argument so

that it reflects a more sophisticated form of IBE. (EP1) would need to be changed to stipulate, for

example, that the explanation needs to fit nicely with our background knowledge, or that it needs

to be the loveliest explanation. While this would fix the first premise to be a more faithful

representation of IBE, it actually renders the EIA invalid. As shown above, our present stock of

GMEs do not satisfy the criteria such that making use of IBE is legitimate. To circumvent this,

the second premise, (EP2), which states that GMEs exist, would also need to be changed to

assert that there exist GMEs which do fit our background knowledge, or are the loveliest

explanations available. It is no longer enough to just produce a GME, and this means that the

status of (EP2) is again cast into doubt. What we actually need is a GME that satisfies the criteria

for being the best explanation with respect to IBE. With these changes the debate would shift

away from the indispensable explanatory role of mathematics to other complicated issues such as

the relationship between prediction and understanding. This is certainly a reasonable way to

proceed for defenders of the EIA, but it raises such challenging issues that the argument is no

longer the nice and easy path to mathematical realism that it is meant to be. Suddenly, the EIA

depends on spelling out the relationship between mathematical and physical objects such that we

can make sense of how mathematical explanations lead to understanding. This is a tall order and

it is unclear if it is even possible to achieve without prior metaphysical assumptions.

The second option for the EIA is to note that all the objections that have been raised here seem to

hinge on the assumption that there are real and important differences between mathematical and

physical entities. For example, the inability to generate further mathematical-related predictions

was due to the assumed abstract and noncausal nature of mathematical entities, which is, of

course, different from the concrete and causal nature of physical entities. This assumed

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difference seems fine, but actually part of this assumption is that this is a difference in kind rather

than in degree. Similarly, the bad fit between GMEs and our background knowledge is because

of the difference between standard physical explanations and abstract, noncasual, mathematical

explanations. Again, this assumes that the explanations are significantly different in kind, and not

just different along the same continuum of scientific explanation. Rejecting these assumptions

would be a radical move, but it would open up the possibility that we could make further

predictions and test for the existence of mathematical objects, and for GMEs to

unproblematically fit with our background knowledge. The reason why is that there is no longer

a cleft between the mathematical entities, predictions, and explanations in our best scientific

theories and the physical entities, predictions and explanations. Surely they are different in some

important ways, but fundamentally they are all of the same kind. The obvious problem with this

second path is that abandoning the assumption that mathematical entities are different in kind

from physical entities would be unappealing to the bulk of scientific realists and mathematicians

alike. This would make it a nonstarter and not a good way in which to craft an overarching

argument for realism.

Another problem with treating mathematical entities and explanations exactly like physical ones

is that such a move seems ad hoc in nature. There would need to be significant additional reasons

to abandon the belief that they are different beyond simply rescuing the EIA. Fortunately, there

is one set of beliefs that would provide independent reasons for maintaining the equality of

mathematical and physical entities and explanations: Quine’s thesis of confirmational holism

which Quine believed faithfully reflected the actual practice of science. Confirmational holism

has two important features. The first is that theories are confirmed in a holistic manner. We do

not separate entities within our theories by their properties; what matters is simply what role

these entities play. Secondly, there is no real difference in kind between the unobservable entities

utilized in our best scientific theories. Terms such as abstract versus concrete merely indicate a

difference in degree. If the EIA adopts confirmational holism, then even when using a

sophisticated version of IBE we will be able to infer mathematical realism. No longer is there

any problem with our GMEs fitting with our background knowledge. Our best scientific theories

are to be treated holistically, so any scientific explanation, be it mathematical or otherwise, is just

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another part of science itself. The differences that we had previously identified assumed that we

could separate GMEs from regular scientific explanations, but this goes against the spirit of

conformational holism. Similarly, there is no difference in using IBE to infer abstract

mathematical entities compared to concrete unobservable physical entities, as these entities are

not different in kind. They are all unobservable posits in our theories, and hence there is no

significant difference at all in the inference either.

Confirmational holism also helps us escape the challenge that our GMEs are not lovely

explanations – that they do not contribute to our understanding. It is a mistake to look at specific

explanations in isolation. We need to consider how the scientific theories that our GMEs are a

part of contribute to our understanding of the world. When thought of this way, then it is clear

that GMEs are lovely explanations. Even the electron spin example which is certainly an

excellent predictor contributes to our understanding as the concept of spin itself is critical in

understanding observed behaviour of things like electrons. Before we were able to isolate the

mathematical formalism from the rest of the theory and allege that we did not understand how

this aspect of the explanation functioned. When we adopt confirmational holism, this criticism is

no longer possible. This all points to confirmational holism being a perfectly reasonable and

effective way at ensuring that using IBE to infer mathematical realism is entirely justified.

I feel that fully embracing confirmational holism is the only reasonable way forward for the EIA.

However, at the same time it also spells the end for the argument itself. Unlike the two above

suggestions, adopting confirmational holism is not merely altering the premises of the EIA.

Rather, it requires supplementing the argument with an entirely new inferential tool. By adopting

confirmational holism the EIA simply collapses to the Quinean indispensability argument. The

EIA differentiates and enhances itself from the original Quinean version by pointing to IBE as

the inferential process that leads to realism. The catch is that, as I have argued, IBE alone cannot

presently deliver the desired conclusion. Confirmational holism is needed to circumvent the

problem that any sophisticated expression of IBE does not justify inferring mathematical realism.

If confirmational holism is adopted, then in actual fact the inferential process leading to realism

is not IBE at all, but rather a product of our holistic beliefs. The differences between the EIA and

the QIA vanish entirely. In no way then is the EIA actually enhanced at all. It points to the exact

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same inferential process, depends on the exact same claim that mathematics is explanatorily

indispensable52, and is thus susceptible to the exact same criticisms that we examined in chapter

2. The Enhanced Indispensability Argument collapses entirely to the Quinean indispensability

argument, and ultimately the appeals to inference to the best explanation have done nothing to

strengthen indispensability arguments for mathematical realism.

Perhaps there is some other inferential tool that we can add to the EIA to rescue it other than

confirmational holism. Although I will not make the case for it here, I believe that confirmational

holism is necessary for any indispensability argument that aims to lead us to mathematical

realism.53 IBE forces us to examine the precise role of the entities in our explanation. Moreover,

any justified use of IBE also requires us to examine how well our supposed explanations fit with

what we already know, and how they lead to a greater understanding of the world. The trouble

with GMEs is that this type of examination will always lead to trouble due to the differences

between mathematical and physical entities. This difference is not only due to the abstract,

noncausal nature of mathematics, but also stems from our lack of understanding of the

relationship, if any, between mathematics and the physical world. These problems cannot be

avoided when we look at GMEs and mathematical entities in isolation. Thus, the only way to get

around these issues is to not look in isolation at all, and to not consider mathematical entities

truly different from physical. This is exactly what confirmational holism allows us to do, and is

why the EIA needs it to work.

What does this mean for mathematical realism? The failure of the EIA notwithstanding, by

confirmational holism alone and the positive results from chapters 3 and 4, it seems that we

ought to be mathematical realists. The pressing question then is whether or not we should believe

that confirmation is applied in a holistic manner. In chapter 2 I argued that the criticisms of

52 Quine never pointed to being explanatorily indispensable in his writings. As mentioned in chapter 2, Colyvan

shows that it is simple to alter Quine’s argument to take explanation into account.

53 Bangu (2012) argues at length that confirmational holism is central to any indispensability argument. He claims

that the best an indispensability argument can do is to establish a conditional conclusion: if one is a scientific realist

and a Quinean holist, then one should also be a mathematical realist. The challenge, which he admits is unaddressed,

is why we should ever want to be a Quinean holist in the first place.

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confirmational holism are powerful enough to reject it, and I stand by that analysis here. Looking

at it more broadly, we can see now the trouble with indispensability arguments that depend on

confirmational holism such as the QIA and EIA. All the nominalists needs to do to avoid the

conclusion of mathematical realism is to reject confirmational holism as an inferential tool in our

best scientific theories. The tradeoff for the nominalist is minimal. Adopting holism leads to

unwanted ontological commitments, whereas rejecting it they are able to maintain the sparse

ontology they desire while at the same time the nominalist arguably does not lose any significant

understanding of the practice of science. The upshot to this is that now philosophers of science

can look to examine and understand the explanatory role of mathematics in science free from the

trouble of worrying about ontological commitment. The downside is that this may spell the end

of the recent revival in interest of indispensability arguments.

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Chapter 6 Conclusion

In a brief survey on mathematical explanation, Mancosu suggests three ‘philosophical pay-offs’

that could result from an increased understanding of mathematical explanations of physical facts.

First, in the direction of a better understanding of the applicability of

mathematics to the world. Indeed, understanding the ‘unreasonable

effectiveness’ of mathematics in discovering and accounting for the laws

of the physical world (Wigner, Steiner) can only be resolved if we

understand how mathematics helps in scientific explanation. Second, the

study of mathematical explanations of scientific facts will serve as a test

for theories of scientific explanation, in particular those which assume that

explanation is causal explanation… Third, philosophical benefits might

also emerge in the metaphysical arena by improved exploration of various

forms of the indispensability argument. Whether any such argument is

going to be successful remains to be seen but the discussion will yield

philosophical benefits in forcing for instance the nominalist to take a stand

on how he can account for the explanatoriness of mathematics in the

empirical sciences. (Mancosu, 2008, p. 138)

The principle aim of this dissertation was to analyze and assess the Enhanced Indispensability

Argument (EIA) for mathematical realism. Ultimately the EIA was found to be lacking as it

relied on a naïve understanding of inference to the best explanation (IBE). Regardless, as

Mancosu suggests, several important philosophical pay-offs were arrived at along the way. The

first important result was to generate a set of criteria for a genuine mathematical explanation

(GME) that nominalists and realists alike could agree to. This step, which is part of Mancosu’s

third pay-off, has been overlooked in the literature and is a major factor in why there has been no

agreement whatsoever on the status of GMEs. The key to success for this generation was in

refining and understanding the ways in which nominalists reject supposed examples of GME via

the indexing argument. With the criteria in hand, presenting an example of GME turned out to be

a much easier task as it immediately became clear what sort of features in the explanation were

needed in order to circumvent the nominalists’ arguments. The electron spin example, which

explains the splitting of a beam of electrons passing through a Stern-Gerlach apparatus, was thus

shown to satisfy all the criteria of a GME.

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From there we were able to address a second overlooked area in the literature; we looked

towards accounts of scientific explanation to corroborate the claim that GMEs exist. Strevens’

kairetic account was an ideal account to make use of due to its commitment to a two-level

approach to explanation that could identify difference-makers in any domain. In addition to

tackling our stock of mathematical explanations, this process was interesting for two other

reasons. First, it presented an interesting stress test for the kairetic account. Strevens’ belief that

the kairetic criterion can be applied to other domains, such as noncausal or mathematical, is

appealing but up till now untested. GMEs represent a serious challenge for any account of

scientific explanation, and attempting to appropriate the kairetic account in order to analyze

GMEs without a doubt pushes the limits of Strevens’ account of scientific explanation. Secondly,

by corroborating GMEs as a legitimate form of scientific explanation, this paves the way forward

for supporters of noncausal explanation to both put pressure on exclusively causal accounts of

scientific explanation. All this work speaks to Mancosu’s second philosophical pay-off; GMEs

serve as a challenging and arguably necessary test for accounts of scientific explanations.

Once the electron spin example was shown to be a GME we were able to free ourselves from the

burden of proof and focus on the more interesting question of how it is that mathematics

explains. We briefly surveyed some of the promising suggestions for how to make sense of the

explanatory role that mathematics plays in scientific explanations. Though this work was quite

preliminary, it represents progress in Mancosu’s first philosophical pay-off of an increased

understanding of the applicability of mathematics through understanding mathematical

explanation. Moving forward, this area of research looks to have the greatest impact in terms of

our understanding of mathematics in science.

In my opinion, one of the greatest impediments to understanding applicability via mathematical

explanation in science comes from the staunch opposition by nominalists due to their fear of

ontological commitment. This was a key motivating factor in my methodological approach of

treating mathematical explanation and metaphysical conclusions as independently as possible.

To that end, taking a precise look at IBE was essential as it showed that the nominalist has

nothing to fear from arguments such as the EIA. This analysis has two immediate benefits. The

first is that no longer is the idea of mathematical entities playing a genuine explanatory role in

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scientific explanations equivalent to mathematical realism. This separation means that further

research can continue on the applicability of mathematics in explanation free from metaphysical

biases. Secondly, understanding how IBE works made clear that our present indispensability

arguments critically depend on confirmational holism to succeed. This speaks directly to

Mancosu’s third pay-off of an increased understanding of indispensability arguments and what

they truly need to succeed.

1 Moving Forward

I have just suggested that a promising area for future research is to continue the work started in

section 4.5 where we looked at the difference-making role of mathematics in GMEs. I will close

by considering two other areas for further development. Throughout this dissertation I have

remained purposefully silent on two important overarching themes: naturalism and mathematical

realism. I will now briefly address these topics.

Quine’s naturalism plays an incredibly important role in the Quinean indispensability argument

(QIA) and for all indispensability arguments that stem from it such as the EIA. Naturalism serves

as the backdrop for Quine’s motivations for being a mathematical realist, the inferential

framework that leads to the realist conclusion, and also as a first line of defense against many

potential objections to the QIA and mathematical realism. Beyond this foundational role,

naturalism also specifies the arena in which the debate for mathematical realism takes place.

Science is the sole arbiter of our ontology. The actual practice of mathematics is simply not

taken into account as Quinean naturalism explicitly rules out looking anywhere else for

guidance. But is such a restriction necessary? Are we cutting ourselves off from potentially

fruitful avenues of exploration? For the naturalist it seems reasonable to want to rule out certain

other domains, such as religion, voodoo, or astrology. These things are often found to be in

conflict with the practice of science, be it in their conclusions or in their methodologies.

However, mathematics is different in that its practice is acceptable and in fact is indispensable to

the practice of science, whereas the other domains are certainly not. In short, mathematics seems

special.

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The disrespect of mathematical practice was highlighted in chapter 2 but played no role in our

analysis of the EIA. However, it does stand to reason that if one finds this disrespect to be

offensive, then any sort of indispensability argument that depends on the naturalistic perspective

that science is the sole arbiter of our beliefs is simply a nonstarter. Maddy is one such person

who finds the lack of respect towards mathematical practice inherent in Quinean naturalism to be

objectionable; so much so that she proposes a mathematical naturalism to operate alongside her

scientific naturalism.54

What I propose here is a mathematical naturalism that extends the same

respect to mathematical practice that the Quinean naturalist extends to

scientific practice. It is, after all, those methods – the actual methods of

mathematics – not the Quinean replacements, that have led to the

remarkable successes of modern mathematics. Where Quine holds that

science is ‘not answerable to any supra-scientific tribunal, and not in need

of any justification beyond observation and the hypothetico-deductive

method’ (Quine, 1981a, p. 72), the mathematical naturalist adds that

mathematics is not answerable to any extra-mathematical tribunal and not

in need of any justification beyond proof and axiomatic method. Where

Quine takes science to be independent of first philosophy, my naturalist

takes mathematics to be independent of both first philosophy and natural

science (including the naturalized philosophy that is continuous with

science) – in short, from any external standard. (Maddy, 1997, p. 184)

Since mathematical practice answers to no ‘extra-mathematical’ tribunal, then issues such as

deciding on the continuum hypothesis or new axioms of set theory are made solely within the

set-theoretic community, which Maddy takes to be representative of the mathematical

community as a whole. This also means that the question of whether or not mathematical objects

exist is also only answerable within mathematics. Indispensability arguments have no say in

issues of ontology regarding mathematics. This, however, ends up being problematic. Science is

54 Maddy’s scientific naturalism is similar to Quine’s in that she believes that science is not answerable to any

external or ‘supra-scientific’ tribunal. However, her position differs in three important ways. First, we have already

seen that Maddy rejects confirmational holism. Second, she argues that her naturalism is a much more subtle

position compared to Quine’s. For example, Maddy considers her naturalism more of an approach, and that belief in

scientific results does not stem from “some general meta-thesis about the reliability of science... but the detailed

scientific evidence specific to each individual case.” (Maddy, 2002, p. 61) The last point of departure between

Maddy and Quine is in their treatment of mathematics. For more details see Maddy (2007)

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clear in telling us that many physical objects, such as tables and chairs, and even unobservables

like protons and quarks, exist. Science also tells us many properties of these objects – they exist

objectively and are located in space-time. But mathematics tells us almost nothing about

mathematical objects. At best, mathematics only tells us that a small number of specific

mathematical objects exist as stated in mathematical axioms. Yet mathematics tells us nothing

about what this existence entails; whether or not they are spatio-temporal, objective, etc. Maddy

notes that so little is specified that realist and anti-realist positions, such as fictionalism and

formalism, are all perfectly compatible with mathematical practice. (Maddy, 1997, p. 192) She

also claims that from a methodological perspective, questions about the existence of

mathematical objects have historically proven to be essentially useless for the development and

expansion of mathematics. In this light, questions regarding existence and ontology should be

dismissed as they turn out to be irrelevant to the development of successful mathematics, and to

regular mathematical practice.

In a more recent work, Maddy (2007) is careful to distinguish between the methodological and

the metaphysical question. Her original argument was that issues of existence and ontology play

no role in the advancement and practice of mathematics. However, Maddy does recognize that

there are still interesting and important metaphysical questions that remain, such as ‘does

mathematics have a subject matter?’, or ‘are mathematical claims true or false?’. Interestingly,

Maddy admits that the answers to these questions will not stem from mathematical practice, but

rather will be inspired by traditional philosophy of mathematics. She considers three possible

positions that purport to answers the metaphysical questions and assesses how well each fits into

her overall naturalistic approach to mathematics and science.

The first position Maddy calls Robust Realism. Robust Realism is actually one of any of a

collection of realist positions such as ontological platonism, epistemological platonism,

structuralism, etc. By my definition, a robust realist accepts all three of the realist theses and can

adopt further theses as he sees fit. Maddy is quick to reject Robust Realism as a good answer as

it conflicts with mathematical practice which is sacrosanct according to her mathematical

naturalism. Her evidence for this is the case of the continuum hypothesis and whether or not we

should accept V = L as a set theoretic axiom. Recall that the set theoretic community wishes to

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reject V = L as doing so would maximize the potentialility of set theory. What would the robust

realist say? Maddy alleges that the robust realist would need to appeal to the fact that V = L must

be objectively false, as there is an objective world of sets that we are trying to discover. But this

is not the type of reasoning that actually goes on in mathematical practice. Set theorists dislike

V = L on grounds of its limiting nature. The problem is, “granting that selecting for

maximization generates theories we like, what reason do we have to think it likely to generate

theories that are true?” (Maddy, 2007, p. 366) The robust realist is adding metaphysical

assertions on top of the methodological concerns of the mathematician. In essence, they are

adding some ‘extra-mathematical’ interpretation of mathematics that is just not supported by

practice, and thus should be rejected by the mathematical naturalist.

Thin Realism is the second position that Maddy considers. The thin realist takes the practice of

set theory at face value and accepts that the axioms and theorems of set theory, and hence the

sets generated by the axioms, are true and exist because set theory tells us so. But what about

questions regarding independence, abstractness, etc.? Set theory and mathematical practice say

nothing about this. Answers to these sorts of questions are typically found within science. The

thin realist looks towards science and finds that scientific practice does not ascribe spatio-

temporal or causal properties to things like sets (nor do they deny them either). Sets, then, “have

the properties ascribed to them by set theory and lack the properties set theory and natural

science ignore as irrelevant. There is nothing more to be said about them.” (Maddy, 2007, p.

369) Maddy contends that Thin Realism has the advantage over Robust Realism in that it does

not need to inject anything over and above mathematical practice, such as an appeal to objective

truth or reality. What we know about mathematical objects just comes from the practice of

mathematics and science. Issues such as whether or not the continuum hypothesis is true or false

will be strictly determined by whether or not one day we have well motivated mathematical

reasons for adopting one or the other. It has nothing to do with objective truth value of the

continuum hypothesis. Of course, Thin Realism will fail at giving satisfactory answers to many

of the metaphysical questions that we are interested in. It cannot tell us anything insightful into

the properties of mathematical objects, their nature, etc., beyond simply pointing to what set

theory tells us. This, as we know, amounts to basically nothing.

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Finally, Maddy considers Arealism. Arealism is just like anti-realism but they differ in their

reasons for adopting their positions. Maddy aims to distinguish between a priori arguments

against realism that are typically associated with anti-realism, and reasons that are motivated

from within a naturalistic framework and free from prior prejudice. Ultimately their beliefs are

the same, but it is how they arrive at these beliefs that are different. The Arealist does not take

the axioms and theorems of set theory to be true, and thus maintains that sets and all other

mathematical entities do not exist. Is this compatible with mathematical practice? Haven’t we

already acknowledged that certain axioms of mathematics assert that some mathematical objects

exist? The Arealist needs to somehow explain how they can take back the existential claim of

these axioms, and also provide an account of how it is that something that is false can be so

useful. Fortunately we have seen several options already such as fictionalism and the weaseling

argument. Maddy provides another:

set theoretic claims, including existence claims, generated by its most

effective methods should be adopted as appropriate means towards

theories that serve our goals, but natural science is the final arbiter of truth

and existence, and it confirms neither the truth of mathematics nor the

existence of sets. (Maddy, 2007, p. 384)

Natural science is where we should look, and thus we do not need to believe in the truth of the

axioms of set theory.

The final task is to compare the respective positions and see which fares best. Robust Realism is

already off the table as it conflicts with mathematical practice. But when adjudicating between

Thin Realism and Arealism, Maddy comes to a surprising conclusion.

If this is a fair description of the state of debate between the Thin Realist

and the Arealist, then it’s hard to see that there is any fact of the matter

here about which we can be right or wrong... [T]he decision between Thin

Realism and Arealism appears to hinge on matters of convenience, taste,

and preference in the bestowing of these honorific terms (true, exists,

science, knowledge). (Maddy, 2007, p. 389)

Thin Realism and Arealism are perfectly compatible with mathematical and scientific practice.

They are equally supported by our mathematical naturalism. In fact, the actual differences

between Thin Realism and Arealism are quite trivial. The thin realist observes that science says

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nothing about the causal or spatio-temporal nature of mathematical objects, so he excludes these

properties from mathematical objects. All he allows for is existence due to the axioms of set

theory. The Arealist also notes that science says nothing about the existence of mathematical

objects either, so she believes they do not exist. Asides from existence, they agree on every other

aspect in that there is simply nothing more to say about mathematical objects at all. What

choosing between the two positions boils down to is whether or not we think science should be

the sole arbiter of existence, in which case we are Arealists, or whether we think that

mathematics can also determine existence, in which case we are Thin Realists. The difference

between the two positions is minimal, and furthermore, Maddy claims that there is no objective

way to judge between them.

Although I am quite sympathetic towards Maddy’s goal of treating mathematics with greater

respect within a naturalistic framework, I find her analysis quite unsatisfactory. Maddy rules out

Robust Realism as it appeals to the objective truth or falsity of V = L as justification instead of

purely methodological concerns. However, this need not be the case. It is perfectly reasonable

for a robust realist in the face of an independent question to look towards mathematical practice

for guidance. This makes sense as it is this very practice that has been refined, improved, and so

successful over so many years in discovering objective mathematical truths. The robust realist

need not appeal to objective notions of truth prior to there being good mathematical reasons for

adopting or rejecting V = L. In fact, Gödel, a Robust Realist if there ever was one, weighed in on

V = L and ultimately sided against the axiom. His reasons seem to be the exact types of reasons

that Maddy says that we should use, and that Robust Realists do not. “[The] axiom [of

constructibility] states a minimum property. Note that only a maximum property would seem to

harmonize with the concept of set.”55 (Gödel, 1983a, pp. 478–479) Regardless, Maddy does not

consider other realist positions, such as plenitudinous platonism56, that would be happy to accept

55 Maddy (1988) suggests that it is possible that Gödel’s position here was actually motivated by his disbelief in the

continuum hypothesis, and that this disbelief stems entirely from an unacceptable appeal to mathematical intuition

and truth. Whether or not this is actually the case is debateable.

56 Plenitudinous platonism claims that all logically consistent mathematical theories are true and refer to real

mathematical objects. See Balaguer (1998) for details.

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multiple independent sets of axioms for set theory, so long as each set is consistent, or, the

situation could resemble Euclidian and non-Euclidian axioms of geometry where all are equally

valid. I see no reason to conclude that all Robust Realist positions conflict with mathematical

practice.

Another issue is that Thin Realism and Arealism seem to both violate the mathematical

naturalism that Maddy seeks to protect. The limited realism that Thin Realists adopt adds

properties of mathematical objects that are informed by science. If science suddenly says that

mathematical objects are causal, then the Thin Realist would have to agree. This suspiciously

resembles an ‘extra-mathematical’ tribunal for mathematical objects. Granted, it is not so

grievous in that such input from science regarding the properties of mathematical objects would

never contradict what mathematical practice says, as mathematical practice says next to nothing.

Still, given Maddy's efforts to treat mathematics as an independently respected practice, it is

peculiar that we look externally to what science says. The conclusion of Arealism is also

suspicious. The Arealist believes, like Quine, that science is the sole arbiter of existence. But the

Arealist notes that nothing in science implies the existence of mathematical objects (although it

does not imply non-existence either).57 Again, if science changes its tune in the future by

explicitly stating that mathematical objects exist, then the Arealist would be forced to change

their position. But isn’t this the very type of situation that we were trying to avoid in the first

place? The point of mathematical naturalism is that mathematics is understood through

mathematical practice alone. Appealing to science for discovering properties of mathematical

objects, or answering truth or existence claims is simply unnaturalistic.

Maddy wants to have her cake and eat it too. As a committed scientific naturalist, she

acknowledges that questions regarding existence and ontology are legitimate questions. At the

same time she is committed to the importance of respecting mathematical practice as much as we

respect scientific practice. In her analysis of mathematical practice she discovers that

57 Recall that Maddy already rejects the QIA as she rejects confirmational holism so we cannot say that

indispensability points to the existence of mathematical objects.

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mathematics says next to nothing regarding the existence of mathematical objects. It would stand

to reason then that that is the end of that. There is simply nothing to say about the existence of

mathematical objects. Here, Maddy’s commitment to ontological questions betrays her. She

insists that the big metaphysical questions regarding mathematics are still reasonable. However,

seeing as we know that mathematical practice cannot help us, our only recourse is to look

towards scientific practice. Her conclusion is that two positions fit with both mathematical and

scientific practice: Thin Realism and Arealism. There is no fact of the matter about which one is

true as both are equally supported. So, what it all comes down to in the end are our personal

preferences.58 Do we prefer the existence claims of mathematics to hold, or do we prefer letting

science rule the day? This is a shocking admission as nothing could seem further from the

naturalistic position that Maddy so wishes to defend. Recall that Maddy’s choice of the term

Arealism was meant to highlight that arguments for Arealism are not a priori but based on

mathematical and scientific practice. But in the end, what leads one to Arealism ends up being a

preference that has nothing to do with practice, and everything to do with a priori feelings! The

Thin Realist fares no better as ultimately the justification for their beliefs also stem from a priori

beliefs. This appears to be incredibly unnaturalistic.

Given that mathematical practice does not conclusively decide questions regarding existence and

ontology of mathematical objects, Maddy is left in a difficult dilemma. On the one hand she can

accept that mathematical practice says nothing and that there is nothing more to say. Ontological

questions are essentially meaningless within mathematics. The downside to this is that she must

abandon her naturalistic commitment that all metaphysical questions are meaningful and

important. Alternatively, Maddy can affirm her commitment to metaphysical questions, but this

leads to an un-naturalistic conclusion where the answers all depend on a priori personal beliefs

and external, ‘extra-mathematical’ tribunals.

58 One could cite this as another example of underdetermination where both Thin Realism and Arealism are equally

supported so there is no need to choose between them. This certainly is compatible with Maddy's naturalism, but

does it pay enough respect to her commitment to ontological inquiry? I believe that it does not.

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There is a third option that evades the horns of the dilemma. This third option has the added

advantages of respecting mathematical practice and remaining true to our naturalistic beliefs that

questions of ontology matter. A problem with the QIA is that it looks only at science and not

towards mathematical practice for guidance in answering metaphysical questions regarding

mathematical objects. The problem with Maddy’s mathematical naturalism is that the pendulum

has swung too far to the other extreme. In looking solely at mathematical practice there is no way

to answer questions of existence. Her solution is to take a cursory look at science to provide

answers, which I have argued is no solution at all. Surely there is value in looking at both

scientific and mathematical practice together in order to help us make sense of our mathematical

knowledge. Taking lessons from both disciplines and appreciating the important role that

mathematics plays in science, and that science plays in mathematics, opens up many avenues of

exploration in order to understand the nature of mathematics. Separating science and

mathematics and arguing that one or the other is the sole arbiter is a legacy from Quine’s hard

line naturalism, but we need not be forced into it now. The third option then is to free ourselves

from the confines of the naturalistic tendency to treat mathematics and science separately, and

also from any talk of ‘sole-arbiters’ with regards to ontological issues.

What I propose is that the best way to understand mathematical practice and ontological issues is

to treat mathematics and science together. In essence, naturalism should not cover only scientific

practice, nor should there be a separate naturalism for mathematical practice. Instead, we need to

consider both disciplines using the exact same standards and methods. The immediate benefits of

this joint approach are obvious. No longer would there be tension between respecting

mathematical versus scientific practice as this brand of naturalism would treat them both

together. There would not be any of the issues that Maddy faces where her mathematical

naturalism says next to nothing about ontological matters and she is forced to look externally for

answers. Beyond this, focus will be clearly placed on the integration and interplay of

mathematics and specific scientific domains such as physics. Such integration goes beyond mere

applicability, but can speak towards methodological approaches and interesting topics such as

reduction. Of course there are major obstacles that would need to be overcome for this project to

get off the ground. First, how can we treat mathematics and science together when, according to

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the standard view, mathematics has always been developed entirely independently from the

needs and demands of science? Secondly, what should we do regarding the abstract unapplied

areas of mathematical research? These, and many more issues are daunting, but it is my belief

that the Quinean form of naturalism is a relic from the past, and that removing this cleft between

mathematics and science represents a fruitful way forward in understanding our best

mathematical-scientific practices.

My suggestion that naturalism should refer to both the practice of science and mathematics

together betrays my metaphysical beliefs. My commitment to mathematical realism does not

stem solely from the indispensability of mathematics in science, as it does for Quine, Colyvan,

Bangu, and other indispensabilists. At the same time, my belief in mathematical objects does not

come solely from the practice of mathematics, as it does for epistemological platonists such as

Gödel. Instead, my belief in mathematical realism comes from both mathematical and scientific

practice together. Because of this I find traditional platonist arguments as well as the modern

indispensability arguments unsatisfactory as they each are instructing individuals to look at a

singular domain for the answers to ontological questions in mathematics. I feel that this is

misguided, and looking towards both mathematics and mathematical applications in science is

reasonable and makes for a stronger argument.

Looking towards both scientific and mathematical practice has been suggested before, although

it seems to have been forgotten. The QIA is closely associated with Putnam as he did much to

advance and defend the argument. In fact, he supported the argument so much that many call it

the Quine-Putnam Indispensability Argument. I have purposefully avoided this label as it

obfuscates the differences between Putnam and Quine’s take on the argument. One difference

already discussed is in the type of conclusion generated by the argument. Quine is vague and

non-committal, whereas Putnam advocates a form of semantic realism. Another important

difference that is almost entirely overlooked is the naturalistic component. Putnam has no

problem drawing on inspiration from both mathematical and scientific practice. In fact, for him

the QIA only has force once you do.

In my view, there are two supports for realism in the philosophy of

mathematics: mathematical experience and physical experience... If there

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is no interpretation under which most of mathematics is true, if we are

really just writing down strings of symbols at random, or even by trial and

error, what are the chances that our theory would be consistent, let alone

mathematically fertile? (Putnam, 1979, p. 73)

When considering mathematical practice, Putnam says that we cannot help but notice the

apparent consistency and fertility of the discipline. This leads him to believe that there must be

some interpretation for which mathematics is true. This inference is definitely not rock solid.

Putnam is making a type of no-miracles argument; it would be a miracle if mathematics was not

true under any interpretation given how our experience of mathematics seems to be consistent

and is certainly fruitful. We do not want to explain this via a miracle, so the only recourse is to

believe that there must be an interpretation for which mathematics is true as it is the best

explanation of our experience. As discussed in chapter 5, this type of IBE leaves much to be

desired, but for now let us grant Putnam’s conclusion.

At this point, Putnam turns towards the applications of mathematics to try to determine which

interpretation makes mathematics true.

The interpretation under which mathematics is true has to square with the

application of mathematics outside of mathematics... I argued in detail that

mathematics and physics are integrated in such a way that it is not possible

to be a realist with respect to physical theory and a nominalist with respect

to mathematical theory... Mathematical experience says that mathematics

is true under some interpretation; physical experience says that that

interpretation is a realistic one. (Putnam, 1979b, p. 74)

It is scientific practice that leads Putnam to conclude that the realist interpretation of

mathematics is the correct one. The realist interpretation is what makes mathematics true and

fruitful. The route to this conclusion is through an indispensability argument. Given that we are

scientific realists, and that mathematics is indispensable to the practice of science, then we must

also take a realist attitude towards mathematics as to do otherwise would be ‘intellectually

dishonest’. For Putnam, an analysis of both mathematical and scientific practice work together to

lead to mathematical realism.

Putnam’s indispensability argument can be formulated as follows.

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(P-P1) Mathematics must be true under some interpretation.

(P-P2) We ought to have ontological commitment to all those entities

that are indispensable to our best scientific theories.

(P-P3) Mathematical entities are indispensable to our best scientific

theories.

Therefore:

(P-C) The correct interpretation of mathematics is the realist

interpretation.

The justification for (P-P1) is a no-miracles argument which I will not critique here. (P-P3) is a

standard indispensability premise, but could easily be altered to specifically point to mathematics

being explanatorily indispensable. What makes Putnam’s argument interesting and unique is his

second premise. (P-P2) is a softer, more open version of Quinean naturalism that removes the

clause that only those entities that are indispensable to science be considered. Removing this

restriction is what allows us to look towards mathematical practice as an aid for understanding

mathematical knowledge and mathematical entities, and this allows us to make use of his first

premise which is a conclusion drawn from mathematics alone. It is this move of rejecting the

idea that science (or mathematics) is the sole arbiter of ontological questions in mathematics that

I find immediately appealing about Putnam’s version of the argument.

Certainly there are problems with this argument. It still critically depends on confirmational

holism which I have consistently rejected, the conclusion of (P-P1) is based on a suspect

inference, and the relationship between true interpretations and realism needs to be made clear in

a non-question-begging way. Regardless, there are several important points that are worth

highlighting from this examination of Putnam. The first point is a historical one: it is a costly

mistake to lump Putnam too closely with Quine. Surely Putnam agreed with Quine in several

important ways, but as we have seen their differences are significant. Secondly, mathematical

and scientific practice together can contribute towards leading us to a realist conclusion. Our

naturalism need not be so overly restrictive as to bar potentially fruitful avenues of exploration.

This point is made even more salient since the focus has shifted towards mathematical

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explanations in science. Finally, arguments for mathematical realism should look to draw from

as many areas of mathematics as possible in order to make a broad and convincing case. Moving

forward, the goal would be to learn from the lessons of Putnam and create a stronger argument

for mathematical realism in line with the joint naturalism suggested above.

2 Final Thoughts

The recent surge in interest for mathematical explanations was most certainly sparked by

indispensability arguments for mathematical realism. I hope that I have shown that while

indispensability arguments are interesting, the worth of studying them goes far beyond looking at

a simple argument for mathematical realism. Understanding the role that mathematics plays in

scientific explanations is an exciting and fruitful area of research. This understanding helps us

make better sense of the practice of science, scientific explanation, and inference to the best

explanation regardless of metaphysical worries. Finally, looking at mathematical explanation

leads to at least three areas of further research that could significantly advance the field. Focus

can now be placed on understanding how it is that mathematics explains physical facts, the key

explanatory role of mathematics can motivate the rejection of exclusive forms of naturalism, and

lastly we can draw on all this to craft a superior argument for mathematical realism that makes

use of both mathematical and scientific practice together.

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