1

Sorin Bangu

Wynn’s Experiments and the Later Wittgenstein’s Philosophy of

Mathematics [forthcoming in Iyyun. The Philosophical Quarterly of Jerusalem]

1. This essay explores the possible connections between two issues.1 The first has to do with

recent work in experimental psychology on the early development of arithmetical abilities. The

experiments I’ll be discussing have been designed and performed by the psychologist Karen

Wynn, and are part of a series of several somewhat similar experimental studies on infants (some

of the most notable include Starkey, Spelke, and Gelman 1983 and 1990). My focus here,

however, is on Wynn’s now famous 1992 paper (Wynn 1992), not only for reasons of space, but

also because she, more than other psychologists, doesn’t shy away from drawing explicit

philosophical conclusions from her findings.2

The second issue is a cryptic remark by Wittgenstein, namely, that arithmetical

propositions emerge by ‘hardening’ certain empirical regularities ‘into rules’. This idea has been

noticed and investigated by Mark Steiner in a series of papers over the last fifteen years or so

(see Steiner 1996, 2000, 2009); since he also gives credit for this reading to Robert Fogelin, from

now on I’ll be referring to it as the ‘Steiner / Fogelin interpretation’ (‘SF’ for short). What

follows is my own take on the SF reading, and thus it shouldn’t be assumed that these two

authors will agree with all I have to say here.

1 This paper is based on a talk I gave at the Van Leer Jerusalem Institute in December 2011, as part of the workshop

Mathematical Knowledge and Its Application organized in honor of Mark Steiner. I benefited from comments and

suggestions from the audience, especially from Yemima Ben-Menahem, Mark Steiner, and Saul Kripke. I also thank

Denise Cummins for a long conversation on this kind of experiments and Karen Wynn for clarifying some issues

over email. Alex Radulescu, Dirk Schlimm, and Sorin Costreie made insightful remarks on one of the last drafts. I

am of course entirely responsible for the final version of the paper. 2 The specialist literature discussing experiments of this ilk is huge and growing, and here I have neither the

intention nor the competence to offer a comprehensive presentation of all ideas advanced in these debates. I first

came across these experiments in Maddy (2007), which contains a very informative, philosophically oriented,

account of these controversies, with numerous references to the main contributions. De Cruz, Neth, and Schlimm

(2010, esp. sect. 2.3) also offers guidance through the more recent developments in this area.

2

The most important Wittgenstein passage for my purposes is in his Remarks on the

Foundations of Mathematics (RFM hereafter):

It is as if we had hardened the empirical proposition into a rule. And now we

have, not an hypothesis that gets tested by experience, but a paradigm with which

experience is compared and judged. And so a new kind of judgment. (VI-22)

As the context makes it clear, ‘empirical proposition’ refers to an empirical regularity. Also

important, in RFM (VII-1) Wittgenstein notes the role of the ‘physical and psychological facts’

in grounding the application of mathematical propositions.

In a nutshell, the connection I envisage is simply the following: Wynn’s empirical

discoveries — of certain regularities of behavior, or ‘psychological facts’ — fit surprisingly well

into (the SF version of) Wittgenstein’s account of arithmetic, and thus increase its credibility.

I begin with a brief presentation of the experiments, sprinkled with some comments on

their methodology and conclusions. Then, after I sketch the SF interpretation, I elaborate on what

I take to be the relation between these experimental results and the SF reading.

2. This is how Wynn summarizes her work:

Studies recently conducted in my infant cognition laboratory suggest that young

human infants can calculate the results of both additions and subtractions (Wynn

1992). These studies tested 5-month-old infants’ knowledge that 2 is composed of

1 and 1. (1992a, 321; emphasis added)

According to Wynn then, the results of these experiments are nothing short of revolutionary:

they simply refute empiricism as a philosophical account of mathematical knowledge.3

3 Wynn’s (1992a) paper is suggestively titled “Evidence against Empiricist Accounts of the Origins of Numerical

Knowledge.” See also Wynn (1992b). Yet a more stubborn empiricist might reply that the experiments only show

that at 5 months human beings have acquired more arithmetical abilities than we would otherwise expect — and not

that these abilities are truly innate. But note, in fairness to Wynn, the problem with such a retort: it would render the

3

Here are more details. A group of thirty-two full-term, normal infants (mean age 5

months and 1 day) was divided into two smaller equal groups, called the ‘1+1’ and the ‘2 — 1’

group. Every infant in the ‘1+1’ group was shown a single object (a small doll) being brought

into an empty display area (a depiction of the following descriptions is in Wynn 1992, 749).

Then a small screen rotated up, occluding the doll from the infant’s sight. Next, the experimenter

brought another, identical doll into the display area, such that the infant could see this operation

being performed: a hand placed the doll behind the screen, out of the infant’s view. Thus, the

infants could clearly see the succession of operations, but not their result. Then the infant was

shown the following displays: first, a display containing two items, and next, a display

containing only one item. The variable of interest the researchers measured was the looking time,

i.e., how long the infants have looked at each display.

As it turned out, the infants’ looking times at the two situations was, on average,

different. They looked longer at (what we would take to be) the odd outcome: the situation in

which the screen comes down and only one item is revealed. The mean values reported in

Wynn’s 1992 paper are 13.36 seconds mean looking time for the ‘incorrect’ display and 12.80

seconds for the ‘correct’ one.4 This procedure has been repeated several times for every child.

Wynn’s central methodological commitment here is the reliability of the looking time

procedure, which is part of the widely accepted violation of expectation paradigm.5 The basic

assumption backing the violation of expectation idea is that infants’ behavior, just like that of the

adults, gives indications about what they take to be an unexpected event. In the infant case, the

behavior is of course non-verbal and consists in fixing the gaze for a longer time on a certain

event — which thus is taken to be ‘unexpected’, or ‘surprising’. It should be added that the

experiments also involved pre-testing preparation, during which infants were habituated, or

familiarized, with the test events. Upon performing the experiments, infants’ looking reliably

longer at the ‘incorrect’ display was taken to constitute evidence that the infants did not expect

empiricist doctrine practically irrefutable, since that would require doing experiments on babies as soon as they are

delivered. (I owe these points to Alex Radulescu.) 4 The experiment was replicated with fewer infants and the mean looking times were 12.08 seconds at the ‘incorrect’

display v. 9.45 seconds at the ‘correct’ one. Another similar experiment was performed, but the infants were shown

3 v. 2 objects. The mean values were 11.89s v. 9.96s, respectively. 5 Yet see Munakata (2000) for some skepticism over it.

4

this event to occur (while not looking longer at the ‘correct’ display shows that this event was

expected.)

Similar results have been obtained for the infants in the ‘2 — 1’ group. They were

presented with a sequence of events depicting (what we would call) a subtraction of one item

from two items (also shown graphically in Wynn 1992, 749). The looking time at the display was

also recorded and, again, the infants’ looking times at the two situations was, on average,

different. They looked significantly longer at the ‘wrong’ display.6

Thus, the two groups showed significantly different looking patterns. Infants in the ‘1+1’

group looked longer when the result was 1 than when it was 2, while infants in the ‘2 — 1’ group

showed the reverse pattern. (A pre-test showed that infants in the two groups did not differ in

their baseline looking patterns to 1 and 2 items.) Summarizing, (i) infants in the two groups

demonstrated different patterns of looking in the test trials, but not in the pre-test trials, and (ii)

infants in the ‘1+1’ group looked longer when the addition appeared to result in a single item

than when it resulted in 2 items, while infants in the ‘2 — 1’ group looked longer when the

subtraction appeared to result in 2 items than when it resulted in a single item.

3. What is my view on these results? I believe that they are robust, and that they constitute

genuine discoveries of important empirical correlations, or behavioral regularities, which are

clearly not learned. Although not everyone is convinced that such regularities actually hold,7

many early childhood psychologists working on this topic do accept them. Reassuringly, the

experiments have been replicated many times since 1992.8

6 The mean looking times for the first run of the experiment were 13.73s (for the ‘2–1= 2’ case) v. 10.54s (for the

‘2–1=1’). The replication of this experiment also yielded 10.98s v. 8.05s. The initial 1992 paper gets these latter

numbers in the wrong order, but Wynn corrected the mistake in a subsequent errata list; see Nature 361 (Jan. 1993):

374. 7 For more recent dissenting views on Wynn’s findings, see Huntley-Fener et al. (2002), Cohen and Marks (2002),

Carey (2002), Mix (2002). Wynn (2002) responds to some of these criticisms. 8 As Carey (2002, n. 1) notes: “Wynn’s experiment has been replicated and extended in many independent

laboratories — with 4-month-old infants by Cohen and Marks, this volume; Simon, Hespos, and Rochat (1995);

Koechlin, Dehaene, and Mehler (1998); with older infants, Chiang and Wynn (2000), Huntley-Fenner, Carey, and

Solimando (under review), Uller et al. (1999), Feigenson, Carey, and Spelke (in press) [...] In spite of the failure to

5

Despite agreeing with Wynn that the findings are genuine, and not an artifact of the

experimental set up, I am not persuaded by the philosophical interpretation attached to them. I

have serious reservations to conclude that they corroborate the hypothesis which she takes them

to corroborate, namely, that “infants possess true numerical concepts” (1992a, 749), that they

“possess a genuine system of numerical knowledge” (1995, 35), or that infants “compute” what

1+1 is (2002, 207). More generally, I think it is definitely premature to claim the demise of

empiricism as a philosophy (of mathematics). There are just too many ways in which the

empiricists could defend their position — and thus it is somewhat surprising that Wynn’s

(philosophical) conclusions haven’t received more attention in the philosophy of mathematics

literature (perhaps simply because they aren’t sufficiently known, or weren’t taken seriously.9)

For one thing, Wynn writes quite casually about what philosophers take to be a notoriously

difficult matter — the ‘possession of a concept’, or of ‘knowledge’ — especially of the abstract

kind; for another, the expectation talk seems very problematic as well. In particular, I doubt that

there is any testable method to identify, unambiguously, what the infants’ expectations actually

are, since various and incompatible alternatives are easily conceivable. On the face of things, this

seems rather an interesting case of under-determination: the behavioral data collected in these

experiments support a variety of hypotheses of the type ‘infants look longer at a certain display

because X’, where X can be many things — and not only ‘the outcome consistent with the

correct arithmetical result is not obtained’.10 However, I will not articulate these concerns here.

These reservations notwithstanding, I believe that these experimental results still are of

philosophical relevance — it’s just that their import is not what Wynn takes it to be, namely, that

humans posses innate arithmetical knowledge (and, more generally, that empiricism is refuted).

As I suggested above, and will elaborate subsequently, my proposal is that their significance can

replicate by Wakeley, Rivera, and Langer (2000), I do not take the replicability of these findings to be in doubt.”

Wynn (2000) responds to Wakeley et al. (2000), who in turn reply to her in Wakeley et al. (2000a). 9 Exceptions are Giaquinto (1992) and Schwartz (1995). More recently, the above-mentioned Maddy (2007),

DeCruz, Neth, and Schlimm (2010), as well as Leng, Paseau, and Potter (2007), although the essay on these

experiments included in this collection is co-authored by a psychologist (Capelletti) and a philosopher (Giardino). 10 See also DeCruz, Neth, and Schlimm (2010, 70–71): “A methodological problem with the looking time procedure

is that one cannot be sure what causes the longer looking times.”

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be appreciated by approaching them from a rather unexpected angle: (some of) the later

Wittgenstein’s views on the nature of arithmetical propositions.

So, in what follows, I will set aside Wynn’s interpretations of these findings. In order to

see how Wittgenstein’s (SF) account fits in, I will focus primarily on what goes on in these

experiments at a very basic level of description. What I will retain from the reported findings is

what seem to be uncontroversial, minimally theoretically-loaded facts: what is demonstrated is

the existence of robust, innate patterns of behavior, specifically, that the infants single out certain

arrangements of concrete objects. Certain outcomes of the manipulations of concrete objects —

precisely the ones that we, arithmetically astute adults, take to be relevant — prompt, on a

regular basis, discernible reactions in the infant subjects. This will be the empirical regularities of

interest in what follows. To see their philosophical significance we need to outline the SF

interpretation.

4. As is well known, it is notoriously difficult to even sketch an appropriate characterization of

the later Wittgenstein’s philosophy of mathematics.11 This is so not only because he might not

have had one in the traditional sense (recently, Potter [2012, 136] describes it as “inconclusive”

and “fragmentary”),12 but also because the very frame of reference is missing: most of his

remarks are directed against virtually all academic schools. Fogelin (1995, 211), for instance,

characterizes his position via a double negative: “anti-Platonism without conventionalism.”

11 Portions of the following sections are borrowed from my entry in Internet Encyclopedia of Philosophy on the

“Later Wittgenstein’s Philosophy of Mathematics” (http://www.iep.utm.edu/wittmath/) 12 Here I shall leave aside two important issues: First, the fact that no materials conveying his later views on

mathematics have ever been given green light for publication by Wittgenstein himself. Second, the debate over the

division of Wittgenstein’s thinking (yet see section 7 below). When it comes to mathematics, the standard

demarcation (two Wittgensteins — the ‘early’ one of the Tractatus, and the ‘later’ one of the PI and RFM) is

questioned by Gerrard (1991, 1996), who distinguishes two lines of thought within the post-Tractarian period: a

middle one, or “the calculus conception,” to be found in Philosophical Grammar (PG), and a truly later one, “the

language-game conception.” Stern (1991), however, worries about too finely dividing Wittgenstein’s thinking,

concerned with the more recent tendency to add even the fourth, post-PI, period, which Wittgenstein devoted to

philosophical psychology and is illustrated in his Remarks on the Philosophy of Psychology (RPP).

7

I will not attempt to propose a new label here; yet, it seems to me, the best starting point

to discuss his conception is its relation to conventionalism.13 How exactly do his views relate to

this doctrine is, once again, hard to pin down; what is relatively clearer, I submit, is that his

conventionalism is not the ‘full-blooded’ conventionalism that Dummett (1959) attributed to

him.

To begin with, for Wittgenstein, arithmetical identities (such as ‘3 x 3 = 9’) are not

descriptive propositions, but a certain kind of rules. And, importantly, these rules are not

whimsical inventions; in a sense (to be explicated later on), the rules in place are the only ones

that could have been adopted, or, as Steiner (2009, 12) put it “the only rules available.” The

typical (anti-)conventionalist worry — that they ultimately have an arbitrary character — is

answered when it is added that the rules are grounded in objective, publicly ascertainable

empirical regularities (Fogelin 1995; Steiner 1996, 2000, 2009); or, as Wittgenstein says, the

empirical regularities are “hardened” into rules (RFM VI-22).

To unpack this idea, we can rely on Fogelin’s point that arithmetical propositions provide

us with “rules for (establishing) the identity of descriptions” (Fogelin 1995, 214). More

concretely, consider the multiplication 3 x 3. Suppose we describe it to a child by the following

arrangement:

(a)

■ ■ ■

■ ■ ■

■ ■ ■

Next, suppose we produce another arrangement:

(b)

■ ■ ■ ■ ■ ■ ■ ■ ■

and tell her that this arrangements describes the same operation. So — we hope — the child is

convinced that taken together, three batches of three objects amount to nine objects. The identity

13 For a recent comprehensive investigation of conventionalism, see Ben-Menahem (2006).

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‘3 x 3 = 9’ is the rule that these descriptions, and others like them, are correct. One might think

of these arrangements as ‘proving’ to the child that 3 x 3 = 9.

But suppose that a few days later the child no longer remembers all the details of this

story and describes the ‘4 x 4’ multiplication as follows:

(c)

■ ■ ■ ■ 1.

■ ■ ■ ■ 2.

■ ■ ■ ■ 3.

■

4.

She counts the squares, and reports the result: 4 x 4 = 13. We protest, and the child gets

confused. Her puzzlement14 originates in her belief that she generated arrangement (c) by doing

the same thing we did initially. To her, arrangement (c) is like (a) (or (b)): she considered four

batches of four objects and then counted them!

This simple example signals a more serious problem. We recognize here one version of

the well-known rule-following ‘paradoxes’ in PI highlighted by Kripke (1982) (and Fogelin

(1976)): “any course of action can be made out to accord with the rule”, and thus “no course of

action could be determined by a rule” (PI 201). One just can’t draw a list containing everything

that is, and isn’t, allowed in the manipulation of our little black squares, thus clearing up all

possible confusions. (Obviously, explicitly ruling out the superposition of squares wouldn’t help

too much, as one might misunderstand what a superposition is; and so on.) Yet, when dealing

with this situation in the presence of normal people (or, better: educated adults), the (a)-type and

(b)-type arrangements predominate, while those of the (c)-type are very rare — and even more so

after training.

14

For more real examples of this, see sections 2.1, 3.1, and 4.1 in Schlimm and Lengnink (2010).

9

I attribute to Wittgenstein the view that it is just a brute fact of (human) nature that we are

able to avoid confusions like the one just presented, especially after the teachers intervene and

signal the mistake we made. More generally, it is a brute fact of human nature that we are able to

follow rules and orders. It is a given that there is a point when we stop ‘interpreting’ them — and

just act. Thus, crucial here is our ability to act “blindly,” as Wittgenstein puts it (PI §219; italics

in original). But we have to be careful how we understand this point. We don’t act blindly in the

sense that we don’t reflect upon what we are doing; this would be false, since we often proceed

quite carefully when following an order (Fogelin 1995, 159). The sense of Wittgenstein’s remark

is that when we understood an order, or a rule, this is typically shown by our blindness to

“distractions” (Stern 2004, 155), to all that is irrelevant, and thus we avoid all possible

confusions, such as the one illustrated by arrangement (c).

5. This brings us back to the arithmetical issue, and prepares the stage for a key-point in

understanding Wittgenstein: if we were not able to act blindly (in the sense just explained), if the

confusions were dominant, then the arithmetical practice would not exist at all. Thus, this

particular form of rigidity, or blindness, is constitutive to arithmetic.

Typically, by training, a child becomes able to recognize that despite some similarities,

arrangement (c) is not the result of doing ‘the same thing’ we did with (a) (Fogelin 1995, 215).

The recognition process is gradual: some children ‘get it’ faster than others, and there is no

guarantee that all the children will eventually reach this stage in understanding. So, one (in fact:

the academic skeptic) might ask, what about those who don’t get it at all, and are stubbornly

deviant in claiming that arrangement (c) is exactly what they were supposed to construct when

finding out what 4 x 4 is? The answer Wittgenstein would give, I believe, is: not much can be

done about them, and they end up as victims of social exclusion. The immersion in the

community of ‘mathematicians’, the continual checking and correcting, the social pressure

(through low grades!), helps the child eventually master the technique of distinguishing what’s

allowed from what is not, to differentiate what must be ‘turned a blind eye on’ from what really

matters.

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Thus, the arithmetical training consists in inculcating in children a certain technique to

deploy when presented with arithmetical questions of the kind discussed here. They understand

multiplication when they are able to behave in a certain way: to establish, and recognize what is

(or counts) as ‘the same’ from what is (does) not.

6. At this stage, two aspects should be distinguished. The first is descriptive. In terms of what

people actually do when asked to follow an order (‘calculate m times n!’), the ‘normal’ situations

(arrangements of (a)-type and (b)-type) prevail, while the deviant ones (arrangements of the (c)-

type) are the exception. This is an empirical regularity: (with training) most people get the results

right most of the time.

The second aspect is normative. This empirical regularity becomes a candidate to be

codified into a special form. By doing this, we thus grant it a new status, that of a norm, or rule

— an arbiter of ‘right’ and ‘wrong’. We say that it is incorrect to claim that 4 x 4 = 13, or to take

an arrangement like (c) to stand for the 4 x 4 calculation. By doing this, the empirical regularity,

the uniform application of the technique, is thus ‘hardened’ into a rule.15 When discussing

multiplication in Lectures on the Foundations of Mathematics (LFM X, p. 95), Wittgenstein says

the following:

I say to (the trainee), “You know what you’ve done so far. Now do the same sort

of thing for these two numbers.”—I assume he does what we usually do. This is

an experiment—and one which we may later adopt as a calculation.

What does that mean? Well, suppose that 90 per cent do it all one way. I say,

“This is now going to be the right result.” The experiment was to show what the

most natural way is—which way most of them go. Now everybody is taught to do

it—and now there is a right and wrong. Before there was not.

15

Note that when the empirical regularities become less robust (for instance, when we operate with very large

numbers) the hardening is done by way of calculation (or, more generally, proof). This is an important aspect of the

regularity-hardening idea, but elaborating on it would take me too far afield. See Steiner (2000, esp. sect. 4).

11

Yet a complication occurs at this point. This emphasis on conceiving arithmetic as essentially

presupposing human practices should not be taken as an indication that Wittgenstein endorses a

form of subjectivism, of the sort right-is-what-I-(my-community)-take-to-be-right. So, on one

hand there is a sense in which Wittgenstein actually agrees with the line taken by G. F. Hardy,

Frege, and other Platonists insisting on the objectivity and necessity of mathematics. What he

opposes is not objectivity per se, but the ‘philosophical’ (pseudo-)explanation of it; to wit, the

idea that mathematics is transcendent, that an extra-layer of ‘mathematical reality’ is what

ultimately sanctions arithmetical identities.

The account he proposes is an alternative to this radical subjectivism, but an element of

human agreement can still be discerned in it. One might say that arithmetical identities emerge,

qua arithmetical identities, from a change in attitude — of the whole community — toward the

role of certain contingent, yet extremely robust, behavioral regularities: they are instituted as

rules. However, although the arithmetical propositions owe their origin and relevance to the

existence of such regularities, as arithmetical propositions they are no longer taken to be

expressing mere regularities — that is, they are no longer descriptive propositions open to

falsification. Qua arithmetical identities, they belong to a different order, as they became norms,

or rules.

To indicate the change of status, Wittgenstein uses a couple of suggestive metaphors. The

first is the road-building process:

It is like finding the best place to build a road across the moors. We may first send

people across, and see which is the most natural way for them to go, and then

build the road that way. (LFM, p. 95)

Here the road-building process is a hardening process (almost literally!). We begin in the

descriptive realm: as a matter of fact, we note that several paths across the moors have been

taken over time, and that one of them was most often travelled. It is this path that gradually

emerges as the most suited for crossing, and the one that the lasting road will follow — if a

decision to build a road is made. The passage into the normative realm is completed once the

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road has been built. To stress, the very building of the road is a human, communal, genuine

decision — it might well be decided otherwise, that no road shall be built. But, once the decision

to build is made, where to build it is, in a non-Platonic sense, pre-determined: it will be built

following the ‘most natural’ path. Anyone trying to cross the moors will take the solidified road

now, and thus will follow its path. Note, importantly, that, by hypothesis, people crossing the

moors would have followed this path anyway, assuming they are not ‘deviant’ travelers — hence

the absence of disagreement on what path to take. (And this is why the absence of disagreement

over the result of a calculation is considered by Wittgenstein a fact of major relevance; see RFM

VI-21, VI-39, etc.)

The second metaphor is legalistic. To turn an empirical regularity into a mathematical

(arithmetical) proposition is to “deposit [it] in the archives” (LFM, p. 104). As is the case with

the archives, what they do contain today has previously been in circulation in the past, had a

genuine life as a descriptive proposition, open to verification and refutation. On the other hand,

what ends up in the archives is protected, not in circulation anymore — that is, it is no longer

open to change and dispute. The relations between the archived items are frozen, solidified,

hardened. Note the normative function of the archives: it is illegal to tamper with them.

A related metaphor is that of the physical process of condensation. Just as an amount of

vapor enters a qualitatively different form of aggregation turning into a drop of liquid water, so

the empirical-behavioral regularities are turned into arithmetical rules. What happens is that

these regularities are ‘condensed’, or, to use Wittgenstein’s own word, “hardened”, into rules,

and thus acquire a new status (indicated by archiving):

Because they all agree in what they do, we lay it down as a rule, and put it in the

archives. Not until we do that have we got to mathematics. One of the main

reasons for adopting this as a standard, is that it’s the natural way to do it, the

natural way to go — for all these people. (LFM, p. 107)

The fundamental contingent regularity in this context is behavioral agreement. This type of

agreement consists in (almost) all of us having, roughly, the same natural reactions when

presented with the same ‘mathematical’ situations and contexts: arranging, sorting, recognizing

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shapes, performing one-to-one correspondences, i.e., counting, and so on. (The relevance of this

type of agreement is more general though; at PI 206 and 207 Wittgenstein suggests that these

regularities form the basis of language itself.)

7. To stress a point I made earlier, speaking in terms of behavioral agreement when it comes to

understanding the arithmetical enterprise should not lead us to believe that Wittgenstein is in the

business of undermining the objectivity of mathematics. This claim is worth dwelling upon, as it

is Dummett’s (1959) early influential line of interpretation, summarized in describing

Wittgenstein as a “full-blooded conventionalist” — or even “anarchist” (this is Dummett’s

(1978b, 64) famous label, when comparing Wittgenstein, “in his later phase,” with the

“Bolsheviks” Brouwer and Weyl.) According to Dummett, Wittgenstein maintains that at any

step in a calculation we could go any way we want, and the only reason that we go the way we

usually go is the agreement between us, as the members of a community: in essence, a

convention we all accept. (And which, since it might be changed, is entirely contingent.

Dummett (1978b, 67) writes: “What makes a […mathematical] answer correct is that we are able

to agree in acknowledging it as correct.”)

Thus, on this view, one can say that a mathematical identity is true by convention; i.e., it

is taken, accepted as true by all calculators because a convention binds them — nothing more,

and nothing less. However, textual evidence can be adduced against this reading. Wittgenstein

does not regard the agreement among the members of the community’s opinions on

mathematical propositions as establishing their truth-value. Convincing passages illustrating this

point can be easily found in his later works. Here is one in LFM, but there are others:

“Mathematical truth isn’t established by their all agreeing that it’s true” (p. 107); also, “it has

often been put in the form of an assertion that the truths of logic are determined by a consensus

of opinions. Is this what I am saying? No” (p. 107).

Therefore, it is simply not the case that Wittgenstein believes that the truth-value of a

mathematical identity is established by a convention adopted by (even large) groups of people.

Being a precondition of the existence of mathematical practice, this specific kind of non-

linguistic, behavioral agreement (in action) is constitutive of the practice; it must already be in

place before we can speak of ‘mathematics’. The regularities of behavior (eventually ‘hardened’)

must already hold. So, we don’t ‘go on’ in calculations and make up rules as we wish, just as we

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don’t build the road wherever we fancy it: it is the existent regularities of behavior (the preferred

path across the moors) that constrain (or, better: ‘constrain’) us. (As an exegetical aside, it looks

as if Wittgenstein only gradually came to realize this. According to Steiner (2009), remarks

about the major role of empirical regularities are absent in PG and PR; so, he proposes that

understanding the importance of the empirical regularities for mathematics amounts, for

Wittgenstein, to a “silent revolution,” in addition to the well-known “overt revolution,” i.e., the

repudiation of the Tractatus.)

Thus, not much is left of the ‘full-blooded conventionalist’ interpretation. We don’t really

have much freedom. Steiner (2009, 12) explains: the rules obtained from these regularities

are the only rules available. (The only degree of freedom is to avoid laying down

these rules, not to adopt alternative rules. It is only in this sense that the

mathematician is an inventor, not a discoverer.)

We can now see more clearly how this view contrasts with Dummett’s: mathematical identities

are not true by convention, since they themselves are conventions, or rules, elevated to this new

status (‘archived’) from their initial condition of empirical regularities (Steiner (1996, 196).

Note, moreover, that while the behavioral agreement constitutes the background for the

arithmetical practice, Wittgenstein separates it from the content of this practice. His view is that

the latter, i.e., the relations between the already ‘archived’ items, are governed by necessity, not

contingency; the existence of the former, however, is entirely contingent. This distinction

corresponds, roughly, to the one drawn in LFM, p. 241: “We must distinguish between a

necessity in the system and a necessity of the whole system.” The ‘whole system’ of arithmetic is

necessary in a social sense,16 since it promotes social cohesion (for one thing, it simplifies the

exchange of goods), although we can of course imagine a form of life with no use for (our)

arithmetic in this exchange (as some of Wittgenstein’s funny scenarios in RFM demonstrate).

This is a weaker form of necessity because, again, to lay down the rules, extracted from

16 If I understood him correctly, Martin Kusch proposed the notion of ‘social necessity’, as different from logical,

metaphysical, and physical necessity, in a talk on Wittgenstein (and Einstein) on measurement, given at Cambridge

University in 2010.

15

regularities, is a genuine decision, made mostly for its social utility. On the other hand, the

necessity ‘in the system’ corresponds to what Wittgenstein calls in RFM the ‘hardness of the

logical must’ (VI-49): it is no longer an empirical matter, but the expression of a convention, or

rule, that certain expressions — such as ‘four times four is thirteen’ — are excluded from use,

which simply means that in normal circumstances one is just paralyzed upon trying to act in

accordance with this ‘identity’.

8. At this point we can return to the main theme here, and ask how this account connects with the

innate behavioral regularities discovered by Wynn.

Let’s first remark that one interesting aspect of the SF reading just presented is that the

nature of the empirical behavioral regularities relevant for Wittgenstein is left open. Clearly, the

type of regularity this reading emphasizes is the one established by training. As we recall, the

key-regularity Wittgenstein draws on was that ‘similar people receiving similar training will

react in similar ways’. Arithmetic, mathematics in general, is not possible without this kind of

behavioral agreement. But note that not much is said about the role (if any) of the regularities of

behavior which are not acquired by training — but are innate. This is not meant as a criticism of

the SF reading, since Wittgenstein himself talks about abilities acquired by training, and not so

much, or not explicitly, about innate abilities. And, in fact, this point — the absence of explicit

textual reference to the innate aspects of human psychology — imposes a limitation on the

attempt to establish a connection between his regularity-hardening idea and the Wynn-style

discoveries.

While this might pose a problem for the project here, more encouraging is the

observation that there are several occurrences of the word ‘natural’ in important passages in

LFM (recall the already quoted passages on p. 95 and p. 107, where Wittgenstein talks about

“the natural way” to do things.) It would surely be a stretch to read ‘natural’ as meaning ‘innate’,

but the proposal I develop here doesn’t need to push things that far. All we need to accept is a

much weaker point, namely, that such a reading isn’t ruled out, and that the empirical regularities

relevant for his account of arithmetic might include innate ones. ‘Which ones, specifically?’ is a

question for empirical psychology to answer, and, as it happens, it did, recently, some of them

being the experiments presented above.

16

No doubt, the innate Wynn-type behavioral regularities are different from the regularities

achieved by training. The former are primitive and pre-reflective; all infants do the same thing —

look (the same way — longer) at the same display — not because of social pressure and

immersion in common practices, that is, not because they share in a ‘form of life’. (Rather, one

might remark here, what makes possible, later on, for people to participate in such a form of life

is that they all share innate behavioral dispositions.) There are other differences as well.

Applying a technique as a result of training is, as explained, acting blindly — but it is still acting,

in the sense that reflectivity is not suspended. Yet the infants in these experiments don’t quite

act. They only re-act, uniformly. This type of regularity is most deeply entrenched in the domain

of pure biology, pure instinct.

So, can this form of innate behavioral regularity play any role in Wittgenstein’s account?

As I said at the outset, I’m inclined to answer affirmatively, and the first step to see how it can

play such a role is to return to Wittgenstein’s discussion of what kind of agreement is needed for

the very existence of arithmetical practice.

9. In LFM, Wittgenstein distinguishes between agreement in verbal, discursive behavior and

agreement in action. Thus, on the one hand he talks about agreement in the “opinions” of the

members of a community, or the regularity that, for instance, if asked to voice their views, they

would all say roughly the same thing. On the other, he emphasizes that what is truly relevant, and

deeper, is agreement in “what [people] do” (LFM, p. 107), or regularity “of action”: in certain

mathematically related situations we all do the same thing. RFM, VI-30C reads: “The agreement

of people in calculation is not an agreement in opinions or convictions.” Moreover, LFM (pp.

183–84) contains passages like this:

There is no opinion at all; it is not a question of opinion. They are determined by a

consensus of action: a consensus of doing the same thing, reacting in the same

way. There is a consensus but it is not a consensus of opinion. We all act the same

way, walk the same way, count the same way.

The key idea that arithmetical propositions supervene on the natural, regular behavior of

calculators appears here once again. So now the stage is set to state the proposal more explicitly:

17

Wittgenstein’s emphasis on the constitutive role of the regularities of behavior, and on the

natural-instinctive agreement in (re)actions, are indications that he would not have been

surprised by findings such as Wynn’s — had he known about them. He would have been

interested in the discovery of these primitive, natural, innate regularities of behavior, and would

have regarded them as substantiating his invocation of certain ‘psychological facts’ (RFM VII-1)

as grounding the acquisition of mathematical abilities.

10. Several final clarifications of the relation between these facts and Wittgenstein’s (SF)

account are in order.

To begin with, it is important to emphasize that my claim is not that Wittgenstein’s

account of arithmetic ultimately rests on psychological facts like the ones documented in these

recent experiments. As we saw, in his conception, the fundamental (psychological) facts are the

regularities inculcated in the calculators by training in applying a technique. And there is no

tension in saying that although the modes of behavior Wynn (and others) have discovered are

innate, they are not fundamental: the truly fundamental notion for (the SF reading of) the later

Wittgenstein remains that of rule-following — and infants’ behavior can’t be described as an

illustration of rule-following. Infants do display a form of regular behavior, but it is an open

question whether exhibiting this behavior in infancy is necessary (or sufficient) to be able to be

successfully trained in arithmetical matters later on. In other words: even if it turns out that such

innate regularities don’t exist, as long as the regularities inculcated by training are robust (and

they are, as we can easily witness), Wittgenstein’s account still stands. But, if they do exist (and I

believe they do), this prompts the scientific-empirical question-proposal to which I hinted at

above, namely, that the two types of regularities are related in a deeper way: do the taught-and-

learned regular arithmetical abilities on which Wittgenstein’s account relies somehow build on

innate dispositions (or even neural hard-wiring)?17 Is the developed ability to act blindly (in

Wittgenstein’s sense) grounded in, or in any way related to, the natural-innate, ‘brute’ blindness

exhibited by infants in the experiments? I take it to be an interesting research hypothesis that

those infants who don’t behave ‘normally’ in the experiments (don’t look longer at the

corresponding displays) might be among those who, later on, will be deviant arithmetically — as 17 As Wittgenstein once remarked, certain animals are harder to train (to fetch, for instance) than others.

18

children or adults, they will be those hard to train in mathematics, i.e., will have difficulties to

turn a (Wittgensteinean) ‘blind eye’ on all irrelevancies when operating arithmetically.

Although not fundamental (in the sense just explicated), the discovery of these innate

regularities should be regarded as welcome when assessed from a broader Wittgensteinean

perspective. They fit quite neatly within the (SF) regularity-hardening account of arithmetic —

and thus increase one’s confidence that it might actually be on the right track. However, note,

finally, my reluctance to make a stronger claim, namely, that these data confirm, or corroborate

Wittgenstein’s account. Saying this might easily be misread as implying that I understand

Wittgenstein’s (SF) account to constitute some kind of (scientific) theory — say, about the innate

nature of mathematical knowledge — and this despite his well-known insistence to the contrary

(PI 124, 126, etc.) But then, how should we understand his regularity-hardening idea? What is

the status of this account, anyway? What is its purpose?

The answer to this query has to be formulated having in mind the later Wittgenstein’s

more general philosophy-as-therapy project. We have to recall that for him philosophizing is not

an occupation that reaches its goal when a doctrine or theory has been built and defended, but

rather an endless activity whose purpose is the liberation of one’s mind from the grip of

‘illegitimate questions’.18 Thus, I submit, for Wittgenstein an account of arithmetic, such as the

regularity-hardening one, is a “speculation — something prior even to the formulation of an

hypothesis” (LC, p. 44; emphasis in original), where the notion of ‘speculation’ should be traced

back to his understanding of the Freudian method: “Take Freud’s view that anxiety is always a

repetition in some way of the anxiety we felt at birth. He does not establish this by reference to

evidence — for he could not do so. But this is an idea which has a marked attraction. […] When

people do accept or adopt this, then certain things seem much clearer and easier for them” (LC,

p. 43; emphasis added).19

18 Such an idea goes back to Heinrich Hertz’s insight in Principles of Mechanics, one of Wittgenstein’s formative

readings: “the question [...] will not have been answered; but our minds, no longer vexed, will cease to ask

illegitimate questions” (Hertz 1899/2003, 8). The question Hertz discussed was about ‘the ultimate nature of force’. 19 Maddy (1997, 164) discusses Wittgenstein’s relation with Freudian methods in connection with his later

philosophy of mathematics. Hacker (2008), however, objecting to what the late G. Baker (2004) has argued, is rather

skeptical of the proposal that this relation holds the key to illuminating the later Wittgenstein’s thinking.

19

In a similar fashion, the regularity-hardening account is not meant as a (scientific, i.e.,

psychological, or sociological) theory, or as a collection of confirmable claims, but as an attempt

to open up a possibility (which just hasn’t occurred to us). Once the ‘speculation’ advanced by

the account is entertained, this comes with the benefit of “making it easier for (those tormented

by philosophical questions about the nature of mathematics) to go certain ways: it makes certain

ways of behaving and thinking natural to them” (LC, pp. 44–45).20

University of Bergen

Bibliography

Baker, G. P. 2004. Wittgenstein’s Methods — Neglected Aspects. Oxford: Blackwell. Ben-Menahem, Y. 2006. Conventionalism. New York: Cambridge University Press. Cappelletti M., and V. Giardino. 2007. “The Cognitive Basis of Mathematical Knowledge,” 74–83 in M. Leng, A. Passeau, M. Potter, eds. 2007. Carey, S. 2002. “Evidence for Numerical Abilities in Young Infants: A Fatal Flaw?” Developmental Science 5, 2:202–205. Chiang, W.C., and K. Wynn. 2000. “Infants’ Representations and Teaching of Objects: Implications from Collections.” Cognition 77:169–195.

Cohen, L. B., and K. S. Marks. 2002. “How Infants Process Addition and Subtraction Events.” Developmental Science 5, 2:186–201.

DeCruz, H., H. Neth, and D. Schlimm. 2010. “The Cognitive Basis of Arithmetic,” 39–86 in B. Löwe & T. Müller, eds., Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. London: College Publications.

Dummett, M. 1959. “Wittgenstein’s Philosophy of Mathematics.” The Philosophical Review 68: 324–348. Dummett, M. 1978a. “Platonism,” 202–15 in his Truth and Other Enigmas. Cambridge, MA: Harvard University Press.

20 When it comes to the questions asked at the end of the previous paragraph (about the very point of this account),

Steiner doesn’t really address them, and Fogelin will probably answer by indicating what he takes to be

Wittgenstein’s project: “not a criticism of the results of philosophizing, but an interrogation of its source” (1994,

110).

20

Dummett, M. 1978b. “Reckonings: Wittgenstein on Mathematics.” Encounter 50, 3:63–68. Feigenson, L., S. Carey, and E. Spelke. 2002. “Infants’ Discrimination of Number vs. Continuous Extent.” Cognitive Psychology 44:33–66. Fogelin, R. J. 1995. (1st ed. 1976.) Wittgenstein. 2nd ed. New York: Routledge & Kegan Paul. Gerrard, S. 1991. “Wittgenstein’s Philosophies of Mathematics.” Synthese 87:125–142. Gerrard, S. 1996. “A Philosophy of Mathematics between Two Camps,” 171–197 in Sluga and Stern, eds., 1996. Giaquinto, M. 1992. “Infant Arithmetic: Wynn’s Hypothesis Should Not Be Dismissed.” Mind & Language 7, 4:364–366.

Goodman, N. 1955. Fact, Fiction, & Forecast. Cambridge, MA: Harvard University Press. Hacker, P. M. S. 2008. “Gordon Baker’s Late Interpretation of Wittgenstein’s,” in G. Kahane, E. Kanterian, and O. Kuusela (eds.), Wittgenstein and His Interpreters: Essays in Memory of Gordon Baker. Oxford: Blackwell. Hertz, H. 1899 / 2003. The Principles of Mechanics, ed. Robert S. Cohen, trans. D. E. Jones, and J. T. Walley. New York: reprint of Dover Phoenix editions, 1956. Huntley-Fenner, G., S. Carey, and A. Solimando. 2002. “Objects are Individuals but Stuff Doesn’t Count: Perceived Rigidity and Cohesiveness Influence Infants’ Representations of Small Groups of Discrete Entities,” Cognition 85:203–221. Koechlin, E., S. Dehaene, and J. Mehler. 1998. “Numerical Transformations in Five-month-old Infants.” Mathematical Cognition 3:89–104. Kripke, S. A. 1982. Wittgenstein on Rules and Private Language. Cambridge, Mass.: Harvard University Press. Leng, M., A. Passeau, and M. Potter, eds. 2007. Mathematical Knowledge. Cambridge: Cambridge University Press. Maddy, P. 1997. Naturalism in Mathematics. Oxford: Clarendon. Maddy, P. 2007. Second Philosophy. New York: Oxford University Press. Morton, A., and S. Stich. 1996. Benacerraf and His Critics. Oxford: Blackwell. Mix, K. 2002. “Trying to Build on Shifting Sand: Commentary on Cohen and Marks.” Developmental Science 5, 2:205–206.

21

Munakata, Y. 2000. “Challenges to the Violation-of-Expectation Paradigm: Throwing the Conceptual Baby Out With the Perceptual Processing Bathwater?” Infancy 1, 4: 471–477. Potter, M. 2011. “Wittgenstein on Mathematics,” 122–137 in M. McGinn, and O. Kuusela, eds., The Oxford Handbook of Wittgenstein. Oxford: Oxford University Press. Schlimm, D., and K. Lengnink. 2010. “Learning and Understanding Numeral Systems: Semantic Aspects of Number Representations from an Educational Perspective,” 235–264 in B. Löwe and T. Müller, eds., Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. London: College Publications. Schwartz, R. 1995. “Is Mathematical Competence Innate?” Philosophy of Science 62, 2:227–240. Shapiro, S., ed. 2005. The Oxford Handbook of Philosophy of Logic and Mathematics. Oxford: Oxford University Press. Simon, T., S. Hespos, and P. Rochat. 1995. “Do Infants Understand Simple Arithmetic? A Replication of Wynn (1992).” Cognitive Development 10:253–269. Sluga, H., and D. Stern, eds. 1996. The Cambridge Companion to Wittgenstein. Cambridge: Cambridge University Press. Spelke, E. S. 1976. “Infants’ Intermodal Perception of Events.” Cognitive Psychology 8:533–560. Spelke, E. S. 1987. “The Development of Intermodal Perception,” in P. Salapatek and L. B. Cohen, eds., Handbook of Infant Perception. New York: Academic Press. Starkey, P., E. Spelke, and R. Gelman. 1983. “Detection of Intermodal Numerical Correspondence by Human Infants.” Science 222:179–181. Starkey, P., E. Spelke, and R. Gelman. 1990. “Numerical Abstraction by Human Infants.” Cognition 36:97–127. Steiner, M. 1996. “Wittgenstein: Mathematics, Regularities, Rules,” 190–212 in Benacerraf and His Critics, ed. A. Morton and S. Stich. Oxford: Blackwell. Steiner, M. 2000. “Mathematical Intuition and Physical Intuition in Wittgenstein’s Later Philosophy.” Synthese 125:333–340. Steiner, M. 2009. “Empirical Regularities in Wittgenstein’s Philosophy of Mathematics.” Philosophia Mathematica 17, 1:1–34.

22

Stern, D. 1991. “The ‘Middle Wittgenstein’: From Logical Atomism to Practical Holism.” Synthese 87:203–226. Stern, D. 2004. Wittgenstein’s Philosophical Investigations: An Introduction. Cambridge: Cambridge University Press. Uller, C., G. Huntley-Fenner, S. Carey, and L. Klatt. 1999. “What Representations might Underlie Infant Numerical Knowledge?” Cognitive Development 14:1–36.

Wakeley, A., S. Rivera, and J. Langer. 2000. “Can Young Infants Add and Subtract?” Child Development 71, 6:1525–1534.

Wakeley, A., S. Rivera, and J. Langer. 2000a. “Not Proved: Reply to Wynn.” Child Development 71, 6:1537–1539.

Wittgenstein, L. 1922. Tractatus Logico-Philosophicus. Translated by D.F. Pears and B.F. McGuinness. London: Routledge and Kegan Paul, 1961. Wittgenstein, L. 1953. (PI) Philosophical Investigations. The German text, with a revised English translation by G. E. M. Anscombe. 3rd ed. Oxford: Blackwell, 2001. Wittgenstein, L. 1956. (RFM) Remarks on the Foundations of Mathematics. Edited by G. H. von Wright, R. Rhees, and G. E. M. Anscombe, translated by G. E. M Anscombe. Cambridge: MIT Press, 1991. Wittgenstein, L. 1938. (LC) Lectures and Conversations on Aesthetics, Psychology and Religious Belief. Edited by C. Barrett. Oxford: Blackwell, 1967. Wittgenstein, L. 1974. (PG) Philosophical Grammar. Edited by Rush Rhees, translated by Anthony Kenny. Oxford: Basil Blackwell.

Wittgenstein, L. 1975. (PR) Philosophical Remarks. Edited by Rush Rhees, translated by Raymond Hargreaves and Roger White. Oxford: Basil Blackwell.

Wittgenstein, L. 1976. (LFM) Wittgenstein’s Lectures on the Foundations of Mathematics. Edited by C. Diamond. Ithaca, N.Y.: Cornell University Press. Wrigley, M. 1977. “Wittgenstein’s Philosophy of Mathematics.” Philosophical Quarterly 27, 106:50–59. Wynn, K. 1992. “Addition and Subtraction by Human Infants.” Nature 358:749–750. Wynn, K. 1992a. “Evidence against Empiricist Accounts of the Origins of Numerical Knowledge.” Mind and Language 7:315–332.

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Wynn, K. 1992b. “Issues Concerning a Nativist Theory of Numerical Knowledge.” Mind and Language 7:367–381. Wynn, K. 1995. “Origins of Numerical Knowledge.” Mathematical Cognition 1, 1:35–60. Wynn, K. 2000. “Findings of Addition and Subtraction in Infants Are Robust and Consistent: Reply to Wakeley, Rivera, and Langer.” Child Development 71, 6:1535–1536.

Wynn, K. 2002. “Do Infants Have Numerical Expectations or Just Perceptual Preferences?” Developmental Science 5, 2:207–209.