are ordinary objects abstracta? - st andrews
TRANSCRIPT
Are Ordinary Objects Abstracta?. . ⁰
Introduction
. Metaphysics and Compositional Structure
Metaphysicians care about what things exist and how they are structured. One key way things
are structured is by the composition relation. Composition is tied up in particularly foundational
debates regarding both the existence of composites and their parthood structure. van Inwagen [?]
put these issues at the forefront of the metaphysical debate by asking two questions.
General Composition Question What is the nature of composition?
Special Composition Question When do composite objects exist?
Regarding the latter, he writes:
“Suppose one had certain material objects, the Xs, at one’s disposal; what would one
have to do — what could one do — to get the Xs to compose something?” (van
Inwagen [?], p. )
Not all answers to the special question are answers to the general question; indeed most of the
discussion has focused merely on giving necessary and sufficient conditions for composition to
occur, rather than on the nature of the relation.¹
e two extreme answers to the special composition question have been surprisingly popular.
ey are: mereological universalism and mereological nihilism.
⁰is paper is a very rough draft. Please forgive the typos and undefined references, etc. Also please don’t cite orshare without checking with me.
¹ere are exceptions. Perhaps two of the most prominent answers are Lewis [?] — who thought the nature ofcomposition was analogous to the identity relation in various ways — and Fine [?] — who thinks there are manycomposition relations, which all are building operations such that their wholes contain their parts.
Universalism ere is a concrete whole made up of the Xs whenever the are some Xs.
Nihilism ere is a concrete whole made up of the Xs whenever the Xs contain only a single
thing.
e universalist picture is such that the conditions under which composition occurs are trivially
satisfied.² Whenever there are objects at all, there are composites. One of the immediate objec-
tions to universalism is that it is ontologically profligate. And we have often been told that bloated
ontologies are to be avoided. Another problem is the simple commonsense incredulity about the
existence of gerrymandered composite objects. Are there really trout-turkeys? Is there a ‘fusion’
composed of my nose and the Eiffel Tower? In the face of such costs, the universalist picture has
proven particularly resilient.
It’s main rival is nihilism, the view that everything is mereologically simple having no parts.
e conditions under which composition occurs are never (non-trivially) satisfied, according to
the nihilist. e composition relation just is the standard identity relation. Nihilism has in its
favour a parsimonious ontology. ere are only simples, there are no genuine composites at all.
Of course, with this parsimonious ontology comes the violation of the commonsensical view that
there are tables and chairs and humans and the like. Moreover, there is the difficulty of saying
just why the common sense ontology seems ‘correct’ even if the nihilist thinks it to be strictly
speaking false. So nihilists are faced with the task of attempting to accommodate the semantics
of ordinary language claims by paraphrasing them into a strict philosophical language. Various
attempts have been made, of course, but many still face problems.³
²Modulo questions about when pluralities exist. For the purposes of this paper, we will simply be presupposingplural comprehension.
³e most popular paraphrase is van Inwagen’s [?] arrangement strategy according to which sentences like ‘e ballbroke the window’ are to be paraphrased as something like ‘Particles arranged ball-wise scattered particles arrangedwindow-wise’ (cf. Merricks [?]). Irreducibly plural quantification can help paraphrase sentences like ‘e chair isheavier than the table’ is paraphrased as ‘ere are xx and yy such that the xx are arranged chair-wise, the yy arearranged table-wise, and the xx are heavier than the yy.’ (as in Hossack [?]). Uzquiano [?] argues, however, thisstrategy faces problems with simple cardinality comparisons: ‘e chairs outnumber the tables’ cannot be reduced to acardinality comparison of the xx arranged chair-wise to the yy arranged table-wise. After all there may collectively beless chair-particles than table particles.
ere are many other answers to the special composition question, of course. e main aim
of this paper is to explore a neglected alternative to the two extreme answers. As we will see, the
view is equally extreme but it seems to have all the advantages of universalism and nihilism while
avoiding many of the major defects of either.
. An Analogy
Before setting out to explore the view in detail, let us pause to consider an analogy to help motivate
the view.
Consider another debate central to contemporary metaphysics: the debate over possible worlds.
ere has been an attempt to answer what we might call the ‘Leibnizian Equivalence Question’.
Leibnizian Equivalence Question Under what conditions is “It’s possible that φ” true?
Here are two fairly extreme answers:
Modal Realism “It’s possible that φ” is true whenever there is a concrete possible world in which
“φ” is true.
Necessitism “It’s possible that φ” is true whenever “φ” is actually true.
e concrete modal realist faces many of the same problems that the universalist faces. ere is the
ontological bloat of countenancing concrete possible worlds as existents. ere is the common-
sense incredulity of the existence of such strange entities. ere are strange and gerrymandered
possibilities: worlds in which uncountably many duplicates of George Washington are dancing in
a conga line. But like universalism, the expressive power of modal realism has led some (thankfully
not many!) to embrace the costs.
Similarly, the actualist necessitist faces similar problems to the nihilist’s. While it counte-
nances only the actual existents of our world, it faces problems with the semantics of ordinary
modal discourse. Paraphrasing such discourse is difficult without any merely possible entities to
appeal to.
Of course, there is a well-known third way forward: ersatzism. One can accept the existence of
worlds, but reject that worlds are concrete; worlds may be thought of as abstracta (e.g. sets). is
view has no ontological bloat, at least so long as we already accept abstracta. It provides a smooth
semantics and easy paraphrase of modal discourse with individual worlds serving as proxies for
the realist’s concrete worlds. Likewise, it can avoid the incredulous stare.
e analogy should be obvious: an ersatzist view about composite objects is clearly an available
position in conceptual space. e view accepts the existence of wholes, but rejects that composite
objects are concrete. One implementation of the view would be straightforward: ordinary objects
are sets and composition is set-theoretic relation. e view could avoid the ontological bloat of
universalism, so long as we already accept abstracta. In fact, versions of the view might have
an ontology as minimal as the nihilists. It would also be ideologically parsimonious — there is
no need for any additional fundamental relation of composition or parthood. Any incredulity
regarding gerrymandered fusions of universalism would now be diffused: gerrymandered sets are
not so strange. Moreover, it would allow for a smoother semantics for ordinary assertions than
nihilism; paraphrasing becomes significantly easier once individual objects — namely sets — can
serve as proxies for composites.
Despite having many of the benefits and few costs, the view has been almost completely
neglected and virtually unexplored in the recent literature. e view is not without its historical
roots: Carnap [?] attempted to give a translation of scientific sentences regarding ordinary objects
into a language containing just terms referring to elementary experiences (erleb), a similarity
relation (Er), and the set-membership relation (∈). Carnap qua logicist thought of membership as
a logical relation, distinguishing him sharply from the other main precursor to the view: Quine [?,
?]. Quine thought that the language of ordinary objects was intertranslatable with the language
of places (spatio-temporal locations), and places were simply sets of points. Hence, the parthood
relation could be reduced to the set-theoretic subset relation.
e goal of this paper is to explore and develop a workable version of ordinary object er-
satzism. I sketch a number of ways of developing ersatzism about ordinary objects (§§–), each
an improvement of the last. Over the course of these developments, I argue that ersatzism can
accommodate any part-whole structure, and any properties of wholes. It is worth considering
what the background meta-ontological commitments of such a view might be, and I will suggest
that the picture fits well within a broadly neo-Fregean conception of ontology (§). I close by
considering some important objections (§).
Classical Ersatzism
Universalism has found its strongest support from classical extensional mereology (CEM), the dom-
inant formal theory of parts and wholes. CEM has an axiom (or axiom schema) of ‘Unrestricted
Fusion’ which states that for any (non-empty) plurality, the xs, there exists a whole composed
of the xs. According to classical mereologists, the world has the structure of a complete Boolean
algebra (Tarski [?]).⁴ Many classical mereologists also accept atomism, the thesis that every ob-
ject is composed of simple parts. On this view, the world has the structure of a complete atomic
Boolean algebra.
Classical Ersatzism I
Our first attempt at a basic ersatzist view of objects is simply an attempt to mimic atomistic
models of CEM with sets. ankfully, there is a standard algebraic result that allow us to easily
represent models of CEM via sets.
CABA Representation Every complete atomic Boolean algebra is isomorphic to a powerset Boolean
algebra.
⁴Here and below I will ignore the issues regarding ‘empty objects’, or bottom elements of complete Boolean algebras.Standardly, mereologists reject the existence of such objects (see Simons [?, p. ]). Some have argued in its favor (e.g.[?], [?], [?], [?]). Others (e.g. Forrest [?]) countenance a bottom element for algebraic simplicity.
is result tells us that every atomistic model of CEM is structurally identical to powerset of its
atoms.
e classical ersatzist who accepts atomism has a very simple theory. We start by taking
the entities that a mereological nihilist accepts — a fundamental level — as our atoms, A. We
then treat any possible subset of A as an object. e set of all such sets is our model with the
subset relation corresponding to the parthood relation, and set union operation corresponding to
composition. is structure, the powerset Boolean algebra of A, is isomorphic to the models of
the atomist who accepts CEM.
is is all well and good for atomists. However, some CEM is compatible with gunk —
objects whose parts all have proper parts. Such structures cannot be handled by this approach.
And insofar as we think it is metaphysically possible that the world might be gunky, we should
be able to accommodate such structures in an ersatzist way.
. Classical Ersatzism II
As in the case of classical atomistic mereology, there are other more general results that show that
atomless structures have set-theoretic representations too.
Stone Representation Every Boolean algebra is isomorphic to a field of sets i.e. a subalgebra of
a powerset Boolean algebra (Stone [?]).
It follows, then, that for every way CEM says the world might be (whether atomic or not), there
will be a set-theoretic representation of it. e actual representation theorem however is less meta-
physically perspicuous than CABA representation. Before, it was clear how the ersatzist constructs
objects from subsets of atoms. In this case, it is much less clear what the underlying metaphysics
of the representing sets are. Which subalgebras of powerset Boolean algebras represent gunky
worlds? How should we think of the ‘atoms’ of the carrier powerset in that case?
Previously we treated our fundamental entities as the simples of the nihilist, and built our
set-theoretic objects out of them. But insofar allowing for gunk requires us to avoid commitment
to mereological simples, we’ll need an alternative fundamental level of entities to build our set-
theoretic objects out of. e way forward is to think of the fundamental level as consisting of
spacetime points with a specific distribution of properties.⁵
How do we get from propertied spacetime to objects?
A partition P of propertied spacetime is such that: (i) P covers all of spacetime (⋃P ) = A);
and (ii) the members of P are pairwise disjoint (pi ∩ pj = ∅ for any pi and pj in P s.t. i ≠ j).
One constraint it might seem natural for to adopt might be plenitude:
Plentiude Every partition is a metaphysically acceptable carving of the world into objects
Plenitude also guarantees that there is a finest carving — the one that treats each spacetime point
as a simple. Since the set of all singletons of points in spacetime is a partition, then Plenitude will
commit us to atomism yet again. Similarly, since every subset of P is a member of some partition
on P , then the mereological structure is isomorphic again to the full powerset Boolean algebra.
e key to utilising Stone Representation to get models of gunk is to claim that some but not
all partitions are metaphysically acceptable carvings. But which partitions are the good ones? We
need an answer to the question of which regions of spacetime are possible locations for ordinary
objects. Some answers will give us gunk. To see an example, first we endow the set P of spacetime
points with a topology — a classification of sets as open and closed regions. A topological space is
any set A together with a set of its subsets O ⊆ ℘(A) satisfying the following conditions:
. ∅ and A are both in O
. If O ⊆ O, then ⋃O is in O.
. If O1, . . . ,On are in O, then O1 ∩ . . . ∩On is in O.
e members of the collection O are the open sets. e complement of an open set is closed.
We’ll call the set of all closed sets C. Given a topological space, we can use notions of the interior
⁵ese properties should be thought of as fundamental properties, perhaps those given to us by fundamental physics.How these properties relate to properties ordinary objects will be a question that detains us at some length in §.
(written ‘i(X)’), and closure (written ‘c(X)’) of a region.⁶ A region X is open iff it is identical
with its interior, i(X) = X and a region is closed iff it is identical with its closure, c(X) = X .
Furthermore, we say that an open set X is regular open iff i(c(X)) =X ; similarly, a closed set X
is regular closed iff c(i(X)) =X .⁷
Once we have a topological structure on space, we can restrict our Plenitude principle to only
certain sorts of regions. If we allow only regular open regions as possible locations of objects,
we can recapture ‘gunk’ (indeed, these structures mimic Tarski’s [?] original models for gunky
geometry).⁸ at’s because every regular open region has another regular open as a proper subset.
We’ll need to tweak the definitions of partitions a bit.⁹
Partition* A set P of non-empty sets is a partition* of A if: (i) c(⋃P ) = A; and (ii) the members
of P are regular open sets such that pi ∩ pj = ∅ for any pi and pj in P s.t. i ≠ j.
We can then answer the special composition question in a fairly straightforward way:
Composition* ‘Xs compose y’ is true iff c1 and c2 are partitions* on A, and i(c(⋃Xc1)) =
i(c(yc2)).
We vindicate the possibility of gunk, since for every object x in some partition ci, there will be
some partition cj in which there is a y such that y under cj is a proper subset of x under ci. For
example, imagine a sphere with radius without its surface. Now, there are uncountably many
more surfaceless spheres with smaller radii contained in it. Is the centre point of the sphere in the
domain? No. ere are no point-sized regions, since any such region will be closed and so not an
admissible location.¹⁰
⁶e interior of a set X is the largest open set contained in X : i(X) = ⋃{O ∈ O ∶ O ⊆X}. e closure of a set Xis the smallest closed set containing X : c(X) = ⋂{C ∈ C ∶X ⊆ C}.
⁷Not all open sets are regular open, e.g. the complement of a point x in Euclidean three-dimensional space withthe standard topology; likewise not all closed sets are regular closed, e.g. the surface of a sphere in the same space.
⁸Cartwright [?], Uzquiano [?].⁹ese definitions are taken from Cotnoir [?].
¹⁰It is worth highlighting that this particular example required an additional stipulation that objects can only belocated in regular open regions. is stipulation rules out atomistic objects. Are there variants of this view that wouldallow for gunky and atomistic objects? Can there be hybrid models? Given the Stone Representation theorem, it wouldappear so. But whether there will be any principled metaphysical reasons for such models, it is hard to say.
In this case, our ersatzist view is that composite objects are regular open sets of propertied
spacetime points, which parthood as the subset relation. Our second version of classical ersatzism
works well, as far as it goes. However, it still lacks some of the flexibility one might want. Ordinary
object ersatzism as a philosophical view is neutral with respect to which mereology is correct.
But all the structures admitted here are really very ‘classical’. Non-standard or ‘non-classical’
approaches to mereology are not accommodated.
ere are also a handful of problems regarding properties of ordinary objects. e first is
an issue about inheritance: how are the properties had by spacetime points be inherited by the
objects that occupy them? We are not told what properties objects have, or how the properties of
points contribute to the properties of the objects themselves. is seems like a serious barrier to
any kind of semantics for ordinary language.
An additional problem is analogous to a problem for modal ersatzists: the problem of alien
properties. Let an alien property be any property had by an ordinary object that is not had by any
of its parts. Can there be any such properties? It is hard to see how. On classical ersatzism I,
the only propertied objects are atoms. On classical ersatzism II, the only propertied objects are
points. Even if we had an answer to the question about how properties are inherited, we would
still face the further question.
So in the interest of moving toward a fully general account of erstazism, we will attempt to
tackle these problems in the next section.
Non-Classical Ersatzism
Our final version of the ersatzist view requires that we reimagine and reconceptualize our re-
sources once again. Non-classical ersatzism is still going to be the view that composite objects are
set-theoretic entities and composition and parthood are set-theoretic relations. However, these
structures will need to be much more complicated.
. Mereology Reconcieved
We can think of mereological structures as frames. Abstractly, a frame is a pair ⟨D,<⟩, where D
is a non-empty domain of points, and < an accessibility relation. You should think of D as our
domain of objects, and < is our (proper) parthood relation. Axioms for parthood are going to be
given by placing constraints on accessibility. Frames are very flexible, and the hope is that it will
provide enough flexibility to allow for virtually any mereological structure.
But we aren’t merely going to be looking at frames. To solve the inheritance and aliens prob-
lems, we are going to need properties in the mix. Let us call a structure propertied if it is tells
us for any object, whether each one-place predicate is true of that object or not. at is, we will
include an interpretation function ν, a map from objects in D to properties p, q, r, etc. Proper-
tied structures correspond to interpreted frames ⟨D,<, ν⟩, i.e. Kripke models. (I restrict focus to
monadic properties as a simplifying assumption. Generalising to n-ary relations is a project for a
expert ersatzists to explore.)
Kripke models are familiar structures from the semantics for modal logics. In our case, we are
thinking of points not as possible worlds but as objects, and accessibility as the parthood relation.
What do modal operators do in such a set-up? I rely on two main primitive modal operators: F
and P .
“some fusion of d is p” νd(Fp) = 1 iff there’s some d∗ s.t. d < d∗ and νd∗(p) = 1.
“some proper part of d is p” νd(Pp) = 1 iff there’s some d∗ s.t. d∗ < d and νd∗(p) = 1.
We can define two other modal operators as follows: Gp ∶= ¬F¬p (read: “every fusion containing
d is p”) and Hp ∶= ¬P¬p (read: “every proper part of d is p”).
We have yet to implement any constraints on our accessibility relation. ere are all sorts
of various axioms we might wish to impose, depending on which mereology we are aiming to
model. Here are some of the more salient conditions:
Structure Condition Axiom
G(p→ q)→ (Gp→ Gq)
p→ GPp
Asymmetry ∀x∀y(x < y → y /< x) none
Transitivity ∀x∀y((x < y ∧ y < z)→ x < z) Gp→ GGp
Universal Object ∃x∀y(y ≤ x) G� ∨ FG�
Empty Object ∃x∀y(x ≤ y) H� ∨ PH�
Gunk ∀x∃y(y < x) Hp→ Pp
Junk ∀x∃y(x < y) Gp→ Fp
Wellfoundedness ∀X∃x(x ≺X ∧ ∀y(y < x→ y /≺X)) H(Hp→ p)→Hp
Upper Bounds ∀x∀y∃z(x ≤ z ∧ y ≤ z) FGp→ GFp
Lower Bounds ∀x∀y∃z(z ≤ x ∧ z ≤ x) PHp→HPp
Least Upper Bounds ∀X∃z(z ≤X ∧ ∀y(y ≤X → z ≤ y) ??Once we have selected a class of constraints on the accessibility relations (i.e. selected a partic-
ular mereological theory) and given our intended models an interpretation (said of all the objects
in the model whether they satisfy any given property or not), we are now in something like the
position of the ordinary theorist who accepts composite objects into their ontology.
e advanced ersatzist needs to be able to recover this expressive power without the additional
ontology. We will do that by reconstructing interpreted mereologies via sets.
. An Ersatzist Reconstruction
To reconstruct these structures in a ersatzist-friendly way, we can use a technique developed by
Meyer [?] to mimic a full propertied structure from a selection of points in the model. Here is
the recipe in brief.
. Start with a propertied mereological model (an interpreted frame).
. Pick any ‘fundamental’ object @; let a be the set of all properties true of @.
. Notice that a is maximally consistent: every property is either true of @ or false of it
(including modal properties).
. us, a specifies what is going on at all other points. We can extract this information using
two relations ⊳ and ⊲.
d ⊲ d′ iff ⌜Pφ⌝ ∈ d′ for all φ ∈ d
d ⊳ d′ iff ⌜Fφ⌝ ∈ d′ for all φ ∈ d
. We can show the following:
⌜Pφ⌝ ∈ a iff there is a maximally consistent d, s.t. d ⊲ a and φ ∈ d.
⌜Fφ⌝ ∈ a iff there is a maximally consistent d′, s.t. d′ ⊳ a and φ ∈ d′
. Generate a new set-theoretic domain Ia which is the minimal closure of a under the ⊲ and
⊳ relations.
. We now set a model to be M = ⟨Ia,a,⊲,⊳, ∈⟩ and prove the following theorem:
As a result of this process, we can prove the following theorem:
eorem A sentence is a theorem of a class of interpreted mereological frames iff it is true in all
models generated from choices of @.¹¹
Notice that for virtually all composition structures we wish to accommodate, there is a set-
theoretic way of building it from a fundamental level. We simply build these structural features
into the frame conditions; alternatively we stipulate certain ‘modal’ axioms be true. Given this
flexibility regarding frame conditions, this goes a long way toward solving the generality problem.
Secondly, we can now see how to express when wholes inherit properties of their parts. Fol-
lowing Goodman [?, pp. –], we can introduce the following terminology:
¹¹See Meyer [?], Proposition ..
Dissective a property p is dissective (inherited downward) iff for all d, νd(Hp) = 1 .
Expansive a property p is expansive (inherited upward) iff for all d, νd(Gp) = 1.
And these modal facts will be encoded in the set-theoretic models generated from our various
choices of fundamental objects.
Finally, we can also see how we’ve allowed for alien properties. We can think of modal sen-
tences like Fp as telling us the conditions of an object when it is part of a whole that has some
property p. (Note: p might not be had by any part of the whole.) ese conditions may involve
arrangements/positions of particles, or having certain causal powers, etc. We need to be given
such a story for each macro-level property of wholes, and that is no trivial task.¹²
e Metaontological Picture
ere are a number of philosophical worries for the view that ordinary objects are abstracta. One
worry is conceptual: on what possible conception of abstracta would ordinary objects count?
Ordinary objects are paradigm cases of concreta. If we are no longer allowed that, then it is not
clear we even understand the distinction.
So, providing a coherent conception of abstract/concrete distinction according to which or-
dinary objects might plausibly fall on the abstract side of that distinction is a necessary precursor
for the view to have any traction at all. But in order for the ersatzist approach to have any of the
advantages over its rivals mentioned in §, we actually do need to be committed to their existence.
is section is an attempt to address both of these pressing issues in a unified way. It should be
¹²Compare Sider [?, p. ]
“‘hands” or “books” may be built out of physical properties in our world, they could be built out of “alien”properties in some other world (in a way that perhaps water couldn’t be). Definitions of these whichtake into account such possibilities will be fairly abstract (Sider a, pp.–). For example, thenatural properties might be set-theoretic constructions (what goes into the sets will depend on the make-up of the world) playing an “abstractly characterized” role in a network of laws (themselves abstractlyspecified). Even so, if such a story can be given, there is no barrier to sets being the bearers of such(alien) properties.
said that much of this discussion is purely optional. ose few who might be tempted by the
ersatzist thought are not committed to the metaontological picture outlined here. In fact, the
metaontological story fits very nicely with classical ersatzists, who wish to recover the models of
CEM. In other cases, perhaps other approaches would do better. However, this approach is fairly
natural and is worth considering as a potential companion to erstazism about ordinary objects.
. Neo-Fregean Ontology
Neo-Fregeanism arose as a view in the philosophy of mathematics. e view attempts to use
abstraction principles to provide a foundation for the epistemology of mathematics. Abstraction
principles generally have the following structure:
Abstraction ∀X∀Y [f(X) = f(Y )↔X ∼ Y ]
Here f is some function mapping the Xs and Y s to individuals, and ∼ is an equivalence relation.
In the case of arithmetic, the abstraction principle is Hume’s Principle.
Hume’s Principle ∀F∀G[#xFx =#xGx↔ Eq(F,G)].
Hume’s Principle says that the number of F s equals the number of Gs iff F and G are equinu-
merous. Here ‘#xFx’ is a function from a concept (the F s) to a number. And ‘Eq(F,G)’ is an
abbreviation for the second-order formula stating that there exists a bijection between the Fs and
Gs.¹³ is is an equivalence relation; and indeed all abstraction principles include an equivalence
on the RHS. It turns out that all of arithmetic can be derived from this principle. So, insofar as
the principle is epistemically secure, so too is arithmetic.
But neo-Fregeanism is also an approach to ontology;¹⁴ it is intended to vindicate Platonism
about numbers and other abstracta. ere are a few key ideas that are characteristic of neo-Fregean
ontology. I will focus on two.
¹³∃R(∀x∀y∀z∀w[(Rxy∧Rzw)→ (x = z ↔ y = w)]∧∀x[Fx→ ∃y(Gy∧Rxy)]∧∀y[Gy → ∃x(Fx∧Rxy)])¹⁴See for example Eklund [?], Sider [?], and Hawley [?]
Reconceptualization e LHS and RHS of abstraction principles involve a reconceptualization
of the same state of affairs.
e reconceptualization thesis is best seen via an example. Take the following example from
Frege:
e judgement ‘line a is parallel to line b’, or using symbols
a//b
can be taken as an identity. If we do this, we obtain the concept of a direction and
say, ‘the direction of a is identical to the direction of b’. us we replace the symbol
// with the more general symbol =, through removing what is specific in content of
the former and dividing it between a and b. We carve up the content in a different
way from the original way, and this yields us a new concept. (Grundlagen §)
e thought is that if we already have the concept of what it is for two lines to be parallel, then the
abstraction to directions ‘recarves’ the content of the parallelism concept, giving us the concept
of a direction.
But the key to Frege’s view is that […] we have the option, by laying down the
Direction abstraction, of reconceptualizing, as it were, the type of state of affairs
which is described on the right. at type of state of affairs is initially given to us as the
obtaining of a certain equivalence relation — parallelism — among lines; but we have
the option, by stipulating that the abstraction is to hold, of so reconceiving such states
of affairs that they come to constitute the identity of a new kind of thing, directions,
of which, by this very stipulation, we introduce the concept. e concept of direction
is thus so introduced that that two lines are parallel constitutes the identity of their
direction. It is in no sense a further substantial claim that their directions exist and
are identical under the described circumstances. (Wright [?, pp. –])
So the abstraction principle is a stipulative definition of the concept of direction. It is an implicit
definition, but a definition nonetheless. is, in itself, is not enough to get us the existence of
directions, however. What guarantees that the expression ‘the direction of a’ is really a singular
term, and that it refers? Likewise for Hume’s Principle: what assurance do we have that ‘#xFx’
is a singular term referring to numbers?
For this we need to turn to our second key thesis:
Syntactic Priority If an expression behaves like a singular term, then it is a singular term; truth
is ontologically prior to reference.
e idea is that the syntactic role played by an expression completely determines its grammatical
type. is idea is not so crazy, but it becomes a lot more radical when we consider it in conjunction
with the controversial view that truth is prior to reference. e thought is that if any singular
term appears in a true sentence, then it thereby has a referent. Here is Dummett:
If a word functions as a proper name, then it is a proper name. If we have fixed the
sense of sentences in which it occurs, then we have done all there is to be done towards
fixing the sense of the word. If its syntactic function is that of a proper name, then
we have fixed the sense, and with it the reference, of a proper name. If we can find a
true statement of identity in which the identity sign stands between the name and a
phrase of the form ‘the x such that Fx’, then we can determine whether the name has
a reference by finding out, in the ordinary way, the truth-value of the corresponding
sentence of the form ‘ere is one and only one x such that Fx’. ere is no further
philosophical question whether the name — i.e., every name of that kind — really
stands for something or not. (Dummett [?, pp. –])
And again, Wright:
[…] when it has been established, by the sort of syntactic criteria sketched, that a
given class of terms are functioning as singular terms, and when it has been verified
that certain appropriate sentences containing them are, by ordinary criteria, true,
then it follows that those terms do genuinely refer. (Wright [?, pp. –])
ese claims are fairly radical. ey advocate for a metaontological in which referents of terms
are very easy to come by, because they are given to us merely by abstraction; in particular, we are
ontologically commitment whenever a conceptual recarving of a certain kind of content — a true
equivalence — results in the claim of identity between two singular terms.
. Neo-Fregean Mereology
ese two neo-Fregean theses are enough to get the account off the ground. For our particular
application of the view, we will need to introduce one new equivalence relation ≈ which for our
purposes will represent the relation of between two pluralities when they ‘are the same portion
of reality’.¹⁵ is relation is a plural generalisation of the singular co-location relation. So, some
objects are the same portion of reality as some others whenever they collectively cover all and only
the same spatiotemporal points (or regions). inking in terms of our second classical ersatzist
approach, we can see that some objects the Xs in a partition might be the same portion of reality
as another plurality of objects the Y s in a different partition even when e.g. the X ’s partition is
much more fine-grained than Y ’s partition. Where X is the location of the Xs,
⌜X ≈ Y ⌝ is true iff ⋃X = ⋃Y
Now, ≈ is generally equivalence relation on pluralities that holds between portions of reality
independent of the different ways of partitioning it into objects. (Indeed, set union ⋃ is collapses
the partitioned subset into a single region of spacetime.)
Given this equivalence relation, we can formulate the abstraction principle for it.
Fusions ∀X∀Y (σ(X) = σ(Y )↔X ≈ Y )
¹⁵For a different use of the same relation, see [?].
Here ‘σ(X)’ is a function taking us from a plurality, the Xs, to an object, their fusion. e fusion
abstraction principle says that the fusion of the Xs is the fusion of the Y s just if the Xs and Y s
are the same portion of reality.
First, by appealing to reconceptualization, we suggest that the fusion facts are just a conceptual
recarving of the same portion of reality facts. Secondly, by appealing to syntactic priority, we have
it that ‘σ(X)’ is a singular term. Moreover, insofar as these ‘fusion of ’ expressions feature in true
sentences, we can rest assured that they refer.
If we accept the abstract objects that result from these abstraction principles — as the neo-
Fregean should — what is the mereology that results? e answer requires thinking in a bit more
detail about ≈. Let us first define a notion of overlap.
Overlap ⌜x ○ y⌝ is true iff x ∩ y ≠ ∅
On this definition, two things overlap whenever they are partly the same portion of reality, that
is, whenever the intersection of their locations is non-empty. Set-theoretically, this entails that
X ≈ Y iff ∀z(z ○X ↔ z ○ Y ).¹⁶ So we can think of our ‘same portion of reality’ relation has
holding between pluralities whenever any object overlaps an X iff it overlaps a Y .
From here, we can prove that the objects generated from the fusion abstraction principle sat-
isfy the axioms of CEM. Given the usual definition of parthood in terms of overlap — namely,
x ≤ y ∶≡ ∀z(z ○ x → z ○ y)— we know that parthood is reflexive, and transitive. For antisym-
metry, suppose x ≤ y and y ≤ x. en ∀z(z ○ x ↔ z ○ y) which simply means that x ≈ y.¹⁷
Strong supplementation follows by definition of parthood; ∀z[(z ○ x → z ○ y) → x ≤ y] is
just the contrapositive of strong supplementation. e last thing to prove is that composition is
unrestricted.
¹⁶Here an below I am using z ○X to mean that z ○ x for some x among the Xs.¹⁷To get the full numerical identity of mutual parts, we need the plausible additional claim that σ(x) = x for all
objects x. is is not something we can show from the abstraction principle itself. As such, for mutual parts x and y,all we can prove is that σ(x) = σ(y). e abstraction principle by itself is compatible with co-located objects.
Unrestricted composition, in this case, follows from the neo-Fregean assumptions. at is,
since it is a logical truth for any plurality that X ≈X ,¹⁸ e fusion abstraction immediately gives
us that the singular term ‘σ(X)’ refers, and hence unrestricted composition holds.
is shows that the neo-Fregean approach to ersatzism about ordinary objects fits extremely
well with classical extensional mereology. It also provides a clear sense in which composite objects
can be thought to be abstracta.¹⁹ One might wonder whether this metaontological perspective
on ersatzism will actually deliver the advantages that were outlined in §. It seems clear that it
will have semantic advantages over nihilism. But it looks like it will not have the advantage of
ontological parsimony over the universalist. at advantage was supposed to be secured by the
fact that the ordinary objects are to be thought of as abstracta to which we are already committed. In
the case here, what guarantee is there that these abstracts are sets, or for that matter any other kind
of abstract object we might have been committed to. ere is clearly an ontological commitment
here. In the same way neo-Fregeans about arithmetic are Platonists about numbers, neo-Fregeans
about fusions should accept the existence of composite objects. However, there is at least one
sense in which the ontological commitment is supposed to be unproblematic. Recall Wright [?,
] regarding parallelism:
[T]hat two lines are parallel constitutes the identity of their direction. It is in no
sense a further substantial claim that their directions exist and are identical under
the described circumstances.
¹⁸We are not considering empty pluralities; if there were any such thing, presumably it would fail to have a locationand hence ‘X ≈X ’ would be false. (ose wishing for an empty fusion might want to accept the existence of an emptyplurality together with the thesis that it is located at the empty region ∅.)
¹⁹It is worth noting that Dummett’s way of drawing the abstract/concrete distinction is by appealing to differentways of achieving reference. For Dummett, while abstract objects are objects of abstraction, concrete objects are objects ofostention. e ordinary object ersatzist will likely want to accept that we can achieve reference to composite objects byostention; but if composite objects are abstract, then this seems to cause trouble for the Dummettian distinction. Butthere is other, independent, trouble for this way of drawing the distinction: it is perfectly consistent with abstractionprinciples for the abstracts themselves to be identical to concreta. is has come to be known as the Julius Ceasarproblem, as it is consistent with Hume’s Principle that the number one is identical to Julius Caesar. ([?]) It would seemas if the Julius Caesar problem is not so much of a problem here but a help. If we think fusions are located where theirparts are, then this can help motivate the identity of the fusion of an object with itself. (See previous footnote.)
In the case of the fusion abstraction principle, the thought is that existence of the composites are
an immediate consequence of the abstraction principle — the existence and identity of composites
is not some further substantive claim. So there may be some reason to think that the ontological
commitments are mitigated.²⁰
Objections and Replies
Objection Sets are not located in spacetime. Objects are located in spacetime. erefore, objects
are not sets.
Reply It is important to distinguish pure sets, and impure sets. Pure sets have only other sets as
elements. Impure sets may have objects (‘urelements’) somewhere down the chain of members.
It is clear that pure sets are not located in spacetime; however is it clear that impure sets are
not located? An ersatzist should hold that an impure set is located wherever its members are
(Maddy [?]).²¹
Objection Sets exist necessarily, while ordinary objects typically exist only contingently.
Reply Again the distinction between pure and impure sets comes to the rescue. Pure sets exist
necessarily, but an impure set exists only in possible worlds where its members do (e.g. Socrates’
singleton cannot exist without Socrates himself ). Moreover, most metaphysicians who accept
mereology think that this structure is necessary. If some xs compose y, then it does so in every
²⁰Universalists also attempt to mitigate the ontological commitments of their views as well. Lewis [?] and Arm-strong [?] both appeal to the view that composition is identity on this point. It is highly controversial whether the viewthat many parts can be identical to one whole is even coherent, and if so whether it actually secures any kind of on-tological innocence (for more on this controversy, see the essays in Cotnoir and Baxter [?]). e neo-Fregean is notcommitted to a many-one identity claim; they merely claim the identity between two facts with different conceptualcontent. is is also controversial (for a robust defence of the idea, see Rayo [?]; for criticisms, see [?]), but a fullconsideration of whether the neo-Fregean has any advantage over the universalist on this point would take us too farafield.
²¹See also the debate between Cook [?] and Effingham [?].
possible world in which the xs exist. Composition is not contingent (Cameron [?] notwithstand-
ing).
Objection Sets have their members essentially, but objects have their parts only accidentally.
Reply is objection presupposes a rejection of the controversial doctrine of mereological essen-
tialism, the view that a whole has its parts essentially. Ersatzists could bite the bullet here (as
others do). Alternatively, one could appeal to counterpart theory to rescue ordinary claims about
an object changing its parts.
Objection Two sets with the same members are identical. But many philosophers think objects
with the same parts might be distinct (e.g. the statue and the clay).
Reply is objection presupposes a rejection of the controversial doctrine of mereological exten-
sionalism, the view that objects with the same parts are identical. Ersatzism can easily bite the
bullet here, too. But surprisingly, the above approach to Ersatzism is perfectly compatible with
non-extensional structure.
Objection Knowledge of sets is a priori, whereas our knowledge of objects is typically had a
posteriori.
Reply Two replies. First, is it so obvious that our knowledge of impure sets is had a priori? Of
course, our knowledge of impure sets qua sets is a priori. But insofar as any properties of impure
sets depend on the properties of its members, it would seem as though some of our knowledge
might be a posteriori. (Consider that Socrates’ singleton has the property of being such that its only
member is snub-nosed.) Second, is it so obvious that our knowledge of the metaphysical structure
of objects is had a posteriori? Many of us think of metaphysical inquiry as (partly) an a priori
endeavour.
Objection Erstazism assumes that all (would-be) facts about parts and wholes are reducible to
facts about fundamental objects; this is fully reductionist. Can we be at all confident that this
reduction can be carried out?
Reply Indeed, the final proposal assumes that there are no properties of wholes not reducible
to properties of their parts. But this is no worse off than the nihilist, who wishes to do away
with composite wholes because they do no theoretical work. But it is also no worse off than many
universalists (e.g. Armstrong [?], Lewis [?], and Varzi [?]) who believe that the whole is nothing
over and above its parts. Can any version of ersatzism accommodate any view that allows emergent
properties of complex wholes?
Objection Erstazism proposes we identify a set of fundamental entities with an ordinary object.
But given an ordinary object, isn’t it indeterminate which such set it would be? Aren’t there many
equally good candidates for such objects?
Reply is is a particularly difficult problem; after all, it was originally raised as an argument
in favour of nihilism (Unger ). One might locate the source of indeterminacy in language,
and hence accept the existence of the many. Alternatively, one might attempt to replicate vague
objects via fuzzy sets or some other formal machinery. (ere is no shortage of options, even if
there aren’t many good ones.)
Objection But I’m sitting in a chair, not a set!
Reply Fair enough. But perhaps “common sense tells us to taste our food before we salt it and
to cut the cards. It does not tell us there are chairs.” (van Inwagen [?, p. ])
References