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Abstract Objects
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Stanford Encyclopedia One doesn’t go far in the study of what there is without encountering the
view that every entity falls into one of two categories: concrete or
of Philosophy abstract. The distinction is supposed to be of fundamental significance for
metaphysics (especially for ontology), epistemology, and the philosophy
of the formal sciences (especially for the philosophy of mathematics); it is
also relevant for analysis in the philosophy of language, the philosophy of
mind, and the philosophy of the empirical sciences. This entry surveys (a)
Edward N. Zalta Uri Nodelman attempts to say how the distinction should be drawn and (b) some of main
Principal Editor Senior Editor theories of, and about, abstract objects.
Colin Allen Hannah Kim Paul Oppenheimer
Associate Editor Associate Editor Associate Editor 1. Introduction
Faculty Sponsors: R. Lanier Anderson & Thomas Icard 1.1 About the Expression ‘Object’
Editorial Board: https://plato.stanford.edu/board.html 1.2 About the Abstract/Concrete Distinction
Library of Congress ISSN: 1095-5054
2. Historical Remarks
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3.3 The Way of Abstraction
Stanford Encyclopedia of Philosophy 3.4 The Way of Abstraction Principles
Copyright c 2022 by the publisher
The Metaphysics Research Lab 3.5 The Ways of Negation
Department of Philosophy 3.6 The Way of Encoding
Stanford University, Stanford, CA 94305
3.7 The Ways of Weakening Existence
Abstract Objects
3.8 Eliminativism
Copyright c 2022 by the authors
José L. Falguera, Concha Martı́nez-Vidal, and Gideon Rosen 4. Further Reading
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 Abstract Objects José L. Falguera, Concha Martı́nez-Vidal, and Gideon Rosen

Other Internet Resources two categories nor is there a definite list of items that fall under one or the
Related Entries other category (assuming neither is empty).

The first challenge, then, is to articulate the distinction, either by defining

1. Introduction the terms explicitly or by embedding them in a theory that makes their
connections to other important categories more explicit. In the absence of
The abstract/concrete distinction has a curious status in contemporary such an account, the philosophical significance of the contrast remains
philosophy. It is widely agreed that the ontological distinction is of uncertain, for the attempt to classify things as abstract or concrete by
fundamental importance, but as yet, there is no standard account of how it appeal to intuition is often problematic. Is it clear that scientific theories
should be drawn. There is a consensus about how to classify certain (e.g., the general theory of relativity), works of fiction (e.g., Dante’s
paradigm cases. For example, it is usually acknowledged that numbers and Inferno), fictional characters (e.g., Bilbo Baggins) or conventional entities
the other objects of pure mathematics, like pure sets, are abstract (if they (e.g., the International Monetary Fund or the Spanish Constitution of
exist), whereas rocks, trees, and human beings are concrete. In everyday 1978) are abstract?
language, it is common to use expressions that refer to concrete entities as
well as those that apparently refer to abstractions such as democracy, It should be stressed that there may not be one single “correct” way of
happiness, motherhood, etc. Moreover, formulations of mathematical explaining the abstract/concrete distinction. Any plausible account will
theories seem to appeal directly to abstract entities, and the use of classify the paradigm cases in the standard way or give reasons for
mathematical expressions in the empirical sciences seems indispensable to proceeding otherwise, and any interesting account will draw a clear and
the formulation of our best empirical theories (see Quine 1948; Putnam philosophically significant line in the domain of objects. Yet there may be
1971; and the entry on indispensability arguments in the philosophy of many equally interesting ways of accomplishing these two goals, and if we
mathematics). Finally, apparent reference to abstract entities such as sets, find ourselves with two or more accounts that do the job rather well, there
properties, concepts, propositions, types, and possible worlds, among may be no point in asking which corresponds to the real abstract/concrete
others, is ubiquitous in different areas of philosophy. distinction. This illustrates a general point: when technical terminology is
introduced in philosophy by means of examples, but without explicit
Though there is a pervasive appeal to abstract objects, philosophers have definition or theoretical elaboration, the resulting vocabulary is often
nevertheless wondered whether they exist. The alternatives are: platonism, vague or indeterminate in reference. In such cases, it usually is pointless to
which endorses their existence, and nominalism, which denies the seek a single correct account. A philosopher may find herself asking
existence of abstract objects across the board. (See the entries on questions like, ‘What is idealism?’ or ‘What is a substance?’ and treating
nominalism in metaphysics and platonism in metaphysics.) But the these questions as difficult questions about the underlying nature of a
question of how to draw the distinction between abstract and concrete certain determinate philosophical category. A better approach may be to
objects is an open one: it is not clear how one should characterize these recognize that in many cases of this sort, we simply have not made up our
minds about how the term is to be understood, and that what we seek is

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not a precise account of what this term already means, but rather a 1.2 About the Abstract/Concrete Distinction
proposal for how it might fruitfully be used for philosophical analysis.
Anyone who believes that something in the vicinity of the Though we’ve spoken as if the abstract/concrete distinction must be an
abstract/concrete distinction matters for philosophy would be well advised exhaustive dichotomy, we should be open to the possibility that the best
to approach the project of explaining the distinction with this in mind. sharpening of it will entail that some objects are neither abstract nor
concrete. Holes and shadows, if they exist, do not clearly belong in either
So before we turn to the discussion of abstract objects in earnest, it will category; nor do ghosts, Cartesian minds, fictional characters,[2] immanent
help if we clarify how some of the key terms will be used in what follows. universals, or tropes. The main constraint on an account of the distinction
is that it draws a philosophically significant line that classifies at least
1.1 About the Expression ‘Object’ many of the standard examples in the standard ways. It is not a constraint
that everything be shoehorned into one category or the other.
Frege famously distinguished two mutually exclusive ontological
domains, functions and objects. According to his view, a function is an Finally, some philosophers see the main distinction not as between
‘incomplete’ entity that maps arguments to values, and is denoted by an abstract and concrete objects but as between abstract objects and ordinary
incomplete expression, whereas an object is a ‘complete’ entity and can be objects, where the distinction is a modal one – ordinary objects are
denoted by a singular term. Frege reduced properties and relations to possibly concrete while abstract objects (like the number 1) couldn’t be
functions and so these entities are not included among the objects. Some concrete (Zalta 1983, 1988). In any case, in the following discussion, we
authors make use of Frege’s notion of ‘object’ when discussing abstract shall assume that the abstract/concrete distinction is a division among
objects (e.g., Hale 1987). But though Frege’s sense of ‘object’ is existing objects, and that any plausible explanation of the distinction
important, it is not the only way to use the term. Other philosophers should aim to characterize a distinction among such objects.
include properties and relations among the abstract objects. And when the
background context for discussing objects is type theory, properties and 2. Historical Remarks
relations of higher type (e.g., properties of properties, and properties of
relations) may be all be considered ‘objects’. This latter use of ‘object’ is 2.1 The Provenance of the Distinction
interchangeable with ‘entity.’[1] Throughout this entry, we will follow this
last usage and treat the expressions ‘object’ and ‘entity’ as having the same The contemporary distinction between abstract and concrete is not an
meaning. (For further discussion, see the entry on objects.) ancient one. Indeed, there is a strong case for the view that, despite
occasional exceptions, it played no significant role in philosophy before
the 20th century. The modern distinction bears some resemblance to
Plato’s distinction between Forms and Sensibles. But Plato’s Forms were
supposed to be causes par excellence, whereas abstract objects are
generally supposed to be causally inert. The original ‘abstract’/‘concrete’

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distinction was a distinction among words or terms. Traditional grammar mine? In that case, the appearance of a common mathematical subject
distinguishes the abstract noun ‘whiteness’ from the concrete noun ‘white’ matter would be an illusion.) In The Foundations of Arithmetic (1884),
without implying that this linguistic contrast corresponds to a Frege concludes that numbers are neither external concrete things nor
metaphysical distinction in what these words stand for. In the 17th century, mental entities of any sort.
this grammatical distinction was transposed to the domain of ideas. Locke
speaks of the general idea of a triangle which is “neither Oblique nor Later, in his essay “The Thought” (1918), Frege claims the same status for
Rectangle, neither Equilateral, Equicrural nor Scalenon [Scalene]; but all the items he calls thoughts—the senses of declarative sentences—and also,
and none of these at once,” remarking that even this idea is not among the by implication, for their constituents, the senses of subsentential
most “abstract, comprehensive and difficult” (Essay, IV.vii.9). Locke’s expressions. Frege does not say that senses are abstract. He says that they
conception of an abstract idea as one that is formed from concrete ideas by belong to a third realm distinct both from the sensible external world and
the omission of distinguishing detail was immediately rejected by from the internal world of consciousness. Similar claims had been made
Berkeley and then by Hume. But even for Locke there was no suggestion by Bolzano (1837), and later by Brentano (1874) and his pupils, including
that the distinction between abstract ideas and concrete or particular ideas Meinong and Husserl. The common theme in these developments is the
corresponds to a distinction among objects. “It is plain, …” Locke writes, felt need in semantics and psychology, as well as in mathematics, for a
“that General and Universal, belong not to the real existence of things; but class of objective (i.e., non-mental) non-physical entities. As this new
are Inventions and Creatures of the Understanding, made by it for its own realism was absorbed into English-speaking philosophy, the traditional
use, and concern only signs, whether Words or Ideas” (III.iii.11). term ‘abstract’ was enlisted to apply to the denizens of this third realm. In
this vein, Popper (1968) spoke of the ‘third world’ of abstract, objective
The abstract/concrete distinction in its modern form is meant to mark a entities, in the broader sense that includes cultural products such as
line in the domain of objects or entities. So conceived, the distinction arguments, theories, and works of art.
becomes a central focus for philosophical discussion primarily in the 20th
century. The origins of this development are obscure, but one crucial As we turn to an overview of the current debate, it is therefore important
factor appears to have been the breakdown of the allegedly exhaustive to remember that the use of the terms platonist (for those who affirm the
distinction between mental and material objects, which had formed the existence of abstract objects) and nominalist (for those who deny
main division for ontologically-minded philosophers since Descartes. One existence) is somewhat lamentable, since these words have established
signal event in this development is Frege’s insistence that the objectivity senses in the history of philosophy. These terms stood for positions that
and aprioricity of the truths of mathematics entail that numbers are neither have little to do with the modern notion of an abstract object. Modern
material beings nor ideas in the mind. If numbers were material things (or platonists (with a small ‘p’) need not accept any of the distinctive
properties of material things), the laws of arithmetic would have the status metaphysical and epistemological doctrines of Plato, just as modern
of empirical generalizations. If numbers were ideas in the mind, then the nominalists need not accept the distinctive doctrines of the medieval
same difficulty would arise, as would countless others. (Whose mind nominalists. Moreover, the literature also contains mention of anti-
contains the number 17? Is there one 17 in your mind and another in platonists, many of whom see themselves as fictionalists about abstracta,

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though this doesn’t help if it turns out that the best analysis of fictions is to language); and (iii) the entities required for the truth on an empirical
regard them as abstract objects. So the reader should therefore be aware theory are those in the range of the variables bounded by its first-order
that terminology is not always well-chosen and that the terms so used quantifiers (i.e., the entities in the domain of the existential quantifier ‘∃x ’
sometimes stand for doctrines that are more restricted than the traditional and the universal quantifier ‘∀x ’). He concluded that in addition to the
doctrines that go by the same name. Henceforth, we simply use platonism concrete entities contemplated by our best empirical science, we must
for the thesis that there exists at least one abstract object, and nominalism accept the existence of mathematical entities, even if they are abstract (see
for the thesis that the number of abstract objects is exactly zero (Field also Quine 1960, 1969, 1976).
1980).
Quine’s argument initiated a debate that is still alive. Various nominalist
2.2 An Initial Overview of the Contemporary Debate responses questioned one or another of the premises in his argument. For
instance, Field (1980) challenged the idea that mathematics is
Before we survey the various proposals for drawing the abstract/concrete indispensable for our best scientific theories—i.e., rejecting (i) above—
distinction, we should briefly say why the distinction has been thought to and thus faced the task of rewriting classical and modern physics in
matter. Among philosophers who take the distinction seriously, it is nominalistic terms in order to sustain the challenge. Others have taken on
generally supposed that while concrete objects clearly exist, abstract the somewhat less daunting task of accepting (i) but rejecting (ii) and (iii);
entities are problematic in distinctive ways and deny the existence of they’ve argued that even if our best scientific theories, in regimented form,
abstract entities altogether. In this section we briefly survey the arguments quantify over mathematical entities, this doesn’t entail a commitment to
for nominalism and the responses that platonists have offered. If the mathematical entities (see Azzouni 1997a, 1997b, 2004; Balaguer 1996,
abstract objects are unified as a class, it is because they have some feature 1998; Maddy 1995, 1997; Melia 2000, 2002; Yablo 1998, 2002, 2005,
that generates what seems to be a distinctive problem—a problem that 2009; Leng 2010.) Colyvan (2010) coined the expression ‘easy-roaders’
nominalists deem unsolvable and which platonists aim to solve. Before we for this second group, since they avoided the ‘hard road’ of paraphrasing
ask what the unifying feature might be, it may therefore help to our best scientific theories in non-mathematical terms.
characterize the various problems it has been thought to generate.
By contrast, some mathematical platonists (Colyvan 2001; Baker 2005,
The contemporary debate about platonism developed in earnest when 2009) have refined Quine’s view by advancing the so-called ‘Enhanced
Quine argued (1948) that mathematical objects exist, having changed his Indispensability Argument’ (though see Saatsi 2011 for a response). Some
mind about the nominalist approach he had defended earlier (Goodman & participants describe the debate in terms of a stalemate they hope to
Quine 1947). Quine’s 1948 argument involves three key premises, all of resolve (see Baker 2017, Baron 2016, 2020, Knowles & Saatsi 2019, and
which exerted significant influence on the subsequent debate: (i) Martínez-Vidal & Rivas-de-Castro 2020, for discussion).[3]
mathematics is indispensable for empirical science; (ii) we should be
Aside from the debate over Quine’s argument, both platonism and
ontologically committed to the entities required for the truth of our best
nominalism give rise to hard questions. Platonists not only need to provide
empirical theories (all of which should be expressible in a first-order

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a theory of what abstract objects exist, but also an account of how we attempt to explain knowledge of abstracta on the same model that is used
cognitively access and come to know non-causal, abstract entities. This to explain knowledge of concrete objects. They argue that not only a
latter question has been the subject of a debate that began in earnest in certain plenitude principle for abstract objects (namely, the comprehension
Benacerraf (1973), which posed just such a dilemma for mathematical principle for abstract objects put forward in Zalta 1983, 1988—see below)
objects. Benacerraf noted that the causal theory of reference doesn’t seem yields unproblematic ‘acquaintance by description’ to unique abstract
to make it possible to know the truth conditions of mathematical objects but also that their approach actually comports with naturalist
statements, and his argument applies to abstract entities more generally. beliefs. Balaguer (1995, 1998) also suggests that a plenitude principle is
On the other hand, nominalists need to explain the linguistic uses in which the best way forward for the platonist, and that our knowledge of the
we seem to appeal to such entities, especially those uses in what appear to consistency of mathematical theories suffices for knowledge of
be good explanations, such as those in scientific, mathematical, linguistic, mathematical objects. And there are views that conceive of abstract
and philosophical pursuits (see Wetzel 2009, 1–22, for a discussion of the objects as constituted by human—or, in general, intelligent—subjects, or
many places where abstract types are used in scientific explanations). Even as abstract artifacts (see Popper 1968; Thomasson 1999).
though nominalists argue that there are no abstracta, the very fact that
there is disagreement about their existence suggests that both platonists A number of nominalists have been persuaded by Benacerraf’s (1973)
and nominalists acknowledge the distinction between the abstract and epistemological challenge about reference to abstract objects and
concrete to be a meaningful one. concluded that sentences with terms making apparent reference to them—
such as mathematical statements—are either false or lack a truth value.
On the platonist side, various proposals have been raised to address the They argue that those sentences must be paraphrasable without vocabulary
challenge of explaining epistemic access to abstract entities, mostly in that commits one to any sort of abstract entity. These proposals sometimes
connection with mathematical objects. Some, including Gödel (1964), suggest that statements about abstract objects are merely instrumental;
allege that we access abstract objects in virtue of a unique kind of they serve only to help us establish conclusions about concrete objects.
perception (intuition). Maddy (1990, 1997) developed two rather different Field’s fictionalism (1980, 1989) has been influential in this regard. Field
ways of understanding our knowledge of mathematics in naturalistic ways. reconstructed Newtonian physics using second-order logic and
Other platonists have argued that abstract objects are connected to quantification over (concrete) regions of space-time. A completely
empirical entities, either via abstraction (Steiner 1975; Resnik 1982; different tactic for avoiding the commitment to abstract, mathematical
Shapiro 1997) or via abstraction principles (Wright 1983; Hale 1987); objects is put forward in Putnam (1967) and Hellman (1989), who
we’ll discuss some of these views below. There are also those who speak separately reconstructed various mathematical theories in second-order
of existent and intersubjective abstract entities as a kind of mental modal logic. On their view, abstract objects aren’t in the range of the
representation (Katz 1980). existential quantifier at the actual world (hence, we can’t say that they
exist), but they do occur in the range of the quantifier at other possible
A rather different line of approach to the epistemological problem was worlds, where the axioms of the mathematical theory in question are true.
proposed in Linsky & Zalta 1995, where it is suggested that one shouldn’t

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These nominalistic approaches must contend with various issues, of a new kind of entity, we must have a linguistic framework for talking
course. At the very least, they have to successfully argue that the tools about those entities. He then distinguished two kinds of existence
they use to avoid commitments to abstract objects don’t themselves questions: internal questions within the framework about the existence of
involve such commitment. For example, Field must argue that space-time the new entities and external questions about the reality of the framework
regions are concrete entities, while Putnam and Hellman must argue that itself. If the framework deals with abstract entities such as numbers, sets,
by relying on logical possibility and modal logic, there is no commitment propositions, etc., then the internal question can be answered by logical
to possible worlds considered as abstract objects. In general, any analysis of the rules of the language, such as whether it includes terms for,
nominalist account that makes essential use of set theory or model- or implies claims that quantify over, abstract objects. But, for Carnap, the
theoretic structures must convincingly argue that the very use of such external question, about whether the abstract entities really exist, is a
analytic tools doesn’t commit them to abstract objects. (See Burgess & pseudo-question and should be regarded as nothing more than the
Rosen 1997 for a systematic survey of different proposals about the pragmatic question of whether the framework is a useful one to adopt, for
existence of abstract objects.) scientific or other forms of enquiry. We’ll discuss Carnap’s view in more
detail in subsection 3.7.1.
Another nominalistic thread in the literature concerns the suggestion that
sentences about (posited) abstract objects are quasi-assertions, i.e., not Some have thought that Carnap’s view offers a deflationist view of
evaluable as to truth or falsehood (see Yablo 2001 and Kalderon 2005). objects, since it appears that the existence of objects is not language
Still others argue that we should not believe sentences about abstracta independent. After Carnap’s seminal article, several other deflationist
since their function, much like the instrumentalism discussed earlier, is to approaches were put forward (Putnam 1987, 1990; Hirsch 2002, 2011;
ensure empirical adequacy for observational sentences (Yablo 1998). This Sider 2007, 2009; Thomasson 2015), many of them claiming to be a
may involve differentiating between apparent content, which involves vindication of Carnap’s view. However, there are deflationist proposals
posited abstract objects, and real content, which only concerns concrete that run counter to Carnap’s approach, among them, deflationary
objects (Yablo 2001, 2002, 2010, 2014). (For more on these fictionalist nominalism (Azzouni 2010) or agnosticism about abstract objects (Bueno
accounts, see Kalderon 2005, Ch. 3, and the entry on fictionalism.) 2008a, 2008b, 2020). Additionally, philosophers inspired by Frege’s work
have argued for a minimal notion of an object (Rayo 2013, Rayo 2020
A final group of views in the literature represents a kind of agnosticism [Other Internet Resources]; and Linnebo 2018). We’ll discuss some of
about what exists or about what it is to be an object, be it abstract or these in greater detail below, in subsection 3.7.2. A final agnostic position
concrete. These views don’t reject an external material world, but rather that has emerged is one that rejects a strict version of platonism, but
begin with some question as to whether we can have experience, suggests that neither a careful analysis of mathematical practice (Maddy
observation, and knowledge of objects directly, i.e., independent of our 2011), nor the enhanced version of the indispensability argument (Leng
theoretical frameworks. Carnap (1950 [1956]), for example, started with 2020) suffice to decide between nominalism and moderate versions of
the idea that our scientific knowledge has to be expressed with respect to a platonism. Along these lines, Balaguer (1998) concluded that the question
linguistic framework and that when we wish to put forward a theory about doesn’t have an answer, since the arguments for ‘full-blooded’ platonism

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can be matched one-for-one by equally good arguments by the anti- abstract/concrete distinction and theorize about abstract objects. Even if
platonist. there is no single, acceptable account, these various ways of drawing the
distinction and theorizing about abstract objects do often cast light on the
For additional discussion about the basic positions in the debate about questions we’ve been discussing, especially when the specific proposals
abstract and concrete objects, see Szabó 2003 and the entries on are integrated into a supplementary (meta-)ontological project. For each
nominalism in metaphysics and platonism in metaphysics, nominalism in method of drawing the distinction and specific proposal adopting that
the philosophy of mathematics and platonism in the philosophy of method acquires a certain amount of explanatory power, and this will help
mathematics. us to compare and contrast the various ideas that are now found in the
literature.
3. What is an Abstract Object?
3.1 The Way of Example and the Way of Primitivism
As part of his attempt to understand the nature of possible worlds, Lewis
(1986a, 81–86) categorizes different ways by which one can draw the According to the way of example, it suffices to list paradigm cases of
abstract/concrete distinction.[4] These include: the way of example (which abstract and concrete entities in the hope that the sense of the distinction
is simply to list the paradigm cases of abstract and concrete objects in the will somehow emerge. Clearly, a list of examples for each category would
hope that the sense of the distinction will somehow emerge); the way of be a heuristically promising start in the search for some criterion (or list of
conflation (i.e., identifying abstract and concrete objects with some criteria) that would be fruitful for drawing the distinction. However, a
already-known distinction); the way of negation (i.e., saying what abstract simple list would be of limited significance since there are too many ways
objects are by saying what they are not, e.g., non-spatiotemporal, non- of extrapolating from the paradigm cases to a distinction that would cover
causal, etc.); and the way of abstraction (i.e., saying that abstract objects the unclear cases, with the result that no clear notion has been explained.
are conceptualized by a process of considering some known objects and
omitting certain distinguishing features). He gives a detailed examination For example, pure sets are paradigm examples of abstract entities. But the
of the different proposals that typify these ways and then attempts to show case of impure sets is far from clear. Consider the unit set whose sole
that none of them quite succeeds in classifying the paradigms in accord member is Joe Biden (i.e., {Joe Biden}), the Undergraduate Class of 2020
with prevailing usage. Given the problems he encountered when analyzing or The Ethics Committee, etc. They are sets, but it is not clear that they are
the various ways, Lewis became pessimistic about our ability to draw the abstract given that Joe Biden, the members of the class and committee are
distinction cleanly. concrete. Similarly, if one offers the characters of Sherlock Holmes stories
as examples to help motivate the primitive concept abstract object, then
Despite Lewis’s pessimism about clarifying the abstract/concrete one has to wonder about the object London that appears in the novels.
distinction, his approach for categorizing the various proposals, when
extended, is a useful one. Indeed, in what follows, we’ll see that there are The refusal to characterize the abstract/concrete distinction while
a number of additional ways that categorize attempts to characterize the maintaining that both categories have instances might be called the way of

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primitivism, whenever the following condition obtains: a few predicates no doubt that some authors have used the terms in this way. (Thus Quine
are distinguished as primitive and unanalyzable, and the explanatory 1948 uses ‘abstract entity’ and ‘universal’ interchangeably.) This sort of
power rests on the fact that other interesting predicates can be defined in conflation is however rare in recent philosophy.
terms of the primitives and that interesting claims can be judged true on
the basis of our intuitive understanding of the primitive and defined 3.3 The Way of Abstraction
notions. Thus, one might take abstract and concrete as primitive notions. It
wouldn’t be an insignificant result if one could use this strategy to explain Another methodology is what Lewis calls the way of abstraction.
why abstract objects are necessarily existent, causally ineffiacious, non- According to a longstanding tradition in philosophical psychology,
spatiotemporal, intersubjective, etc. (see Cowling 2017: 92–97). abstraction is a distinctive mental process in which new ideas or
conceptions are formed by considering the common features of several
But closer inspection of this method reveals some significant concerns. To objects or ideas and ignoring the irrelevant features that distinguish those
start with, when a distinction is taken as basic and unanalyzable, one objects. For example, if one is given a range of white things of varying
typically has to offer some intuitive instances of the primitive predicates. shapes and sizes; one ignores or abstracts from the respects in which they
But it is not always so easy to do this. For example, when mathematicians differ, and thereby attains the abstract idea of whiteness. Nothing in this
take set and membership as primitives and then assert some principles of tradition requires that ideas formed in this way represent or correspond to
set theory, they often illustrate their primitives by offering some examples a distinctive kind of object. But it might be maintained that the distinction
of sets, such as The Undergraduate Class of 2020 or The Ethics between abstract and concrete objects should be explained by reference to
Committee, etc. But these, of course, aren’t quite right, since the members the psychological process of abstraction or something like it. The simplest
of the class and committee may change while the class and committee version of this strategy would be to say that an object is abstract if it is (or
remain the same, whereas if the members of a set change, one has a might be) the referent of an abstract idea; i.e., an idea formed by
different set. A similar concern affects the present proposal. If one offers abstraction. So conceived, the way of abstraction is wedded to an
sets or the characters of the Sherlock Holmes novels as examples to help outmoded philosophy of mind.
motivate the primitive concept abstract object, then one has to wonder
about impure sets such as the unit set whose sole member is Aristotle (i.e., It should be mentioned, though, that the key idea behind the way of
{Aristotle}) and the object London that appears in the novels. abstraction has resurfaced (though transformed) in the structuralist views
about mathematics that trace back to Dedekind. Dedekind thought of
3.2 The Way of Conflation numbers by the way of abstraction. Dedekind suggested that when
defining a number-theoretic structure, “we entirely neglect the special
According to the way of conflation, the abstract/concrete distinction is to character of the elements, merely retaining their distinguishability and
be identified with one or another metaphysical distinction already familiar taking into account only the relations to one another” (Dedekind 1888
under another name: as it might be, the distinction between sets and [1963, 68]). This view has led some structuralists to deny that numbers are
individuals, or the distinction between universals and particulars. There is abstract objects. For example, Benacerraf concluded that “numbers are not

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objects at all, because in giving the properties (that is, necessary and The number of Fs = the number of Gs if and only if there are just as
sufficient) of numbers you merely characterize an abstract structure—and many Fs as Gs
the distinction lies in the fact that the ‘elements’ of the structure have no
properties other than those relating them to other ‘elements’ of the same These biconditionals (or abstraction principles) appear to have a special
structure” (1965, 70). We shall therefore turn our attention to a variant of semantic status. While they are not strictly speaking definitions of the
the way of abstraction, one that has led a number of philosophers to functional expression that occurs on the left hand side, they would appear
conclude that numbers are indeed abstract objects. to hold in virtue of the meaning of that expression. To understand the term
‘direction’ is (in part) to know that the direction of a and the direction of b
3.4 The Way of Abstraction Principles are the same entity if and only if the lines a and b are parallel. Moreover,
the equivalence relation that appears on the right hand side of the
In the contemporary philosophical literature, a number of books and biconditional would appear to be semantically and perhaps
papers have investigated a form of abstraction that doesn’t depend on epistemologically prior to the functional expressions on the left (Noonan
mental processes. We may call this the way of abstraction principles. 1978). Mastery of the concept of a direction presupposes mastery of the
Wright (1983) and Hale (1987) have developed an account of abstract concept of parallelism, but not vice versa.
objects on the basis of an idea they trace back to certain suggestive
The availability of abstraction principles meeting these conditions may be
remarks in Frege (1884). Frege notes (in effect) that many of the singular
exploited to yield an account of the distinction between abstract and
terms that appear to refer to abstract entities are formed by means of
concrete objects. When ‘f ’ is a functional expression governed by an
functional expressions. We speak of the shape of a building, the direction
abstraction principle, there will be a corresponding kind Kf such that:
of a line, the number of books on the shelf. Of course, many singular terms
formed by means of functional expressions denote ordinary concrete x is a Kf if and only if, for some y, x = f (y).
objects: ‘the father of Plato’, ‘the capital of France’. But the functional
terms that pick out abstract entities are distinctive in the following respect: For example,
where f (a) is such an expression, there is typically an equation of the
form: x is a cardinal number if and only if for some concept F, x = the
number of Fs.
f (a) = f (b) if and only if Rab
The simplest version of the way of abstraction principles is then to say
where R is an equivalence relation, i.e., a relation that is reflexive, that:
symmetric and transitive, relative to some domain. For example:
x is an abstract object if (and only if) x is an instance of some kind Kf
The direction of a = the direction of b if and only if a is parallel to b whose associated functional expression ‘f ’ is governed by a suitable
abstraction principle.

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The strong version of this account—which purports to identify a necessary style. If so, the abstractionist approach does not provide a necessary
condition for abstractness—is seriously at odds with standard usage. Pure condition for abstractness as that notion is standardly understood.
sets are usually considered paradigmatic abstract objects. But it is not clear
that they satisfy the proposed criterion. According to a version of naïve set More importantly, there is some reason to believe that it fails to supply a
theory, the functional expression ‘set of’ is indeed characterized by a sufficient condition. A mereological fusion of concrete objects is itself a
putative abstraction principle. concrete object. But the concept of a mereological fusion is governed by
what appears to be an abstraction principle:
The set of Fs = the set of Gs if and only if, for all x, x is F if and only
if x is G. The fusion of the Fs = the fusion of the Gs if and only if the Fs and
Gs cover one another,
But this principle, which is a version of Frege’s Basic Law V, is
inconsistent and so fails to characterize an interesting concept. In where the Fs cover the Gs if and only if every part of every G has a part
contemporary mathematics, the concept of a set is not introduced by an in common with an F. Similarly, suppose a train is a maximal string of
abstraction principle, but rather axiomatically. Though attempts have been railroad carriages, all of which are connected to one another. We may
made to investigate abstraction principles for sets (Cook 2003), it remains define a functional expression, ‘the train of x’, by means of an
an open question whether something like the mathematical concept of a ‘abstraction’ principle: The train of x = the train of y if and only if x and y
set can be characterized by a suitably restricted abstraction principle. (See are connected carriages. We may then say that x is a train if and only if for
Burgess 2005 for a survey of recent efforts in this direction.) Even if such some carriage y, x is the train of y. The simple account thus yields the
a principle is available, however, it is unlikely that the epistemological consequence that trains are to be reckoned abstract entities.
priority condition will be satisfied. That is, it is unlikely that mastery of
It is unclear whether these objections apply to the more sophisticated
the concept of set will presuppose mastery of the equivalence relation that
abstractionist proposals of Wright and Hale, but one feature of the simple
figures on the right hand side. It is therefore uncertain whether the way of
account sketched above clearly does apply to these proposals and may
abstraction principles will classify the objects of pure set theory as
serve as the basis for an objection to this version of the way of abstraction
abstract entities (as it presumably must).
principles. The neo-Fregean approach seeks to explain the
On the other hand, as Dummett (1973) has noted, in many cases the abstract/concrete distinction in semantic terms: We said that an abstract
standard names for paradigmatically abstract objects do not assume the object is an object that falls in the range of a functional expression
functional form to which the definition adverts. Chess is an abstract entity, governed by an abstraction principle, where ‘f ’ is governed by an
but we do not understand the word ‘chess’ as synonymous with an abstraction principle when that principle holds in virtue of the meaning of
expression of the form ‘f (x) ’, where ‘f ’ is governed by an abstraction ‘f ’. This notion of a statement’s holding in virtue of the meaning of a word
principle. Similar remarks would seem to apply to such things as the is notoriously problematic (see the entry the analytic/synthetic distinction).
English language, social justice, architecture, and Charlie Parker’s musical But even if this notion makes sense, one may still complain: The

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abstract/concrete distinction is supposed to be a metaphysical distinction; Abstraction functions have two key features. First, for each abstraction
abstract objects are supposed to differ from concrete objects in some function f there is an equivalence relation R such that it lies in the nature
important ontological respect. It should be possible, then, to draw the of f that f (x) = f (y) iff Rxy . Intuitively, we are to think that R is
distinction directly in metaphysical terms: to say what it is in the objects metaphysically prior to f , and that the abstraction function f is defined (in
themselves that makes some things abstract and others concrete. As Lewis whole or in part) by this biconditional. Second, each abstraction function
writes, in response to a related proposal by Dummett: is a generating function: its values are essentially values of that function.
Many functions are not generating functions. Paris is the capital of France,
Even if this … way succeeds in drawing a border, as for all I know but it is not essentially a capital. The number of solar planets, by contrast,
it may, it tells us nothing about how the entities on opposite sides is essentially a number. The notion of an abstraction function may be
of that border differ in their nature. It is like saying that snakes are defined in terms of these two features:
the animals that we instinctively most fear—maybe so, but it tells
us nothing about the nature of snakes. (Lewis 1986a: 82) f is an abstraction function if and only if

The challenge is to produce a non-semantic version of the abstractionist 1. for some equivalence relation R, it lies in the nature of f that
criterion that specifies directly, in metaphysical terms, what the objects f (x) = f (y) if and only if Rxy; and
whose canonical names are governed by abstraction principles all have in 2. for all x , if x is a value of f , then it lies in the nature of x that
common. there is (or could be) some object y such that x = f (y).

One response to this difficulty is to transpose the abstractionist proposal We may then say that:
into a more metaphysical key (see Rosen & Yablo 2020). The idea is that
each Fregean number is, by its very nature, the number of some Fregean x is an abstraction if and only if, for some abstraction function f , there
concept, just as each Fregean direction is, by its very nature, at least is or could be an object y such that x = f (y),
potentially the direction of some concrete line. In each case, the abstract
and that:
object is essentially the value of an abstraction function for a certain class
of arguments. This is not a claim about the meanings of linguistic x is an abstract object if (and only if) x is an abstraction.
expressions. It is a claim about the essences or natures of the objects
themselves. (For the relevant notion of essence, see Fine 1994.) So for This account tells us a great deal about the distinctive natures of these
example, the Fregean number two (if there is such a thing) is, essentially, broadly Fregean abstract objects. It tells us that each is, by its very nature,
by its very nature, the number that belongs to a concept F if and only if the value of a special sort of function, one whose nature is specified in a
there are exactly two Fs. More generally, for each Fregean abstract object simple way in terms of an associated equivalence relation. It is worth
x , there is an abstraction function f , such that x is essentially the value of f stressing, however, that it does not supply much metaphysical information
for every argument of a certain kind. about these items. It doesn’t tell us whether they are located in space,

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whether they can stand in causal relations, and so on. It is an open artifacts owe their existence to the mind. What can this mean? One
question whether this somewhat unfamiliar version of the promising approach is to say that an object should be reckoned mind-
abstract/concrete distinction lines up with any of the more conventional dependent when, by its very nature, it exists at a time if and only if it is the
ways of drawing the distinction outlined above. An account along these object or content of some mental state or process at that time. This counts
lines would be at odds with standard usage, but may be philosophically tables and chairs as mind-independent, since they might survive the
interesting all the same. In any case, the problem remains that this annihilation of thinking things. But it counts paradigmatically mental
metaphysical version of the way of abstraction principles leaves out items, like a purple afterimage of which a person X may become aware, as
paradigmatic cases of abstract objects such as the aforementioned game of mind-dependent, since it presumably lies in the nature of such items to be
chess. objects of conscious awareness whenever they exist. However, it is not
clear that this account captures the full force of the intended notion.
3.5 The Ways of Negation Consider, for example, the mereological fusion of X’s afterimage and Y ’s
headache. This is surely a mental entity if anything is. But it is not
According to the way of negation, abstract objects are defined as those necessarily the object of a mental state. (The fusion can exist even if no
which lack certain features possessed by paradigmatic concrete objects. one is thinking about it.) A more generous conception would allow for
Many explicit characterizations in the literature follow this model. Let us mind-dependent objects that exist at a time in virtue of mental activity at
review some of the options. that time, even if the object is not the object of any single mental state or
act. The fusion of X’s afterimage and Y ’s headache is mind-dependent in
3.5.1 The Combined Criterion of Non-Mental and Non-Sensible the second sense but not the first. That is a reason to prefer the second
account of mind-dependence.
According to the account implicit in Frege’s writings:
If we understand the notion of mind-dependence in this way, it is a
An object is abstract if and only if it is both non-mental and non- mistake to insist that abstract objects be mind-independent. To strike a
sensible. theme that will recur, it is widely supposed that sets and classes are
abstract entities—even the impure sets whose urelements are concrete
Here the first challenge is to say what it means for a thing to be ‘non- objects. Any account of the abstract/concrete distinction that places set-
mental’, or as we more commonly say, ‘mind-independent’. The simplest theoretic constructions like {Alfred, {Betty, {Charlie, Deborah}}} on the
approach is to say that a thing depends on the mind when it would not (or concrete side of the line will be seriously at odds with standard usage.
could not) have existed if minds had not existed. But this entails that tables With this in mind, consider the set whose sole members are X’s afterimage
and chairs are mind-dependent, and that is not what philosophers who and Y’s headache, or some more complex set-theoretic object based on
employ this notion have in mind. To call an object ‘mind-dependent’ in a these items. If we suppose, as is plausible, that an impure set exists at a
metaphysical context is to suggest that it somehow owes its existence to time only when its members exist at that time, this will be a mind-
mental activity, but not in the boring ‘causal’ sense in which ordinary

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dependent entity in the generous sense. But it is also presumably an objects, in a broad sense, are concrete), there are mental entities that are
abstract entity. neither concrete nor abstract. As mentioned above (section 1.2), there is no
need to insist that the distinction is an exhaustive one. Second, while the
A similar problem arises for so-called abstract artifacts, like Jane Austen’s approach may well draw an important line, it inherits one familiar
novels and the characters that inhabit them. Some philosophers regard problem, namely, that of saying what it is for a thing to be a physical
such items as eternally existing abstract entities that worldly authors object (Crane and Mellor 1990; for discussion, see the entry on
merely describe but do not create. But of course the commonsensical view physicalism). In one sense, a physical entity is an entity in which physics
is that Austen created Pride and Prejudice and Elizabeth Bennett, and might take an interest. But physics is saturated with mathematics, so in
there is no good reason to deny this (Thomasson 1999; cf. Sainsbury this sense a great many paradigmatically abstract objects—e.g. π—will
2009). If we take this commonsensical approach, there will be a clear count as physical. The intended point is that abstract objects are to be
sense in which these items depend for their existence on Austen’s mental distinguished, not from all of the objects posited by physics, but from the
activity, and perhaps on the mental activity of subsequent readers.[5] These concrete objects posited by the physics. But if that is the point, it is
items may not count as mind-dependent in either of the senses canvassed unilluminating in the present context to say that abstract objects are non-
above, since Pride and Prejudice can presumably exist at a time even if no physical.
one happens to be thinking at that time. (If the world took a brief
collective nap, Pride and Prejudice would not pop out of existence.) But
3.5.2 The Non-Spatiality Criterion
they are obviously mind-dependent in some not-merely-causal sense. And
yet they are still presumably abstract objects. For these reasons, it is
Contemporary purveyors of the way of negation typically amend Frege’s
probably a mistake to insist that abstract objects be mind-independent.
criterion by requiring that abstract objects be non-spatial, causally
(For more on mind-dependence, see Rosen 1994, and the entry platonism
inefficacious, or both. Indeed, if any characterization of the abstract
in the philosophy of mathematics.)
deserves to be regarded as the standard one, is this:
Frege’s proposal in its original form also fails for other reasons. Quarks
An object is abstract if and only if it is non-spatial and causally
and electrons are usually considered neither sensible nor mind-dependent.
inefficacious.
And yet they are not abstract objects. A better version of Frege’s proposal
would hold that: This standard account nonetheless presents a number of perplexities.

An object is abstract if and only if it is both non-physical and non- First of all, one must consider whether there are abstract objects that have
mental. one of the two features but not the other. For example, consider an impure
set, such as the unit set of Plato (i.e., {Plato} ). It has some claim to being
Two remarks on this last version are in order. First, it opens the door to
abstract because it is causally inefficacious, but some might suggest that it
thinking that besides abstract and concrete entities (assuming that physical
has a location in space (namely, wherever Plato is located). Or consider a

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work of fiction such as Kafka’s The Metamorphosis. It, too, has some Other abstract objects appear to stand in a more interesting relation to
claim to being abstract because it (or at least its content) is non-spatial. spacetime. Consider the game of chess. Some philosophers will say that
But one might suggest that works of fiction as paradigmatic abstract chess is like a mathematical object, existing nowhere and ‘no when’—
objects seem to have causal powers, e.g., powers to affect us. either eternally or outside of time altogether. But the most natural view is
that chess was invented at a certain time and place (though it may be hard
In the remainder of this subsection, we focus on the first criterion in the to say exactly where or when); that before it was invented it did not exist
above proposal, namely, the non-spatial condition. But it gives rise to a at all; that it was imported from India into Persia in the 7th century; that it
subtlety. It seems plausible to suggest that, necessarily, if something x is has changed over the years, and so on. The only reason to resist this
causally efficacious, then (since x is a cause or has causal powers) x, or natural account is the thought that since chess is clearly an abstract object
some part of x , has a location in time. So if something has no location in —it’s not a physical object, after all!—and since abstract objects do not
time, it is causally inefficacious. The theory of relativity implies that space exist in space and time—by definition!—chess must resemble the cosine
and time are nonseparable, i.e., combined into a single spacetime function in its relation to space and time. And yet one might with equal
manifold. So the above proposal might be restated in terms of a single justice regard the case of chess and other abstract artifacts as
condition: an object is abstract if and only if it is non-spatiotemporal. counterexamples to the hasty view that abstract objects possess only trivial
Sometimes this revised proposal is the correct one for thinking about spatial and temporal properties.
abstract objects, but our discussion in the previous section showed that
abstract artifacts and mental events may be temporal but non-spatial. Should we then abandon the non-spatiotemporality criterion? Not
Given the complexities here, in what follows we use spatiotemporality, necessarily. Even if there is a sense in which some abstract entities possess
spatiality, or temporality, as needed. non-trivial spatiotemporal properties, it might still be said that concrete
entities exist in spacetime in a distinctive way. If we had an account of this
Some of the archetypes of abstractness are non-spatiotemporal in a distinctive manner of spatiotemporal existence characteristic of concrete
straightforward sense. It makes no sense to ask where the cosine function objects, we could say: An object is abstract (if and) only if it fails to exist
was last Tuesday. Or if it makes sense to ask, the sensible answer is that it in spacetime in that way.
was nowhere. Similarly, for many people, it makes no good sense to ask
when the Pythagorean Theorem came to be. Or if it does make sense to One way to implement this approach is to note that paradigmatic concrete
ask, the only sensible answer for them is that it has always existed, or objects tend to occupy a relatively determinate spatial volume at each time
perhaps that it does not exist ‘in time’ at all. It is generally assumed that at which they exist, or a determinate volume of spacetime over the course
these paradigmatic ‘pure abstracta’ have no non-trivial spatial or temporal of their existence. It makes sense to ask of such an object, ‘Where is it
properties; that they have no spatial location, and they exist nowhere in now, and how much space does it occupy?’ even if the answer must
particular in time. sometimes be somewhat vague. By contrast, even if the game of chess is
somehow ‘implicated’ in space and time, it makes no sense to ask how
much space it now occupies. (To the extent that this does make sense, the

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only sensible answer is that it occupies no space at all, which is not to say An object is abstract (if and) only if it either fails to occupy space at
that it occupies a spatial point.) And so it might be said: all, or does so only in virtue of the fact some other items—in this case,
its urelements—occupy that region.
An object is abstract (if and) only if it fails to occupy anything like a
determinate region of space (or spacetime). But of course Peter himself occupies a region in virtue of the fact that his
parts—his head, hands, etc.—together occupy that region. So a better
This promising idea raises several questions. First, it is conceivable that version of the proposal would say:
certain items that are standardly regarded as abstract might nonetheless
occupy determinate volumes of space and time. Consider, for example, the An object is abstract (if and) only if it either fails to occupy space at
various sets composed from Peter and Paul: {Peter, Paul}, all, or does so of the fact that some other items that are not among its
{Peter, {Peter, {{Paul}}}}, etc. We don’t normally ask where such things parts occupy that region.
are, or how much space they occupy. And indeed many philosophers will
say that the question makes no sense, or that the answer is a dismissive This approach appears to classify the cases fairly well, but it is somewhat
‘nowhere, none’. But this answer is not forced upon us by anything in set artificial. Moreover, it raises a number of questions. What are we to say
theory or metaphysics. Even if we grant that pure sets stand in only the about the statue that occupies a region of space, not because its parts are
most trivial relations to space, it is open to us to hold, as some arrayed in space, but rather because its constituting matter occupies that
philosophers have done, that impure sets exist where and when their region? And what about the unobserved electron, which according to some
members do (Lewis 1986a). It is not unnatural to say that a set of books is interpretations of quantum mechanics does not really occupy a region of
located on a certain shelf in the library, and indeed, there are some space at all, but rather stands in some more exotic relation to the spacetime
theoretical reasons for wanting to say this (Maddy 1990). On a view of it inhabits? Suffice it to say that a philosopher who regards ‘non-spatiality’
this sort, we face a choice: we can say that since impure sets exist in as a mark of the abstract, but who allows that some abstract objects may
space, they are not abstract objects after all; or we can say that since have non-trivial spatial properties, owes us an account of the distinctive
impure sets are abstract, it was a mistake to suppose that abstract objects relation to spacetime, space, and time that sets paradigmatic concreta
cannot occupy space. apart.

One way to finesse this difficulty would be to note that even if impure sets Perhaps the crucial question about the ‘non-spatiality’ criterion concerns
occupy space, they do so in a derivative manner. The set {Peter, Paul} the classification of the parts of space itself. If they are considered
occupies a location in virtue of the fact that its concrete elements, Peter concrete, then one might ask where the spatiotemporal points or regions
and Paul, together occupy that location. The set does not occupy the are located. And a similar question arises for spatial points and regions,
location in its own right. With that in mind, it might be said that: and for temporal instants or intervals. So, the ontological status of
spatiotemporal locations, and of spatial and temporal locations, is
problematic. Let us suppose that space, or spacetime, exists, not just as an

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object of pure mathematics, but as the arena in which physical objects and Concrete objects, whether mental or physical, have causal powers;
events are somehow arrayed. It is essential to understand that the problem numbers and functions and the rest make nothing happen. There is no such
is not about the numerical coordinates that represent these points and thing as causal commerce with the game of chess itself (as distinct from its
regions (or instants and intervals) in a reference system; the issue is about concrete instances). And even if impure sets do in some sense exist in
the points and regions (or instants and intervals). Physical objects are space, it is easy enough to believe that they make no distinctive causal
located ‘in’ or ‘at’ regions of space; as a result, they count as concrete contribution to what transpires. Peter and Paul may have effects
according to the non-spatiality criterion. But what about the points and individually. They may even have effects together that neither has on his
regions of space itself? There has been some debate about whether a own. But these joint effects are naturally construed as effects of two
commitment to spacetime substantivalism is consistent with the concrete objects acting jointly, or perhaps as effects of their mereological
nominalist’s rejection of abstract entities (Field 1980, 1989; Malament aggregate (itself usually regarded as concretum), rather than as effects of
1982). If we define the abstract as the ‘non-spatial’, this debate amounts to some set-theoretic construction. Suppose Peter and Paul together tip a
whether space itself is to be reckoned ‘spatial’. To reject that these points, balance. If we entertain the possibility that this event is caused by a set, we
regions, instants, and intervals, are concrete because they are not located, shall have to ask which set caused it: the set containing just Peter and
entails featuring them as abstract. However, to think about them as Paul? Some more elaborate construction based on them? Or is it perhaps
abstract sounds a bit weird, given their role in causal processes. Perhaps, it the set containing the molecules that compose Peter and Paul? This
is easier to treat them as concrete if we want to establish that concrete proliferation of possible answers suggests that it was a mistake to credit
entities are spatiotemporal—or spatial and temporal. sets with causal powers in the first place. This is good news for those who
wish to say that all sets are abstract.
The philosopher who believes that there is a serious question about
whether the parts of space-time count as concrete would thus do well to (Note, however, that some writers identify ordinary physical events—
characterize the abstract/concrete distinction in other terms. Still—as causally efficacious items par excellence—with sets. For David Lewis, for
mentioned above—the philosopher who thinks that it is defensible that example, an event like the fall of Rome is an ordered pair whose first
parts of space are concrete might use non-spatiality to draw the distinction member is a region of spacetime, and whose second member is a set of
if she manages to provide a way of accounting for how impure sets relate such regions (Lewis 1986b). On this account, it would be disastrous to say
to space differs from the way concreta do. both that impure sets are abstract objects, and that abstract objects are non-
causal.)
3.5.3 The Causal Inefficacy Criterion
The biggest challenge to characterizing abstracta as causally inefficacious
entities is that causality itself is a notoriously problematic and difficult to
According to the most widely accepted versions of the way of negation:
define idea. It is undoubtedly one of the most controversial notions in the
An object is abstract (if and) only if it is causally inefficacious. history of thought, with all kinds of views having been put forward on the
matter. Thus, causally efficacious inherits any unclarity that attaches to

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causality. So, if we are to move the discussion forward, we need to take hitting the window is an event in which the rock ‘participates’ in a certain
the notion of causation—understood as a relation among events—as way, and it is because the rock participates in events in this way that we
sufficiently clear, even though in fact it is not. Having acknowledged this credit the rock itself with causal efficacy. But what is it for an object to
no doubt naïve assumption, several difficulties arise for the suggestion that participate in an event? Suppose John is thinking about the Pythagorean
abstract objects are precisely the causally inefficacious objects. Theorem and you ask him to say what’s on his mind. His response is an
event—the utterance of a sentence; and one of its causes is the event of
The idea that causal inefficacy constitutes a sufficient condition for John’s thinking about the theorem. Does the Pythagorean Theorem
abstractness is somewhat at odds with standard usage. Some philosophers ‘participate’ in this event? There is surely some sense in which it does. The
believe in ‘epiphenomenal qualia’: objects of conscious awareness (sense event consists in John’s coming to stand in a certain relation to the
data), or qualitative conscious states that may be caused by physical theorem, just as the rock’s hitting the window consists in the rock’s
processes in the brain, but which have no downstream causal coming to stand in a certain relation to the glass. But we do not credit the
consequences of their own (Jackson 1982; Chalmers 1996). These items Pythagorean Theorem with causal efficacy simply because it participates
are causally inefficacious if they exist, but they are not normally regarded in this sense in an event which is a cause.
as abstract. The proponent of the causal inefficacy criterion might respond
by insisting that abstract objects are distinctively neither causes nor The challenge is therefore to characterize the distinctive manner of
effects. But this is perilous. Abstract artifacts like Jane Austen’s novels (as ‘participation in the causal order’ that distinguishes the concrete entities.
we normally conceive them) come into being as a result of human activity. This problem has received relatively little attention. There is no reason to
The same goes for impure sets, which come into being when their concrete believe that it cannot be solved, though the varieties of philosophical
urelements are created. These items are clearly effects in some good sense; analysis for the notion of causality make the task full of pitfalls. Anyway,
yet they remain abstract if they exist at all. It is unclear how the proponent in the absence of a solution, this standard version of the way of negation
of the strong version of the causal inefficacy criterion (which views causal must be reckoned a work in progress.
inefficacy as both necessary and sufficient for abstractness) might best
respond to this problem. 3.5.4 The Discernibility / Non-Duplication Criteria
Apart from this worry, there are no decisive intuitive counterexamples to
Some philosophers have supposed that, under certain conditions, there are
this account of the abstract/concrete distinction. The chief difficulty—and
numerically different but indiscernible concrete entities, i.e., that there are
it is hardly decisive—is rather conceptual. It is widely maintained that
distinct concrete objects x and y that exemplify the same properties. If this
causation, strictly speaking, is a relation among events or states of affairs.
can be sustained, then one might suggest that distinct abstract objects are
If we say that the rock—an object—caused the window to break, what we
always discernible or, in a weaker formulation, that distinct abstract
mean is that some event or state (or fact or condition) involving the rock
objects are never duplicates.
caused the break. If the rock itself is a cause, it is a cause in some
derivative sense. But this derivative sense has proved elusive. The rock’s

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Cowling (2017, 86–89) analyzes whether the abstract/concrete distinction has pointed out that “if two individuals are indiscernible then so are their
thus rendered is fruitful, though criteria in this line are normally offered as unit sets” (1986a, 84). If this is correct, {A} and {B} would be
glosses on the universal/particular distinction. As part of his analysis, he indiscernible, but (at least for some philosophers) distinct abstract objects,
deploys two pairs of (not uncontroversial) distinctions: (i) between contrary to the discernibility criterion.[8] It is possible to counter-argue
qualitative and non-qualitative properties, and (ii) between intrinsic and that we could happily accept impure sets as concrete; after all, it was
extrinsic properties. Roughly, a non-qualitative property is one that always a bit unclear how they should be classified. Obviously, this has the
involves specific individuals (e.g., being the teacher of Alexander the problematic consequence of having some sets—pure sets—as abstract and
Great, being Albert Einstein, etc.), while qualitative properties are not other sets—impure sets—as concrete. But the idea that abstract objects
(e.g., having mass, having a shape, having a length, etc.). Intrinsic have distinctive intrinsic natures allows one to establish a criterion less
properties are those an object has regardless of what other objects are like strong than that of discernibility; if an entity has a distinctive intrinsic
and regardless of its relationships with other objects (e.g., being made of nature, it cannot have a duplicate. So, the next criterion of non-duplication
copper). By contrast, an object’s extrinsic properties are those that depend can be put forward:
on other entities (e.g., being the fastest car).[6]
x is an abstract object iff it is impossible for there to be an object
With these distinctions in mind, it seems impossible that there be distinct which is a duplicate of x but distinct from x .
abstract entities which are qualitatively indiscernible; each abstract entity
is expected to have a unique, distinctive qualitative intrinsic nature (or But there is a more serious counterexample to this criterion, namely,
property), which is giving reason for its metaphysical being. This wouldn’t immanent universals. These are purportedly concrete objects, for they are
be the case for any concrete entity given the initial assumption in this universals wholly present where their instances are. But this criterion
section. Therefore, the following criterion of discernability could be renders them abstract. Take the color scarlet; it is a universal wholly
pondered: present in every scarlet thing. Each of the scarlets in those things is an
immanent universal. These are non-duplicable, but at least according to
x is an abstract object iff it is impossible for there to be an object Armstrong (1978, I, 77, though see 1989, 98–99), they are
which is qualitatively indiscernible from x but distinct from x. paradigmatically concrete: spatiotemporally located, causally efficacious,
etc. Despite how promising they initially seemed, the criteria of
However, one can develop a counterexample to the above proposal, by discernibility and non-duplicability do not appear to capture the
considering two concrete objects that are indiscernible with respect to their abstract/concrete distinction.
intrinsic qualitative properties. Cowling (2017) considers the case of a
possible world with only two perfectly spherical balls, A and B, that share
the same intrinsic qualitative properties and that are floating at a certain
distance from each other. So A and B are distinct concrete objects but
indiscernible in terms of their intrinsic qualitative properties.[7] But Lewis

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3.6 The Way of Encoding but they are not abstract, since abstract objects, like the number one,
couldn’t be concrete. Indeed, Zalta’s theory implies that abstract objects
One of the most rigorous proposals about abstract objects has been (A!) aren’t possibly concrete, since he defines them to be objects that
developed by Zalta (1983, 1988, and in a series of papers). It is a formal, aren’t ordinary (1993, 404):
axiomatic metaphysical theory of objects (both abstract and concrete), and
also includes a theory of properties, relations, and propositions. The theory O!x =df ◊E!x
explicitly defines the notion of an abstract object but also implicitly A!x =df ¬O!x
characterizes them using axioms.[9] There are three central aspects to the
Thus, the ordinary objects include all the concrete objects (since E!x
theory: (i) a predicate E! which applies to concrete entities and which is
implies ◊E!x), as well as possible objects that aren’t in fact concrete but
used to define a modal distinction between abstract and ordinary objects;
might have been. On this theory, therefore, being abstract is not the
(ii) a distinction between exemplifying relations and encoding properties
negation of being concrete. Instead, the definition validates an intuition
(i.e., encoding 1-place relations); and (iii) a comprehension schema that
that numbers, sets, etc., aren’t the kind of thing that could be concrete.
asserts the conditions under which abstract objects exist.
Though Zalta’s definition of abstract seems to comport with the way of
(i) Since the theory has both a quantifier ∃ and a predicate E!, Zalta offers primitivism—take concrete as primitive, and then define abstract as not
two interpretations of his theory (1983, 51–2; 1988, 103–4). On one possibly concrete—it differs in that (a) axioms are stated that govern the
interpretation, the quantifier ∃ simply asserts there is and the predicate E! conditions under which abstract objects exist (see below), and (b) the
asserts existence. On this interpretation, a formula such as ∃x¬E!x, which features commonly ascribed to abstract objects are derived from principles
is implied by the axioms described below, asserts “there is an object that that govern the property of being concrete. For example, Zalta accepts
fails to exist”. So, on this interpretation, the theory is Meinongian because principles such as: necessarily, anything with causal powers is concrete
it endorses non-existent objects. But there is a Quinean interpretation as (i.e., ◻∀x(Cx → E!x)). Then since abstract objects are, by definition,
well, on which the quantifier ∃ asserts existence and the predicate E! concrete at no possible world, they necessarily fail to have causal powers.
asserts concreteness. On this interpretation, the formula ∃x¬E!x asserts
(ii) The distinction between exemplifying and encoding is a primitive one
and is represented in the theory by two atomic formulas: F n x1 … xn
“there exists an object that fails to be concrete”. So, on this interpretation,
(x1 , … , xn exemplify F n ) and xF 1 (x encodes F 1 ). While both ordinary
the theory is Platonist, since it doesn’t endorse non-existents but rather
asserts the existence of non-concrete objects. We’ll henceforth use the
and abstract objects exemplify properties, only abstract objects encode
Quinean/Platonist interpretation.
properties;[10] it is axiomatic that ordinary objects necessarily fail to
In the more expressive, modal version of his theory, Zalta defines ordinary encode properties (O!x → ◻¬∃FxF). Zalta’s proposal can be seen a
objects (O!) to be those that might be concrete. The reason is that Zalta positive metaphysical proposal distinct from all the others we have
holds that possible objects (i.e., like million-carat diamonds, talking considered; a positive proposal that uses encoding as a key notion to
donkeys, etc.) are not concrete but rather possibly concrete. They exist, characterize abstract objects. On this reading, the definitions and axioms

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of the theory convey what is meant by encoding and how it works. relations, and propositions. Here we describe only the theory of properties.
Intuitively, an abstract object encodes the properties by which we define or It is governed by two principles: a comprehension principle for properties
conceive of it, but exemplifies some properties contingently and others and a principle of identity. The comprehension principle asserts that for
necessarily. Thus, the number 1 of Dedekind-Peano number theory any condition on objects expressible without encoding subformulas, there
encodes all and only its number-theoretic properties, and whereas it is a property F such that necessarily, an object x exemplifies F if and only
contingently exemplifies the property being thought about by Peano, it if x is such that ϕ , i.e., ∃F◻∀x(Fx ≡ ϕ) , where ϕ has no encoding
necessarily exemplifies properties such as being abstract, not having a subformulas and no free Fs. The identity principle asserts that properties F
shape, not being a building, etc. The distinction between exemplifying and and G are identical just in case F and G are necessarily encoded by the
encoding a property is also used to define identity: ordinary objects are same objects, i.e., F = G =df ◻∀x(xF ≡ xG) . This principle allows one to
identical whenever they necessarily exemplify the same properties while assert that there are properties that are necessarily equivalent in the
abstract objects are identical whenever they necessarily encode the same classical sense, i.e., in the sense that ◻∀x(Fx ≡ Gx), but which are
properties. distinct.[11]

(iii) The comprehension principle asserts that for each expressible Since α = β is defined both when α and β are both individual variables or
condition on properties, there is an abstract object that encodes exactly the both property variables, Zalta employs the usual principle for the
properties that fulfill (satisfy) that condition. Formally: substitution of identicals. Since all of the terms in his system are rigid,
∃x(A!x & ∀F(xF ≡ ϕ)), where ϕ has no free xs. Each instance of this substitution of identicals preserves truth even in modal contexts.
schema asserts the existence of an abstract object of a certain sort. So, for
example, where ‘s ’ denotes Socrates, the instance The foregoing principles implicitly characterize both abstract and ordinary
∃x(A!x & ∀F(xF ≡ Fs)) asserts that there is an abstract object that objects. Zalta’s theory doesn’t postulate any concrete objects, though,
encodes exactly the properties that Socrates exemplifies. Zalta uses this since that is a contingent matter. But his system does include the Barcan
object to analyze the complete individual concept of Socrates. But any formula (i.e., ◊∃xFx → ∃x◊Fx), and so possiblity claims like “there
condition ϕ on conditions on properties with no free occurrences of x can might have been talking donkeys” imply that there are (non-concrete)
be used to form an instance of comprehension. In fact, one can prove that objects at our world that are talking donkeys at some possible world. Since
the object asserted to exist is unique, since there can’t be two distinct Zalta adopts the view that ordinary properties like being a donkey
abstract objects that encode exactly the properties satisfying ϕ . necessarily imply concreteness, such contingently nonconcrete objects are
ordinary.
The theory that emerges from (i)–(iii) is further developed with additional
axioms and definitions. One axiom asserts that if an object encodes a Zalta uses his theory to analyze Plato’s Forms, concepts, possible worlds,
property, it does so necessarily (xF → ◻xF). So the properties that an Fregean numbers and Fregean senses, fictions, and mathematical objects
object encodes are not relative to any circumstance. Moreover, Zalta and relations generally. However, some philosophers see his
supplements his theory of abstract objects with a theory of properties, comprehension principle as too inclusive, for in addition to these objects,

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it asserts that there are entities like the round square or the set of all sets The traditional platonist conception is a realist one: abstract objects exist
which are not members of themselves. The theory doesn’t assert that in just the same full-blooded sense that objects in the natural world exist—
anything exemplifies being round and being square—the theory preserves they are mind-independent, rather than artifacts of human endeavor or
the classical form of predication without giving rise to contradictions. But dependent on concrete objects in any way. But a number of deflationary,
it does assert that there is an abstract object that encodes being round and metatontological views, now established in the literature, are based on the
being square, and that there is an abstract object that encodes the property idea that the problems traditional platonists face have to do with “some
of being a set that contains all and only non-self-membered sets. Zalta very general preconceptions about what it takes to specify an object”
would respond by suggesting that such objects are needed not only to state rather than with “the abstractness of the desired object” (Linnebo 2018,
truth conditions, and explain the logical consequences, of sentences 42). These views suggest that abstract objects exist in some weaker sense.
involving expressions like “the round square” and “the Russell set”, but Various approaches therefore articulate what may be called the ways of
also to analyze the fictional characters of inconsistent stories and weakening existence. One clear precedent is due to Carnap 1950 [1956],
inconsistent theories (e.g., Fregean extensions). whose deflationary approach may go the furthest; Carnap rejects the
metaphysical pursuit of what “really exists” (even in the case of concrete
It should be noted that Zalta’s comprehension principle for abstract objects objects) since he maintains that the question “Do Xs really exist?” are
is unrestricted and so constitutes a plenitude principle. This allows the pseudo-questions (if asked independently of some linguistic framework).
theory to provide objects for arbitrary mathematical theories. Where τ is a
term of mathematical theory T , the comprehension principle yields a But there are other ways to suggest that abstract objects have existence
unique object that encodes all and only the properties F that are attributed conditions that demand little of the world. For example, Linsky & Zalta
to τ in T (Linsky & Zalta 1995, Nodelman & Zalta 2014).[12] Zalta’s (1995, 532) argued that the mind-independence and objectivity of abstract
theory therefore offers significant explanatory power, for it has multiple objects isn’t like that of physical objects: abstract objects aren’t subject to
applications and advances solutions to a wide range of puzzles in different an appearance/reality distinction, they don’t exist in a ‘sparse’ way that
fields of philosophy.[13] requires discovery by empirical investigation, and they aren’t complete
objects (e.g., mathematical objects are defined only by their mathematical
3.7 The Ways of Weakening Existence properties). They use this conception to naturalize Zalta’s comprehension
principle for abstract objects.
Many philosophers have supposed that abstract objects exist in some thin,
deflated sense. In this section we consider the idea that the Other deflationary accounts develop some weaker sense in which abstract
abstract/concrete distinction might be defined by saying that abstract objects exist (e.g., as ‘thin’ objects). We further describe some of these
objects exist in some less robust sense than the sense in which concrete proposals below and try to unpack the ways in which they characterize the
objects exist. weakened, deflationary sense of existence (even when such
characterizations are not always explicit).

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3.7.1 The Criterion of Linguistic Rules Carnap’s paper (1950 [1956]) considers a variety of linguistic frameworks,
such as those for: observable things (i.e., the spatiotemporally ordered
Carnap held that claims about the “real” existence of entities (concrete or system of observable things and events), natural numbers and integers,
abstract) do not have cognitive content. They are pseudo-statements. propositions, thing properties, rational and real numbers, and
However, he admitted: (a) that there are sentences in science that use spatiotemporal coordinate systems. Each framework is established by
terms that designate mathematical entities (such as numbers); and (b) that developing a language that typically includes expressions for one or more
semantic analysis seems to require entities like properties and kinds of entities in question, expressions for properties of the entities in
propositions. Since mathematical entities, properties, and propositions are question (including a general category term for each kind of entity in
traditionally considered abstract, he wanted to clarify how it is possible to question), and variables ranging over those entities. Thus, a framework for
accept a language referring to abstract entities without adopting what he the system of observable things has expressions that denote such things
considered pseudo-sentences about such entities’ objective reality. (‘the Earth’, ‘the Eiffel Tower’, etc.), expressions for properties of such
Carnap’s famous paper (1950 [1956]) contained an attempt to show that, things (‘planet’, ‘made of metal’, etc.), and variables ranging over
without embracing Platonism, one can use a language referring to abstract observables. The framework for natural numbers has expressions that
entities. denote them (‘0’, ‘2+5’), expressions for properties of the numbers
(‘prime’, ‘odd’), including the general category term ‘number’), and
To achieve these goals, Carnap begins by noting that before one can ask variables ranging over numbers.
existence questions about entities of a determinate kind, one first has to
have a language, or a linguistic framework, that allows one to speak about For Carnap, each statement in a linguistic framework should have a truth
the kinds of entity in question. He then distinguishes ‘internal’ existence value that can be determined either by analytical or empirical methods. A
questions expressed within such a linguistic framework from ‘external’ statement’s truth value is analytically determinable if it is logically true (or
existence questions about a framework. Only the latter ask whether the false), or if it’s truth is determinable exclusively from the rules of the
entities of that framework are objectively real. As we’ll see below, Carnap language or on the basis of semantic relationships among its component
thought that internal existence questions within a framework can be expressions. A statement is empirically determinable when it is
answered, either by empirical investigation or by logical analysis, confirmable (or disconfirmable) in the light of the perceived evidence.
depending on the kind of entity the framework is about. By contrast, Note that the very attempt to confirm an empirical statement about
Carnap regards external questions (e.g., ‘Do Xs exist?’, expressed either physical objects on the basis of the evidence requires that one adopt the
about, or independent of, a linguistic framework) as pseudo-questions: language of the framework of things. Carnap warns us, however, that “this
though they appear to be theoretical questions, in fact they are merely must not be interpreted as if it meant … acceptance of a belief in the
practical questions about the utility of the linguistic framework for reality of the thing world; there is no such belief or assertion or
science. assumption because it is not a theoretical question” (1950 [1956, 208]).
For Carnap, to accept an ontology “means nothing more than to accept a
certain form of language, in other words to accept rules for forming

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statements and for testing, accepting, or rejecting them” (1950 [1956, All of the existence assertions just discussed are therefore internal to their
208]). respective linguistic frameworks. Carnap thinks that the only sense that
can be given to talk of “existence” is an internal sense. Internal questions
Carnap takes this approach to every linguistic framework, no matter about the existence of things or abstract objects are not questions about
whether it is a framework about physical, concrete things, or a framework their real metaphysical existence.[15] Hence, it seems more appropriate to
about abstract entities such as numbers, properties, concepts, propositions, describe his view as embodying a deflationary notion of object. For
etc. For him, the pragmatic reasons for accepting a given linguistic Carnap concludes “the question of the admissibility of entities of a certain
framework are that it has explanatory power, unifies the explanation of type or of abstract entities in general as designata is reduced to the
disparate kinds of data and phenomena, expresses claims more efficiently, question of the acceptability of the linguistic framework for those entities”
etc. And we often choose a framework for a particular explanatory (1950 [1956, 217]).
purpose. We might therefore choose a framework with expressions about
abstract entities to carry out an explication (i.e., an elucidation of Thus, for each framework (no matter whether it describes empirical
concepts), or to develop a semantics for natural language. For Carnap, the objects, abstract objects, or a mix of both), one can formulate both simple
choice between platonism or nominalism is not a legitimate one; both are and complex existential statements. According to Carnap, each simple
inappropriate attempts to answer an external pseudo-question. existential statement is either empirical or analytic. If a simple statement is
empirical, its truth value can be determined by a combination of empirical
As sketched earlier, the truth of such existence claims as ‘there are tables’ inquiry and consideration of the linguistic rules governing the framework;
and ‘there are unicorns’, which are expressed within the framework for if the simple existential statement is analytic, then its truth value can be
observable entities, is to be determined empirically, since empirical determined simply by considering the linguistic rules governing the
observations and investigations are needed. These statements are not true framework. Whereas the simple existential statements that require
in virtue of the rules of the language. By contrast, existence claims such as empirical investigation assert the existence of possible concrete entities
‘there are numbers’ (‘∃xNx ’) expressed within the framework of number (like ‘tables’ or ‘unicorns’), the simple existential statements that are
theory, or ‘there is a property F such that both x and y are F’ analytic assert the existence of abstract entities. Let us call this criterion
(‘∃F(Fx & Fy) ’) expressed within the framework of property theory, can for asserting the existence of abstract objects the criterion of linguistic
be determined analytically. For these statements either form part of the rules.
rules of the language (e.g., expressed as axioms that govern the terms of
the language) or are derivable from the rules of the language. When these The case of mixed frameworks poses some difficulties for the view.
statements are part of the rules that make up the linguistic framework, they According to the Criterion of Linguistic Rules,
are considered analytic, as are the existential statements that follow from
those rules.[14] x is abstract iff “x exists” is analytic in the relevant language.

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But this criterion suggests that impure sets, object-dependent properties, objects, she prefers a simpler kind of realism (see Thomasson 2015, 145–
abstract artifacts, and the rest are not abstract. For this criterion appears to 158). She argues that everyday uses of existential statements provide
draw a line between certain pure abstract entities and everything else. The acceptable ontological commitments when those assertions are supported
truth of simple existence statements about {Bob Dylan} or Dickens’ A either by empirical evidence or merely by the rules of use that govern
Christmas Carol, which usually are considered abstract entities, does not general terms (e.g., sortal terms); in both cases she says that “application
depend solely on linguistic rules. The same goes for simple and complex conditions” for a general term are fulfilled (see Thomasson 2015, 86, 89–
existential statements with general terms such as ‘novel’, ‘legal statute’, 95). She, too, therefore offers a criterion of linguistic rules for accepting
etc. abstract objects. Given her defense of simple realism, it appears that she
takes both observable objects and theoretical entities in science as
In the end, though, Carnap doesn’t seem to be either a realist or nominalist concrete.
about objects (abstract or concrete). Carnap rejects the question whether
these objects are real in a metaphysical sense. But, contrary to the
3.7.2 The Criterion of Minimalism
nominalist, he rejects the idea that we can truly deny the real existence of
abstract objects (i.e., a denial that is external to a linguistic framework).
In what follows, two ways of formulating criteria for the abstract/concrete
This attitude, which settles the question of which framework to adopt on
distinction are considered. The views start with the idea that our concept
pragmatic grounds (e.g., which framework best helps us to make sense of
of an object allows for objects whose existence places very few demands
the data to be explained), is the reason why we’ve labeled his view as a
on reality over and above the demands imposed by claims that do not
way of weakening existence. See the entry on Carnap for further details.
mention abstract objects. Those philosophers who maintain this
Proposals by other philosophers are related to Carnap’s view. Resnik philosophical thesis are what Linnebo (2012) calls metaontological
(1997, Part Two) has put forward a postulational epistemology for the minimalists. Their proposals are typically put forward in connection with
existence of mathematical objects. According to this view, all one has to issues in the philosophy of mathematics, but then applied to other
do to ensure the existence of mathematical objects is to use a language to domains.
posit mathematical objects and to establish a consistent mathematical
Parsons (1990), Resnik (1997), and Shapiro (1997) contend that, in the
theory for them.[16] Nevertheless, their existence does not result from their
case of mathematical theories, coherence suffices for the existence of the
being posited. Instead, we recognize those objects as existent because a
objects mentioned in those theories.[17] They do not offer an explicit
consistent mathematical theory for them has been developed. Resnik
criterion for distinguishing abstract and concrete objects. Nevertheless,
requires both a linguistic stipulation for considering mathematical objects
their proposals implicitly draw the distinction; abstract objects are those
and a coherency condition for recognizing them as existent. Thomasson
objects that exist in virtue of the truth of certain modal claims. In
(2015, 30–34) advocates for an approach which she takes to be inherited
particular, the existence of mathematical objects is “grounded in” pure
from Carnap. She calls it easy ontology. Since she is not trying to find
modal truths. For example, numbers exist “in virtue of” the fact that there
ultimate categories or a definitive list of basic (abstract or concrete)

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could have been an ω-sequence of objects; sets exist because there might It is important for Linnebo that sufficiency be asymmetric. He wouldn’t
be entities that satisfy the axioms of one or another set theory, etc. Since accept mutual sufficiency, i.e., principles of the form Rab ⇔ f (a) = f (b),
these pure modal truths are necessary, this explains why pure abstract since these would imply that both sides are equivalent as a matter of
objects exist necessarily. It also explains a sense in which they are meaning. Instead, the point is that the seemingly unproblematic claim Rab
insubstantial: their existence is grounded in truths that do not (on the face renders the claim f (a) = f (b) unproblematic, and this is best expressed by
of it) require the actual existence of anything at all.[18] sufficiency statements of the form Rab ⇒ f (a) = f (b), on which the left
side grounds the right side. So Linnebo’s notion of reconceptualization is
Linnebo (2018) advances a proposal about how to conceive abstract not the Fregean notion of recarving of content.
objects by revising our understanding of Fregean biconditional principles
of abstraction (see subsection 3.4). Some philosophers take these Fregean Moreover, in a sufficiency statement, Linnebo doesn’t require that the
abstraction principles to be analytic sentences. For example, Hale & relation R be an equivalence relation; he requires only that R be symmetric
Wright (2001; 2009) consider the two sides of an abstraction principle as and transitive. It need not be reflexive, for the domain might contain
equivalent as a matter of meaning; they ‘carve up content’ in different entities x such that ¬Rxx (e.g., in the case of the sufficiency statement for
ways (to use Frege’s metaphor). But Linnebo (2018, 13–14) rejects this directions, not every object x in the domain is such that x is parallel with
view and the view that such biconditional principles are analytic. x —being parallel is restricted to lines). Linnebo calls such symmetric and
transitive relations unity relations. When a sufficiency statement—
He suggests instead that we achieve reference to abstract (and other Rab ⇒ f (a) = f (b)—holds, then new objects are identified. The new
objects) by means of a sufficiency operator, ⇒ , which he takes to be a objects are specified in terms of the less problematic entities related by R;
strengthening of the material conditional. He starts with conditional for example, directions become specified by lines that are parallel.
principles of the form “if Rab, then f (a) = f (b) ” (e.g., “if a and b are According to Linnebo, the parallel lines become specifications of the new
parallel, then the direction of a = the direction of b”) and takes the right- objects. A unity relation R is therefore the starting point for developing a
hand side to be reconceptualization of the left-hand side. He represents sufficient (but not necessary and sufficient) condition for reference.
these claims as ϕ ⇒ ψ , where the new operator ‘⇒’ is meant to capture
the intuitive idea that ϕ is (conceptually) sufficient for ψ , or all that is Sometimes the new objects introduced by conditional principles do not
required for ψ is ϕ . For ϕ to be sufficient for ψ , sufficiency must be make demands on reality; when that happens, they are said to be thin (for
stronger than metaphysically implies but weaker than analytically implies example, directions only require that there be parallel lines). However,
(see Linnebo 2018, 15). The notion Linnebo considers is a ‘species of when the new objects introduced by sufficiency statements make more
metaphysical grounding’. Hence, sufficiency statements allow us to substantial demands on reality, the objects are considered thick. Suppose
conceptualize statements mentioning abstract objects (or other problematic Rab asserts a and b are spatiotemporal parts of the same cohesive and
objects) in terms of metaphysically less problematic or non-problematic naturally bounded whole. Then a and b become specifications for physical
objects. bodies via the following principle: Rab ⇒ Body(a) = Body(b) . In this
case, the principle “makes a substantial demand on the world” because it

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requires checking that there are spatiotemporal parts constituting a abstract objects of a mixed nature; namely, those that are thin relative to
continuous stretch of solid stuff (just looking at the spatiotemporal parts other objects. For example, the type of the letter ‘A’ is abstract because it
does not suffice to determine whether they constitute to a body; see is thin and has a shallow nature, but it is thin with respect the tokens of the
Linnebo 2018, 45). letter ‘A’.

However, Linnebo does not identify being abstract with being thin (2012, This view, as Linnebo himself admits, faces some problems. One of them
147), for there are thin objects in a relative sense that are not abstract, is that the methodologies used by working mathematicians, such as
namely those that make no substantial demands on the world beyond those classical logic, impredicative definitions, and taking arbitrary
introduced in terms of some antecedently given objects. The mereological subcollections of infinite domains, seem to presuppose objects that are
sum of your left hand and your laptop makes no demand on the world more independent, i.e., objects that don’t have a shallow nature (2018,
beyond the demands of its parts.[19] Instead, he suggests that abstract 197; for a discussion of independence, see Section 4.1 of the entry on
objects are those that are thin and that have a shallow nature. The notion platonism in mathematics). Another problem (2018, 195) is that in order
of shallow nature is meant to capture “the intuitive idea that any question for an object to count as having a shallow nature, an intrinsic unity
that is solely about Fs has an answer that can be determined on the basis relation has to be available. An investigation is required to establish that
of any given specifications of these Fs” (2018, 192–195). For example, there is such an intrinsic unity relation in each case. It is far from clear that
directions have a shallow nature because any question about directions a conditional principle with an intrinsic unity relation is available for each
(e.g., are they orthogonal, etc.?) can be determined solely on the basis of of problematic cases mentioned in this entry, such as chess, legal
the lines that specify them. Shapes have a shallow nature because any institutions or the English language. Finally, Linnebo doesn’t discuss the
question about them (e.g., are they triangular, circular, etc.?) can be question of whether sets of concrete urelements are themselves abstract or
determined solely on the basis of their underlying concrete figures. By concrete. At present, there may be an important question left open by his
contrast, mereological sums of concrete objects are not shallow because theory that other theories of abstract objects answer.
there are questions about them that cannot be answered solely on the basis
of their specifications; for instance, the weight of the mereological sum of 3.8 Eliminativism
your laptop and your left hand depends not only on their combination but
also on the gravitational field in which they are located.[20] We come finally to proposals that reject the abstract/concrete distinction.
We can consider three cases. First, there are the nominalists who both
Linnebo thus contrasts abstract objects, which are thin and have a shallow reject abstract entities and reject the distinction as illegitimate. They focus
nature, with concrete objects, which do not have a shallow nature. Linnebo on arguing against the formulations of the distinction proposed in the
extends this view in several ways. He constructs an account of literature. A second group of eliminativists reject real objects of any kind,
mathematical objects that goes beyond the way of abstraction principles thereby dismissing the distinction as irrelevant; these are the ontological
by providing a reconstruction of set theory in terms of ‘dynamic nihilists. A final group of eliminativists agree that there are prototypical
abstraction’ (2018, ch. 3). This form of minimalism also allows for cases of concrete objects and abstract objects, but conclude that a rigorous

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 Abstract Objects José L. Falguera, Concha Martı́nez-Vidal, and Gideon Rosen

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Press. object | physicalism | Platonism: in metaphysics | Platonism: in the

–––, 2014, Aboutness, Princeton: Princeton University Press. philosophy of mathematics | states of affairs | types and tokens
Zalta, Edward N., 1983, Abstract Objects: An Introduction to Axiomatic
Metaphysics, Dordrecht: D. Reidel. Acknowledgments
–––, 1988, Intensional Logic and Metaphysics of Intentionality,
Cambridge, MA: The MIT Press. This entry was revised, updated, and expanded in 2021 by José L.
–––, 1993, “Twenty-Five Basic Theorems in Situation and World Theory,” Falguera and Concha Martínez-Vidal. The author of the previous version
Journal of Philosophical Logic, 22: 385–428. of this entry, Gideon Rosen, remains credited on this entry since
–––, 2006, “Essence and Modality,” Mind, 115(459): 659–693. significant content in Sections 1, 2.1, 3.5.1–3.5.3, and 4 has been retained
–––, 2020, “Typed Object Theory,” in Falguera & Martínez-Vidal (eds.) from the previous version.
2020, pp. 59–88.
Notes to Abstract Objects
Academic Tools
1. See Swoyer 2007 and Cowling 2017 for further discussion of abstract
How to cite this entry. and concrete entities.
Preview the PDF version of this entry at the Friends of the SEP
Society. 2. Some (van Inwagen 1977, 1983; Kripke 1973 [2013]), analyze these
Look up topics and thinkers related to this entry at the Internet names as referring to theoretical entities of literary criticism or abstract
Philosophy Ontology Project (InPhO). objects, but only in so far as they occur in the context of statements about
Enhanced bibliography for this entry at PhilPapers, with links fictions (e.g., when we critically reflect on the characters of a story), while
to its database. others (Thomasson 1999) regard all the uses of these names as signifying
abstract artifacts (and thus ontologically dependent on concrete objects).
Other Internet Resources
3. Indispensability arguments in other areas of philosophical enquiry, such
Rayo, A., 2020, “The Ultra-Thin Conception of Objecthood,” as ethics, have also been developed. See Enoch 2011, 2016; Leibowitz
unpublished manuscript. & Sinclair 2016.

Related Entries 4. Lewis was trying to determine whether any of the ways for defining
abstract objects would categorize nonactual possible worlds (as he
analytic/synthetic distinction | Carnap, Rudolf | fictionalism | Frege, understood them) as abstract.
Gottlob | mathematics, philosophy of: indispensability arguments in the |
mathematics, philosophy of: nominalism | nominalism: in metaphysics | 5. Donato and Falguera 2020 make a similar case in connection with the
idea that scientific models are abstract artifacts.

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 Abstract Objects José L. Falguera, Concha Martı́nez-Vidal, and Gideon Rosen

6. The two distinctions allow us to consider four kinds of properties: even for the fictionalist version). His view is that there may be no fact of
qualitative intrinsic (e.g., being human); extrinsic qualitative (e.g., being the matter since the best arguments for platonism and the best arguments
the largest planet); intrinsic non-qualitative (e.g., being Barack Obama); for nominalism/fictionalism cancel each other out and are inconclusive.
non-qualitative extrinsic (e.g., being 5 cm taller than Napoleon
Bonaparte). 13. See, for example, Zalta 2006 for a discussion of how object theory
handles the case of the unit set of Socrates, i.e., {Socrates} .
7. Note, however, the limitations of the example, since the molecular
compositions of A and B cannot be the same. 14. These rules include syntactic and semantic rules, rules of reasoning,
and methods of proof.
8. Cowling (2017, 89) also discusses a ‘non-duplication’ criterion.
15. When the concept of reality occurs in internal questions about the
9. Some reminiscences of this proposal can be traced in the work of existence of things, then it “is an empirical non-metaphysical concept”
Meinong (1904, 1915) and in that of Mally (1912); more recently in (Carnap 1950 [56]).
Parsons’ attempt (1980) to reconstruct Meinong’s theory axiomatically.
16. See Fine 2005 for a different postulational proposal, namely his
10. The fact that only abstract objects encode properties cannot be used to procedural postulationist account of mathematical knowledge.
define abstract objects because it leaves out the null abstract object, an
abstract object which encodes no properties. 17. This criterion goes back to Hilbert, who suggested that mathematical
existence is guaranteed by consistency.
11. Zalta’s theory of properties extends to a theory of relations and
propositions. The comprehension principle for relations and propositions 18. See Rosen 2011 for an analysis of the notion of grounding in relation
is just a generalization of the principle for properties, extended to the cases to these issues.
where n ≥ 2 and n = 0. Relation and proposition identity are then defined
19. Thin objects in an absolute sense (i.e., pure sets, directions, numbers)
in terms of property identity. See Zalta 1983, 1988, and 1993.
simply make no substantial demand on the world (see Linnebo 2018, 4).
12. Balaguer (1998) also formulates a plenitude principle for what he calls
20. Linnebo does attempt to make the notion of having a shallow nature
‘full-blooded’ platonism. But his principle is primarily about mathematical
precise. He appeals to the notion of an intrinsic relation and the notion of a
objects and is a conditional one; his principle essentially asserts that every
relation being reducible on a sortal F. Intrinsic relations are similar to
possible mathematical object exists, though see Colyvan & Zalta 1999 for
intrinsic properties (which were discussed earlier, in Section 3.5.4); they
criticism. Balaguer (2020) asserts a similar principle for wrong-like
are relations that things bear to one another in virtue of how they are and
properties. In any case, Balaguer does not advocate for platonism, not
how they are related to each other, as opposed to how they are related to
even one based on a plenitude principle (though he agrees that the latter is
things outside of them and how things outside of them are (see the entry
the best version of platonism). Nor does he advocate for nominalism (not
on intrinsic vs. extrinsic properties). (For example, the relation of being

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parallel is intrinsic because of how the lines are related to each other, and
the relation of similarity, which may hold between concrete geometrical
figures, is intrinsic because of the shape of the figures.) The definition of a
relation being reducible on a sortal F occurs in 2018, 192–195, and we
omit the explanation here. But with these notions, Linnebo says that an
object x has a shallow nature if and only if there is a sortal F, such that x
is an F and all the intrinsic relations on x are reducible on F.

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68 Stanford Encyclopedia of Philosophy