## 1. Introduction



The abstract/concrete distinction has a curious status in contemporary
philosophy. It is widely agreed that the ontological distinction is of
fundamental importance, but as yet, there is no standard account of
how it should be drawn. There is a consensus about how to classify
certain paradigm cases. For example, it is usually acknowledged that
numbers and the other objects of pure mathematics, like pure sets, are
abstract (if they exist), whereas rocks, trees, and human beings are
concrete. In everyday language, it is common to use expressions that
refer to concrete entities as well as those that apparently refer to
abstractions such as democracy, happiness, motherhood, etc. Moreover,
formulations of mathematical theories seem to appeal directly to
abstract entities, and the use of mathematical expressions in the
empirical sciences seems indispensable to the formulation of our best
empirical theories (see Quine 1948; Putnam 1971; and the entry on
 indispensability arguments in the philosophy of mathematics).
 Finally, apparent reference to abstract entities such as sets,
properties, concepts, propositions, types, and possible worlds, among
others, is ubiquitous in different areas of philosophy.



Though there is a pervasive appeal to abstract objects, philosophers
have nevertheless wondered whether they exist. The alternatives are:
*platonism*, which endorses their existence, and
*nominalism*, which denies the existence of abstract objects
across the board. (See the entries on
 nominalism in metaphysics
 and
 platonism in metaphysics.)
 But the question of how to draw the distinction between abstract and
concrete objects is an open one: it is not clear how one should
characterize these two categories nor is there a definite list of
items that fall under one or the other category (assuming neither is
empty).



The first challenge, then, is to articulate the distinction, either by
defining the terms explicitly or by embedding them in a theory that
makes their connections to other important categories more explicit.
In the absence of such an account, the philosophical significance of
the contrast remains uncertain, for the attempt to classify things as
abstract or concrete by appeal to intuition is often problematic. Is
it clear that scientific theories (e.g., the general theory of
relativity), works of fiction (e.g., Dante's *Inferno*),
fictional characters (e.g., Bilbo Baggins) or conventional entities
(e.g., the International Monetary Fund or the Spanish Constitution of
1978) are abstract?



It should be stressed that there may not be one single
"correct" way of explaining the abstract/concrete
distinction. Any plausible account will classify the paradigm cases in
the standard way or give reasons for proceeding otherwise, and any
interesting account will draw a clear and philosophically significant
line in the domain of objects. Yet there may be many equally
interesting ways of accomplishing these two goals, and if we find
ourselves with two or more accounts that do the job rather well, there
may be no point in asking which corresponds to the real
abstract/concrete distinction. This illustrates a general point: when
technical terminology is introduced in philosophy by means of
examples, but without explicit definition or theoretical elaboration,
the resulting vocabulary is often vague or indeterminate in reference.
In such cases, it usually is pointless to seek a single correct
account. A philosopher may find herself asking questions like,
'What is idealism?' or 'What is a substance?'
and treating these questions as difficult questions about the
underlying nature of a certain determinate philosophical category. A
better approach may be to 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 not a precise account of what
this term already means, but rather a *proposal* for how it
might fruitfully be used for philosophical analysis. Anyone who
believes that something in the vicinity of the abstract/concrete
distinction matters for philosophy would be well advised to approach
the project of explaining the distinction with this in mind.



So before we turn to the discussion of abstract objects in earnest, it
will help if we clarify how some of the key terms will be used in what
follows.


### 1.1 About the Expression 'Object'



Frege famously distinguished two mutually exclusive ontological
domains, *functions* and *objects*. According to his
view, a function is an 'incomplete' entity that maps
arguments to values, and is denoted by an incomplete expression,
whereas an object is a 'complete' entity and can be
denoted by a singular term. Frege reduced properties and relations to
functions and so these entities are not included among the objects.
Some authors make use of Frege's notion of 'object'
when discussing abstract objects (e.g., Hale 1987). But though
Frege's sense of 'object' is important, it is not
the only way to use the term. Other philosophers include properties
and relations among the abstract objects. And when the background
context for discussing objects is type theory, properties and
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
interchangeable with
 'entity.'[1]
 Throughout this entry, we will follow this last usage and treat the
expressions 'object' and 'entity' as having
the same meaning. (For further discussion, see the entry on
 objects.)


### 1.2 About the Abstract/Concrete Distinction



Though we've spoken as if the abstract/concrete distinction must
be an exhaustive dichotomy, we should be open to the possibility that
the best 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 category; nor do ghosts, Cartesian minds,
fictional
 characters,[2]
 immanent universals, or tropes. The main constraint on an account of
the distinction is that it draws a philosophically significant line
that classifies at least many of the standard examples in the standard
ways. It is not a constraint that everything be shoehorned into one
category or the other.



Finally, some philosophers see the main distinction not as between
abstract and concrete objects but as between abstract objects and
*ordinary* objects, where the distinction is a modal one
- ordinary objects are possibly concrete while abstract objects
(like the number 1) *couldn't* be concrete (Zalta 1983,
1988). In any case, in the following discussion, we shall assume that
the abstract/concrete distinction is a division among existing
objects, and that any plausible explanation of the distinction should
aim to characterize a distinction among such objects.


## 2. Historical Remarks


### 2.1 The Provenance of the Distinction



The contemporary distinction between abstract and concrete is not an
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' distinction was a
distinction among words or terms. Traditional grammar distinguishes
the abstract noun 'whiteness' from the concrete noun
'white' without implying that this linguistic contrast
corresponds to a metaphysical distinction in what these words stand
for. In the 17th century, this grammatical distinction was transposed
to the domain of ideas. Locke speaks of the general idea of a triangle
which is "neither Oblique nor Rectangle, neither Equilateral,
Equicrural nor Scalenon [Scalene]; but all and none of these at
once," remarking that even this idea is not among the most
"abstract, comprehensive and difficult" (*Essay*,
IV.vii.9). Locke's conception of an abstract idea as one that is
formed from concrete ideas by the omission of distinguishing detail
was immediately rejected by Berkeley and then by Hume. But even for
Locke there was no suggestion that the distinction between abstract
ideas and concrete or particular ideas corresponds to a distinction
among *objects*. "It is plain, ..." Locke
writes, "that General and Universal, belong not to the real
existence of things; but are Inventions and Creatures of the
Understanding, made by it for its own use, and concern only signs,
whether Words or Ideas" (III.iii.11).



The abstract/concrete distinction in its modern form is meant to mark
a line in the domain of objects or entities. So conceived, the
distinction becomes a central focus for philosophical discussion
primarily in the 20th century. The origins of this development are
obscure, but one crucial factor appears to have been the breakdown of
the allegedly exhaustive distinction between mental and material
objects, which had formed the main division for ontologically-minded
philosophers since Descartes. One signal event in this development is
Frege's insistence that the objectivity and aprioricity of the
truths of mathematics entail that numbers are neither material beings
nor ideas in the mind. If numbers were material things (or properties
of material things), the laws of arithmetic would have the status of
empirical generalizations. If numbers were ideas in the mind, then the
same difficulty would arise, as would countless others. (Whose mind
contains the number 17? Is there one 17 in your mind and another in
mine? In that case, the appearance of a common mathematical subject
matter would be an illusion.) In *The Foundations of
Arithmetic* (1884), Frege concludes that numbers are neither
external *concrete* things nor mental entities of any sort.



Later, in his essay "The Thought" (1918), Frege claims the
same status for the items he calls *thoughts*--the senses
of declarative sentences--and also, by implication, for their
constituents, the senses of subsentential expressions. Frege does not
say that senses are *abstract*. He says that they belong to a
*third realm* distinct both from the sensible external world
and from the internal world of consciousness. Similar claims had been
made by Bolzano (1837), and later by Brentano (1874) and his pupils,
including Meinong and Husserl. The common theme in these developments
is the felt need in semantics and psychology, as well as in
mathematics, for a class of objective (i.e., non-mental) non-physical
entities. As this new *realism* was absorbed into
English-speaking philosophy, the traditional 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 entities, in the
broader sense that includes cultural products such as arguments,
theories, and works of art.



As we turn to an overview of the current debate, it is therefore
important to remember that the use of the terms *platonist*
(for those who affirm the existence of abstract objects) and
*nominalist* (for those who deny existence) is somewhat
lamentable, since these words have established senses in the history
of philosophy. These terms stood for positions that have little to do
with the modern notion of an abstract object. Modern platonists (with
a small 'p') need not accept any of the distinctive
metaphysical and epistemological doctrines of Plato, just as modern
nominalists need not accept the distinctive doctrines of the medieval
nominalists. Moreover, the literature also contains mention of
*anti-platonists*, many of whom see themselves as
*fictionalists* about abstracta, though this doesn't help
if it turns out that the best analysis of fictions is to regard them
as abstract objects. So the reader should therefore be aware that
terminology is not always well-chosen and that the terms so used
sometimes stand for doctrines that are more restricted than the
traditional doctrines that go by the same name. Henceforth, we simply
use *platonism* for the thesis that there exists at least one
abstract object, and *nominalism* for the thesis that the
number of abstract objects is exactly zero (Field 1980).


### 2.2 An Initial Overview of the Contemporary Debate



Before we survey the various proposals for drawing the
abstract/concrete distinction, we should briefly say why the
distinction has been thought to matter. Among philosophers who take
the distinction seriously, it is generally supposed that while
concrete objects clearly exist, abstract entities are problematic in
distinctive ways and deny the existence of abstract entities
altogether. In this section we briefly survey the arguments for
nominalism and the responses that platonists have offered. If the
abstract objects are unified as a class, it is because they have some
feature that generates what seems to be a distinctive problem--a
problem that nominalists deem unsolvable and which platonists aim to
solve. Before we ask what the unifying feature might be, it may
therefore help to characterize the various problems it has been
thought to generate.



The contemporary debate about platonism developed in earnest when
Quine argued (1948) that mathematical objects exist, having changed
his mind about the nominalist approach he had defended earlier
(Goodman & Quine 1947). Quine's 1948 argument involves
three key premises, all of which exerted significant influence on the
subsequent debate: (i) mathematics is indispensable for empirical
science; (ii) we should be ontologically committed to the entities
required for the truth of our best empirical theories (all of which
should be expressible in a first-order language); and (iii) the
entities required for the truth on an empirical theory are those in
the range of the variables bounded by its first-order quantifiers
(i.e., the entities in the domain of the existential quantifier
'\(\exists x\)' and the universal quantifier
'\(\forall x\)'). He concluded that in addition to the
concrete entities contemplated by our best empirical science, we must
accept the existence of mathematical entities, even if they are
abstract (see also Quine 1960, 1969, 1976).



Quine's argument initiated a debate that is still alive. Various
nominalist responses questioned one or another of the premises in his
argument. For instance, Field (1980) challenged the idea that
mathematics is indispensable for our best scientific
theories--i.e., rejecting (i) above--and thus faced the task
of rewriting classical and modern physics in nominalistic terms in
order to sustain the challenge. Others have taken on the somewhat less
daunting task of accepting (i) but rejecting (ii) and (iii);
they've argued that even if our best scientific theories, in
regimented form, quantify over mathematical entities, this
doesn't entail a commitment to mathematical entities (see
Azzouni 1997a, 1997b, 2004; Balaguer 1996, 1998; Maddy 1995, 1997;
Melia 2000, 2002; Yablo 1998, 2002, 2005, 2009; Leng 2010.) Colyvan
(2010) coined the expression 'easy-roaders' for this
second group, since they avoided the 'hard road' of
paraphrasing our best scientific theories in non-mathematical
terms.



By contrast, some mathematical platonists (Colyvan 2001; Baker 2005,
2009) have refined Quine's view by advancing the so-called
'Enhanced Indispensability Argument' (though see Saatsi
2011 for a response). Some participants describe the debate in terms
of a stalemate they hope to resolve (see Baker 2017, Baron 2016, 2020,
Knowles & Saatsi 2019, and Martinez-Vidal
& Rivas-de-Castro 2020, for
 discussion).[3]




Aside from the debate over Quine's argument, both platonism and
nominalism give rise to hard questions. Platonists not only need to
provide a theory of what abstract objects exist, but also an account
of how we cognitively access and come to know non-causal, abstract
entities. This latter question has been the subject of a debate that
began in earnest in Benacerraf (1973), which posed just such a dilemma
for mathematical objects. Benacerraf noted that the causal theory of
reference doesn't seem to make it possible to know the truth
conditions of mathematical statements, and his argument applies to
abstract entities more generally. On the other hand, nominalists need
to explain the linguistic uses in which we seem to appeal to such
entities, especially those uses in what appear to be good
explanations, such as those in scientific, mathematical, linguistic,
and philosophical pursuits (see Wetzel 2009, 1-22, for a
discussion of the many places where abstract *types* are used
in scientific explanations). Even though nominalists argue that there
are no abstracta, the very fact that there is disagreement about their
existence suggests that both platonists and nominalists acknowledge
the distinction between the abstract and concrete to be a meaningful
one.



On the platonist side, various proposals have been raised to address
the challenge of explaining epistemic access to abstract entities,
mostly in connection with mathematical objects. Some, including
Godel (1964), allege that we access abstract objects in virtue of
a unique kind of perception (intuition). Maddy (1990, 1997) developed
two rather different ways of understanding our knowledge of
mathematics in naturalistic ways. Other platonists have argued that
abstract objects are connected to empirical entities, either via
abstraction (Steiner 1975; Resnik 1982; Shapiro 1997) or via
abstraction principles (Wright 1983; Hale 1987); we'll discuss
some of these views below. There are also those who speak of existent
and intersubjective abstract entities as a kind of mental
representation (Katz 1980).



A rather different line of approach to the epistemological problem was
proposed in Linsky & Zalta 1995, where it is suggested that
one shouldn't attempt to explain knowledge of abstracta on the
same model that is used to explain knowledge of concrete objects. They
argue that not only a certain plenitude principle for abstract objects
(namely, the comprehension principle for abstract objects put forward
in Zalta 1983, 1988--see below) yields unproblematic
'acquaintance by description' to unique abstract objects
but also that their approach actually comports with naturalist
beliefs. Balaguer (1995, 1998) also suggests that a plenitude
principle is the best way forward for the platonist, and that our
knowledge of the consistency of mathematical theories suffices for
knowledge of mathematical objects. And there are views that conceive
of abstract objects as constituted by human--or, in general,
intelligent--subjects, or as abstract artifacts (see Popper 1968;
Thomasson 1999).



A number of nominalists have been persuaded by Benacerraf's
(1973) epistemological challenge about reference to abstract objects
and concluded that sentences with terms making apparent reference to
them--such as mathematical statements--are either false or
lack a truth value. They argue that those sentences must be
paraphrasable without vocabulary that commits one to any sort of
abstract entity. These proposals sometimes suggest that statements
about abstract objects are merely instrumental; they serve only to
help us establish conclusions about concrete objects. Field's
*fictionalism* (1980, 1989) has been influential in this
regard. Field reconstructed Newtonian physics using second-order logic
and quantification over (concrete) regions of space-time. A completely
different tactic for avoiding the commitment to abstract, mathematical
objects is put forward in Putnam (1967) and Hellman (1989), who
separately reconstructed various mathematical theories in second-order
*modal* logic. On their view, abstract objects aren't in
the range of the 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 worlds, where the axioms of the
mathematical theory in question are true.



These nominalistic approaches must contend with various issues, of
course. At the very least, they have to successfully argue that the
tools they use to avoid commitments to abstract objects don't
themselves involve such commitment. For example, Field must argue that
space-time regions are concrete entities, while Putnam and Hellman
must argue that by relying on logical possibility and modal logic,
there is no commitment to possible worlds considered as abstract
objects. In general, any nominalist account that makes essential use
of set theory or model-theoretic structures must convincingly argue
that the very use of such analytic tools doesn't commit them to
abstract objects. (See Burgess & Rosen 1997 for a systematic
survey of different proposals about the existence of abstract
objects.)



Another nominalistic thread in the literature concerns the suggestion
that sentences about (posited) abstract objects are quasi-assertions,
i.e., not evaluable as to truth or falsehood (see Yablo 2001 and
Kalderon 2005). Still others argue that we should not believe
sentences about abstracta since their function, much like the
instrumentalism discussed earlier, is to ensure empirical adequacy for
observational sentences (Yablo 1998). This may involve differentiating
between apparent content, which involves posited abstract objects, and
real content, which only concerns concrete objects (Yablo 2001, 2002,
2010, 2014). (For more on these fictionalist accounts, see Kalderon
2005, Ch. 3, and the entry on
 fictionalism.)



A final group of views in the literature represents a kind of
agnosticism about what exists or about what it is to be an object, be
it abstract or concrete. These views don't reject an external
material world, but rather begin with some question as to whether we
can have experience, observation, and knowledge of objects directly,
i.e., independent of our theoretical frameworks. Carnap (1950 [1956]),
for example, started with the idea that our scientific knowledge has
to be expressed with respect to a linguistic framework and that when
we wish to put forward a theory about a new kind of entity, we must
have a linguistic framework for talking about those entities. He then
distinguished two kinds of existence questions: internal questions
*within the framework* about the existence of the new entities
and external questions about the reality of the framework itself. If
the framework deals with abstract entities such as numbers, sets,
propositions, etc., then the internal question can be answered by
logical analysis of the rules of the language, such as whether it
includes terms for, or implies claims that quantify over, abstract
objects. But, for Carnap, the external question, about whether the
abstract entities really exist, is a pseudo-question and should be
regarded as nothing more than the pragmatic question of whether the
framework is a useful one to adopt, for scientific or other forms of
enquiry. We'll discuss Carnap's view in more detail in
 subsection 3.7.1.



Some have thought that Carnap's view offers a
*deflationist* view of objects, since it appears that the
existence of objects is not language independent. After Carnap's
seminal article, several other deflationist approaches were put
forward (Putnam 1987, 1990; Hirsch 2002, 2011; Sider 2007, 2009;
Thomasson 2015), many of them claiming to be a vindication of
Carnap's view. However, there are deflationist proposals that
run counter to Carnap's approach, among them, deflationary
nominalism (Azzouni 2010) or agnosticism about abstract objects (Bueno
2008a, 2008b, 2020). Additionally, philosophers inspired by
Frege's work have argued for a minimal notion of an object (Rayo
2013, Rayo 2020 [Other Internet Resources]; and Linnebo
2018). We'll discuss some of these in greater detail below, in
subsection 3.7.2. A final agnostic
position that has emerged is one that rejects a strict version of
platonism, but suggests that neither a careful analysis of
mathematical practice (Maddy 2011), nor the enhanced version of the
indispensability argument (Leng 2020) suffice to decide between
nominalism and moderate versions of platonism. Along these lines,
Balaguer (1998) concluded that the question doesn't have an
answer, since the arguments for 'full-blooded' platonism
can be matched one-for-one by equally good arguments by the
anti-platonist.



For additional discussion about the basic positions in the debate
about abstract and concrete objects, see Szabo 2003 and the
entries on
 nominalism in metaphysics
 and
 platonism in metaphysics,
 nominalism in the philosophy of mathematics and
 platonism in the philosophy of mathematics.


## 3. What is an Abstract Object?



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 abstract/concrete
 distinction.[4]
 These include: *the way of example* (which is simply to list
the paradigm cases of abstract and concrete objects in the hope that
the sense of the distinction will somehow emerge); *the way of
conflation* (i.e., identifying abstract and concrete objects with
some already-known distinction); *the way of negation* (i.e.,
saying what abstract objects are by saying what they are not, e.g.,
*non*-spatiotemporal, *non*-causal, etc.); and *the
way of abstraction* (i.e., saying that abstract objects are
conceptualized by a process of considering some known objects and
omitting certain distinguishing features). He gives a detailed
examination of the different proposals that typify these ways and then
attempts to show that none of them quite succeeds in classifying the
paradigms in accord with prevailing usage. Given the problems he
encountered when analyzing the various ways, Lewis became pessimistic
about our ability to draw the distinction cleanly.



Despite Lewis's pessimism about clarifying the abstract/concrete
distinction, his approach for categorizing the various proposals, when
extended, is a useful one. Indeed, in what follows, we'll see
that there are a number of additional *ways* that categorize
attempts to characterize the abstract/concrete distinction and
theorize about abstract objects. Even if there is no single,
acceptable account, these various ways of drawing the distinction and
theorizing about abstract objects do often cast light on the questions
we've been discussing, especially when the specific proposals
are integrated into a supplementary (meta-)ontological project. For
each method of drawing the distinction and specific proposal adopting
that method acquires a certain amount of explanatory power, and this
will help us to compare and contrast the various ideas that are now
found in the literature.


### 3.1 The Way of Example and the Way of Primitivism



According to the *way of example*, it suffices to list paradigm
cases of abstract and concrete entities in the hope that the sense of
the distinction will somehow emerge. Clearly, a list of examples for
each category would be a heuristically promising start in the search
for some criterion (or list of criteria) that would be fruitful for
drawing the distinction. However, a simple list would be of limited
significance since there are too many ways of extrapolating from the
paradigm cases to a distinction that would cover the unclear cases,
with the result that no clear notion has been explained.



For example, pure sets are paradigm examples of abstract entities. But
the case of impure sets is far from clear. Consider the unit set whose
sole member is Joe Biden (i.e., {Joe Biden}), the Undergraduate Class
of 2020 or The Ethics Committee, etc. They are sets, but it is not
clear that they are abstract given that Joe Biden, the members of the
class and committee are concrete. Similarly, if one offers the
characters of Sherlock Holmes stories as examples to help motivate the
primitive concept abstract object, then one has to wonder about the
object London that appears in the novels.



The refusal to characterize the abstract/concrete distinction while
maintaining that both categories have instances might be called the
*way of primitivism*, whenever the following condition obtains:
a few predicates are distinguished as primitive and unanalyzable, and
the explanatory power rests on the fact that other interesting
predicates can be defined in terms of the primitives and that
interesting claims can be judged true on the basis of our intuitive
understanding of the primitive and defined 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 why
abstract objects are necessarily existent, causally ineffiacious,
non-spatiotemporal, intersubjective, etc. (see Cowling 2017:
92-97).



But closer inspection of this method reveals some significant
concerns. To start with, when a distinction is taken as basic and
unanalyzable, one typically has to offer some intuitive instances of
the primitive predicates. But it is not always so easy to do this. For
example, when mathematicians take *set* and *membership*
as primitives and then assert some principles of set theory, they
often illustrate their primitives by offering some examples of sets,
such as The Undergraduate Class of 2020 or The Ethics Committee, etc.
But these, of course, aren't quite right, since the members of
the class and committee may change while the class and committee
remain the same, whereas if the members of a set change, one has a
different set. A similar concern affects the present proposal. If one
offers sets or the characters of the Sherlock Holmes novels as
examples to help 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., \(\{\textrm{Aristotle}\}\))
and the object London that appears in the novels.


### 3.2 The Way of Conflation



According to the *way of conflation*, the abstract/concrete
distinction is to be identified with one or another metaphysical
distinction already familiar under another name: as it might be, the
distinction between sets and individuals, or the distinction between
universals and particulars. There is no doubt that some authors have
used the terms in this way. (Thus Quine 1948 uses 'abstract
entity' and 'universal' interchangeably.) This sort
of conflation is however rare in recent philosophy.


### 3.3 The Way of Abstraction



Another methodology is what Lewis calls the *way of
abstraction*. According to a longstanding tradition in
philosophical psychology, abstraction is a distinctive mental process
in which new ideas or conceptions are formed by considering the common
features of several objects or ideas and ignoring the irrelevant
features that distinguish those objects. For example, if one is given
a range of white things of varying shapes and sizes; one ignores or
*abstracts from* the respects in which they differ, and thereby
attains the abstract idea of whiteness. Nothing in this tradition
requires that ideas formed in this way represent or correspond to a
distinctive kind of object. But it might be maintained that the
distinction between abstract and concrete objects should be explained
by reference to the psychological process of abstraction or something
like it. The simplest version of this strategy would be to say that an
object is abstract if it is (or might be) the referent of an abstract
idea; i.e., an idea formed by abstraction. So conceived, the *way
of abstraction* is wedded to an outmoded philosophy of mind.



It should be mentioned, though, that the key idea behind the *way
of abstraction* has resurfaced (though transformed) in the
structuralist views about mathematics that trace back to Dedekind.
Dedekind thought of numbers by the *way of abstraction*.
Dedekind suggested that when defining a number-theoretic structure,
"we entirely neglect the special character of the elements,
merely retaining their distinguishability and taking into account only
the relations to one another" (Dedekind 1888 [1963, 68]). This
view has led some structuralists to deny that numbers are abstract
objects. For example, Benacerraf concluded that "numbers are not
objects at all, because in giving the properties (that is, necessary
and sufficient) of numbers you merely characterize an *abstract
structure*--and 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
structure" (1965, 70). We shall therefore turn our attention to
a variant of the *way of abstraction*, one that has led a
number of philosophers to conclude that numbers are indeed abstract
objects.


### 3.4 The Way of Abstraction Principles



In the contemporary philosophical literature, a number of books and
papers have investigated a form of abstraction that doesn't
depend on mental processes. We may call this the *way of
abstraction principles*. Wright (1983) and Hale (1987) have
developed an account of abstract objects on the basis of an idea they
trace back to certain suggestive remarks in Frege (1884). Frege notes
(in effect) that many of the singular terms that appear to refer to
abstract entities are formed by means of functional expressions. We
speak of *the shape of* a building, *the direction of* a
line, *the number of* books on the shelf. Of course, many
singular terms formed by means of functional expressions denote
ordinary concrete objects: 'the father of Plato',
'the capital of France'. But the functional terms that
pick out abstract entities are distinctive in the following respect:
where \(f(a)\) is such an expression, there is typically an equation
of the form:



\[
f(a)\! =\! f(b) \text{ if and only if } Rab
\]


where \(R\) is an equivalence relation, i.e., a relation that is
reflexive, symmetric and transitive, relative to some domain. For
example:



The direction of \(a\) = the direction of \(b\) if and only if \(a\)
is parallel to \(b\)



The number of \(F\text{s}\) = the number of \(G\text{s}\) if and only
if there are just as many \(F\text{s}\) as \(G\text{s}\)



These biconditionals (or *abstraction principles*) appear to
have a special semantic status. While they are not strictly speaking
*definitions* of the functional expression that occurs on the
left hand side, they would appear 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 \(b\)
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 biconditional would appear to be semantically and
perhaps epistemologically prior to the functional expressions on the
left (Noonan 1978). Mastery of the concept of a direction presupposes
mastery of the concept of parallelism, but not vice versa.



The availability of abstraction principles meeting these conditions
may be exploited to yield an account of the distinction between
abstract and concrete objects. When '\(f\)' is a
functional expression governed by an abstraction principle, there will
be a corresponding kind \(K\_{f}\) such that:



\(x\) is a \(K\_{f}\) if and only if, for some \(y\), \(x\! =\!
f(y)\).



For example,



\(x\) is a cardinal number if and only if for some concept \(F\),
\(x\) = the number of \(F\text{s}\).



The simplest version of the *way of abstraction principles* is
then to say that:



\(x\) is an abstract object if (and only if) \(x\) is an instance of
some kind \(K\_{f}\) whose associated functional expression
'\(f\)' is governed by a suitable abstraction
principle.



The strong version of this account--which purports to identify a
necessary condition for abstractness--is seriously at odds with
standard usage. Pure sets are usually considered paradigmatic abstract
objects. But it is not clear that they satisfy the proposed criterion.
According to a version of naive set theory, the functional
expression '*set of*' is indeed characterized by a
putative abstraction principle.



The set of \(F\text{s}\) = the set of \(G\text{s}\) if and only if,
for all \(x\), \(x\) is \(F\) if and only if \(x\) is \(G\).



But this principle, which is a version of Frege's Basic Law V,
is inconsistent and so fails to characterize an interesting concept.
In contemporary mathematics, the concept of a set is not introduced by
an abstraction principle, but rather axiomatically. Though attempts
have been made to investigate abstraction principles for sets (Cook
2003), it remains an open question whether something like the
mathematical concept of a set can be characterized by a suitably
restricted abstraction principle. (See Burgess 2005 for a survey of
recent efforts in this direction.) Even if such a principle is
available, however, it is unlikely that the epistemological priority
condition will be satisfied. That is, it is unlikely that mastery of
the concept of set will presuppose mastery of the equivalence relation
that figures on the right hand side. It is therefore uncertain whether
the *way of abstraction principles* will classify the objects
of pure set theory as abstract entities (as it presumably must).



On the other hand, as Dummett (1973) has noted, in many cases the
standard names for paradigmatically abstract objects do not assume the
functional form to which the definition adverts. Chess is an abstract
entity, but we do not understand the word 'chess' as
synonymous with an expression of the form '\(f(x)\)',
where '\(f\)' is governed by an abstraction principle.
Similar remarks would seem to apply to such things as the English
language, social justice, architecture, and Charlie Parker's
musical style. If so, the abstractionist approach does not provide a
necessary condition for abstractness as that notion is standardly
understood.



More importantly, there is some reason to believe that it fails to
supply a *sufficient* condition. A mereological fusion of
concrete objects is itself a concrete object. But the concept of a
mereological fusion is governed by what appears to be an abstraction
principle:



The fusion of the \(F\text{s}\) = the fusion of the \(G\text{s}\) if
and only if the \(F\text{s}\) and \(G\text{s}\) cover one
another,



where the \(F\text{s}\) *cover* the \(G\text{s}\) if and only
if every part of every \(G\) has a part in common with an \(F\).
Similarly, suppose a train is a maximal string of railroad carriages,
all of which are connected to one another. We may define a functional
expression, 'the train of \(x\)', by means of an
'abstraction' principle: The train of \(x\) = the train of
\(y\) if and only if \(x\) and \(y\) are connected carriages. We may
then say that \(x\) is a train if and only if for some carriage \(y\),
\(x\) is the train of \(y\). The simple account thus yields the
consequence that trains are to be reckoned abstract entities.



It is unclear whether these objections apply to the more sophisticated
abstractionist proposals of Wright and Hale, but one feature of the
simple account sketched above clearly does apply to these proposals
and may serve as the basis for an objection to this version of the
*way of abstraction principles*. The neo-Fregean approach seeks
to explain the abstract/concrete distinction in *semantic*
terms: We said that an abstract object is an object that falls in the
range of a functional expression governed by an abstraction principle,
where '\(f\)' is *governed* by an abstraction
principle when that principle holds in virtue of the *meaning*
of '\(f\)'. This notion of a statement's holding in
virtue of the meaning of a word is notoriously problematic (see the
entry
 the analytic/synthetic distinction).
 But even if this notion makes sense, one may still complain: The
abstract/concrete distinction is supposed to be a metaphysical
distinction; abstract objects are supposed to differ from concrete
objects in some important ontological respect. It should be possible,
then, to draw the distinction directly in metaphysical terms: to say
what it is *in the objects themselves* that makes some things
abstract and others concrete. As Lewis writes, in response to a
related proposal by Dummett:



> 
> 
> Even if this ... way succeeds in drawing a border, as for all I know
> it may, it tells us nothing about how the entities on opposite sides
> of that border differ in their nature. It is like saying that snakes
> are the animals that we instinctively most fear--maybe so, but it
> tells us nothing about the nature of snakes. (Lewis 1986a: 82)
> 
> 
> 



The challenge is to produce a non-semantic version of the
abstractionist criterion that specifies directly, in metaphysical
terms, what the objects whose canonical names are governed by
abstraction principles all have in common.



One response to this difficulty is to transpose the abstractionist
proposal 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 concept, just as each Fregean
direction is, by *its* very nature, at least potentially the
direction of some concrete line. In each case, the abstract 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 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 example, the Fregean number two (if there is such a
thing) is, essentially, by its very nature, the number that belongs to
a concept \(F\) if and only if there are exactly two \(F\text{s}\).
More generally, for each Fregean abstract object \(x\), there is an
abstraction function \(f\), such that \(x\) is essentially the value
of \(f\) for every argument of a certain kind.



Abstraction functions have two key features. First, for each
abstraction function \(f\) there is an equivalence relation \(R\) such
that it lies in the nature of \(f\) that \(f(x)\! =\! f(y)\) iff
\(Rxy\). Intuitively, we are to think that \(R\) is metaphysically
prior to \(f\), and that the abstraction function \(f\) is
*defined* (in whole or in part) by this biconditional. Second,
each abstraction function is a *generating* function: its
values are essentially values of that function. Many functions are not
generating functions. Paris is the capital of France, but it is not
essentially a capital. The number of solar planets, by contrast, is
essentially a number. The notion of an abstraction function may be
defined in terms of these two features:



\(f\) is an *abstraction function* if and only if

1. for some equivalence relation \(R\), it lies in the nature of \(f\)
that \(f(x)\! =\! f(y)\) if and only if \(Rxy\); and
2. for all \(x\), if \(x\) is a value of \(f\), then it lies in the
nature of \(x\) that there is (or could be) some object \(y\) such
that \(x\! =\! f(y)\).




We may then say that:



\(x\) is an *abstraction* if and only if, for some abstraction
function \(f\), there is or could be an object \(y\) such that \(x\!
=\! f(y)\),



and that:



\(x\) is an abstract object if (and only if) \(x\) is an
abstraction.



This account tells us a great deal about the distinctive natures of
these broadly Fregean abstract objects. It tells us that each is, by
its very nature, the value of a special sort of function, one whose
nature is specified in a simple way in terms of an associated
equivalence relation. It is worth stressing, however, that it does not
supply much *metaphysical* information about these items. It
doesn't tell us whether they are located in space, whether they
can stand in causal relations, and so on. It is an open question
whether this somewhat unfamiliar version of the abstract/concrete
distinction lines up with any of the more conventional ways of drawing
the distinction outlined above. An account along these lines would be
at odds with standard usage, but may be philosophically interesting
all the same. In any case, the problem remains that this metaphysical
version of *the way of abstraction principles* leaves out
paradigmatic cases of abstract objects such as the aforementioned game
of chess.


### 3.5 The Ways of Negation



According to the *way of negation*, abstract objects are
defined as those which *lack* certain features possessed by
paradigmatic concrete objects. Many explicit characterizations in the
literature follow this model. Let us review some of the options.


#### 3.5.1 The Combined Criterion of Non-Mental and Non-Sensible



According to the account implicit in Frege's writings:



An object is abstract if and only if it is both non-mental and
non-sensible.



Here the first challenge is to say what it means for a thing to be
'non-mental', or as we more commonly say,
'mind-independent'. The simplest approach is to say that a
thing depends on the mind when it would not (or could not) have
existed if minds had not existed. But this entails that tables and
chairs are mind-dependent, and that is not what philosophers who
employ this notion have in mind. To call an object
'mind-dependent' in a metaphysical context is to suggest
that it somehow owes its existence to mental activity, but not in the
boring 'causal' sense in which ordinary artifacts owe
their existence to the mind. What can this mean? One promising
approach is to say that an object should be reckoned mind-dependent
when, by its very nature, it exists *at a time* if and only if
it is the object or content of some mental state or process *at
that time*. This counts tables and chairs as mind-independent,
since they might survive the annihilation of thinking things. But it
counts paradigmatically mental items, like a purple afterimage of
which a person \(X\) may become aware, as mind-dependent, since it
presumably lies in the nature of such items to be objects of conscious
awareness whenever they exist. However, it is not clear that this
account captures the full force of the intended notion. 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 necessarily the object of a mental state. (The fusion
can exist even if no one is thinking about *it*.) A more
generous conception would allow for mind-dependent objects that exist
at a time in virtue of mental activity at 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 the second sense but not the first. That is a reason
to prefer the second account of mind-dependence.



If we understand the notion of mind-dependence in this way, it is a
mistake to insist that abstract objects be mind-independent. To strike
a theme that will recur, it is widely supposed that sets and classes
are abstract entities--even the *impure* sets whose
urelements are concrete objects. Any account of the abstract/concrete
distinction that places set-theoretic constructions like
\(\{\textrm{Alfred}, \{\textrm{Betty}, \{\textrm{Charlie},
\textrm{Deborah}\}\}\}\) on the concrete side of the line will be
seriously at odds with standard usage. With this in mind, consider the
set whose sole members are X's afterimage and Y's
headache, or some more complex set-theoretic object based on these
items. If we suppose, as is plausible, that an impure set exists at a
time only when its members exist at that time, this will be a
mind-dependent entity in the generous sense. But it is also presumably
an abstract entity.



A similar problem arises for so-called *abstract artifacts*,
like Jane Austen's novels and the characters that inhabit them.
Some philosophers regard such items as eternally existing abstract
entities that worldly authors merely describe but do not create. But
of course the commonsensical view is that Austen created *Pride and
Prejudice* and Elizabeth Bennett, and there is no good reason to
deny this (Thomasson 1999; cf. Sainsbury 2009). If we take this
commonsensical approach, there will be a clear sense in which these
items depend for their existence on Austen's mental activity,
and perhaps on the mental activity of subsequent 
readers.[5] 
These items
may not count as mind-dependent in either of the senses canvassed
above, since *Pride and Prejudice* can presumably exist at a
time even if no 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 they are obviously mind-dependent in some
not-merely-causal sense. And yet they are still presumably abstract
objects. For these reasons, it is probably a mistake to insist that
abstract objects be mind-independent. (For more on mind-dependence,
see Rosen 1994, and the entry
 platonism in the philosophy of mathematics.)



Frege's proposal in its original form also fails for other
reasons. Quarks and electrons are usually considered neither sensible
nor mind-dependent. And yet they are not abstract objects. A better
version of Frege's proposal would hold that:



An object is abstract if and only if it is both non-physical and
non-mental.



Two remarks on this last version are in order. First, it opens the
door to thinking that besides abstract and concrete entities (assuming
that physical objects, in a broad sense, are concrete), there are
mental entities that are 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 approach may well draw an
important line, it inherits one familiar problem, namely, that of
saying what it is for a thing to be a *physical* object (Crane
and Mellor 1990; for discussion, see the entry on
 physicalism).
 In one sense, a physical entity is an entity in which physics might
take an interest. But physics is saturated with mathematics, so in
this sense a great many paradigmatically abstract objects--e.g.
\(\pi\)--will count as physical. The intended point is that
abstract objects are to be distinguished, not from *all* of the
objects posited by physics, but from the *concrete objects*
posited by the physics. But if that is the point, it is unilluminating
in the present context to say that abstract objects are
non-physical.


#### 3.5.2 The Non-Spatiality Criterion



Contemporary purveyors of the *way of negation* typically amend
Frege's criterion by requiring that abstract objects be
non-spatial, causally inefficacious, or both. Indeed, if any
characterization of the abstract deserves to be regarded as the
standard one, is this:



An object is abstract if and only if it is non-spatial and causally
inefficacious.



This standard account nonetheless presents a number of
perplexities.



First of all, one must consider whether there are abstract objects
that have one of the two features but not the other. For example,
consider an impure set, such as the unit set of Plato (i.e.,
\(\{\textrm{Plato}\}\)). It has some claim to being abstract because
it is causally inefficacious, but some might suggest that it has a
location in space (namely, wherever Plato is located). Or consider a
work of fiction such as Kafka's *The Metamorphosis*. It,
too, has some claim to being abstract because it (or at least its
content) is non-spatial. But one might suggest that works of fiction
as paradigmatic abstract objects seem to have causal powers, e.g.,
powers to affect us.



In the remainder of this subsection, we focus on the first criterion
in the above proposal, namely, the non-spatial condition. But it gives
rise to a subtlety. It seems plausible to suggest that, necessarily,
if something \(x\) is causally efficacious, then (since \(x\) is a
cause or has causal powers) \(x\), or some part of \(x\), has a
location in time. So if something has no location in time, it is
causally inefficacious. The theory of relativity implies that space
and time are nonseparable, i.e., combined into a single
*spacetime* manifold. So the above proposal might be restated
in terms of a single condition: an object is abstract if and only if
it is non-spatiotemporal. Sometimes this revised proposal is the
correct one for thinking about abstract objects, but our discussion in
the previous section showed that abstract artifacts and mental events
may be temporal but non-spatial. Given the complexities here, in what
follows we use spatiotemporality, spatiality, or temporality, as
needed.



Some of the archetypes of abstractness are non-spatiotemporal in a
straightforward sense. It makes no sense to ask where the cosine
function was last Tuesday. Or if it makes sense to ask, the sensible
answer is that it 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 ask, the only sensible answer for them is that it
has always existed, or perhaps that it does not exist 'in
time' at all. It is generally assumed that these paradigmatic
'pure abstracta' have no non-trivial spatial or temporal
properties; that they have no spatial location, and they exist nowhere
in particular in time.



Other abstract objects appear to stand in a more interesting relation
to spacetime. Consider the game of chess. Some philosophers will say
that chess is like a mathematical object, existing nowhere and
'no when'--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 to say exactly where or
when); that before it was invented it did not exist at all; that it
was imported from India into Persia in the 7th century; that it has
changed over the years, and so on. The only reason to resist this
natural account is the thought that since chess is clearly an abstract
object--it's not a physical object, after all!--and
since abstract objects do not exist in space and time--by
definition!--chess must resemble the cosine function in its
relation to space and time. And yet one might with equal justice
regard the case of chess and other abstract artifacts as
counterexamples to the hasty view that abstract objects possess only
trivial spatial and temporal properties.



Should we then abandon the non-spatiotemporality criterion? Not
necessarily. Even if there is a sense in which some abstract entities
possess 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 distinctive *manner* of
spatiotemporal existence characteristic of concrete objects, we could
say: An object is abstract (if and) only if it fails to exist in
spacetime *in that way*.



One way to implement this approach is to note that paradigmatic
concrete objects tend to occupy a relatively determinate spatial
volume at each time at which they exist, or a determinate volume of
spacetime over the course of their existence. It makes sense to ask of
such an object, 'Where is it now, and how much space does it
occupy?' even if the answer must 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 only sensible answer is that it occupies no space at all,
which is not to say that it occupies a spatial point.) And so it might
be said:



An object is abstract (if and) only if it fails to occupy anything
like a determinate region of space (or spacetime).



This promising idea raises several questions. First, it is conceivable
that certain items that are standardly regarded as abstract might
nonetheless occupy determinate volumes of space and time. Consider,
for example, the various sets composed from Peter and Paul:
\(\{\textrm{Peter}, \textrm{Paul}\},\) \(\{\textrm{Peter},
\{\textrm{Peter}, \{\{\textrm{Paul}\}\}\}\},\) etc. We don't
normally ask where such things 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 'nowhere, none'. But this
answer is not forced upon us by anything in set theory or metaphysics.
Even if we grant that *pure* sets stand in only the most
trivial relations to space, it is open to us to hold, as some
philosophers have done, that impure sets exist where and when their
members do (Lewis 1986a). It is not unnatural to say that a set of
books is located on a certain shelf in the library, and indeed, there
are some theoretical reasons for wanting to say this (Maddy 1990). On
a view of this sort, we face a choice: we can say that since impure
sets exist in space, they are not abstract objects after all; or we
can say that since impure sets are abstract, it was a mistake to
suppose that abstract objects cannot occupy space.



One way to finesse this difficulty would be to note that even if
impure sets occupy space, they do so in a derivative manner. The set
\(\{\textrm{Peter}, \textrm{Paul}\}\) occupies a location in virtue of
the fact that its concrete elements, Peter and Paul, together occupy
that location. The set does not occupy the location *in its own
right*. With that in mind, it might be said that:



An object is abstract (if and) only if it either fails to occupy space
at all, or does so only in virtue of the fact some other
items--in this case, its urelements--occupy that
region.



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 version of the proposal would say:



An object is abstract (if and) only if it either fails to occupy space
at all, or does so of the fact that some other items *that are not
among its parts* occupy that region.



This approach appears to classify the cases fairly well, but it is
somewhat artificial. Moreover, it raises a number of questions. What
are we to say about the statue that occupies a region of space, not
because its *parts* are arrayed in space, but rather because
its constituting *matter* occupies that region? And what about
the unobserved electron, which according to some interpretations of
quantum mechanics does not really *occupy* a region of space at
all, but rather stands in some more exotic relation to the spacetime
it inhabits? Suffice it to say that a philosopher who regards
'non-spatiality' as a mark of the abstract, but who allows
that some abstract objects may have non-trivial spatial properties,
owes us an account of the *distinctive relation* to spacetime,
space, and time that sets paradigmatic concreta apart.



Perhaps the crucial question about the 'non-spatiality'
criterion concerns the classification of the parts of space itself. If
they are considered concrete, then one might ask where the
spatiotemporal points or regions are located. And a similar question
arises for spatial points and regions, 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 object of pure
mathematics, but as the arena in which physical objects and events are
somehow arrayed. It is essential to understand that the problem is not
about the numerical coordinates that represent these points and
regions (or instants and intervals) in a reference system; the issue
is about the points and regions (or instants and intervals). Physical
objects are located 'in' or 'at' regions of
space; as a result, they count as concrete according to the
non-spatiality criterion. But what about the points and regions of
space itself? There has been some debate about whether a commitment to
spacetime substantivalism is consistent with the nominalist's
rejection of abstract entities (Field 1980, 1989; Malament
1982). If we define the abstract as the 'non-spatial',
this debate amounts to whether space itself is to be reckoned
'spatial'. To reject that these points, regions, instants,
and intervals, are concrete because they are not *located*,
entails featuring them as abstract. However, to think about them as
abstract sounds a bit weird, given their role in causal processes.
Perhaps, it is easier to treat them as concrete if we want to
establish that concrete entities are spatiotemporal--or spatial
and temporal.



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 characterize the abstract/concrete distinction in other terms.
Still--as mentioned above--the philosopher who thinks that
it is defensible that parts of space are concrete might use
non-spatiality to draw the distinction if she manages to provide a way
of accounting for how impure sets relate to space differs from the way
concreta do.


#### 3.5.3 The Causal Inefficacy Criterion



According to the most widely accepted versions of the *way of
negation*:



An object is abstract (if and) only if it is causally
inefficacious.



Concrete objects, whether mental or physical, have causal powers;
numbers and functions and the rest make nothing happen. There is no
such thing as causal commerce with the game of chess itself (as
distinct from its concrete instances). And even if impure sets do in
some sense exist in space, it is easy enough to believe that they make
no *distinctive* causal contribution to what transpires. Peter
and Paul may have effects individually. They may even have effects
together that neither has on his own. But these joint effects are
naturally construed as effects of two concrete objects acting jointly,
or perhaps as effects of their mereological aggregate (itself usually
regarded as concretum), rather than as effects of some set-theoretic
construction. Suppose Peter and Paul together tip a balance. If we
entertain the possibility that this event is caused by a set, we shall
have to ask which set caused it: the set containing just Peter and
Paul? Some more elaborate construction based on them? Or is it perhaps
the set containing the molecules that compose Peter and Paul? This
proliferation of possible answers suggests that it was a mistake to
credit sets with causal powers in the first place. This is good news
for those who wish to say that all sets are abstract.



(Note, however, that some writers identify ordinary physical
events--causally efficacious items par excellence--with
sets. For David Lewis, for example, an event like the fall of Rome is
an ordered pair whose first member is a region of spacetime, and whose
second member is a set of such regions (Lewis 1986b). On this account,
it would be disastrous to say both that impure sets are abstract
objects, and that abstract objects are non-causal.)



The biggest challenge to characterizing abstracta as causally
inefficacious entities is that *causality* itself is a
notoriously problematic and difficult to define idea. It is
undoubtedly one of the most controversial notions in the history of
thought, with all kinds of views having been put forward on the
matter. Thus, *causally efficacious* inherits any unclarity
that attaches to *causality*. So, if we are to move the
discussion forward, we need to take the notion of
causation--understood as a relation among events--as
sufficiently clear, even though in fact it is not. Having acknowledged
this no doubt naive assumption, several difficulties arise for
the suggestion that abstract objects are precisely the causally
inefficacious objects.



The idea that causal inefficacy constitutes a *sufficient*
condition for abstractness is somewhat at odds with standard usage.
Some philosophers believe in 'epiphenomenal qualia':
objects of conscious awareness (sense data), or qualitative conscious
states that may be caused by physical processes in the brain, but
which have no downstream causal consequences of their own (Jackson
1982; Chalmers 1996). These items are causally inefficacious if they
exist, but they are not normally regarded as abstract. The proponent
of the causal inefficacy criterion might respond by insisting that
abstract objects are distinctively neither causes *nor
effects*. But this is perilous. Abstract artifacts like Jane
Austen's novels (as we normally conceive them) come into being
*as a result* of human activity. The same goes for impure sets,
which come into being when their concrete urelements are created.
These items are clearly *effects* in some good sense; yet they
remain abstract if they exist at all. It is unclear how the proponent
of the strong version of the causal inefficacy criterion (which views
causal inefficacy as both necessary and sufficient for abstractness)
might best respond to this problem.



Apart from this worry, there are no decisive intuitive counterexamples
to this account of the abstract/concrete distinction. The chief
difficulty--and it is hardly decisive--is rather conceptual.
It is widely maintained that causation, strictly speaking, is a
relation among events or states of affairs. If we say that the
rock--an object--caused the window to break, what we mean is
that some event or state (or fact or condition) *involving* the
rock 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 hitting the window is an event in which the rock
'participates' in a certain way, and it is because the
rock participates in events in this way that we credit the rock itself
with causal efficacy. But what is it for an object to
*participate* in an event? Suppose John is thinking about the
Pythagorean 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 John's thinking about the theorem.
Does the Pythagorean Theorem 'participate' in this event?
There is surely *some* sense in which it does. The event
consists in John's coming to stand in a certain relation to the
theorem, just as the rock's hitting the window consists in the
rock's coming to stand in a certain relation to the glass. But
we do not credit the Pythagorean Theorem with causal efficacy simply
because it participates in this sense in an event which is a
cause.



The challenge is therefore to characterize the distinctive manner of
'participation in the causal order' that distinguishes the
concrete entities. This problem has received relatively little
attention. There is no reason to believe that it cannot be solved,
though the varieties of philosophical analysis for the notion of
*causality* make the task full of pitfalls. Anyway, in the
absence of a solution, this standard version of the *way of
negation* must be reckoned a work in progress.


#### 3.5.4 The Discernibility / Non-Duplication Criteria



Some philosophers have supposed that, under certain conditions, there
are numerically different but indiscernible concrete entities, i.e.,
that there are distinct concrete objects \(x\) and \(y\) that
exemplify the same properties. If this can be sustained, then one
might suggest that distinct abstract objects are always discernible
or, in a weaker formulation, that distinct abstract objects are never
duplicates.



Cowling (2017, 86-89) analyzes whether the abstract/concrete
distinction thus rendered is fruitful, though criteria in this line
are normally offered as glosses on the universal/particular
distinction. As part of his analysis, he deploys two pairs of (not
uncontroversial) distinctions: (i) between qualitative and
non-qualitative properties, and (ii) between intrinsic and extrinsic
properties. Roughly, a non-qualitative property is one that involves
specific individuals (e.g., *being the teacher of Alexander the
Great*, *being Albert Einstein*, etc.), while qualitative
properties are not (e.g., *having mass*, *having a
shape*, *having a length*, etc.). Intrinsic properties are
those an object has regardless of what other objects are like and
regardless of its relationships with other objects (e.g., *being
made of copper*). By contrast, an object's extrinsic
properties are those that depend on other entities (e.g., *being
the fastest
 car*).[6]




With these distinctions in mind, it seems impossible that there be
distinct abstract entities which are qualitatively indiscernible; each
abstract entity is expected to have a unique, distinctive qualitative
intrinsic nature (or property), which is giving reason for its
metaphysical being. This wouldn't be the case for any concrete
entity given the initial assumption in this section. Therefore, the
following criterion of discernability could be pondered:



\(x\) is an abstract object iff it is impossible for there to be an
object which is qualitatively indiscernible from \(x\) but distinct
from \(x\).



However, one can develop a counterexample to the above proposal, by
considering two concrete objects that are indiscernible with respect
to their 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 has pointed out that "if two individuals are
indiscernible then so are their unit sets" (1986a, 84). If this
is correct, \(\{A\}\) and \(\{B\}\) would be indiscernible, but (at
least for some philosophers) distinct abstract objects, contrary to
the discernibility criterion.[8] It is possible to counter-argue that we could
happily accept impure sets as concrete; after all, it was always a bit
unclear how they should be classified. Obviously, this has the
problematic consequence of having some sets--pure sets--as
abstract and other sets--impure sets--as concrete. But the
idea that abstract objects have distinctive intrinsic natures allows
one to establish a criterion less strong than that of discernibility;
if an entity has a distinctive intrinsic nature, it cannot have a
duplicate. So, the next criterion of non-duplication can be put
forward:



\(x\) is an abstract object iff it is impossible for there to be an
object which is a duplicate of \(x\) but distinct from \(x\).



But there is a more serious counterexample to this criterion, namely,
immanent universals. These are purportedly concrete objects, for they
are universals wholly present where their instances are. But this
criterion renders them abstract. Take the color scarlet; it is a
universal wholly 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 Armstrong (1978, I, 77, though see 1989,
98-99), they are paradigmatically concrete: spatiotemporally
located, causally efficacious, etc. Despite how promising they
initially seemed, the criteria of discernibility and non-duplicability
do not appear to capture the abstract/concrete distinction.


### 3.6 The Way of Encoding



One of the most rigorous proposals about abstract objects has been
developed by Zalta (1983, 1988, and in a series of papers). It is a
formal, axiomatic metaphysical theory of objects (both abstract and
concrete), and also includes a theory of properties, relations, and
propositions. The theory explicitly defines the notion of an
*abstract object* but also implicitly characterizes them using
 axioms.[9]
 There are three central aspects to the theory: (i) a predicate \(E!\)
which applies to concrete entities and which is used to define a modal
distinction between *abstract* and *ordinary* objects;
(ii) a distinction between *exemplifying* relations and
*encoding* properties (i.e., encoding 1-place relations); and
(iii) a comprehension schema that asserts the conditions under which
abstract objects exist.



(i) Since the theory has both a quantifier \(\exists\) and a predicate
\(E!\), Zalta offers two interpretations of his theory (1983,
51-2; 1988, 103-4). On one interpretation, the quantifier
\(\exists\) simply asserts *there is* and the predicate \(E!\)
asserts *existence*. On this interpretation, a formula such as
\(\exists x \neg E!x\), which is implied by the axioms described
below, asserts "there is an object that fails to exist".
So, on this interpretation, the theory is Meinongian because it
endorses non-existent objects. But there is a Quinean interpretation
as well, on which the quantifier \(\exists\) asserts
*existence* and the predicate \(E!\) asserts
*concreteness*. On this interpretation, the formula \(\exists x
\neg E!x\) asserts "there exists an object that fails to be
concrete". So, on this interpretation, 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
Quinean/Platonist interpretation.



In the more expressive, modal version of his theory, Zalta defines
*ordinary* objects \((O!)\) to be those that might be concrete.
The reason is that Zalta holds that possible objects (i.e., like
million-carat diamonds, talking donkeys, etc.) are not concrete but
rather possibly concrete. They exist, but they are not abstract, since
abstract objects, like the number one, couldn't be concrete.
Indeed, Zalta's theory implies that abstract objects \((A!)\)
aren't possibly concrete, since he defines them to be objects
that aren't ordinary (1993, 404):



\[\begin{align}
O!x &=\_\mathit{df} \Diamond E!x \\
A!x &=\_\mathit{df} \neg O!x
\end{align}\]


Thus, the ordinary objects include all the concrete objects (since
\(E!x\) implies \(\Diamond E!x\)), as well as possible objects that
aren't in fact concrete but might have been. On this theory,
therefore, *being abstract* is not the negation of *being
concrete*. Instead, the definition validates an intuition that
numbers, sets, etc., aren't the kind of thing that could be
concrete. Though Zalta's definition of *abstract* seems
to comport with the *way of primitivism*--take
*concrete* as primitive, and then define *abstract* as
*not possibly concrete*--it differs in that (a) axioms are
stated that govern the conditions under which abstract objects exist
(see below), and (b) the features commonly ascribed to abstract
objects are *derived* from principles that govern the property
of being concrete. For example, Zalta accepts principles such as:
necessarily, anything with causal powers is concrete (i.e., \(\Box
\forall x(Cx \to E!x)\)). Then since abstract objects are, by
definition, concrete at no possible world, they necessarily fail to
have causal powers.



(ii) The distinction between *exemplifying* and
*encoding* is a primitive one and is represented in the theory
by two atomic formulas: \(F^nx\_1\ldots x\_n\) \((x\_1,\ldots ,x\_n\)
*exemplify* \(F^n\)) and \(xF^1\) \((x\) *encodes*
\(F^1).\) While both ordinary and abstract objects exemplify
properties, only abstract objects encode
 properties;[10]
 it is axiomatic that ordinary objects necessarily fail to encode
properties \((O!x \to \Box \neg \exists FxF).\) Zalta's proposal
can be seen a positive metaphysical proposal distinct from all the
others we have considered; a positive proposal that uses encoding as a
key notion to characterize abstract objects. On this reading, the
definitions and axioms of the theory convey what is meant by
*encoding* and how it works. Intuitively, an abstract object
*encodes* the properties by which we define or conceive of it,
but exemplifies some properties contingently and others necessarily.
Thus, the number 1 of Dedekind-Peano number theory encodes all and
only its number-theoretic properties, and whereas it contingently
exemplifies the property *being thought about by Peano*, it
necessarily exemplifies properties such as *being abstract*,
*not having a shape*, *not being a building*, etc. The
distinction between exemplifying and encoding a property is also used
to define identity: ordinary objects are identical whenever they
necessarily exemplify the same properties while abstract objects are
identical whenever they necessarily encode the same properties.



(iii) The comprehension principle asserts that for each expressible
condition on properties, there is an abstract object that
*encodes* exactly the properties that fulfill (satisfy) that
condition. Formally: \(\exists x(A!x \:\&\: \forall F(xF \equiv
\phi))\), where \(\phi\) has no free \(x\)s. Each instance of this
schema asserts the existence of an abstract object of a certain sort.
So, for example, where '\(s\)' denotes Socrates, the
instance \(\exists x(A!x \:\&\: \forall F(xF \equiv Fs))\) asserts
that there is an abstract object that encodes exactly the properties
that Socrates exemplifies. Zalta uses this object to analyze the
complete individual concept of Socrates. But any condition \(\phi\) on
conditions on properties with no free occurrences of \(x\) can be used
to form an instance of comprehension. In fact, one can prove that the
object asserted to exist is unique, since there can't be two
distinct abstract objects that encode exactly the properties
satisfying \(\phi\).



The theory that emerges from (i)-(iii) is further developed
with additional axioms and definitions. One axiom asserts that if an
object encodes a property, it does so necessarily \((xF \to \Box
xF).\) So the properties that an object encodes are not relative to
any circumstance. Moreover, Zalta supplements his theory of abstract
objects with a theory of properties, relations, and propositions. Here
we describe only the theory of properties. It is governed by two
principles: a comprehension principle for properties and a principle
of identity. The comprehension principle asserts that for any
condition on objects expressible without encoding subformulas, there
is a property \(F\) such that necessarily, an object \(x\) exemplifies
\(F\) if and only if \(x\) is such that \(\phi\), i.e., \(\exists
F\Box \forall x(Fx \equiv \phi)\), where \(\phi\) has no encoding
subformulas and no free \(F\text{s}\). The identity principle asserts
that properties \(F\) and \(G\) are identical just in case \(F\) and
\(G\) are necessarily encoded by the same objects, i.e., \(F\! =\! G
=\_\mathit{df} \Box \forall x(xF \equiv xG)\). This principle allows
one to assert that there are properties that are necessarily
equivalent in the classical sense, i.e., in the sense that \(\Box
\forall x(Fx \equiv Gx)\), but which are
 distinct.[11]




Since \(\alpha\! =\! \beta\) is defined both when \(\alpha\) and
\(\beta\) are both individual variables or both property variables,
Zalta employs the usual principle for the substitution of identicals.
Since all of the terms in his system are rigid, substitution of
identicals preserves truth even in modal contexts.



The foregoing principles implicitly characterize both abstract and
ordinary objects. Zalta's theory doesn't postulate any
concrete objects, though, since that is a contingent matter. But his
system does include the Barcan formula (i.e., \(\Diamond \exists xFx
\to \exists x\Diamond Fx\)), and so possiblity claims like
"there might have been talking donkeys" imply that there
are (non-concrete) objects at our world that are talking donkeys at
some possible world. Since Zalta adopts the view that ordinary
properties like being a donkey necessarily imply concreteness, such
*contingently nonconcrete* objects are ordinary.



Zalta uses his theory to analyze Plato's Forms, concepts,
possible worlds, Fregean numbers and Fregean senses, fictions, and
mathematical objects and relations generally. However, some
philosophers see his comprehension principle as too inclusive, for in
addition to these objects, it asserts that there are entities like
*the round square* or *the set of all sets which are not
members of themselves*. The theory doesn't assert that
anything *exemplifies* being round and being square--the
theory preserves the classical form of predication without giving rise
to contradictions. But it does assert that there is an abstract object
that encodes being round and being square, and that there is an
abstract object that encodes the property of being a set that contains
all and only non-self-membered sets. Zalta would respond by suggesting
that such objects are needed not only to state truth conditions, and
explain the logical consequences, of sentences involving expressions
like "the round square" and "the Russell set",
but also to analyze the fictional characters of inconsistent stories
and inconsistent theories (e.g., Fregean extensions).



It should be noted that Zalta's comprehension principle for
abstract objects is unrestricted and so constitutes a
*plenitude* principle. This allows the theory to provide
objects for arbitrary mathematical theories. Where \(\tau\) is a term
of mathematical theory \(T\), the comprehension principle yields a
unique object that encodes all and only the properties \(F\) that are
attributed to \(\tau\) in \(T\) (Linsky & Zalta 1995,
Nodelman & Zalta
 2014).[12]
 Zalta's theory therefore offers significant explanatory power,
for it has multiple applications and advances solutions to a wide
range of puzzles in different fields of
 philosophy.[13]



### 3.7 The Ways of Weakening Existence



Many philosophers have supposed that abstract objects exist in some
thin, deflated sense. In this section we consider the idea that the
abstract/concrete distinction might be *defined* by saying that
abstract objects exist in some less robust sense than the sense in
which concrete objects exist.



The traditional platonist conception is a realist one: abstract
objects exist in just the same full-blooded sense that objects in the
natural world exist--they are mind-independent, rather than
artifacts of human endeavor or dependent on concrete objects in any
way. But a number of deflationary, metatontological views, now
established in the literature, are based on the idea that the problems
traditional platonists face have to do with "some very general
preconceptions about what it takes to specify an object" rather than
with "the abstractness of the desired object" (Linnebo 2018,
42). These views suggest that abstract objects exist in some weaker
sense. Various approaches therefore articulate what may be called the
ways of weakening existence. One clear precedent is due to Carnap 1950
[1956], whose deflationary approach may go the furthest; Carnap
rejects the metaphysical pursuit of what "really exists"
(even in the case of concrete objects) since he maintains that the
question "Do \(X\text{s}\) really exist?" are
pseudo-questions (if asked independently of some linguistic
framework).



But there are other ways to suggest that abstract objects have
existence conditions that demand little of the world. For example,
Linsky & Zalta (1995, 532) argued that the mind-independence
and objectivity of abstract objects isn't like that of physical
objects: abstract objects aren't subject to an
appearance/reality distinction, they don't exist in a
'sparse' way that requires discovery by empirical
investigation, and they aren't complete objects (e.g.,
mathematical objects are defined only by their mathematical
properties). They use this conception to *naturalize*
Zalta's comprehension principle for abstract objects.



Other deflationary accounts develop some weaker sense in which
abstract objects exist (e.g., as 'thin' objects). We
further describe some of these proposals below and try to unpack the
ways in which they characterize the weakened, deflationary sense of
existence (even when such characterizations are not always
explicit).


#### 3.7.1 The Criterion of Linguistic Rules



Carnap held that claims about the "real" existence of
entities (concrete or abstract) do not have cognitive content. They
are pseudo-statements. However, he admitted: (a) that there are
sentences in science that use terms that designate mathematical
entities (such as numbers); and (b) that semantic analysis seems to
require entities like properties and propositions. Since mathematical
entities, properties, and propositions are traditionally considered
abstract, he wanted to clarify how it is possible to accept a language
referring to abstract entities without adopting what he considered
pseudo-sentences about such entities' objective reality.
Carnap's famous paper (1950 [1956]) contained an attempt to show
that, without embracing Platonism, one can use a language referring to
abstract entities.



To achieve these goals, Carnap begins by noting that before one can
ask 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 the kinds of entity in question. He then
distinguishes 'internal' existence questions expressed
within such a linguistic framework from 'external'
existence questions about a framework. Only the latter ask whether the
entities of that framework are objectively real. As we'll see
below, Carnap thought that internal existence questions within a
framework can be answered, either by empirical investigation or by
logical analysis, depending on the kind of entity the framework is
about. By contrast, Carnap regards external questions (e.g., 'Do
\(X\text{s}\) exist?', expressed either about, or independent
of, a linguistic framework) as pseudo-questions: though they appear to
be theoretical questions, in fact they are merely practical questions
about the utility of the linguistic framework for science.



Carnap's paper (1950 [1956]) considers a variety of linguistic
frameworks, such as those for: observable things (i.e., the
spatiotemporally ordered system of observable things and events),
natural numbers and integers, propositions, thing properties, rational
and real numbers, and spatiotemporal coordinate systems. Each
framework is established by developing a language that typically
includes expressions for one or more kinds of entities in question,
expressions for properties of the entities in question (including a
general category term for each kind of entity in question), and
variables ranging over those entities. Thus, a framework for the
system of observable things has expressions that denote such things
('the Earth', 'the Eiffel Tower', etc.),
expressions for properties of such things ('planet',
'made of metal', etc.), and variables ranging over
observables. The framework for natural numbers has expressions that
denote them ('0', '2+5'), expressions for
properties of the numbers ('prime', 'odd'),
including the general category term 'number'), and
variables ranging over numbers.



For Carnap, each statement in a linguistic framework should have a
truth value that can be determined either by analytical or empirical
methods. A statement's truth value is analytically determinable
if it is logically true (or false), or if it's truth is
determinable exclusively from the rules of the language or on the
basis of semantic relationships among its component expressions. A
statement is empirically determinable when it is confirmable (or
disconfirmable) in the light of the perceived evidence. Note that the
very attempt to confirm an empirical statement about physical objects
on the basis of the evidence requires that one adopt the language of
the framework of things. Carnap warns us, however, that "this
must not be interpreted as if it meant ... acceptance of a
*belief* in the reality of the thing world; there is no such
belief or assertion or 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 statements and for testing,
accepting, or rejecting them" (1950 [1956, 208]).



Carnap takes this approach to every linguistic framework, no matter
whether it is a framework about physical, concrete things, or a
framework about abstract entities such as numbers, properties,
concepts, propositions, etc. For him, the pragmatic reasons for
accepting a given linguistic framework are that it has explanatory
power, unifies the explanation of disparate kinds of data and
phenomena, expresses claims more efficiently, etc. And we often choose
a framework for a particular explanatory purpose. We might therefore
choose a framework with expressions about abstract entities to carry
out an *explication* (i.e., an elucidation of concepts), or to
develop a semantics for natural language. For Carnap, the choice
between platonism or nominalism is not a legitimate one; both are
inappropriate attempts to answer an external pseudo-question.



As sketched earlier, the truth of such existence claims as
'there are tables' and 'there are unicorns',
which are expressed within the framework for observable entities, is
to be determined empirically, since empirical observations and
investigations are needed. These statements are not true in virtue of
the rules of the language. By contrast, existence claims such as
'there are numbers' ('\(\exists xNx\)')
expressed within the framework of number theory, or 'there is a
property \(F\) such that both \(x\) and \(y\) are \(F\)'
('\(\exists F(Fx \:\&\:
Fy)\)') expressed within the framework of property
theory, can be determined analytically. For these statements either
form part of the 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 statements are part of the rules that make up
the linguistic framework, they are considered *analytic*, as
are the existential statements that follow from those
rules.[14]




All of the existence assertions just discussed are therefore
*internal* to their respective linguistic frameworks. Carnap
thinks that the only sense that can be given to talk of
"existence" is an internal sense. Internal questions about
the existence of things or abstract objects are not questions about
their real metaphysical
 existence.[15]
 Hence, it seems more appropriate to describe his view as embodying a
deflationary notion of object. For Carnap concludes "the
question of the admissibility of entities of a certain type or of
abstract entities in general as designata is reduced to the question
of the acceptability of the linguistic framework for those
entities" (1950 [1956, 217]).



Thus, for each framework (no matter whether it describes empirical
objects, abstract objects, or a mix of both), one can formulate both
simple and complex existential statements. According to Carnap, each
simple existential statement is either empirical or analytic. If a
simple statement is empirical, its truth value can be determined by a
combination of empirical inquiry and consideration of the linguistic
rules governing the framework; if the simple existential statement is
analytic, then its truth value can be determined simply by considering
the linguistic rules governing the framework. Whereas the simple
existential statements that require empirical investigation assert the
existence of possible concrete entities (like 'tables'
or 'unicorns'), the simple existential statements that are
analytic assert the existence of abstract entities. Let us call this
criterion for asserting the existence of abstract objects the
*criterion of linguistic rules*.



The case of mixed frameworks poses some difficulties for the view.
According to the Criterion of Linguistic Rules,



\(x\) is abstract iff "\(x\) exists" is analytic in the
relevant language.



But this criterion suggests that *impure* sets,
object-dependent properties, abstract artifacts, and the rest are not
abstract. For this criterion appears to draw a line between certain
pure abstract entities and everything else. The truth of simple
existence statements about \(\{\textrm{Bob Dylan}\}\) or
Dickens' *A Christmas Carol*, which usually are
considered abstract entities, does not depend solely on linguistic
rules. The same goes for simple and complex existential statements
with general terms such as 'novel', 'legal
statute', etc.



In the end, though, Carnap doesn't seem to be either a realist
or nominalist about objects (abstract or concrete). Carnap rejects the
question whether these objects are real in a metaphysical sense. But,
contrary to the 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). This attitude, which settles the
question of which framework to adopt on pragmatic grounds (e.g., which
framework best helps us to make sense of the data to be explained), is
the reason why we've labeled his view as a *way of weakening
existence*. See the entry on
 Carnap
 for further details.



Proposals by other philosophers are related to Carnap's view.
Resnik (1997, Part Two) has put forward a *postulational
epistemology* for the existence of mathematical objects. According
to this view, all one has to do to ensure the existence of
mathematical objects is to use a language to posit mathematical
objects and to establish a consistent mathematical theory for
 them.[16]
 Nevertheless, their existence does not result from their being
posited. Instead, we recognize those objects as existent because a
consistent mathematical theory for them has been developed. Resnik
requires both a linguistic stipulation for considering mathematical
objects and a coherency condition for recognizing them as existent.
Thomasson (2015, 30-34) advocates for an approach which she
takes to be inherited from Carnap. She calls it *easy
ontology*. Since she is not trying to find ultimate categories or
a definitive list of basic (abstract or concrete) objects, she prefers
a simpler kind of realism (see Thomasson 2015, 145-158). She
argues that everyday uses of existential statements provide acceptable
ontological commitments when those assertions are supported either by
empirical evidence or merely by the rules of use that govern general
terms (e.g., sortal terms); in both cases she says that
"application conditions" for a general term are fulfilled
(see Thomasson 2015, 86, 89-95). She, too, therefore offers a
*criterion of linguistic rules* for accepting abstract objects.
Given her defense of simple realism, it appears that she takes both
observable objects and theoretical entities in science as
concrete.


#### 3.7.2 The Criterion of Minimalism



In what follows, two ways of formulating criteria for the
abstract/concrete distinction are considered. The views start with the
idea that our concept of an object allows for objects whose existence
places very few demands on reality over and above the demands imposed
by claims that do not mention abstract objects. Those philosophers who
maintain this philosophical thesis are what Linnebo (2012) calls
metaontological *minimalists*. Their proposals are typically
put forward in connection with issues in the philosophy of
mathematics, but then applied to other domains.



Parsons (1990), Resnik (1997), and Shapiro (1997) contend that, in the
case of mathematical theories, coherence suffices for the existence of
the objects mentioned in those
 theories.[17]
 They do not offer an explicit criterion for distinguishing abstract
and concrete objects. Nevertheless, their proposals implicitly draw
the distinction; abstract objects are those objects that exist in
virtue of the truth of certain modal claims. In particular, the
existence of mathematical objects is "grounded in" pure
modal truths. For example, numbers exist "in virtue of"
the fact that there could have been an \(\omega\)-sequence of objects;
sets exist because there might be entities that satisfy the axioms of
one or another set theory, etc. Since these pure modal truths are
necessary, this explains why pure abstract objects exist necessarily.
It also explains a sense in which they are insubstantial: their
existence is grounded in truths that do not (on the face of it)
require the actual existence of anything at
 all.[18]




Linnebo (2018) advances a proposal about how to conceive abstract
objects by revising our understanding of Fregean biconditional
principles of abstraction (see
 subsection 3.4).
 Some philosophers take these Fregean abstraction principles to be
analytic sentences. For example, Hale & Wright (2001; 2009)
consider the two sides of an abstraction principle as equivalent as a
matter of meaning; they 'carve up content' in different
ways (to use Frege's metaphor). But Linnebo (2018, 13-14)
rejects this view and the view that such biconditional principles are
analytic.



He suggests instead that we achieve reference to abstract (and other
objects) by means of a *sufficiency operator*, \(\Rightarrow\),
which he takes to be a strengthening of the material conditional. He
starts with *conditional* principles of the form "if
\(Rab\), then \(f(a) \! =\! f(b)\)" (e.g., "if \(a\) and
\(b\) are parallel, then the direction of \(a\) = the direction of
\(b\)") and takes the right-hand side to be reconceptualization
of the left-hand side. He represents these claims as \(\phi
\Rightarrow \psi\), where the new operator
'\(\Rightarrow\)' is meant to capture the intuitive idea
that \(\phi\) *is* (*conceptually*) *sufficient
for* \(\psi\), or *all that is required for*
\(\psi\) *is* \(\phi\). For \(\phi\) to be sufficient for
\(\psi\), sufficiency must be stronger than
*metaphysically implies* but weaker than *analytically
implies* (see Linnebo 2018, 15). The notion Linnebo considers is a
'species of metaphysical grounding'. Hence, sufficiency
statements allow us to conceptualize statements mentioning abstract
objects (or other problematic objects) in terms of metaphysically less
problematic or non-problematic objects.



It is important for Linnebo that sufficiency be asymmetric. He
wouldn't accept mutual sufficiency, i.e., principles of the form
\(Rab \Leftrightarrow f(a) \! =\! f(b)\), since these would imply that
both sides are equivalent as a matter of meaning. Instead, the point
is that the seemingly unproblematic claim \(Rab\) renders the claim
\(f(a) \! =\! f(b)\) unproblematic, and this is best expressed by
sufficiency statements of the form \(Rab \Rightarrow f(a) \! =\!
f(b)\), on which the left side *grounds* the right side. So
Linnebo's notion of reconceptualization is not the Fregean
notion of recarving of content.



Moreover, in a sufficiency statement, Linnebo doesn't require
that the relation \(R\) be an equivalence relation; he requires only
that \(R\) be symmetric and transitive. It need not be reflexive, for
the domain might contain entities \(x\) such that \(\neg Rxx\) (e.g.,
in the case of the sufficiency statement for directions, not every
object \(x\) in the domain is such that \(x\) is parallel with
\(x\)--being parallel is restricted to lines). Linnebo calls such
symmetric and transitive relations *unity* relations. When a
sufficiency statement--\(Rab \Rightarrow f(a) \! =\!
f(b)\)--holds, then new objects are identified. The new objects
are specified in terms of the less problematic entities related by
\(R\); for example, directions become specified by lines that are
parallel. According to Linnebo, the parallel lines become
*specifications* of the new objects. A unity relation \(R\) is
therefore the starting point for developing a sufficient (but not
necessary and sufficient) condition for reference.



Sometimes the new objects introduced by conditional principles do not
make demands on reality; when that happens, they are said to be
*thin* (for example, directions only require that there be
parallel lines). However, when the new objects introduced by
sufficiency statements make more substantial demands on reality, the
objects are considered *thick*. Suppose \(Rab\) asserts \(a\)
and \(b\) are spatiotemporal parts of the same cohesive and naturally
bounded whole. Then \(a\) and \(b\) become specifications for physical
bodies via the following principle: \(Rab \Rightarrow \mathrm{Body}(a)
\! =\! \mathrm{Body}(b)\). In this case, the principle "makes a
substantial demand on the world" because it requires checking
that there are spatiotemporal parts *constituting a continuous
stretch of solid stuff* (just looking at the spatiotemporal parts
does not suffice to determine whether they constitute to a body; see
Linnebo 2018, 45).



However, Linnebo does not identify being abstract with being thin
(2012, 147), for there are thin objects in a relative sense that are
not abstract, namely those that make no substantial demands on the
world beyond those introduced in terms of some antecedently given
objects. The mereological sum of your left hand and your laptop makes
no demand on the world beyond the demands of its
 parts.[19]
 Instead, he suggests that abstract objects are those that are thin
and that *have a shallow nature*. The notion of *shallow
nature* is meant to capture "the intuitive idea that any
question that is solely about \(F\text{s}\) has an answer that can be
determined on the basis of any given specifications of these
\(F\text{s}\)" (2018, 192-195). For example, directions
have a shallow nature because any question about directions (e.g., are
they orthogonal, etc.?) can be determined solely on the basis of the
lines that specify them. Shapes have a shallow nature because any
question about them (e.g., are they triangular, circular, etc.?) can
be determined solely on the basis of their underlying concrete
figures. By contrast, mereological sums of concrete objects are
*not* shallow because 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 your laptop and your
left hand depends not only on their combination but also on the
gravitational field in which they are
 located.[20]




Linnebo thus contrasts abstract objects, which are thin and have a
shallow nature, with concrete objects, which do not have a shallow
nature. Linnebo extends this view in several ways. He constructs an
account of mathematical objects that goes beyond the *way of
abstraction principles* by providing a reconstruction of set
theory in terms of 'dynamic abstraction' (2018,
ch. 3). This form of minimalism also allows for abstract objects
of a mixed nature; namely, those that are *thin relative to*
other objects. For example, the type of the letter 'A' is
abstract because it is thin and has a shallow nature, but it is thin
with respect the tokens of the letter 'A'.



This view, as Linnebo himself admits, faces some problems. One of them
is that the methodologies used by working mathematicians, such as
classical logic, impredicative definitions, and taking arbitrary
subcollections of infinite domains, seem to presuppose objects that
are more independent, i.e., objects that don't have a shallow
nature (2018, 197; for a discussion of independence, see 
Section 4.1 of the entry on
 platonism in mathematics). 
Another problem (2018, 195) is that
in order for an object to count as having a *shallow nature*,
an *intrinsic* unity relation has to be available. An
investigation is required to establish that there is such an intrinsic
unity relation in each case. It is far from clear that a conditional
principle with an *intrinsic* unity relation is available for
each of problematic cases mentioned in this entry, such as chess,
legal institutions or the English language. Finally, Linnebo
doesn't discuss the question of whether sets of concrete
urelements are themselves abstract or concrete. At present, there may
be an important question left open by his theory that other theories
of abstract objects answer.


### 3.8 Eliminativism



We come finally to proposals that reject the abstract/concrete
distinction. We can consider three cases. First, there are the
nominalists who both reject abstract entities and reject the
distinction as illegitimate. They focus on arguing against the
formulations of the distinction proposed in the literature. A second
group of eliminativists reject real objects of any kind, thereby
dismissing the distinction as irrelevant; these are the ontological
nihilists. A final group of eliminativists agree that there are
prototypical cases of concrete objects and abstract objects, but
conclude that a rigorous philosophical distinction can't be made
clearly enough to have any explanatory power (see Sider 2013, 287).
This recalls Lewis' pessimism (1986a, 81-86) about the
possibility of establishing a distinction that is sufficiently clear
to be theoretically interesting.


## 4. Further Reading



Berto & Plebani (2015) provide an useful introduction to ontology
and metaontology. Putnam (1971) makes the case for abstract objects on
scientific grounds. Bealer (1993) and Tennant (1997) present *a
priori* arguments for the necessary existence of abstract
entities. Fine (2002) systematically studies of abstraction principles
in the foundations of mathematics. Wetzel (2009) examines the
type-token distinction, argues that types are abstract objects while
the tokens of those types are their concrete instances, and shows how
difficult it is to paraphrase away the many references to types that
occur in the sciences and natural language. Zalta (2020) develops a
type-theoretic framework for higher-order abstract objects (which
includes abstract properties and abstract relations in addition to
ordinary properties and relations) and offers both comparisons to
other type theories and applications in philosophy and linguistics.
Moltmann (2013) investigates the extent to which abstract objects are
needed when developing a semantics of natural language; in this book,
and also in her article (2020), she defends a
'core-periphery' distinction and suggests that natural
language ontology contains references to abstract objects only in its
periphery. Falguera and Martinez-Vidal (2020) have edited a
volume in which contributors present positions and debates about
abstract objects of different kinds and categories, in different
fields in philosophy.