One doesn’t go far in the study of what there is withoutencountering the view that every entity falls into one of twocategories:concrete orabstract. The distinction issupposed to be of fundamental significance for metaphysics (especiallyfor ontology), epistemology, and the philosophy of the formal sciences(especially for the philosophy of mathematics); it is also relevantfor analysis in the philosophy of language, the philosophy of mind,and the philosophy of the empirical sciences. This entry surveys (a)attempts to say how the distinction should be drawn and (b) some ofmain theories of, and about, abstract objects.

• 1. Introduction
  • 1.1 About the Expression ‘Object’
  • 1.2 About the Abstract/Concrete Distinction
• 2. Historical Remarks
  • 2.1 The Provenance of the Distinction
  • 2.2 An Initial Overview of the Contemporary Debate
• 3. What is an Abstract Object?
  • 3.1 The Way of Example and the Way of Primitivism
  • 3.2 The Way of Conflation
  • 3.3 The Way of Abstraction
  • 3.4 The Way of Abstraction Principles
  • 3.5 The Ways of Negation
  • 3.6 The Way of Encoding
  • 3.7 The Ways of Weakening Existence
  • 3.8 Eliminativism
• 4. Further Reading
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

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

Though there is a pervasive appeal to abstract objects, philosophershave nevertheless wondered whether they exist. The alternatives are:platonism, which endorses their existence, andnominalism, which denies the existence of abstract objectsacross the board. (See the entries onnominalism in metaphysics andplatonism in metaphysics.) But the question of how to draw the distinction between abstract andconcrete objects is an open one: it is not clear how one shouldcharacterize these two categories nor is there a definite list ofitems that fall under one or the other category (assuming neither isempty).

The first challenge, then, is to articulate the distinction, either bydefining the terms explicitly or by embedding them in a theory thatmakes their connections to other important categories more explicit.In the absence of such an account, the philosophical significance ofthe contrast remains uncertain, for the attempt to classify things asabstract or concrete by appeal to intuition is often problematic. Isit clear that scientific theories (e.g., the general theory ofrelativity), works of fiction (e.g., Dante’sInferno),fictional characters (e.g., Bilbo Baggins) or conventional entities(e.g., the International Monetary Fund or the Spanish Constitution of1978) are abstract?

It should be stressed that there may not be one single“correct” way of explaining the abstract/concretedistinction. Any plausible account will classify the paradigm cases inthe standard way or give reasons for proceeding otherwise, and anyinteresting account will draw a clear and philosophically significantline in the domain of objects. Yet there may be many equallyinteresting ways of accomplishing these two goals, and if we findourselves with two or more accounts that do the job rather well, theremay be no point in asking which corresponds to the realabstract/concrete distinction. This illustrates a general point: whentechnical terminology is introduced in philosophy by means ofexamples, 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 correctaccount. A philosopher may find herself asking questions like,‘What is idealism?’ or ‘What is a substance?’and treating these questions as difficult questions about theunderlying nature of a certain determinate philosophical category. Abetter 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 beunderstood, and that what we seek is not a precise account of whatthis term already means, but rather aproposal for how itmight fruitfully be used for philosophical analysis. Anyone whobelieves that something in the vicinity of the abstract/concretedistinction matters for philosophy would be well advised to approachthe project of explaining the distinction with this in mind.

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

1.1 About the Expression ‘Object’

Frege famously distinguished two mutually exclusive ontologicaldomains,functions andobjects. According to hisview, a function is an ‘incomplete’ entity that mapsarguments to values, and is denoted by an incomplete expression,whereas an object is a ‘complete’ entity and can bedenoted by a singular term. Frege reduced properties and relations tofunctions 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 thoughFrege’s sense of ‘object’ is important, it is notthe only way to use the term. Other philosophers include propertiesand relations among the abstract objects. And when the backgroundcontext for discussing objects is type theory, properties andrelations of higher type (e.g., properties of properties, andproperties of relations) may be all be considered‘objects’. This latter use of ‘object’ isinterchangeable with ‘entity.’^[1] Throughout this entry, we will follow this last usage and treat theexpressions ‘object’ and ‘entity’ as havingthe same meaning. (For further discussion, see the entry onobjects.)

1.2 About the Abstract/Concrete Distinction

Though we’ve spoken as if the abstract/concrete distinction mustbe an exhaustive dichotomy, we should be open to the possibility thatthe best sharpening of it will entail that some objects are neitherabstract nor concrete.^[2] Holes and shadows, if they exist, do not clearly belong in eithercategory; nor do ghosts, Cartesian minds, fictional characters,^[3] immanent universals, or tropes. The main constraint on an account ofthe distinction is that it draws a philosophically significant linethat classifies at least many of the standard examples in the standardways. It is not a constraint that everything be shoehorned into onecategory or the other.

Finally, some philosophers see the main distinction not as betweenabstract and concrete objects but as between abstract objects andordinary 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 thatthe abstract/concrete distinction is a division among existingobjects, and that any plausible explanation of the distinction shouldaim 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 anancient one. Indeed, there is a strong case for the view that, despiteoccasional exceptions, it played no significant role in philosophybefore the 20th century. The modern distinction bears some resemblanceto Plato’s distinction between Forms and Sensibles. ButPlato’s Forms were supposed to be causesparexcellence, whereas abstract objects are generally supposed to becausally inert. The original‘abstract’/‘concrete’ distinction was adistinction among words or terms. Traditional grammar distinguishesthe abstract noun ‘whiteness’ from the concrete noun‘white’ without implying that this linguistic contrastcorresponds to a metaphysical distinction in what these words standfor. In the 17th century, this grammatical distinction was transposedto the domain of ideas. Locke speaks of the general idea of a trianglewhich is “neither Oblique nor Rectangle, neither Equilateral,Equicrural nor Scalenon [Scalene]; but all and none of these atonce,” 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 isformed from concrete ideas by the omission of distinguishing detailwas immediately rejected by Berkeley and then by Hume. But even forLocke there was no suggestion that the distinction between abstractideas and concrete or particular ideas corresponds to a distinctionamongobjects. “It is plain, …” Lockewrites, “that General and Universal, belong not to the realexistence of things; but are Inventions and Creatures of theUnderstanding, 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 marka line in the domain of objects or entities. So conceived, thedistinction becomes a central focus for philosophical discussionprimarily in the 20th century. The origins of this development areobscure, but one crucial factor appears to have been the breakdown ofthe allegedly exhaustive distinction between mental and materialobjects, which had formed the main division for ontologically-mindedphilosophers since Descartes. One signal event in this development isFrege’s insistence that the objectivity and aprioricity of thetruths of mathematics entail that numbers are neither material beingsnor ideas in the mind. If numbers were material things (or propertiesof material things), the laws of arithmetic would have the status ofempirical generalizations. If numbers were ideas in the mind, then thesame difficulty would arise, as would countless others. (Whose mindcontains the number 17? Is there one 17 in your mind and another inmine? In that case, the appearance of a common mathematical subjectmatter would be an illusion.) InThe Foundations ofArithmetic (1884), Frege concludes that numbers are neitherexternalconcrete things nor mental entities of any sort.

Later, in his essay “The Thought” (1918), Frege claims thesame status for the items he callsthoughts—the sensesof declarative sentences—and also, by implication, for theirconstituents, the senses of subsentential expressions. Frege does notsay that senses areabstract. He says that they belong to athird realm distinct both from the sensible external worldand from the internal world of consciousness. Similar claims had beenmade by Bolzano (1837), and later by Brentano (1874) and his pupils,including Meinong and Husserl. The common theme in these developmentsis the felt need in semantics and psychology, as well as inmathematics, for a class of objective (i.e., non-mental) non-physicalentities. As this newrealism was absorbed intoEnglish-speaking philosophy, the traditional term‘abstract’ was enlisted to apply to the denizens of thisthird realm. In this vein, Popper (1968) spoke of the‘third world’ of abstract, objective entities, in thebroader 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 thereforeimportant to remember that the use of the termsplatonist(for those who affirm the existence of abstract objects) andnominalist (for those who deny existence) is somewhatlamentable, since these words have established senses in the historyof philosophy. These terms stood for positions that have little to dowith the modern notion of an abstract object. Modern platonists (witha small ‘p’) need not accept any of the distinctivemetaphysical and epistemological doctrines of Plato, just as modernnominalists need not accept the distinctive doctrines of the medievalnominalists. Moreover, the literature also contains mention ofanti-platonists, many of whom see themselves asfictionalists about abstracta, though this doesn’t helpif it turns out that the best analysis of fictions is to regard themas abstract objects. So the reader should therefore be aware thatterminology is not always well-chosen and that the terms so usedsometimes stand for doctrines that are more restricted than thetraditional doctrines that go by the same name. Henceforth, we simplyuseplatonism for the thesis that there exists at least oneabstract object, andnominalism for the thesis that thenumber 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 theabstract/concrete distinction, we should briefly say why thedistinction has been thought to matter. Among philosophers who takethe distinction seriously, it is generally supposed that whileconcrete objects clearly exist, abstract entities are problematic indistinctive ways and deny the existence of abstract entitiesaltogether. In this section we briefly survey the arguments fornominalism and the responses that platonists have offered. If theabstract objects are unified as a class, it is because they have somefeature that generates what seems to be a distinctive problem—aproblem that nominalists deem unsolvable and which platonists aim tosolve. Before we ask what the unifying feature might be, it maytherefore help to characterize the various problems it has beenthought to generate.

The contemporary debate about platonism developed in earnest whenQuine argued (1948) that mathematical objects exist, having changedhis mind about the nominalist approach he had defended earlier(Goodman & Quine 1947). Quine’s 1948 argument involves threekey premises, all of which exerted significant influence on thesubsequent debate: (i) mathematics is indispensable for empiricalscience; (ii) we should be ontologically committed to the entitiesrequired for the truth of our best empirical theories (all of whichshould be expressible in a first-order language); and (iii) theentities required for the truth on an empirical theory are those inthe 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 theconcrete entities contemplated by our best empirical science, we mustaccept the existence of mathematical entities, even if they areabstract (see also Quine 1960, 1969, 1976).

Quine’s argument initiated a debate that is still alive. Variousnominalist responses questioned one or another of the premises in hisargument. For instance, Field (1980) challenged the idea thatmathematics is indispensable for our best scientifictheories—i.e., rejecting (i) above—and thus faced the taskof rewriting classical and modern physics in nominalistic terms inorder to sustain the challenge. Others have taken on the somewhat lessdaunting task of accepting (i) but rejecting (ii) and (iii);they’ve argued that even if our best scientific theories, inregimented form, quantify over mathematical entities, thisdoesn’t entail a commitment to mathematical entities (seeAzzouni 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 thissecond group, since they avoided the ‘hard road’ ofparaphrasing our best scientific theories in non-mathematicalterms.

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 Saatsi2011 for a response). Some participants describe the debate in termsof a stalemate they hope to resolve (see Baker 2017, Baron 2016, 2020,Knowles & Saatsi 2019, and Martínez-Vidal &Rivas-de-Castro 2020, for discussion).^[4]

Aside from the debate over Quine’s argument, both platonism andnominalism give rise to hard questions. Platonists not only need toprovide a theory of what abstract objects exist, but also an accountof how we cognitively access and come to know non-causal, abstractentities. This latter question has been the subject of a debate thatbegan in earnest in Benacerraf (1973), which posed just such a dilemmafor mathematical objects. Benacerraf noted that the causal theory ofreference doesn’t seem to make it possible to know the truthconditions of mathematical statements, and his argument applies toabstract entities more generally.^[5] On the other hand, nominalists need to explain the linguistic uses inwhich we seem to appeal to such entities, especially those uses inwhat 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 abstracttypes are used in scientific explanations). Even thoughnominalists argue that there are no abstracta, the very fact thatthere is disagreement about their existence suggests that bothplatonists and nominalists acknowledge the distinction between theabstract and concrete to be a meaningful one.

On the platonist side, various proposals have been raised to addressthe challenge of explaining epistemic access to abstract entities,mostly in connection with mathematical objects. Some, includingGödel (1964), allege that we access abstract objects in virtue ofa unique kind of perception (intuition). Maddy (1990, 1997) developedtwo rather different ways of understanding our knowledge ofmathematics in naturalistic ways. Other platonists have argued thatabstract objects are connected to empirical entities, either viaabstraction (Steiner 1975; Resnik 1982; Shapiro 1997) or viaabstraction principles (Wright 1983; Hale 1987); we’ll discusssome of these views below. There are also those who speak of existentand intersubjective abstract entities as a kind of mentalrepresentation (Katz 1980).

A rather different line of approach to the epistemological problem wasproposed in Linsky & Zalta 1995, where it is suggested that oneshouldn’t attempt to explain knowledge of abstracta on the samemodel that is used to explain knowledge of concrete objects. Theyargue that not only a certain plenitude principle for abstract objects(namely, the comprehension principle for abstract objects put forwardin Zalta 1983, 1988—see below) yields unproblematic‘acquaintance by description’ to unique abstract objectsbut also that their approach actually comports with naturalistbeliefs. Balaguer (1995, 1998) also suggests that a plenitudeprinciple is the best way forward for the platonist, and that ourknowledge of the consistency of mathematical theories suffices forknowledge of mathematical objects. And there are views that conceiveof 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 objectsand concluded that sentences with terms making apparent reference tothem—such as mathematical statements—are either false orlack a truth value. They argue that those sentences must beparaphrasable without vocabulary that commits one to any sort ofabstract entity. These proposals sometimes suggest that statementsabout abstract objects are merely instrumental; they serve only tohelp us establish conclusions about concrete objects. Field’sfictionalism (1980, 1989) has been influential in thisregard. Field reconstructed Newtonian physics using second-order logicand quantification over (concrete) regions of space-time. A completelydifferent tactic for avoiding the commitment to abstract, mathematicalobjects is put forward in Putnam (1967) and Hellman (1989), whoseparately reconstructed various mathematical theories in second-ordermodal logic. On their view, abstract objects aren’t inthe range of the existential quantifier at the actual world (hence, wecan’t say that they exist), but they do occur in the range ofthe quantifier at other possible worlds, where the axioms of themathematical theory in question are true.

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

In defending nominalism, philosophers have often tried to account forhow language about abstract objects in our theories can still yieldknowledge about concrete objects—for instance, by paraphrasingstatements that appear to refer to abstract entities. This approachtypically assumes that expressions about abstract objects, and thesentences in which such expressions occur, are meaningful, even whiledenying the existence of such objects. Challenging this assumption,Himelright (2023) offers an argument—drawing on Lewis’scritique (1986a, 174–191) of ‘magical ersatz’theories of possible worlds—which contends that terms like“property,” “proposition,” and“set,” along with the sentences that include them, are infact meaningless. If sound, this argument would not only supportnominalism but also render unnecessary the nominalist strategiesdesigned to account for the discourse about abstract objects sincesuch discourse would lack sense altogether.

Another nominalistic thread in the literature concerns the suggestionthat sentences about (posited) abstract objects are quasi-assertions,i.e., not evaluable as to truth or falsehood (see Yablo 2001 andKalderon 2005). Still others argue that we should not believesentences about abstracta since their function, much like theinstrumentalism discussed earlier, is to ensure empirical adequacy forobservational sentences (Yablo 1998). This may involve differentiatingbetween apparent content, which involves posited abstract objects, andreal content, which only concerns concrete objects (Yablo 2001, 2002,2010, 2014). (For more on these fictionalist accounts, see Kalderon2005, Ch.3, and the entry onfictionalism.)

A final group of views in the literature represents a kind ofagnosticism about what exists or about what it is to be an object, beit abstract or concrete. These views don’t reject an externalmaterial world, but rather begin with some question as to whether wecan 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 hasto be expressed with respect to a linguistic framework and that whenwe wish to put forward a theory about a new kind of entity, we musthave a linguistic framework for talking about those entities. He thendistinguished two kinds of existence questions: internal questionswithin the framework about the existence of the new entitiesand external questions about the reality of the framework itself. Ifthe framework deals with abstract entities such as numbers, sets,propositions, etc., then the internal question can be answered bylogical analysis of the rules of the language, such as whether itincludes terms for, or implies claims that quantify over, abstractobjects. But, for Carnap, the external question, about whether theabstract entities really exist, is a pseudo-question and should beregarded as nothing more than the pragmatic question of whether theframework is a useful one to adopt, for scientific or other forms ofenquiry. We’ll discuss Carnap’s view in more detail in subsection3.7.1.

Some have thought that Carnap’s view offers adeflationist view of objects, since it appears that theexistence of objects is not language independent. After Carnap’sseminal article, several other deflationist approaches were putforward (Putnam 1987, 1990; Hirsch 2002, 2011; Sider 2007, 2009;Thomasson 2015), many of them claiming to be a vindication ofCarnap’s view. However, there are deflationist proposals thatrun counter to Carnap’s approach, among them, deflationarynominalism (Azzouni 2010) or agnosticism about abstract objects (Bueno2008a, 2008b, 2020). Additionally, philosophers inspired byFrege’s work have argued for a minimal notion of an object (Rayo2013, Rayo 2020 [Other Internet Resources]; and Linnebo 2018).We’ll discuss some of these in greater detail below, in subsection3.7.2. A final agnostic position that has emerged is one that rejects astrict version of platonism, but suggests that neither a carefulanalysis of mathematical practice (Maddy 2011), nor the enhancedversion of the indispensability argument (Leng 2020) suffice to decidebetween nominalism and moderate versions of platonism. Along theselines, Balaguer (1998) concluded that the question doesn’t havean answer, since the arguments for ‘full-blooded’platonism can be matched one-for-one by equally good arguments by theanti-platonist.

For additional discussion about the basic positions in the debateabout abstract and concrete objects, see Szabó 2003 and theentries onnominalism in metaphysics andplatonism in metaphysics,nominalism in the philosophy of mathematics andplatonism 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 candraw the abstract/concrete distinction.^[6] These include:the way of example (which is simply to listthe paradigm cases of abstract and concrete objects in the hope thatthe sense of the distinction will somehow emerge);the way ofconflation (i.e., identifying abstract and concrete objects withsome 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.); andtheway of abstraction (i.e., saying that abstract objects areconceptualized by a process of considering some known objects andomitting certain distinguishing features). He gives a detailedexamination of the different proposals that typify these ways and thenattempts to show that none of them quite succeeds in classifying theparadigms in accord with prevailing usage. Given the problems heencountered when analyzing the various ways, Lewis became pessimisticabout our ability to draw the distinction cleanly.

Despite Lewis’s pessimism about clarifying the abstract/concretedistinction, his approach for categorizing the various proposals, whenextended, is a useful one. Indeed, in what follows, we’ll seethat there are a number of additionalways that categorizeattempts to characterize the abstract/concrete distinction andtheorize about abstract objects. Even if there is no single,acceptable account, these various ways of drawing the distinction andtheorizing about abstract objects do often cast light on the questionswe’ve been discussing, especially when the specific proposalsare integrated into a supplementary (meta-)ontological project. Foreach method of drawing the distinction and specific proposal adoptingthat method acquires a certain amount of explanatory power, and thiswill help us to compare and contrast the various ideas that are nowfound in the literature.

3.1 The Way of Example and the Way of Primitivism

According to theway of example, it suffices to list paradigmcases of abstract and concrete entities in the hope that the sense ofthe distinction will somehow emerge. Clearly, a list of examples foreach category would be a heuristically promising start in the searchfor some criterion (or list of criteria) that would be fruitful fordrawing the distinction. However, a simple list would be of limitedsignificance since there are too many ways of extrapolating from theparadigm 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. Butthe case of impure sets is far from clear. Consider the unit set whosesole member is Joe Biden (i.e., {Joe Biden}), the Undergraduate Classof 2020 or The Ethics Committee, etc. They are sets, but it is notclear that they are abstract given that Joe Biden, the members of theclass and committee are concrete. Similarly, if one offers thecharacters of Sherlock Holmes stories as examples to help motivate theprimitive concept abstract object, then one has to wonder about theobject London that appears in the novels.

The refusal to characterize the abstract/concrete distinction whilemaintaining that both categories have instances might be called theway of primitivism, whenever the following condition obtains:a few predicates are distinguished as primitive and unanalyzable, andthe explanatory power rests on the fact that other interestingpredicates can be defined in terms of the primitives and thatinteresting claims can be judged true on the basis of our intuitiveunderstanding of the primitive and defined notions. Thus, one mighttake abstract and concrete as primitive notions. It wouldn’t bean insignificant result if one could use this strategy to explain whyabstract objects are necessarily existent, causally inefficacious,non-spatiotemporal, intersubjective, etc. (see Cowling 2017:92–97).

But closer inspection of this method reveals some significantconcerns. To start with, when a distinction is taken as basic andunanalyzable, one typically has to offer some intuitive instances ofthe primitive predicates. But it is not always so easy to do this. Forexample, when mathematicians takeset andmembershipas primitives and then assert some principles of set theory, theyoften 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 ofthe class and committee may change while the class and committeeremain the same, whereas if the members of a set change, one has adifferent set. A similar concern affects the present proposal. If oneoffers sets or the characters of the Sherlock Holmes novels asexamples to help motivate the primitive conceptabstractobject, then one has to wonder about impure sets such as the unitset 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 theway of conflation, the abstract/concretedistinction is to be identified with one or another metaphysicaldistinction already familiar under another name: as it might be, thedistinction between sets and individuals, or the distinction betweenuniversals and particulars. There is no doubt that some authors haveused the terms in this way. (Thus Quine 1948 uses ‘abstractentity’ and ‘universal’ interchangeably.) This sortof conflation is however rare in recent philosophy.

3.3 The Way of Abstraction

Another methodology is what Lewis calls theway ofabstraction. According to a longstanding tradition inphilosophical psychology, abstraction is a distinctive mental processin which new ideas or conceptions are formed by considering the commonfeatures of several objects or ideas and ignoring the irrelevantfeatures that distinguish those objects. For example, if one is givena range of white things of varying shapes and sizes; one ignores orabstracts from the respects in which they differ, and therebyattains the abstract idea of whiteness. Nothing in this traditionrequires that ideas formed in this way represent or correspond to adistinctive kind of object. But it might be maintained that thedistinction between abstract and concrete objects should be explainedby reference to the psychological process of abstraction or somethinglike it. The simplest version of this strategy would be to say that anobject is abstract if it is (or might be) the referent of an abstractidea; i.e., an idea formed by abstraction. So conceived, thewayof abstraction is wedded to an outmoded philosophy of mind.

It should be mentioned, though, that the key idea behind thewayof abstraction has resurfaced (though transformed) in thestructuralist views about mathematics that trace back to Dedekind.Dedekind thought of numbers by theway 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 onlythe relations to one another” (Dedekind 1888 [1963, 68]). Thisview has led some structuralists to deny that numbers are abstractobjects. For example, Benacerraf concluded that “numbers are notobjects at all, because in giving the properties (that is, necessaryand sufficient) of numbers you merely characterize anabstractstructure—and the distinction lies in the fact that the‘elements’ of the structure have no properties other thanthose relating them to other ‘elements’ of the samestructure” (1965, 70). We shall therefore turn our attention toa variant of theway of abstraction, one that has led anumber of philosophers to conclude that numbers are indeed abstractobjects.

3.4 The Way of Abstraction Principles

In the contemporary philosophical literature, a number of books andpapers have investigated a form of abstraction that doesn’tdepend on mental processes. We may call this theway ofabstraction principles. Wright (1983) and Hale (1987) havedeveloped an account of abstract objects on the basis of an idea theytrace back to certain suggestive remarks in Frege (1884). Frege notes(in effect) that many of the singular terms that appear to refer toabstract entities are formed by means of functional expressions. Wespeak ofthe shape of a building,the direction of aline,the number of books on the shelf. Of course, manysingular terms formed by means of functional expressions denoteordinary concrete objects: ‘the father of Plato’,‘the capital of France’. But the functional terms thatpick out abstract entities are distinctive in the following respect:where \(f(a)\) is such an expression, there is typically an equationof the form:

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

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 onlyif there are just as many \(F\text{s}\) as \(G\text{s}\)

These biconditionals (orabstraction principles) appear tohave a special semantic status. While they are not strictly speakingdefinitions of the functional expression that occurs on theleft hand side, they would appear to hold in virtue of the meaning ofthat expression. To understand the term ‘direction’ is (inpart) to know that the direction of \(a\) and the direction of \(b\)are the same entity if and only if the lines \(a\) and \(b\) areparallel. Moreover, the equivalence relation that appears on the righthand side of the biconditional would appear to be semantically andperhaps epistemologically prior to the functional expressions on theleft (Noonan 1978). Mastery of the concept of a direction presupposesmastery of the concept of parallelism, but not vice versa.

The availability of abstraction principles meeting these conditionsmay be exploited to yield an account of the distinction betweenabstract and concrete objects. When ‘\(f\)’ is afunctional expression governed by an abstraction principle, there willbe 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 theway of abstraction principles isthen to say that:

\(x\) is an abstract object if (and only if) \(x\) is an instance ofsome kind \(K_{f}\) whose associated functional expression‘\(f\)’ is governed by a suitable abstractionprinciple.

The strong version of this account—which purports to identify anecessary condition for abstractness—is seriously at odds withstandard usage. Pure sets are usually considered paradigmatic abstractobjects. But it is not clear that they satisfy the proposed criterion.According to a version of naïve set theory, the functionalexpression ‘set of’ is indeed characterized by aputative 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 byan abstraction principle, but rather axiomatically. Though attemptshave been made to investigate abstraction principles for sets (Cook2003), it remains an open question whether something like themathematical concept of a set can be characterized by a suitablyrestricted abstraction principle. (See Burgess 2005 for a survey ofrecent efforts in this direction.) Even if such a principle isavailable, however, it is unlikely that the epistemological prioritycondition will be satisfied. That is, it is unlikely that mastery ofthe concept of set will presuppose mastery of the equivalence relationthat figures on the right hand side. It is therefore uncertain whethertheway of abstraction principles will classify the objectsof pure set theory as abstract entities (as it presumably must).

On the other hand, as Dummett (1973) has noted, in many cases thestandard names for paradigmatically abstract objects do not assume thefunctional form to which the definition adverts. Chess is an abstractentity, but we do not understand the word ‘chess’ assynonymous 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 Englishlanguage, social justice, architecture, and Charlie Parker’smusical style. If so, the abstractionist approach does not provide anecessary condition for abstractness as that notion is standardlyunderstood.

More importantly, there is some reason to believe that it fails tosupply asufficient condition. A mereological fusion ofconcrete objects is itself a concrete object. But the concept of amereological fusion is governed by what appears to be an abstractionprinciple:

The fusion of the \(F\text{s}\) = the fusion of the \(G\text{s}\) ifand 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 onlyif 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 functionalexpression, ‘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 maythen 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 theconsequence that trains are to be reckoned abstract entities.

It is unclear whether these objections apply to the more sophisticatedabstractionist proposals of Wright and Hale, but one feature of thesimple account sketched above clearly does apply to these proposalsand may serve as the basis for an objection to this version of theway of abstraction principles. The neo-Fregean approach seeksto explain the abstract/concrete distinction insemanticterms: We said that an abstract object is an object that falls in therange of a functional expression governed by an abstraction principle,where ‘\(f\)’ isgoverned by an abstractionprinciple when that principle holds in virtue of themeaningof ‘\(f\)’. This notion of a statement’s holding invirtue of the meaning of a word is notoriously problematic (see theentrythe analytic/synthetic distinction). But even if this notion makes sense, one may still complain: Theabstract/concrete distinction is supposed to be a metaphysicaldistinction; abstract objects are supposed to differ from concreteobjects in some important ontological respect. It should be possible,then, to draw the distinction directly in metaphysical terms: to saywhat it isin the objects themselves that makes some thingsabstract and others concrete. As Lewis writes, in response to arelated proposal by Dummett:

Even if this … way succeeds in drawing a border, as for all Iknow it may, it tells us nothing about how the entities on oppositesides of that border differ in their nature. It is like saying thatsnakes 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 theabstractionist criterion that specifies directly, in metaphysicalterms, what the objects whose canonical names are governed byabstraction principles all have in common.

One response to this difficulty is to transpose the abstractionistproposal into a more metaphysical key (see Rosen & Yablo 2020).The idea is that each Fregean number is, byits very nature,the number of some Fregean concept, just as each Fregean direction is,byits very nature, at least potentially the direction ofsome concrete line. In each case, the abstract object isessentially the value of an abstraction function for acertain class of arguments. This is not a claim about the meanings oflinguistic expressions. It is a claim about the essences or natures ofthe objects themselves. (For the relevant notion of essence, see Fine1994.) So for example, the Fregean number two (if there is such athing) is, essentially, by its very nature, the number that belongs toa concept \(F\) if and only if there are exactly two \(F\text{s}\).More generally, for each Fregean abstract object \(x\), there is anabstraction function \(f\), such that \(x\) is essentially the valueof \(f\) for every argument of a certain kind.

Abstraction functions have two key features. First, for eachabstraction function \(f\) there is an equivalence relation \(R\) suchthat it lies in the nature of \(f\) that \(f(x)\! =\! f(y)\) iff\(Rxy\). Intuitively, we are to think that \(R\) is metaphysicallyprior to \(f\), and that the abstraction function \(f\) isdefined (in whole or in part) by this biconditional. Second,each abstraction function is agenerating function: itsvalues are essentially values of that function. Many functions are notgenerating functions. Paris is the capital of France, but it is notessentially a capital. The number of solar planets, by contrast, isessentially a number. The notion of an abstraction function may bedefined in terms of these two features:

We may then say that:

\(x\) is anabstraction if and only if, for some abstractionfunction \(f\), 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 anabstraction.

This account tells us a great deal about the distinctive natures ofthese broadly Fregean abstract objects. It tells us that each is, byits very nature, the value of a special sort of function, one whosenature is specified in a simple way in terms of an associatedequivalence relation. It is worth stressing, however, that it does notsupply muchmetaphysical information about these items. Itdoesn’t tell us whether they are located in space, whether theycan stand in causal relations, and so on. It is an open questionwhether this somewhat unfamiliar version of the abstract/concretedistinction lines up with any of the more conventional ways of drawingthe distinction outlined above. An account along these lines would beat odds with standard usage, but may be philosophically interestingall the same. In any case, the problem remains that this metaphysicalversion ofthe way of abstraction principles leaves outparadigmatic cases of abstract objects such as the aforementioned gameof chess.

3.5 The Ways of Negation

According to theway of negation, abstract objects aredefined as those whichlack certain features possessed byparadigmatic concrete objects. Many explicit characterizations in theliterature 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 andnon-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 athing depends on the mind when it would not (or could not) haveexisted if minds had not existed. But this entails that tables andchairs are mind-dependent, and that is not what philosophers whoemploy this notion have in mind. To call an object‘mind-dependent’ in a metaphysical context is to suggestthat it somehow owes its existence to mental activity, but not in theboring ‘causal’ sense in which ordinary artifacts owetheir existence to the mind. What can this mean? One promisingapproach is to say that an object should be reckoned mind-dependentwhen, by its very nature, it existsat a time if and only ifit is the object or content of some mental state or processatthat time. This counts tables and chairs as mind-independent,since they might survive the annihilation of thinking things. But itcounts paradigmatically mental items, like a purple afterimage ofwhich a person \(X\) may become aware, as mind-dependent, since itpresumably lies in the nature of such items to be objects of consciousawareness whenever they exist. However, it is not clear that thisaccount captures the full force of the intended notion. Consider, forexample, 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 fusioncan exist even if no one is thinking aboutit.) A moregenerous conception would allow for mind-dependent objects that existat a time in virtue of mental activity at that time, even if theobject is not the object of any single mental state or act. The fusionof \(X\)’s afterimage and \(Y\)’s headache ismind-dependent in the second sense but not the first. That is a reasonto prefer the second account of mind-dependence.

If we understand the notion of mind-dependence in this way, it is amistake to insist that abstract objects be mind-independent. To strikea theme that will recur, it is widely supposed that sets and classesare abstract entities—even theimpure sets whoseurelements are concrete objects. Any account of the abstract/concretedistinction that places set-theoretic constructions like\(\{\textrm{Alfred}, \{\textrm{Betty}, \{\textrm{Charlie},\textrm{Deborah}\}\}\}\) on the concrete side of the line will beseriously at odds with standard usage. With this in mind, consider theset whose sole members are X’s afterimage and Y’sheadache, or some more complex set-theoretic object based on theseitems. If we suppose, as is plausible, that an impure set exists at atime only when its members exist at that time, this will be amind-dependent entity in the generous sense. But it is also presumablyan abstract entity.

A similar problem arises for so-calledabstract artifacts,like Jane Austen’s novels and the characters that inhabit them.Some philosophers regard such items as eternally existing abstractentities that worldly authors merely describe but do not create. Butof course the commonsensical view is that Austen createdPride andPrejudice and Elizabeth Bennett, and there is no good reason todeny this (Thomasson 1999; cf. Sainsbury 2009). If we take thiscommonsensical approach, there will be a clear sense in which theseitems depend for their existence on Austen’s mental activity,and perhaps on the mental activity of subsequent readers.^[7] These items may not count as mind-dependent in either of the sensescanvassed above, sincePride and Prejudice can presumablyexist at a time even if no one happens to be thinking at that time.(If the world took a brief collective nap,^[8]Pride and Prejudice would not pop out of existence.) Butthey are obviously mind-dependent in some not-merely-causal sense. Andyet they are still presumably abstract objects. For these reasons, itis probably a mistake to insist that abstract objects bemind-independent. (For more on mind-dependence, see Rosen 1994, andthe entryplatonism in the philosophy of mathematics.)

Frege’s proposal in its original form also fails for otherreasons. Quarks and electrons are usually considered neither sensiblenor mind-dependent. And yet they are not abstract objects. A betterversion of Frege’s proposal would hold that:

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

Two remarks on this last version are in order. First, it opens thedoor to thinking that besides abstract and concrete entities (assumingthat physical objects, in a broad sense, are concrete), there aremental entities that are neither concrete nor abstract. As mentionedabove (section 1.2), there is no need to insist that the distinctionis an exhaustive one. Second, while the approach may well draw animportant line, it inherits one familiar problem, namely, that ofsaying what it is for a thing to be aphysical object (Craneand Mellor 1990; for discussion, see the entry onphysicalism). In one sense, a physical entity is an entity in which physics mighttake an interest. But physics is saturated with mathematics, so inthis sense a great many paradigmatically abstract objects—e.g.\(\pi\)—will count as physical. The intended point is thatabstract objects are to be distinguished, not fromall of theobjects posited by physics, but from theconcrete objectsposited by the physics. But if that is the point, it is unilluminatingin the present context to say that abstract objects arenon-physical.

3.5.2 The Non-Spatiality Criterion

Contemporary purveyors of theway of negation typically amendFrege’s criterion by requiring that abstract objects benon-spatial, causally inefficacious, or both. Indeed, if anycharacterization of the abstract deserves to be regarded as thestandard one, is this:

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

This standard account nonetheless presents a number ofperplexities.

First of all, one must consider whether there are abstract objectsthat 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 becauseit is causally inefficacious, but some might suggest that it has alocation in space (namely, wherever Plato is located). Or consider awork of fiction such as Kafka’sThe Metamorphosis. It,too, has some claim to being abstract because it (or at least itscontent) is non-spatial. But one might suggest that works of fictionas 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 criterionin the above proposal, namely, the non-spatial condition. But it givesrise to a subtlety. It seems plausible to suggest that, necessarily,if something \(x\) is causally efficacious, then (since \(x\) is acause or has causal powers) \(x\), or some part of \(x\), has alocation in time. So if something has no location in time, it iscausally inefficacious. The theory of relativity implies that spaceand time are nonseparable, i.e., combined into a singlespacetime manifold. So the above proposal might be restatedin terms of a single condition: an object is abstract if and only ifit is non-spatiotemporal. Sometimes this revised proposal is thecorrect one for thinking about abstract objects, but our discussion inthe previous section showed that abstract artifacts and mental eventsmay be temporal but non-spatial. Given the complexities here, in whatfollows we use spatiotemporality, spatiality, or temporality, asneeded.

Some of the archetypes of abstractness are non-spatiotemporal in astraightforward sense. It makes no sense to ask where the cosinefunction was last Tuesday. Or if it makes sense to ask, the sensibleanswer is that it was nowhere. Similarly, for many people, it makes nogood sense to ask when the Pythagorean Theorem came to be. Or if itdoes make sense to ask, the only sensible answer for them is that ithas always existed, or perhaps that it does not exist ‘intime’ at all. It is generally assumed that these paradigmatic‘pure abstracta’ have no non-trivial spatial or temporalproperties; that they have no spatial location, and they exist nowherein particular in time.

Other abstract objects appear to stand in a more interesting relationto spacetime. Consider the game of chess. Some philosophers will saythat chess is like a mathematical object, existing nowhere and‘no when’—either eternally or outside of timealtogether. But the most natural view is that chess was invented at acertain time and place (though it may be hard to say exactly where orwhen); that before it was invented it did not exist at all; that itwas imported from India into Persia in the 7th century; that it haschanged over the years, and so on. The only reason to resist thisnatural account is the thought that since chess is clearly an abstractobject—it’s not a physical object, after all!—andsince abstract objects do not exist in space and time—bydefinition!—chess must resemble the cosine function in itsrelation to space and time. And yet one might with equal justiceregard the case of chess and other abstract artifacts ascounterexamples to the hasty view that abstract objects possess onlytrivial spatial and temporal properties.

Should we then abandon the non-spatiotemporality criterion? Notnecessarily. Even if there is a sense in which some abstract entitiespossess non-trivial spatiotemporal properties, it might still be saidthat concrete entities exist in spacetimein a distinctiveway. If we had an account of this distinctivemanner ofspatiotemporal existence characteristic of concrete objects, we couldsay: An object is abstract (if and) only if it fails to exist inspacetimein that way.

One way to implement this approach is to note that paradigmaticconcrete objects tend to occupy a relatively determinate spatialvolume at each time at which they exist, or a determinate volume ofspacetime over the course of their existence. It makes sense to ask ofsuch an object, ‘Where is it now, and how much space does itoccupy?’ even if the answer must sometimes be somewhat vague. Bycontrast, even if the game of chess is somehow‘implicated’ in space and time, it makes no sense to askhow much space it now occupies. (To the extent that this does makesense, 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 mightbe said:

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

This promising idea raises several questions. First, it is conceivablethat certain items that are standardly regarded as abstract mightnonetheless 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’tnormally ask where such things are, or how much space they occupy. Andindeed many philosophers will say that the question makes no sense, orthat the answer is a dismissive ‘nowhere, none’. But thisanswer is not forced upon us by anything in set theory or metaphysics.Even if we grant thatpure sets stand in only the mosttrivial relations to space, it is open to us to hold, as somephilosophers have done, that impure sets exist where and when theirmembers do (Lewis 1986a). It is not unnatural to say that a set ofbooks is located on a certain shelf in the library, and indeed, thereare some theoretical reasons for wanting to say this (Maddy 1990). Ona view of this sort, we face a choice: we can say that since impuresets exist in space, they are not abstract objects after all; or wecan say that since impure sets are abstract, it was a mistake tosuppose that abstract objects cannot occupy space.

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

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

But of course Peter himself occupies a region in virtue of the factthat hisparts—his head, hands, etc.—togetheroccupy that region. So a better version of the proposal would say:

An object is abstract (if and) only if it either fails to occupy spaceat all, or does so of the fact that some other itemsthat are notamong its parts occupy that region.

This approach appears to classify the cases fairly well, but it issomewhat artificial. Moreover, it raises a number of questions. Whatare we to say about the statue that occupies a region of space, notbecause itsparts are arrayed in space, but rather becauseits constitutingmatter occupies that region? And what aboutthe unobserved electron, which according to some interpretations ofquantum mechanics does not reallyoccupy a region of space atall, but rather stands in some more exotic relation to the spacetimeit inhabits? Suffice it to say that a philosopher who regards‘non-spatiality’ as a mark of the abstract, but who allowsthat some abstract objects may have non-trivial spatial properties,owes us an account of thedistinctive 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. Ifthey are considered concrete, then one might ask where thespatiotemporal points or regions are located. And a similar questionarises for spatial points and regions, and for temporal instants orintervals. So, the ontological status of spatiotemporal locations, andof spatial and temporal locations, is problematic. Let us suppose thatspace, or spacetime, exists, not just as an object of puremathematics, but as the arena in which physical objects and events aresomehow arrayed. It is essential to understand that the problem is notabout the numerical coordinates that represent these points andregions (or instants and intervals) in a reference system; the issueis about the points and regions (or instants and intervals). Physicalobjects are located ‘in’ or ‘at’ regions ofspace; as a result, they count as concrete according to thenon-spatiality criterion. But what about the points and regions ofspace itself? There has been some debate about whether a commitment tospacetime substantivalism is consistent with the nominalist’srejection of abstract entities (Field 1980, 1989; Malament 1982). Ifwe define the abstract as the ‘non-spatial’, this debateamounts to whether space itself is to be reckoned‘spatial’. To reject that these points, regions, instants,and intervals, are concrete because they are notlocated,entails featuring them as abstract. However, to think about them asabstract sounds a bit weird, given their role in causal processes.Perhaps, it is easier to treat them as concrete if we want toestablish that concrete entities are spatiotemporal—or spatialand temporal.

The philosopher who believes that there is a serious question aboutwhether the parts of space-time count as concrete would thus do wellto characterize the abstract/concrete distinction in other terms.Still—as mentioned above—the philosopher who thinks thatit is defensible that parts of space are concrete might usenon-spatiality to draw the distinction if she manages to provide a wayof accounting for how impure sets relate to space differs from the wayconcreta do.

3.5.3 The Causal Inefficacy Criterion

According to the most widely accepted versions of theway ofnegation:

An object is abstract (if and) only if it is causallyinefficacious.

Concrete objects, whether mental or physical, have causal powers;numbers and functions and the rest make nothing happen. There is nosuch thing as causal commerce with the game of chess itself (asdistinct from its concrete instances). And even if impure sets do insome sense exist in space, it is easy enough to believe that they makenodistinctive causal contribution to what transpires. Peterand Paul may have effects individually. They may even have effectstogether that neither has on his own. But these joint effects arenaturally construed as effects of two concrete objects acting jointly,or perhaps as effects of their mereological aggregate (itself usuallyregarded as concretum), rather than as effects of some set-theoreticconstruction. Suppose Peter and Paul together tip a balance. If weentertain the possibility that this event is caused by a set, we shallhave to ask which set caused it: the set containing just Peter andPaul? Some more elaborate construction based on them? Or is it perhapsthe set containing the molecules that compose Peter and Paul? Thisproliferation of possible answers suggests that it was a mistake tocredit sets with causal powers in the first place. This is good newsfor those who wish to say that all sets are abstract.

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

The biggest challenge to characterizing abstracta as causallyinefficacious entities is thatcausality itself is anotoriously problematic and difficult to define idea. It isundoubtedly one of the most controversial notions in the history ofthought, with all kinds of views having been put forward on thematter. Thus,causally efficacious inherits any unclaritythat attaches tocausality. So, if we are to move thediscussion forward, we need to take the notion ofcausation—understood as a relation among events—assufficiently clear, even though in fact it is not. Having acknowledgedthis no doubt naïve assumption, several difficulties arise forthe suggestion that abstract objects are precisely the causallyinefficacious objects.

The idea that causal inefficacy constitutes asufficientcondition for abstractness is somewhat at odds with standard usage.Some philosophers believe in ‘epiphenomenal qualia’:objects of conscious awareness (sense data), or qualitative consciousstates that may be caused by physical processes in the brain, butwhich have no downstream causal consequences of their own (Jackson1982; Chalmers 1996). These items are causally inefficacious if theyexist, but they are not normally regarded as abstract. The proponentof the causal inefficacy criterion might respond by insisting thatabstract objects are distinctively neither causesnoreffects. But this is perilous. Abstract artifacts like JaneAusten’s novels (as we normally conceive them) come into beingas a result of human activity. The same goes for impure sets,which come into being when their concrete urelements are created.These items are clearlyeffects in some good sense; yet theyremain abstract if they exist at all. It is unclear how the proponentof the strong version of the causal inefficacy criterion (which viewscausal inefficacy as both necessary and sufficient for abstractness)might best respond to this problem.

Apart from this worry, there are no decisive intuitive counterexamplesto this account of the abstract/concrete distinction. The chiefdifficulty—and it is hardly decisive—is rather conceptual.It is widely maintained that causation, strictly speaking, is arelation among events or states of affairs. If we say that therock—an object—caused the window to break, what we mean isthat some event or state (or fact or condition)involving therock caused the break. If the rock itself is a cause, it is a cause insome 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 therock participates in events in this way that we credit the rock itselfwith causal efficacy. But what is it for an object toparticipate in an event? Suppose John is thinking about thePythagorean Theorem and you ask him to say what’s on his mind.His response is an event—the utterance of a sentence; and one ofits causes is the event of John’s thinking about the theorem.Does the Pythagorean Theorem ‘participate’ in this event?There is surelysome sense in which it does. The eventconsists in John’s coming to stand in a certain relation to thetheorem, just as the rock’s hitting the window consists in therock’s coming to stand in a certain relation to the glass. Butwe do not credit the Pythagorean Theorem with causal efficacy simplybecause it participates in this sense in an event which is acause.

The challenge is therefore to characterize the distinctive manner of‘participation in the causal order’ that distinguishes theconcrete entities. This problem has received relatively littleattention. There is no reason to believe that it cannot be solved,though the varieties of philosophical analysis for the notion ofcausality make the task full of pitfalls. Anyway, in theabsence of a solution, this standard version of theway ofnegation must be reckoned a work in progress.

3.5.4 The Discernibility / Non-Duplication Criteria

Some philosophers have supposed that, under certain conditions, thereare numerically different but indiscernible concrete entities, i.e.,that there are distinct concrete objects \(x\) and \(y\) thatexemplify the same properties. If this can be sustained, then onemight suggest that distinct abstract objects are always discernibleor, in a weaker formulation, that distinct abstract objects are neverduplicates.

Cowling (2017, 86–89) analyzes whether the abstract/concretedistinction thus rendered is fruitful, though criteria in this lineare normally offered as glosses on the universal/particulardistinction. As part of his analysis, he deploys two pairs of (notuncontroversial) distinctions: (i) between qualitative andnon-qualitative properties, and (ii) between intrinsic and extrinsicproperties. Roughly, a non-qualitative property is one that involvesspecific individuals (e.g.,being the teacher of Alexander theGreat,being Albert Einstein, etc.), while qualitativeproperties are not (e.g.,having mass,having ashape,having a length, etc.). Intrinsic properties arethose an object has regardless of what other objects are like andregardless of its relationships with other objects (e.g.,beingmade of copper). By contrast, an object’s extrinsicproperties are those that depend on other entities (e.g.,beingthe fastest car).^[9]

With these distinctions in mind, it seems impossible that there bedistinct abstract entities which are qualitatively indiscernible; eachabstract entity is expected to have a unique, distinctive qualitativeintrinsic nature (or property), which is giving reason for itsmetaphysical being. This wouldn’t be the case for any concreteentity given the initial assumption in this section. Therefore, thefollowing criterion of discernability could be pondered:

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

However, one can develop a counterexample to the above proposal, byconsidering two concrete objects that are indiscernible with respectto their intrinsic qualitative properties. Cowling (2017) considersthe case of a possible world with only two perfectly spherical balls,\(A\) and \(B\), that share the same intrinsic qualitative propertiesand that are floating at a certain distance from each other. So \(A\)and \(B\) are distinct concrete objects but indiscernible in terms oftheir intrinsic qualitative properties.^[10] But Lewis has pointed out that “if two individuals areindiscernible then so are their unit sets” (1986a, 84). If thisis correct, \(\{A\}\) and \(\{B\}\) would be indiscernible, but (atleast for some philosophers) distinct abstract objects, contrary tothe discernibility criterion.^[11] It is possible to counter-argue that we could happily accept impuresets as concrete; after all, it was always a bit unclear how theyshould be classified. Obviously, this has the problematic consequenceof having some sets—pure sets—as abstract and othersets—impure sets—as concrete. But the idea that abstractobjects have distinctive intrinsic natures allows one to establish acriterion less strong than that of discernibility; if an entity has adistinctive intrinsic nature, it cannot have a duplicate. So, the nextcriterion of non-duplication can be put forward:

\(x\) is an abstract object iff it is impossible for there to be anobject 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 theyare universals wholly present where their instances are. But thiscriterion renders them abstract. Take the color scarlet; it is auniversal wholly present in every scarlet thing. Each of the scarletsin 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: spatiotemporallylocated, causally efficacious, etc. Despite how promising theyinitially seemed, the criteria of discernibility and non-duplicabilitydo not appear to capture the abstract/concrete distinction.

3.6 The Way of Encoding

One of the most rigorous proposals about abstract objects has beendeveloped by Zalta (1983, 1988, and in a series of papers). It is aformal, axiomatic metaphysical theory of objects (both abstract andconcrete), and also includes a theory of properties, relations, andpropositions. The theory explicitly defines the notion of anabstract object but also implicitly characterizes them using axioms.^[12] There are three central aspects to the theory: (i) a predicate \(E!\)which applies to concrete entities and which is used to define a modaldistinction betweenabstract andordinary objects;(ii) a distinction betweenexemplifying relations andencoding properties (i.e., encoding 1-place relations); and(iii) a comprehension schema that asserts the conditions under whichabstract 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 assertsthere is and the predicate \(E!\)assertsexistence. On this interpretation, a formula such as\(\exists x \neg E!x\), which is implied by the axioms describedbelow, asserts “there is an object that fails to exist”.So, on this interpretation, the theory is Meinongian because itendorses non-existent objects. But there is a Quinean interpretationas well, on which the quantifier \(\exists\) assertsexistence and the predicate \(E!\) assertsconcreteness. On this interpretation, the formula \(\exists x\neg E!x\) asserts “there exists an object that fails to beconcrete”. So, on this interpretation, the theory is Platonist,since it doesn’t endorse non-existents but rather asserts theexistence of non-concrete objects. We’ll henceforth use theQuinean/Platonist interpretation.

In the more expressive, modal version of his theory, Zalta definesordinary objects \((O!)\) to be those that might be concrete.The reason is that Zalta holds that possible objects (i.e., likemillion-carat diamonds, talking donkeys, etc.) are not concrete butrather possibly concrete. They exist, but they are not abstract, sinceabstract 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 objectsthat aren’t ordinary (1993, 404):

Thus, the ordinary objects include all the concrete objects (since\(E!x\) implies \(\Diamond E!x\)), as well as possible objects thataren’t in fact concrete but might have been. On this theory,therefore,being abstract is not the negation ofbeingconcrete. Instead, the definition validates an intuition thatnumbers, sets, etc., aren’t the kind of thing that could beconcrete. Though Zalta’s definition ofabstract seemsto comport with theway of primitivism—takeconcrete as primitive, and then defineabstract asnot possibly concrete—it differs in that (a) axioms arestated that govern the conditions under which abstract objects exist(see below), and (b) the features commonly ascribed to abstractobjects arederived from principles that govern the propertyof 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, bydefinition, concrete at no possible world, they necessarily fail tohave causal powers.

(ii) The distinction betweenexemplifying andencoding is a primitive one and is represented in the theoryby 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 exemplifyproperties, only abstract objects encode properties;^[13] it is axiomatic that ordinary objects necessarily fail to encodeproperties \((O!x \to \Box \neg \exists FxF).\) Zalta’s proposalcan be seen a positive metaphysical proposal distinct from all theothers we have considered; a positive proposal that uses encoding as akey notion to characterize abstract objects. On this reading, thedefinitions and axioms of the theory convey what is meant byencoding and how it works. Intuitively, an abstract objectencodes 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 andonly its number-theoretic properties, and whereas it contingentlyexemplifies the propertybeing thought about by Peano, itnecessarily exemplifies properties such asbeing abstract,not having a shape,not being a building, etc. Thedistinction between exemplifying and encoding a property is also usedto define identity: ordinary objects are identical whenever theynecessarily exemplify the same properties while abstract objects areidentical whenever they necessarily encode the same properties.

(iii) The comprehension principle asserts that for each expressiblecondition on properties, there is an abstract object thatencodes exactly the properties that fulfill (satisfy) thatcondition. Formally: \(\exists x(A!x \:\&\: \forall F(xF \equiv\phi))\), where \(\phi\) has no free \(x\)s. Each instance of thisschema asserts the existence of an abstract object of a certain sort.So, for example, where ‘\(s\)’ denotes Socrates, theinstance \(\exists x(A!x \:\&\: \forall F(xF \equiv Fs))\) assertsthat there is an abstract object that encodes exactly the propertiesthat Socrates exemplifies. Zalta uses this object to analyze thecomplete individual concept of Socrates. But any condition \(\phi\) onconditions on properties with no free occurrences of \(x\) can be usedto form an instance of comprehension. In fact, one can prove that theobject asserted to exist is unique, since there can’t be twodistinct abstract objects that encode exactly the propertiessatisfying \(\phi\).

The theory that emerges from (i)–(iii) is further developed withadditional axioms and definitions. One axiom asserts that if an objectencodes a property, it does so necessarily \((xF \to \Box xF).\) Sothe properties that an object encodes are not relative to anycircumstance. Moreover, Zalta supplements his theory of abstractobjects with a theory of properties, relations, and propositions. Herewe describe only the theory of properties. It is governed by twoprinciples: a comprehension principle for properties and a principleof identity. The comprehension principle asserts that for anycondition on objects expressible without encoding subformulas, thereis a property \(F\) such that necessarily, an object \(x\) exemplifies\(F\) if and only if \(x\) is such that \(\phi\), i.e., \(\existsF\Box \forall x(Fx \equiv \phi)\), where \(\phi\) has no encodingsubformulas and no free \(F\text{s}\). The identity principle assertsthat 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 allowsone to assert that there are properties that are necessarilyequivalent in the classical sense, i.e., in the sense that \(\Box\forall x(Fx \equiv Gx)\), but which are distinct.^[14]

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 ofidenticals preserves truth even in modal contexts.

The foregoing principles implicitly characterize both abstract andordinary objects. Zalta’s theory doesn’t postulate anyconcrete objects, though, since that is a contingent matter. But hissystem does include the Barcan formula (i.e., \(\Diamond \exists xFx\to \exists x\Diamond Fx\)), and so possibility claims like“there might have been talking donkeys” imply that thereare (non-concrete) objects at our world that are talking donkeys atsome possible world. Since Zalta adopts the view that ordinaryproperties like being a donkey necessarily imply concreteness, suchcontingently nonconcrete objects are ordinary.

Zalta uses his theory to analyze Plato’s Forms, concepts,possible worlds, Fregean numbers and Fregean senses, fictions, andmathematical objects and relations generally. However, somephilosophers see his comprehension principle as too inclusive, for inaddition to these objects, it asserts that there are entities likethe round square orthe set of all sets which are notmembers of themselves. The theory doesn’t assert thatanythingexemplifies being round and being square—thetheory preserves the classical form of predication without giving riseto contradictions. But it does assert that there is an abstract objectthat encodes being round and being square, and that there is anabstract object that encodes the property of being a set that containsall and only non-self-membered sets. Zalta would respond by suggestingthat such objects are needed not only to state truth conditions, andexplain the logical consequences, of sentences involving expressionslike “the round square” and “the Russell set”,but also to analyze the fictional characters of inconsistent storiesand inconsistent theories (e.g., Fregean extensions).

It should be noted that Zalta’s comprehension principle forabstract objects is unrestricted and so constitutes aplenitude principle. This allows the theory to provideobjects for arbitrary mathematical theories. Where \(\tau\) is a termof mathematical theory \(T\), the comprehension principle yields aunique object that encodes all and only the properties \(F\) that areattributed to \(\tau\) in \(T\) (Linsky & Zalta 1995, Nodelman& Zalta 2014).^[15] Zalta’s theory therefore offers significant explanatory power,for it has multiple applications and advances solutions to a widerange of puzzles in different fields of philosophy.^[16]

3.7 The Ways of Weakening Existence

Many philosophers have supposed that abstract objects exist in somethin, deflated sense. In this section we consider the idea that theabstract/concrete distinction might bedefined by saying thatabstract objects exist in some less robust sense than the sense inwhich concrete objects exist.

The traditional platonist conception is a realist one: abstractobjects exist in just the same full-blooded sense that objects in thenatural world exist—they are mind-independent, rather thanartifacts of human endeavor or dependent on concrete objects in anyway. But a number of deflationary, metaontological views, nowestablished in the literature, are based on the idea that the problemstraditional platonists face have to do with “some very generalpreconceptions about what it takes to specify an object” ratherthan with “the abstractness of the desired object”(Linnebo 2018, 42). These views suggest that abstract objects exist insome weaker sense. Various approaches therefore articulate what may becalled the ways of weakening existence. One clear precedent is due toCarnap 1950 [1956], whose deflationary approach may go the furthest;Carnap rejects the metaphysical pursuit of what “reallyexists” (even in the case of concrete objects) since hemaintains that the question “Do \(X\text{s}\) reallyexist?” are pseudo-questions (if asked independently of somelinguistic framework).

But there are other ways to suggest that abstract objects haveexistence conditions that demand little of the world. For example,Linsky & Zalta (1995, 532) argued that the mind-independence andobjectivity of abstract objects isn’t like that of physicalobjects: abstract objects aren’t subject to anappearance/reality distinction, they don’t exist in a‘sparse’ way that requires discovery by empiricalinvestigation, and they aren’t complete objects (e.g.,mathematical objects are defined only by their mathematicalproperties). They use this conception tonaturalizeZalta’s comprehension principle for abstract objects.

Other deflationary accounts develop some weaker sense in whichabstract objects exist (e.g., as ‘thin’ objects). Wefurther describe some of these proposals below and try to unpack theways in which they characterize the weakened, deflationary sense ofexistence (even when such characterizations are not alwaysexplicit).

3.7.1 The Criterion of Linguistic Rules

Carnap held that claims about the “real” existence ofentities (concrete or abstract) do not have cognitive content. Theyare pseudo-statements. However, he admitted: (a) that there aresentences in science that use terms that designate mathematicalentities (such as numbers); and (b) that semantic analysis seems torequire entities like properties and propositions. Since mathematicalentities, properties, and propositions are traditionally consideredabstract, he wanted to clarify how it is possible to accept a languagereferring to abstract entities without adopting what he consideredpseudo-sentences about such entities’ objective reality.Carnap’s famous paper (1950 [1956]) contained an attempt to showthat, without embracing Platonism, one can use a language referring toabstract entities.

To achieve these goals, Carnap begins by noting that before one canask existence questions about entities of a determinate kind, onefirst has to have a language, or alinguistic framework, thatallows one to speak about the kinds of entity in question. He thendistinguishes ‘internal’ existence questions expressedwithin such a linguistic framework from ‘external’existence questions about a framework. Only the latter ask whether theentities of that framework are objectively real. As we’ll seebelow, Carnap thought that internal existence questions within aframework can be answered, either by empirical investigation or bylogical analysis, depending on the kind of entity the framework isabout. By contrast, Carnap regards external questions (e.g., ‘Do\(X\text{s}\) exist?’, expressed either about, or independentof, a linguistic framework) as pseudo-questions: though they appear tobe theoretical questions, in fact they are merely practical questionsabout the utility of the linguistic framework for science.

Carnap’s paper (1950 [1956]) considers a variety of linguisticframeworks, such as those for: observable things (i.e., thespatiotemporally ordered system of observable things and events),natural numbers and integers, propositions, thing properties, rationaland real numbers, and spatiotemporal coordinate systems. Eachframework is established by developing a language that typicallyincludes expressions for one or more kinds of entities in question,expressions for properties of the entities in question (including ageneral category term for each kind of entity in question), andvariables ranging over those entities. Thus, a framework for thesystem 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 overobservables. The framework for natural numbers has expressions thatdenote them (‘0’, ‘2+5’), expressions forproperties of the numbers (‘prime’, ‘odd’),including the general category term ‘number’), andvariables ranging over numbers.

For Carnap, each statement in a linguistic framework should have atruth value that can be determined either by analytical or empiricalmethods. A statement’s truth value is analytically determinableif it is logically true (or false), or if it’s truth isdeterminable exclusively from the rules of the language or on thebasis of semantic relationships among its component expressions. Astatement is empirically determinable when it is confirmable (ordisconfirmable) in the light of the perceived evidence. Note that thevery attempt to confirm an empirical statement about physical objectson the basis of the evidence requires that one adopt the language ofthe framework of things. Carnap warns us, however, that “thismust not be interpreted as if it meant … acceptance of abelief in the reality of the thing world; there is no suchbelief or assertion or assumption because it is not a theoreticalquestion” (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 matterwhether it is a framework about physical, concrete things, or aframework about abstract entities such as numbers, properties,concepts, propositions, etc. For him, the pragmatic reasons foraccepting a given linguistic framework are that it has explanatorypower, unifies the explanation of disparate kinds of data andphenomena, expresses claims more efficiently, etc. And we often choosea framework for a particular explanatory purpose. We might thereforechoose a framework with expressions about abstract entities to carryout anexplication (i.e., an elucidation of concepts), or todevelop a semantics for natural language. For Carnap, the choicebetween platonism or nominalism is not a legitimate one; both areinappropriate 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, isto be determined empirically, since empirical observations andinvestigations are needed. These statements are not true in virtue ofthe 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 aproperty \(F\) such that both \(x\) and \(y\) are \(F\)’(‘\(\existsF(Fx \:\&\: Fy)\)’)expressed within the framework of property theory, can be determinedanalytically. For these statements either form part of the rules ofthe language (e.g., expressed as axioms that govern the terms of thelanguage) or are derivable from the rules of the language. When thesestatements are part of the rules that make up the linguisticframework, they are consideredanalytic, as are theexistential statements that follow from those rules.^[17]

All of the existence assertions just discussed are thereforeinternal to their respective linguistic frameworks. Carnapthinks that the only sense that can be given to talk of“existence” is an internal sense. Internal questions aboutthe existence of things or abstract objects are not questions abouttheir real metaphysical existence.^[18] Hence, it seems more appropriate to describe his view as embodying adeflationary notion of object. For Carnap concludes “thequestion of the admissibility of entities of a certain type or ofabstract entities in general as designata is reduced to the questionof the acceptability of the linguistic framework for thoseentities” (1950 [1956, 217]).

Thus, for each framework (no matter whether it describes empiricalobjects, abstract objects, or a mix of both), one can formulate bothsimple and complex existential statements. According to Carnap, eachsimple existential statement is either empirical or analytic. If asimple statement is empirical, its truth value can be determined by acombination of empirical inquiry and consideration of the linguisticrules governing the framework; if the simple existential statement isanalytic, then its truth value can be determined simply by consideringthe linguistic rules governing the framework. Whereas the simpleexistential statements that require empirical investigation assert theexistence of possible concrete entities (like ‘tables’ or‘unicorns’), the simple existential statements that areanalytic assert the existence of abstract entities. Let us call thiscriterion for asserting the existence of abstract objects thecriterion 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 therelevant language.

But this criterion suggests thatimpure sets,object-dependent properties, abstract artifacts, and the rest are notabstract. For this criterion appears to draw a line between certainpure abstract entities and everything else. The truth of simpleexistence statements about \(\{\textrm{Bob Dylan}\}\) orDickens’A Christmas Carol, which usually areconsidered abstract entities, does not depend solely on linguisticrules. The same goes for simple and complex existential statementswith general terms such as ‘novel’, ‘legalstatute’, etc.

In the end, though, Carnap doesn’t seem to be either a realistor nominalist about objects (abstract or concrete). Carnap rejects thequestion whether these objects are real in a metaphysical sense. But,contrary to the nominalist, he rejects the idea that we can truly denythe real existence of abstract objects (i.e., a denial that isexternal to a linguistic framework). This attitude, which settles thequestion of which framework to adopt on pragmatic grounds (e.g., whichframework best helps us to make sense of the data to be explained), isthe reason why we’ve labeled his view as away of weakeningexistence. See the entry onCarnap for further details.

Proposals by other philosophers are related to Carnap’s view.Resnik (1997, Part Two) has put forward apostulationalepistemology for the existence of mathematical objects. Accordingto this view, all one has to do to ensure the existence ofmathematical objects is to use a language to posit mathematicalobjects and to establish a consistent mathematical theory for them.^[19] Nevertheless, their existence does not result from their beingposited. Instead, we recognize those objects as existent because aconsistent mathematical theory for them has been developed. Resnikrequires both a linguistic stipulation for considering mathematicalobjects and a coherency condition for recognizing them as existent.Thomasson (2015, 30–34) advocates for an approach which shetakes to be inherited from Carnap. She calls iteasyontology. Since she is not trying to find ultimate categories ora definitive list of basic (abstract or concrete) objects, she prefersa simpler kind of realism (see Thomasson 2015, 145–158). Sheargues that everyday uses of existential statements provide acceptableontological commitments when those assertions are supported either byempirical evidence or merely by the rules of use that govern generalterms (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 acriterion of linguistic rules for accepting abstract objects.Given her defense of simple realism, it appears that she takes bothobservable objects and theoretical entities in science asconcrete.

3.7.2 The Criterion of Minimalism

In what follows, two ways of formulating criteria for theabstract/concrete distinction are considered. The views start with theidea that our concept of an object allows for objects whose existenceplaces very few demands on reality over and above the demands imposedby claims that do not mention abstract objects. Those philosophers whomaintain this philosophical thesis are what Linnebo (2012) callsmetaontologicalminimalists. Their proposals are typicallyput forward in connection with issues in the philosophy ofmathematics, but then applied to other domains.

Parsons (1990), Resnik (1997), and Shapiro (1997) contend that, in thecase of mathematical theories, coherence suffices for the existence ofthe objects mentioned in those theories.^[20] They do not offer an explicit criterion for distinguishing abstractand concrete objects. Nevertheless, their proposals implicitly drawthe distinction; abstract objects are those objects that exist invirtue of the truth of certain modal claims. In particular, theexistence of mathematical objects is “grounded in” puremodal 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 ofone or another set theory, etc. Since these pure modal truths arenecessary, this explains why pure abstract objects exist necessarily.It also explains a sense in which they are insubstantial: theirexistence is grounded in truths that do not (on the face of it)require the actual existence of anything at all.^[21]

Linnebo (2018) advances a proposal about how to conceive abstractobjects by revising our understanding of Fregean biconditionalprinciples of abstraction (see subsection3.4). Some philosophers take these Fregean abstraction principles to beanalytic sentences. For example, Hale & Wright (2001; 2009)consider the two sides of an abstraction principle as equivalent as amatter of meaning; they ‘carve up content’ in differentways (to use Frege’s metaphor). But Linnebo (2018, 13–14)rejects this view and the view that such biconditional principles areanalytic.

He suggests instead that we achieve reference to abstract (and otherobjects) by means of asufficiency operator, \(\Rightarrow\),which he takes to be a strengthening of the material conditional. Hestarts withconditional 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 reconceptualizationof the left-hand side. He represents these claims as \(\phi\Rightarrow \psi\), where the new operator‘\(\Rightarrow\)’ is meant to capture the intuitive ideathat \(\phi\)is (conceptually)sufficientfor \(\psi\), orall that is required for \(\psi\)is \(\phi\). For \(\phi\) to be sufficient for \(\psi\),sufficiency must be stronger thanmetaphysically implies butweaker thananalytically implies (see Linnebo 2018, 15). Thenotion Linnebo considers is a ‘species of metaphysical grounding’.^[22] Hence, sufficiency statements allow us to conceptualize statementsmentioning abstract objects (or other problematic objects) in terms ofmetaphysically less problematic or non-problematic objects.

It is important for Linnebo that sufficiency be asymmetric. Hewouldn’t accept mutual sufficiency, i.e., principles of the form\(Rab \Leftrightarrow f(a) \! =\! f(b)\), since these would imply thatboth sides are equivalent as a matter of meaning. Instead, the pointis that the seemingly unproblematic claim \(Rab\) renders the claim\(f(a) \! =\! f(b)\) unproblematic, and this is best expressed bysufficiency statements of the form \(Rab \Rightarrow f(a) \! =\!f(b)\), on which the left sidegrounds the right side. SoLinnebo’s notion of reconceptualization is not the Fregeannotion of recarving of content.

Moreover, in a sufficiency statement, Linnebo doesn’t requirethat the relation \(R\) be an equivalence relation; he requires onlythat \(R\) be symmetric and transitive. It need not be reflexive, forthe domain might contain entities \(x\) such that \(\neg Rxx\) (e.g.,in the case of the sufficiency statement for directions, not everyobject \(x\) in the domain is such that \(x\) is parallel with\(x\)—being parallel is restricted to lines). Linnebo calls suchsymmetric and transitive relationsunity relations. When asufficiency statement—\(Rab \Rightarrow f(a) \! =\!f(b)\)—holds, then new objects are identified. The new objectsare specified in terms of the less problematic entities related by\(R\); for example, directions become specified by lines that areparallel. According to Linnebo, the parallel lines becomespecifications of the new objects. A unity relation \(R\) istherefore the starting point for developing a sufficient (but notnecessary and sufficient) condition for reference.

Sometimes the new objects introduced by conditional principles do notmake demands on reality; when that happens, they are said to bethin (for example, directions only require that there beparallel lines). However, when the new objects introduced bysufficiency statements make more substantial demands on reality, theobjects are consideredthick. Suppose \(Rab\) asserts \(a\)and \(b\) are spatiotemporal parts of the same cohesive and naturallybounded whole. Then \(a\) and \(b\) become specifications for physicalbodies via the following principle: \(Rab \Rightarrow \mathrm{Body}(a)\! =\! \mathrm{Body}(b)\). In this case, the principle “makes asubstantial demand on the world” because it requires checkingthat there are spatiotemporal partsconstituting a continuousstretch of solid stuff (just looking at the spatiotemporal partsdoes not suffice to determine whether they constitute to a body; seeLinnebo 2018, 45).

However, Linnebo does not identify being abstract with being thin(2012, 147), for there are thin objects in a relative sense that arenot abstract, namely those that make no substantial demands on theworld beyond those introduced in terms of some antecedently givenobjects. The mereological sum of your left hand and your laptop makesno demand on the world beyond the demands of its parts.^[23] Instead, he suggests that abstract objects are those that are thinand thathave a shallow nature. The notion ofshallownature is meant to capture “the intuitive idea that anyquestion that is solely about \(F\text{s}\) has an answer that can bedetermined on the basis of any given specifications of these\(F\text{s}\)” (2018, 192–195). For example, directionshave a shallow nature because any question about directions (e.g., arethey orthogonal, etc.?) can be determined solely on the basis of thelines that specify them. Shapes have a shallow nature because anyquestion about them (e.g., are they triangular, circular, etc.?) canbe determined solely on the basis of their underlying concretefigures. By contrast, mereological sums of concrete objects arenot shallow because there are questions about them thatcannot be answered solely on the basis of their specifications; forinstance, the weight of the mereological sum of your laptop and yourleft hand depends not only on their combination but also on thegravitational field in which they are located.^[24]

Linnebo thus contrasts abstract objects, which are thin and have ashallow nature, with concrete objects, which do not have a shallownature. Linnebo extends this view in several ways. He constructs anaccount of mathematical objects that goes beyond theway ofabstraction principles by providing a reconstruction of settheory in terms of ‘dynamic abstraction’ (2018, ch. 3).This form of minimalism also allows for abstract objects of a mixednature; namely, those that arethin relative to otherobjects. For example, the type of the letter ‘A’ isabstract because it is thin and has a shallow nature, but it is thinwith respect the tokens of the letter ‘A’.

This view, as Linnebo himself admits, faces some problems. One of themis that the methodologies used by working mathematicians, such asclassical logic, impredicative definitions, and taking arbitrarysubcollections of infinite domains, seem to presuppose objects thatare more independent, i.e., objects that don’t have a shallownature (2018, 197; for a discussion of independence, see Section 4.1of the entry onplatonism in mathematics). Another problem (2018, 195) is that in order for an object to countas having ashallow nature, anintrinsic unityrelation has to be available. An investigation is required toestablish that there is such an intrinsic unity relation in each case.It is far from clear that a conditional principle with anintrinsic unity relation is available for each of problematiccases mentioned in this entry, such as chess, legal institutions orthe English language. Finally, Linnebo doesn’t discuss thequestion of whether sets of concrete urelements are themselvesabstract or concrete. At present, there may be an important questionleft open by his theory that other theories of abstract objectsanswer.

3.8 Eliminativism

We come finally to proposals that reject the abstract/concretedistinction. We can consider three cases. First, there are thenominalists who both reject abstract entities and reject thedistinction as illegitimate. They focus on arguing against theformulations of the distinction proposed in the literature. A secondgroup of eliminativists reject real objects of any kind, therebydismissing the distinction as irrelevant; these are the ontologicalnihilists. A final group of eliminativists agree that there areprototypical cases of concrete objects and abstract objects, butconclude that a rigorous philosophical distinction can’t be madeclearly enough to have any explanatory power (see Sider 2013, 287).This recalls Lewis’ pessimism (1986a, 81–86) about thepossibility of establishing a distinction that is sufficiently clearto be theoretically interesting.

4. Further Reading

Berto & Plebani (2015) provide an useful introduction to ontologyand metaontology. Liggins (2024) provides a systematic presentation ofplatonist and nominalist positions—focused primarily onmathematical objects, but extendable to propositions andproperties—using as a guiding thread the assumption that mostarguments in favor of the existence of abstract objects share a commonargumentative pattern. Van Inwagen & Craig (2024) offer a debateon the existence of abstract objects—particularlynumbers—exploring various arguments, including theologicalconsiderations. Putnam (1971) makes the case for abstract objects onscientific grounds. Bealer (1993) and Tennant (1997) presentapriori arguments for the necessary existence of abstractentities. Fine (2002) systematically studies of abstraction principlesin the foundations of mathematics. Wetzel (2009) examines thetype-token distinction, argues that types are abstract objects whilethe tokens of those types are their concrete instances, and shows howdifficult it is to paraphrase away the many references to types thatoccur in the sciences and natural language. Zalta (2020) develops atype-theoretic framework for higher-order abstract objects (whichincludes abstract properties and abstract relations in addition toordinary properties and relations) and offers both comparisons toother type theories and applications in philosophy and linguistics.Moltmann (2013) investigates the extent to which abstract objects areneeded 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 naturallanguage ontology contains references to abstract objects only in itsperiphery. Falguera and Martínez-Vidal (2020) have edited avolume in which contributors present positions and debates aboutabstract objects of different kinds and categories, in differentfields in philosophy.

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Acknowledgments

This entry was revised, updated, and expanded in 2021 by JoséL. Falguera and Concha Martínez-Vidal. The author of the 2020version of this entry, Gideon Rosen, remains credited on this entrysince significant content in Sections 1, 2.1, 3.5.1–3.5.3, and 4has been retained from his text.