Platonism about mathematics (ormathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons andplanets are made true or false by the objects with which they areconcerned and these objects’ perfectly objective properties, soare statements about numbers and sets. Mathematical truths aretherefore discovered, not invented.

The most important argument for the existence of abstract mathematicalobjects derives from Gottlob Frege and goes as follows (Frege 1953).The language of mathematics purports to refer to and quantify overabstract mathematical objects. And a great number of mathematicaltheorems are true. But a sentence cannot be true unless itssub-expressions succeed in doing what they purport to do. So thereexist abstract mathematical objects that these expressions refer toand quantify over.

Frege’s argument notwithstanding, philosophers have developed avariety of objections to mathematical platonism. Thus, abstractmathematical objects are claimed to be epistemologically inaccessibleand metaphysically problematic. Mathematical platonism has been amongthe most hotly debated topics in the philosophy of mathematics overthe past few decades.

• 1. What is Mathematical Platonism?
  • 1.1 Historical remarks
  • 1.2 The philosophical significance of mathematical platonism
  • 1.3 Object realism
  • 1.4 Truth-value realism
  • 1.5 The mathematical significance of platonism
• 2. The Fregean Argument for Existence
  • 2.1 The structure of the argument
  • 2.2 Defending Classical Semantics
  • 2.3 Defending Truth
  • 2.4 The notion of ontological commitment
  • 2.5 From Existence to Mathematical Platonism?
• 3. Objections to Mathematical Platonism
  • 3.1 Epistemological access
  • 3.2 A metaphysical objection
  • 3.3 Other metaphysical objections
• 4. Between object realism and mathematical platonism
  • 4.1 How to understand independence
  • 4.2 Plenitudinous platonism
  • 4.3 Lightweight semantic values
  • 4.4 Inspiration from Aristotle
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. What is Mathematical Platonism?

Mathematical platonism can be defined as the conjunction of thefollowing three theses:

Existence.

There are mathematical objects.

Abstractness.

Mathematical objects are abstract.

Independence.

Mathematical objects are independent of intelligent agents andtheir language, thought, and practices.

Some representative definitions of ‘mathematicalplatonism’ are listed in the supplement

Some Definitions of Platonism

and document that the above definition is fairly standard.

Platonism in general (as opposed to platonism about mathematicsspecifically) is any view that arises from the above three claims byreplacing the adjective ‘mathematical’ by any otheradjective.

The first two claims are tolerably clear for present purposes.Existence can be formalized as ‘\(\existsxMx\)’, where ‘\(Mx\)’ abbreviates the predicate‘\(x\) is a mathematical object’ which is true of all andonly the objects studied by pure mathematics, such as numbers, sets,and functions.Abstractness says that everymathematical object is abstract, where an object is said to beabstract just in case it is non-spatiotemporal and (therefore)causally inefficacious. (For further discussion, see the entry onabstract objects.)

Independence is less clear than the other two claims.What does it mean to ascribe this sort of independence to an object?The most obvious gloss is probably the counterfactual conditionalthat, had there not been any intelligent agents, or had theirlanguage, thought, or practices been different, there would still havebeen mathematical objects. However, it is doubtful that this glosswill do all the work thatIndependence is supposed todo (see Section4.1). For now,Independence will be left somewhatschematic.

1.1 Historical remarks

Platonism must be distinguished from the view of the historical Plato.Few parties to the contemporary debate about platonism make strongexegetical claims about Plato’s view, much less defend it.Although the view that we are calling ‘platonism’ isinspired by Plato’s famous theory of abstract and eternal Forms(see the entry onPlato’s metaphysics and epistemology), platonism is now defined and debated independently of its originalhistorical inspiration.

Not only is the platonism under discussion not Plato’s,platonism as characterized above is a purely metaphysical view: itshould be distinguished from other views that have substantiveepistemological content. Many older characterizations of platonism addstrong epistemological claims to the effect that we have someimmediate grasp of, or insight into, the realm of abstract objects.(See e.g., Rees 1967.) But it is useful (and nowadays fairly standard)to reserve the term ‘platonism’ for the purelymetaphysical view described above. Many philosophers who defendplatonism in this purely metaphysical sense would reject theadditional epistemological claims. Examples include Quine and otherphilosophers attracted to the so-calledindispensabilityargument, which seeks to give a broadly empirical defense ofmathematical platonism. (See the entry onindispensability arguments in the philosophy of mathematics.)

Finally, the above definition of ‘mathematical platonism’excludes the claim that all truths of pure mathematics are necessary,although this claim has traditionally been made by most platonists.Again, this exclusion is justified by the fact that some philosopherswho are generally regarded as platonists (for instance, Quine and someadherents of the aforementioned indispensability argument) reject thisadditional modal claim .

1.2 The philosophical significance of mathematical platonism

Mathematical platonism has considerable philosophical significance. Ifthe view is true, it will put great pressure on the physicalist ideathat reality is exhausted by the physical. For platonism entails thatreality extends far beyond the physical world and includes objectsthat aren’t part of the causal and spatiotemporal order studiedby the physical sciences.^[1] Mathematical platonism, if true, will also put great pressure on manynaturalistic theories of knowledge. For there is little doubt that wepossess mathematical knowledge. The truth of mathematical platonismwould therefore establish that we have knowledge of abstract (and thuscausally inefficacious) objects. This would be an important discovery,which many naturalistic theories of knowledge would struggle toaccommodate.

Although these philosophical consequences are not unique tomathematical platonism, this particular form of platonism isunusually well suited to support such consequences. For mathematics isa remarkably successful discipline, both in its own right and as atool for other sciences.^[2] Few contemporary analytic philosophers are willing to contradict anyof the core claims of a discipline whose scientific credentials are asstrong as those of mathematics (Lewis 1991, pp. 57–9). So ifphilosophical analysis revealed mathematics to have some strange andsurprising consequences, it would be unattractive simply to reject mathematics.^[3] A form of platonism based on a discipline whose scientificcredentials are less impressive than those of mathematics would not bein this fortunate situation. For instance, when theology turns out tohave some strange and surprising philosophical consequences, manyphilosophers do not hesitate to reject the relevant parts oftheology.

1.3 Object realism

Letobject realism be the view that there exist abstractmathematical objects. Object realism is thus just the conjunction ofExistence andAbstractness.^[4] Object realism stands opposed tonominalism, which incontemporary philosophy is typically defined as the view that thereare no abstract objects. (In more traditional philosophical usage theword ‘nominalism’ refers instead to the view that thereare no universals. See Burgess & Rosen 1997, pp. 13–25 andthe entry onabstract objects.)

Because object realism leaves outIndependence, thisview is logically weaker than mathematical platonism. Thephilosophical consequences of object realism are thus not as strong asthose of platonism. Many physicalists would accept non-physicalobjects provided that these are dependent on or reducible to physicalobjects. They may for instance accept objects such as corporations,laws, and poems, provided that these are suitably dependent orreducible to physical objects. Moreover, there appears to be nomystery about epistemic access to non-physical objects that we havesomehow made or ‘constituted’. If corporations, laws, andpoems are made or ‘constituted’ by us, presumably we gainknowledge of them in the process of making or‘constituting’ them.

Some views in the philosophy of mathematics are object realist withoutbeing platonist. One example are traditional intuitionist views, whichaffirm the existence of mathematical objects but maintain that theseobjects depend on or are constituted by mathematicians and their activities.^[5] Some further examples of views that are object realist without beingplatonist will be discussed in Section4.

1.4 Truth-value realism

Truth-value realism is the view that every well-formedmathematical statement has a unique and objective truth-value that isindependent of whether it can be known by us and whether it followslogically from our current mathematical theories. The view also holdsthat most mathematical statements that are deemed to be true are infact true. So truth-value realism is clearly ametaphysicalview. But unlike platonism it is not anontological view. Foralthough truth-value realism claims that mathematical statements haveunique and objective truth-values, it is not committed to thedistinctively platonist idea that these truth-values are to beexplained in terms of an ontology of mathematical objects.

Mathematical platonism clearly motivates truth-value realism byproviding an account of how mathematical statements get theirtruth-values. But the former view does not entail the latter unlessfurther premises are added. For even if there are mathematicalobjects, referential and quantificational indeterminacy may deprivemathematical statements of a unique and objective truth-value.Conversely, truth-value realism does not by itself entailExistence and thus implies neither object realism norplatonism. For there are various accounts of how mathematicalstatements can come to possess unique and objective truth-values whichdo not posit a realm of mathematical objects.^[6]

In fact, many nominalists endorse truth-value realism, at least aboutmore basic branches of mathematics, such as arithmetic. Nominalists ofthis type are committed to the slightly odd-sounding view that,although the ordinary mathematical statement

(1)

There are primes numbers between 10 and 20.

is true, there are in fact no mathematical objects and thus inparticular no numbers. But there is no contradiction here. We mustdistinguish between the language \(L_M\) in which mathematicians maketheir claims and the language \(L_P\) in which nominalists and otherphilosophers make theirs. The statement (1) is made in \(L_M\). Butthe nominalist’s assertion that (1) is true but that there areno abstract objects is made in \(L_P\). The nominalist’sassertion is thus perfectly coherent provided that (1) is translatednon-homophonically from \(L_M\) into \(L_P\). And indeed, when thenominalist claims that the truth-values of sentences of \(L_M\) arefixed in a way that doesn’t appeal to mathematical objects, itis precisely this sort of non-homophonic translation she has in mind.The view mentioned in the previous note provides an example.

This shows that for the claimExistence to have itsintended effect, it must be expressed in the language \(L_P\) used byus philosophers. If the claim was expressed in the language \(L_M\)used by mathematicians, then nominalists could accept the claim whilestill denying that there are mathematical objects, contrary to thepurpose of the claim.

A small but important tradition of philosophers urge that the debateabout platonism should be replaced by, or at least transformed into, adebate about truth-value realism. One reason offered in support ofthis view is that the former debate is hopelessly unclear, while thelatter is more tractable (Dummett 1978a, pp. 228–232 and Dummett1991b, pp. 10–15). Another reason offered is that the debateabout truth-value realism is of greater importance to both philosophyand mathematics than the one about platonism.^[7]

1.5 The mathematical significance of platonism

Working realism is the methodological view that mathematicsshould be practicedas if platonism was true (Bernays 1935,Shapiro 1997, pp. 21–27 and 38–44). This requires someexplanation. In debates about the foundations of mathematics platonismhas often been used to defend certain mathematical methods, such asthe following:

1. Classical first-order (or stronger) languages whose singular termsand quantifiers appear to be referring to and ranging overmathematical objects. (This contrasts with the languages thatdominated earlier in the history of mathematics, which relied moreheavily on constructive and modal vocabulary.)
2. Classical rather than intuitionistic logic.
3. Non-constructive methods (such as non-constructive existenceproofs) and non-constructive axioms (such as the Axiom ofChoice).
4. Impredicative definitions (that is, definitions that quantify overa totality to which the object being defined would belong).
5. ‘Hilbertian optimism’, that is, the belief that everymathematical problem is in principle solvable.^[8]

According to working realism, these and other classical methods areacceptable and available in all mathematical reasoning. But workingrealism does not take a stand on whether these methods require anyphilosophical defense, and if so, whether this defense must be basedon platonism. In short, where platonism is an explicitly philosophicalview, working realism is first and foremost a view within mathematicsitself about the correct methodology of this discipline. Platonism andworking realism are therefore distinct views.

However, there may of course be logical relations between the twoviews. Given the origin of working realism, it is not surprising thatthe view receives strong support from mathematical platonism. Assumethat mathematical platonism is true. Then clearly the language ofmathematics ought to be as described in (i). Secondly, provided it islegitimate to reason classically about any independently existing partof reality, (ii) would also follow. Thirdly, since platonism ensuresthat mathematics is discovered rather than invented, there would be noneed for mathematicians to restrict themselves to constructive methodsand axioms, which establishes (iii). Fourth, there is a powerful andinfluential argument due to Gödel (1944) that impredicativedefinitions are legitimate whenever the objects being defined existindependently of our definitions. (For instance, ‘the tallestboy in the class’ appears unproblematic despite beingimpredicative.) If this is correct, then (iv) would follow. Finally,if mathematics is about some independently existing reality, thenevery mathematical problem has a unique and determinate answer, whichprovides at least some motivation for Hilbertian optimism. (See,however, the discussion ofplenitudinous platonism in Section4.2.)

The truth of mathematical platonism would therefore have importantconsequences within mathematics itself. It would justify the classicalmethods associated with working realism and encourage the search fornew axioms to settle questions (such as the Continuum Hypothesis)which are left open by our current mathematical theories.

However, working realism does not in any obvious way imply platonism.Although working realism says that we are justified in using theplatonistic language of contemporary mathematics, this falls short ofplatonism in at least two ways. As the above discussion of truth-valuerealism showed, the platonistic language of mathematics can beanalysed in such a way as to avoid reference to and quantificationover mathematical objects. Moreover, even if a face-value analysis ofthe language of mathematics could be justified, this would supportobject realism but not platonism. An additional argument would beneeded for the third component of platonism, namely,Independence. The prospects for such an argument arediscussed in Section4.1.

2. The Fregean Argument for Existence

We now describe a template of an argument for the existence ofmathematical objects. Since the first philosopher who developed anargument of this general form was Frege, it will be referred to asthe Fregean argument. But the template is general andabstracts away from most specific aspects of Frege’s own defenseof the existence of mathematical objects, such as his view thatarithmetic is reducible to logic. Fregean logicism is just one way inwhich this template can be developed; some other ways will bementioned below.

2.1 The structure of the argument

The Fregean argument is based on two premises, the first of whichconcerns the semantics of the language of mathematics:

Classical Semantics.

The singular terms of the language of mathematics purport to referto mathematical objects, and its first-order quantifiers purport torange over such objects.

The word ‘purport’ needs to be explained. When a sentence\(S\) purports to refer or quantify in a certain way, this means thatfor \(S\) to be true, \(S\) must succeed in referring or quantifyingin this way.

The second premise does not require much explanation:

Truth.

Most sentences accepted as mathematical theorems are true(regardless of their syntactic and semantic structure).

Consider sentences that are accepted as mathematical theorems and thatcontain one or more mathematical singular terms. ByTruth, most of these sentences are true.^[9] Let \(S\) be one such sentence. ByClassicalSemantics, the truth of \(S\) requires that its singularterms succeed in referring to mathematical objects. Hence there mustbe mathematical objects, as asserted byExistence.^[10]

2.2 Defending Classical Semantics

Classical Semantics claims that the language ofmathematics functions semantically much like language in generalfunctions (or at least has traditionally been assumed to function):the semantic functions of singular terms and quantifiers are to referto objects and to range over objects, respectively. This is a broadlyempirical claim about the workings of a semi-formal language used bythe community of professional mathematicians. (In the widely adoptedterminology of Burgess & Rosen 1997, pp. 6–7,Classical Semantics is ahermeneutic claim;that is, it is a descriptive claim about how a certain language isactually used, not a normative claim about how this language ought tobe used.) Note also thatClassical Semantics iscompatible with most traditional views on semantics; in particular, itis compatible with all the standard views on the meanings ofsentences, namely that they are truth-values, propositions, or sets ofpossible worlds.

Classical Semantics enjoys strongprimafacie plausibility. For the language of mathematics stronglyappears to have the same semantic structure as ordinarynon-mathematical language. As Burgess (1999) observes, the followingtwo sentences appear to have the same simple semantic structure of apredicate being ascribed to a subject (p. 288):

• Evelyn is prim.
• Eleven is prime.

This appearance is also borne out by the standard semantic analysesproposed by linguists and semanticists.

Classical Semantics has nevertheless been challenged,for instance by nominalists such as Hellman (1989) and by Hofweber(2005 and 2016). (See also Moltmann (2013) for some challengesconcerned with arithmetical vocabulary in natural language, as well asSnyder (2017) for discussion.) This is not the place for an extendeddiscussion of such challenges. Let me just note that a lot of work isneeded to substantiate this sort of challenge. The challenger willhave to argue that the apparent semantic similarities betweenmathematical and non-mathematical language are deceptive. And thesearguments will have to be of the sort that linguists andsemanticists—with no vested interest in the philosophy ofmathematics—could come to recognize as significant.^[11]

2.3 Defending Truth

Truth can be defended in a variety of different ways.Common to all defenses is that they first identify some standard bywhich the truth-values of mathematical statements can be assessed andthen argue that mathematical theorems meet this standard.

One option is to appeal to a standard that is more fundamental thanthat of mathematics itself. Logicism provides an example. Frege andother logicists first claim that any theorem of pure logic is true.Then they attempt to show that the theorems of certain branches ofmathematics can be proved from pure logic and definitions alone.

Another option is to appeal to the standards of empirical science. TheQuine-Putnam indispensability argument provides an example. First itis argued that any indispensable part of empirical science is likelyto be true and therefore something we are justified in believing. Thenit is argued that large amounts of mathematics are indispensable toempirical science. If both claims are correct, it follows thatTruth is likely to be true and that belief inTruth therefore is justified. (See the entry onindispensability arguments in the philosophy of mathematics.)

A third option is to appeal to the standards of mathematics itself.Why should one have to appeal to non-mathematical standards, such asthose of logic or empirical science, in order to defend the truth ofmathematical theorems? When we defend the truth of the claims of logicand physics, we do not need to appeal to standards outside ofrespectively logic and physics. Rather we assume that logic andphysics provide their ownsui generis standards ofjustification. Why should mathematics be any different? This thirdstrategy has received a lot of attention in recent years, often underthe heading of ‘naturalism’ or ‘mathematicalnaturalism’. (See Burgess & Rosen 1997, Maddy 1997, and, forcritical discussion, see the entry onnaturalism in the philosophy of mathematics.)

Here is an example of how a naturalistic strategy can be developed.Call the attitude that mathematicians take to the theorems ofmathematics ‘acceptance’. Then the following claims seemplausible:

(2)

Mathematicians are justified in accepting the theorems ofmathematics.

(3)

Accepting a mathematical statement \(S\) involves taking \(S\) tobe true.

(4)

When a mathematician accepts a mathematical statement \(S\), thecontent of this attitude is in general the literal meaning of\(S\).

From these three claims it follows that mathematical experts arejustified in taking the theorems of mathematics to be literal truths.By extension the rest of us too are justified in believingTruth. Note that the experts with whom (2) isconcerned need not themselves believe (3) and (4), let alone bejustified in any such belief. What matters is that the three claimsare true. The task of establishing the truth of (3) and (4) may fallto linguists, psychologists, sociologists, or philosophers, butcertainly not to mathematicians themselves.

Admittedly, fictionalists about mathematics will try to resist (3) or(4). See Field (1982), Yablo (2005), Leng (2010), as well as the entryonfictionalism in the philosophy of mathematics.

2.4 The notion of ontological commitment

Versions of the Fregean argument are sometimes stated in terms of thenotion of ontological commitment. Suppose we operate with the standardQuinean criterion of ontological commitment:

Quine’s Criterion.

A first-order sentence (or collection of such sentences) isontologically committed to such objects as must be assumed to be inthe range of the variables for the sentence (or collection ofsentences) to be true.

Then it follows fromClassical Semantics that manysentences of mathematics are ontologically committed to mathematicalobjects. To see this, consider a typical mathematical theorem \(S\),which involves some normal extensional occurrence of either singularterms or first-order quantifiers. ByClassicalSemantics, these expressions purport to refer to or rangeover mathematical objects. For \(S\) to be true, these expressionsmust succeed in doing what they purport to do. Consequently, for \(S\)to be true, there must be mathematical objects in the range of thevariables. ByQuine’s Criterion this means that\(S\) is ontologically committed to mathematical objects.

Quine and many others takeQuine’s Criterion tobe little more than a definition of the term ‘ontologicalcommitment’ (Quine 1969 and Burgess 2004). But the criterion hasnevertheless been challenged. Some philosophers deny that singularterms and first-order quantifiers automatically give rise toontological commitments. Perhaps what is “required of theworld” for the sentence to be true involves the existence ofsome but not all of the objects in the range of the quantifiers (Rayo2008). Or perhaps we should sever the link between the first-orderexistential quantifier and the notion of ontological commitment(Azzouni 2004, Hofweber 2000 and 2016).

One response to these challenges is to observe that the Fregeanargument was developed above without any use of the term‘ontological commitment’. Any challenge to the definitionof ‘ontological commitment’ provided byQuine’s Criterion therefore appears irrelevantto the version of the Fregean argument developed above. However, thisresponse is unlikely to satisfy the challengers, who will respond thatthe conclusion of the argument developed above is too weak to have itsintended effect. Recall that the conclusion,Existence, is formalized in our philosophicalmeta-language \(L_P\) as ‘\(\exists xMx\)’. So thisformalization will fail to have its intended effect unless thismeta-language sentence is of the sort that incurs ontologicalcommitment. But that is precisely what the challengers dispute. Thiscontroversy cannot be pursued further here. For now, we simply observethat the challengers need to provide an account of why theirnon-standard notion of ontological commitment is better andtheoretically more interesting than the standard Quinean notion.

2.5 From Existence to Mathematical Platonism?

Suppose we acceptExistence, perhaps based on theFregean argument. As we have seen, this is not yet to acceptmathematical platonism, which is the result of adding toExistence the two further claimsAbstractness andIndependence. Arethese two further claims defensible?

By the standards of philosophy,Abstractness hasremained relatively uncontroversial. Among the few philosophers tohave challenged it are Maddy (1990) (concerning impure sets) andBigelow (1988) (concerning sets and various kinds of numbers). Thisrelative lack of controversy means that few explicit defenses ofAbstractness have been developed. But it is not hardto see how such a defense might go. Here is one idea. It is aplausibleprima facie constraint on any philosophicalinterpretation of mathematical practice that it should avoid ascribingto mathematics any features that would render actual mathematicalpractice misguided or inadequate. This constraint makes it hard todeny that the objects of pure mathematics are abstract. For if theseobjects had spatiotemporal locations, then actual mathematicalpractice would be misguided and inadequate, since pure mathematiciansought then to take an interest in the locations of their objects, justas zoologists take an interest in the locations of animals. The factthat pure mathematicians take no interest in this question suggeststhat their objects are abstract.

Independence says that mathematical objects, if thereare any, are independent of intelligent agents and their language,thought, and practices. We will discuss what this thesis might amountto, and how it might be defended, in Section4.

3. Objections to Mathematical Platonism

A variety of objections to mathematical platonism have been developed.Here are the most important ones.

3.1 Epistemological access

The most influential objection is probably the one inspired byBenacerraf (1973). What follows is an improved version ofBenacerraf’s objection due to Field (1989).^[12] This version relies on the following three premises.

If these three premises are correct, it will follow that mathematicalplatonism undercuts our justification for believing inmathematics.

But are the premises correct? The first two premises are relativelyuncontroversial. Most platonists are already committed to Premise 1.And Premise 2 seems fairly secure. If the reliability of some beliefformation procedure could not even in principle be explained, then theprocedure would seem to work purely by chance, thus undercutting anyjustification we have for the beliefs produced in this way.

Premise 3 is far more controversial. Field defends this premise byobserving that “the truth-values of our mathematical assertionsdepend on facts involving platonic entities that reside in a realmoutside of space-time” (Field 1989, p. 68) and thus are causallyisolated from us even in principle. However, this defense assumes thatany adequate explanation of the reliability in question must involvesome causal correlation. This has been challenged by a variety ofphilosophers who have proposed more minimal explanations of thereliability claim. (See Burgess & Rosen 1997, pp. 41–49 andLewis 1991, pp. 111–112; cf. also Clarke-Doane 2016. See Linnebo2006 for a critique.)^[13]

3.2 A metaphysical objection

Another famous article by Benacerraf develops a metaphysical objectionto mathematical platonism (Benacerraf 1965, cf. also Kitcher 1978).Although Benacerraf focuses on arithmetic, the objection naturallygeneralizes to most pure mathematical objects.

Benacerraf opens by defending what is now known as a structuralistview of the natural numbers, according to which the natural numbershave no properties other than those they have in virtue of beingpositions in an \(\omega\)-sequence. For instance, there is nothingmore to being the number 3 than having certain intrastructurallydefined relational properties, such as succeeding 2, being half of 6,and being prime. No matter how hard we study arithmetic and settheory, we will never know whether 3 is identical with the fourth vonNeumann ordinal, or with the corresponding Zermelo ordinal, orperhaps, as Frege suggested, with the class of all three-memberedclasses (in some system that allows such classes to exist).

Benacerraf now draws the following conclusion:

Therefore, numbers are not objects at all, because in giving theproperties …of numbers you merely characterize an abstractstructure—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. (Benacerraf 1965, p. 291)

In other words, Benacerraf claims that there can be no objects whichhave nothing but structural properties. All objects must have somenon-structural properties as well. (See Benacerraf 1996 for some laterreflections on this argument.)

Both of the steps of Benacerraf’s argument are controversial.The first step—that natural numbers have only structuralproperties—has been defended by a variety of mathematicalstructuralists (Parsons 1990, Resnik 1997, and Shapiro 1997, as wellas Schiemer & Wigglesworth 2019 for the notion of a structuralproperty). But this step is denied by logicists and neo-logicists, whoclaim that the natural numbers are intrinsically tied to thecardinalities of the collections that they number (Wright 2000). Andthe second step—that there can be no objects with onlystructural properties—is explicitly rejected by all of thestructuralists who defend the first step. (For some voices sympatheticto the second step, see Hellman 2001 and MacBride 2005. See alsoLinnebo 2008 for discussion.)

3.3 Other metaphysical objections

In addition to Benacerraf’s, a variety of metaphysicalobjections to mathematical platonism have been developed. One of themore famous examples is an argument of Nelson Goodman’s againstset theory. Goodman (1956) defends thePrinciple ofNominalism, which states that whenever two entities have the samebasic constituents, they are identical. This principle can be regardedas a strengthening of the familiar set theoretic axiom ofextensionality. The axiom of extensionality states that if two sets\(x\) and \(y\) have the same elements—that is, if \(\forall u(u\in x \leftrightarrow u \in y)\)—then they are identical. ThePrinciple of Nominalism is obtained by replacing the membershiprelation with its transitive closure.^[14] The principle thus states that if \(x\) and \(y\) are borne \(\in^*\)by the same individuals—that is, if \(\forall u(u \in^*\) \(x\leftrightarrow u \in^*\) \(y)\)—then \(x\) and \(y\) areidentical. By endorsing this principle, Goodman disallows theformation of sets and classes, allowing only the formation ofmereological sums and the application to the standard mereologicaloperations (as described by his “calculus ofindividuals”).

However, Goodman’s defense of the Principle of Nominalism is nowwidely held to be unconvincing, as witnessed by the widespreadacceptance by philosophers and mathematicians of set theory as alegitimate and valuable branch of mathematics.

4. Between object realism and mathematical platonism

Object realism says there exist abstract mathematical objects, whereasplatonism addsIndependence, which says thatmathematical objects are independent of intelligent agents and theirlanguage, thought, and practices. This final section surveys somelightweight forms of object realism that stop short of full-fledgedplatonism. Such intermediate views are attracting an increasing amountof interest.

4.1 How To Understand Independence

A natural gloss onIndependence is the followingcounterfactual conditional:

Counterfactual Independence.

Had there not been any intelligent agents, or had their language,thought, or practices been suitably different, there would still havebeen mathematical objects.

This counterfactual conditional is accepted by most analyticphilosophers. To see why, consider the role that mathematics plays inour reasoning. We often reason about scenarios that aren’tactual. Were we to build a bridge across this canyon, say, how strongwould it have to be to withstand the powerful gusts of wind? Sadly,the previous bridge collapsed. Would it have done so had the steelgirders been twice as thick? This form of reasoning aboutcounterfactual scenarios is indispensable both to our everydaydeliberations and to science. The permissibility of such reasoning hasan important consequence. Since the truths of pure mathematics canfreely be appealed to throughout our counterfactual reasoning, itfollows that these truths are counterfactually independent of ushumans, and all other intelligent life for that matter. That is, hadbeen there been no intelligent life, these truths would still haveremained the same.

Pure mathematics is in this respect very different from ordinaryempirical truths. Had intelligent life never existed, this articlewould not have been written. More interestingly, pure mathematics alsocontrasts with various social conventions and constructions, withwhich it is sometimes compared (Hersh 1997, Feferman 2009, Cole 2013).Had intelligent life never existed, there would have been no laws,contracts, or marriages—yet the mathematical truths would haveremained the same.

Thus, ifIndependence is understood merely ascounterfactual independence, then anyone who accepts object realismshould also accept platonism.

It is doubtful, however, that this understanding ofIndependence fully captures the intended content ofthe thesis. ForIndependence is meant to substantiatean analogy between mathematical objects and ordinary physical objects.Just as electrons and planets exist independently of us, so do numbersand sets. And just as statements about electrons and planets are madetrue or false by the objects with which they are concerned and theseobjects’ perfectly objective properties, so are statements aboutnumbers and sets. (See the quotes from Dummett 1978b and Maddy 1990 inthe supplement.) In short, we have the following thesis:

Robust Independence.
Mathematical objects are metaphysically on a par with ordinaryphysical objects.

Let us now consider some views that reject the stronger thesis ofRobust Independence. These views are thus lightweight forms of objectrealism, which stop short of full-blown platonism.

4.2 Plenitudinous platonism

One lightweight form of object realism is the “full-bloodedplatonism” of Balaguer 1998. This view is characterized by aplenitude principle to the effect that any mathematicalobjects that could exist actually do exist. For instance, since theContinuum Hypothesis is independent of the standard axiomatization ofset theory, there is a universe of sets in which the hypothesis istrue and another in which it is false. And neither universe ismetaphysically privileged (Hamkins 2012). By contrast, traditionalplatonism asserts that there is a unique universe of sets in which theContinuum Hypothesis is either determinately true or determinately false.^[15]

One alleged benefit of this plenitudinous view is in the epistemologyof mathematics. If every consistent mathematical theory is true ofsome universe of mathematical objects, then mathematical knowledgewill, in some sense, be easy to obtain: provided that our mathematicaltheories are consistent, they are guaranteed to be true of someuniverse of mathematical objects.

However, “full-blooded platonism” has received muchcriticism. Colyvan and Zalta 1999 criticize it for undermining thepossibility of reference to mathematical objects, and Restall 2003,for lacking a precise and coherent formulation of the plenitudeprinciple on which the view is based. Martin (2001) proposes thatdifferent universes of sets be amalgamated to yield a single maximaluniverse, which will be privileged by fitting our conception of setbetter than any other universe of sets.

A different version of plenitudinous platonism is developed in Linsky& Zalta 1995 and a series of further articles. (See, for instance,Linsky & Zalta 2006 and other articles cited therein.) Traditionalplatonism goes wrong by “conceiv[ing] of abstract objects on themodel of physical objects” (Linsky & Zalta 1995, p. 533),including in particular the idea that such objects are“sparse” rather than plenitudinous. Linsky & Zaltadevelop an alternative approach on the basis of the secondauthor’s “object theory”. The main feature of objecttheory is a very general comprehension principle which asserts theexistence of a plenitude of abstract objects: for any collection ofproperties, there is an abstract object which “encodes”precisely these properties. In object theory, moreover, two abstractobjects are identical just in case they encode precisely the sameproperties. Object theory’s comprehension principle and identitycriterion are said to “provide the link between our cognitivefaculty of understanding and abstract objects” (ibid.,p. 547). (See Ebert & Rossberg 2007 for critical discussion.)

4.3 Lightweight semantic values

Assume that object realism is true. For convenience, assume alsoClassical Semantics. These assumptions ensure thatthe singular terms and quantifiers of mathematical language refer toand range over abstract objects. Given these assumptions, should onealso be a mathematical platonist? In other words, do the objects thatmathematical sentences refer to and quantify over satisfy eitherversion of the independence thesis?

It will be useful to restate our assumptions in more neutral terms. Wecan do this by invoking the notion of asemantic value, whichplays an important role in semantics and the philosophy of language.In these fields it is widely assumed that each expression makes somedefinite contribution to the truth-value of sentences in which theexpression occurs. This contribution is known as thesemanticvalue of the expression. It is widely assumed that (at least inextensional contexts) the semantic value of a singular term is justits referent.

Our assumptions can now be stated neutrally as the claim thatmathematical singular terms have abstract semantic values and that itsquantifiers range over the kinds of item that serve as semanticvalues. Let’s focus on the claim about singular terms. What isthe philosophical significance of this claim? In particular, does itsupport some version ofIndependence? The answer willdepend on what is required for a mathematical singular term to have asemantic value.

Some philosophers argue that not very much is required (Frege 1953,Dummett 1981, Dummett 1991a, Wright 1983, Hale & Wright 2000, Rayo2013, and Linnebo 2012 and 2018). It suffices for the term \(t\) tomake some definite contribution to the truth-values of sentences inwhich it occurs. The whole purpose of the notion of a semantic valuewas to represent such contributions. It therefore suffices for asingular term to possess a semantic value that it makes some suchsuitable contribution.

This may even open the way for a form of non-eliminative reductionismabout mathematical objects (Dummett 1991a, Linnebo 2018). Although itis perfectly true that the mathematical singular term \(t\) has anabstract object as its semantic value, this truth may obtain in virtueof more basic facts which do not mention or involve the relevantabstract object. For example, a singular term may refer to a directionin virtue of being associated with an appropriately oriented line andsubject to the criterion of identity stating that two lines specifyone and the same direction just in case they are parallel. Thus,although it is perfectly true that the term refers to an abstractobject, this truth obtains in virtue of some more basic facts that donot mention or involve that abstract object. Linnebo (2018, ch. 11)argues that this approach to mathematical objects neverthelessvalidates Counterfactual Independence.

There is no reason, however, why an approach of this sort should becommitted to Robust Independence. On the contrary, such approachesentail some important disanalogies between mathematical and physicalobjects. For a direction to exist, for example, it suffices that thereexists an appropriate oriented line that specifies that the direction.Since this line can be located anywhere, the existence of thedirection does not impose any constraints on any particular region ofspacetime. By contrast, the existence of a physical object imposessubstantial constraints on the particular region of spacetime wherethe object is located.

In short, if some lightweight account of semantic values isdefensible, we can accept object realism andCounterfactualIndependence without committing ourselves to a more robustform of platonism.

4.4 Inspiration from Aristotle

We conclude by describing two further examples of lightweight forms ofobject realism that reject the platonistic analogy betweenmathematical objects and ordinary physical objects. Both examples areinspired by Aristotle.

First, perhaps mathematical objects exist only in a potential manner,which contrasts with the actual mode of existence of ordinary physicalobjects. This idea is at the heart of the ancient notion of potentialinfinity (Lear 1980, Linnebo & Shapiro 2019). According toAristotle, the natural numbers arepotentially infinite inthe sense that, however large a number we have produced (byinstantiating it in the physical world), it is possible to produce aneven larger number. But Aristotle denies that the natural numbers areactually infinite: this would require the physical world tobe infinite, which he argues is impossible.

Following Cantor, most mathematicians and philosophers now defend theactual infinity of the natural numbers. This is made possible in partby denying the Aristotelian requirement that every number needs to beinstantiated in the physical world. When this is denied, the actualinfinity of the natural numbers no longer entails the actual infinityof the physical world.

However, a form of potentialism about the hierarchy of sets continuesto enjoy considerable support, especially in connection with theiterative conception of sets (Parsons 1977, Jané 2010, Linnebo2013, Studd 2013). No matter how many sets have been formed, it ispossible to form even more. If true, this would mean that sets have apotential form of existence which distinguishes them sharply fromordinary physical objects.^[16]

Second, perhaps mathematical objects are ontologically dependent orderivative in a way that distinguishes them from independentlyexisting physical objects. According to what Rosen (2011) calls“qualified realism”, mathematical facts are grounded inother facts that do not involve mathematical objects. For example, thenatural numbers exist, but their existence and their properties andrelations are grounded in facts more fundamental than the arithmeticalones, e.g., in facts about provability or about structuralpossibilities. The view can also be given a more Aristotelian spin bytaking simple arithmetical truths to be grounded in facts aboutappropriately numerous pluralities that instantiate the relevantnumbers (Schwartzkopff 2011, Donaldson 2017). For example, \(2+2=4\)is grounded in any fact about a pair and a disjoint pair making aquadruple. There are other versions of the view as well. For example,Kit Fine (1995) and others argue that a set is ontologically dependenton its elements. (This view is also closely related to theset-theoretic potentialism mentioned above.)

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Acknowledgments

Thanks to Philip Ebert, Leon Horsten, James Ladyman, Hannes Leitgeb,David Liggins, Alexander Paseau, and Philip Welch for comments anddiscussion. Thanks also to an audience at the ECAP6 in Krakow, whereparts of this material were presented. This article was written duringa period of leave funded by an AHRC Research Leave Grant (numberAH/E003753/1). I gratefully acknowledge their support.