Mathematical pluralism

2.1 OT and Its analysis of classical mathematics

where $$\phi$$ is any condition in which $$x$$ doesn't occur free.

This asserts that there is an abstract object of type $$t$$ that encodes just the properties $$F$$ such that $$\phi$$.8 As we'll see below, mathematical individuals will be identified as abstracta of type $$i$$ and mathematical properties and relations will be identified as abstracta of type $$\langle i\rangle$$, $$\langle i,i\rangle$$, $$\langle i,i,i\rangle$$, etc. Note that principle (1) is an unrestricted comprehension principle and, as such, is a plenitude principle – no matter what properties are used to define an abstract object of some type $$t$$, the principle guarantees that there is an object of type $$t$$ that encodes just those properties and no others.

Moreover, identity is defined at each type, so that $$x\! =\! y$$ just in case either $$x$$ and $$y$$ are both ordinary objects of type $$t$$ and necessarily exemplify the same properties, or $$x$$ and $$y$$ are both abstract objects of type $$t$$ and necessarily encode the same properties. Given the 2nd disjunct of this definiens for $$x\! =\! y$$, each instance of (1) yields a unique abstract object that encodes just the properties such that $$\phi$$ – there couldn't be two distinct abstract objects that encode exactly the properties such that $$\phi$$, since distinct abstract objects have to differ by one of their encoded properties. Thus, descriptions of the form $$\iota x(A!x\hspace*{0.222222em}\mathrm{\&}\forall F(\textit{xF}\equiv \phi ))$$ are canonical – the description is well-defined (has a denotation) for each $$\phi$$ (with no free occurrences of $$x$$).

To analyze mathematics, OT distinguishes natural mathematics and theoretical mathematics. Natural mathematical objects are referenced in everyday language, such as when we say “the number of planets is eight”, “the class of insects is larger than the class of humans”, “lines $$a$$ and $$b$$ have the same direction” or “figures $$a$$ and $$b$$ have the same shape”. The natural mathematical objects referenced in these sentences are analyzed directly in OT without appealing to any mathematical theories.9

By contrast, theoretical mathematical objects and relations assume distinctive mathematical principles. These are often, but not always, expressed in the form of axioms that govern distinctive mathematical primitives. OT exhibits the first form of mathematical pluralism by using the canonical descriptions just discussed to analyze the objects and relations described by arbitrary mathematical theories. The analysis proceeds by assigning, for each mathematical theory $$T$$, (i) a unique denotation to the distinguished non-logical terms (individual terms and relation terms) of $$T$$ and (ii) truth conditions to the theorems of $$T$$.10

We next consider any classical mathematical theory $$T$$ and formalize it in a non-modal, higher-order logic without function terms (or definite descriptions) but with (closed) relational $$\lambda$$-expressions. The $$\lambda$$-expressions allow one to represent complex properties; for example, in 2nd-order Peano Arithmetic (henceforth ‘PA'), we use $$[\lambda x\; Px\:\&\: x\,{<}\,4]3$$ to represent the claim that 3 exemplifies the property being prime and less than 4. Then (a) for each non-logical term $$\tau$$ (constant or predicate) of $$T$$, we add $$\tau _{T}$$ to OT, and (b) whenever $$\phi$$ is any closed theorem of $$T$$, we add to OT the analytic truth $$T\models \phi ^*$$, where $$\phi ^*$$ is just like $$\phi$$ except that every non-logical term $$\tau$$ in $$\phi$$ has been replaced by $$\tau _{T}$$. For example, “0 is a number” is asserted in PA and so becomes imported into OT as the claim $$\mathrm{PA}\models N_\mathrm{PA}0_\mathrm{PA}$$. This formal claim was defined in the previous paragraph and can be read as the analytic truth “In PA, $$0_\mathrm{PA}$$ exemplifies being a PA-number”. We then begin our analysis by identifying a mathematical theory $$T$$ as an abstract object that encodes all of the truths of $$T$$.11 In general, for theories presented axiomatically, facts of the form $$T\vdash \phi$$ become imported as facts of the form $$T\models \phi ^*$$. But if, for example, one were to identify a theory (i.e., a body of truths) non-axiomatically, then we can introduce a proper name, say ‘$$\mathfrak {T}$$’, for that body of truths and extend object theory with analytic truths of the form $$\mathfrak {T}\models \phi ^*$$ for each such truth $$\phi$$ in $$\mathfrak {T}$$.

And principles analogous to (5) hold when $$\Pi$$ is an $$n$$-place relation term of $$T$$ for $$n\! \ne \! 2$$. For example, ‘being a number’ (‘$$N$$') is a 1-place relation term of PA and would be subject to an identification similar to (6), though expressed using the identity principle for 1-place mathematical relations. This analysis makes it clear that OT's pluralism extends to both the individual and relation terms of a theory $$T$$; few mathematical pluralists identify mathematical relations as well as individuals.

ZF encodes (the proposition): nothing that exemplifies the property of being a ZF-set bears the ZF-membership relation to the ZF-emptyset.

This states the truth conditions of the target sentence in terms of objects and relations that have been antecedently identified as abstract entities within the background ontology of OT. This analysis applies to any theory $$T$$ and its theorems.

But now remove the ‘In $$\mathrm{ZF}$$’ operator from (7) to obtain the bare (‘unprefixed') mathematical sentence “No set is a member of the null set”. OT treats this sentence, stated in some context, as ambiguous; it has both true and false readings. If we take the context to be ZF, then the false reading is the pure exemplification formula $$\lnot \exists x(Sx \:\&\: x\in \emptyset )$$, in which the index on ‘0’, ‘$$S$$’, and ‘$$\in$$’ to ZF has been suppressed. OT stipulates that such unprefixed exemplification readings of theoretical mathematics are not true – this is a key to OT's form of pluralism.12

 $$\in$$ encodes the property: being a relation $$R$$ such that no set bears $$R$$ to $$\emptyset$$.

These readings are provable in OT. For example, (9) follows from (4) and the result of importing the proof-theoretic fact $$\mathrm{ZF}\vdash [\lambda z \: \lnot \exists x(Sx \:\&\:x\in z)]\emptyset$$. (11) follows from (6) and the result of importing the proof-theoretic fact that $$\mathrm{ZF}\vdash [\lambda R \: \lnot \exists x(Sx \:\&\:xR\emptyset )]{\in }$$. So the conjunction of (9) – (11) is derivable, and if we use the conjunction to define a single, tertiary encoding claim, then we have fully represented the true reading of the unprefixed claim “No set is a member of the null set” when considered relative to ZF.14

2.2 The multiverse and non-classical mathematics

Thus, each distinct set theory yields a distinct universe of sets and a distinct membership relation. As long as set theories $$T$$ and $$T^{\prime }$$ have different theorems (and aren't mere alphabetic variants), the notion of ‘set’ each implicitly defines is different. There isn't one Urconcept of membership systematized by the one true set theory. Rather, each set theory defines a different conception of ‘set’ and ‘membership’.15

This captures the multiverse view in Hamkins (2012) (416) quoted earlier. Hamkins contrasts his view with the ‘universe’ view, on which there is a single conception of set and a single set-theoretic universe in which every set-theoretic assertion has a definite truth-value (416). He then argues that, on his multiverse view, each set-theoretic universe “exists independently in the same Platonic sense that proponents of the universe view regard their universe to exist” (416–17). But our analysis also makes it clear that the distinct universes embody distinct conceptions of ‘membership’.

There are, of course, points of difference between the present view and Hamkins’ multiverse view. Some are minor differences, while others are more significant. Hamkins regards the multiverse view as a ‘higher-order realism’ and a platonism about universes (417), though clearly the OT analysis extends this to realism and platonism about abstract objects generally, at least in the interpretation of the formalism we've assumed thus far for the purposes of exposition. A more significant difference concerns Hamkins’ view that “the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated, and … identify a set concept with the model of set theory to which it gives rise” (417). OT does identify set concepts by description but not with models of set theory. Model theory already assumes set theory and so the statements of model theory constitutes part of the data OT attempts to explain. We'll return to this issue in Section 2.3, where we investigate whether one can, as Hamkins suggests, identify a set concept with the model of set theory to which it gives rise.16

It is now easy to extend OT's analysis to constructive, intuitionistic, finitistic, etc., mathematical theories. Some of these theories (e.g., finitist theories) are expressed in classical logic but with axioms that are weaker than classical mathematical theories, while others (e.g., intuitionistic, constructive theories) use non-classical logic. In the former case, we use the analysis described above. In the latter case, we consider the deductive system as a whole, i.e., the system that results by combining the non-logical axioms and the logic. Some non-classical theories use the same non-logical axioms as classical theories but are just formulated within a non-classical logic. So, for example, Heyting Arithmetic (HA) uses the same language and non-logical axioms as PA but asserts the latter in the context of intuitionistic predicate logic (IQC). So, although we could regard the proof-theoretic claim $$\mathrm{HA}\vdash \phi$$ as having the form $$T \vdash _L \phi$$, where $$T$$ = PA and $$L$$ = IQC, we can equally well regard HA as a single deductive system comprising the logical axioms and rules of IQC and the non-logical axioms of PA. So the claim $$\mathrm{HA}\vdash \phi$$ becomes a claim of the form $$T_L \vdash \phi$$. Then we can use the methods outlined above to analyze the terms and truth conditions of HA. And if the consistent theory $$T$$ in question asserts non-classical axioms within a non-classical logic $$L$$, we again consider the theory to be the body of theorems as a whole system $$T_L$$ and use the same method to analyze sentences $$\phi$$ such that $$T_L\vdash \phi$$.

2.3 What's missing from other accounts of pluralism

What's distinctive about OT as a form of mathematical pluralism is its comprehension principle labeled (1) above. Hilbert's early view is an informal conditional (roughly, “if the theory is consistent, its objects and relations exist”). (1) explains why Hilbert's conditional is true and also tells us about the nature of mathematical objects and relations that exist. Given Carnap's interest in semantics, one might expect his work (1950 [1956]) to contain an explicit statement of the principle that guarantees the internal existence of the appropriate objects for each logical framework. (1) is such a principle; without it, we lack a semantic interpretation of the terms for arbitrary frameworks and can't therefore say why the answer to the internal question, ‘Do Xs exist?’, for arbitrary frameworks, is always ‘yes’. Resnik's postulational view isn't unrelated to Carnap's view, since a mathematical language is needed to posit the objects in question. But Resnik admits that his view “raises many questions concerning how positing can generate knowledge about preexisting entities – especially how it can do this when the entities are mathematical ones” (1989, 8). Principle (1) connects postulation with existence and provides denotations for mathematical terms; it addresses the open problem of ‘aboutness’ stated at the end of Resnik (1989) paper (26).17

Field and Balaguer both agree that mathematical platonism needs an explicit plenitude principle, and Balaguer (1998a), (7) attempts to formulate one. His ‘full-blooded platonism’ (FBP) is the thesis that every mathematical object that could possibly exist does exist. So FBP is clearly pluralistic. But the FBP plenitude principle faces the ‘non-uniqueness problem’, namely, it doesn't provide unique denotations to the individual constants and relation terms of a mathematical theory. Since FBP invokes possible mathematical objects but not objects that are ‘partial’ (e.g., in the sense of encoding only the properties attributed them in a theory), it is subject to questions such as: what does the symbol ‘$$\emptyset$$’ of ZF denote? Does it denote (a) an empty set such that AC is true or such that AC is false, or (b) an empty set such that CH is true or such that CH is false, … ?

Here, the ‘U-platonists’ are those who claim that mathematical theories describe unique collections of abstract mathematical objects and the ‘FBP-NUP-ists’ are full-blooded platonists who adopt non-uniqueness platonism. But the suggested truth conditions are not virtually identical to the compositional ones a U-platonist would give for ‘3 is prime’. The contrast with OT is clear – assuming the background theory of numbers PA, OT analyzes the denotation of ‘3’ as the abstract individual $$3_\textrm {PA}$$, analyzes the denotation of ‘is prime’ (‘$$P$$') as the abstract property $${P}_{\mathrm{PA}}$$, and resolves the ambiguous predication in terms of two truth conditions, one on which $$3_\textrm {PA}$$ exemplifies $${P}_{\mathrm{PA}}$$ (false) and one on which $$3_\textrm {PA}$$ encodes $$P_\textrm {PA}$$ (true). Moreover, the OT analysis doesn't invoke a quantifier “there is an object such that” that doesn't appear in the target sentence ‘3 is prime’, nor property expressions like ‘being 3’ or ‘desiderata for being 3’. And OT treats ‘is prime’ in the same way as ‘3’ – as denoting something abstract. Balaguer has to abandon the idea of de re truth conditions and de re knowledge of mathematical claims. Jonas (2023) (§4.2.2) notes that such a result leaves it unclear as to “which one of the countless copies of the numbers 13 and 17 are involved in scientific explanation”.

To see how OT supplies components missing from both structuralist and inferentialist accounts of mathematics, we begin with structuralism, i.e., the view that mathematics is about structures (Hellman, 1989, vii; Parsons 1990, 303; Shapiro, 1997, 5). If a structuralist philosophy of mathematics is to be free of ‘ontological danglers’, then it must supply a mathematics-free theory of both structures and the elements of structures. We cannot rest with set theory or category theory as our background theory of structures, as that simply turns mathematical pluralism into mathematical foundationalism and leaves us with the question, what is our philosophical account of the foundational theory?19 So, what are structures? Nodelman and Zalta (2014) (49–53) answer: (a) intuitively, structures are defined by a partial body of propositions that assert which mathematical objects stand in which mathematical relations, and (b) in OT, the structure $$T$$ can be identified as $$T$$ itself, since $$T$$ encodes the partial group of propositions that are true in $$T$$. This analysis of structures is mathematics-free.

Moreover, OT has something to say about what the ‘indeterminate elements’ of structures are supposed to be (Dedekind, 1888 [68]; Benacerraf, 1965, 70; Shapiro, 1997, 5–6).20 The abstract objects of OT that serve as mathematical individuals and relations encode only mathematical properties. Since every mathematical entity is thereby identified entirely by its encoded properties, its ‘special character’, as given by its exemplified properties, is ‘entirely ignored’. The classical structuralist philosophies of mathematics lack an alternative theory of such elements (or places) in a structure.

Clearly an inferentialist can use these rules to explain the truth of any arithmetical sentence that follows from them. And the rules do indicate what role the non-logical expressions have in the transition from premises to conclusion. However, if Warren's metasemantic claim implies that in a semantics for the language of number theory, each of these distinct non-logical expressions could be assigned a distinct, theoretically-describable inferential role, then the Peano rules don't yet accomplish this. The rules don't provide distinct, meaning-constituting rules for each distinct non-logical symbol; for example, the rules don't specify, in theoretical terms, what the meaning is of the constant symbol ‘0’ or of the predicate symbol ‘$$N$$’. Of course, one might be able to use set theory or other forms of mathematics to give a theoretical description of the total inferential pattern of usage for the symbols ‘0’, ‘N’, etc., but OT gives a distinct, mathematics-free, description of the inferential role of each non-logical expression.

In OT, the inferential role of an individual symbol $$\kappa$$ of $$T$$ is precisely captured as $$\kappa _T$$, as defined by (3), and the inferential role of a relation symbol $$\Pi$$ of $$T$$ is captured as $$\Pi _T$$, as defined by (5). (3) and (5) reify distinct subpatterns existing within the entire body of theorems of $$T$$ and so objectify the inferential roles of $$\kappa$$ and $$\Pi$$ in $$T$$. $$\emptyset _\textrm {ZF}$$ in (4) objectifies the inferential role of $$\emptyset$$ in ZF, and $$\in _\textrm {ZF}$$ in (6) objectifies the inferential role of $$\in$$ in ZF. Without some theoretical description of the inferential role on a per symbol basis, one can't give compositional truth conditions for mathematical  theorems. Of course, inferentialism may simply abandon compositionality given its anti-realist approach to meaning, but OT preserves compositionality: the denotations of the terms in a theorem play a role in the truth conditions of the theorem. The truth conditions it offers, in (8) and (9) – (11) for example, yield a content for mathematical  theorems that ‘code up’ proof-theoretic facts. They specify truth conditions that make use of objectified inferential roles.

These theoretical identifications of the inferential roles of the non-logical symbols of mathematical theories provide a heretofore missing component of the ‘meaning as use’ doctrine as applied to mathematics. The classical works on inferentialism in the philosophy of mathematics (Wittgenstein, 1956; Sellars, 1953, Sellars, 1974; Dummett, 1991) do not offer a theoretical account of the meaning of such symbols. And the recent developments of proof-theoretic semantics are limited to the inferential role of logical constants.22

Finally, we consider the multiverse view in Hamkins (2012). Hamkins appears to rely on an existence principle that asserts: different conceptions of sets are instantiated in different set-theoretic universes (216). So how do we apply such a principle to produce truth conditions for the various axiom systems for set theory? A related concern about the view is Hamkins’ identification of a set concept with “the model of set theory to which it gives rise” (2012, 417). Assuming this can be made precise, it raises the question: doesn't any attempt to specify a model of set theory presuppose some conception of set? In any case, the appeal to model theory seems to presuppose more mathematics, the language of which is precisely what is in question. Interestingly, OT's analysis seems to be consistent with Hamkins’ claim (417) that “Often the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated, ….” We've seen that in OT, any distinctive body of set-theoretic truths or theorems describes a set concept and, hence, a universe of sets. That understanding is incorporated into the OT analysis of mathematics generally.23

3.1 Inconsistent theories in paraconsistent logic

Beall (1999) RFBP gives rise to the same problem posed for FBP above, namely, the inability to assign unique denotations to the non-logical individual and relation terms of mathematical theories. But the version of RFBP available in OT is immune. Let $$L$$ be some paraconsistent logic, and let $$T$$ be, say, naive set theory, or one of the theories in discussed in Mortensen (1995) or Mortensen (2009). Then we can apply OT as we did for consistent, non-classical mathematics (Section 2.2). We consider the deductive system $$T_L$$, i.e., $$T$$ added to the logic $$L$$, and then add $$T_L\models \phi ^*$$ to OT whenever $$T_L\vdash \phi$$. Then we identify the denotation of an individual term $$\kappa$$ in $$T_L$$ with the abstract object that encodes the properties $$F$$ such that $$T_L\models F\kappa$$, and identify the denotation of a relation term $$\Pi$$ in $$T_L$$ with the abstract property that encodes the properties of relations attributed to $$\Pi$$ in $$T_L$$. Given this analysis, the objects of $$T_L$$ are mathematically impossible but not trivial with respect to $$T_L$$ – they encode some incompatible properties that are expressible in $$T_L$$ but they don't encode every property expressible in $$T_L$$. And truth conditions for the claims of $$T_L$$ can be stated in exactly the way described above. Thus, OT overcomes the non-uniqueness problem for Beall 1999.

If we take the Quinean interpretation of the quantifiers of OT, then there is no dilemma; one can be Platonist and admit inconsistent objects.25

3.2 Inconsistent theories in classical logic

Though OT's analysis implies, for example, that all the individual terms of $$\mathfrak {G}$$ denote the same individual, it doesn't imply that they all have the same sense. The senses of expressions are also representable in OT – as abstract objects that encode properties. OT assumes that these senses vary from person to person and even from time to time (Zalta, 1988, Ch. 9–12). The reason Frege didn't see the contradiction is that his sense (representation) of ‘0’ and his sense (representation) of ‘1’ were distinct – his senses of the terms encoded different properties, at least prior to being informed about the paradox. Had he cognitively associated the same representations with ‘0’ and ‘1’ and concluded that 0 and 1 were characterized by the same properties, he would have judged that $$0=1$$ and that his system led to an absurdity.

This same analysis extends to the primitive predicates of $$\mathfrak {G}$$, such as the identity predicate. OT analyzes ‘$$=_\mathfrak {G}$$’ as the mathematically trivial relation that encodes all of the properties of identity expressible in (the OT representation of) $$\mathfrak {G}$$. For example, $$=_\mathfrak {G}$$ encodes being a binary relation F that relates 0 and 1, i.e., $$[\lambda F\: 0F1]$$ (suppressing indices). Thus, for any classically inconsistent $$T$$ (such $$\mathfrak {G}$$), OT predicts that, for each $$n$$, the $$n$$-ary relation terms of $$T$$ all have the same mathematically trivial inferential role – this is, in part, what makes such theories mathematically uninteresting.

4.1 Metaphilosophy of mathematical language

Linsky and Zalta (1995) explain in some detail how the main principles of platonism and naturalism are preserved in OT.26 We can extend this reconciliation a bit here, by first remembering that the quantifier ‘$$\exists \alpha$$’ in OT, on this interpretation, is Quinean and implies the existence of the entities over which $$\alpha$$ ranges. Then OT becomes committed to the existence of abstract individuals and abstract relations. While this preserves the main thesis of Platonism, note that we can apply OT to assert the existence of theoretical mathematical entities only once a mathematical theory $$T$$ is identified and analyzed. That is, the data are truths of the form “In mathematical theory $$T$$, …” that emerge from mathematical practice. Thus, the existence claims that OT outputs from this data in some sense depends on natural world patterns that are inherent in mathematical practice. Instances of (1) objectify those patterns. This is a form of naturalism (more on this in footnote 39 below).

Some of the main ideas underlying fictionalism can be sustained as well. By taking truths of the form “In mathematical theory $$T$$, …” as basic, OT agrees with Field's view (1989, 3) that “the sense in which ‘$$2+2=4$$’ is true is pretty much the same as the sense in which ‘Oliver Twist lived in London’ is true”. But we need not agree that “Oliver Twist lived in London” and ‘$$2+2=4$$’ are true only in the story- or theory-relative sense, for they can be given true encoding readings. Furthermore, Colyvan and Zalta (1999)(347–348) develop an interpretation of OT under which one can derive the claim that mathematical objects don't exist. Their suggestion is to adopt the Meinongian interpretation of the quantifier ‘$$\exists \alpha$$’ as ‘there are’ (not ‘there exist'), as in “there are fictional characters that inspire us even though they don't exist”. Then if one interprets the predicate ‘$$E!$$’ as ‘exists’, abstract objects become entities that couldn't possibly exist, since the definition is $$A!x \equiv _\mathit {df} \lnot \Diamond E!x$$. And since mathematical objects, as identified above, are abstract, it follows a fortiori that they don't exist.

Recall that in addition to offering true readings, OT offers a false reading for unprefixed mathematical claims such as ‘2 is prime’. It therefore validates the intuition that such claims of mathematics are false (Field, 1980; Leng, 2010). So under this interpretation, OT preserves the fictionalist claims (a) that “In PA, $$2+2=4$$” is true, (b) that “$$2+2=4$$” is false [at least on one reading], and (c) that none of 2, 4, $$\emptyset$$, $$\omega$$, $$\pi$$, etc., exist.27

Further on, he again suggests that we should read the existential quantifier as some.28 Since this position has been ably defended, apply it to OT's formalism and the result is Azzouni-Priest-Routley nominalism.

Indeed, OT helps us to make sense of ideas that, at present, are somewhat metaphorical, namely, that mathematical objects are ‘ultrathin’ (Azzouni, 2004, 127; Rayo, forthcoming) and are objects whose “existence does not make a substantial demand upon the world” (Linnebo, 2018, 4). Azzouni suggests that mathematicians just need to write down axioms and the resulting ‘posits’ have no epistemic ‘burdens’ (cf. Resnik, 1989). And Rayo (forthcoming) develops a conception of ‘ultrathin’ objects on which they arise in virtue of language-based networks. This notion of thinness is evident in OT, in several ways. The mere statement of a mathematical theory $$T$$ triggers OT to articulate a distinctive group of mathematical objects and relations for $$T$$. These objects and relations are thin along the encoding dimension, for they have only a partial (i.e., not complete) complement of encoded properties (namely, only the properties attributed to them in their respective theories). With OT in the background, no additional ‘demands upon the world’ are needed for the terms of $$T$$ to acquire content. Indeed, all one has to do to become acquainted with a mathematical entity such as $$0_\mathrm{PA}$$, $$\emptyset _\mathrm{ZF}$$, $$\in _\mathrm{ZF}$$, or $$\in _\mathrm{ZFC}$$, etc., is to understand its defining description, as given by theoretical identity claims  similar to (4), (6), (12), and (13). We don't need a special faculty, or an ‘information pathway’ for acquiring knowledge of abstract objects; we just need the faculty of the understanding (Linsky & Zalta, 1995, 547).

We've already seen, in Section 2.3, how OT supplies components missing from structuralism and inferentialism. Given this discussion, we can then interpret the OT formalism in a way that preserves the central insights of both philosophies of mathematics, starting with structuralism. The OT analysis is that mathematical theories are structures, where these are identified without any mathematical assumptions other than analytic truths about mathematical theories. Let me reiterate that by identifying mathematical individuals and relations as abstracta that encode only their mathematical properties and no others, OT neglects their ‘special character’ (i.e., neglects their exemplified properties). And, as previously noted, this analysis complements the standard (non-modal) structuralist views, which only discuss ‘places in structures’, but rarely talk about ‘relational places’, i.e., the places that ‘partial’ or ‘indeterminate’ relations occupy in a structure.

After an extended discussion (2006, 117–20), he concludes not only that abstractness is not a mathematical property but that it isn't therefore an essential property of natural numbers. From the point of view of OT, this conclusion is a consequence of the analysis in Section 2.1. The essential properties of numbers are just the mathematical properties they encode, not the properties (such as being abstract, not being a building, having no causal powers, etc.) they necessarily exemplify.29

Turning now to inferentialism, we again restrict our attention to axiomatic theories, since inferentialism presupposes some sort of deductive relationships among the truths of $$T$$. But with this restriction, we can be brief, since the discussion in Section 2.3 already provides the essentials. Given any axiomatic theory $$T$$, we can interpret the schematic and specific principles (3) – (6) as picking out the inferential roles of the non-logical, mathematical terms of $$T$$. These principles identify a specific role for each non-logical term of $$T$$.

While this interpretation preserves the basic insight of inferentialism, the more interesting fact is how it reconciles the referential and use-theoretic approaches to the meaning of mathematical language and renders them consistent (cf. Murzi & Steinberger, 2017). The point has already been made: the objectified inferential roles can serve as the denotations of mathematical terms in a compositional semantics. Thus, the formalism of OT suggests that the traditional opposition between inferentialism, on the one hand, and referential theories such as Platonism and structuralism, is partly a matter of focus – there is no inherent inconsistency.

To see how the basic insight of formalism is preserved, let us ignore many of the differences between Hilbertian formalism,30 term formalism,31 and game formalism.32 That's because in each case, the essential idea is that mathematics is about (formula and symbol) types and not tokens. That is, on any version of formalism, mathematics is not about any particular marks on the page or about any particular sound waves emanating from the mouths of mathematicians, but rather about the types that the marks or sound waves are tokens of. The ‘formal rules’ that the principles of $$T$$ represent apply to types, not to tokens.

To preserve this insight, we use OT to identify types as abstract objects that encode properties. A type encodes just the distinctive properties that the tokens of that type exemplify. For example, a pure symbol type encodes just the shape and/or sound properties needed to identify tokens of that type. In the case of a mathematical theory $$T$$, the formal objects denoted by the terms and predicates of $$T$$ are not pure symbol types, but symbol types as abstracted from the role they play in the formulas true in $$T$$. Under this interpretation, the individual terms of mathematics denote individual-symbol types that encode the abstract property-symbol types denoted by the predicates.33 For example, $$\emptyset _\mathrm{ZF}$$, as identified in (6), becomes a symbol-type that encodes just those property-symbol types $$F$$ whenever the formula type “In ZF, $$F\emptyset$$” constitutes part of the data. Since ZF is given axiomatically, this data comes from sentence types of the form “$$\textrm {ZF}\vdash F\emptyset$$”.

We've already discussed how principles (1), (3), and (5) answer the question of how the constants and predicates of each framework come to denote the right objects and relations, so that the Carnapian internal question “Do Xs exist?” is always true, or provably true, within the framework. This fact about OT suffices to show how it preserves the basic insight of Carnapianism.

No metaphilosophy of mathematics would be complete without some discussion of logicism. But my discussion here will be only a sketch, since this is a topic of ongoing research. Many philosophers now believe that logicism is a non-starter, since mathematics has strong existence claims and logic has very weak ones, making any reduction of mathematics to logic impossible. Indeed, logicism is a non-starter if one's conception of logic makes it impossible for strong existence claims to be logical truths and relative interpretability is the standard of reduction. But if one (a) develops a conception of logic that allows 2nd-order comprehension and (1) above to be logically true, and (b) uses an alternative, but equally precise, standard of reduction (on which each well-defined term is assigned a unique denotation and the theorems are assigned readings on which they are true), then not only can OT be viewed as part of logic but mathematics becomes reducible to logic plus analytic truths. The key to this conception of logic is the idea that 2nd-order comprehension and (1) are required for having logically complex thoughts (including mathematical thoughts) and for the validity of complex reasoning (including mathematical reasoning). Since this is the topic of another paper, we'll leave the matter here.34

4.2 Paraphrasing mathematical language

OT shares with deductivism the idea that the fundamental truths of a mathematical theory $$T$$ are statements under the scope of an operator: “In $$T$$, …” in the case of OT, and “If the conjunction of the axioms of $$T$$ hold, then …” in the case of deductivism.35 But the similarity ends there, especially when we consider the sophisticated variant of deductivism embodied by modal structuralism (MS). OT doesn't preserve the ideas underlying MS because the two theories are attempts to address different problems. OT takes mathematical language at face value, as containing constants and predicates that have a semantic content (at our world). It attempts to preserve the tradition in which (axiomatic) mathematical theories are formally represented in a classical, non-modal predicate calculus extended with (a) the non-logical constants and non-logical predicates, and (b) non-logical axioms that are categorically stated. OT, which is based on an ambiguity in predication, assigns these categorical predications two readings (a true encoding reading and a false exemplification reading), as outlined above.

This methodology doesn't attempt to analyze the axioms of $$T$$ as categorical predications or universal claims. Indeed, it is consistent with MS that none of the constants or predicates of $$T$$ have denotations, much less denote specifically mathematical objects or relations. Moreover, since the Barcan Formula (BF) is invalid in the S5 modal logic assumed in MS (Hellman, 1989, 17), one cannot validly infer $$\exists \vec{x}\;\exists \vec{F}\Diamond (\wedge T(\vec{x},\vec{F}))$$ from $$\Diamond \exists \vec{x}\;\exists \vec{F}(\wedge T(\vec{x},\vec{F}))$$. So mathematical theories are not about structures or indeed about anything (such theories are not committed to objects and relations standing in the right structural relationships), though it is possible that they are about something.36 In many ways, MS is a form of mathematical eliminativism rather than a form of mathematical pluralism, since the distinctive primitive notions employed by mathematicians are all eliminated in favor of variables and modally quantified conditionals.

But supposing OT and MS are comparable theories, it is still difficult to compare them. Here are some questions that can be raised. One cluster concerns the status of the possibility claims that MS must assert to complete the analysis of mathematical theories. MS has to add at least one special axiom of the form $$\Diamond \exists \vec{x}\;\exists \vec{F}(\wedge T(\vec{x},\vec{F}))$$, for each mathematical theory $$T$$ that it analyzes. So, the question is, can one actually state MS generally? Does MS include the universal claim: $$\forall T\Diamond \exists \vec{x}\;\exists \vec{F}(\wedge T(\vec{x},\vec{F}))$$? Is MS analyzing $$T$$ or $$T$$ + $$\Diamond \exists \vec{x}\;\exists \vec{F}(\wedge T(\vec{x},\vec{F}))$$? If the latter, then why don't we see explicit modal claims in mathematical practice? These questions may prove to be difficult to answer, especially if the possibility claims added to MS have to be customized so as to be the weakest claims that can do the job.37

By contrast, OT doesn't have to add modal claims for each new mathematical theory it analyzes – it just takes the theory-prefaced statements as the data and the rest falls out from OT comprehension (1), the identification principles (3) and (5), and the various readings of the unprefixed mathematical claims that this methodology makes possible.38