##### 3.2.1. Object platonism

We, of course, have one particularly prominent example of a naturalist who endorses object platonism, viz., Quine.^{35} Indispensability considerations lead Quine to conclude that we're ontologically committed to a range of abstract, mathematical objects: numbers, functions, sets, etc. This much is well known. Perhaps less well known is Quine's attitude about just how much of pure mathematics the indispensability argument yields. Quine (1998b) indicates that a great deal of pure mathematics has only recreational value:

Pure mathematics, in my view, is firmly embedded as an integral part of our system of the world. Thus my view of pure mathematics is oriented strictly to application in empirical science. Parsons has remarked, against this attitude, that pure mathematics extravagantly exceeds the needs of application. It does indeed, but I see these excesses as a simplistic matter of rounding out. . . . I recognize indenumerable infinities only because they are forced on me by the simplest systematizations of more welcome matters. Magnitudes in excess of such demands, e.g., or inaccessible numbers, I look upon only as mathematical recreation and without ontological rights.^{36} (p. 400)

The upshot is that mathematics that goes beyond what's needed for applications in the empirical sciences also goes beyond the reach of the indispensability argument. Consequently, for Quine much of pure of mathematics isn't even truth apt, since for an object platonist truth aptness depends on ontological standing and for Quine much of pure mathematics has no ontological standing. Moreover, there is reason to think that the upper limit of truth aptness on this view is actually quite small. It is widely held that the empirical sciences need no more mathematics than functional analysis, which requires at most entities found in *V*_{ω+ω}.^{37} As it happens the cardinality of *V*_{ω+ω} is , to which Quine denies ontological standing. So we have here a case that, on Quine's view, mathematical claims concerning entities of rank^{38} greater than fail to be truth apt. But truth aptness is a precondition for knowledge. So on Quine's view much of pure mathematics is excluded from our pool of mathematical knowledge.

This is revisionary, as it fails to respect (PI*). Just as with rival intuitionism, Quine must explain why the set theory that he would leave out of the pool of mathematical knowledge should be left out, and thatexplanation is going to proceed in terms of concerns alien to set theory. Those concerns, having to do with explaining and predicting experience, will be of type (ii) and (maybe) type (iii). But they will not be of type (i), so they will not be set theory-type concerns. Hence, Quine might be a naturalist and his naturalism might extend to a portion of pure mathematics, but he's not a naturalist about mathematics.

One way of thinking about what's going on here is by considering how the indispensability argument succeeds (let's suppose) in naturalizing some of mathematics and how it fails to naturalize all of mathematics. The indispensability argument relies on the Quine–Duhem thesis, according to which confirmation accrues only to relatively large bodies of theory and whatever is required by a body of theory that receives confirmation shares in that confirmation. Since certain parts of mathematics are required for doing science, and science is well confirmed, so are certain parts of mathematics. The indispensability argument succeeds in confirming only those parts of mathematics required by empirical science; for definiteness let's fix this at those parts of mathematics concerning entities of rank no greater than . The naturalistic credentials of this success appear to be underwritten by an empiricist conviction that perceptual experience is the ultimate source of confirmation. Naturalistic epistemology aims to account for our knowledge in terms of naturally explicable facts and faculties, facts and faculties studied by or otherwise compatible with the empirical sciences, broadly construed to include the social sciences. The final arbiter of the empirical sciences is perceptual experience. So naturalistic justification (confirmation) at bottom rests on perceptual experience. Naturalistic justification is in some sense a matter of there being a sufficiently strong connection to perceptual experience. The empirical sciences are directly confirmed because they directly confront, explain, and predict experience. The parts of mathematics confirmed by the indispensability argument are indirectly confirmed by their (direct) role in the empirical sciences. The indispensability argument fails to confirm those parts of mathematics concerning entities of rank greater than because those parts of mathematics *prima facie* fail to be sufficiently strongly connected to experience.

This suggests that the shortcoming of a Quinean approach as a strategy for naturalizing mathematics as a whole might be overcome by arguing for a sufficiently strong connection between those parts of mathematics concerning entities of rank greater than and experience. Colyvan (2001) gestures at such an argument:

As for the charge that the [Quinean] indispensability argument leaves too much mathematics unaccounted for (i.e. any mathematics that does not find its way into empirical science), this seems to misrepresent the amount of mathematics that has directly or indirectly found its way into empirical science. On a holistic view of science, even the most abstract reach of mathematics is applicable to empirical science so long as it has applications in some further branch of mathematics, which may in turn have applications in some further branch until eventually one of these find applications in empirical science. Indeed, once put this way it's hard to imagine what part of mathematics could possibly be unapplied. (p. 107)

Call a part of mathematics that concerns entities of rank up to but not exceeding *α mathematics of rank α*. Here, a part of mathematics P ‘concerning’ an entity *e* is understood to mean that *e* is in the domain of quantification on the standard interpretation of the language of P. For present purposes, I identify mathematical entities with their canonical set-theoretic surrogates. So, for example, the natural numbers are identified with finite von Neumann ordinals and arithmetic is of rank *ω*. Then the argument suggested in this passage runs as follows.

- (1′)
For any parts of mathematics

,

,

of rank

*α*,

*β*, and

*γ*, respectively: if

has indispensable applications in

and

has indispensable applications in

, then

has indispensable applications in

.

- (2′)
Any part of mathematics that has indispensable applications in mathematics of rank

through a chain of indispensable applications as licensed by (1′) has indispensable applications in empirical science.

- (3′)
It's likely that for every

*α* there is a part of mathematics of rank

*α* that has indispensable applications in mathematics of rank

through a chain of indispensable applications as licensed by (1′).

- (4′)
So, it's likely that for every *α* there is a part of mathematics of rank *α* that has indispensable applications in empirical science.

- (5′)
Any mathematics that has indispensable applications in empirical science shares in the confirmation of empirical science.

- (6′)
Therefore, it's likely that mathematics of every rank shares in the confirmation of empirical science.

Call the sense in which a part of mathematics of rank greater than is indispensable to empirical science in virtue of its participation in a chain of indispensable applications culminating in an indispensable application to a part of mathematics of rank *extended indispensability*, and call the above argument the *extended indispensability argument*. If (6′) is correct, then the extended indispensability argument arguably justifies us in believing that there are mathematical entities of rank *α*, for every ordinal *α*, thus clearing the way for truth aptness for mathematics in its entirety. At least this is the hope of an extended indispensability theorist hoping to naturalize mathematics.

There are problems with the extended indispensability argument. I focus on the most serious.^{39} The most reasonable way to understand one part of mathematics having ‘applications in’ another is in terms of one part of mathematics being used in proving results in another part of mathematics.^{40} But if we understand one part of mathematics having applications in another part of mathematics proof theoretically in this way, then work on predicativist mathematics by Solomon Feferman and others^{41} shows that (3′) is quite likely false.^{42}

The predicativist program pursued by Feferman *et al.* began in earnest with Weyl (1918/1984), where Weyl aimed to develop analysis taking as given only: the system of natural numbers (*ω*, 0, Sc), where ‘Sc’ denotes the the successor relation; arithmetical subsets of *ω* (i.e. subsets of *ω* definable from a formula all quantifiers of which range over natural numbers); and notions of inductive definition and proof.^{43} Feferman has produced a system W formalizing Weyl's system, the details of which need not detain us.^{44} For present purposes, two features of W are important.

(W1) W is proof-theoretically reducible to and a conservative extension of Dedekind–Peano Arithmetic (DPA), which is to say, for any sentence *σ* in the language of arithmetic: (i) DPA proves that any proof of *σ* in W can be effectively transformed into a proof of *σ* in DPA and (ii) *σ* is provable in W only if it's already provable in DPA.

(W2) W suffices for all mathematics, which is indispensable to our best science.

Since (W1) and (W2) yield that whatever mathematical results science needs can be proved in DPA, these features of W imply that mathematics of rank *ω* suffices to prove all the results which find indispensable application in science. In other words, the indispensability of mathematics beyond rank *ω* is merely apparent.^{45} This is, of course, considerably less ontologically committed than even the interpretation of the indispensability argument that accords ontological standing only to those entities of rank no greater than . Challenges to (W2) have been raised. Those challenges tend to involve ‘questions at the margin’ involving ‘the possible essential use in physical applications of such objects as nonmeasurable sets or nonseparable spaces, which are not accounted for in W’ (Feferman, 1992, p. 297).^{46} However, plausible responses to these challenges have been given.^{47} In short, the theoretical models in which the problematic sets and spaces arise, as well as the question of applying these models in practice, are highly controversial. Thus, their indispensability is at best an open question.

Some might think that the predicativist program won't bear the weight I've put on it, that, e.g., the responses just noted are inadequate, plausible though they may be. So let's set aside predicativism and the problems it raises for the extended indispensability argument. Colyvan's strategy faces other difficulties.

First, it follows from the Reflection Theorem^{48} that no matter how much set theory is needed to prove the results which are indispensable to empirical science in the extended sense, it will be only a fraction of the whole of set theory. An exact statement of the theorem is unnecessary. The key point is that reflection implies:

(R) For any finite list *φ*_{1}, *φ*_{2}, . . . , *φ*_{n} of axioms of ZFC, there is an ordinal *α* such that *V*_{α} models {*φ*_{1}, *φ*_{2}, . . . , *φ*_{n}}.

The amount of set theory which is indispensable to empirical science in the extended sense is limited by (R). To see this, let *S*= {*τ*_{1}, *τ*_{2}, . . . , *τ*_{k}} be the set of mathematical theorems which are directly indispensable (i.e. indispensable apart from the extended sense) to empirical science. *S* is finite. At any given time, there are only finitely many statements of empirical science^{49} and arguments for those statements use only finitely many mathematical results.^{50} Moreover, since each *τ*_{i} has a proof from the axioms of set theory, there is a set *AX*(*S*) of set-theoretic axioms from which every member of *S* is provable. But now, since proofs are finite and there are only finitely many *τ*_{i}, *AX*(*S*) is finite and, by (R), there is an ordinal *α* such that *V*_{α} models *AX*(*S*). Thus, only entities up to rank *α* are required for empirical science, which implies that (4′) is false.^{51}

Second, even if the arguments of the preceding two paragraphs fail and the extended indispensability argument is correct, naturalized mathematics does not automatically follow. Being justified in believing that there are mathematical entities of every rank isn't the same as being justified in believing all currently accepted mathematics. For instance, one might be justified in believing that all ordinals exist without being justified in believing that some sets are not ordinals. So the correctness of (6′) is compatible with revisionism, and fairly radical revisionism at that. This is so even given a stronger form of (6′) without the qualifier ‘likely’. At a minimum, then, it's dubious that extending the indispensability argument along the lines suggested by Colyvan will work to bring all of mathematics into the naturalistic fold. Moreover, it's not at all clear how else one might naturalize mathematics along Quinean lines.

Colyvan has recently been defending recreational mathematics, by which he means mathematics without (extended) indispensable application in our best science, and the naturalist might hope that the arguments used in this defense would be helpful here. Colyvan's basic idea is that ‘[m]athematical recreation is an important part of mathematical practice’ and that ‘like other forms of theoretical investigation, [it] should not be thought of as second class or *mere* recreation’ (Colyvan, 2007, p. 116). We needn't worry over the details, because there are at least two reasons independent of them that the naturalist's hope for help would be in vain. First, Colyvan's view in (2007) (call this the *recreational view*) is an extension of the view based on the extended indispensability argument we've been criticizing. So the critique the naturalist would like help with applies equally well to the recreational view. Second, though Colyvan confers ontological rights to much more mathematics on the basis of the extended indispensability argument than Quine does on the basis of the standard indispensability argument, the mathematics which escapes the extended indispensability argument, viz., recreational mathematics, is still without ontological rights. Colyvan ‘[accepts] that applied mathematics should be treated realistically and with unapplied [i.e. recreational] mathematics we have no reason to treat it this way’ (Colyvan, 2007, p. 116). Hence, recreational mathematics is still excluded from the pool of mathematical knowledge, and so the same problem that led to Colyvan's extension of the indispensability argument arises anew: excluding significant parts of pure mathematics in this way violates (PI*). So the recreational view provides no help to the naturalist.

There is another object platonist approach to naturalizing mathematics that deserves attention. On this approach, ordinary scientific standards of theory choice legitimize mathematics.^{52} The idea is that theoretical virtues such as simplicity, ontological parsimony, fruitfulness, explanatory power, etc.^{53} often deployed in choosing between empirically equivalent scientific theories^{54} ratify accepting contemporary mathematics understood platonistically. So, for example, where an indispensability theorist counsels accepting whatever mathematics finds indispensable application in our best science and rejecting the rest (often by appeal to the virtue of ontological parsimony), the *scientific standards approach* counsels accepting indispensably applicable mathematics plus whatever non-applicable mathematics enhances the simplicity, fruitfulness, explanatory power, etc. of our best science, even at the cost of significantly enlarging our ontology. In short, the scientific standards approach uses a more balanced application of theoretical virtues than approaches (like the indispensability approach) that appear to allow ontological parsimony to trump other theoretical virtues.^{55}

It's not hard to see why the scientific standards approach might be promising for naturalizing mathematics. If it works as advertised it appears to be more or less unconstrained in the mathematics it can deliver, unlike indispensability arguments (standard or extended). Thus it seems well suited to avoid revisionism, thereby respecting (BA2). And as it delivers mathematics on scientific grounds, it seems a good bet to respect (BA1) by countenancing only naturalistic metaphysics. Whether or not this promise is fulfilled is another question.

Notice that the central claim of the scientific standards approach, viz., that contemporary mathematics is ratified by ordinary standards of science, is ambiguous. If we understand science to include mathematics, then the claim is trivially correct and (BA2) is respected. Set aside potential worries raised by the triviality of the claim on this reading. There remains considerable tension with (BA1).

Contemporary mathematics includes theorems such as:

(L) There is a cardinal *λ* such that λ=ℵ_{λ}.^{56}

Witnesses to (L) are so large and so far removed from experience (and *pace* Colyvan's extended indispensability argument also from any mathematics used in organizing or explaining experience) that they have led at least one otherwise realist philosopher of mathematics to ‘suspect that, however it may have been at the beginning of the [set theory] story, by the time we have come thus far the wheels are spinning and we are no longer listening to a description of anything that is the case’ (Boolos, 1998, p. 132). On this reading of the scientific standards approach's central claim, there is no reason to be confident that (BA1) isn't violated. At best, the approach owes us an account of what makes theorems like (L) true which respects (BA1). But this is (largely) what the scientific standards approach was introduced to help with. So including mathematics as part of science in the central claim of the scientific standards approach doesn't get the naturalist anywhere.

If, on the other hand, mathematics is not included as part of science in the central claim of the scientific standards approach, the naturalist still doesn't obviously gain anything. According to Burgess, perhaps the most prominent proponent of the scientific standards approach, scientific standards legitimize mathematical entities on grounds of convenience as well as indispensability, ruling out only mathematics which is gratuitous from the standpoint of our best science (construed now to exclude mathematics).^{57} How much of mathematics does convenience for science get us? This is a highly non-trivial question, but it's reasonable to think that in order to avoid violating (BA2) it would need to get us at least a minimal non-artificial model of ZFC (either Gödel's *L* or *V*_{κ} for the least strongly inaccessible *κ*).^{58} I find it dubious that considerations of convenience for non-mathematical science get us so much. But for the sake of argument, let's grant that they do. Then we have:

(C) As a matter of scientific convenience, we are entitled to accept that there is at least a minimal non-artificial model of ZFC.

But (C) itself is problematic. If we're not going to come back around to recreational mathematics, convenience has to carry some real justificatory weight here. It must be the case that being entitled to accept that *p* as a matter of scientific convenience is systematically correlated with *p*'s being true. Otherwise, the scientific standards approach doesn't even provide us an epistemology of mathematics, let alone a naturalistic epistemology of mathematics. However, if convenience carries real justificatory weight in this way, then worries of the kind discussed in connection with witnesses of (L) above are back in play. And, as before, the approach owes us an account of what makes theorems like (L) true which respects (BA1). So again the scientific standards approach seems to have gotten the naturalist nowhere.

There's more to be said concerning the scientific standards approach, but I take it this is sufficient to raise serious questions about its usefulness to the naturalist. This being the case, in the interest of brevity we move on to other forms of alethic realism.^{59}

##### 3.2.3. Fregean logicism

The situation with Fregean logicism, which I understand to encompass Frege's logicism as well as the neo-Fregean logicism most prominently advocated by Hale and Wright,^{60} is more complicated. A naturalistic epistemology of mathematics, where mathematics is understood in accordance with Fregean logicism, would arguably run afoul of a widely accepted tenet of naturalism, viz., that nothing is knowable *a priori*. If we resist Quine's view that second-order logic is actually disguised set theory, then mathematical knowledge on a Fregean logicist understanding of mathematics depends on knowing logical or conceptual facts, and such facts are arguably knowable *a priori*. Of course, if we accept Quine's view regarding second-order logic, then we have trouble of a different sort: mathematical knowledge according to Fregean logicism would then depend on set-theoretic knowledge (plus knowledge concerning Hume's Principle, the statement that for all concepts *F* and *G*, the number of *F*s equals the number of *G*s just in case there is a 1–1 correspondence between the *F*s and the *G*s), i.e. on mathematical knowledge. In addition, whether or not Hume's Principle is *a priori* is one of the chief worries of neo-Fregean logicism, with neo-Fregeans arguing in favor of *apriority*.^{61} Let's put the question of naturalism and *a priori* knowability aside. There remain questions with respect to both revisionism and metaphysics.

It will be helpful to have a precise way of representing mathematical practices. Philip Kitcher represents a mathematical practice *P* by a quintuple 〈*L*^{P}, *K*^{P}, *Q*^{P}, *A*^{P}, *V*^{P}〉.^{62} Here *L*^{P} is the language used by practitioners of *P*; *K*^{P} is the set of statements accepted by the practitioners of *P*; *Q*^{P} is the set of live research questions of interest to practitioners of *P*; *A*^{P} is the set of argument strategies deployed by the practitioners of *P* to obtain or justify the members of *K*^{P}; and *V*^{P} is the set of views concerning metamathematical issues of *P* (proper methods of proof and definition in mathematics, scope of mathematics, relative importance of sub-disciplines of mathematics, and so on). We can think of positive concerns of a practice *P* as being recorded in *Q*^{P}. Negative concerns of *P*, i.e. constraints on theorizing recognized by practitioners of *P*, show up in *V*^{P}. In the interest of perspicuity, we add to *P* a component *N*^{P} representing such constraints. So we adapt Kitcher's approach and represent a practice *P* by a sextuple 〈*L*^{P}, *K*^{P}, *Q*^{P}, *A*^{P}, *V*^{P}, *N*^{P}〉. (Representing practices in this way should not be taken to endorse the view that theories or practices are set-theoretic objects. For one thing, the identity conditions of theories and practices are not extensional. For another, taking theories or practices to be sets would invite a charge of circularity in the case of set theory.) Given this representational apparatus, a violation of (PI*) with respect to a practice *P* is a recommendation to revise *P* where the revision contributes neither to answering a member of *Q*^{P} nor to satisfying a member of *N*^{P}.

A recommendation to reconstructively revise mathematics may or may not be legitimate according to (PI*). As we have seen, rival intuitionism provides an example of an illegitimate reconstructive revisionism. But a recommendation to reconstructively revise might be legitimate according to (PI*). For example, set theory was reconstructively revised when the Axiom of Replacement was added to the axioms of set theory. But this was legitimate according to (PI*), since the outcome of this revision was of antecedent interest to set theory. In short, the payoffs of adopting Replacement (e.g., the provable existence of cardinals ≥ℵ and a nice theory of ordinal numbers) are set-theoretic payoffs.^{65}

On the other hand, any recommendation to hermeneutically revise mathematics will run afoul of (PI*). Hermeneutic revision involves a change in the semantics of mathematical language. Call a semantics for a practice *P thin* if its referential claims are taken at face value by practitioners of *P*. So, for example, a semantics for *P*_{Acc} is thin if statements such as ‘“” refers to *i*’ typically don't elicit attempts to spell out what *i*‘really is.’ Call a semantics for a practice *P thick* if its referential claims are not taken at face value by the practitioners of *P*, i.e. if questions about the nature of referents of terms occurring in statements of *P* are seriously entertained by practitioners of *P*. *P*_{Acc} pretty clearly contains a thin semantics for its language. Mathematicians take all sorts of referential claims at face value in doing their work. But they just as clearly don't press on to inquire just what numbers, sets, functions, etc. really are. Which is to say that *P*_{Acc} does not contain a thick semantics for its language. Hermeneutic revision involves a thick semantic change; it concerns what mathematical language really means. Thus any recommendation to hermeneutically revise *P*_{Acc} is a recommendation to ‘thicken’ the semantics for the language of *P*_{Acc}. I submit that such thickening augments the membership of both and . For instance, a hermeneutic revision according to which arithmetic is really about *F*s carries with it a recommendation to take seriously questions such as ‘Is analysis also about *F*s?’ and ‘Are any *F*-facts not arithmetic facts,’ which is a recommendation to change the membership of . Similarly a hermeneutic revision as described carries with it a recommendation to take seriously constraints on theorizing such as ‘Avoid results that are inconsistent with arithmetic being about *F*s,’ which is a recommendation to change the membership of . None of these changes to *P*_{Acc} contributes to answering a member of or satisfying a member of . Otherwise, *P*_{Acc} would contain a thick semantics. So recommending these changes violates (PI*). Hence, any hermeneutic revision violates (PI*).

Nearly every case of revisionism we have encountered so far in this paper has been a case of reconstructive revisionism. Frege's logicism, however, is subject to the charge of hermeneutic revisionism.^{66} According to Frege's logicism, statements of arithmetic are really statements about classes of concepts. For example, the statement ‘1 ≠ 0’ for the Fregean logicist means that there is no 1–1 correspondence between the class of concepts equinumerous with the concept «*y*=*Nx* : *x*≠*x*» and the class of concepts equinumerous with the concept «*x*≠*x*» (where ‘*Nx* : *φ*(*x*)’ is read ‘the number of *x*'s such that *φ*(*x*)’). Thus an epistemology of mathematics based on Frege's logicism violates (PI*) and so is inconsistent with a naturalistic epistemology of mathematics.

Whether or not neo-Fregean logicism similarly falls to the charge of hermeneutic revisionism is less clear. In a certain sense, neo-Fregeanism implements a reconstrual strategy: mathematical claims are taken to be claims of second-order logic augmented with Hume's Principle. But given the tight connection between mathematics and logic – by contemporary lights logic is part of mathematics – one might respond that this is at worst a harmless analysis of parts of mathematics in terms of another part of mathematics. And such an analysis isn't obviously of no antecedent interest to mathematics. One might worry that Hume's Principle isn't part of logic, and so that this response to the charge of hermeneutic revisionism falls short of complete of success. However, the real problem with the response under consideration is that it makes the neo-Fregean epistemology of mathematics depend on prior availability of an epistemology of second-order logic, which in the current circumstances is viciously circular: the epistemology of mathematics depends on the epistemology of second-order logic, which, according to the response under consideration, is itself part of mathematics. So at best there are serious questions concerning neo-Fregeanism's anti-revisionist credentials. Thus, neither variety of Fregean logicism fares well with respect to (BA2).

There are also problems with Fregean logicism as regards (BA1). First, it is *prima facie* unlikely that the metaphysics of Frege's logicism qualifies as naturalistic, given that its central entities (e.g. concepts and extensions) inhabit the third realm. As to neo-Fregean logicism, its metaphysical commitments are the metaphysical commitments of second-order logic plus Hume's Principle. How these commitments are cashed out affects whether or not neo-Fregean logicism is a viable candidate for naturalizing mathematics. Here, again, if we take Quine's way and consider second-order logic as set theory in disguise, we run into a circularity problem. We cannot go Frege's way and take the commitments of second-order logic to be Fregean concepts (i.e. properties), since then we're for all intents and purposes back to Frege's logicism and its attendant problems. Of course, one might go a third way, arguing that second-order logic isn't part of mathematics and that we shouldn't be Fregeans about its ontological commitments. In this case we would need an account of the ontological commitments of second-order logic, but if those commitments turned out to be naturalistically acceptable then neo-Fregean logicism might be thought to offer a way of naturalizing mathematics. Moreover, an epistemology of second-order logic, if naturalistically acceptable, would also answer the worries of the previous paragraph. Thus far, then, neo-Fregeanism stands as at least a candidate for naturalizing mathematics. Evaluating the prospects for naturalistic accounts of the metaphysics and epistemology of second-order logic is more than I can do here.^{67} However, there is one well-known difficulty for neo-Fregeanism that might upset its candidacy even if the concerns already canvassed don't: the problem of the scope of neo-Fregean mathematics.

Ignoring philosophical worries for a moment, here is what we know about how much of modern mathematics can be accommodated in a neo-Fregean framework, i.e. with respect to the scope of neo-Fregeanism. Owing to work of Boolos, Richard Heck, Crispin Wright, and others^{68} we know that the axioms of second-order Dedekind–Peano Arithmetic (DPA^{2}) are provable in second-order logic plus Hume's Principle (so-called *Frege Arithmetic* (FA)). Following Boolos, this result is known as *Frege's Theorem*. Hale (2001) and Shapiro (2000a) provide ways to obtain the real numbers on the basis of FA plus additional axioms of the same sort as Hume's Principle (so-called *abstraction principles*), and hence to develop real and complex analysis within DPA^{2}.^{69} But, of course, mathematics extends well beyond real and complex analysis, and for neo-Fregeanism to be maintained as a candidate for naturalizing mathematics it needs to encompass the whole of mathematics. In particular, it needs to encompass set theory. Otherwise, it turns out to be a reconstructive revisionist position which fails to respect (PI*).

Without going too far into detail, there are two approaches to extending the Fregean program to set theory.^{70} One approach adopts various abstraction principles in place of Frege's ill-fated Basic Law V (the statement that for any concepts *F* and *G*, *F* and *G* have the same extension if and only if all and only the same objects fall under *F* as fall under *G*). The other instead restricts Basic Law V, so as to avoid Russell's paradox. The former approach has been pushed furthest by Kit Fine, in the form of his general theory of abstraction.^{71} But even ignoring a lingering difficulty with the so-called *bad company* objection,^{72} the natural limit of Fine's theory is third-order Dedekind–Peano Arithmetic (DPA^{3}), which is equiconsistent with ZF^{-}+℘(ω) the theory one gets by removing the full powerset axiom from ZF (ZF^{-}) and putting an axiom ensuring the existence of the powerset of *ω* (℘(ω)) in its place. The sense in which DPA^{3} is the ‘natural’ limit of Fine's theory is that DPA^{3} is what one gets by restricting Fine's theory to second-order logic, as is customary for neo-Fregeans.^{73} But setting aside this restriction and allowing variables of every finite type would only yield a theory of consistency strength less than that of Zermelo set theory, Z. In either case, we get much less than the whole of modern mathematics.^{74} Hence, the first approach to extending neo-Fregeanism to all of mathematics presently comes up short.

The second approach takes us much further. The set theory Burgess calls *Fregeanized Bernays* set theory (FB) takes us beyond second-order ZFC (ZFC^{2}) to get (in addition) some small large cardinals.^{75} One might argue that we should be satisfied with this, that it covers all of modern mathematics. I would disagree, but let's bracket concerns about whether or how much of the large cardinal hierarchy a theory needs to accommodate to be satisfactory. If a theory accommodates at least ZFC, we'll say it's satisfactory as far as not recommending a reconstructive revision of mathematics. FB is satisfactory in this sense. The question is whether or not FB satisfactorily provides for a naturalistic epistemology of mathematics. A fairly cursory look at FB answers this question negatively.

FB is formulated in monadic second-order logic with primitive notions *extension-of* and *falling-under*. For the former, we have a symbol in the language of FB, ‘€’, so that ‘€*xF*’ translates ‘*x* is the extension of *F*’. The latter requires no special symbol in the language, as we already have the syntactic device of concatenation: ‘*Fx*’ translates ‘*x* falls under *F*’. The notions of sethood and membership, for which we use the symbols ‘S’ and ‘∈’, respectively, are defined by the following *axioms of subordination*:^{76}

(AS1) S*x*∃*X*€*xX*

(AS2) *x*∈*y*∃*Y*(€*yY*∧*Yx*)

That sethood and membership are subordinated to extension-of and falling-under in this way is crucial to the Fregean credentials of FB. As Burgess notes, the primacy of extension-of and falling-under is a chief feature of FB making it ‘similar to Frege's original theory and different from mainstream axiomatic set theories such as ZFC’ (Burgess, 2005, p.185).

The problem, of course, is that this primacy arguably commits FB to Fregean concepts, thus making it hermeneutically revisionist (and so incompatible with (PI*)), or making the naturalistic status of its metaphysics questionable, or both. To be sure, one might define ‘€’ in terms of ‘S’ and ‘∈’.^{77} However, this would still leave us with falling-under as a primitive notion, which *prima facie* carries commitment to Fregean concepts. Moreover, taking ‘S’ and ‘∈’ as primitive, as this strategy does, once again invites a charge of circularity: the epistemology of mathematics relies on an antecedent epistemology of set theory, i.e. of mathematics. All in all, then, it seems unlikely that Fregean logicism is suitable for naturalizing the epistemology of mathematics.

##### 3.2.4. Modal structuralism

According to modal structuralism, mathematical statements are statements about possible structures, as opposed to particular types of mathematical objects. Talk of possible structures is couched in second-order S5, and modal structuralism gives an explicit procedure for reconstruing statements of mathematics as statements of second-order S5. Statements of arithmetic, for instance, are construed as statements about possible *ω*-sequences.^{78} Given a statement *σ* of (informal) arithmetic, *σ* can be formalized in a version of DPA^{2} the language of which consists solely of a unary function symbol ‘**s**’.^{79} Let *σ** be such a formalization of *σ*. Then the *modal-structural interpretation* (msi) of *σ* is:

Hellman (1989) extends the modal-structuralist approach to real analysis and set theory, including some large cardinals. Granting the success of these extensions, modal structuralism accommodates at least ZFC and so is satisfactory as far as not recommending a reconstructive revision of mathematics. However, it does not fare so well with respect to hermeneutic revisionism. As we have seen, modal structuralism construes mathematical statements as statements about possible structures; it's a view about what mathematical claims *really* mean, what mathematicians are *really* talking about. This is precisely to engage in hermeneutic revisionism, which is illegitimate according to (PI*). But for the unconvinced, there are yet other problems.

First, as with Fregean logicism above, there are issues to do with *apriority*. For modal structuralism mathematical knowledge depends on knowing the appropriate msi's, i.e. statements of pure second-order S5. So for modal structuralism mathematical knowledge depends on knowing modal facts, and modal facts are arguably knowable *a priori*. So if one takes *a priori* knowledge to be incompatible with naturalism, modal structuralism isn't naturalistically acceptable. Bracket this worry, as we did with Fregean logicism. Still (and second) one might reasonably worry about an epistemology of mathematics that depends on an epistemology of modality. Naturalistic or not, do we really have that much better grasp of modal knowledge than we do of mathematical knowledge? It's not at all clear that we would be gaining much by basing our epistemology of mathematics on an epistemology of modality. Moreover, Hellman himself suggests that the epistemology of modality sufficient for a modal-structuralist set theory is unlikely to be naturalistic.^{81}

Finally, the naturalistic standing of the metaphysics of modal structuralism is far from clear. Modal structuralism is intended to be neutral between realism and nominalism.^{82} Hellman argues that:

(Eq) *σ* is true iff *σ*_{msi},

where the relevant conception of truth is realist, and that this is recognizable by both mathematical realists and modal structuralists, each on their own terms.^{83} The idea is that the realist can have her preferred reading of *σ* in the left-hand side of (Eq), the nominalist can have a metaphysically innocent reading of *σ* in the right-hand side of (Eq), and the two can agree on the truth-value of *σ*. On the face of it, this is good news for anyone who would like to press modal structuralism into service in naturalizing mathematics. A nominalistic metaphysics is almost certainly naturalistic. But there is more to it than what we see on the surface.

If one is to make any progress with naturalizing mathematics via modal structuralism, the metaphysical commitments of a mathematical claim *σ* had better be those incurred by the right-hand side of (Eq), i.e. by *σ*_{msi}. Since *σ*_{msi} is a statement of second-order S5, the metaphysical commitments of *σ*_{msi}, and hence of the modal-structural approach, are just those of second-order S5. Hellman argues that these commitments can be cashed out noministically, *taking logico-mathematical modality as primitive*. That is, the nominalism yielded by modal structuralism is a modal nominalism.^{84} The question, of course, is: Even granting that second-order logic can be satisfactorily nominalized, why think that the relevant modality can similarly be nominalized? I'll not attempt to answer this question here; rather I leave it as one more significant worry concerning the suitability of modal structuralism as an approach to naturalizing mathematics.

Before concluding I want to briefly address an objection one might raise in connection with my use of hermeneutic revisionism in this and the immediately preceding subsection, viz., that though it is in some sense revisionary, hermeneutic revision isn't revisionary in the sense relevant to violating (PI*). The idea is that revising the semantics of mathematics is unlikely to affect mathematical practice much at all, and it's really revisions that affect practice which are at issue in (PI*). So hermeneutic revision needn't conflict with (PI*). This being the case, the naturalist can accept hermeneutic revision. It seems to me there are at least two things to say in response to this worry.

First, I introduced the representational apparatus for mathematical practice in §3.2.3. to help make perspicuous that revising the semantics of mathematics induces a revision in the non-semantic aspects of the practice. Semantic thickening induces revisions to both and . If this is right, then the objection simply misfires. Of course, one might be unconvinced that semantic revision induces non-semantic revision. This leads to the second response. Even if hermeneutic revision is compatible with (PI*), that does little to undermine my contention that neither Fregean logicism nor modal structuralism is well suited to a naturalistic epistemology of mathematics. This, because my arguments for this contention only partly rely on these views being hermeneutically revisionist. In the first place, there are serious questions as to whether the metaphysics of either view is naturalistic. In the second place, Fregean logicism threatens to be reconstructively revisionist.^{85} In either case, regardless of the status of hermeneutic revisionism vis-à-vis (PI*), serious work remains to be done before either Fregean logicism or modal structuralism is in a position to satisfy the needs of a naturalistic epistemology of mathematics.