Nominalism and Mathematical Objectivity

Abstract

We observe that Putnam’s model-theoretic argument against determinacy of the concept of second-order quantification or that of the set is harmless to the nominalist. It serves as a good motivation for the nominalist philosophy of mathematics. But in the end it can lead to a serious challenge to the nominalist account of mathematical objectivity if some minimal assumptions about the relation between mathematical objectivity and logical objectivity are made. We consider three strategies the nominalist might take to meet this challenge, and we argue that all these strategies are untenable.

1 Introduction

In this paper, we will present an argument against nominalists’ account of mathematical objectivity via logical objectivity. The basic argument is based upon the following three relatively uncontroversial observations: first, following Putnam’s (1980) model-theoretic argument, nominalists deny determinacy of higher set theory and second-order logic; second, nominalists accept the objectivity of proofs and basic arithmetic, in part by way of compromise with mathematical practice in, e.g., higher set theory, but the latter in turn have to appeal to determinacy of one mathematical concept: finitude. Third, two further logical facts cast doubt on determinacy of the concept of finitude: (a1) it is known that the concept of finitude has non-standard models; (a2) the standard model of finitude can only be fixed by ascending to the second-order logic or set theory.¹But these together are incompatible, because non-determinacy of second order logic yields non-determinacy of finiteness which yields non-determinacy of proof and logic. The nominalist cannot get what she wants.

There are at least three promising strategies to address this problem. The moderate nominalist, by appealing to a physical ω-sequence to fix the extension of finiteness,² denies that the determinacy of the notion of finitude depends on that of higher-order concepts; the radical nominalist at one extreme tries to fix determinacy of the notion of finitude by accepting determinacy of second-order logic, whereas at another extreme chooses to accept the radical indeterminacy throughout all mathematics, but denies that the indeterminacy of the notion of finitude poses a serious problem for the nominalist position. We will argue that all these three strategies are untenable. To get a determinate notion of finitude and thereby get an account for mathematical objectivity in general, the nominalist still has a lot of job to be done.

Before proceeding, let us make a few clarifications. First, in this paper, we use “objectivity” and “determinacy” more or less equivalently. Now defining mathematical objectivity in terms of its determinacy or in reverse is philosophically controversial, but from methodological considerations, we think this treatment is unproblematic: we are not interested in the proper definition of mathematical objectivity, the main question of this paper is: if we see mathematical objectivity from the determinacy perspective, what would happen to nominalism? So the conclusion of the paper is also conditional: if we treat mathematical objectivity as mathematical determinacy, then the nominalist cannot get what she wants in the sense we have indicated above. On the other side, we do think that some intuitive sense can be made for the notion of objectivity in terms of determinacy in mathematics: in normal mathematical practices, when it is said that mathematics is to be objective, then when we are answering a mathematical question, there has to be a determinately correct answer to the question. We will say more about it in Sect. 3, but readers should be ensured that when using “objectivity” and “determinacy” interchangeably, we are using these notions that nominalists would be happy to accept.

Second, in this paper, we use “nominalism” in the broad sense, it includes not only traditional nominalism like fictionalism, modal-structuralism, if-thenism, and figuralism, it might also include some nominalist forms of pluralism.³It might be thought that traditional nominalism is only concerned with mathematical objects, but not with mathematical objectivity. The paper will show that this is a misunderstanding about nominalism. It is a general challenge to the nominalist that she should account for why mathematics seems to be an objective discipline after she rejects mathematical objects and holds the nonstandard semantics for mathematical sentences.

The paper consists of following sections: from Sect. 2 to 4, the three observations will be elaborated and discussed. Section 5 will give a powerful argument against nominalists’ account of mathematical objectivity in terms of logical objectivity based on the three observations. The next three Sects. 6, 7, and 8 will take a closer view of the three strategies to address the already-mentioned problem in favor of nominalism: we will argue that all these three strategies are untenable. Section 9 is the conclusion of paper.

2 Observation 1

Observation 1 (the extension of the concept of set and that of second-order quantification are radically indeterminate) arises from Putnam’s (1980) well-known model-theoretic argument against determinacy of set theory plus Field’s (1994a, 1998a, 1998b, 2001, 2015) argument against determinacy of second-order logic. We will call it Putnam-Field’s argument (or PFA for short).

The core idea of PFA is to apply the Löwenheim-Skolem Theorem and the Compactness Theorem to first-order set theory to generate non-standard interpretations of primitive predicates thereof. According to the Löwenheim-Skolem Theorem, every first-order theory has non-standard models. For example, a first-order set theory with an infinite domain will have a model with a countable domain. However, the latter interpretation would be naturally regarded as unintended by a realist. But given the abstract nature of mathematical objects, PFA forces the realist to answer in terms of which the former is intended while the latter is not. If the realist cannot provide a no-question-begging answer to this question, then our concepts of set and membership are referentially indeterminate.

Two strategies might be used by a realist to rescue the determinacy of set theory from PFA. One strategy is to restrict our concept of the set by using other set theories. For example, by using Morse-Kelly set theory rather than the standard Zermelo-Fraenkel (ZF) set theory, the realist can distinguish two variables and let the quantifiers range over sets and classes respectively; through this distinction, he can dispense with the unrestricted conception of set, supplementing it with the conception of classes. One advantage of this distinction is that the predicates “sets” and “$\in{{\mathfrak{}}_{{\text{s}}}}$” will have standard interpretations (up to isomorphism) once the extensions of predicates “classes” and “${\mathfrak{\in}}_{{\text{c}}}$” have been fixed. But Field argues that “the arguments for indeterminacy are arguments for the indeterminacy of ‘class’ and class membership as well as ‘set’ and set membership.” (Field 1994a, p. 393) That is to say, we cannot pin down a unique structure for predicates “classes” and “${\mathfrak{\in}}_{{\text{c}}}$”, because the Löwenheim-Skolem’s Theorem is still applicable in this case. So appealing to Morse-Kelly set theory is of no help to the realist to answer Putnam-Field’s objection.

Another strategy is to resort to second-order set theory. A second-order set theory is a theory that results from ZF set theory by employing a second-order replacement axiom instead of the first-order scheme, with quantifier variables understood either as plural variables in the sense of Boolos (1985) or as logical classes corresponding to sets and proper classes. It is a well-known logical fact that second-order set theory is quasi-categorical: for any two models of the theory, either they are isomorphic, or the first is isomorphic to the initial segment of the second, or the second is isomorphic to the initial segment of the first.⁴ In light of this logical result, PFA is no longer a challenge to the realist. This is a nice result, which actually is often employed by realists (Shapiro 1991; McGee 1997; Parsons 2008) to address Putnam’s problem. However, the approach to meeting Putnam’s challenge through determinacy of second-order set theory is dashed by another argument of Field’s (1994a, 1998b).⁵

It is known that second-order quantification has two types of interpretations: one is the full interpretation, according to which the quantifiers range over all subsets of the first-order domain, and another is the general interpretation, according to which the quantifiers range over only the specified subsets of the first-order domain. Now only in the first case could the second-order quantifiers have the standard interpretation, while in the second case, given that the Löwenheim-Skolem Theorem applies, non-standard models would still not disappear. The realist has to choose the full interpretation while excluding the general one, but Field asks: “in virtue of what would our phrase ‘every subcollection’ mean every subcollection, rather than every F subcollection?” (Field 1994a, 395) Therefore, if Putnam’s argument works in the first-order case, it works equally in the second-order case, we cannot have a determinate notion of second-order quantification without begging the question.

In summary, PFA leads to three consequences: first, traditional realism would be rejected if the argument is right. Put in another way, if PFA is correct, and if a good version of mathematical realism maintains objectivity of set theory and second-order logic, then at least this version of mathematical realism should be rejected. Second, by rejecting this form of mathematical realism, PFA serves as a good motivation for the nominalist position,⁶ for according to nominalists, neither mathematical objects nor determinate mathematical facts exist. For the sake of argument, we will assume in the following sections that the nominalist would accept PFA. Third, it is known that the Löwenheim-Skolem Theorem and the Compactness Theorem apply equally to first-order set theory and first-order arithmetic (e.g., first-order Peano Arithmetic). So if PFA is cogent, it can also be used to reject the determinacy of our concepts of natural numbers. This might be hardly acceptable, but this time it would act as a more serious challenge to nominalism, for as we will see below, nominalists need the determinate notion of natural numbers to account for mathematical objectivity in general if there are any.

3 Observation 2

Mathematics seems to be a highly strict and objective discipline: it has highly constrained standards for what is correct and what is wrong. For instance, if we try to figure out the number of the prime numbers between 10² and 10⁵, one of the following mathematical sentences is objectively correct, while the other is objectively incorrect:

• (1) The number of the prime numbers between 10² and 10⁵ is 9567.

• (2) The number of the prime numbers between 10² and 10⁵ is 9566.

Generally, when it is said that mathematics is to be objective, then when we are answering a mathematical question, there has to be a correct answer to the question. Now there are at least two ways to account for this kind of correctness (or mathematical objectivity in general): one is through mathematical objects, one is through logic or logical objectivity.⁷

The way through which mathematical objects account for mathematical objectivity can be transformed into such an argument: since mathematical sentences are about mathematical objects, there is only one objective fact about these objects, therefore, there is only one objectively correct answer to certain mathematical problems. For instance, both (1) and (2) are about natural numbers, and since there is only one objective fact about the relations between natural numbers, that’s why (1) is correct while (2) is incorrect.⁸

The way through logic or logical objectivity to account for mathematical objectivity can also be spelled out in an argument form: given the objective standard of proof plus logical objectivity, if we treat correct mathematical sentences as the conclusions logically derived from mathematical premises, then there will be only one objectively correct answer to certain mathematical problems. For instance, the reason why (1) but not (2) is correct is that only (1) is the conclusion logically derived from axioms in some arithmetical system (e.g., in PA) according to the objective proof procedures.

It is not beyond controversy that the first way is not available for the nominalists, for they deny that there are any mathematical objects or mathematical facts. So a good choice for nominalists is to employ the second strategy. In fact, such a strategy often appears in many nominalists’ writings. For instance, according to many fictionalists (Field 1980, 1989; Balaguer 1998, 2009; Chihara 1990; Yablo 2001, 2002; Leng 2010; Ye 2010), (1) is correct just because it is derived from some axioms in some fiction; while according to the modal nominalist (Hellman 1989, 1996), (1) is correct just because it is derived from an axiom describing a possible existent physical object; according to some traditional nominalists (Nelson and Quine 1947), (1) is correct just because its paraphrased counterpart is a logical derivation from some paraphrased premises.

Now to make the relevant mathematical sentences objectively correct, all these derivations (or their corresponding informal proofs) should be objective in the sense that the logical relations between mathematical premises and mathematical conclusions should be determinate, no matter whether they appear in fictions or possible structures or only after paraphrasing. This amounts to saying that the notion of logical consequence (or their corresponding informal proofs) should be determinate, which again amounts to saying that the notion of finitude should be determinate because logical consequences (or their corresponding informal proofs)⁹are just finite sequences of strings of symbols.

We will flesh this observation out in more detail below, but before that, let us make some clarifications about several concepts we use above.

First, by saying that nominalists had better take logical objectivity to explain the difference between (1) and (2) and mathematical objectivity in general, we do not exclude other options they might make. They can either take an evolutionary or a cognitive approach such as in Pantsar (2021), or use imagination plus mathematical applicability in the physical world as suggested by an anonymous referee.¹⁰ But all these options rest on a different notion of mathematical objectivity, which is different from the one with which this paper is concerned. For us, mathematics is objective in the minimal sense means that there seems to be an objectively determinate mathematical solution to certain mathematical problems. The notion of objective correctness should accord with the internal standards mathematicians set in real mathematical practices, it should not be altered arbitrarily by one specific philosophical position. For instance, 2 + 2 = 4 is correct, while 2 + 2 = 5 is incorrect, this should not be altered from whatever philosophical considerations. However, this does not mean philosophers have nothing to do with mathematical objectivity. They can argue that when one mathematician finds a correct answer to a certain problem, whether she discovers a new mathematical fact or just derives a mathematical conclusion using certain logical rules within certain systems, it can never turn out to be wrong later. So in general, if this minimal notion of mathematical objectivity is acceptable to nominalists, we think a wise choice for nominalists is to take the second approach, namely to take logical objectivity (and its corresponding informal proof procedures) to account for it.¹¹

Second, some people might object that if nominalists take the second approach, then this is nearly to say that they do not need mathematical objectivity at all. To see what the objector actually means, let us suppose with the nominalists that the way mathematics is objective just means that the mathematical conclusion can be derived from some axioms. But now our objector might say: “the axioms are relative: they are relative in the sense that they are just correct within some fictions or some structures; but given that there are multiple structures or fictions to choose from, there should be multiple axiom systems.” The consequence of this pluralism is that mathematics is not objective at all, for there would then not be a unique answer to one mathematical problem: according to one axiom system for one fiction or structure, “2 + 2 = 4” is correct, while according to another axiom system for another fiction or structure, “2 + 2 = 5” is correct.

Let us give two quick responses to this objection here and leave more complex cases in Sect. 6. First, this objection begs the question. As we said in the first clarification, it’s not us but mathematicians who decide which mathematical sentence is correct and which is wrong. The philosopher’s job is just to account for the correctness condition in an appropriate way to accord with mathematical practices. Second, if this radical anti-objectivism for elementary arithmetic makes sense, it would suffice to make nominalism absurd.¹² It is known that many of nominalists’ notions such as logical consequences, proofs, and consistency depend at least on the determinacy of arithmetic. In other words, without maintaining that arithmetic is objective, the nominalists could not express their own positions. A similar kind of response is also expressed by Field, a prominent nominalist, as follows:

“Indeed, nearly everyone believes that the choices between an undecidable sentence and its negation is objective not only for the simple sorts of number-theoretic statements just described,¹³ but for elementary number-theoretic statements more generally. This belief would be a very hard one to give up, since many claims about provability and consistency are in effect undecidable number-theoretic claims, so that an anti-objectivist about elementary number theory would need to hold that even claims about provability and consistency often lack objectivity, this is position that few will want to swallow.” (Field 2001, p. 318).

So a rational nominalist should not only take logical consequences (and the standard of their corresponding informal proofs) as objective to account for mathematical objectivity, she should also accept at least elementary number theory as objective. In effect, they are just two sides of one coin: without the objective standards of logic or proofs, it would be hard for them to account for mathematical objectivity including arithmetical objectivity in the minimal sense we clarified in the last second paragraph; on the other hand, without arithmetical objectivity or determinacy, the objectivity of logic or proofs would be a nonstarter, mathematical objectivity in the minimal sense would be not accountable for as well.

If this argument is correct, then we can observe that the determinacy of finitude is unavoidable to the nominalists in accounting for mathematical objectivity (in the minimal sense): On the one hand, if they want to maintain the objectivity of elementary arithmetic, then they need the determinacy of the notion of finitude, for it is a fundamental component of determinacy of elementary arithmetic that natural numbers have finitely many predecessors; on the other hand, if they want to use logical objectivity rather than mathematical facts to account for mathematical objectivity in the minimal sense, then they need the determinacy of finiteness as well, for it is a basic metalogical fact that logical consequences (and the corresponding informal proofs) are just finitely long sequences of strings of symbols.¹⁴

4 Observation 3

In this section, following Shapiro (1991) and Bueno (2005), we will elaborate our last observation based on the proof of two well-known logical facts: first, the notion of finiteness is not determinate in first-order theory (Proposition 1); second, only by ascending to second-order logic can the standard model of the quantifier “there are finitely many” be fixed (Proposition 2).

Proposition 1

The first-order formula ∃_finxAx has a non-standard interpretation.¹⁵

Proof

Define the formula '_finxAx in terms of set-theoretical terms in the following way: ∃_finxAx iff {x: A (x)} is in 1–1 correspondence with the predecessors of some natural number n. According to Compactness Theorem, the latter has non-standard models, so does the former.¹⁶

Remark 1

Field (1994a) argues that Putnam’s model-theoretic argument cannot carry over to the notion of finitude because Compactness Theorem has no application in this case without obviating the constraints laid down for the interpretations of physical predicates; in other words, if Compactness Theorem obtains in the case of finiteness, then the same theorem would obtain in the case of physical predicates such as “events” and “one second earlier”, which in turn would impose nonstandard interpretations to those predicates, but this is absurd. Bueno (2005) replied that this objection is question-begging. We agree with Bueno’s objection and will take a closer look at it in Sect. 8. Leaving these philosophical debates aside, at least from the logical point of view, Proposition 1 (and its proof) is not questionable.¹⁷

Proposition 2

Let S be a set of first-order sentences and let $\Phi$(x) be any first-order formula containing x free. The formula $\Phi$(x) in the language of pure second-order logic such that in every standard model Μ of SOL, Μ ⊨ $\Phi$(x)[S/x] iff the extension of $\Phi$is finite.

Proof

See Shapiro (1991), pp. 100–101.

Remark 2

Some people might object that if there are other means to fix the standard model of the formula ∃_finxAx, it will reduce the significance of both propositions proved above. This should be indeed the case if, for example, one follows McGee (1997) and proves that we have the intended conception of natural numbers with the first-order Induction axiom scheme plus certain metaphysical assumptions; or if one follows Halbach and Horsten (2005), argues that once our computational activities guarantee that the first-order operators “ + ” and “${\mathfrak{\times}}$” are recursive, then every model of first-order arithmetic with two operators will be isomorphic to the standard models of PA (Tennenbaum Theorem). But both strategies depend heavily on mathematical objects, which are unavailable to the nominalists. On the other hand, whether those approaches are successful is still highly controversial.¹⁸

On the other hand, Bueno (2005) noticed that if second-order logic cannot rule out Henkin semantics, then the notion of finitude would have a non-standard interpretation (Proposition 3). This reasoning amounts to the fact that if second-order logic is indeterminate, or more precisely, if second-order quantification is indeterminate, then the notion of finitude is indeterminate.

Proposition 3

Let S be a set of first-order sentences and let $\Phi$(x) be any first-order formula containing x free. The formula $\Phi$(x) in the language of pure second-order logic such that in the Henkin model Μ^* of SOL, Μ^* ⊨ $\Phi$(x)[S/x], then the extension of $\Phi$is infinite.

Proof

Omitted here, it is similar to the proof of Proposition 1. For the detailed proof see Shapiro (1991), 101–102.

5 A Pseudo Inconsistency Argument

Let us put all our three observations together: according to observation 1, our set theory and second-order logic are radically indeterminate, and this can serve as a motivation for nominalists to reject one form of mathematical realism on the one hand and deny objectivity in higher mathematics on the other hand. According to observation 2, nominalists need to presuppose the determinacy of finitude, otherwise the determinacy of the notions of proofs, logical consistency, logical consequences, etc., would be affected; in other words, if these notions were not determinate, then their efforts to provide an account for the seeming mathematical objectivity in the minimal sense via logical objectivity would be in vain. But according to observation 3, if the second-order logic (or ZF set theory) is indeterminate, then the notion of finitude would be indeterminate as well.

All of this leads to a powerful argument against nominalists’ determinacy of finitude and their account of mathematical objectivity in terms of logical objectivity in general:

• 1. The nominalist’s notion of finiteness should be determinate. (Observation 2)

• 2. Second-order logic or first-order set theory is radically indeterminate. (Observation 1)

• 3. If second-order logic or first-order set theory is indeterminate, then the nominalist’s notion of finiteness is indeterminate. (Observation 3)

• 4. So, the nominalist’s notion of finiteness is indeterminate. (2, 3, modus ponens)

• 5. So, nominalist’ s notion of finiteness should be determinate but (in fact) indeterminate. (1, 4)¹⁹

Call this argument PICA (Pseudo Inconsistency Argument), it’s not a real inconsistency in the sense that we are confronting with the Humean problem here: we cannot establish the factual from the obligatory (or is from should) in premise 1. But it is nearly to be inconsistent in the sense that most nominalists not only should embrace the determinacy of the notion of finitude but also actually maintain that. For example, Hellman (1989), Field (1998a), and Balaguer (2009) all admit that we have a categorical notion of finitude. But if this is the case, then the premise 1 would turn into its alethic version, i.e., the nominalist’s notion of finiteness is determinate. As a consequence, we would have the following conclusion in correspondence with conclusion 5:

Conclusion 5*: The nominalist’s notion of finiteness is determinate and indeterminate.

Bueno (2005) actually draw conclusion 5*. He observed that the nominalist’s mathematical notion and logical notion stand and fall together, there is no possibility that one is indeterminate while the other is determinate. But this is not our conclusion, we are not intended to draw such a strong conclusion in this paper. Because we are more interested in mathematical objectivity in the minimal sense and nominalist’s account for it, we think the weaker PICA is enough for this paper.²⁰

In the following sections, we will consider the logical spaces generated by PICA, there are at least three alternatives a nominalist can take: she might reject premise 1 by accepting that the notion of finiteness (or natural numbers) is radically indeterminate, but providing reasons to explain why this indeterminacy is not a problem for her; she might reject premise 2 by defending the determinacy of second-order logic and set theory but without subjecting itself to the challenge from PFA; and finally, she might reject premise 3 and provide an alternative explanation for arithmetical determinacy in favor of nominalism. We will argue that all these alternatives are untenable.

6 Alternative 1

The first option a nominalist might take is to deny premise 1 of PICA in the previous section and entertain the radical anti-objectivism throughout all branches of mathematics. So she might not only agree with Putnam that first-order set theory and second-order logic are indeterminate but also accept the consequence of PFA that elementary number theory is indeterminate, thereby accepting that the notion of finitude has no determinate reference. But she will insist that entertaining the radical anti-objectivism poses no insurmountable problem to nominalism at all; unless supporting this position would introduce mathematical objects, there is nothing wrong with it.

In Sect. 3, we have criticized this position from two fronts. On the one hand, this position is in conflict with mathematical practice; on the other hand, the insistence on anti-objectivism throughout all mathematics, in particular, the insistence on indeterminacy of finitude would lead to the indeterminacy of many other core concepts such as proofs, consistency and logical consequence, which in turn would affect the formulation of nominalist positions. In this section, we will examine two more refined anti-objectivist positions that seem to sidestep both of the above criticisms, but (we will point out that) in the end still face serious problems.

The first strategy in defense of radical anti-objectivism appears in Field (1994a), we quote it in the following:

“Radical indeterminacy has it that at any time, a person’s concept of finitude is exhausted by the maximal mathematical theory of finitude that he or she implicitly accepts at that time: more specifically, any mathematical claim about finitude that has different truth values in different models of someone’s maximal theory has no determinate truth value on that person’s current conception of finitude. (Since claims about natural numbers are in effect claims about linear orderings in which every object has only finitely many predecessors, the same holds for mathematical claims about natural numbers.)” (Field 1994a, p. 401).

Since this radical indeterminacy assumes a Platonist perspective, according to which when people really accept a mathematical theory, they believe it to be true, this notion of acceptance is not available to nominalists, let us make a slight modification: even though a nominalist does not believe mathematical theories to be true, she does “make a distinction between mathematical theories that are mathematically acceptable and those that are not,” (Field 1994a, p. 407) and a natural candidate for a nominalist to accept a maximal mathematical theory is its consistency. Correspondingly, a radical version of anti-objectivism for nominalism can be formulated as such:

Radical indeterminacy has it that at any time, a person’s concept of finitude is exhausted by the maximal mathematical theory of finitude that he or she implicitly accepts at that time: more specifically, any mathematical claim about finitude that is consistent with such-and-such systems but inconsistent with so-and-so system has no determinate consistency condition on that person’s current conception of finitude.

But just as the platonist radical anti-objectivism has to face the Gödel phenomenon, this nominalist radical anti-objectivism has to face it as well.

Let us start expounding on the Gödel phenomenon that the Platonist has to face. Suppose the position described by Field above were correct, we implicitly accept some maximal mathematical theory, more concretely, suppose that this maximal mathematical theory is the elementary arithmetic such as PA and it is consistent. According to the anti-objectivist, for the maximal consistent mathematical theory, if its Gödel sentence is undecidable, then the sentence would not have the determinate truth value. So the Gödel sentence of PA has no determinate truth value. But this is wrong: even the Gödel sentence to real arithmetic is independent of elementary arithmetic, it is true. In other words, if we implicitly accept the elementary arithmetic (i.e., PA) in question as our maximal mathematical theory, then we should accept that it is consistent as well. If we accept that PA is consistent, then we should accept its consistent sentence. But in PA, the consistency of PA is proof-theoretically equal to its Gödel sentence, so we should accept it as well. But if Gödel’s sentence is true, then the radical anti-objectivism seems to be an incoherent position.

Move to the nominalist’s radical anti-objectivism, then we see that it is faced with the similar Gödel phenomenon the Platonist faces, the only difference is that she accepts Gödel’s sentence not because it is true but because it is consistent with the elementary arithmetic she has already accepted.²¹ In other words, the nominalist has to accept the following instance of reflection principle:

(RF) For elementary arithmetic, for example, PA, go from accepting PA to accepting PA + CON_PA, where CON_PA is the formalization of the property of consistency of PA.

But if our above reformulation of anti-objectivism is true, then this form of RF would have to be rejected, but this is hard to accept. Field (1994a) argues that the reasoning in (RF) is fallacious because we cannot infer the consistency of a theory from the theory itself.²² A nominalist can accept PA and CON_PA, but she cannot accept PA + CON_PA, because accepting the latter amounts to accepting its consistency, but this is impossible. However, this objection does not get to the heart of the above argument, for on the one hand, if a nominalist accepts PA and CON_PA, there is no reason for her not to accept their conjunction, it’s just a logical rule; on the other hand, suppose that she doesn’t accept the conjunction just because that the consistency of it does not derive from it, then this conflicts with Gödel’s Second Theorem, which says that CON_PA is consistent with PA, if PA is consistent.

There is another version of anti-objectivism also mentioned by Field (1994a), which says that we do not need the determinate notion of finitude at all. After all:

“Any purported proof we come across will have less than (say) 10¹⁰⁰ component formulas (each with less than say 10¹⁰⁰ symbols). The indeterminacy we are contemplating, motivated as it is by Putnam’s model-theoretic argument in (Putnam 1980), will not be so radical as to question that this is definitely finite, so it will not question that there is a definite matter of fact as to whether it is proof. Slightly more generally, the model-theoretic argument does not question that there is a perfectly definite fact as to whether any given sentence (of less than 10¹⁰⁰ symbols) is 10¹⁰⁰-provable, that is, provable (in the given proof procedure) with a proof of less than 10¹⁰⁰ component formulas each with less than 10¹⁰⁰ symbols. And isn’t this all the determinacy we have a right to be confident of?” (Field 1994a, p. 411).

If we take fictionism as an exemplar of nominalism here, the above position can be spelled out respectively as follows: finitude in general is just illusory, it is a fictional object at most, since the alleged arbitrarily long finite proofs that can never be realized in the physical world can only be fictional entities. Perhaps determinacy for all concrete proofs in the physical world and determinacy for the correctness of these concrete proofs will be sufficient to account for objectivity and determinacy of our mathematical practices and applications in the real world. Why would a fictionalist need determinacy for fictional entities in fictions?

Let us mention two problems with this version of fictionalism: the minor problem is that it presupposes the first version of anti-objectivism, that is to say, it presupposes that for undecidable sentences such as the consistency sentences to a fiction, the conjunction of the fiction with its consistency sentence is indeterminate, which as we argued before, is inconsistent.

The major problem with this approach is that it changed the subject. It is true that in normal practices, our notion of finitude is determinate and Putnam’s model-theoretic argument would not cast doubt on this part of determinacy. But it has nothing to do with our concerns. First, just as in normal mathematical practices, we are sure that the full interpretation for the second-order quantifier is what we need, this does not fix the determinacy of the notion of the quantifier in question. Likewise, merely pointing out that most proofs are humanly checkable does not fix the determinacy of the notion of finitude. Second, if a fictionalist acknowledges that there is a minimal notion of mathematical objectivity as we argued before, and if she tries to account for this in terms of the notion of proofs, then as we have repeated several times, she has to first determine the extension of the notion of finitude. But just as having an intuition of the consistency of PA is not enough for proving its consistency, having a finitely long proofs is not enough for determine the extension of the finitude itself.

7 Alternative 2

In Section 2, we argued that PFA can serve as a good reason for nominalists to deny determinacy in higher mathematics on the one hand, and reject one form of mathematical realism on the other. But this attitude is not shared by every nominalist. For instance, in Hellman’s view, the first desideratum of a philosophy of mathematics should “rest on the view that the queen merits the full respect of her handmaid: it is desirable to uphold the objectivity of as much mathematics as possible,”²³by “objectivity” he means “the determinateness as to truth value-true or false- and truth for the error-free portions; moreover, such truth is to be understood classically as implying independence of the particular mathematical investigators.” So it is desirable to consider whether modal nominalism advocated by Hellman succeeds in avoiding the challenge generated by PICA.

Modal nominalism combines two things from the philosophy of mathematics, one is Dedekind’s structuralism, and one is Putnam’s modal approach to mathematics. According to structuralism, there is nothing substantial about singular mathematical objects other than the structure in which they occur; according to modalism, mathematical sentences can be paraphrased as modal claims, through which mathematical structures can also be eliminated. For instance, an arithmetic sentence ϕ is true can be paraphrased as follows²⁴:

$$ \left( {\text{M}} \right) + \left( {^{\prime}X} \right)\left( {^{\prime}f} \right)\left( {{{\mathfrak{n}}{\text{PA}}}^{2} } \right)^{X} \left( {s/f} \right)\$ *\left[ {\left( {\& X} \right)\left( {\& f} \right)\left( {\left( {{{\mathfrak{n}}{\text{PA}}}^{2} } \right)^{X} \left( {s/f} \right)^{\prime \prime } \phi^{X} \left( {s/f} \right)} \right)} \right] $$

This paraphrase can easily be extended to third-order arithmetic (real analysis) and even to Zermelo’s hierarchical set theory. So if it is successful,²⁵ it can account for mathematical objectivity in the most general sense, especially, it can be used to fix the extension of finitude. But the problem with this approach is that it presupposes not only PA² in the above paraphrase for sentence ϕ, but it also presupposes that interpretation of the second-order quantifier in PA² is standard, which according to our observation 2, is question-begging.

Maybe with some clever tricks, modal nominalists can remove these presuppositions and get rid of PFA. For example, (i) she can replace the second-order quantification with plural quantification. (ii) she can dispense with the second-order quantification by making facts about the relations or properties fixed. We will argue below that (i) is still question-begging, (ii) is not available to nominalists.

(i) Instead of taking second-order quantification, a modal nominalist might resort to Boolos’s (1985, 1998) plural quantification. With this logic in hand, she can interpret the quantifier in (M) as a quantifier ranging over “some objects”, “some atoms” or “some points” rather than over a set, thereby avoiding inducing much unflattering ontology on the one hand; on the other hand, it may also help fix the extension of the quantifiers in question, for it is almost a tacit assumption that Boolos’s plural logic is immune to non-standard interpretations.

But this tacit assumption has recently been challenged by Florio and Linnebo (2016). According to their observation, besides Boolos’s canonical semantics, there is another non-standard Henkin semantics for plural logic. The difference between Henkin semantics and traditional Boolos’s semantics is that besides the canonical domain U, the former has an additional domain D, in which the superpluralities or plural properties can be accommodated, but cannot find their place in the latter semantics. If Linnebo and Florio’s observation is correct, “set-based and plurality-based semantics are on a par with respect to worries about indeterminacy” (Florio and Linnebo 2016, p. 567), then appealing to plural logic is of no help to modal nominalist to defeat PFA.²⁶

(ii) How about dispensing with second-order logic by making the relations and properties in (M) fixed? For instance, let us follow Berry (2018) and paraphrase ϕ as²⁷:

$$\begin{aligned}&\left( {{\mathrm{SM}}} \right) \left[ {{\mathrm{PA}} \left( {{\mathbf{N}}/{\mathrm{P}}} \right)\left( {s/R} \right)} \right]\\ &\quad \wedge\Box\left[ {\left( {{\mathrm{PA}} \left( {{\mathbf{N}}/{\mathrm{P}}} \right)\left( {s/R} \right)\rightarrow \phi \left( {{\mathbf{N}}/{\mathrm{P}}} \right)\left( {s/R} \right)} \right)} \right]\end{aligned}$$

In contrast with (M), (SM) paraphrases an arithmetic sentence ϕ as a statement that it is logically possible to have a mathematical object, i.e., an ω–sequence with a particular relation R acting as a successor, given that the facts about the number N and the successor s are fixed, and it is necessary that if an ω–sequence is formed, then ϕ is true of them. With this strategy, modality is retained, but second-order quantification is substituted with an arbitrary property and relation P and R, whose extensions are already determined by known mathematical facts.

But this strategy is helpless for modal nominalism for two reasons²⁸:first, taking the facts about numbers and successors for granted is ontologically committed, it is committed to the facts about numbers and the relations between numbers; second, it is a second-order approach in disguise, for making the facts about numbers and successors fixed needs the first-order induction scheme apply to arbitrary predicates including truth predicate T, but the arithmetical strength of an instance of first-order induction scheme with truth predicate perhaps is as strong as certain second-order arithmetic. So in the end we doubt that Berry’s simplified modal structuralism is a second-order approach, but if this is the case, it would not be able to avoid the challenge generated from PFA.

8 Alternative 3

Premise 3 of ICA asserts that if the second-order logic (or first-order set theory) is indeterminate, then our notion of finiteness is indeterminate. This accords with Bueno’s observation that “the notion of finiteness can only be properly characterized in pure second-order logic; finiteness is ultimately a second-order notion.” But we expect that this observation would be challenged by some nominalists. They might think that determinacy of finiteness can be fixed by means other than second-order logic, even granted that the extension of second-order quantifiers is indeterminate. This is the basic idea of the third option we are going to consider in this section.

In various places, Field proposes that if “the physical world is as we typically think it is, our physical beliefs are enough to determine the extension of ‘finite’” (Field 1994a, 416). To see how it works, consider a formula $\Phi$(Z) that is defined as follows:

Z is a set of events which (i) has an earliest member and a last member; and (ii) is such that any two of its members occur at least one second apart.

To make the formula $\Phi$(Z) satisfied, Field uses two cosmological assumptions:

• (A) Time is infinite in extent;

• (B) Time is Archimedean.

Assumption (A) means that there is no finite bound on the size of set Z, and assumption (B) means that only finite sets satisfy the events described by $\Phi$(Z). So if these two assumptions are correct, it is easy to infer that the notion of finitude has a determinate extension in the sense that the following would obtain (in some impure set theory)²⁹:

$$\left( {\mathrm{F}} \right)\;{{\mathrm{F}}}\;xA\left( x \right){\mathfrak{\equiv}}^{\ } \exists Y^{ } \exists Z^{\ } \exists f\left[ {\Phi \left( Z \right)\;\& x\;{\mathrm{is in}}\;Y\;{\mathrm{such that}}\;A\left( x \right)\& f\;{\mathrm{is}}\;1 - 1\;{\mathrm{function mapping }}Y\;{\mathrm{into}}\;Z} \right].$$

One might object that the concept of finitude in (F) still has non-standard models: there might be an infinite set satisfying $\Phi$(Z), which in turn satisfies the formulae FxA(x). However, according to assumption (B), since only finite sets are intended in the sense only these sets comply with the constraints laid down for the interpretations of the physical predicates, the infinite sets would be automatically excluded.

Needless to say, this proposal is ingenious, and if successful, would be good news for nominalism. On the one hand, it would fix the extension of finitude while maintaining the determinacy of elementary arithmetic, on the other hand, it could follow Putnam, rejecting the determinacy in higher mathematics. But the bad news is that this proposal still begs the question: it presupposes that the mathematical notion finitude in the physical world is determinate.³⁰ It is question-begging in another way: according to our previous description, Field seems to have smuggled the mathematical finitude into physical cases, he used a mathematical notion of finitude to understand the physical one, which in turn would fix the extension of mathematical finitude if it has the determinate extension as he hoped.

The second problem with this proposal is that it bases our mathematical determinacy on a physical hypothesis,³¹ it is at least weird for two reasons: first, it is hard to relate the physical ω–sequence with a mathematical notion; second, to assume that time has the Archimedean property is empirically controversial, it subjects our mathematical concept to physical fortune.

9 Conclusion

In this paper, we argue that nominalists should account for the objectivity of elementary arithmetic in the minimal sense through logical objectivity, for otherwise, nominalism would appear very absurd. Then we argue that accounting for the objectivity of logic and elementary arithmetic depends on offering a determinate notion of finitude because formulas and proofs can be arbitrarily long finite strings of symbols and natural numbers normally are considered to have finitely many predecessors. Subsequently we examine several strategies taken by the nominalist for addressing the problem, from straightly denying objectivity to offering a nominalist but determinate notion of finitude, we have observed that they are all untenable. We expect that the problem is insurmountable because the notion of finitude is actually a second-order notion, but if nominalists follow Putnam to hold that second-order logic is indeterminate, then they have to accept that finitude is indeterminate as well.

Maybe someday a nominalist comes up and put forwards that there is something deeply wrong with PFA, second-order logic is determinate, therefore the notion of finitude is determinate. But to achieve that, a lot of work still needs to be done!