The Caesar-problem Problem

Abstract

1. Introduction

Neologicism aims to provide a foundation for arithmetic on the basis of Hume’s Principle (HP).¹ HP states that the (cardinal) number of the concept |$ F $| is identical to the (cardinal) number of the concept |$ G $| just in case |$ F $| and |$ G $| are equinumerous, that is, the |$ F $|s and the |$ G $|s can be put into one-to-one correspondence — formally:

where |$ \# $| is the term-forming operator ‘the number of’, and |$ \approx $| is the second-orderly definable relation of equinumerosity. Frege Arithmetic is the theory of full second-order logic with HP as its sole non-logical axiom, and is sufficient to derive the axioms of second-order Peano Arithmetic (PA²) via definitions of the arithmetical primitives. This result is now known as Frege’s Theorem.

On the philosophical side, in particular the so-called Scottish Neologicism of Crispin Wright and Bob Hale traditionally relies on the following desiderata.² On the one hand, neologicists claim that HP can be known, or at least blamelessly believed, a priori as a result of its stipulation. Frege’s Theorem then shows that the basic truths of arithmetic are a priori: they can be known independently of experience. Thus, Neologicism vindicates an epistemological foundation for arithmetic. On the other hand, Number is a pure sortal concept:³ not only has it identity and application conditions to distinguish numbers among themselves and numbers from objects of other sorts, but also it characterizes the nature of the objects that fall under it. Thus, Neologicism vindicates a form of Platonism about numbers. This article will concern exclusively this particular brand of Neologicism, so we will drop the label ‘Scottish’ throughout.

Famously, sortality has been questioned because of the so-called Caesar Problem (CP), which amounts to HP’s incapability to establish the truth value of mixed identity statements such as ‘Caesar = |$ \#F $|’. Neologicists contend they can solve CP by determining that Caesar is not identical with any number. In this article, we will argue that neologicists face a Caesar-problem Problem (CPP): if neologicists solve CP by establishing that the number of |$ F $| is not identical with Caesar in the way they suggest, i.e., by appealing to principles concerning pure sortal concepts, apriority and sortality cannot be retained simultaneously.

Precisely, Neologicism faces a dilemma: if CP is not solved, it cannot be argued that Number is a pure sortal concept, thereby undermining neologicist Platonism about numbers. But if CP is indeed solved as neologicists propose, it is unclear that HP retains apriority, thus jeopardising the neologicist epistemological foundation of arithmetic. CP appears to be more pernicious than usually considered, since it undermines the neologicist project. In general, in light of CPP, a reconciliation of neologicist epistemology of arithmetic with a Platonist ontology seems highly unlikely.

We will proceed as follows. In §2, we will reconstruct the overall architecture of the neologicist program. In §3, we will introduce the Caesar Problem, assess its impact on Neologicism, and present the neologicist solution to it. In §4, we will pose the Caesar-problem Problem. We will examine various responses that neologicists might provide and argue that none of them is successful (§§5–7). We will conclude that the Caesar Problem uncovers a fatal tension in Neologicism (§8).

2. Apriority and Sortality

Neologicists claim that the following philosophical desiderata hold:⁴

• (Apriority)

    HP is a priori.

• (Sortality)

    HP introduces a pure sortal concept.

Let us start from HP’s apriority. In order to argue for it, neologicists openly rely upon what they call the traditional connection between implicit definitions and apriority. Neologicists hold that HP implicitly defines the cardinality operator |$ \# $|⁠, in the sense that it is “determinative of the concept [Number] it thereby serves to explain” [Hale and Wright, 2001a, p. 14].⁵ On these bases, neologicists argue that HP is also a priori:

if the stipulation has the effect that [‘|$ \# $|’] and hence [HP] are fully understood — […] then nothing will stand in the way of an intelligent disquotation: the knowledge that ‘[HP] is true’ will extend to knowledge that [HP]. [Hale and Wright, 2000, p. 126–127, modified]

Neologicists, therefore, appear to claim that HP is epistemically analytic in Boghossian’s sense [Boghossian, 1996] — grasping its meaning is sufficient to have a justified belief that the content it expresses is true.⁶

Furthermore, Neologicism is a project aiming to found PA² on solid epistemological grounds, as “[i]ts official aim is to demonstrate the possibility of a certain uniform mode of a priori knowledge of the basic laws of arithmetic” [Wright, 2016, p. 161].⁷ To this purpose, neologicists must also motivate that:

(1) if HP’s apriority holds, then PA² is a priori.

It is reasonable to think that, in order to motivate (1), neologicists have to hold something along the lines of a Transmission Principle:

• (TRM)

  If HP is a priori and second-order logic is apriority-preserving, then what follows from HP and second-order logic is a priori.

From TRM, it follows that, if neologicists motivate that HP is a priori, and indeed second-order logic is apriority-preserving, PA² will be a priori too, since PA² follows from HP — and second-order logic.⁸ Although neologicists do not state TRM explicitly, we argue that they have to hold it.⁹ We will delve further into the importance of TRM for the apriority of PA² in Neologicism in §4.

Finally, let us come to sortality. A concept is sortal if and only if it is provided with both an identity criterion, stating necessary and sufficient conditions to distinguish among objects falling under it, and an application criterion, stating necessary and sufficient conditions to distinguish the objects falling under it from objects that do not. Furthermore, a pure sortal characterizes the nature of the objects that fall under it.¹⁰ So specified, the pure sortality of Number yields a form of Platonism, according to which not only does HP deliver the existence of numbers, but the latter are essentially the objects HP dictates [Hale and Wright, 2001a, pp. 9–11].

In what follows, we will see how the neologicist philosophical framework is threatened by the Caesar Problem.

3. The Caesar Problem

The Caesar Problem (CP) originates with Frege¹¹ and concerns HP’s incapability to determine the truth value of mixed identity statements: though HP delivers all the identity and distinctness truths concerning numbers expressed by formulæ of the form |$ \#F=\#G $| and |$ \#F\neq\#G $| respectively, it does not determine the truth value of identity statements such as the JC-sentence

(JC) Julius Caesar = |$ \#F $|⁠.

In general, HP stays silent as to the truth value of statements of the form |$ \#F=q $|⁠, if |$ q $| is not given as a number-term governed by HP.

Admittedly, the success of the neologicist program hinges on the neologicists’ ability to solve CP [Hale and Wright, 2001a, pp. 25–26]. Neologicism is indeed a “logicist-cum-platonist” project [Hale and Wright, 2001a, p. 223]. Its logicist side consists in proving the axioms of PA² from an implicit definition, i.e., HP, and (second-order) logic. As emphasised by Wright, however,

[We] will be able to know the truth-conditions — and indeed the truth value — of [numerical identities on the basis of HP] provided, and so far as I can see, only provided [we have] a general solution to the Caesar problem: some principled account of the distinction between numbers and things of other kinds which allows [us] to determine that no concrete object can be a number. [Wright, 1998, p. 360]

If CP remains unsolved, HP cannot be deemed a good (implicit) definition of the concept Number, as not every identity statement containing number-terms has a determinate truth value. Therefore, the inability to solve CP would get in the way of the neologicist arguments for HP’s apriority and, ultimately, the apriority of PA².

At the same time, the Platonist side of Neologicism is established by a proof from HP that numbers do exist.¹² However, CP is the “ineluctable nemesis” [Hale and Wright, 2001a, p. 249] of mathematical Platonism; as recently specified by Wright:

[t]o solve the Caesar problem is not to show that terms introduced by good abstractions refer. But it is — or so I have suggested — to meet a necessary condition for showing that. [Wright, 2020, p. 306]

Neologicists have contended repeatedly that they can provide a solution to CP [Hale and Wright, 2001b; Wright, 1983; 2020]. In fact, neologicists claim that, even though HP does not explicitly provide application conditions for the concept Number, such conditions can be derived indirectly from the identity conditions that HP does provide [Wright, 1983, p. 114]. To this end, neologicists rely on the Principle of Sortal Inclusion (SIP), a “substantial metaphysical claim” that neologicists deem as independently plausible [Hale and Wright, 2001b, p. 370]:

(SIP) A sort of objects |$ F $| is included within a sort |$ G $| only if the content of a suitable range of identity statements about |$ G $|s — those linking terms denoting |$ G $|s that are candidates to be |$ F $|s — is the same as that of statements asserting satisfaction of the criterion of identity for the corresponding |$ F $|s. [Hale, 1994, p. 131, reprint p. 198]¹³

On the basis of SIP, if Caesar were in fact a number, facts about his identity would be determined by the same facts determining identity among cardinal numbers, i.e., equinumerosity between concepts. Since the identity of persons is not determined by relations among concepts, Caesar cannot be a number, and, so, the JC-sentence is false.¹⁴

In general, it may be argued that, by the identity conditions delivered by HP and the application conditions obtained from HP and SIP, CP is solved. Still, as Hale and Wright [2001b] point out, sortal inclusion does not in general imply satisfaction of SIP. This is because of two issues. The first one concerns different sortals under which the same kind of objects may fall.¹⁵ The second, which is more general, is that, if there is no way to distinguish between different sortals, then there is no way to distinguish between the objects falling under them.¹⁶ A mere difference in truth conditions between relevant identity statements concerning objects as put forward by SIP would not account for what Hale and Wright call the grundgedanke: “because they are canonically decided by references to inequivalent considerations, statements of numerical and of, say, personal identity must be reckoned to concern different categories of objects” [Hale and Wright, 2001b, p. 385]. This would bring about a characterization of sortal overlap provided not by the coincidence in truth conditions of statements about different sorts of objects, but, crucially, by the “restrictions on the allowable sources of their differences” [Hale and Wright, 2001b, p. 385]. In a nutshell, what is needed is an identity (or equivalence) criterion for sortals, which would in turn shed light on the distinction among the objects falling under them.

As mentioned in §2 above, neologicists argue that pure sortals are concepts that characterize the intrinsic nature of the objects that fall under them. The overall issue now becomes: “what counts as establishing — or wherein consists — the truth of an identity statement depends upon what (pure) sort of thing the claim of identity concerns” [Hale and Wright, 2001b, p. 388].

This leads to the second piece of metaphysical machinery neologicists put into place, in order to distinguish between different (pure) sortals. Since different sortals may share their identity conditions, but differ with respect to applications, it is just natural to move the investigation on a solution to CP up a level of philosophical complexity: by investigating maximally extensive sortals, i.e., categories, namely sortals all of whose sub-sortals share the same identity conditions, and such that any object to which that category does not apply must not fall under sortals not associated with those identity conditions. Therefore, “granting that Number and Person are each pure sortals, numbers essentially belong to whatever category subsumes Number and persons essentially belong to whatever category subsumes Person” [Hale and Wright, 2001b, p. 389]. This amounts to providing an identity criterion for sortals in terms of categories.

In this respect, categories provide a solution to CP by establishing that, in general, sortals not satisfying the same identity conditions as Number are subsumed under different categories:

(SI^#) Some |$ F $|s are |$ G $|s only if |$ F $| and |$ G $| are each sub-sortals of one and the same category. [Hale and Wright, 2001b, p. 396]¹⁷

By SI^#, neologicists have a way of properly distinguishing between different (pure) sortals, and then between objects falling under them on the basis of SIP. This strategy, neologicists claim, would finally solve CP, since from HP, SIP, and SI^#, it would follow that Caesar is not identical with any cardinals.^18,19

4. The Caesar-problem Problem

We will now argue that Neologicism faces what we call the Caesar-problem Problem (CPP): if neologicists solve CP as they propose, then apriority and sortality cannot be retained simultaneously. This threatens to undermine Neologicism as a foundational project for arithmetic based on HP.

Foremost, we want to emphasize that not only does HP fail to settle ‘mixed’ identity statements, but more importantly, as we will argue in what follows, according to CPP, HP cannot settle these statements if the two neologicist desiderata must be retained.

Furthermore, neologicists stress that their solution has the following features. First, the solution to CP must follow from HP — along with further principles.²⁰ Neologicists indeed argue that a criterion of application for Number can be indirectly obtained from HP — see §3 above. Wright claims, even more strikingly, that “HP is effectively all we have to go on — that there is no relevant additional, independent, and independently supported principle that we have overlooked” [Wright, 2020, p. 306, emphasis original].²¹ This is best understood as a claim that HP is the only principle about numbers that is needed to solve CP, even though other principles, e.g., SIP or SI^#, are relied upon.

Secondly, CP must be solved in the negative, i.e., by establishing that Caesar |$ \neq\#F $|⁠. This is achieved by HP and other principles such as SIP or SI^#.

However, if HP along with further principles entails that Caesar |$ \neq\ \#F $| for any |$ F $|⁠, and what follows from HP is a priori (modulo TRM), then it must be also a priori that Caesar is not a number.

Nonetheless, it does seem a posteriori that Caesar is not a number, since it is arguably a posteriori that Caesar ever existed.²² As pointed out in [Boccuni and Woods, 2020, p. 413] for example,

solving [CP] requires giving up on HP[’s] privileged epistemic status. Why? Because in showing that HP determines that 2 is not a Roman, we would learn something about the nature of both numbers and Roman generals. If HP entails that |$ \#F $|⁠, for any |$ F $|⁠, is not Caesar, it also entails Caesar is not a number. But HP is supposed to be some form of logical or conceptual truth and these are not supposed to inform us about the underlying nature of the objects they applied to. Especially not Romans.

In general, HP is supposed to provide an implicit definition of Number. Neologicists argue that a priori knowledge of truths concerning the objects falling under Number (Peano Axioms) can be attained on the basis of HP. However, HP should not grant us knowledge of truths concerning other sorts of individuals — for example Caesar.²³ In particular, HP should not put one in a position to know a priori that no Roman conqueror is a number.

If all the above holds, and the negation of the JC-sentence follows from HP and second-order logic plus further principles, and is indeed a posteriori, then either HP is not a priori or second-order logic is not apriority-preserving, modulo TRM.

Neologicists would therefore be left with a dilemma. If it must follow from HP and second-order logic (along with further principles) that Caesar |$ \neq\#F $|⁠, then (also) a posteriori truths follow from HP and second-order logic, and therefore, modulo TRM, either HP is not a priori or second-order logic is not apriority-preserving. Thus, in order to solve CP, retain (pure) sortality, and argue for HP as a good implicit definition of Number, neologicists must relinquish either HP’s apriority or the apriority-preserving status of second-order inferences. Either way, neologicist epistemology of arithmetic would not be in good standing: the alleged apriority of PA² could not be justified by HP, and its derivability from HP by second-order logical inferences. Vice versa, to retain the apriority of HP (and the apriority transmission of second-order inferences), neologicists must renounce their solution to CP, and so (pure) sortality. This would have the consequence that HP is not a good implicit definition of the (pure) sortal concept Number, jeopardising neologicist Platonism. The Caesar-problem Problem (CPP), as we call it, consists in the impossibility of retaining (pure) sortality and apriority at the same time, given CP and the neologicist solution to it:²⁴

(CPP) If CP is not solved, then Number is not a (pure) sortal — and HP is not a good (implicit) definition of Number. If CP is solved (as neologicists do), then either HP is not a priori or second-order logic is not apriority-preserving.

We envisage three potential objections to CPP:

• (i)

  rejecting TRM, as TRM is crucial for deeming a priori the negation of the JC-sentence, and, consequently, jeopardising the apriority either of HP or of second-order inferences;

• (ii)

  the aposteriority of the negation of the JC-sentence depends upon the principles other than HP (and second-order logic) from which it follows: if the latter are a posteriori, the apriority of HP and second-order inferences can still be secured;

• (iii)

  in order to solve CP, it is not necessary for HP, second-order logic, and further principles to establish that Caesar |$ \neq\ \#F $|⁠, but it is sufficient that they simply provide the means to distinguish between numbers and non-numbers.

We will address objections (ii) and (iii) in §§5 and 6 respectively, and show that (ii) does not solve CPP and (iii) requires the assumption of sortality. But for the remainder of this section, we invite the reader to bear with us, while we investigate why neologicists cannot reject TRM.

Rejecting TRM implies that some truths, such as the negation of the JC-sentence, following from HP and second-order logic (with or without additional principles), are a posteriori, despite HP being a priori and second-order logic preserving apriority. We contend, however, that TRM cannot be rejected altogether, otherwise neologicists could not argue for the apriority of PA² on the basis of its derivation from HP via second-order logic. In fact, rejecting TRM would require a criterion to distinguish, independently of HP, between a priori and a posteriori identities of the form |$ \#F\neq q $|⁠, possibly following from HP and second-order logic, whether with further premisses or not.

Neologicists might retort that TRM requires qualification, since the required notion of “follow from” must have both a lower bound, such that the a priori status of HP is transmitted at least to Peano axioms,²⁵ and an upper bound, so that not every truth, e.g., the negation of the JC-sentence, that can be obtained by inferences involving HP is deemed a priori.²⁶ In what follows, we will provide a few characterizations, which seem the most natural. At the same time, we reckon the burden of providing a precise qualification lies with neologicists.

Suppose neologicists qualify TRM so that, of all truths following from HP and second-order logic, only numerical truths are a priori. Obviously, this restriction is too weak, since it fails to rule out a posteriori truths such as ‘The number of the planets = 9’.²⁷ Consequently, this approach does not secure the epistemic status of PA².

Analogously, a qualified TRM cannot carve out, of all truths following from HP and second-order logic, only arithmetical truths as a priori. This strategy requires a distinction between arithmetical and non-arithmetical truths, so that, e.g., ‘The number of the planets = 9’ is deemed non-arithmetical as expected. Still, the only principle concerning arithmetical identities in this framework is HP — neither SIP nor |$ N^{d} $| do the job. Nonetheless, in this scenario, distinguishing between arithmetical and non-arithmetical identities is precisely what HP alone cannot do, since it cannot tell numbers from non-numbers.

Furthermore, neologicists may reformulate TRM by restricting apriority to the truths following solely from HP and second-order logic. In this respect, the negation of the JC-sentence does not follow solely from HP and second-order logic; therefore it does not follow that it is a priori. At the same time, PA² does follow from HP and second-order logic alone, securing its a priori status.

We believe the advantages of this strategy are only seeming. First, neologicists must justify such a restriction. The burden of proof rests with them, and if they do not provide any, the above restriction is ad hoc. Secondly, suppose that an abstraction principle for finite cardinals is added to HP and second-order logic — for instance, so-called Finite Hume,²⁸ stating that if |$ F,G $| are finite concepts, then the number of |$ F $| is identical with the number of |$ G $| if and only if |$ F,G $| are equinumerous. HP and Finite Hume agree on all ascriptions of cardinals to finite concepts, thereby plausibly individuating the same abstracts, i.e., natural numbers. Also, assume that the identity of Finite Hume’s number 2 with HP’s number 2 follows from HP, Finite Hume, and second-order logic, but not solely from HP and second-order logic. Neologicists would be compelled to argue that such an identity cannot be deemed a priori, since it does not follow solely from (either) HP (or Finite Hume) and second-order logic. Restricting TRM exclusively to the consequences of HP and second-order logic not only appears ad hoc, but also implies puzzling consequences for neologicists.²⁹

Finally, let us consider a further qualification of TRM, restricting apriority only to the truths following from HP and second-order logic. In this scenario, following from HP and second-order logic is a necessary condition for a truth to be a priori. Consequently, its apriority cannot be established by its derivability from HP and second-order logic. In particular, the apriority of PA² cannot be established by the derivation of Frege’s Theorem.³⁰ Furthermore, given any (alleged) a priori truth, such as, e.g., ‘All bachelors are unmarried males’, restricting TRM as above would imply that ‘All bachelors are unmarried males’ should follow from HP and second-order logic, which is hardly the case.

It appears that the only principle that guarantees the apriority of PA² on the basis of its derivability from HP and second-order logic is TRM without qualifications. The bad news is that CPP applies.

5. Compatibilism

We will refer to the following reply as compatibilism. According to this view, Caesar |$ \neq\#F $| is a posteriori, because HP entails it only in conjunction with additional premisses, e.g., SIP or SI^#. The latter are not themselves a priori, so that (pure) sortality and apriority can be retained. Let us see how the compatibilist strategy could be implemented by neologicists.

At first glance, one might expect that the only premise needed, along with HP, for Caesar |$ \neq\#F $| to follow is SIP — or SI^#.³¹ Neologicists provide no justification either for SIP or SI^#, besides stating that they are independently plausible and so up for grabs for a philosophical solution to CP.

Let us assume, for the sake of argument, that SIP (or SI^#) is not a priori. Even so, by HP and SIP (or SI^#), the conclusion that no human is a number can be drawn, but the conclusion that Caesar is no number cannot. Indeed, HP, SIP, and SI^# might entail that Number and Person do not overlap. However, they are not sufficient to conclude that any particular individual, e.g., Caesar, is a person, and, therefore, neither are those sufficient to conclude that Caesar is not a number. To this extent, if the fact that Caesar is a person is not assumed or somehow proved, we cannot conclude that, to paraphrase SIP, “the content of a suitable range of identity statements” on numbers is not the same as that of statements “asserting satisfaction of the criterion of identity” for Caesar. Thus, HP cannot establish Caesar |$ \neq\#F $| even when paired with SIP and SI^#. Consequently, further assumptions are required to establish the (pure) sortality of Number. These assumptions seem to concern specifically what Caesar is.

Upon reflection, it is not surprising that we still need truths about Caesar in order to conclude something about Caesar, namely that Caesar is not a number.³² In this case, truths about Caesar are singularly necessary (and jointly sufficient with HP, SIP, and SI^#) for Caesar |$ \neq\#F $| to follow. Thus, even if SIP and SI^# were indeed a priori, similar to HP, it might still be a posteriori that Caesar |$ \neq\#F $| — even though it could then be a priori that no person is a number [Hale and Wright, 2001b, pp. 380–383].

We argue, however, that the compatibilist response is not successful. If claims about Caesar are indeed necessary to solve CP and establish the (pure) sortality of Number, then HP’s status as a good implicit definition of Number as a sortal and numbers having the nature that HP ascribes them depend upon truths about individuals of different sorts. But if so, it seems that the argument from CPP still holds. In fact, if the (pure) sortality of Number depends upon truths about, e.g., humans, possibly among other kinds of objects, then once again, the apriority of the truths about numbers, and of HP, is in jeopardy. If truths such as “Caesar is human” are necessary in order to establish that Caesar is not identical with any number, then those very truths are crucial to establish that Number is a sortal concept, and therefore that numbers are essentially the objects they are (on the basis of Number’s purity). That would solve CP. But it would also imply that the truths of arithmetic (and numbers being essentially what they are) would depend upon, e.g., Caesar’s humanity. If the latter is not a priori, then HP (and PA²) would depend on a posteriori truths, and their apriority would be in jeopardy.

6. Compromising Neologicism

Since the compatibilist strategy does not work, neologicists might explore an alternative path. In particular, they might lower their expectations regarding what HP should establish concerning mixed identity statements, aiming to avoid CPP, while still supporting Neologicism as a foundational view. Let us call this strategy compromising Neologicism.

Precisely, it might be argued that CPP sets the bar too high for Neologicism:³³ for (pure) sortality to be vindicated in a way that is still compatible with apriority, it is not necessary for HP and further principles to establish that Caesar |$ \neq\#F $|⁠, but it is sufficient that they simply provide the means to distinguish between numbers and non-numbers.³⁴ It is enough that HP and further principles entail, for any individual |$ q $|⁠, that if |$ q $| is not such-and-such, then it is not a number, even if HP and further principles do not entail that a particular individual, e.g., Caesar, is not a number.³⁵ We take Hale’s and Wright’s response to CP in terms of SIP and SI^# to be a paradigmatic case of the compromising strategy.

Neologicists might claim that this result is already secured by SIP, particularly when applied specifically to Number — i.e., principle |$ N^{d} $| in fn.14 above — and SI^#. If that is the case and SIP and SI^# are themselves a priori, then no further CPP arises.

However, SIP must be restricted to pure sortals.³⁶ Unless Number is indeed a pure sortal, it cannot be ‘fed into’ SIP, so as to obtain |$ N^{d} $| as a special case. Therefore, in order to argue that we are asking Neologicism for too much, neologicists must assume or establish HP, SIP, SI^#, and the pure sortality of Number. Obviously, HP, SIP, and SI^# are assumed, but if the pure sortality of Number is to be established, Neologicism faces CPP again. In fact, sortality does not hold, in light of CP. Since sortality, let alone pure sortality, is not established unless CP is somehow solved, instead of arguing in favour of (pure) sortality by solving CP, neologicists would have to assume that the (pure) sortality of Number is built into the stipulation of HP,³⁷ and then employ it to solve CP. This move, we reckon, would result in turning Neologicism into a primarily metaphysical rather than epistemological project. While this outcome is not necessarily untenable, it is clearly different from the one Neologicism openly aims for. More generally, the metaphysical assumptions neologicists make along the way in order to respond to CPP can be seen as the theoretical cost of their response to the Caesar Problem.

In addition, further concerns may arise from this strategy. Assuming sortality, one still faces the challenge of showing that |$ q $| is not such and such, in order to establish that it is not identical with any number — for any ‘|$ q $|’ that is not of the appropriate form. The question then arises: how exactly is one supposed to specify ‘such and such’, and how is one supposed to show that it is the case that |$ q $| is not such and such? Probably the answer to the first question is: ‘such and such’ is the concept Number. If that is conceded, how do we address the second question, if we have no information at all on what |$ q $| is? Not having such information is the whole point, after all. Probably also that |$ q $| is not a number has to be assumed. If all the above is the case, surely ‘|$ q\neq\#F $|’ will come out true and a priori — by TRM and assuming that all assumptions involved are a priori. Nevertheless, first, it is far from clear that an individual |$ q $| not being such and such is a priori knowable. Secondly, even if that were the case, we cannot help to think that this strategy would be rather ad hoc.

7. Divide and Conquer

In the previous sections, we posed the Caesar-problem Problem. If HP’s apriority were retained as a desideratum of the neologicist program, then neologicists could not solve CP as they envisage and, thus, the (pure) sortality of Number could not be established. Vice versa, if (pure) sortality were to be retained by solving CP in the negative, HP would not satisfy apriority. We also investigated possible replies, and concluded that, ultimately, none of them is satisfactory.

It seems, therefore, that neologicists must give up either on apriority or on (pure) sortality, in order to retain the other. We will call this strategy divide and conquer: neologicists can either renounce HP’s apriority and develop Neologicism as a purely metaphysical project; or they can recant the (pure) sortality of Number and develop their project as a purely epistemological one. In this section, we will investigate the consequences of each strategy for Neologicism.

If neologicists choose to retain (pure) sortality, they must solve CP in the negative, by establishing that Caesar |$ \neq\#F $|⁠. Neologicists argue that they achieve that by a general principle concerning the overlapping of sortals, i.e., SIP. Alternatively, HP itself can be understood as a metaphysical claim along the lines of [Donaldson, 2017] and [Rosen and Yablo, 2020]. This grants a solution to CP:

if the objects “introduced” by [Hume’s Principle] must be abstract objects [in the sense that by their very nature, they are the values of the cardinality function], … this means that ordinary objects like Julius Caesar are excluded, since it is perfectly clear that such things are not abstract objects in this sense. [Rosen and Yablo, 2020, p. 130]

Unlike the neologicist reply to CP,³⁸ this solution commits to what we may call ‘inflationism’ about abstraction, since it ascribes a metaphysical content to HP that goes beyond the one that is expressed by the material biconditional.

However, as CPP shows, neologicists would not be able to establish HP as a foundation of arithmetic on secure epistemological grounds. As neologicists stressed repeatedly, “abstractionism is first and foremost an epistemological project; its official aim is to demonstrate the possibility of a certain uniform mode of a priori knowledge of the basic laws of arithmetic” [Wright, 2016, p. 162]. Therefore, they cannot renounce HP’s apriority without renouncing Neologicism itself.³⁹

If neologicists opt to retain apriority instead, they must provide a conception of numbers that does not require (pure) sortality. A possibility is a form of ‘deflationism’ about abstraction similar to [Antonelli, 2010], [Boccuni and Woods, 2020], and, more recently, [Schindler, 2021]. According to deflationism, numbers lack inherent nature; by contrast, any thing can serve as the semantic value of a numerical term introduced by HP, provided that it belongs to a domain that contains at least denumerably many objects. This leads to a dissolution of CP:

The answer to the question, “What prevents the number of the planets from being equal to Julius Caesar?” is: nothing. … Nothing much is to be made of this, for the corresponding abstraction principle, HP, is silent about it. [Antonelli, 2010, p. 202]

If there is no CP, there is no CPP either. However, deflationism diverges further than inflationism from Hale’s and Wright’s neologicist conception of numbers as objects.

Neologicism positions itself between inflationism and deflationism. Neologicists claim that, on the one hand, HP introduces a pure sortal concept, while, on the other, it is also a priori. CPP shows that neologicists cannot retain both of these desiderata simultaneously; so they may be forced to choose between inflationism and deflationism. Either way, CP undermines crucial aspects of the neologicist project.⁴⁰

8. Closing Remarks

Neologicism aims to provide a consistent foundation for arithmetic on the basis of Hume’s Principle. Philosophically, (Scottish) Neologicism consists of two main claims (among others), according to which HP is a priori, and introduces the pure sortal concept Number. Because of apriority, neologicists argue for the apriority of second-order Peano arithmetic PA².

In this paper, we argued that Neologicism, as a foundational view consisting of (pure) sortality and apriority, is undermined by the so-called Caesar Problem.

As a matter of fact, neologicists claim that they can solve CP in the negative. Still, we argued, their solution to CP leads to what we called the Caesar-problem Problem, according to which the two main philosophical claims of Neologicism cannot be retained simultaneously.

We argued that, in light of CPP, neologicists face a dilemma: either they renounce HP’s apriority and, therefore, the apriority of PA², or they relinquish their solution to CP and, therefore, the (pure) sortality of Number. Either way, Neologicism would be undermined in its crucial philosophical features as a foundational view on arithmetic.

Acknowledgements

We are grateful to two anonymous reviewers, whose comments significantly improved the quality of our paper. Previous versions of this work were presented at: the CRESA Work in Progress Seminar of San Raffaele University; the Science & More Talks in Turin; the Synergia Seminars of the University of Urbino; the IUSS Philosophy Seminars in Pavia; the 14th Conference of the Italian Society for Analytic Philosophy; and the Philosophy Department at Kansas State University. We are grateful to the audiences for valuable feedback and comments. We are particularly grateful to Andrea Sereni, Ludovica Conti, Bahram Assadian, and Fiona Doherty for commenting on previous drafts of this paper.

Footnotes

References