FOR BETTER AND FOR WORSE. ABSTRACTIONISM, GOOD COMPANY, AND PLURALISM

Abstract

A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, we start by discussing the Good Company Problem for neo-logicists recently raised by Paolo Mancosu (2016), concerning the existence of rival abstractive definitions of cardinal number which are nonetheless equally able to reconstruct Peano Arithmetic. We survey Mancosu’s envisaged possible replies to this predicament, and suggest as a further path the adoption of some form of mathematical pluralism concerning abstraction principles. We then explore three possible ways of substantiating such pluralism—Conceptual Pluralism, Domain Pluralism, Pluralism about Criteria—showing how each of them can be related to analogous proposals in the philosophy of logic. We conclude by considering advantages, concerns, and theoretical ramifications for these varieties of mathematical pluralism.

1 Introduction

Neo-logicism (cf. ([22]; [56])) provides one of the major philosophical reconstructions of arithmetic. The project consists in the attempt at resuscitating a form of Fregean logicism, by abandoning the ill-fated Basic Law V, and defending Hume’s Principle (HP) as the correct definition of the concept of cardinal number (hence also of natural number, i.e. finite cardinal). Frege’s Theorem (see e.g. [25]) establishes that HP, added as the single non-logical axiom to a system of full impredicative second-order logic, delivers a consistent theory—Frege Arithmetic—which allows for the derivation of the axioms of second-order Dedekind-Peano Arithmetic (PA $_2$ ). If HP delivers a priori knowledge of the concept of cardinal number, it is possible to argue for the viability of an a priori route to arithmetical knowledge, and to prove the axioms of arithmetic to be analytic in Frege’s sense (derivable from logic and definitions alone).

This project is hostage to the solution of a vast array of concerns. One of the most pressing is to explain how HP can deliver a tenable definition of cardinal number even if it shares the logical form of many other incompatible principles. HP, i.e. the principle stating that the number of F’s is equal to the number of G’s if and only if the F’s and the G’s are equinumerous (namely can be put into one-one correspondence), i.e. formally

(HP) $$ \begin{align} \forall F \forall G \ (\# (F) = \# (G) \leftrightarrow (F \approx G)) \end{align} $$

is an instance of so-called abstraction principles, i.e. principles stating that the abstracts of two entities are identical if those entities stand into a suitable equivalence relation, formally

(ABS) $$ \begin{align} \forall \alpha \forall \beta \ (\Sigma (\alpha) = \Sigma (\beta) \leftrightarrow (\alpha \equiv \beta)). \end{align} $$

It is well-known that HP shares the same logical form with a number of principles which cannot be individually or jointly accepted. This jeopardizes the idea that principles of the form of ABS can by this very fact alone deliver acceptable definitions. While the inconsistent BLV bears its untenability on its sleeves, several other principles can be devised, which are consistent but are either inconsistent with HP, or else are pairwise inconsistent with other consistent principles. The Bad Company Objection consists in asking why HP should be accepted as a good definition when so many bad companions are available.

Recently, Mancosu [35] has raised a cognate concern. Backed by a thorough investigation of other definitions by abstraction of the concept of natural number that may be available, and may have been available at Frege’s time, he has shown that competing definitions can all be able to deliver a reconstruction of arithmetic even while offering diverging characterizations of the notion of cardinal number. The Good Company Problem is the concern that the status of HP as the correct definition of cardinal number is put in jeopardy by the existence of rival definitions which are both consistent and equiconsitent with it (and, also, which meet all the requirements for good abstractions that the Bad Company debate has identified).

Our aim will be to review the possible replies to the Good Company Problem considered by Mancosu, and to suggest that other options can be explored. More specifically, we will advance ways of dealing with the objection by adopting a pluralist stance with regards to abstraction principles (which can be extended to mathematical definitions more generally). Pluralism has been variously defended concerning logic, and forms of mathematical pluralism have been advanced in the past. However, a detailed exploration of pluralist views in mathematics, of their motivation and their import, seems still to be lacking. ¹ We believe that approaching the Good Company Problem offers an ideal wedge into this debate, and lends itself to the investigation of different forms of abstractionist pluralism which can then be independently studied.

In order to pursue this aim, in Section 2 we first (Section 2.1) rehearse the Bad Company Problem, and then (Section 2.2) consider Mancosu’s formulation of the Good Company Problem. In Section 3 we then consider Mancosu’s taxonomy of possible solutions to the Good Company Problem, and explore three different forms of abstractionist pluralism, also by comparison with analogous debates on logical and mathematical pluralism: Conceptual Pluralism (Section 3.1), Domain Pluralism (Section 3.2), and Pluralism about Criteria (Section 3.3). In the last section (Section 4), we discuss possible concerns and ramifications. It is our contention not only that such pluralist views concerning abstraction and mathematics are worth exploring, but also that they give a fruitful perspective on how apparently unrelated debates in the philosophy of logic and in the philosophy of mathematics can inform each other and provide unexpected theoretical vantage points.

2 Abstractionism, from Bad to Good to Worse

Let us then first rehearse the Bad Company Problem (Section 2.1). We will then present (Section 2.2) the cognate Good Company Problem. This lays the ground for the subsequent exploration of pluralist solutions to the latter and the frameworks within which they can be advanced. ²

2.1 The Bad Company Problem

The Bad Company Problem ³ consists in distinguishing “good” from “bad” abstractions. Neo-logicists seek to provide a foundation for arithmetic based on HP, but many principles of the same form are unacceptable, being either inconsistent or jointly unsatisfiable with HP. An example of inconsistent abstraction is Frege’s Basic Law V (BLV), which states that the extension of F and the extension of G are the same if and only if every F is a G and vice versa:

(BLV) $$ \begin{align} \forall F \forall G \ (ext (F) = ext (G) \leftrightarrow \forall x \ (F(x) \leftrightarrow G(x))). \end{align} $$

An example of consistent abstraction that is nevertheless incompatible with HP is the Nuisance Principle (NP; cf. [57], pp. 289–90), which states that two concepts have the same “nuisance” just in case they differ finitely—viz., iff the concept $F \land \neg G$ is finite and the concept $\neg F \land G$ is also finite:

(NP) $$ \begin{align} \forall F \forall G \ (\nu (F) = \nu (G) \leftrightarrow (\mathrm{Fin}(F \land \neg G) \land \mathrm{Fin}(\neg F \land G))). \end{align} $$

NP has only finite models; therefore, it is jointly unsatisfiable with HP, which requires an infinite domain. ⁴

Over the last forty years a plethora of increasingly strong criteria for acceptable abstraction has been suggested. These include, among others:

Consistency: an abstraction principle AP is acceptable only if it is consistent.

Semantic Field-Conservativeness: an abstraction principle AP is acceptable only if it is (semantically Field-)conservative, viz. if for any theory T to which AP can be consistently added, and for any sentence $\phi $ of the language of T whose quantifiers have been restricted to the ontology of the original theory, $\{T + AP\} \models \phi $ only if $T \models \phi $ .

Strong Stability: an abstraction principle AP is acceptable only if it is strongly stable, viz. only if there is a cardinal k such that AP is $\gamma $ -satisfiable, namely satisfiable in a model of cardinality $\gamma $ , if and only if $\gamma \geq \kappa $ . ⁵

However, there is as yet no consensus that the list is complete and effective.

Things get even worse if one desires to extend neo-logicism beyond arithmetic (e.g. to real analysis and set theory). The abstraction principles that would be most suited for this task are often unacceptable. An example is George Boolos’ New V, which assigns extensions to concepts in the same way as BLV unless those concepts are “(Too) Big,” viz. they are of the same size as the universe; in this case, New V assigns a dummy extension to all the (too) Big concepts:

(NewV) $$ \begin{align} \forall F \forall G \ (ext (F) = ext (G) \leftrightarrow [(\mathrm{Big} (F) \land \mathrm{Big} (G)) \lor (\forall x \ (F(x) \leftrightarrow G(x)))]). \end{align} $$

New V is sufficient to recover a remarkable portion of ZF (without Infinity and Powerset), but it is not strongly stable ([49], pp. 304–9). More in general, Uzquiano [51] shows that no abstraction principle that complies with Strong Stability can recover ZFC. As noted by Studd ([50], pp. 595–6) neo-logicists seem to face a dilemma: either the criteria that they introduce are permissive enough to include promising cases of abstraction such as New V, but, then, they do not avoid Bad Company; or those criteria are restrictive enough to avoid Bad Company, but, then, they exclude New V and other promising principles.

2.2 From Bad to Good

New V is what Weir ([53], p. 17) calls a Distraction Principle. A Distraction is a principle stating that two concepts F and G have the same extension* if and only if either they are both “Bad”, where “Bad” is a second-order property for which equinumerosity is a congruence ⁶ (e.g. “Big”), or every F is a G and vice versa:

(D) $$ \begin{align} \forall F \forall G \ (ext^* (F) = ext^* (G) \leftrightarrow [(\mathrm{Bad} (F) \land \mathrm{Bad} (G)) \lor (\forall x \ (F(x) \leftrightarrow G(x)))]). \end{align} $$

There are many pairs of Distractions that are individually consistent but jointly incompatible, or individually incompatible with HP.

Let us note that a Distraction can also be formulated for HP. This would be an abstraction principle stating that two concepts have the same cardinal number* if and only if either F and G are both “Large,” where “Large” is a second-level property for which equinumerosity is a congruence, or F and G are equinumerous:

(HPD) $$ \begin{align} \forall F \forall G \ (\#^* (F) = \#^* (G) \leftrightarrow [(\mathrm{Large} (F) \land \mathrm{Large} (G)) \lor (\mathrm{Fin} (F) \land \mathrm{Fin} (G) \land F \approx G)]. \end{align} $$

Each such principle assigns the same cardinal number* to each Large concept, and cardinal numbers to non-Large concepts in the same way as HP.

Also the principle presented by Mancosu [35], p. 165, as capturing Peano’s num function can be subsumed under this general form. As Mancosu reminds us:

Peano connected the definition of ‘having the same number of objects’ for classes to that of one to one correspondence by giving first the explicit definition of a function num(a) which yields values in the natural numbers (including zero) or a single value ‘infinity’ denoted by the symbol $\infty $ . The definition num(a), for a class, is given as follows:

• $num (a) = 0$ iff a is the empty class;

• $num (a) = n$ iff a is not the empty class and for any element x in a, $num (a - \{ x \}) = n - 1$ ;

• $num (a) = \infty $ iff for every n in the natural numbers (including zero), $n \not = num(a)$ .

The principle capturing this, which Mancosu calls Peano’s Principle (PP), assigns one and the same cardinal number to every infinite concept and cardinal numbers to finite concepts in the same way as HP:

(PP) $$ \begin{align} \forall F \forall G \ (\#^{P} (F) = \#^{P} (G) \leftrightarrow [(\mathrm{Inf} (F) \land \mathrm{Inf} (G)) \lor (\mathrm{Fin} (F) \land \mathrm{Fin} (G) \land F \approx G)], \end{align} $$

where ‘Inf’ abbreviates a second-order predicate which is true of all and only the infinite concepts, and ‘Fin’ abbreviates a second-order predicate which is true of all and only the finite ones.

Another Distraction for HP is Boolos’ Principle (BP; [35], pp. 171–2), which assigns (i) the same cardinal number a to each concept F which is both infinite and co-infinite (viz. such that the extension of $\neg F$ is also infinite); (ii) a different cardinal number b to each concept G which is infinite but co-finite (viz. such that the extension of $\neg G$ is finite); and (iii) cardinal numbers to finite concepts in the same way as HP:

(BP) $$ \begin{align} &\forall F \forall G \ (\#^{B} (F) = \#^{B} (G) \leftrightarrow ([\mathrm{Inf} (F) \land \mathrm{Inf} (G) \land \neg \mathrm{Cof} (F) \land \neg \mathrm{Cof} (G)] \lor \nonumber\\ &\quad [\mathrm{Inf} (F) \land \mathrm{Inf} (G) \land \mathrm{Cof} (F) \land \mathrm{Cof} (G)] \lor [(\mathrm{Fin} (F) \land \mathrm{Fin} (G) \land F \approx G)]), \end{align} $$

where ‘Cof’ abbreviates a second-order formula which is true of all and only the co-finite concepts. Finally, Mancosu generalizes BP to each finite number n. ⁷

According to Mancosu ([35], p. 173), one major motivation for the adoption of BP is that it, as opposed to HP and PP, partly reflects those conceptions of infinity which respect a part-whole principle, according to which sets which are proper subsets of other sets should be counted as having smaller size that the latter. As Mancosu states:

Notice that PP defeats all infinite instances of part–whole, for if A and B are infinite, with A strictly included in B, then $\clubsuit $ (A) = $\clubsuit $ (B). By assigning the same number to all infinite sets, PP cannot respect part-whole relations between infinite concepts. That’s not the case with BP, however. Indeed, the even numbers are classed in the equivalence class of infinite sets with infinite complement. But $N - \{1\}$ which contains all the even numbers is classed in a different class of equivalence. But of course, there are plenty of instances in which sets and proper subsets are assigned the same object by BP, for instance the even numbers and the multiples of four.

Mancosu calls PP, BP, and all the (infinitely many) instances of BP-n the Good Company of HP. All these principles are similar to HP, but differ from it on the assignment of cardinal numbers to infinite concepts: while HP assigns cardinal numbers to those concepts according to their (equi)numerosity, PP assigns the same cardinal number to each infinite concept, BP assigns to them one or the other of two infinite cardinals, and each instance of BP-n assigns to them one of three infinite cardinal numbers.

Are PP, BP, and all the BP-ns also a case of bad company? Mancosu’s answer is “Yes and No” ([35], pp. 184–5). On the one hand, these principles are all strongly stable (and, therefore, semantically Field-conservative and consistent), and jointly satisfiable with HP. Therefore, PP, BP, and all the BP-ns form a good rather than a bad company. Moreover, each one of these principles is sufficient to derive, in fully impredicative second-order logic and via the usual definitions, the Axioms of PA $_2$ . There are, therefore, (infinitely) many Frege’s Theorems, each one starting from a different good companion. On the other hand, these principles cannot all be jointly true, or analytic, or definitional, or constitutive of the same concept of cardinal number. For instance, according to HP, the set of the naturals, $\mathbb {N}$ , and the set of the reals, $\mathbb {R}$ , do not have the same (‘HP’-)cardinal number, since they are not equinumerous; while according to PP, $\mathbb {N}$ and $\mathbb {R}$ do have the same (‘PP’-)cardinal number, since both are infinite. ⁸

The Good Company Problem consists in asking which one among the good companions is the correct one, viz. which one underlies the concept of cardinal number (assuming there is only one such concept ⁹ ). However, each good companion complies with the neo-logicist criteria. According to Mancosu ([35], p. 186), this casts doubt upon the claim that any of those principles can be analytic of the ordinary notion of number:

we seem to be stuck in the following dilemma. On the one hand, HP and every one of the good companions can be argued to be a priori, analytic, and therefore true, using the line of argument used by the neo-logicists for Hume’s Principle. Indeed, Hume’s Principle shares many properties with the other abstraction principles just listed. They are all consistent, mutually compatible, and in good standing (hence the good company). From each one of them one can derive a suitable version of the axioms for PA $_2$ […] Any argument in favor of the analyticity (or status as conceptual truths) of HP would seem to also apply to any other of them. ¹⁰

3 Varieties of Pluralist Abstractionism

Whether the Good Company Problem counts as a proper objection to neo-logicism will depend on which form of the latter position is defended; Mancosu ([35], p. 188) envisions three possible responses the neo-logicist might embrace when faced with this predicament:

1. a. a conservative neo-logicist will argue for HP being the only correct abstraction principle;

2. b. a moderate neo-logicist might turn to a weaker and finite version of HP—for example, a version of which only applies to finite concepts (i.e. HP Finite see below);

3. c. a liberal neo-logicist, finally, might claim that any abstraction principle sufficient to derive the axioms of $\mbox {PA}_{2}$ is good enough.

In what follows, we argue that these do not exhaust the whole taxonomy of possible reactions. We will advance a further approach inspired by pluralism about logic; in a nutshell, we will suggest a viable abstractionist view which endorses a host of rival abstraction principles as equally correct. Let us quickly go through the three possible responses above and see how the latter can be added.

A conservative neo-logicist needs to provide arguments or criteria that will isolate HP amid its good companions; leveraging on usual properties—consistency, Field-conservativity, irenicity, stability, etc.—won’t do, as all of these are displayed by good companions as well. Hence, according to Mancosu ([35], p. 190), the conservative is left with two options: she could either (i) “give an a priori argument” or (ii) “come up with new criteria” to single out HP as correct vis-à-vis its good companions. As regards (ii), he mentions, by way of example, the idea that “HP enjoys a naturalness and simplicity that is manifestly not shared by the other principles. All but one of them are far too ad hoc and disjunctive to merit any claim to be analytic of our pre-theoretic concept of cardinality.” Mancosu, however, doubts that either of the two strategies is viable. The reason for this is, at least in part, that the charge of non-naturaleness and non-simplicity against HP’s good companions seems to him to presuppose the existence of some “pre-theoretic concept of cardinality” that HP would capture better. But the Good Company objection may exactly suggest that we have no such pre-theoretic concept to capture; more specifically, it raises doubts that any pre-theoretic conception of cardinality extends to the infinite.

A way of avoiding this predicament is to fall back on a moderate neo-logicism, i.e. to the claim that we should retreat to a version of HP that does not account for infinite cardinalities. One such version has been made popular by Heck [24] as HP Finite (HPF). HPF shares the same form of HP but is restricted to finite concepts:

(HPF) $$ \begin{align} \forall F \forall G \ ((\mathrm{Fin}(F) \lor \mathrm{Fin}(G)) \rightarrow [\#(F) = \# (G) \leftrightarrow F \approx G]). \end{align} $$

As Heck ([24], p. 594) emphasizes, HPF makes “no further claim about the conditions under which infinite concepts have the same number. (For all that HPF says, all infinite concepts could have the same number, so long as no finite concept also has that number.).” Since, as a matter of fact, HPF is silent on the identity and distinctness of infinite cardinalities, it doesn’t strictly speaking count among the good companions.

As a third option, the liberal neo-logicist submits that, as long as a principle is sufficient to derive the arithmetical axioms of PA $_2$ , that principle is good enough. On the liberal perspective, “there was never a claim to the uniqueness of HP in this particular context” ([35], p. 188), and the Good Company Problem seems to be dissolved, since at least for practical purposes the disagreement of all such principles for what concerns infinite cardinalities becomes irrelevant (but cf. Section 4.1 for more on this).

A fourth option, which is not accounted for by Mancosu’s taxonomy, consists in adopting some form of pluralism and accepting that all or at least more than one among the disputed abstraction principles are equally correct or legitimate (despite their disagreement on infinite cardinalities). ¹¹ In this section, we will examine three forms of pluralism about abstraction closely connected to akin ways of substantiating forms of logical and mathematical pluralism:

1. 1. Conceptual pluralism (Section 3.1)—inspired by specific forms of logical pluralism (LP);

2. 2. Domain pluralism (Section 3.2)—built upon a particular version of mathematical pluralism;

3. 3. Pluralism about Criteria (Section 3.3)—a new kind of pluralism about abstractions, which in turn informs some novel takes on LP.

We will now consider these three options in turn.

3.1 Conceptual pluralism

Let’s start with conceptual pluralism, a first version of which we base on the form of logical pluralism advanced by Beall and Restall (e.g. [4]; [5]; [6]). The authors start building their case for LP by reflecting on a schematic principle which is supposed to capture the core content of the notion of logical consequence, and that they label “Generalized Tarski Thesis”:

(GTT) An argument is valid $_{\textit {x}}$ if and only if in every case $_{\textit {x}}$ in which the premises are true, so is the conclusion.

We can thus obtain instances of GTT “by a specification of the cases $_{\textit {x}}$ in GTT, and a specification of the relation is true in a case” ([6], p. 35). Such instances will be said to be admissible if the judgments they provide on logical consequence satisfy three key features associated with the notion—viz. necessity, normativity, and formality (for more details on these features, see the discussion below in Section 3.3). According to Beall and Restall, each so defined admissible instance of GTT provides us with an admissible logic; then, LP is the thesis that there are at least two different admissible instances of GTT.

Different specifications can of course diverge on particular inferences; consider, for instance, the argument from “a is red” to “a is colored” ([4]). On the one hand, if we consider cases to be possible worlds the argument is valid, for there are no possible worlds in which a is red but it is not colored. On the other, if cases are Tarskian models the argument is not valid, for it is of the form $Ra \vDash Ca$ and there are many substitutions of the non-logical vocabulary (e.g. ‘R’ = ‘is a rabbit’, ‘C’ = ‘is cyan’) that might make the premise true and the conclusion false.

In an analogous fashion, the conceptual pluralist can argue that the core content of the notion of cardinal number can similarly be conveyed by a (schematic) principle of equality of numbers:

(EN) The number of F is identical with the number of G if and only if F and G are same-sized $_x$ ,

and that different specifications of same-sized $_x$ in EN correspond to different precisifications of the notion of cardinal number. Mancosu ([35], sect. 4.3) discusses several historical examples which can be helpful here. For instance, if we follow the definition of Peano’s num function presented above (see Section 2.2), then two finite collections will be assigned the same number if (as was also the case for Schröder) they are paired one-to-one, while all infinite collections will be assigned the same number, i.e. $\infty $ . On the other hand, if we follow Cantor and Frege (and, indirectly, Hume) in believing that equinumerosity is what characterizes identity of number, then any two collections, be they finite or infinite, will be assigned the same number if and only if they can be paired one-to-one. The analogy with the form of logical pluralism just reviewed should then be clear: just as different precisifications of cases $_{\textit {x}}$ in GTT can diverge on the assignment of validity to particular inferences, so different precisifications of same-sized $_x$ in EN can result in different assignments of numbers to the same concepts. Following Peano, for instance, both $\mathbb {N}$ and $\mathbb {R}$ will be assigned $\infty $ as their number. Following Frege and Cantor, on the other hand, since $\mathbb {N}$ and $\mathbb {R}$ cannot be paired one-to-one, they will be assigned different (infinite) numbers. In both cases any concept F will be assigned the same cardinal number as any other concept G equinumerous with F (including F itself), although on different grounds when it comes to infinite concepts: in Peano’s precisification, because F and G are both infinite and thus assigned the very same number, while in Frege’s and Cantor’s precisification because F and G are equinumerous.

A slightly different version of conceptual pluralism can be built upon some of the ideas presented by Shapiro [48]; in his book, Shapiro highlights several ways in which one could be a pluralist about logic. In Ch. 2, he shows how LP could stem from the term ‘logical consequence’ being polysemous ([48], sect. 2.2). Indeed, there are many definitions of the notion; for example, some of them are modal – namely, ‘ $\Phi $ is a logical consequence of $\Gamma $ just in case $\Phi $ holds in every possible world in which every member of $\Gamma $ holds’ – others are linguistic or semantic – namely, ‘ $\Phi $ is a logical consequence of $\Gamma $ just in case $\Phi $ holds in every interpretation of the language in which every member of $\Gamma $ holds’. Some of these definitions will be reducible to one another, but others won’t; hence, a form of pluralism might be defended on the basis of the fact that “there are different, mutually incompatible articulations or sharpenings of the intuitive notion or notions of logical consequence” ([48], p. 25).

In a parallel manner, a conceptual pluralist might argue that the locution ‘same-sized’ is polysemous—that is, there are many, connected, partially overlapping, and equally legitimate sharpenings of the intuitive notion of being same-sized. More precisely, the pluralist might submit that there is no fact of the matter as to which one between Peano’s and Frege-Cantor’s definitions is the correct one: the relation between the (informal) notion of being same-sized and the equivalence relation embedded in the right-hand side of the corresponding abstraction is one-many. ¹²

3.2 Domain pluralism

The second form of pluralism that we will consider is modeled upon some forms of mathematical pluralism which have already been advanced in the literature. Common to these is the idea that different, competing, and incompatible mathematical theories can both be true ([3], p. 380). For example, mathematical pluralists can submit that standard and non-standard arithmetics are both true, that Eucledian geometry and non-Eucledian geometries are both true, and that ZFC + CH (viz., ZFC set theory with the Continuous Hypothesis) and ZFC + $\neg $ CH (viz., ZFC set theory with the negation of the Continuous Hypothesis) are both true.

Recent years have seen different implementations of this brand of pluralism. Let us mention three of them. Balaguer [2] originally defended a form of mathematical pluralism that he calls “platitudinous platonism”, viz. the view that any mathematical objects of consistent mathematical theories do exist, and, therefore, that every consistent mathematical theory describes, and is true of, some domain of mathematical entities. Friend [20] defends another form of mathematical pluralism building on a principle of tolerance between rival mathematical theories. Finally, Priest [37] submits that different mathematical practices, like different games, have different rules, including different inferential rules.

Only Balaguer’s version of mathematical pluralism will be relevant here. Balaguer argues for two claims:

1. I. Full-blooded Platonism (or Platitudinous Platonism): any mathematical objects that could exist actually do exist, and every consistent mathematical theory describes, and is true of, some domain of mathematical entities;

2. II. Non-uniqueness: mathematical terms refer simultaneously to different abstract mathematical objects belonging to different universes; even if our mathematical theories describe collections of mathematical objects, they do not describe a unique collection of such objects.

Let us call the conjunction of I and II Domain Mathematical Pluralism. For example, ZFC + CH and ZFC + $\neg $ CH are both consistent; by Full-blooded Platonism, they therefore describe, and are true of, distinct set-theoretic universes ([2], p. 58). The question is now which one of these two universes is described by the language of ZFC (which is consistent both with CH and with $\neg $ CH). Non-uniqueness says that there is no unique set-theoretic universe which is described by ZFC, or by our mathematical theories in general. By contrast, set-theoretic terms “divide their reference” among different sets belonging to different universes. For example, the term ‘ $\aleph _1$ ’ refers to some cardinals of which CH is true, ¹³ and it refers to some other cardinals (in other universes) of which CH is false. ¹⁴

Let’s now recall the different abstraction principles considered in the context of the Good Company Problem. It is natural to think that HP and PP introduce the same domain of finite cardinal numbers (viz., that the HP-numeral ‘1’ has the same referent as the PP-numeral ‘1’), and that those principles differ only on the cardinal numbers of infinite concepts (for example, PP assigns a unique cardinal number to all infinite concepts, a cardinal which could be distinct from any cardinal number introduced by HP, even from those cardinal numbers which HP assigns to infinite concepts on the grounds of their being equinumerous with concepts of different infinite cardinalities). Mancosu points out that this intuition is at odds with the Principle of Uniform Identity (PUI) formulated by Cook and Ebert ([13], p. 129), which they find “rather intuitive” and implicit in the neo-logicist stance. PUI states, roughly, that if the abstract $_1$ of some concept F (namely, the abstract assigned to F by an abstraction principle AP $_1$ ) is identical with the abstract $_2$ of F (namely, the abstract assigned to that same concept F by a different abstraction principle AP $_2$ ), then for any G, abstract $_1$ (G) = abstract $_2$ (G). Suppose that HP and PP assign the same cardinal numbers to finite concepts; then PUI implies that HP and PP must also assign the same cardinal numbers to infinite ones.

Mancosu notices in passing that this conflict between PUI, on the one hand, and the intuition that HP and PP both introduce the same finite cardinals, on the other, may be more pressing for the liberal than for the conservative and moderate neo-logicist ([35], p. 192). The liberal submits that any principle that is sufficient for a derivation of the Peano Axioms is good for the purposes of neo-logicism. The underlying thought is that, no matter which one among the good companions we begin with, the arithmetical truths that we can derive from them would be about the same numbers, i.e. the finite cardinals. For a simple and vivid representation of an example of this, consider the following diagram:

However, since HP and PP do not introduce the same infinite cardinals, PUI entails that HP and PP introduce different abstract objects, including different natural numbers. Again, consider the following diagram by way of example:

As Mancosu concludes, “either we give up the intuition that HP and PP ‘share’ even a single abstract or the principle of uniform identity leads to a contradiction when applied to simple cases (such as HP and PP)” ([35], p. 196).

Domain pluralism would allow one to explain away our intuition that HP and PP introduce the same natural numbers without giving up on PUI. Indeed, it can be argued that each good abstraction introduces its own domain of cardinal numbers, so that there are no two distinct principles, allegedly both introducing the same finite cardinals and disagreeing on infinite ones (as depicted by the second diagram above); hence PUI would not be violated. Sure, there would be (possibly infinitely) many numbers one—the HP-number one, the PP-number one, etc.—, (possibly infinitely) many numbers two, and so on; however, ordinary arithmetical terms would divide their reference among all of them. For example, let F and G be two infinite but not equinumerous concepts, and let a precisification of a numeral n be an assigment of one of the eligible referents of n as its unique referent. The sentence ‘the number of F = the number of G’ is then true under some precisifications of ‘the number of F’ and ‘the number of G’ (e.g. in which PP-numbers are taken as unique referents), but not under others (e.g. in which HP-numbers are taken as unique referents). By contrast, arithmetical sentences are super-true, viz. true under any precisification, and so it is harmless, for practical purposes, not to distinguish between different precisifications ([1], p. 80). On this perspective, one can thus dispense with any concern raised by the idea that HP and PP introduce the same cardinal numbers, but only when restricted to finite concepts. We have now a plurality of distinct domains, inhabited by distinct objects; it just happens that our ordinary arithmetical terms divide their reference plurally, capturing in each such universe those finite numbers which satisfy the Peano Axioms. ¹⁵

3.3 Pluralism about criteria

In order to introduce a third and final version of abstractionist pluralism, let us recall that the Good Company Problem emerges from there being many abstraction principles similar to HP that (1) differ from it on the assignment of cardinal numbers to infinite concepts, and (2) all satisfy the neo-logicist criteria for acceptability. From (2) it follows that neo-logicists cannot distinguish HP from its good companions by means of their standard criteria.

One may argue that, nonetheless, they do not necessarily need to do so. After all, if one takes seriously Wright ([57], p. 303)’s idea that “success for an abstraction is success for an explanation” of a concept, and not for any empirically adequate “description” of such concept, then one is in principle at liberty to claim that HP, PP, and the like each successfully explain slightly different concepts of cardinal number (cf. [35], pp. 183–91), although based on different criteria of legitimacy for abstraction principles. We find examples of such different criteria in the forms of abstractionism discussed so far. The conservative selects HP because it best satisfies a set of criteria which includes simplicity and naturalness. The moderate submits that HPF is the principle that best fits ordinary arithmetical reasoning. The liberal takes the derivability of PA $_2$ axioms to be the only necessary condition an abstraction principle has to ensure.

Now, we have already seen two forms of abstractionist pluralism. According to conceptual pluralists more than one abstraction principle is legitimate in that more than one corresponds to an acceptable precisification of the concept of number. For domain pluralists, any abstraction which can consistently characterize a domain of mathematical objects is legitimate. Hence, all these proposals select one or more suitable abstractions on the basis of some single legitimacy criterion or unique set of such criteria. A third route seems thus to be open: to claim for a form of abstractionist pluralism according to which different abstraction principles introduce rival and still equally acceptable concepts (like the concept of cardinal number) depending on which criteria one chooses to adopt, where a plurality of different criteria (or sets of them) can count as equally legitimate even if they in the end yield different concepts. Actually, on the face of it, this should appear as quite a natural option for one wishing to defend some robust form of pluralism in any given domain.

More precisely, since the conservative cannot appeal to the standard neo-logicist criteria against bad companions—given that both PP and the BP-ns all comply with those same criteria—she might contend that even though each good companion is consistent, conservative, stable, etc., HP has extra virtues which are not shared by any of its rivals. In his review of [35], Hale ([21], p. 165) writes ¹⁶ for example that:

$[ $ Some $]$ might argue that while “good” alternatives to Hume are not incorrect, HP is to be preferred over weaker principles such as Finite Hume because it settles cardinality questions quite generally, whereas they answer them only partially, and preferred over BP and its countable infinity of refinements on grounds of naturalness and simplicity, while the choice of any one of BP’s progeny would be arbitrary and ad hoc. Further, [some] can and should separate the question of providing a general analysis of cardinality from providing a foundation for elementary arithmetic. They can agree that full Hume is not needed for the latter and that a much weaker principle such as Finite Hume suffices, and they can, in my view, concede that full Hume is not implicit in “ordinary arithmetic thought,” but argue that the fact that full Hume is stronger than is required for that purpose is no objection to its furnishing a good explication of cardinality in general, and does not prevent it from providing a foundation for arithmetic via its entailment of Finite Hume.

Therefore, if we take this to describe a conservative strategy (see fn. 16), then according to Hale the superiority of an abstraction principle vis-à-vis its good companions might be defended by appeal to the following criterion:

Conservative Success (CS): An abstraction principle AP is successful iff:

1. (i) AP is good (i.e. complies with standard criteria against Bad Company);

2. (ii) AP is the simplest, most natural, and least ad hoc explication of the concept of cardinal number.

As seen, Hale contends that only HP satisfies CS:

Let’s turn to the moderate position; the moderate seemingly adopts the following criterion:

Moderate Success (MS): An abstraction principle AP is successful iff:

1. (i) AP is good (i.e. complies with standard criteria against Bad Company) and

2. (ii) AP is implicit in ordinary arithmetical thought.

As seen, only HPF is arguably selected by MS:

Finally, the liberal contends that any principle which is sufficient for a derivation of PA $_2$ “will do” for the purposes of neo-logicism:

Liberal Success (LS): An abstraction principle AP is successful iff:

1. (i) AP is good (i.e. complies with standard criteria against Bad Company) and

2. (ii) AP is sufficient to derive the axioms of $\text {PA}_{2}$ .

Each one of the principles listed by Mancosu satisfies LS: ¹⁷

Therefore, each one of the three brands of neo-logicism mentioned by Mancosu, i.e. the conservative, the moderate, and the liberal, employs different criteria or sets of criteria, i.e. CS, MS, and LS respectively. ¹⁸ These three brands of neo-logicism need not disagree on every particular case: for example, both the conservative and the liberal claim that HP is correct (even though the liberal doesn’t agree that HP is the only correct one). At the same time, they can also agree on which abstraction principle(s) satisfy given (sets of) criteria. For example, the moderate might agree with the conservative that HP is simpler and more natural than HPF, in some relevant sense, and the conservative might agree with the moderate that HPF best fits the available data. Both the conservative and the moderate can acknowledge, in addition, that both HP and PP are sufficient to derive the Peano Axioms. However, they disagree on which criterion should be given priority.

A pluralism about criteria for abstractions requires a further step away from traditional neo-logicism. According to such pluralist the conservative, the moderate, and the liberal might all be right, viz. their criteria might all be legitimate, even if they diverge on which abstraction principles count as acceptable:

As noted before, the same principle can be selected by two different criteria. For example, HP complies both with the set comprising naturalness and simplicity, and with deductive strength. The conservative and the liberal indeed disagree on which criteria are legitimate and on how many principles might possibly be correct, but they both agree on the acceptability of HP. At the same time, HPF complies both with deductive strength and with adequacy to the data. Again, also the moderate and the liberal do disagree on the criteria, but not on HPF in particular.

A pluralist about criteria would be able to defend a form of pluralism about abstraction even if each legitimate criterion were shown to be satisfied by exactly one principle (provided that different criteria select different principles). Suppose for example that someone argued that naturalness and simplicity on the one side, and adequacy to the data on the other side, are the only (sets of) legitimate criteria. In this case, each criterion would be satisfied by exactly one abstraction principle, namely by HP and HPF respectively. However, the proponent of this view would still be a pluralist, since she would still argue that both HP and HPF are acceptable, even though based on different criteria. ¹⁹

Note that the same strategy might be adopted in order to defend a—so far neglected—form of logical pluralism. Surprisingly enough, in fact, even pluralism in logic has solely been advanced as the view that more than one logic complies either with (i) a given legitimacy criterion or (ii) a certain set of criteria.

An instance of (i) is Shapiro’s eclective perspective—according to which logic is relative to a particular mathematical structure ([48], ch. 3). In his book, Shapiro describes some mathematical theories that essentially employ intuitionistic logic (IL)—i.e. Heyting arithmetic augmented with each instance of an intuitionistic version of Church’s thesis, intuitionistic analysis, and smooth infinitesimal analysis—and argues that, given their fruitfulness, it would be dogmatic to abandon them just because their logic is not classical. Since there are also good mathematical theories that employ classical logic (CL), and one day there might be ones that invoke a paraconsistent (PL), relevant (RL), or quantum logic (QL), pluralism is the best option for an adequate account of validity. So, according to Shapiro, as long as a logic is in some sense indispensable to a legitimate mathematical theory, that logic fulfills the only relevant criterion.

An example of (ii) is instead provided by Beall and Restall. As mentioned, instances of GTT are considered admissible if the judgments they provide on logical consequence are necessary, normative, and formal; these criteria are unpacked by Beall and Restall ([6], pp. 14–23) as follows:

• * necessity: the premises of a valid argument necessitate its conclusion;

• * normativity: one would be wrong in accepting the premises of a valid argument while rejecting its conclusion;

• * formality: a logic is ([33]): I-formal if it provides constitutive norms for thought as such; II-formal if it is indifferent to the identities of objects; III-formal if it abstracts from the semantic content of thought.

On a natural reading of Beall and Restall’s position, logics have to satisfy all of these conditions in order to be legitimate; therefore, the collection of acceptable logics is selected on the grounds of a unique set of criteria.

According to Shapiro, the logics that comply with his criterion are the classical, the intuitionistic and (possibly) the paraconsistent, the relevant and the quantum ones:

The logics that fulfill Beall and Restall’s threefold set of criteria are the classical, the intuitionistic and the relevant ones:

Schematically, then, the ways in which multiple logics have been deemed jointly legitimate by logical pluralists so far fall under either of the following schema:

Here we want to briefly highlight the possibility of a different perspective, which consists in allowing for a plurality of legitimacy criteria or sets of legitimacy criteria themselves. In this novel framework, the shift from the acceptance of multiple logics to the acceptance of multiple (sets of) criteria would allow one to defend a form of pluralism even if there was, for each criterion, only one logic satisfying that criterion, with distinct logics satisfying each criterion:

or, for each set of criteria, only one logic satisfying that set of criteria, with distinct logics satisfying each set of criteria:

Before proceeding, let us dispel a possible worry. Criteria $\textit {c}_{1},\ \textit {c}_{2},\ \ldots \ \textit {c}_{\textit {n}}$ do not tell us whether $\textit {L}_{1},\ \textit {L}_{2}$ , etc. each are or are not logics; rather, they tell us whether something already acknowledged to be a logic by prior independent conditions is legitimate or not, i.e. whether it also happens to be acceptable or desirable under a particular selection of additional criteria. In other words, they are not criteria for the logicality (vs. the non-logicality) of a given theory, but criteria for assessing the legitimacy or non-legitimacy of a logical theory; as such, their application is conditionalized on $\textit {L}_{1}$ already fulfilling the former. Hence, before feeding a possible candidate $\textit {L}_{1}$ to the criterial pluralist’s model, that candidate has to be recognized as satisfying basic requirements for being a logical theory (however difficult it can be to pin down exactly what these are), exactly as something has to priorly and independently qualify as a good (vs. bad) abstraction principle before it can be evaluated as legitimate or non-legitimate according to a selection of legitimacy criteria (see fn. 18 above). To give some examples, a theory which validates tonk but abandons $\wedge $ -introduction will promptly be recognized by the model as illegitimate, but would still count as a logical theory. Clearly, logical theories need not be deductive. A system in which the premises of a good argument do not necessitate its conclusion would not be validated by criteria selecting only legitimate deductive systems, but would still count as a logical theory (unless one objects that necessitation of conclusions from premises is a criteria for logicality itself, and not just of acceptability).

If we focus on the very last diagram, we can stress the difference between this position and Beall and Restall’s logical pluralism. One might think that the latter should be counted as an instance of pluralism about criteria, as Beall and Restall do make use of a set of requirements in their evaluation—viz., necessity, normativity, and formality. However this is not the case since, in their view, such requirements are arguably meant to be regarded as individually necessary (and jointly sufficient) ²⁰ ; that is, in order to be legitimate, a candidate logic has to provide judgements on validity which satisfy all three conditions. The situation is different for the pluralist about criteria. She does not define a set of conditions and then checks which logics (if any) comply with all of them; rather, she accepts a series of criteria as individually sufficient, and thus might endorse different logical theories on separate grounds.

This comes with a couple of intriguing advantages. Firstly, as stated, a pluralist about criteria might be able to defend a form of LP even if not all—or, for that matter, none—of the proposed candidates happened to satisfy all of the proposed conditions. Indeed, say this brand of pluralist too endorses normativity, necessity and formality as the relevant criteria; she might sanction a logic $\textit {L}_{1}$ in virtue of its providing us with a necessary (but not formal) concept of logical consequence, and $\textit {L}_{2}$ for its behaving the other way around. She’d thus be left with multiple acceptable logics; by contrast, Beall and Restall’s pluralism would warrant neither logic. Of course, these two perspectives would collapse into one if the chosen criteria co-implied each others, as a theory would satisfy one condition if and only if it satisfied the other, hence traditional LP and pluralism about criteria would in the end output the same set of acceptable logics. This hints at the second advantage: a pluralist about criteria, as opposed to a traditional pluralist, might be able to endorse somehow incompatible criteria. Given a collection of prima facie desirable conditions, we cannot always put them together—either because they ask for a priori conflicting properties, or because, de facto, no candidate is able to jointly satisfy all of the requirements. In both cases, a pluralist about criteria might retain a form of LP, while a traditional pluralist would be left with an empty set of legitimate theories.

4 Remarks, objections, and ramifications

Let us take stock. Starting from a consideration of the Good Company Problem as raised by Mancosu, we have proposed ways of dealing with the ensuing predicament by advancing three different versions of mathematical pluralism based on abstraction principles. All three positions seem prima facie viable options to those keen on adapting pluralist perspectives to the philosophy of mathematics. However, as they stand they leave many issues open and possible questions and objections unanswered. Let us examine some of these in what follows.

4.1 Liberal vs. Pluralist

The pluralist and the liberal clearly have more in common than what they each share with the conservative and the moderate. So much so, that a legitimate concern is whether the pluralist response indeed differs at all from the liberal, and if so to what extent. ²¹ Mancosu ([35], p. 188) describes the liberal neo-logicist reaction to the Good Company Problem as follows:

The liberal neo-logicist might claim: who cares if ordinary arithmetic can be recaptured from many different principles? There was never a claim to the uniqueness of HP in this particular context. What counts is that each such principle is of the appropriate form (abstraction), and each abstraction in good standing that allows us to derive $\text {PA}_{2}$ will do, thereby yielding a priori knowledge of $\text {PA}_{2}$ . This is obviously MacBride’s ‘modal’ position.

As Mancosu earlier recalls (cf. [35], pp. 181–3), in responding to a criticism moved by Black [8] (and, on similar grounds, by Heck [24]) MacBride (correctly, according to us) stresses the “modal” and “reconstructive” character of the neo-logicist epistemology, as opposed to an hermeneutic one. Whereas the latter would have it that the role of HP, and the corresponding derivation of Peano Arithmetic, is that of “‘making explicit the ideas which already underlay our use of the natural number system’ ([8], p. 236)” ([32], p. 99) (or, to quote a similar response to Heck, of showing that “what we had in mind all along when we reasoned arithmetically is a priori”; cf. [31], p. 157), a modal and reconstructive interpretation has it that:

neo-Fregean epistemology (so envisaged) offers an account of how it is possible to acquire knowledge of the fundamental laws of arithmetic. […] But it is not thereby committed […] to saying this route is ‘the’ only one available. It is consistent with our coming to recognize arithmetical truths in one way that we could have, and perhaps do, come to recognize their truths by different means. ²²

One may suspect that this final statement encodes nothing but a pluralist view, so that what we have been exploring here provides in the end no additional content over the liberal view. Against this, we want to make a twofold rejoinder. First, that on a natural understanding, the liberal (modal, reconstructive) view is indeed incompatible with a pluralist stance. Second, that even if one opposes such a natural reading, the liberal (modal, reconstructive) position as sketched by MacBride and advanced by Mancosu is too underspecific to sustain a pluralist view, so that at least our proposal can be seen as a way of establishing how such a would-be liberal pluralist view could be cashed-out in details.

As far as the first claim is concerned, consider the following. According to Mancosu’s liberal (see quotation above), “each abstraction in good standing that allows us to derive $\text {PA}_{2}$ will do.” How should we interpret this “will do”? This, in the reconstructive spirit we agreed upon, need not mirror in any way the actual process by which we come to learn such axioms and statements, nor need be the only possible reconstruction available: indeed, any of the principles which the Good Company objection trades upon provide the liberal with equally accessible paths to arithmetical knowledge. ²³ Arguably, each such principle will do in the sense that it will deliver—in the modal fashion described—a possible way of our coming to know arithmetical axioms and statements a priori.

Bear in mind, however, that no such principle introduces by itself the concept of natural number. They all introduce, albeit differently, some concept of cardinal number, which, equally differently, is also meant to account for infinite cardinalities. The fact that Peano Arithmetic can be interpreted by any such principle need not distract from the fact that all such concepts of cardinal numbers are different, and treat infinity differently. Now this may seem to cause no obstacle to the liberal, and in a sense it is so. Indeed, the liberal somehow mimics instrumentalist views in the philosophy of science. She accepts that more than one rival theory can deliver an appropriate description of a target class of phenomena (in the analogy, arithmetic) while suspending judgment both on the truth of any of these theories and on what they entail for what the world is actually like beyond that class of phenomena (in the analogy, the theory of transfinite cardinals). Somehow, rather than being a theoretical stance, the liberal one is just an attitude, and a very tolerant one at that. The liberal just focuses on the underdetermination for the explanation of her preferred domain (arithmetic), but does not commit herself to the truth of any one specific explanation, especially since such explanations entail contradictory accounts of other domains. Her “will do” appears to be strictly pragmatic in nature.

Quite ironically, the modal-reconstructive character of the liberal is to this extent indeed rather distant from the modal-reconstructive character of standard neo-logicism, which combines a reconstructive epistemology with the attempt of capturing the only true account of arithmetical knowledge (as difficult as it may be to keep the two things together). This liberal is instead an easy prey to the same criticism that Quine [41] aimed at Carnap [11]’s proposal of rational reconstructions: if there is more than one legitimate reconstruction, then there are infinitely many, and no one has more right to correctness than the other.

Things become more interesting if the liberal is pressed to abandon her tolerant attitude and to endorse some robustly theoretical view. When, the liberal may be asked, can we attribute truth to any of the available reconstructions? And, more to the point, can we attribute truth to more than one? On a natural reading, the answer to this latter question should be in the negative. For while the liberal is at ease in claiming that any reconstruction “will do,” pragmatically, in so far as the deliverance of Peano Arithmetic is involved, she could not agree that more than one such reconstruction “will do,” theoretically (or, if you want, semantically), once the full import of their consequences is envisaged, since all such reconstructions point to a different and incompatible analysis of the concept of cardinal number—and they all should be true a priori. To put it vividly, if the liberal has to be some kind of neo-logicist, then she has to be a pragmatic liberal with a silenced monist (when not conservative) theoretical stance in the background, according to which only one analysis of the concept of cardinal number, among the many, is correct.

On this reading, the liberal and the pluralist clearly disagree. While the liberal accepts that, for the limited purposes at hand (Peano Arithmetic) any (suitable) principle can pragmatically be adopted, the pluralist claims that, globally, many principles can be adopted at once (in one or the other of the senses explored above), even if they respectively deliver conflicting pictures of the mathematical universe (and for some brands of pluralism, such as instances of pluralism about criteria, the requirement that each such principle is true a priori can even be dropped). That is the main difference that we see between the liberal and our pluralist.

It may be retorted that, despite natural, this is not the only available reading of the liberal stance. On a different reading, once pressed to take position on which one of the pragmatically equal principles she is adopting for Peano Arithmetic is true, the liberal can reply: all of them. But how can she defend such answer? On the one hand, she could reply that all principles define concepts under which the same finite cardinals fall but distinct transfinite cardinals fall, similarly to what is depicted in the first diagram in Section 3.2 (for an imperfect analogy, just consider the concepts of “biped” and “featherless”: humans fall under both, although both have additional members which are not humans). On the other hand, she could claim that each principle defines its own concept of cardinal number, and also its own concept of finite cardinal, similarly to what is depicted in the second diagram in Section 3.2. Or again, she could claim that each principle corresponds to different precisifications of a unique and yet vague or underdetermined concept of cardinal number. All these possible answers, however, are not provided simply by endorsing a liberal stance. They are different ways of implementing a pluralist position which can be compatible with this second reading of the liberal, but are all ways that need be explored in their own right, as we attempted above.

In short, then: the liberal view, as it is described by Mancosu and inspired by MacBride, appears to be more of a pragmatic attitude than a theoretical stance; if it is pushed so as to be fully explicit on its theoretical commitments, then the most natural reading makes the liberal lean towards the monist, hence the pluralist views surveyed above are outright conflicting with it; if, despite this natural reading, the liberal is allowed to give a pluralist answer, then the many options at her disposal will have to be presented in full, and our proposals can be seen at least as a way of cashing them out properly (a way, moreover, that make such options continuous with a wider debate on mathematical and logical pluralism).

4.2 The anti-exceptionalist import of the Good Company Problem

We have shown above how LPs arising from there being different specifications or definitions of key disciplinary notions can inspire parallel positions within the philosophy of mathematics (Section 3.1). Also, we have discussed how the idea of a pluralism about abstraction criteria can in turn highlight the existence of a currently blank spot in the logical pluralism’s spectrum (Section 3.3). Connections between the philosophy of mathematics and the philosophy of logic are thus neatly emphasized when we look to pluralist solutions to the Good Company Problem. But similarities need not to halt here. Indeed, the monist abstractionist stances envisioned by Mancosu as alternative responses to the Good Company Problem—viz. conservative and moderate positions—bear close resemblances to a recently debated view in the philosophy of logic—namely, anti-exceptionalism (AE).

Anti-exceptionalism stems from a broadly Quinean take on logic (esp. [40]), and has recently been defended and discussed by a number of authors ([15]; [26]; [34]; [39]; [44]; [54]; [55]; [61]). According to AE, logical laws do not possess the series of peculiar features that have traditionally been thought to distinguish the discipline from most other areas of inquiry—most notably, they are neither a priori, nor analytic, nor necessary. Furthermore, not only the status of logic’s claims is on par with that of science’s, but so is the nature of its inquiry; theory-choice in logic is to be carried out by means of an abductive methodology, i.e. logical theories get to be supported, revised and compared with respect to a set of traditional criteria, such as adequacy to the data, simplicity, strength, conservativeness, and fruitfulness. Among these supra-theoretical properties, adequacy to the data is arguably the most important ([38], p. 217); unfortunately, it is also one of the murkiest, as it is unclear what should count as evidence in favor of or against a given logic, and opinions among anti-exceptionalists are here split. The account most relevant to the present purposes is Priest’s. According to him, the data for the assessment of a logical theory are provided by “our intuitions about the validity or otherwise of vernacular inferences” ([39], sect. 2.5; cf. also [38], p. 217). As Priest puts it, some inferences, such as:

$$ \begin{align*} \begin{array}{c}{\textit{If John is in Rome he is in Italy}} \\ \dfrac{\textit{John is in Rome}}{\textit{John is in Italy}} (1)\end{array}\end{align*} $$

strike us as correct; others, such as:

$$ \begin{align*} \frac{\textit{John is either in Rome or in Florence}}{\textit{John is in Florence}} (2) \end{align*} $$

strike us as incorrect—“any account that gets things the other way around is not adequate to the data” ([39], sect. 2.5).

Hence, according to Priest, the goal of a logical theory is that of capturing the notion of logical consequence embedded in ordinary reasoning (although he thinks that relevant intuitions have to be purged from systematic mistakes, such as those elicited by the Selection Task and akin puzzles). Russell likely holds a similar position, in that she thinks that a logical theory explains and unifies pre-theoretic intuitions ([45], pp. 788–89); arguably, Read can be put in the same group. ²⁴

Recall now the conservative’s stance (Section 3). According to her, HP is the uniquely best candidate; being unable to single it out by means of usual abstractionist criteria, she needs to look somewhere else. According to Mancosu, one of the strategies at her disposal is in fact that of claiming that abstraction principles, in order to be legitimate, should capture a concept actually employed in ordinary mathematical practice, and that HP is the only principle apt to fulfill such task when it comes to the concept of natural (finite cardinal) number. Hence, similarly to what many anti-exceptionalists do with respect to validity, she would thus be relying on HP’s adequacy to the data and consider these to be provided by the pre-theoretical conception(s) of the notion of cardinal number implicit in our everyday arithmetical reasoning. Notice also that the moderate neo-logicist would retreat to HPF for these very reasons:

Heck has claimed that one should keep the connection between ordinary arithmetical principles and abstractions tight and has recommended adopting HPF as a viable such principle, which can be justified by appeal to reflection on our ordinary arithmetical reasoning. […T]he a priori way to $\text {PA}_{2}$ is given by HPF and the latter principle is taken to be the principle underlying our arithmetical knowledge. ([35], pp. 189–190)

Adequacy to our pre-theoretic concepts is, however, not the only virtue of HP the conservative might turn to; as quoted above (Section 3), among the more general and less technical merits she might make use of, Mancosu ([35], p. 190) also mentions naturalness, simplicity and non ad-hocness. An analogy with what happens in the anti-exceptionalist debate for logic can neatly be drawn here as well, as these are some of the traditional criteria involved both in the general abductive methodology for science, and in the adoption of an abductive methodology for logic in particular. Among these, simplicity is arguably the most representative, though its precise meaning is regarded as unclear. What does it mean for a (logical) theory $\textit {T}_{1}$ to be simpler than another (logical) theory $\textit {T}_{2}$ ? As both Priest and Hjortland notice, the criterion is not univocal, and can be unpacked in several ways ([26], fn. 23 lists some). Despite the ambiguity of the criterion, Williamson—as Quine before him—thinks that classical logic has the edge:

Once we assess logics abductively, it is obvious that classical logic has a head start on its rivals, none of which can match its combination of simplicity and strength. [54], p. 337

So here we have, again, another close parallelism between abstractionism and the philosophy of logic, since lines of thought which might be employed by conservative neo-logicists closely resemble paths followed by anti-exceptionalists in logical theory-choice.

4.3 Pluralism about criteria and Bad Company

In this section we will suggest that pluralism about criteria (henceforth, PaC) might ramify into an attractive solution to the Bad Company problem.

Note that there are (at least) two ways in which two different abstraction principles might conflict with each other. On the one hand, two principles, for example HP and PP, might be mutually consistent, but offer diverging characterizations of the same concept, e.g. cardinal number. On the other hand, two principles, e.g. HP and NP, might be individually consistent, but mutually incompatible. In this paper we elaborated PaC in reaction to the first kind of predicament (good company). We will now show how it might apply to the second one (bad company).

Indirect support for PaC as applied to the Bad Company problem seems to come from the fact that different criteria for acceptable abstraction have in fact been proposed. Neo-logicists (e.g. [23], pp. 465–6, 481) argue that good abstraction principles comply both with some formal criteria, e.g. Stability, and with an epistemological desideratum, loosely that those principles can be be stipulated “without collateral (a posteriori) epistemic work” ([22], p. 128). Rayo [43] contends that any internally consistent and conservative set of abstraction principles might be accepted. ²⁵ Finally, an abstractionist might also consider mathematical fruitfulness as the sole criterion for acceptability. ²⁶ The pluralist’s take on this dialectical situation would be that all these criteria might count as equally legitimate.

A minimal requirement for the legitimacy of a criterion seems to be that the abstraction principles that comply with it form a good company, namely that the set of principles picked up by that criterion is internally consistent. One might also require that each acceptable abstraction is also conservative, since non-conservative stipulations might have consequences for the ontology of the non-abstract domain; however, a failure of conservativeness might be counterbalanced by other virtues (see e.g. [58], pp. 301–3). However, let $C_1$ and $C_2$ be two criteria that comply with this condition. Suppose also that $C_1$ deems HP as acceptable and NP as unacceptable, while $C_2$ does the opposite. HP and NP would then both be good, even though in different senses of ‘good’. At the same time, it is often assumed that the acceptability of an abstraction principle entails its truth (see in particular [19]); since HP and NP cannot be true together, PaC would be untenable unless it is also required that all the acceptable principles, even if selected by different criteria, form a good company.

A moral that could be drawn from this predicament is that, while pluralism about abstraction can offer a new response to the Good Company Problem, this response cannot be extended as a solution to the Bad Company Problem. Alternatively, the pluralist might try to refine her proposal. One possible way of doing this is, for instance, to take on board a suggestion advanced by [43].

According to Rayo, different linguistic stipulations might give rise to different ways of conceptualizing or ‘carving’ reality; he adds:

Just because different fragments of one’s discourse correspond to different carvings of the world, it doesn’t mean that one is barred from using a single language to formalize one theorizing. A multi-sorted language can use terms and predicates of different sorts to talk about objects corresponding to different carvings. ([43], p. 182)

A (higher-order) many-sorted language features many sorts of first-order variables, each ranging over its own universe, and many sorts of higher-order variables, ranging over subsets of those universes. Intuitively, each sort will be interpreted as the good company picked up by a given legitimacy criterion. More precisely, the domain of the sort corresponding to a criterion C will consist of (i) a domain of “basic”, viz. non-abstract, objects, or a subset thereof; and (ii) all the abstract entities that are introduced by the abstraction principles that comply with C. The many-sorted setting will ensure that different good companies do not interact with each other. This will make each good company conservative with respect to any other; moreover, internal consistency then emerges as a sufficient criterion for acceptability.

This pluralist solution to the Bad Company problem comes at some costs. Neo-logicists submit that good abstraction principles like HP are analytic and a priori; the pluralist about criteria might be forced to give up on analyticity and aprioricity at least in some cases. ²⁷ Moreover, abstraction on any domain whatsoever (or, at least, on domains of an arbitrary cardinality) would no longer be permissible; this seems to be at odds with the neo-Fregeans’—and, before them, Frege’s own—dictum that mathematical concepts should be universally applicable (e.g. [60]; cf. also [43], pp. 80–2). Finally, HP itself will plausibly comply with more than one criterion; the question will then be whether the cardinal numbers that belong to one sort are identical with the ones that belong to other sorts. PaC therefore requires us to step away from traditional neo-logicism; still, “all progress depends on parting with tradition in some way or other” ([36], p. 220). Moreover, abstractionists might recover much more mathematics by relying on principles that belong to different good companies.

4.4 Concluding remarks: unification and analogies

The varieties of abstractionist pluralism explored in this paper are meant to provide a theoretical framework for a given family of responses to the Good Company Problem. One natural objection is that no such pluralism could count as a variety of neo-logicism. To a certain extent (see also the remarks in Section 4.1), the liberal could still be counted as a neo-logicist of sort, just of a very modest kind: she need not abandon the idea that there is just one correct concept of cardinal number, but faced with the Good Company Problem she reluctantly agrees to somehow suspend judgment and to settle pragmatically for whatever brings the desired result home. This being said, it was not our intention to present pluralism as something that neo-logicists could take on board while remaining, in one sense or another, neo-logicists. The thought is rather that when faced with the Good Company Problem, the neo-logicist abstractionist may be forced to abandon some crucial neo-logicist tenets, and agree that another significantly different abstractionist setting is the most promising path to take.

A different objection is that the forms of pluralism that have been discussed depart too much from ordinary arithmetical thought. After all, despite its limitations, neo-logicism (analogously to the original logicism championed by Frege) aimed at rationally reconstructing a portion of mathematics by paying close attention to some essential uses of the concept of cardinal number both in ordinary discourse and in mathematical practice. This criticism, however, seems misplaced. The three forms of pluralism are all motivated by the attempt at doing justice to the plurality of possible conceptions of the notion of cardinal number, which most notably include those historically determined conceptions which Mancosu has reconstructed. The fact that different conceptions of comparison of size have given rise to different conceptions of cardinal numbers (e.g. Cantor-Frege’s vs. Peano’s) is something that one may one want to take into account in a theoretical framework, and pluralism (of any sort) seems a reasonable way to do that.

More specifically, the thing that both the conservative, the moderate and the liberal have in common is that they fail to accommodate properly the different insights which gave (or could give) rise to divergent concepts of cardinals. While the conservative sticks to her guns, and the moderate somehow ignores the problem by resting content of a definition limited to finite cardinals, the liberal just pragmatically avoids the issue entirely. A pluralist stance on abstractions, on the other hand, shares the same merit that Shapiro ([48], p. 25) praises for logical pluralism, i.e. of presenting itself as the best explanation for the different historical or theoretical role played by more than one notion in the cluster of equally legitimate but still incompatible companions. In the case of logical pluralism, this amounts to finding a theoretical framework that gives equal dignity to the different insights which led (or can lead) to rival conceptions of logical consequence, as these have been (or could be) developed in different contexts, mathematical or not (e.g. classical reasoning, constructive mathematics, paraconsistent theories, etc.). In the present case, this would amount to finding a theoretical framework that gives equal dignity to different insights which led (or may lead) to different conceptions of cardinals and comparison of size, especially when it comes to infinity (e.g. those based on the part-whole relation, those based on equinumerosity, etc.).

Abstractionist pluralism, of the varieties which have been explored, has also the additional advantage of unifying debates across different although traditionally related disciplines. Indeed, it shows—provided one agrees that pluralism about logic is a desirable option—that both basic logical and mathematical notions can be given an equal theoretical treatment (or, depending on how much one assimilates mathematics and logic, that they even should be given such an equal treatment).

A separate issue is to consider whether the interaction between logical pluralism and mathematical abstractionist pluralism has additional consequences. For instance, it is easy to notice (analogously to what Shapiro [48] himself does on different grounds) that a change in the background logic can affect the consistency of a given abstraction principle, and therefore can lead to different abstractive definitions of the same mathematical notion. On this regards, if one accepts more than one rival logic as equally legitimate, one may end up with another motivation for endorsing a kind of mathematical abstractionist pluralism, depending on how many rival and still viable abstractive definitions of the same notion are licensed by different background logics. This clearly would gain interest if the forms of pluralism discussed above for the abstractive definition of the notion of cardinal were to be applied to other historically more controversial notions, such as the notion of extension as tentatively defined by Frege’s BLV. The debates on logical pluralism, abstractionist puralism, and consistent subsystems of Frege’s Grundgesetze could then find in such an approach a fruitful intersection.

Finally, one has to concede that analogies, as good as they may be, can break down at some point. For instance, one may raise doubts that the analogy between logical pluralism and mathematical pluralism can in fact be pursued indefinitely. For instance, one may argue that while the principles involved in the Good Company Problem all agree on at least one set of cases (i.e. finite arithmetic), nothing mandates that the different logics admitted by a logical pluralist will end up having any shared set of principles. ²⁸ Here’s where the analogy between the two could stop. Consider, however, that while it is in principle possible for a logical pluralist to accept wholly disjoint sets of logical laws, actual pluralists usually limit the set of legitimate logics to a restricted number which share a core of laws while diverging on others. ²⁹ So the analogy could, not as a matter of principle but as a matter of fact, still be preserved. Related concerns can come up nonetheless. For instance, while it seems unreasonable to expect to find a logic whose laws could be super-valid, i.e. valid in all possible rival logics (cf. [52]), any set of rival but legitimate abstractive definitions of—to stick to our present concern—cardinal number could hardly be acknowledged unless all such definitions were able to license a common core of arithmetical results, as weaker than second-order Peano Arithmetic they could be. It is then surely an interesting matter to explore how much the analogy could really be maintained. Acknowledging this, however, does not seem to diminish in any strong sense the fruitfulness of the analogy itself.