Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality (...) of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious. (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...) which points out that the conceptual resources to grasp Hume's Principle vastly outstrip the conceptual resources employed in arithmetical reasoning. And lastly whether Hume's Principle is in bad company with other unsuccessful implicit definitions. In the last section, an account towards our entitlement to Hume&'s Principle is sketched. (shrink)

Ante rem structuralism is the doctnne that mathematics descubes a realm of abstract (structural) universab. According to its proponents, appeal to the exutence of these universab provides a source distinctive insight into the epistemology of mathematics, in particular insight into the so-called 'access problem' of explaining how mathematicians can reliably access truths about an abstract realm to which they cannot travel andfiom which they recave no signab. Stewart Shapiro offers the most developed version of this view to date. Through an (...) examination of Shapiro's proposed structuralist epistemology for mathematics I argue that ante rem structuralism faib to provide the ingredients for a satisfactory resolution of the access problem for infinite structures (whether small or large). (shrink)

A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietism. For this purpose the notion of entitlement (...) is invoked along the way, suitably modified for the present anti-foundationalist setting. (shrink)

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of (...) the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?". (shrink)

Defining the real numbers by abstraction as ratios of quantities gives prominence to then- applications in just the way that Frege thought we should. But if all the reals are to be obtained in this way, it is necessary to presuppose a rich domain of quantities of a land we cannot reasonably assume to be exemplified by any physical or other empirically measurable quantities. In consequence, an explanation of the applications of the reals, defined in this way, must proceed indirectly. (...) This paper explains the main complications involved and answers the main objections advanced in Batitsky's paper in this issue. (shrink)

In this note we derive Robinson???s Arithmetic from Hume???s Principle in the context of very weak theories of classes and relations.

The philosophy of mathematics has long been an important part of philosophy in the analytic tradition, ever since the pioneering works of Frege and Russell. Richard Dedekind was roughly Frege's contemporary, and his contributions to the foundations of mathematics are widely acknowledged as well. The philosophical aspects of those contributions have been received more critically, however. In the present essay, Dedekind's philosophical reception is reconsidered. At the essay’s core lies a comparison of Frege's and Dedekind's legacies, within and outside of (...) analytic philosophy. While the comparison proceeds historically, it is shaped by current philosophical concerns, especially by debates about neo-logicist and neo-structuralist views. In fact, philosophical and historical considerations are intertwined thoroughly, to the benefit of both. The underlying motivation is to rehabilitate Dedekind as a major philosopher of mathematics, in relation, but not necessarily in opposition, to Frege. (shrink)

We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's Constraint, (...) that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis. (shrink)

According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea (...) of ante rem structures have appeared. Some argue that it is impossible to give identity conditions for places in homogeneous ante rem structures, invoking a version of the identity of indiscernibles. Others raise issues concerning the identity and distinctness of places in different structures, such as the the natural number 2 and the real number 2. The purpose of this paper is to take the measure of these objections, and to further articulate ante rem structuralism to take them into account. (shrink)

Various contributors to recent philosophy of mathematics havetaken Richard Dedekind to be the founder of structuralismin mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the main contemporary variants. In addition, I argue that his writings contain philosophical insights that are worth reexamining and reviving. The discussion focusses on Dedekind''s classic essay Was sind und was sollen die Zahlen?, supplemented by evidence from Stetigkeit und (...) irrationale Zahlen, his scientific correspondence, and his Nachlaß. (shrink)

This chapter discusses various senses in which mathematics is applied to the material world. It distinguishes between canonical and noncanonical applications of mathematics, the former being those for which the mathematics was developed to deal with in the first place. It also distinguishes between empirical and nonempirical applications, thus yielding four different kinds of applications. Examples of each are provided, and philosophical problems connected with each are treated.

Dehaene (this volume) articulates a naturalistic approach to the cognitive foundations of mathematics. Further, he argues that the ‘number line’ (analog magnitude) system of representation is the evolutionary and ontogenetic foundation of numerical concepts. Here I endorse Dehaene’s naturalistic stance and also his characterization of analog magnitude number representations. Although analog magnitude representations are part of the evolutionary foundations of numerical concepts, I argue that they are unlikely to be part of the ontogenetic foundations of the capacity to represent natural (...) number. Rather, the developmental source of explicit integer list representations of number are more likely to be systems such as the object–file representations that articulate mid–level object based attention, systems that build parallel representations of small sets of individuals. (shrink)

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...) was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

On the neo-Fregean approach to the foundations of mathematics, elementary arithmetic is analytic in the sense that the addition of a principle wliich may be held to IMJ explanatory of the concept of cardinal number to a suitable second-order logical basis suffices for the derivation of its basic laws. This principle, now commonly called Hume's principle, is an example of a Fregean abstraction principle. In this paper, I assume the correctness of the neo-Fregean position on elementary aritlunetic and seek to (...) explain one way in which it may be extended to encompass the theory of real numbers, introducing the reals, by means of suitable further abstraction principles, as ratios of quantities. (shrink)

In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly (...) defining the number operator, what advantage, if any, does adopting this course possess over a direct stipulation of the Dedekind-Peano axioms? This paper attempts to answer them. In response to the first, we spell out the links between the recognition of numerical terms as vehicles of singular reference and the conception of numbers as possible objects of singular, or object-directed, thought, and the role of the acknowledgement of numbers as objects in the neo-Fregean attempt to justify the basic laws of arithmetic. In response to the second, we argue that the crucial issue concerns the capacity of either stipulation—of Hume’s Principle, or of the Dedekind-Peano axioms—to found knowledge of the principles involved, and that in this regard there are crucial differences which explain why the former stipulation can, but the latter cannot, play the required foundational role. (shrink)

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of (...) rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies. (shrink)

In recent years, neologicists have demonstrated that Hume's principle, based on the one-to-one correspondence relation, suffices to construct the natural numbers. This formal work is shown to be relevant for empirical research on mathematical cognition. I give a hypothetical account of how nonnumerate societies may acquire arithmetical knowledge on the basis of the one-to-one correspondence relation only, whereby the acquisition of number concepts need not rely on enumeration (the stable-order principle). The existing empirical data on the role of the one-to-one (...) correspondence relation for numerical abilities is assessed and additional empirical tests are proposed. In the final part, it is argued that the fact that the successor relation and the one-to-one correspondence relation can play independent roles in number concept acquisition may be a complication for testing the Whorfian hypothesis. (shrink)

Hale proposes a neo-logicist definition of real numbers by abstraction as ratios defined on a complete ordered domain of quantities (magnitudes). I argue that Hale's definition faces insuperable epistemological and ontological difficulties. On the epistemological side, Hale is committed to an explanation of measurement applications of reals which conflicts with several theorems in measurement theory. On the ontological side, Hale commits himself to the necessary and a priori existence of at least one complete ordered domain of quantities, which is extremely (...) implausible because science treats the logical structure of quantities as subject to experimentally and theoretically motivated refinements and revisions. (shrink)

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it (...) applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of different instances of$AC$and$FC$(§3). We turn to the possible interpretations of$FC$(§4), and advance a Semantic$FC$(§4.1), arguing that while it suits Frege’s definition of natural numbers (4.1.1), it cannot reasonably be imposed on definitions of real numbers (§4.1.2), for reasons only partly similar to those offered by Crispin Wright (§4.1.3). We then rehearse a recent exchange between Bob Hale and Vadim Batitzky to shed light on Frege’s conception of real numbers and magnitudes (§4.2). We argue that an Architectonic version of$FC$is indeed faithful to Frege’s definition of real numbers, and compatible with his views on natural ones. Finally, we consider how attributing different instances of$FC$to Frege and appreciating the role of the Architectonic$FC$can provide a more perspicuous understanding of his foundational program, by questioning common pictures of his logicism (§5). (shrink)

Part I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung]” of his formal system.

In recent philosophy of mathematics avariety of writers have presented ``structuralist''views and arguments. There are, however, a number ofsubstantive differences in what their proponents take``structuralism'' to be. In this paper we make explicitthese differences, as well as some underlyingsimilarities and common roots. We thus identifysystematically and in detail, several main variants ofstructuralism, including some not often recognized assuch. As a result the relations between thesevariants, and between the respective problems theyface, become manifest. Throughout our focus is onsemantic and metaphysical issues, (...) including what is orcould be meant by ``structure'' in this connection. (shrink)

With developments in the 19th and early 20th centuries, structuralist ideas concerning the subject matter of mathematics have become commonplace. Yet fundamental questions concerning structures and relations themselves as well as the scope of structuralist analyses remain to be answered. The distinction between axioms as defining conditions and axioms as assertions is highlighted as is the problem of the indefinite extendability of any putatively all-embracing realm of structures. This chapter systematically compares four main versions: set-theoretic structuralism, a version taking structures (...) as sui generis universals, structuralism based on category theory, and a quasi-nominalist modal-structuralism. While none of the approaches is problem-free, it appears that some synthesis of the category-theoretic approach with modal-structuralism can meet the challenges set out, given the notion of “logical possibility.”. (shrink)

It is known that Hume’s Principle, adjoined to a suitable formulation of second-order logic, gives a theory which is almost certainly consistent4 and suffices for arithmetic in the sense that it yields the Dedekind-Peano axioms as theorems. While Hume’s Principle cannot be taken as a definition in any strict sense requiring that it provide for the eliminative paraphrase of its definiendum in every admissible type of occurrence, we hold that it can be viewed as an implicit definition of a sortal (...) concept of cardinal number and, accordingly, as being analytic of that concept. This, coupled with the fact that Hume’s Principle so conceived requires a prior understanding only of (second-order) logical vocabulary, is enough to justify regarding the resulting account of the foundations of arithmetic as a form of logicism. Whether this claim can ultimately be sustained is of course still very much a matter of controversy. But here, rather than add to the debate on that issue, I should like to assume that it can be, and on that basis, address some further questions. (shrink)

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that (...) our abstraction based theory can be modified to yield much of ordinary mathematics, indeed everything needed to recapture all branches of mathematics short of set theory itself. However, our conclusions are mostly negative. The theory will fall far short of the power of ordinary Zermelo-Fraenkel set theory. It is consistent that our set theory has models that are relatively small, smaller than the first cardinal with an uncountable index. More important, there is a strong tension between the idea that the iterative hierarchy is somehow ineffable, or indefinitely extensible, and the neo-logicist theme of capturing mathematical theories with abstraction principles. (shrink)

In Grundgesetze, Vol. II, §91, Frege argues that ‘it is applicability alone which elevates arithmetic from a game to the rank of a science’. Many view this as an in nuce statement of the indispensability argument later championed by Quine. Garavaso has questioned this attribution. I argue that even though Frege's applicability argument is not a version of ia, it facilitates acceptance of suitable formulations of ia. The prospects for making the empiricist ia compatible with a rationalist Fregean framework appear (...) thus much less dim than expected. Nonetheless, those arguing for such compatibility eventually face an hardly surmountable dilemma. (shrink)

Much of The Reason’s Proper Study is devoted to defending the claim that simply by stipulating an abstraction principle for the “number-of” functor, we can simultaneously fix a meaning for this functor and acquire epistemic entitlement to the stipulated principle. In this paper, I argue that the semantic and epistemological principles Hale and Wright offer in defense of this claim may be too strong for their purposes. For if these principles are correct, it is hard to see why they do (...) not justify platonist strategies that are not in any way “neo-Fregean,” e.g. strategies that treat “the number of Fs” as a Russellian definite description rather than a singular term, or employ axioms that do not have the form of abstraction principles. (shrink)