This book has many virtues. It is concentrated on fundamental questions in the philosophy of mathematics, which it explores with an open mind – or even two open minds; it is richly informed and informative in its clear exposition of the details of nominalistic reconstruction programs, indeed the whole extant gamut of them, some themselves usefully reconstructed; it concludes with a novel insight into the unsuspected value of these programs (to be explained below); and, of special immediate relevance, it is remarkably balanced in its argumentation and self-contained, even to the point of containing its own review! Not verbatim, of course, but implicitly, as a scattered whole, merely awaiting a judicious selection and assembly, with occasional textually inspired critical commentary. Here follows an attempt at such.

A remarkable twentieth century development in mathematics has been the construction of smooth inifintesimal analysis (SIA) and its extension, synthetic differential geometry (SDG), realizing a non-punctiform conception of continua in contrast with the dominant classical (set-theoretic) conception. Here one is concerned with “smooth worlds” in which all functions (on or between spaces) are continuous and have continuous derivatives of all orders. In this setting, the once discredited notion of “infinitesimal quantity” is admitted and placed on a rigorous footing, reviving intuitive and effective methods in analysis prior to the nineteenth century development of the limit method. The infinitesimals introduced, however – unlike the invertible ones of Robinsonian non-standard analysis – are nilsquare and nilpotent. While they themselves are not provably identical to zero, their squares or higher powers are set equal to 0.

A major outstanding issue in the foundations and philosophy of mathematics concerns the indispensability of classical infinitistic mathematics for the empirical sciences. Claims of such indispensability form a modern cornerstone of mathematical platonism and alternative classical realist conceptions as well (e.g. modal structuralism) (cf. e.g., Quine [1953], Putnam [1967, 1971], and Hellman [1989a]). This has posed a corresponding challenge to constructivist views (intuitionistic, Bishop-constructivist [Bishop 1967], and related approaches). How much of the mathematics actually employed in the empirical sciences, especially physics, can be carried out constructively (in the various relevant senses)? It is probably no exaggeration to say that the viability of a constructivist philosophy of mathematics is here at stake (cf. Burgess [1984]).

As is well known, predicative mathematics has long been motivated by skepticism concerning the classical conception of the Cantorian transfinite and, in particular, of the continuum, or the notion of “all subsets of (even) a countably infinite set.” Along with constructivism, predicativism regards as suspect talk of functions which cannot even in principle be given a definite mathematical description. Indeed, a basic predicativity requirement is that any recognized mathematical object be presentable by means of a finite string of symbols from a countable language, where this is understood to include formulas defining sets or functions, with quantification in the formulas restricted to “already acceptable” objects. In contrast with intuitionism, however, the natural number system is treated classically, and classical logic is taken as legitimate. The focus, then, is on principles of set existence. In accordance with Russell’s “vicious circle principle,” sets of natural numbers cannot legitimately be introduced via definitions or formulaic conditions with unrestricted quantifiers over such sets.

Probably there is no position in Goodman’s corpus that has generated greater perplexity and criticism than Goodman’s “nominalism.” As is abundantly clear from Goodman’s writings, it is not “abstract entities” generally that he questions – indeed, he takes sensory qualia as “basic” in his Carnap-inspired constructional system in Structure [Goodman, 1977]] – but rather just those abstracta that are so crystal clear in their identity conditions, so fundamental to our thought, so prevalent and seemingly unavoidable in our discourse and theorizing that they have come to form the generally accepted framework for the most time-honored, exact, sophisticated, refined, central, and secure branch of human knowledge yet devised, mathematics itself! Of all the abstracta to question, why sets? Of course, Goodman gave his “reasons,” the unintelligibility of “generating” an infinitude of “constructed objects” automatically from any given object or objects.

The roots of the view of mathematics known as “if-thenism” or “deductivism,” like its near cousins, logicism and formalism, are to be found in the nineteenth and early twentieth centuries’ “new birth” of mathematics. As summarized nicely by Maddy [to appear], this period witnessed a multifaceted transformation from mathematics as investigating formal properties of aspects of material reality, such as space, time, and motion, to its being an exploration of abstract concepts and structures in their own right quite apart from any material applications that they might have.

As with many “isms,” “structuralism” is rooted in some intuitive views or theses which are capable of being explicated and developed in a variety of distinct and apparently conflicting ways. One such way, the modal-structuralist approach, was partially articulated in Hellman [1989] (hereinafter MWON). That account, however, was incomplete in certain important respects bearing on the overall structuralist enterprise. In particular, it was left open how to treat generally some of the most important structures or spaces in mathematics, for example, metric spaces, topological spaces, differentiable manifolds, and so forth. This may have left the impression that such structures would have to be conceived as embedded in models of set theory, whose modal-structural interpretation depends on a rather bold conjecture, for example, the logical possibility of full models of the second-order ZF axioms.

After reviewing shortcomings of Boolos’ presentation of the “Iterative conception of set,” we formulate simple axioms on “stages” of set formation, using modal logic and the logic of plurals, that imply both the Axiom of Infinity and the Axiom of Replacement. (Two routes to these results are presented, the second less open to charges of circularity than the first.) We then present two other advantages of the Height-Potentialist framework, pertaining to motivating the smallest large cardinals (strongly inaccessible and Mahlo), and to furnishing attractive resolutions of the set-theoretic paradoxes. Our routes to these results are not available from within the Height-Actualist framework.

Often the Burali-Forti paradox is referred to as the paradox of “the largest ordinal,” which goes as follows. Let Ω be the class of all (von Neumann, say) ordinals.

This is a sequel to our article “Predicative foundations of arithmetic” (Feferman and Hellman [1995], reproduced as Chapter 7 in this volume), referred to in the following as PFA; here we review and clarify what was accomplished in PFA, present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by PFA was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons [1992]. Another critique is due to Daniel Isaacson [1987]. Most recently, Alexander George and Daniel Velleman [1998] have examined PFA closely in the context of a general discussion of different philosophical approaches to the foundations of arithmetic.

It is commonplace that opposing philosophical schools – whatever their subject – have difficulty communicating with one another, and, indeed frequently talk past one another rather than engage in rational debate. At times, this seems to have been Brouwer’s own view of the relation between intuitionistic and classical approaches to foundations of mathematics. In such circumstances, it is frequently obscure just wherein genuine disagreement – as opposed to merely apparent or verbal disagreement – actually resides, or even whether there really is at bottom any genuine disagreement at all.

Abstract mathematics, from its earliest times in ancient Greece right up to the present, has always presented a major challenge for philosophical understanding. On the one hand, mathematics is widely considered a paradigm of providing genuine knowledge, achieving a degree of certainty and security as great as or greater than knowledge in any other domain. A part of this, no doubt, is that it proceeds by means of deductive proofs, thereby inheriting the security of necessary truth preservation of deductive logical inference. But proofs have to start somewhere: ultimately there need to be axioms, and these are the starting points, not end points, of logical inference. But what then grounds or justifies axioms?

Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or set-theoretic standpoint, this appears problematic, for, as the story is usually told, impredicative principles seem to play an essential role in the foundations of arithmetic itself. It is the main purpose of this paper to show that this appearance is illusory: as will emerge, a predicatively acceptable axiomatization of the natural number system can be formulated, and both the existence of structures of the relevant type and the categoricity of the relevant axioms can be proved in a predicatively acceptable way.

In his influential paper, “Truth by convention,” Quine subjected the linguistic doctrine of logical truth (LD) to a critique that, to many, has seemed devastating. Having granted the conventionalist (what Quine took to be) his starting points, Quine caught his opponent in a vicious regress: to proceed from the linguistic stipulations to the (full class of) logical truths requires logical rules themselves in addition to any of the stipulations. What Lewis Carroll’s tortoise said to Achilles (on the need to appeal to modus ponens to justify any application of it) seemed an arrow in Carnap’s heel.

Carnap seems never to have taken the critique very seriously. His reply to Quine’s “Carnap and logical truth,” which repeated the upshot of “Truth by convention,” is couched in irony. Quine had found LD “empty” and “without experimental meaning”; moreover, he had found it “implying nothing not already implied by’’ the assertion – which he surely accepted – that logic is obvious.