Neo-Fregeans argue that substantial mathematics can be derived from a priori abstraction principles, Hume's Principle connecting numerical identities with one:one correspondences being a prominent example. The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles, thus not all consistent abstractions can be true. This paper considers and criticizes various further criteria on acceptable abstractions proposed by Wright settling on another one—stability—as the best bet for neo-Fregeans. However, an analogue of the embarrassment of (...) riches objection resurfaces in the metatheory and I conclude by arguing that the neo-Fregean program, at least insofar as it includes a platonistic ontology, is fatally wounded by it. (shrink)

In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levelsUαVα. The recursive definition of theVα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory shows thatVω, the first transfinite level of the hierarchy, is a model of all the axioms ofZFwith the exception of the axiom of infinity. And, in general, one finds that ifκis a strongly inaccessible ordinal, thenVκis a model of all of (...) the axioms ofZF. Doubtless, when cast as a first-order theory,ZFdoes not characterize the structures 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-orderZFbe, as usual, the theory that results fromZFwhen the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-orderZFcan only be satisfied in models of the form 〈Vκ,∈∩〉 forκa strongly inaccessible ordinal. (shrink)

I present a puzzle for absolutely unrestricted quantification. One important advantage of absolutely unrestricted quantification is that it allows us to entertain perfectly general theories. Whereas most of our theories restrict attention to one or another parcel of reality, other theories are genuinely comprehensive taking absolutely all objects into their domain. The puzzle arises when we notice that absolutely unrestricted theories sometimes impose incompatible constraints on the size of the universe.

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...) the pure sets of the other. This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine. (shrink)

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (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)

One of the earliest discussions of the so-called 'bad company' objection to Neo-Fregeanism, I show that the consistency of an arbitrary second-order 'contextual definition' (nowadays known as an 'abstraction principle' is recursively undecidable. I go on to suggest that an acceptable such principle should satisfy a condition nowadays known as 'stablity'.

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set theory plus the axiom of choice (ZFC) is desirable. One advantage of such an axiomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form Vκ, ∈ ∩ (Vκ × Vκ) , for κ a strongly inaccessible ordinal.

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...) measurable cardinals, whether or not those facts are knowable by us. (shrink)

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...) fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits ofion breaks new ground both technically and philosophically.

In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ...

Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...) most powerful presentation yet of a neo-Fregean program. (shrink)

This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; ...

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

In this festschrift for the eminent philosopher Hilary Putnam, a team of distinguished philosophers write on a broad range of topics and thus reflect the remarkably fertile and provocative research of Putnam himself. The volume is not merely a celebration of a man, but also a report on the state of philosophy in a number of significant areas. The essays fall naturally into three groups: a central core on the theme of conventionality and content in the philosophy of mind, language, (...) and science, and two smaller sections on the relationship of ethics and language, and on the philosophy of logic and aesthetics.In this festschrift for the eminent philosopher Hilary Putnam, a team of distinguished philosophers write on a broad range of topics and thus reflect the remarkably fertile and provocative research of Putnam himself. The volume is not merely a celebration of a man, but also a report on the state of philosophy in a number of significant areas. The essays fall naturally into three groups: a central core on the theme of conventionality and content in the philosophy of mind, language, and science, and two smaller sections on the relationship of ethics and language, and on the philosophy of logic and aesthetics. (shrink)

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are (...) measurable cardinals, whether or not those facts are knowable by us. (shrink)