Geoffrey Hellman

This book explores the research of Professor Hilary Putnam, a Harvard professor as well as a leading philosopher, mathematician and computer scientist. It features the work of distinguished scholars in the field as well as a selection of young academics who have studied topics closely connected to Putnam’s work. It includes 12 papers that analyze, develop, and constructively criticize this notable professor's research in mathematical logic, the philosophy of logic and the philosophy of mathematics. In addition, it features a short essay presenting reminiscences and anecdotes about Putnam from his friends and colleagues, and also includes an extensive bibliography of his work in mathematics and logic. The book offers readers a comprehensive review of outstanding contributions in logic and mathematics as well as an engaging dialogue between prominent scholars and researchers. It provides those interested in mathematical logic, the philosophy of logic, and the philosophy of mathematics unique insights into the work of Hilary Putnam.

After some introductory remarks on "experimental metaphysics", a brief survey of the current situation concerning the major types of hidden-variable theories and the inexistence proofs is presented. The category of stochastic, contextual, local theories remains open. Then the main features of a logical analysis of "locality" are sketched. In the deterministic case, a natural "light-cone determination" condition helps bridge the gap that has existed between the physical requirements of the special theory of relativity and formal conditions used in proving the Bell -Wigner theorem. Natural generalization to the stochastic type, taking account of the distinction between epistemic and physical probabilities, leads to a series of independence claims constituting some significant limitations on generalized Bell theorems. In particular, the conditional stochastic independence requirement is seen both to go beyond the demand of compliance with the STR and to be a genuine necessity in deriving any Bell theorem for the stochastic case. The conclusion is also supported that, if determinism is given up, the Bell theorems and experiments do not pose an additional obstacle to unifying relativity theory and quantum mechanics beyond what is already posed by the "instantaneous" collapse of the wave function

Standard presentations of axioms for set theory as truths simpliciter about actual-objects the sets-confront a number of puzzles associated with platonism and foundationalism. In his classic, Zermelo suggested an alternative "many worlds" view. Independently, Putnam proposed something similar, explicitly incorporating modality. A modal-structural synthesis of these ideas is sketched in which obstacles to their formalization are overcome. Extendability principles are formulated and used to motivate many small large cardinals. The use of second-order logic as a coherent and clear framework for set theory is supported

Three questions are highlighted concerning the scope and force of indispensability arguments supporting classical, infinitistic mathematics. The first concerns the need for non-constructive reasoning for scientifically applicable mathematics; the second concerns the need for impredicative set existence principles for finitistic and scientifically applicable mathematics, respectively; and the third concerns the general status of such arguments in light of recent work in mathematical logic, especially that of Friedman et al. and Feferman et al. Some recent results are then presented bearing on the first question on the need for non-constructive analysis, especially for quantum physics. Despite the impressive work of Bishop et al. in constructive analysis, Hilbert's objection to intuitionism still carries significant force, and may be decisive depending in part on one's conception of "physics"

In these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman's modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which sets occur. In the first of three new essays written for this volume, Hellman shows how extendability can be deployed to derive the axiom of Infinity and that of Replacement, improving on earlier accounts; he also shows how extendability leads to attractive, novel resolutions of the set-theoretic paradoxes. Other essays explore advantages and limitations of restrictive systems - nominalist, predicativist, and constructivist. Also included are two essays, with Solomon Feferman, on predicative foundations of arithmetic.

Structuralism is a view about the subject matter of mathematics according to which what matters are structural relationships in abstraction from the intrinsic nature of the related objects. Mathematics is seen as the free exploration of structural possibilities, primarily through creative concept formation, postulation, and deduction. The items making up any particular system exemplifying the structure in question are of no importance; all that matters is that they satisfy certain general conditions—typically spelled out in axioms defining the structure or structures of interest—characteristic of the branch of mathematics in question. Thus, in the basic case of arithmetic, the famous “axioms” of Richard Dedekind (taken over by Giuseppe Peano, as he acknowledged) were conditions in a definition of a “simply infinite system”, with an initial item, each item having a unique next one, no two with the same next one, and all items finitely many steps from the initial one. (The latter condition is guaranteed by the axiom of mathematical induction.) All such systems are structurally identical, and, in a sense to be made more precise, the shared structure is what mathematics investigates. (In other cases, multiple structures are allowed, as in abstract algebra with its many groups, rings, fields, and so forth.) This structuralist view of arithmetic thus contrasts with the absolutist view, associated with Gottlob Frege and Bertrand Russell, that natural numbers must in fact be certain definite objects, namely classes of equinumerous concepts or classes.

In previous technical work ([1] and [2]) on which his present paper [3] draws, Benioff has presented results conforming with the following argument-scheme:First, if we construe Quantum Mechanics as making claims to the effect that infinite outcome sequences (generated by repeated measurements on a system for a given observable in a given state) be random; and second, if a strong definition of “random” is adopted in this construal, then certain models of Zermelo-Fraenkel set theory (ZF) cannot be “carriers for the mathematics of physics”.How interesting is this? One can approach the matter on two levels, the level of the specific technical results, and the level of more general implications of this pattern of argument for understanding connections between mathematics and physics. The second more general level should be our primary focus, but it will pay at the outset to look briefly at the technical level.