Proof and Other Dilemmas: Mathematics and Philosophy (Spectrum) Review

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Proof and Other Dilemmas: Mathematics and Philosophy (Spectrum) ReviewThe following is quoted from a review by John Corcoran to appear in Mathematical Reviews.
This volume is in the Mathematical Association of America's Spectrum Series, which is intended to "appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers". Its sixteen chapters are individual essays each written by a different author. The authors are said to be "leading mathematicians, mathematics educators, and philosophers of mathematics". It also includes a 20-page introduction by one of the editors. The seventeen essays are intended to be "a sampler of current topics in philosophy of mathematics"; the essays by philosophers are said to "provide a much gentler introduction to what philosophers have been discussing over the last 30 years than will be found in a typical book". This should not be taken to mean that these essays give a summary or overview of the last 30 years of philosophy of mathematics. The book also includes a glossary of the "more common philosophical terms (such as epistemology, ontology, etc.)". The content of the book fully justifies the subtitle "Mathematics and Philosophy"; but nothing seems to explain the implication in the main title "Proof and other Dilemmas" that proof is a dilemma, nor is there anything to indicate which "other dilemmas" are intended.
Unfortunately, there are no indexes. There is no easy way to see how the terms in the glossary are actually used in the book or to compare different authors on the same issue or topic. For example, an index would reveal that the realist philosophy of mathematics called "platonism" is widely accepted--both as "the default position among philosophers" (pages xv and 179) and as the view "still dominant among working mathematicians" (page 40); but an index would also reveal that platonism is widely rejected--by leading mathematicians Paul Cohen and Saunders Mac Lane, and also by "most of the famous mathematicians who have expressed themselves on the question" (page 140). An index would greatly improve the usefulness of the book: it would prevent many misleading impressions.
The glossary is neither well-written nor accurate: for example, existential import is confused with ontological commitment, token is confused with occurrence, entailment is confused with implication, and there is no hint of awareness of the multiple meanings that have been attached to the word `implication' and its cognates--to mention a small selection from the 30 entries. Any reader new to philosophy of mathematics is advised to ignore the glossary and to rely instead on one of the several excellent philosophy dictionaries made by philosophers. One favorite is the 1999 Cambridge Dictionary of Philosophy.
The first three essays concern the focus in the title of the book: proof, as in "demonstrative proof" as opposed to "proof theory". All three are subjectivist in that they emphasize subjective belief or "conviction" while ignoring objective cognition--the idea that a proof proves a proposition to be true: a proof produces knowledge in the strict sense, not just persuasion. Moreover, there is no reference to the traditional "truth-and-consequence" conception of proof: that a proposition is proved to be true by showing that it is a logical consequence of known truths, i. e. by deducing the conclusion from established premises--leaving no room for pictures, constructions, diagrams, analogue or digital devices, or anything other than deductive reasoning once the premises have been taken.
Overall the book is not easy to read or easy to use. There are however some generally excellent articles--those by Michael Detlefsen, Stewart Shapiro, and Julian Cole stand out--but even these are heavy going, even for someone familiar with previous writings by the same author. Moreover, in almost every essay there are scattered passages containing informative scholarship, useful insights, and interesting and provocative points.
Proof and Other Dilemmas: Mathematics and Philosophy (Spectrum) OverviewDuring the first 75 years of the twentieth century almost all work in the philosophy of mathematics concerned foundational questions. In the last quarter of the century, philosophers of mathematics began to return to basic questions concerning the philosophy of mathematics such as, what is the nature of mathematical knowledge and of mathematical objects, and how is mathematics related to science? Two new schools of philosophy of mathematics, social constructivism and structuralism, were added to the four traditional views (formalism, intuitionalism, logicism, and platonism). The advent of the computer led to proofs and the development of mathematics assisted by computer, and to questions of the role of the computer in mathematics. This book of 16 essays, all written specifically for this volume, is the first to explore this range of new developments in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers are not yet discussing. The other half, written by philsophers of mathematics, summarize the discussion in that community during the last 35 years. In each case, a connection is made to issues relevant to the teach of mathematics.

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