Realism in Mathematics

Realism in Mathematics

Realism in Mathematics

Realism in Mathematics

Synopsis

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics.

Excerpt

The philosophy of mathematics is a borderline discipline, of fundamental importance to both mathematics and philosophy. Despite this, one finds surprisingly little co-operation between philosophers and mathematicians engaged in its pursuit; more often, widespread disregard and misunderstanding are broken only by alarming pockets of outright antagonism. (The glib and dismissive formalism of many mathematicians is offset by the arrogance of those philosophers who suppose they can know what mathematical objects are without knowing what mathematics says they are.) This might not matter much in another age, but it does today, when the most pressing foundational problems are unlikely to be answered without a concerted co-operative effort. I have tried in this book to do justice to the concerns of both parties, to present the background, the issues, the proposed solutions on a neutral ground where the two sides can meet for productive debate.

For this reason, I've aimed for a presentation accessible to both non-philosophical mathematicians and non-mathematical philosophers and, if I've succeeded, students and interested amateurs should also be served. As far as I can judge, very little philosophical training or background is presupposed here. Mathematical prerequisites are more difficult to avoid, owing to the relentlessly cumulative nature of the discipline, but I've tried to keep them to a minimum. Some familiarity with the calculus and its foundations would be helpful, though surely not necessary. And the relevant set theoretic concepts are referenced to Enderton's excellent introductory textbook (see his (1977)), for the benefit of those innocent of that subject.

The central theme of the book is the delineation and defence of a version of realism in mathematics called 'set theoretic realism'. In this, my deep and obvious debt is to the writings of the great mathematical realists of our day: Kurt Gödel, W. V. O. Quine, and . . .

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