The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

Introduction
MATTHIAS SCHIRN

The Philosophy of Mathematics Today --the title of this collection is meant to convey a twofold objective: first, to document important approaches, tendencies, and results in current philosophy of mathematics, and second, to provide fruitful impulses for future work in several of its areas. In my view, the volume reflects, to a considerable extent, the different methods and viewpoints of current research in the field. If one were to attempt to characterize the essays collected here in terms of a feature they all share, one would probably appeal to their being essentially concerned with foundational issues. Questions regarding the historical development of mathematics or the growth of mathematical knowledge play little role in these essays.1

This volume is divided into five parts: I. Ontology, Models, and Indeterminacy; II. Mathematics, Science, and Method; III. Finitism and Intuitionism; IV. Frege and the Foundations of Arithmetic; and V. Sets, Structure, and Abstraction. Classical positions in the philosophy of mathematics are brought into focus and are critically discussed in Parts III and IV and also, though to a lesser extent, in Part V. The philosophical legacy of Frege and Hilbert predominates in these parts. I welcome this; yet what pleases me more is the fact that the collection maintains a reasonable balance between the treatment of some central historical views and the analysis of topics which have emerged recently in the discussion of the philosophy of mathematics and have come to dominate a large part of it.

____________________
I am very grateful to my friend Daniel Isaacson for his helpful comments on an earlier version of this Introduction. I only regret that Daniel did not finish for publication the paper 'On the Need for New Concepts in Mathematics' which he presented at the conference. My thanks go also to Robert Bublak and Karl-Georg Niebergall for discussing several issues of this volume.
1
Perhaps the most important approach to the history of mathematics in recent years is Philip Kitcher study The Nature of Mathematical Knowledge, OUP, New York, 1984. It offers a novel account of how mathematical knowledge evolves, ascribing to the present mathematical community and to previous communities an epistemological significance with which they are not usually credited. The doctrine that mathematical knowledge is a priori is expressly rejected. Furthermore, Kitcher holds that the growth of mathematical knowledge is far more akin to the growth of scientific knowledge than is usually assumed.

-1-

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