Set Theory and Its Philosophy: A Critical Introduction

Set Theory and Its Philosophy: A Critical Introduction

Set Theory and Its Philosophy: A Critical Introduction

Set Theory and Its Philosophy: A Critical Introduction

Synopsis

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

Excerpt

This book was written for two groups of people, philosophically informed mathematicians with a serious interest in the philosophy of their subject, and philosophers with a serious interest in mathematics. Perhaps, therefore, I should begin by paying tribute to my publisher for expressing no nervousness concerning the size of the likely readership. Frege (1893–1903, I, p.xii) predicted that the form of his Grundgesetze would cause him to

Relinquish as readers all those mathematicians who, if they bump into logical expres
sions such as ‘concept’, ‘relation’, ‘judgment’, think: metaphysica sunt, non leguntur, and
likewise those philosophers who at the sight of a formula cry: mathematica sunt, non
leguntur
.

And, as he rightly observed, ‘the number of such persons is surely not small’.

As then, so now. Any book which tries to form a bridge between mathematics and philosophy risks vanishing into the gap. It is inevitable, however hard the writer strives for clarity, that the requirements of the subject matter place demands on the reader, sometimes mathematical, sometimes philosophical. This is something which anyone who wants to make a serious study of the philosophy of mathematics must simply accept. To anyone who doubts it there are two bodies of work which stand as an awful warning: the philosophical literature contains far too many articles marred by elementary technical misunderstandings, while mathematicians have often been tempted, especially in later life, to commit to print philosophical reflections which are either wholly vacuous or hopelessly incoherent.

Both mathematicians and philosophers, then, need to accept that studying the philosophy of mathematics confronts them with challenges for which their previous training may not have prepared them. It does not follow automatically, of course, that one should try, as I have done here, to cater for their differing needs in a single textbook. However, my main reason for writing the book was that I wanted to explore the constant interplay that set theory seems to exemplify between technical results and philosophical reflection, and this convinced me that there was more than expositional economy to be said for trying to address both readerships simultaneously.

I should know: I have written one myself.

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