### Chapter 1

A cartesian introduction

###### 1 Proofs, applications, and other mathematical activities

Why* is* there a whole field of inquiry, a discipline if you like, called the philosophy of mathematics? This unusual question, the very title of this book, will not begin to be examined with care until Chapter 3, but two summary answers can be stated at once.

First, because of the experience of some demonstrative proofs, the *experience *of proving to one’s complete satisfaction some new and often unlikely fact. Or simply experiencing the power and conviction conveyed by a good proof that one is taught, that one reads, or has explained to one. How can mere words, mere ideas, sometimes mere pictures, have those effects?

Second, because of the richness of applications of mathematics, often derived by thinking at a desk and toying with a pencil. Or more poetically, in the words of the historian of science A. C. Crombie (1994 i, ix), ‘the enigmatic matching of nature with mathematics and of mathematics by nature’.

Thus this book is a series of philosophical thoughts about proofs, applications, and other mathematical activities.

In line with the authors of my second and third epigraphs, Lakatos and Stein, I think of proving and using mathematics as activities, not as static done deeds. But at once a word of caution. One does not need proofs to think mathematically: it is a contingent historical fact that proof is the present (and was once the Euclidean) gold standard of mathematicians. Our common distinction between pure mathematics and applications is likewise by no means inevitable. (These assertions to be established in Chapters 4 and 5.) Thus the philosophical difficulties prompted by proofs and applications have arisen because of the historical trajectory of mathematical practice, and are in no sense ‘essential’ to the subject. What are often presented as very clear ideas, namely proof and application, turn out to be much more fluid than one might have imagined.

Two figures haunt the philosophy of mathematics: Plato and Kant. Plato, as I read him, was bowled over by the experience of demonstrative proof. Kant, as I understand him, crafted a major part of his philosophy to account for the enigmatic ability of mathematics, seemingly the product of human reason, so perfectly to describe the natural world. Thus Plato inaugurated philosophizing about mathematics, and Kant created a whole new problematic. Another version of my answer to the question, ‘Why philosophy of mathematics?’ is, therefore, ‘Plato and Kant’.

A third figure haunts my own philosophical thinking about mathematics: Wittgenstein. I bought my copy of the *Remarks on the Foundation of Mathematics* on 6 April 1959, and have been infatuated ever since. What follows in this book is not what is called ‘Wittgensteinian’, but he hovers in the background. Hence it is well to begin with a warning.

###### 2 On jargon

My first epigraph, ‘For mathematics is after all an anthropological phenomenon’,is taken out of context from a much longer paragraph, some of which is quoted in Chapter 2, §13 (henceforth, §2.13). The long paragraph is itself part of an internal dialogue stretching over a couple of pages. The words I used for the epigraph sound good, but bear in mind the last recorded sentence of Wittgenstein’s 1939 *Lectures on the Foundations of Mathematics *(1976: 293): ‘The seed I’m most likely to sow is a certain jargon.’

A good prediction! Recall some jargon derived from his words:

- Language game
- Form of life
- Family resemblance
- Rule following considerations
- Surveyable/perspicuous
- Hardness of the logical ‘must,’ etc. . . . and:
- Anthropological phenomenon
- What we are supplying are remarks on the natural history of mankind.

Not to mention things he did not exactly say – ‘Don’t ask for the meaning, ask for the use.’ (For what he did say, see §6.23.

These are wonderful phrases. But they should be treated with caution and read in context. They all too easily invite the feeling of understanding. They are often cited as if they were at the end, not in the middle, of a series of thoughts. This book will quote scraps from Wittgenstein quite often. Hence it is prudent to begin with this *caveat emptor*, lest I forget.

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