### 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 theÂ philosophy of mathematics? This unusual question, the very title of thisÂ book, will not begin to be examined with care until Chapter 3, but twoÂ summary answers can be stated at once.

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

Second, because of the richness of applications of mathematics, oftenÂ derived by thinking at a desk and toying with a pencil. Or more poetically,Â in the words of the historian of science A. C. Crombie (1994 i, ix), â€˜theÂ enigmatic matching of nature with mathematics and of mathematics byÂ natureâ€™.

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

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

Two figures haunt the philosophy of mathematics: Plato and Kant. Plato,Â as I read him, was bowled over by the experience of demonstrative proof.Â Kant, as I understand him, crafted a major part of his philosophy to accountÂ for the enigmatic ability of mathematics, seemingly the product of humanÂ reason, so perfectly to describe the natural world. Thus Plato inauguratedÂ philosophizing about mathematics, and Kant created a whole new problematic.Â Another version of my answer to the question, â€˜Why philosophy ofÂ mathematics?â€™ is, therefore, â€˜Plato and Kantâ€™.

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

###### 2 On jargon

My first epigraph, â€˜For mathematics is after all an anthropological phenomenonâ€™,is taken out of context from a much longer paragraph, some of whichÂ is quoted in Chapter 2, Â§13 (henceforth, Â§2.13). The long paragraph is itselfÂ part of an internal dialogue stretching over a couple of pages. The wordsÂ I used for the epigraph sound good, but bear in mind the last recordedÂ sentence of Wittgensteinâ€™s 1939 *Lectures on the Foundations of MathematicsÂ *(1976: 293): â€˜The seed Iâ€™m most likely to sow is a certain jargon.â€™

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 â€˜must,â€™Â 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 â€“ â€˜Donâ€™t ask for the meaning,Â ask for the use.â€™ (For what he did say, see Â§6.23.

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

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