What to read after Mathematics: A Very Short Introduction

I have only just begun this page, but I intend to expand it in due course. I recommended various books in the section entitled "Further reading", and I may say some more about them here. (At the moment I am not in a good position to do so because I have been avoiding popular expositions of mathematics so as not to be inhibited by them.)

On this website I have written several mathematical discussions, informal but mostly aimed at a more experienced audience than the projected audience for Mathematics VSI. (I had in mind first-year mathematics undergraduates at Cambridge.) A full index appears here . In the not too distant future, I shall write some less advanced material, and also recommend other websites. For now, the following discussions should be relatively easy to understand for a pre-university reader.

A dialogue concerning the existence of the square root of two.

How could the existence of the square root of two be in doubt? Well, it isn't exactly, but this dialogue aims to show that there is more to the issue than one might imagine. It complements parts of Chapters 2 and 3 of Mathematics VSI.

What is wrong with thinking of real numbers as infinite decimals?

This discussion presupposes some familiarity with the theory of real numbers, and would therefore not be fully comprehensible to a typical reader of Mathematics VSI. Maybe I will write about the real numbers at some point. However, for now, this discussion extends the ideas of Chapter 4 and may be of some interest.

Paradoxes concerning definability.

There was very little in Mathematics VSI about logic or set theory. This discussion is about a topic that many lay mathematicians find interesting.

Miscellaneous Further Topics

Not everything in the following discussions is at a pre-university level, but some of it should be comprehensible.

Just-do-it proofs.

The definition of `definition'.

What is `solved' when one solves an equation?

Non-Euclidean Geometry.

The following discussions were written to accompany a Cambridge course in geometry. They cover some of the same ground as Chapter 6, but they are not identical to it. Again, I occasionally assume university-level knowledge, so some skimming may be necessary.

What makes non-Euclidean geometry interesting?

What makes it hard?

What makes hyperbolic geometry particularly hard?

What can I do to make it seem less hard?

How can the arc of a circle be considered straight?

Why did people want to prove the parallel postulate?

What is the historical importance of non-Euclidean geometry?

What is geometry?

General articles.

The two cultures of mathematics

This article, which is in pdf format, is a discussion of two very different styles of doing mathematics and a defence of one of them. It is aimed at mathematicians and a typical reader of Mathematics VSI would need to skim a few parts. Nevertheless, it should be possible to understand the general points I make without following all the details.

The importance of mathematics

This is a transcript, slightly modified, of a lecture I gave at the Millennium Meeting of the Clay Mathematics Institute in May 1999 in Paris. It is also in pdf format. I said very little in Mathematics VSI about the justification for pure mathematics - I hope this article redresses the balance. It was aimed at the general public, so all of it was intended to be comprehensible to the non-expert. Unfortunately, it is supposed to come with illustrations, which I do not know how to produce or put online. If you have the right technology you can watch the lecture, with illustrations, on streaming video either here (where there is also the option of ordering the video) or here . Occasionally there is a small overlap with Mathematics VSI, including one sentence that, I now notice, is almost identical.

Does mathematics need a philosophy?

This is a talk I gave to a new society at Cambridge called the Cambridge University Society for the Philosophy of Mathematics and the Mathematical Sciences. It was aimed at an audience of mathematicians (including many undergraduates) and mathematically inclined philosophers. It deals with some of the same issues as Mathematics VSI but assumes a bit more of the reader.