Archive for August, 2008

Princeton Companion latest

August 28, 2008

A quick post to give an update on the Princeton Companion to Mathematics. It’s been off my hands for a few weeks now. If all goes well it will be printed by the middle of September and should appear in bookshops about a month later. The illustration to the right is what the cover will be like. Click here if you want to see it in more or less full size. If you go to this page on the PUP website then you will find a podcast interview that I gave, which contains information that does not appear on this blog.

UPDATE 14TH OCTOBER: here are links to a few blog posts that have discussed the Princeton Companion recently. Terence Tao had a few thoughts on receiving his copy. Brian Hayes did too. Fred Shapiro, editor of The Yale Book of Quotations, has an interest in reference books and commented about the PCM in the New York Times Freakonomics blog. Dmitry Vostokov recommends the PCM for people wanting a broad overview of mathematics. Isallaboutmath points out that the Nautilus shell on the cover does not in fact have anything to do with the golden ratio. And the book has five reviews on Amazon, some more sensible than others.

FURTHER UPDATE (to which I’ll add as the occasion arises): Peter Woit has written a short review on his blog. Ian Stewart has written a review for The Times. Edmund Harriss has reviewed it on his blog. Alexander Bogomolny reviews it here. Does Antonio Cangiano like it? Perhaps you’d better judge for yourself. Having honed your interpretative skills, you’ll be in a better position to appreciate Scott Guthery’s helpful antidote. A different Scott, Scott Aaronson, reviews it on his blog. And Robin Wilson reviews it for the LMS newsletter. A nice review here, in French.

29TH OCTOBER: It’s been a long time coming, but I’ve just noticed that the PCM now appears to be available on Amazon UK, and not just from third-party sellers. According to them, it’s popular amongst geographers. If I knew how, I’d suggest they recategorized it.

Just-do-it proofs

August 16, 2008

This post is another sample Tricks Wiki article, which revisits a theme that I treated on my web page. Imre Leader pointed out to me that I hadn’t completely done justice to the idea, and that the notion of a “just-do-it proof” had some more specific features that I had not sufficiently emphasized. His opinion matters to me since he was the one who told me about the concept. He himself got it from Béla Bollobás: I don’t know whether it goes further back than that. This is a second attempt at explaining it. Imre, if there’s anything you don’t like about this one, then please edit it when it appears on the Tricks Wiki.

Title: Just-do-it proofs.

Quick description: If you are asked to prove that a sequence or a set exists with certain properties, then the best way of doing so may well be not to use any tricks but just to go ahead and do it: that is, you build the set/sequence up one element at a time, and however you do so you find that it is never difficult to continue building. (more…)

How to use Zorn’s lemma

August 12, 2008

I am continuing my series of sample articles for the Tricks Wiki with one that is intended to represent a general class of such articles. It is common practice in lecture courses (at least if the ones I attended as an undergraduate are anything to go by) to state useful theorems, lemmas, propositions, etc., without going to much trouble to explain why they are useful. Of course, there are many ways to pick up this further understanding: taking note of where and how such results are used, doing carefully designed exercises, and so on. Nevertheless, it is often the case that more could be done to help people recognise the signs that indicate that a particular result can be applied.

This article, which is principally aimed at undergraduates early on in a mathematics degree, is inspired by an experience I myself had as an undergraduate. I had a sheet of challenging problems (set by Béla Bollobás) and one of them completely stumped me. (I’m sure several of them stumped me but this is the one that sticks in my mind.) I can’t remember exactly what the question was, but it was something of similar difficulty to that of determining whether an additive function from \mathbb{R} to \mathbb{R} was necessarily linear. My supervision partner solved the problem using Zorn’s lemma, which we had been told about in a lecture, and I just sat there in disbelief because it hadn’t even remotely occurred to me that Zorn’s lemma might be useful. At some point in the intervening years, I “got” Zorn’s lemma and now find it straightforward to see where it is needed. This article is intended to speed up that process for other people. (more…)

A small countability question

August 10, 2008

This is a short post to ask a simple question that arises out of the discussion in a previous post about countability. As is well-known, the familiar statement that a countable union of countable sets is countable requires the axiom of countable choice. Indeed, it comes in in the very first step of the proof, where one says something along the lines of, “For each set A_n let a_{n1},a_{n2},\dots be an enumeration of its elements.” This uses the axiom of choice because if we don’t know anything about the sets A_n then we can’t actually define these enumerations: we just have to assert that a sequence of enumerations exists.

However, if we do have explicit enumerations of the sets A_n then the proof yields for us an explicit enumeration of their union. So one might take the following attitude to this particular application of the axiom of choice: the real theorem is “An explicitly enumerated union of explicitly enumerated sets is explicitly enumerated,” but because we often care only that enumerations should exist and don’t want to keep having to define artificial ones, it is convenient to appeal to the axiom of choice so that we can extend the theorem to the murky world of countable but not explicitly enumerated sets. (more…)