Examples first II

It’s what blogging is all about I suppose, but I have been surprised in several different ways by the comments on my previous post. To begin with, I was so sure of the principle I was advocating that I thought that all I’d have to do was explain it briefly and then anybody who read it would instantly agree with it. That was clearly pretty naive of me, and I certainly didn’t expect that some people would be actively hostile to the idea (though I suspect that their real target was not precisely the same as what I was putting forward). But I was also surprised by the number of interesting further points and qualifications that were made, which I will now try to use to articulate a more nuanced version of the principle.

My favourite pedagogical principle: examples first!

This post is about a very simple idea that can dramatically improve the readability of just about anything, though I shall restrict my discussion to the question of how to write clearly about mathematics. The idea is more or less there in the title: present examples before you discuss general concepts. Before I go any further, I want to make very clear what the point is here. It is not the extremely obvious point that it is good to illustrate what you are saying with examples. Rather, it is to do with where those examples should appear in the exposition. So the emphasis is on the word “first” rather than on the word “examples”.

If this too seems pretty obvious, I invite you to consider how common it is to do the opposite.

A paper on the ArXiV

Today I did something for the first time ever that I should have done many times before: I put a paper on the ArXiV. Since I’ve got a blog I thought I’d use it to give the paper a small plug and, more importantly, to let anybody who might already be familiar with the paper know that I have revised it quite a bit recently, for the better.

The paper itself is called “Hypergraph regularity and the multidimensional Szemerédi theorem.” At the bottom level, the basic idea of the paper is due to Ruzsa, Szemerédi and Rödl. Ruzsa and Szemerédi started the ball rolling with a short and very clever argument that showed that Szemerédi’s famous theorem on arithmetic progressions, in the case of progressions of length 3, could be deduced from Szemerédi’s almost as famous regularity lemma, a remarkable result that allows any graph to be partitioned into a bounded number of pieces, almost all of which “behave randomly.”

When are two proofs essentially the same?

A couple of years ago I spoke at a conference about mathematics that brought together philosophers, psychologists and mathematicians. The proceedings of the conference will appear fairly soon—I will give details when they do. My own article ended up rather too long, because I found myself considering the question of “essential equality” of proofs. Eventually, I cut that section, which was part of a more general discussion of what we mean when we attribute properties to proofs, using informal (but somehow quite precise) words and phrases like “neat”, “genuinely explanatory”, “the correct” (as opposed to merely “a correct”), and so on. It is an interesting challenge to try to be as precise as possible about these words, but I found that even the seemingly more basic question, “When are two proofs the same?” was pretty hard to answer satisfactorily. Since it is also a question on which we all have views (since we all have experience of the phenomenon), it seems ideal for a post. You may have general comments to make, but I’d also be very interested to hear of your favourite examples of different-seeming proofs that turn out, on closer examination, to be based on the same underlying idea (whatever that means).

The exchange lemma and Gaussian elimination

Thanks to this comment, I have finally decided to try to understand in what sense Gaussian elimination and the Steinitz exchange lemma are “basically the same thing”. It’s not at all hard to spot similarities, but it seems to be a little trickier to come up with a purely mechanical process for translating proofs in one language into proofs in the other.

It might be of some interest to know how I approached this post. Rather than working everything out in advance, I started with an incomplete understanding of the connection. I had thought about it enough to convince myself that I could get to the end, but found that as I proceeded there were a few surprises, and the eventual connection was not quite as close as I was expecting. (Actually, this paragraph is slightly misleading. I am writing it while in the middle of writing the rest of the post. I’ve had a few surprises, and though I am fairly sure I’ll get to the end I am not quite sure what the end will look like. [Note added after I’d got to the end: it was nothing like what I expected.])