Explanatory and Unexplanatory Proofs

23 10 2005

I remember one theorem that I proved, and yet I really couldn’t see why it was true. It worried me for years and years… I kept worrying about it, and five or six years later I understood why it had to be true. Then I got an entirely different proof… Using quite different techniques, it was quite clear why it had to be true.

Michael Atiyah, in a 1984 interview with the Mathematical Intelligencer, quoted in Jamie Tappenden’s Proof Style and Understanding in Mathematics I

On the FOM e-mail list there have recently been a few discussions of the notion of explanation in mathematics by Allen Hazen, Richard Heck, and Richard Zach, suggesting that it may or may not be ready yet as a well-defined enough topic to work on. However, I think probably the best way to give it better definition is to gather more examples of it.

So if you, or any of your friends or colleagues, have good examples of proofs that are explanatory (or not), then send them to me at easwaran at berkeley dot edu. I suppose if it’s a very short example, or just a link to some example posted elsewhere, a comment would be good too, so that others can see it. Once I have a few of them, I’ll try to figure out some useful way to make these proofs accessible to others too, and credit the submitters.

Probably the most useful examples would be two proofs of the same result, one of which is clearly a better explanation than the other. For instance, Fürstenberg’s topological proof of the infinitude of primes, despite being remarkably clever, is clearly not as good an explanation of this fact as Euclid’s original proof. Of course, plenty of good examples will probably be like whatever Atiyah was talking about above, and exist in contemporary research, rather than in well-established results. I don’t expect to necessarily understand the relevant proofs, but it’ll still be helpful to have a collection including them, both for other people’s use, and in case I want to check some general or structural relations between the proofs.




2 responses

30 10 2005

Doesn’t your question hinge on the definition of an “explanatory” proof? To a constructivist, only a constructive proof is explanatory; a proof relying on reductio ad absurdem cannot explain anything about the result, he/she would say.

31 10 2005

Well that’s the point – the idea is to come up with examples that people find explanatory or not, and then see what unifying features they have. People who have thought a decent amount about intuitionism one way or another probably have more theoretical corruption going on in deciding on certain examples. But still, we all recognize that certain proofs are more explanatory than others, and we’d like to see what makes them so.

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