Bourbaki and How To Write Numbers, A Trifle


So my attempt at keeping the Reading the Comics posts to Sunday has crashed and burned again. This time for a good reason. As you might have read between the lines on my humor blog, I spent the past week on holiday and just didn’t have time to write stuff. I barely had time to read my comics. I’ll get around to it this week.

In the meanwhile then I’d like to point people to the MathsByAGirl blog. The blog recently had an essay on Nicolas Bourbaki, who’s among the most famous mathematicians of the 20th century. Bourbaki is also someone with a tremendous and controversial legacy, one that I expect to touch on as I catch up on last week’s comics. If you don’t know the secret of Bourbaki then do go over and learn it. If you do, well, go over and read anyway. The author’s wondering whether to write more about Bourbaki’s mathematics and while I’m all in favor of that more people should say.

And as I promised a trifle, let me point to something from my own humor blog. How To Write Out Numbers is an older trifle based on everyone’s love for copy-editing standards. I had forgotten I wrote it before digging it up for a week of self-glorifying posts last week. I hope folks around here like it too.

Oh, one more thing: it’s the anniversary of the publishing of an admirable but incorrect proof of the four-color map theorem. It would take another century to get right. As I said Thursday, the five-color map theorem is easy. it’s that last color that’s hard.

Vacations are grand but there is always that comfortable day or two once you’re back home.

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Kenneth Appel and Colored Maps


Word’s come through mathematics circles about the death of Kenneth Ira Appel, who along with Wolgang Haken did one of those things every mathematically-inclined person really wishes to do: solve one of the long-running unsolved problems of mathematics. Even better, he solved one of those accessible problems. There are a lot of great unsolved problems that take a couple paragraphs just to set up for the lay audience (who then will wonder what use the problem is, as if that were the measure of interesting); Appel and Haken’s was the Four Color Theorem, which people can understand once they’ve used crayons and coloring books (even if they wonder whether it’s useful for anyone besides Hammond).

It was, by everything I’ve read, a controversial proof at the time, although by the time I was an undergraduate the controversy had faded the way controversial stuff doesn’t seem that exciting decades on. The proximate controversy was that much of the proof was worked out by computer, which is the sort of thing that naturally alarms people whose jobs are to hand-carve proofs using coffee and scraps of lumber. The worry about that seems to have faded as more people get to use computers and find they’re not putting the proof-carvers out of work to any great extent, and as proof-checking software gets up to the task of doing what we would hope.

Still, the proof, right as it probably is, probably offers philosophers of mathematics a great example for figuring out just what is meant by a “proof”. The word implies that a proof is an argument which convinces a person of some proposition. But the Four Color Theorem proof is … well, according to Appel and Haken, 50 pages of text and diagrams, with 85 pages containing an additional 2,500 diagrams, and 400 microfiche pages with additional diagrams of verifications of claims made in the main text. I’ll never read all that, much less understand all that; it’s probably fair to say very few people ever will.

So I couldn’t, honestly, say it was proved to me. But that’s hardly the standard for saying whether something is proved. If it were, then every calculus class would produce the discovery that just about none of calculus has been proved, and that this whole “infinite series” thing sounds like it’s all doubletalk made up on the spot. And yet, we could imagine — at least, I could imagine — a day when none of the people who wrote the proof, or verified it for publication, or have verified it since then, are still alive. At that point, would the theorem still be proved?

(Well, yes: the original proof has been improved a bit, although it’s still a monstrously large one. And Neil Robertson, Daniel P Sanders, Paul Seymour, and Robin Thomas published a proof, similar in spirit but rather smaller, and have been distributing the tools needed to check their work; I can’t imagine there being nobody alive who hasn’t done, or at least has the ability to do, the checking work.)

I’m treading into the philosophy of mathematics, and I realize my naivete about questions like what constitutes a proof are painful to anyone who really studies the field. I apologize for inflicting that pain.