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| date | 2001-04-24:20:30 | |
| Reading responses |
Trying to catch up on stuff I've read over the past couple of weeks and haven't had time to comment on. I'm going to write more about the Fergeson book I mentioned in the last entry but one, but not right now. I think Fergeson has raised some really important issues regarding Microsoft, and I'm getting a sinking feeling that I know pretty much how they're going to take over the Web, and I don't see how anyone could possibly stop them, because there's no money in it. Apostolos Doxiadis has written an interesting novel about mathematics in Uncle Petros and Goldbach's Conjecture. Novels that deal with deeply technical subjects--think Moby Dick--are few and far between, and its hard for me to tell how well Doxiadis has portrayed mathematicians to the average person, but for me the story worked very well. When I was a grad student I used to have dinner once a month or so with Barry Hill-Tout, a mathematician. I asked him every couple of months, "So what is it that mathematicians actually do?" After a year or so his answers started to make sense. I subsequently worked closely with a person who dropped out of a math Ph.D. program, and he put the polish on my idea of what mathematicians care about, and Doxiadis portrays people not too unlike the mathematicians I know. Goldbach's Conjecture is one of the great unproven propositions of number theory. It is compellingly simple: "Every even number is the sum of two primes." It is known to be true for every even number up to some astonishing size--there are probably computers even now searching for violations by brute force. But no one has any idea if its true or not. Most mathematicians believe it is, but who knows? The novel's narrator discovers that his anti-social uncle who lives in a cottage and tends his garden outside of Athens is in fact a mathematician who once had great stature. The bulk of the novel is Uncle Petros' story of his own pride and downfall--he decided young that it was all or nothing, he would be the greatest mathematician since Gauss or nothing at all, and he set himself the goal of proving Goldbach's Conjecture. Although an academic, he behaved like a kook, in particular he isolated himself from his colleagues and didn't keep current with the literature. His hubris cost him dearly, although he's in good company. Newton didn't publish until long after the bulk of his work had been done, Cavendish I don't think published anything while he was alive, nor did Leonardo da Vinci, just to name a few. But the modern world, with it's industry of scholarship, is less forgiving, and those who don't contribute to both ends of academic discourse are putting themselves at great risk. Everyone who's ever had an apparently great idea has been tempted to hide it from the world to see if it'll grow. This is often a good idea--many things that appear great in the first flash of insight turn out to be pretty damned stupid when they see the light of day. The difference between a kook and an academic is small, but important: kooks persist in their pursuit of dumb ideas after they know they're dumb. Academics try to squeeze what value they can out of their mistakes and then move on. For Uncle Petros, Kurt Godel's work on undecidability turns out to be deeply disturbing, because you never know when working on an unproven conjecture if it might not be formally undecidable. Apparently it is impossible to tell, using Godel's analysis, if any given proposition is undecidable or not, which is something most mathematicians don't seem to lose any sleep over. Why they don't is a little curious--I know why I don't lose any sleep over it; because my work doesn't involve much formal reasoning from systems like Peano's axioms. I think the whole idea of starting with a tiny set of axioms and deriving every imaginable truth from them is silly, which is one reason I'm not a mathematician (lack of talent is another.) But its interesting that the enterprise of mathematics goes on, with people plunging gamely after elusive propositions through thickets of theorems, even given that there's no assurance the chase has an end. I don't know if anyone has ever even tried to work out the probability of any given proposition being undecidable. It might be possible to do so. As to Goldbach's Conjecture--I think it's false. My argument is simple and certainly wrong, but plausible enough to confuse the naive. Consider a large number, X. Now let it get bigger. As X increases, the number of even numbers smaller than it grows linearly. But the number of prime numbers grows as X/log(X), which is less than X. So the primes get less dense while the evens have a constant density. So it seems that if we let X get big enough, we won't have enough primes smaller than it to add up to all the even numbers smaller than it, What is wrong with this argument is left as an exercise for the student. |