## 04 August 2007

### Who gets credit for quadratic reciprocity?

In today's New York Times crossword, there's a clue "Discoverer of the law of quadratic reciprocity." The correct answer is, according to the crossword, this guy. I originally put in this guy instead. It turns out, according to the Wikipedia article on quadratic reciprocity, that "The theorem was conjectured by [first guy] and Legendre and first satisfactorily proven by [second guy]. [Second guy] called it the 'golden theorem' and was so fond of it that he went on to provide eight separate proofs over his lifetime." (I am deliberately obscuring the links because you might still want to do the crossword.)

Now, who should get the credit for "discovering" a mathematical result? The one who first suspected it might be true, or the one who proved it? I'm of the opinion that in this case they should share the credit (along with Legendre), mostly motivated by the fact that both of the people involved are Really Big Names. There are some examples in which First Guy discovered something and it's not named after him, but Erdos (and I don't think I'm spoiling anything by admitting that Erdos is neither First Guy nor Second Guy) once said that Goldbach's conjecture should be named for Goldbach, not First Guy, because "[First Guy] is so rich and Goldbach is so poor, it would be like taking candy from a baby." I don't know the history in the case of quadratic reciprocity. But I'm motivated here mostly by the fact that it's a lot easier to prove something if you already have reason to suspect it's true. For one thing, if you suspect something is true it is often on the basis of data, and you can look at that data and see how you might generalize the patterns you can see in it. (Quadratic reciprocity is almost certainly such a case, since data is easy to generate.) Secondly, there's a tremendous psychological boost to be gained from knowing that someone whose judgment you trust thinks something is true. Mathematicians are trained to think that only proofs matter, and I suspect there's an extreme strain of this that thinks that we really don't have any idea whether something is true until we've proven it or not; but someone who has given copious hints in the right direction surely deserves some of the credit. The tendency seems to be to give the credit to the person who put the last link in place; the highest-profile example is of course Wiles' proof of Fermat's last theorem. But what Wiles really showed was a special case of the Taniyama-Shimura conjecture; Ribet had already shown that Fermat would follow from this special case. So it seems to me that Ribet definitely deserves some of the credit (for establishing that as the right target for anyone wishing to prove FLT) and probably Taniyama and Shimura as well.

In the case of this particular theorem I realize that there are a very large number of people involved in one way or another, and it's hard to know who exactly to assign credit to, or -- and this is getting really silly -- how much credit to assign to them. Fermat proved that x3 + y3 = z3 has no trivial solutions. What should this count for? On the one hand, it's the first case. On the other hand, there are infinitely many cases. Should Pythagoras get some credit for coming up with his theorem, which inspired the whole thing? (Incidentally, any reasonable scheme of assigning "credit" to every mathematical result ever to all the mathematicians who were in some way involved has a pretty good chance of putting Pythagoras on top -- or at least of putting the Pythagorean theorem on top among theorems.) But even in the case of less famous results there are clearly a lot of people who did something towards them -- the people who first conjectured them, the people who proved some special case, the people who disproved some other special case (thereby helping to establish the boundaries of the result), and so on. Fortunately we don't need to find a way to quantify these contributions; decisions of who to hire or who to give prizes to can be made without them. (I'm not saying that the current hiring system is perfect, but it seems to work well enough.) And do you really want mathematicians in charge of some scheme that assigns a number to their total amount of mathematical contributions? Because let's face it, if you made up such a scheme mathematicians would find a way to beat it.

Except if it involved arithmetic. Mathematicians aren't any good at arithmetic.

(Oh, and this was going to be a post on how mathematically oriented people are good at crosswords. But it's not! Oh well. I'm sure I'll get around to writing that eventually.)

Unknown said...

Only marginally related, but since you're after comments, here's one on mathematical fame and credit, by Hardy: "... on the whole the history of science is fair, and this is particularly true of mathematics. No other subject has such clear-cut or unanimously accepted standards, and the men who are remembered are almost always the men who merit it. Mathematical fame, if you have the cash to pay for it, is one of the soundest and steadiest of
investments." (A Mathematician's Apology, Part 8.)

Anonymous said...

As to mathematicians and arithmetic, supposedly Gauss could do amazing feats of mental arithmetic, and there have been a few other similarly gifted famous mathematicians, though obviously none were more famous than Gauss. Wikipedia tells a nice story of Gauss not only doing difficult mental arithmetic but consulting his mental table of logarithms in the process. Hamilton and von Neumann were adept mental arithmeticians. Von Neumann supposedly could mentally multiply two eight-digit numbers rapidly, having show prodigious arithmetic skills from early childhood, and once at age six asked his mother, who was lost in thought, "What are you calculating?". Going a bit further afield, Feynman is an example of a very mathematically-gifted (he was a Putnam fellow) physicist who reputedly did rapid mental arithmetic.

Michael Lugo said...

anonymous:

you're right that some mathematicians had a reputation for being able to do great feats of mental arithmetic. In fact, someone compiled a list of a hundred or more tellings of the story where the young Gauss quickly found 1 + 2 + ... + 100 = 5050. Euler, of course, is another one who was known for mental arithmetic abilities; those stories are made even more impressive by the fact that he was blind at the end of his life.

I suspect that the facility for mental arithmetic is dying out, though, because calculators are more common; therefore there's a lot less use for skill at mental arithmetic. I'm not bad at it but I suspect that I am no match for my counterparts from an earlier era.

Rather paradoxically, I can do arithmetic quite well until you put dollar signs in front of the numbers. I'm not very good at handling money. (Fortunately, I have enough money and sufficiently cheap tastes that I can handle my financial affairs by the simple strategy of spending less than I earn; this is of course a luxury that is not available to everybody.)

Unknown said...

On mental mathematics: the record for finding the root of a 100-digit number has fallen from 23 minutes in 1970 to 4 seconds today.

Michael Lugo said...

Michael,

I'm a bit suspicious of that -- is it even possible to read a 100-digit number in four seconds?

That being said, I'm curious how much of that is due to better methods and how much of that is due to people who can "think faster".

Unknown said...

I'm curious about the "4 seconds" too. The guy's Wikipedia page says that the time includes "reading, calculating, and displaying the answer." One of the comments on the BBC news article is "Alexis is well known on the mental calculation circuit but not well-loved! He only ever seems to compete in obscure events in which hardly anyone else is interested, and the rules for which are dependent on him." The 13throot.com website (referenced by Wikipedia) is owned by Alexis, and the content is somewhat idiosyncratic. (Most of the edits to the Wikipedia page are anonymous, too...) Still, even with caveats it still seems impressive.

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