19 October 2007

The axiom of choice and the wisdom of crowds

(The two things in the title aren't related, but this post is about both of them.)

I just came across this essay on the Axiom of Choice by Eric Schecter. I'm not a logician, so I don't have any deep commentary. It contains two amusing quotes, though, one by Bertrand Russell:
"To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed."

(the idea here is that you can distinguish between left shoes and right shoes, but not left and right socks), and

The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

where all three of those statements are equivalent.

A classmate of mine, though, just expressed the opinion that logicians "should" be showing that certain problems are "too hard" for current mathematics, by, say, showing that they're equivalent to problems we already know we have no hope of solving with the current state of affairs.

I wonder to what extent it would be possible to use the "wisdom of crowds" to quantify how difficult a given problem is. If one could poll people in a given area and have them say "I think Problem X is harder than Problem Y", you could probably use that information to rank open problems within a given subfield; people's intuitions wouldn't be perfect but if you had enough of them you'd come up with some interesting results. The hard part would be collating the results from various parts of mathematics; if I know Algebra Problem X is harder than Algebra Problem Y, and Analysis Problem Z is harder than Analysis Problem W, then how do you compare X and Z? You'd need links between the various areas to know how to put all that information together, and to get all areas on the same scale. Perhaps what we need is more people offering financial bounties for problems, because hard currency is a scale that everyone can agree on the value of. (Too bad Erdos is dead, and the Clay Foundation only has million-dollar prizes. If some entity were giving out a prize of $1000 for problem A and $2000 for problem B, then that would let me know that the community as a whole considers problem B to be harder than problem A. i don't know about "twice as hard", though; what does that mean?)

The usual "prediction market" methods wouldn't immediately seem to work; people could bet on which problems they expect will be solved first, but just because a problem is solved later doesn't mean it's harder.

Some other interesting things on Schechter's web page include Common Errors in College Math (which I probably should point my students to, come to think of it) and an explicit statement of the "cubic formula", i. e. a solution by radicals to the equation ax3 + bx2 + cx + d. (This formula is a couple lines long significantly longer than the quadratic formula, of course, which is why you don't have it memorized. The "quartic formula" is, if I remember correctly, a couple pages, although it contains a large number of the same subexpressions; in both cases there is a shorter way to write down the formula than the actual formula itself.)

1 comment:

Michael Ejercito said...

The cubic formula is easy to remember, since there are basically three subexpressions to remember.