Thursday, August 31, 2006

The Sokal Affair

I won't bore the reader by describing the well-known hoax, but I did want to provide a link to a website that has collected a lot of documentation related to it.

http://www.physics.nyu.edu/faculty/sokal/

Tuesday, August 29, 2006

Set Theory Proof

Here's a proof that I have always found fascinating.

Theorem: Given any set A, the power set of A has strictly larger cardinality than A.

For two sets to have the same cardinality, there must exist a bijection between the two sets. So, let us assume that there exists such a bijection between A and P(A), denoted f:A->P(A).

For each x in A, either x is in f(x) or x is not in f(x). Consider the subset S of A defined as
S = {x in A | x is not in f(x)}.

Since we have assumed that f:A->P(A) is a bijection and S is in P(A), then there exists some t in A such that f(t) = S.

Either t is in S or t is not in S. Suppose t is in S. Then t satisfies the condition that t is not in f(t) which is S. Therefore t is not in S. Suppose t is not in S. Then t does not satisfy the condition, and t is in f(t)=S. Therefore t is in S.

So in either case, we arrive at the contradiction that t is in S and simultaneously t is not in S. The source of the contradiction is the assumption that there exists a bijection f:A->P(A). Therefore, there exists no such bijection and the cardinality of P(A) is not the same as the cardinality of A. Since the cardinality of P(A) cannot be smaller than the cardinality of A, the conclusion is that P(A) has strictly larger cardinality than A. (Q.E.D.)

Saturday, August 26, 2006

Finite Simple Group (of Order Two)



The lyrics of this song can be found here. More information on the group (from Northwestern University) can be found at http://www.kleinfour.com/.

[Thanks to TMB for sending me this video.]

Friday, August 25, 2006

Fields Medals 2006

Awarded once every 4 years by the International Mathematical Union, the Fields Medal recognizes outstanding contributions to mathematics.

This year's winners...

Andrei Okounkov “for his contributions briding probability; representation theory and algebraic geometry.”

Grigori Perelman
“for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.” (Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow. Perelman is quite the recluse and actually turned down the Fields Medal.)

Terence Tao
“for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.” (At 31, Tao is one of the youngest mathematicians to be awarded the Fields Medal.)

Wendelin Werner
“for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.”

Thursday, August 24, 2006

Busy year for Pluto

The International Astronomical Union defined the term "planet" for the first time. This definition excluded Pluto from planethood, so Pluto was reclassified under the new category of "dwarf planet" along with and Ceres and Eris.

Tuesday, August 15, 2006

At Berkeley

On a bulletin board in the mathematics department at UC Berkeley, somebody had left a flier offering a $300 reward for a solution to the equation x^5 + ax^4 + bx^3 + cx^2 + dx = e in terms of a, b, c, d and e. If you studied college level math, you know this problem is unsolvable. The insolubility of the quintic is proved as part of Galois theory in the second semester of undergraduate abstract algebra. I am not sure if this was a hoax or exactly what the purpose of this flier might have been, but this being Berkeley several people had scribbled sarcastic offers to square the circle (another problem that has been proven to be unsolvable) or to prove that pi is a natural number.