Momentary Language

Paul Cohen (April 2, 1934 – March 23, 2007)

According to the AP, Paul Cohen, the mathematician noted for his revolutionary work in set theory, has died. The American Mathematical Society describes his contribution thus:

He made fundamental contributions to mathematical logic, mathematical analysis, and partial differential equations. In particular in solving Hilbert's first problem, "Cantor's Problem on the Cardinal of the Continuum," in which he proved that the axiom of choice and the generalized continuum hypothesis are independent of Zermelo Fraenkel Set Theory, he introduced the powerful technique of forcing, which today remains one of the basic tools in set theory.

Here's a quote from Cohen's Set Theory and the Continuum Hypothesis (p. 151):

A point of view which [I feel] may eventually come to be accepted is that [the continuum hypothesis] is obviously false. The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now [creating] the set of countable ordinals ... is merely a special and the simplest way of generating a higher cardinal. The set C [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach C. ... This point of view regards C as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.

"Everyone in his heart of hearts agrees with Baudelaire: 'To be a useful person has always seemed to me something particularly horrible ...'"

The Sunday Times of London reprinted a wickedly irreverent and very funny piece by Auden, entitled (by an editor at the Times, no doubt) "So You Think You Can Write." I can't resist giving you three longer excerpts.

First this:

The girl whose boyfriend starts writing her love poems should be on her guard. Perhaps he really does love her, but one thing is certain: while he was writing his poems he was not thinking of her but of his own feelings about her and that is suspicious. Let her remember St Augustine’s confession of his feelings after the death of someone he loved very much: “I would rather have been deprived of my friend than of my grief.”

Then this:

Everyone in his heart of hearts agrees with Baudelaire: “To be a useful person has always seemed to me something particularly horrible,” for, subjectively, to be useful means to be doing not what one wants to do, but what someone else insists on one’s doing. But at the same time, everyone is ashamed to admit in public that he is useless. Thus if a poet gets into conversation with a stranger in a railway coach and the latter asks him: “What is your job?”, he will think quickly and say: “A schoolteacher, a beekeeper, a bootlegger,” because to tell the truth would cause an incredulous and embarrassing silence.

Finally, this:

Happy the lot of the pure mathematician. He is judged solely by his peers and the standard is so high that no colleague can ever win a reputation he does not deserve. No cashier writes articles in the Sunday [New York] Times complaining about the incomprehensibility of modern mathematics and comparing it unfavourably with the good old days when mathematicians were content to paper irregularly shaped rooms or fill bathtubs with the waste-pipe open.

As I was reading that last passage, I couldn't help thinking that philosophers have a similar reason to envy the pure mathematician. (Come to think of it, both the philosopher and the pure mathematician can sympathize with the thoughts expressed in the second passage, the feeling of embarrassment at admitting to being useless.)

Surveying the infinite

In a review in today's Frankfurter Rundschau, Joachim Lange reveals himself to be relatively unimpressed with the world premiere in Halle an der Saale of Ingomar Grünauer's opera, "Cantor: Surveying the Infinite." Nonetheless, any attempt to sing about transfinite ordinals in, say, a sextet, à la Lucia di Lammermoor, seems pretty impressive to me.

Poincare's Conjecture

In what became one of the so-called "Millenium Problems", Henri Poincare conjectured that the 3-sphere is unique in being the simplest 3-dimensional manifold. (More info here.) It is likely that the conjecture has now been proved by the St. Petersburg mathematician Grigori Perelman, who stands to earn $1 million for having solved one of the millenium problems.

Poincare himself proved that the 2-sphere is the simplest manifold in 2 dimensions, respectively. Given that, by 1982, Poincare's Conjecture was proved for n-spheres, where n is greater than or equal to 4, was it at all likely that it would fail to hold for n=3? That is, did we have had some prior justification in believing Poincare's Conjecture to be true, given that we knew that it held for all n other than n=3?