Ars Mathematica

The Edge Foundation has collected over a hundred essays in response to the question “What is your dangerous idea?“.

I haven’t yet read all of them (75,000 words!), but I thought that Steven Strogatz’s idea was worth mentioning. With reference to the four-colour theorem, classification of simple groups, and sphere packing, he worries that mathematics might be getting too hard, that the use of computer programs in mathematical proofs leaves mathematicians with the ability to show something is true without understanding why.

Obviously the use of computers as an aid in proofs is relatively new. But, is it new that there are results where we dont really understand why they are true? I’ve always thought that on the frontiers things are usually not well understood; but, as the body of knowledge grows, new tools are developed and new insights achieved, and what was hard becomes easier. Computer proofs may have skewed this progression somewhat, but do they signal a more fundamental change? Is it worth speculating whether or not, without computers, mathematicians might have continued working on the four-colour theorem and we might have a “real proof” by now?

I don’t think I’m quite ready to accept the idea that we are now reaching the limits of the human brain.

Aside: Professor Strogatz mentions a recent article by Brian Davies, Whither Mathematics, which talks about similar issues. It also talks about using formal verification of computer programs when they are included in a mathematical proof. Until now I’ve not paid much attention to such things, but I guess that if mathematical proofs are requiring computer programs then we’ll need techniques to verify their correctness so they can be verified like more traditional proofs.

Another aside: Not Even Wrong and Cosmic Variance have some comments about a few of the physics related Edge essays.

If anyone’s interested in contributing to Wikipedia, here’s an opportunity. To fix some errant links, I’ve just created a stub page on corecursion. Unfortunately, I know almost nothing about the topic, so if you do, now’s your chance to show off.

Alain Connes and Mathilde Marcolli have posted a new survey paper on Arxiv, A walk in the noncommutative garden. There are many contenders for the title of noncommutative geometry, but Connes’ flavor is the most successful. Most of Connes’ examples are variants of the same basic idea: when a group acts nicely on a space, you can define a new space by collapsing each orbit of the group action to a single point (this construction is known as the quotient space of the action). Unfortunately, most group actions are not nice.

Connes and Marcolli describe an alternative construction. By a theorem of Gelfand, you can study spaces by instead studying its ring of continuous functions (see this Wikipedia article for precise details). Gelfand’s result puts the commutative in commutative geometry. For group actions that have badly behaved quotients, Connes introduced a noncommutative ring that functions as the analogue of the quotient space.

I found two interesting online resources on the Foundations of Mathematics:

• What is Mathematics: Gödel’s Theorem and Around by Karlis Podnieks
• Foundations of Mathematics by Alexander Sakharov

I’d wish everyone a happy new year, but clearly the mathematically inclined all refuse to recognize arbitrary time divisions.

I’ve been tipped, via e-mail, to some other weblog discussions inspired by Math is hard:

• Technicalities
• MathCog Idiocy