IT WAS inevitable that this week鈥檚 award of the Fields medals, mathematics鈥� highest honour, would be dominated by Grigori Perelman鈥檚 refusal to accept his. Perelman鈥檚 work on the Poincar茅 conjecture is of such stature that he would have dominated the headlines whether he had turned up or not. Even so, Perelman鈥檚 no-show is small beer compared with other upheavals taking place in mathematics today.
For one thing, mathematicians are struggling to stomach the implications of computer-assisted proofs. For millennia, mathematics has held a unique position: once the truth of a proposition is established, it exists unassailable forever. No other science can boast such certainty. Now computer-assisted proofs, which use software to analyse details too numerous to check by hand, are introducing uncertainty. How do you prove there isn鈥檛 a programming error? Suddenly mathematicians are finding it hard to decide what even constitutes a proof (see 鈥淏urden of proof鈥�).
One way maths can best take on future problems is to have a clearer picture of those it has already slain. A global effort is under way to digitise paper-based archives so that the entire history of the subject is available electronically. Sadly, the project is bogged down by petty disputes over formatting, but with the right leadership, this so-called retrodigitisation could inject new life into many areas of mathematics.
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There is also a sense that mathematicians should increase their cross-pollination with other fields, not restricting themselves to physics but ranging further afield. This week saw the first award of the Gauss prize, which celebrates the influence of mathematics in other disciplines. The winner was Japan鈥檚 Kiyoshi Ito, whose work on modelling random events has been hugely influential in physics, economics and biology.
In the past, collaboration was thought to reduce an individual鈥檚 chance of bagging the biggest prizes. The International Mathematical Union, which awards the Fields medals, has sent a powerful message with its choice of winners this year by recognising Terence Tao and Wendelin Werner, who have built impressive reputations as serial collaborators.
So far so good, but there is more to do. Mathematics must increase its nurturing of talent from developing nations, for example. And it ought to be a matter of embarrassment that many maths meetings do not have a single woman present. Only by evolving can mathematics hope to face down its demons and reveal new truths. Only by including everyone will it continue to be a powerful cultural force.
