Spectacular advances in number theory
Last weeks we heard about two spectacular results in number theory. As announced in Nature, Yitang Zhang proved that there are infinitely many pairs of consecutive primes which are at most 70 million apart! This is a sensational achievement. Pushing 70 million to 2 will settle the ancient conjecture on twin primes, but this is already an extremely amazing breakthrough. An earlier breakthrough came in 2005 when Daniel Goldston, János Pintz, and Cem Yıldırım proved that the gaps between consecutive primes is infinitely often smaller than .
Update: A description of Zhang’s work and a link to the paper can be found on Emmanuel Kowalski’s bloog Further update: A description of Zhang’s work and related questions and results can be found now in Terry Tao’s blog. Terry Tao also proposed a new polymath project aimed to reading Zhang’s paper and attempting to improve the bounds.
Harald Helfgott proved that every integer is the sum of three primes. Here the story starts with Vinogradov who proved it for sufficiently large integers, but pushing down what “sufficiently large” is, and pushing up the computerized methods needed to take care of “small” integers required much work and ingenuity.
Why is Mathematics possible?
The recent news, and a little exchange of views I had with Boaz Barak, bring us back to the question: “Why is mathematics possible?” This is an old question that David Kazhdan outlined in a lovely 1999 essay “Reflection on the development of mathematics in the twentieth century.” The point (from modern view) is this: We know that mathematical statements can, in general, be undecidable. We also know that a proof for a short mathematical statement can be extremely long. And we also know that even if a mathematical statement admits a short proof, finding such a proof can be computationally intractable. Given all that, what are the reasons that mathematics is at all possible?
It is popular to associate “human creativity” with an answer. The problem with incorrect (or, at least, incomplete) answers is that they often represent missed opportunities for better answers. I think that for the question “why is mathematics possible” there are opportunities (even using computational complexity thinking) to offer better answers.
Please offer your answers.