Sunday, 16 September 2012

Amazing mathematics - the abc conjecture

I heard earlier this week that the solution to an outstanding mathematical problem, the proof of the abc conjecture, has been announced.  (The proof is claimed by Shinichi Mochizuki of Kyoto University, and the ideas are so new and deep that it will take a long time for mathematicians to digest the work and be convinced of the validity of the proof, but the claim is being taken seriously.) Why did this news on Twitter make me happy?

You might think this would have been because it was a problem I have been thinking about for years, that I have attempted to solve myself, that I have lost sleep on.  Or perhaps because I am aware of the ramifications for mathematics, the new vistas that will be opened up, the new opportunities that will arise.

Well, I had heard of the abc conjecture.  Perhaps I once even knew what it was, but I had forgotten.  I have no knowledge of the subject area, the applications (if any) or the methods used.

Yet I am excited by the news of the possible proof!  How is it that a development about a problem which I know nothing about, which I couldn't even describe in the most general terms, can matter to me?

Well, here (from the excellent Wikipedia article on the conjecture) is a statement of the conjecture: it asserts that the answer to the following question is "yes":  "For every ε > 0, are there only finitely many triples of coprime positive integers a + b = c such that c > d (1+ε), where d denotes the product of the distinct prime factors of abc?"  What does this mean?  Basically if the conjecture is true, then (as I understand it) it means that only exceptionally is the product of the prime factors of abc significantly less than c.

Now this is, frankly, quite an obscure statement about numbers, and I cannot envisage any life-changing applications.  What I find wonderful is that human minds like mine can prove this statement.  I cannot even begin to imagine how one would set about proving such a conjecture.  I still have an instinctive belief (which, rationally, I know is rather naive) that mathematical facts are true regardless of the nature of the human brain, the laws of nature, and so on: thirteen would still be a prime if the human race had never existed, if the laws of physics were totally different, if no sentient creature had ever come into being.  That a mind like mine can establish such necessary, deep facts is amazing, a glimpse of something much more true than anything else in our existence.

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