THE PARADOX OF PROOF
by Carolyn Chen
On August 31, 2012, Japanese mathematician Shinichi Mochizuki posted four papers on the Internet.
The titles were inscrutable. The volume was daunting: 512 pages in total. The claim was audacious: he said he had proved the ABC Conjecture, a famed, beguilingly simple number theory problem that had stumped mathematicians for decades.
Then Mochizuki walked away. He did not send his work to the Annals of Mathematics. Nor did he leave a message on any of the online forums frequented by mathematicians around the world. He just posted the papers, and waited.
The Joy of Math – Charles Krauthammer, Time Magazine, April 18, 1988
A quote from the article:
What higher calling can there be than searching for useless and beautiful truths? Number theory is as beautiful and no more useless than mastery of the balance beam or the well-thrown forward pass. And our culture expends enormous sums on those exercises without asking what higher end they serve.
I would rephrase this quote as simply What higher calling can there be than the search for beauty or truth. In any case, discussions about the role of mathematics – or almost anything, for that matter – avoid a serious philosophical problem. If one does X in order to achieve Y, why does one seek Y? Is it because Y is intrinsically worthwhile, or is it because Y is justified by Z? And so on. There are only two possibilities: an infinite regress of X, Y, Z, etc., or a set of things-of-value-in-themselves. Perhaps there is one such thing, perhaps there are several. If it is one, then a modern answer might be money. But of course that can’t be it, because one seeks money to buy food, housing, recordings of the Bach cello suites, etc. I think the answer is several things-of-value-in-themselves. But which ones?
The dichotomy of infinite regress versus things-in-themselves is a well-known and well-worn topic in theology and cosmology. Did God create the universe? Who created him? Etc. Or was God, or the universe created-in-itself?