Saturday, June 12, 2010

Vladimir Arnold RIP

Vladimir Igorevich Arnold , one of my heros during student days passed away on 3rd June. I met him briefly in Bombay when I was a student; it is difficult to imagine that there is only three years difference between us. Here is an old interview and an obituary in NYT.


Rahul Siddharthan said...

Thanks for the interview. I found his comments on the Bourbakists interesting. I read similar comments in Jaynes' book on probability theory. He quotes similar statements from Poincare and Gell-Mann, among others. It is nice to know that if I find modern mathematics incomprehensible, it is not entirely my fault.

(Among the "pathologies" of modern mathematics Jaynes mentions the Hausdorff sphere paradox; where mathematicians see it as a valid proof of a counterintuitive result, Jaynes sees it as a "reductio ad absurdum" that invalidates the mathematical reasoning being used. He insists that one cannot reason on infinite sets (or infinite quantities) directly, but only as well-defined limits of finite sets (just as infinite series can have different "sums" on rearranging terms). He says this was the point of view of all mathematicians -- Gauss, Cauchy, Weierstrass, et al -- up until the 20th century. The "pathologies" of modern mathematics arise from abandoning this principle. I am not enough of an expert to judge, but about the Hausdorff paradox, it does seem to me that he may have a point.

gaddeswarup said...

I have been avoiding this all my life hoping that one day I would get around to understanding how mathematicians actually avoid these contradictions. What all I know is that one has to be careful about taking sets of sets. I never got around studying Axiomatic Set Theory and Model Theory which may provide some information. I think some like B.V. Rao or Srivatsava in ISI, Kolkata may know about it but now different interests have taken over in my case and I will never probably understand it. Early stages of the controversy are described in the last chapters og Bell's book "Men of Mathematics".

gaddeswarup said...

There is an article by Arnold in
"Mathematics:Frontiers and Perspectives" Edited by Arnold, Atiyah, Lax and Mazur, Published by Amer. Math. Soc. 1999, which you may like. He says in the article:
Among other important things Poincare explained that 'only non-interesting problems might be formulated unambiguously and solved completely'.