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Re: Attitudes to mathematics among successful research mathematicians
Posted:
Feb 15, 2013 11:47 PM


On 02/15/2013 07:14 PM, Paul wrote: > Hardy wrote this in A mathematician's apology > "I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively." > > I wonder if this type of attitude is prevalent among successful research mathematicians, even among the world's elite. Do all renowned mathematicians have a strong aesthetic sense of the beauty of mathematics or is the main motivation often that doing well at it allows people to believe that they're more intelligent than others? > > I suspect that many renowned mathematicians don't really have a strong aesthetic appreciation of maths at all, but the motivation is often a competitive motivation, and the desire to be thought to be intelligent. > > This becomes obvious when elegant and remarkable concepts get discovered, and the vast majority of mathematicians are completely indifferent unless they can use these concepts or results in their research. > > For example, how many mathematicians have any motivation to find out about surreal numbers? Few, I would think. If someone is in PDEs for example and doesn't use algebra, and has forgotten undergrad Galois theory, how many of such people can be bothered to open an undergrad textbook and relearn it?
It's not the same when one has to study a topic to pass a course. As an undergraduate, I didn't have too much appreciation for complex analysis. For real analysis, I could see things and I saw a purpose: series, integrals, limits, Lebesgue measure, the fundamental theorem of calculus and so on.
For complex analysis, I was obsessive about details of proofs on line integrals (where the parameterization doesn't matter), or Cauchy's Theorem for triangular regions (just the beginning of more general theorems for circular regions, etc.).
The applications were often in the calculus of residues, and I knew about numerial methods, so what's the point of all this machinery?
Being involved in serious research for a degree or as an assistant professor, there can be a pressure or concern about getting publishable results.
Already in grad school, people usually specialize a lot, probably because it's a good way to get going towards the frontiers of research in some area.
I'd say people in math are often interested in problems and questions in areas that they have some minimal mastery or acquaintance with. But if it's too easy, they might tune out.
I remember a professor in my undergraduate studies who taught analysis, and professed no interest in group theory.
Others aren't interested in 20th century set theory.
It's true that rivalry and envy can happen. I read maybe a young French mathematician writing that he wasn't necessarily delighted when a colleague got a "good result" ...
But there's a lot more to pursuing mathematics than competition and recognition or fame for work done.
In the grips of a problem one has chosen, one can be fascinated by the mysteries of the unknown and emerging insights into "what's going on", leading to higher level unerstanding or "understanding" .
Example (1): Classification of finite simple groups (a huge enterprise, done but reportedly scattered in lots and lots of papers).
Example (2): P =? NP (unsolved) .
Example (3): Alexander Grothendieck in the 1960s (ca. 19581970) working on EGA, SGA and the Weil conjectures:
http://en.wikipedia.org/wiki/Alexander_Grothendieck
I think it's Barry Mazur who wrote that while in Paris, he saw Grothendieck working very hard in his office ...
David Bernier
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