Re: Is the theory of topological vector spaces still alive?

In article <e477922f-1c6b-41ed-86e4-3c41be590f32@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<s.s.akbarov@xxxxxxxxx> wrote:
Mathematicians do not acknowledge their duty to explain simply and
clearly why their field of interests is useful. Gradually they turn
into sportsmen whose aim is to impress the audience by their skill,
and nothing is important besides the skill.

I agree that this kind of abuse is bad and should be fought, though I do not
see it as the same thing as hyper-specialization. Even if a subject has
connections to other fields, that doesn't automatically make it interesting
and beautiful and important. You as a student may still not be convinced
that it is a beautiful and important subject, and powerful experts may still
abuse their power to force others to accept their point of view. It seems
misguided to me to lay the blame on lack of connections to other fields
per se, rather than on the abusive behavior of the people in question.

More interesting to me is the question of whether it's a sign of bad health
when a field becomes dominated by the construction of counterexamples. I
think it certainly can be a bad sign, but even here I would be wary of
over-generalization. Note that people usually don't complain if a subject
contains lots of *examples*. But what makes something a counterexample
rather than an example? It usually means that our naive intuitions about
the subject are wrong. Some areas of mathematics may be particularly
counterintuitive, so that we need an unusual number of (counter)examples
to correct our intuition and map out the conceptual landscape properly.
Devoting a period of time, even a rather long period of time, to building
these counterexamples may in some cases be exactly the right thing to do.

Of course, if the goal shifts from trying to understand natural mathematical
structures to demonstrating one's virtuosity, then something has gone badly
wrong. But again, this can happen in any field, specialized or not.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

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