Re: Where do mathematical ideas come from?


Mathematics Lover wrote:
> Put 100% on applied problems. all pure research ultimately is just a
> generalization of insights
> originating from the solution of applied problems.

A true generalization takes what is available with similar properties
and puts it in a more general framework. It was a bunch of
generalizations that took us from natural numbers and fractions to real
numbers. So, if it were not for generalizations the applied
mathematicians would still be sucking their thumbs when it came to
numbers. I think complex numbers arose in order to make the theory of
equations smoother. That was a generalization which later found
interpretation in other fields.
The "application oriented" Mathematicians' attitude towards Pure
Mathematics has often been the same as yours. I hear that when Cauchy
was doing his work on contour integrals, some powerful "application
oriented" Mathematicians did not want to hear about his (apparently)
empty pursuits. Now were they empty pursuits?
I would not spend all this time responding to you but I had to give an
example of a generalization that was not caused by applied pursuits,
but came to be as a result of Pure Mathematicians putting their house
in order.

Muhammad

.

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