Originally Posted by

**accountabled**
Originally Posted by

**DrRocket**
(...)

I would be willing to guess that if you took a pole of professioinal mathematicians, most would not have heard of more than one or two of the constants in that list, and would not care either.

I understand your logic, however I also do recall from reading James Gleick's book 'Chaos', that the first thing Feigenbaum did after he discovered the new constants, was to try to relate them back to

and

. I guess that's the same thing Riemann tried to do when he found the first non-trivial zero at

. Also when Euler managed to prove the closed form of

he probably immediately started to look for series that would produce some other "mathematical constants".

I believe these constants could therefore well reside at the forefront of the minds of many mathematicians (and most likely with those that are into number theory).

If they reside there I have seen no expression of that fact from any of the many mathematician that I know, including me.

There are several proofs that

, including one that uses Fourier series that I find interesting. But I still don't think of

as a fundamental constant.

Every convergent series produces a constant. And given any number at all it can be realized as an infinite series. It is only when a useful quantity is related to a series in some natural way that anything interesting arises.

BTW I must admit that I have assiduously avoided reading Glieck's book. That is for two reasons. First, I generally only read popularizations written by people who have actually been involved in the research being described. I do not normally read popularizations by professional popularizers. Second, I have a strong bias against may things that go under the name of "chaos". I respect mathematicians who work in topological dynamics, ergodic theory, dynamical systems, etc. Poincare, who initiated many of the ideas with his work in celestial mechanics and stability was clearly a first-rate mathematician. But, I have yet to meet anyone who claimed to work in "chaos theory" who could prove anything, or who really understood mathematics. If you would like to read a real mathematics book that actually covers a bit of chaos in a fully rigorous way I can recommend Bob Devaney's book

*An Introduction to Chaotic Dynamical Systems*.