FOM: Re: groups

[Michael Thayer]

Pen Maddy writes:


>I think this question of what makes 'group' a good mathematical concept is
a
>very important one for philosophy (if not foundations) of mathematics.


I agree completely.

>
>While it is almost certainly not possible to give a perfectly general
>characterization of the good mathematical reasons for preferring
so-and-so's
>to schmo-and-so's

While I think this is right, the sociological idea suggest that a slightly
different question: namely what constitutes and "good" mathematical problem
to work on?
This certainly has elements of sociology, aesthetics etc.  the idea here is
that groups arose in the solution of specific problems, and that if an
entirely different group of problems had been considered central, group
theory would have developed later.

Of course there is also the point that groups are related to symmetries and
symmetries would probably come under Steve's #3 as functions which preserve
structure.  (Of course lattices are tied up with congruence relations and
homomorphisms, and they have never been as close to the mainstream as groups
are, despite the apparent fact that congruence relations were introduced
earlier (by Gauss) than isomorphisms were (I don't know who gets the credit
for this - so my chronology could be wrong)

Michael