FOM: Re: Ferguson's points

[Michael Thayer]

Neil Tennant comments
>Is the
>only justification offered by mathematicians that, once groups are
>defined this way, one finds them all over mathematics (e.g. groups of
>transformations in geometry); and that the group concept leads to
>various interesting theorems about groups?

Just out of curiosity, what sort of justifications for studying a particular
set of axioms are there?

Thus the stated justification is probably best viewed as socialogical rather
thatn philosophical.  If you cant show that your chosen subject has ties
with other mathematics, you will lose professional status.  Arnol'd has
complained in print that this sort of thing explains why some fields are
underrepresnted among Fields medal awards, so a sociological justification
would seem to be helpful.

A more philosophical justifiaction would seem to have to depend on the view
that there are only certain types of mathematical object waiting to be
discovered in Plato's heaven and that going off on a tangent is a bad thing.

I would like to see a more resoned justifiaction than this, but it will be
hard to come by, I think.


>
>3. Perhaps one of the most important foundational problems is the
>problem of justifying a particular logic (and choice of a language
>with certain expressive powers) for different areas of mathematics;
>and, possibly, tailoring the logic to the demands of the particular
>area of mathematics in which one is working.
>
I agree, and found your paper `Transmission of Truth and Transitivity of
Proof' a nice effort in this direction, although I wonder if Steve and
Harvey would consider this f.o.l(ogic) rather than f.o.m(ath).