FOM: groups

[penelope maddy]

I think this question of what makes 'group' a good mathematical concept is a
very important one for philosophy (if not foundations) of mathematics.
There are lots of other consistent notions in the vicinity of 'group' (call
them 'schmoups') that could be defined with equal logical justification, but
mathematicians don't define or study them because they are lacking in
'mathematical interest or importance'.  The trick is to spell out what this
comes to.

For what it's worth, I don't think we are forced to choose between purely
sociological justifications (the NSF funds projects on groups but not
projects on schmoups) and platonistic justifications (groups exist in some
sense that schmoups don't).  Rather, there seem to be objective and rational
mathematical reasons for preferring groups to schmoups.  Others would do
better than I in spelling them out.

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, I do think it should be possible to articulate what
factors are relevant and decisive in particular cases.  Any historical case
in which several options were rejected before the 'right' definition was
found ought to provide an example of how that process works and what count
as good mathematical reasons in that particular context. 

Any thoughts?

Yours,

Pen Maddy