FOM: A pro cat-fom argument

Vaughan Pratt pratt at cs.Stanford.EDU
Thu Feb 19 02:14:44 EST 1998

From: Soren Riis
>A major discovery of Category theory 
>has been that the essence (often expressed as the most general 
>version) of virtually any mathematical result (i.e. theorem 
>plus proof) is best expressed and understood in terms of 
>Category Theory. 

A major discovery of French mathematics has been that mathematical
results are best expressed and understood in French.

Further generalizing, a major discovery of the X point of view has been
that mathematical results are best expressed and understood from the X
point of view.

>The claim is that the "essence" of virtually any mathematical 
>fact is best captured by category theory.

That would explain why the majority of articles in American Math Monthly
these days explain things in category theoretic terms.  Not.

On the other hand perhaps their authors would get their facts right
if they at least thought more categorically.  Opening up February's
American Math Monthly to the sort of article I'd be interested in,
"A Computer Search for Free Actions on Surfaces", one finds the second
paragraph starting out "Movement of a point on a surface can be performed
by a member of a finite group according to a prescribed *action* of
the group.  A group G *acts* on a space X if every element in G induces
a homeomorphism from X onto itself."  It then moves on to orientation
of surfaces.

The bug in this definition of action is that it omits the requirement that
the induced homeomorphisms compose according to the multiplication in G.
Omissions of this kind seem not to bother set theorists one whit---why
bother the reader with a second-order detail that only obscures the
essence of the notion of group action?  However it would greatly bother
a category theorist, to whom functoriality is a first-order detail
without which the concept of group action would be of no practical value.
And the category theorist would be correct there.

As long as set theorists and category theorists can't agree on what's
important, they're unlikely to ever see much virtue in the other side's
point of view.

Vaughan Pratt

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