Mathematics and mathematics only, ...

Mathematics and mathematics only, ...

[Bob Coecke]

Dear Andre and others,

> But presently, I am not convinced that quantum computing can
> contribute significantly to category theory.

What the quantum information `hype' has done is injected some new blood in
foundations of quantum mechanics research, an area which for several
reasons had been suffocated by by the end of the previous century, despite
the universal discomfort of the physics community with quantum mechanics.
(a typical slogan which reflects this is: ``don't ask questions just
compute'')  One would expect that this surge of quantum foundations, which
meanwhile has led to many novel ideas, approaches, and radically different
manners to think about physics in general, will ultimately lead to new
mathematics. Moreover, the natural guise of many of these new ideas is
within category theory, a message that some including myself have been
trying to pass on within the foundations of physics communitee, with
moderate success.

It moreover seems to me that identifying the appropriate mathematical
structure to model a natural phenomenon is arguably as important as
identifying the structure itself as a mathematical entity (cf Riemannian
geometry and its re-discovery by Einstein et al to model relativistic
space-time), even if it doesn't contribute to the area of mathematics in
return.  To my knowledge history has shown that ultimately it will do
something in return, be it only due to the increased activity in
the area, and it is here that money kicks in significantly.

More boldly put, history seems to agree with the fact that in many cases
important progress in fundamental physics fuels important progress in
mathematics and vice versa.  If category theory is, as has been claimed on
this list, the greatest mathematical achievement of the 20th century, then
one would expect it to have led to many important developments outside
mathematics too, which may or may not inject something in to category
theory in return.  By only having `computer science' as a non-pure
mathematics area in a list of now 15 subjects this seems not to be the
case (yet).  [I conceive quantum field theory in category-theoretic guise
as a branch of mathematical physics of the pure mathematics kind]
Doesn't this seriously undermine category theory's claim to fame?

In this context there is the highly unfortunate fact that there are
certain quite prominent people in the category theory community who think
that any deviation from treating category as a branch of pure mathematics
and pure mathematics only is a bad thing!

Best wishes, Bob.




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