Subject: categories: Re: Demystifying the categorial approach

From: "Reinhard Boerger" <[email-masked]>
Date: Tue, 04 Apr 2006 10:29:57 +0200

Hello,

Vaughan Pratt wrote:

> Reinhard Boerger wrote:
>   Of course, category
> > theoty usrather abstract and can hardly be explained to
> > non-mathematians.

Sorry for the misprints.

> Would this not be both true and false of any mathematical subject
> worth its salt?  Such a subject will include deep results and/or
> abstract concepts that are inaccessible even to many mathematicians,
> let alone nonmathematicians.  At the same time it should be possible
> to trace the chains of reasoning motivating those results and concepts
> back to origins that should be easy to motivate for the nonspecialist
> and/or nonmathematician.

I agree but that is not my point. I did not want to judge what is "good" or "deep"
mathematics. Of course, deep results are difficult to uderstand even for specialists,
and the existence of infinitely many primes or the party theorem are definitively not
deep. But their proofs require some amount of mathemtical thinking, which is on the
other hand still comprehensible for non mathematicians. So they can learn how
mathematics works. Then we can tell them about Fermat's Last Theorem or
applications in computerized tomography, certainly without proofs.

> As a case in point, category theory can be motivated by presenting it
> as an approach to axiomatizing sets and functions (and more generally
> algebraic structures and homomorphisms as their structure-respecting
> functions, but one need not start there).  In that approach, instead
> of defining a function to be a binary relation of a certain form, one
> postulates functions as primitives in their own right, axiomatized by
> the laws of composition and the existence of identities.

One point is selling mathematics to nonmathematicians, the other is selling
categories to other mathematicians (not necessarily to physicists). In general,
nonmathematicians don't even know what sets and functions are and why they are
needed in mathematics. I think that category theory makes several things clearer
(and also easier) and reveals connections and similarities between different branches
of mathematics. It is quite nice if it can also be used as a foundation, but for me this
is not the most important aspect.

                                                            Greetings
                                                            Reinhard