ONT Re: Category Theory
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CAT. Discussion Note 6
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JS = John Sowa
MW = Matthew West
Well, if the Ontology Archive ever gets working again,
I am hoping to lead some kind of an e-seminar there,
starting out with MacLane's book, which is the one
that most math folks read if they read only one.
In practice, though, most of this gets learned informally,
in the process of working on some other subject of interest,
with category theory being one of the main tools that gets
used when the going gets tough.
As a person who uses a lot of mappings between different domains
in his work, I think that you would especially benefit from some
of the work that's been done over the years to address precisely
these kinds of tasks.
The official "decade of birth" of category theory is usually given as the 1940's --
though of course you can trace all the main ideas back into the mists, to Riemann
and maybe even to Kant -- but what happened in the 40's was that mathematicians
were getting really bogged down trying to formalize what they do, so long as
they tried to do it in axiomatic set theory and formalized logic, and there
was a sense that most of the real work was falling between the cracks of
what these official doctrines were able to cover.
One of the important things to understand here is these folks were practical people,
with work to do, and if it could not be done effectively and efficiently in the way
that "philosophers of math" said they should be doing it, then they had no choice
but to reflect on the conduct of their own practice and to hammer out their own
tools to the task.
It appears to be a popular misconception, promulgated by the
sort of "philosopher of math" who never does much actual math --
how could they, if they stick to the methods they try to sell
others? -- that your average working theorist spends his days
proving stuff in axiomatic set theory with first order logic.
This is just not how it is -- unless an individual takes up
a special interest in working with both hands tied behind
his or her back.
Of course, some folks will go on to develop baroque and roccoco variations
on just about any subject you give them, if they have a lot of spare time
on their hands, but most of these tools were forged and hammered out to
do solid work on mathematical objects of definite interest.
The fact is that many of the problems that category theory was carved out to solve
are closely analogous to the problems of interrelation between different views and
takes on the world, of the sort that we have before us in the standard ontology job.
It would be a crying shame just to go out and wipe the slate clean and to waste all
of the knowledge that has already been mined, just for the lack of a little effort
to learn the practical methods that were used to mine it.
So I think that there has just got to be a way of explaining
all of this stuff in applicable, practical, sensible terms.
Jon Awbrey
MW: I have been reading (in fits and starts) the book John recommends below.
>
> It is intended as a undergraduate text. When John says it has
> been used in High School I am surprised rather than incredulous.
>
> The book comes in sections each with an exposition of some theory
> and then a number of tutorials with worked examples (and some for
> you to do if you wish) tackling real questions which real students
> asked.
>
> The main thing I find lacking is any sense of what I would use
> this for -- but this is not so unusual with pure maths.
JS: Category theory is usually considered an esoteric subject
> > because (1) it is not taught in high school and (2) most
> > textbooks are written at an advanced level.
> >
> > But there is a textbook of category theory that has been
> > used for teaching high-school students, and it contains
> > large numbers of examples to illustrate the concepts:
> >
> > 'Conceptual Mathematics: A First Introduction to Categories',
> > by F. William Lawvere and Stephen Hoel Schanuel,
> > Cambridge University Press.
> >
> > The first 100 pages could be read by a high-school student
> > with a little help from a teacher or tutor. The material
> > gets deeper and proceeds faster later in the book, but the
> > presentation is completely self-contained, and it could be
> > used for self-study by people who have forgotten most of the
> > mathematics they learned in college.
> >
> > I recommend it to anyone who might be interested in learning
> > the IFF system in order to (1) use it or (2) evaluate it.
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