Famous unsolved problems etc

Famous unsolved problems etc

[Paul Taylor]

I had wanted to give more careful consideration to this question
before posting about it again,  but an emergency has arisen:

My website (paultaylor.eu) is likely to be out of action for the rest
of this week. Apparently the hosting company was badly hacked, and
even their own webpage doesn't exist at the moment.  However, email
gets through to me by a backup route, so please do not try to second
guess my email address.   You can access my papers etc by adding
    site:paultaylor.eu
to a Google query and using their cached versions.

The version of my diagrams package on CTAN was out of date (sorry).
I have uploaded a new release (V3,93) and asked them to install it
quickly.  The package now automatically recognises when it is running
under XeTeX, as it has done with PDFTeX for some time.  I would like
to thank Apostolos Syropoulos for his help in implementing this, which
he wanted to use to combine my diagrams with Greek text in Unicode.

---

I would like to thank all of the people who sent supportive responses
to my autobiographical comments.   However, it seems that people read
these rather more literally than was intended,  whilst incorrectly
interpreting the "inappropriate boyfriend" as a metaphor.  I was
certainly not suggesting that I had a relevation about constructive
categorical logic during puberty (which would have coincided with
the invention of elementary toposes) and that this was suppressed
by my mathematical teachers until my rebellion!

Of course, this was exactly my point:  conceptual and constructive
thinking (which are by no means coincident) require mathematical
and personal maturity.

My reaction to Ellis Cooper's question about "Fundamental Theorems"
was that Ronnie Brown and others had answered this question under
the "famous unsolved problems" heading.   Notwithstanding the
occurrence of the phrases "fundamental theorem of algebra" and
"fundamental theorem of interval analysis" on my (dormant) website,

I would say that they demonstrate a fundamental lack of appreciation
of the breadth of mathematics, and are only appropriate for
schoolteachers to shoehorn the subject into a restrictive curriculum.
The adjoint functor theorem and the Yoneda Lemma are the two obvious
candidates for the title "fundamental theorem of category theory",
but I have to say that I am somewhat alarmed by prospect that the
apparent consensus about this might become "legislated" into
someone's lecture notes, curriculum or textbook.

I think that Ronnie Brown and other people have given the right answer
to the question about famous unsolved problems.  Category theory is
not a collection of miscellaneous problems like combinatorics,
but a way of thinking about mathematics.

Rafael Borowiecki, alias Hasse Riemann, did not seem to be satisfied
with these answers, so what I said to him privately is this:

It would help people to give better answers if he, along with
students and non-academics who ask questions on "categories",
gave some clearer indication of their mathematical background.

If you read the archives of "categories", you will find that people
discuss topics from many areas in mathematics, physics and computer
science.   You may find amongst this something that interests you,
in which case you should look up the web pages and papers of the people
who post on that topic.  Academics are not very responsive to
completely uninformed questions, but they usually are very happy
to help if you show some background.  It is also a good idea to
flatter them by indicating that you have looked at their papers.

Finally, Rafael queried some of the technical points about "illegitimate
presheaves on locales" that I made at the end of my "autobiography".
I will publish my private reply another time, as I have some other
things to say about locale theory.

Sorry for the hurried response.

Paul Taylor



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