Re: Demystifying the categorial approach

Re: Demystifying the categorial approach

[John Stell]

When Venet's commutative diagrams as art appeared in the AMS Notices
(vol 49 no6 2002, pp663-668) this was mystification of the
categorical approach by the artist. Good art but nothing to do with
mathematics. Since many attempts to demystify mathematics have a visual
aspect, the role of art here is interesting, as well as how artists see
this kind of activity.

One aspect of the continuing discussion seems to concern disciplinary
boundaries -- complaints that some computer scientists, physicists,
philosophers etc 'misuse' category theory or deride it. It's interesting
that efforts to explain mathematics to the general public can fall
foul of exactly similar complaints from an opposite quarter.
Often public participation projects involve art in some way and
these are indeed often funded under 'art-science' programmes.
Most such projects are valuable for engaging the
public and succeed in getting people interested, however the art is
usually of no interest to artists concerned with current issues in
contemporary art. This is not to say they are poor projects, but labelling
them as "art-science" creates the false impression there is an engagement
with art in a meaningful way.

On the other hand there is art which references mathematics, but which
has absolutely no mathematical content. For example Venet's work
and others who lift elaborate equations to great visual effect.
Yet other work (e.g. Conrad Shawcross)
involves references to physics in a quite different, but still
non-expository, way.

It seems to be an open question how art might be used to promote or
facilitate a genuine understanding of mathematics (which must
involve the reasoning processes and apreciation of abstract structure).
Perhaps Sol LeWitt (despite his writings and some critics (e.g. Rosalind
Krauss)) indicates a possible way forward. Some of his work (just like
Venet's) has nothing to do with mathematics despite the superficial
apperance; other parts are worth thinking about.

John Stell

Re: Demystifying the categorial approach

[Ronnie Brown]

I have not been able to keep  up with all this correspondence, but would
like to take up Vaughan Pratt's counter to Reinhard Boerger's point saying:
`it can't be done'  which would be better as a question: `How should one do
it?'

The many books and films on maths show there is some kind of hunger to know
what this subject is about, but the books and films often avoid the subject
itself in order to emphasise the weirdness of their chief characters.

The Bangor approach has for years been `advanced mathematics from an
elementary viewpoint', and we have been trying this out on 13 year olds, and
the general public,  for the last 22 years. We have gained a lot from the
exercise, and have used the experience in talks to mathematicians, science
festivals and other scientists.

Here are some references.

136. (with T. Porter), `Category theory and higher dimensional algebra:
potential descriptive tools in neuroscience', Proceedings of the
International Conference on Theoretical Neurobiology, Delhi, February 2003,
edited by Nandini Singh, National Brain Research Centre, Conference
Proceedings 1 (2003) 80-92.

137. (with R.Paton and T. Porter), `Categorical language and hierarchical
models for cell systems', in Computation in Cells and Tissues - Perspectives
and Tools of Thought, Paton, R.; Bolouri, H.; Holcombe, M.; Parish, J.H.;
Tateson, R. (Eds.) Natural Computing Series, Springer Verlag. (2004)
289-303.

I presented the talk for 136. and it went well. One aim was to explain the
concept of colimit, as a way of putting structures together.  A scientist
from the Salk Inst said he kept on thinking about the ideas that night and
could not go to sleep!  This is a better reaction than I get generally from
algebraic topologists (pace Marta's comments)!

Here is another article:

146. (with T. Porter) `Category Theory: an abstract setting for analogy and
comparison', Advanced Studies in Mathematics and Logic (to appear) UWB Math
Preprint 05.10 .
http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/05/cathom05.html#05.10

If you examine our pages on www.popmath.org.uk, you will find discussion of
many topics, such as AIMS.

My whole approach to higher dimensional algebra since 1965 or so has been to
give expression to intuitions such as expressing a big object as a
composition of small objects, i.e. the algebraic inverse to subdivision, and
also explaining the notion of commutative cube.

There are deep ideas (e.g. knots) that can be explained to children, and so
to a general audience, and others for which  this is much more difficult.
There is a hunger among scientists and the general public to get some idea
of what is going on in mathematics, apart from solving some  famous but
weird  problems. The idea of `structure' is a good way to start, perhaps. I
have used Bill Lawvere's tag: The mathematical notion of space is for the
representation of motion, i.e. of change of data. How can space be
structured? How can we calculate and control ideas in that context? How
strange is space?

In 1987 or so,  I gave a lecture on knots to an audience from schools, and a
teacher came up to me afterwards and said: `That is the first time in my
mathematical career that anyone has used the word `analogy' in relation to
mathematics.'  Think on it!

So I have made something of a song and dance on the theme of `analogy' in
the new edition of my book: and the notion of universal property as a way of
making analogies, to obtain understanding and calculation.

Finally, to see an argument about physics and maths, see an article

S. Novikov `The Second Half of the 20th Century and its
Conclusion: Crisis in the Physics and Mathematics Community in Russia and in
the West',   published in American  Math Society Translations,  (2) vol 212,
2004

which might rise some ire among some of us. There are also important points.
It is certainly an aspect of the debate, e.g. `7. Second half of the 20th
century: excessive formalization of mathematics.'
I was sent a pdf file of this. But I do not have a url for this English
translation.


Ronnie Brown
www.bangor.ac.uk/r.brown

----- Original Message -----
From: "Vaughan Pratt" <pratt@cs.stanford.edu>
To: <categories@mta.ca>
Sent: Monday, April 03, 2006 4:09 PM
Subject: categories: Demystifying the categorial approach


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

[balance of quotation omitted...]

Re: Demystifying the categorial approach

[Reinhard Boerger]

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

Demystifying the categorial approach

[Vaughan Pratt]

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

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.

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 should not feel obliged to axiomatize in full the morphisms of Set,
certainly not at the outset, and moreover not if one plans to apply the
same ideas to the morphisms of other categories at some point.  This
central tenet of category theory should be one of the factors guiding
the order of development, with associative composition foremost.

Along similar lines one can point up the parallels with how the very
words with which we speak combine associatively to form phrases and
sentences, and how addition and multiplication of both integers and
reals also obey those laws but differ in furthermore being commutative.
  It should be possible to make such an account of the
category-theoretic axiomatization of functions very accessible both to
non-mathematicians and to eighth-graders with some interest in mathematics.

One should also avoid the common jargon of the category theory
literature.  Everyday language about everyday concepts is to be
preferred to new notions that need to be defined before they can be
understood.

One could also dwell on the politics of the status quo, were it as easy
to explain ZF.  Indeed, when one compares the compositional approach
with how functions are introduced into mathematics founded on ZF, one
has to wonder how the latter came to be preferred over a framework that
axiomatizes functions as primitives.  To any save those that have long
since come to live and breathe the ZF-based definition, it is unnatural,
unmotivated, and hard to explain to nonmathematicians by comparison with
associatively composing functions.

One hypothesis for why set theory is the preferred basis today for the
concept of function is that the primitive-function approach is just too
simple to take seriously.  But that can't be true, as we saw at the
Universal Algebra and Category Theory conference at MSRI 17 years ago,
where every algebraist there understood the motivation for defining
formal languages and relation algebras abstractly as monoids, a really
simple abstraction with a rich literature of implications.  Few of them
however seemed to fully appreciate the power of applying the same
abstraction, subject to essentially the same laws, to the definition of
function, being content with the ZF picture of functions.

Or am I grossly misinterpreting what we all witnessed at that meeting?
Walt Taylor, of McKenzie, McNulty, and Taylor, "Algebras, Lattices,
Varieties: Volume I", spoke on his highly developed theory of varieties
back to back with Fred Linton's talk on monads.  It was like ships
passing in the night.  I pointed this out to George McNulty at lunch
after Fred's talk, but though we both struggled mightily and without
rancour to communicate, we just could not get to square one, in the time
available before the first afternoon talk, with the concepts of monad,
adjunction, or their relevance to what Walt had just spoken about.  And
it's not as though George and I can't communicate at all, as can be
inferred from my profuse acknowledgements of his considerable help at
both the start and end of http://boole.stanford.edu/pub/iowatr.pdf,
followed up by http://boole.stanford.edu/pub/jelia.pdf (work done
entirely in the universal algebra tradition with no mention of
categories, since that was the audience for those papers).  Except for
his being the master and I the student, George and I are very much on
the same wavelength in algebraic matters, it was only category theory
that was a closed channel between us then.  Ralph McKenzie seemed to be
more keen than Walt or George for algebraists to embrace category
theory, but even for the prime mover of tame congruence theory it seemed
to be something of an uphill battle.

It's not just monads and adjunctions that noncategorist mathematicians
have a hard time with.  In his welcoming remarks at the start of the
meeting, MSRI director and formidable geometer Bill Thurston expressed
his discomfort with Set^op, and the incredulity of half the audience was
palpable for a second.

Even today category theory labors under the dual impressions that it is
too abstract to explain to the non-specialist, yet too trivial to take
seriously.  This screwy situation is not unlike the days when oxygen was
perceived only as the absence of phlogiston, oxygen being harder to
"see" than phlogiston despite our constant immersion in it, just as we
are constantly immersed in associatively composing functions.  The
chemists of the pre-oxygen era, who believed that burning wood gave up
phlogiston to the air, would have found quite mystical the idea that the
absence of phlogiston constitutes 20% of air.  Category innocents are in
much the same boat.

Complicating matters is that this works both ways.  It is hard to
understand someone who finds it hard to understand the "obvious,"
whether true or not.   How should a categorist understand a talented
mathematician unable to process the concept of the opposite of a
concrete category?  (Useful trick: define a function to be a binary
relation and consider its converse.  Why didn't Thurston think of that?
  Why didn't someone suggest it to him?)

Marta should look forward to a time when functions are taught as
associatively composing entities as routinely as nitrogen and oxygen are
taught today as the principal constituents of air.  High school math
teachers will look back at the 21st century and marvel at how confused
people were about the nature of functions in those days.

It might of course never happen, with the concept of graph of a function
forever taken as prior to the function concept itself.  One big obstacle
is that the concept of binary relation is sufficiently well motivated in
its own right as to justify being introduced first (but in that case try
to at least get *that* defined via relation algebra, Tarski's program,
which so far has not taken hold).  Another is that there does not seem
to be any knockdown argument making composition prior to application,
and the mathematical world is currently strongly wedded to the opposite.

But the biggest obstacle is that such a sea change can't happen via an
independently written book focused on the need for that change.  Rather
the viewpoint needs to be integrated into the existing literature,
either by adapting existing texts or providing superior alternatives
whose primary purpose is to meet the extant curriculum requirements and
which accepts that getting the definitions right is only a matter of
good hygiene and not a subject in its own right to be added to an
already crowded curriculum.  It has to be an inside job.

Vaughan Pratt