[FOM] Mathematical Aesthetics

[Sean C Stidd]

Wei Zhao wrote:

> I would be interested in reading and knowing what the philosophy of
> mathematics community in general has done regarding the topic of
> mathematical aesthetics. 

Very little indeed, as far as I could tell when I researched the subject
two years ago. Mathematical aesthetics would seem to be a fairly
recondite sub-sub-discipline, and would probably require a writer with
mathematical, philosophical, and art-critical training and perceptiveness
all three (six?) to make progress. Some writers propound theses about the
importance of judgments of beauty, etc., in mathematics, but almost
no-one (at least that I know of) seems to have tried seriously to analyze
what terms of aesthetic praise and approbation are doing in
'metamathematical' discourse, or what mathematical properties they pick
out, if any. I'd love to find out that my research on the subject was
incompetently conducted, however.

> I can't seem to find anything at all on it.

This was my experience also when I looked into the subject. One study I
did find was "The Divine Proportion", H. E. Huntley, New York: Dover,
1970. It's not what one would call a systematic account, though there are
some nice examples in there. (A puzzle: what do mathematical unity and
artistic unity have to do with one another, if anything? The unification
of different concepts sometimes achieved by a good proof and the
conceptual, visual, auditory etc. unity of an artwork are both often
cited as features related to the beauty of the proof/work in question.
This might be one kind of starting point.)

I'd like to echo the request for more. It might even be an important one
for foundational issues broadly construed, at least if one thinks that
'beauty', 'unity', etc. are central to the mathematical enterprise, which
they are as a matter of sociological fact (mathematicians who prove
'beautiful' theorems get more recognition, and that word is among the
highest honorifics mathematicians bestow on proofs), though whether there
is any non-sociological significance to mathematical beauty is no doubt
hotly disputed. Still, it would be interesting to find out e.g. that
canons of 'subjective' mathematical beauty had some kind of connection
with the 'objective' features of mathematical proofs and concepts that
made them 'fruitful' or 'interesting', etc. 

> 1) Is 'mathematical aesthetics' a subject under the philosophy of
> mathematics or under aesthetics? Or is it a mathematical 
> discipline?

Birkhoff wrote a strange article called "The Mathematics of Aesthetics"
(in the familiar Newman 4-volume compilation) which is a sort of attempt
to mathematize the experience of art, within the psychological tradition
in aesthetics. Mathematical aesthetics itself would seem to be part of
the philosophy of mathematics more generally, though surely one would
want to engage in a comparison of the roles aesthetic ideas play in
mathematics with the roles they play in evaluations of artistic/natural
beauty/interest/etc as part of (perhaps indispensably part of) the
process of pursuing it.

> 2) Have there been any major theories/debates/advancements in the
> subject within the context either of modern analytic philosophy or 
> of modern mathematics?

Again, I'd love to hear about them. I know of various very intriguing
comments various philosophers and mathematicians have made on the
subject, but nothing systematic.

Best,

Sean Stidd

"All experiments in physics measure numbers, so all quantities of
physical interest must eventually be reducible to numbers.- " Robert
Wald, General Relativity.