Comments on A Neighborhood of Infinity: Two Kinds of Mathematics

I am no mathematician, but when I was in high scho...

2009-11-19T17:10:47.613-08:00

I am no mathematician, but when I was in high school I remember thinking that the two kinds of math were descriptive and prescriptive. So calculations that described natural phenomena like geometry, motion over time etc. were descriptive. However things like compound interest prescribe something like what you are willing to pay in order to have the use of someone's money.

I can see and agree with what you're saying, sorta...

2006-05-29T22:52:00.000-07:00

I can see and agree with what you're saying, sorta. But I'm not convinced that a theorem is either 'structural' or 'content'. Many theorems I have seen were beautiful in 'content', but then found later on to see how useful they were in 'structure' (think Euler's formula for example). What I'm saying is that many theorems are both structural and content. Moreover, many theoroms are neither structural nor content. These are usually the trivial corollaries, sometimes not worth mentioning.

Maybe we should just grade each theorem out of 10, twice...once for 'structure' and once for 'content'.

Just kidding.

zorbid: your phrase captures some of what I want t...

2006-05-23T12:42:00.000-07:00

zorbid: your phrase captures some of what I want to say - but it's a bit of a mouthful!

Could "theorems about emerging properties" be a be...

2006-05-23T00:37:00.000-07:00

Could "theorems about emerging properties" be a better word for the "content" theorems?

You can read Phil Wadler's paper "Theorems for fre...

2006-05-20T13:30:00.000-07:00

You can read Phil Wadler's paper "Theorems for free!".

(Sorry about the multiple posts, my browser and I were having a disagreement. :) )

augustss,

I don't know the Reynolds paramet...

2006-05-20T10:08:00.000-07:00

augustss,

I don't know the Reynolds parametricity theorem but now you've mentioned it it's inevitable that I'll try to read up on it now.

Actually, this is great. Now people are going to post all kinds of interesting examples of computer science for me to read. Contentious statements are great for prompting people to say interesting things :-)

Your comment about the lack of "content" theorems ...

2006-05-20T09:37:00.000-07:00

Your comment about the lack of "content" theorems arising out of structure in computer science seems a bit misplaced, or at least ignores much of algorithms and computational complexity research.

To a large degree, algorithms is "all content"; structures are minimal, and most theorems are just surprising results. In complexity theory, there is more structure, but even there, there are many many surprising results, dating back to NP-Completeness itself, all the results about space and time and randomness, and many many more.

OK, I understand what you're saying.
And in som...

2006-05-20T09:18:00.001-07:00

OK, I understand what you're saying.
And in some sense CS is about structural content. You want to make an artefact (a program) that you know does something (fulfills its specification), so there's no content there.

Still, there's some content now and then, like Reynold's parametricity theorem.

Another vaguely 'specific' thing that pops out of ...

2006-05-19T18:18:00.000-07:00

Another vaguely 'specific' thing that pops out of category theory that I came across recently is this formulation of the Thompson F group.

There's also a curious 'contentful' fact that I wrote in this Wikipedia entry. It's so unusual for a statement in categorical language (plus a mention of Set Theory) to pin down something as specific as a number like 6 that I felt it warranted a mention on that page. Curiously one of the authors you cite, John Baez, wrote an article about this odd feature of 6 (see references in that Wikipedia entry), though he didn't mention the categorical formulation of the statement. (And JB wrote about the 7-trees in 1 example. There's a pattern here...)

Hmmm...I think that the most curious mathematics is often at the meeting point of the specific and the general. In particular, general definitions that still manage to single out specific objects.

One way of injecting some specificity into a topic...

2006-05-19T15:34:00.000-07:00

One way of injecting some specificity into a topic treated by category theory is by looking at initial or free things. For example, the free monoidal category with duals on one object is the category of tangles. Then any monoidal functor to a category of this type will give you an invariant of tangles (and so knots). This is beginning to feel contentful. Even your Hopf fibration will be found nestled fairly low in amongst the n-categories (see p. 29 of
this).