Re: the mysteries of counting
- From: baez@xxxxxxxxxxxxxx (John Baez)
- Date: Tue, 28 Jun 2005 17:30:06 +0000 (UTC)
In article <d9q972$lbs$1@xxxxxxxxxxxxxxxx>,
Apollonius de Tyane <apolloniusdetyane@xxxxxxxxx> wrote:
>I have something like a question for you, maybe related to this. Do
>you mind if, while asking it, I do an impression of you, and load my
>question down with a lot of expository chatter? Impressions are a form
>of flattery. The highest!
Hmm. Saying I'm "chattering" doesn't sound flattering. But,
since you actually do some nice exposition yourself, you must
know that math is infinitely more interesting when it goes hand
in hand with a bit of what Douxadis calls "paramathematics":
Apostolous Douxadis, Embedding mathematics in the soul:
narrative as a force in mathematics education,
http://www.apostolosdoxiadis.com/files/essays/embeddingmath.pdf
As he says:
"Of course: abstraction, irrelevance, purity, formalism make for
good mathematics.... But sadly, they make for bad mathematics
education. Each one of these concepts - abstract, irrelevance,
purity, formalism - pushes mathematics further away from a growing
human being, a being whose psyche is in the phase of it development
that no soft-brained psychologist but a great mathematician, Alfred
North Whitehead, calls the Romantic Phase."
>The natural numbers categorify to finite sets (is how you would put
>it.)
Right. The set of natural numbers is the decategorification of the
category of finite sets.
>This means that there is a nice way to associate a natural number to a
>finite set, and under this association certain operations on finite
>sets are...associated...with certain operations on natural numbers.
"Associated", especially guarded by ellipses like that, sounds a bit
spooky and vague. But in fact decategorification is a completely
systematic process! In fact, it's a 2-functor:
Decat: Cat -> Set
assigning to each category C its set Decat(C) of isomorphism classes, to each
functor F: C -> D the corresponding function Decat(F): Decat(C) -> Decat(D),
and to each natural transformation the identity. (The category Set becomes
a 2-category in a trivial way, with only identity 2-morphisms.)
When we apply this to FinSet we get N, the set of natural numbers.
The product and coproduct on FinSet give x and + for natural numbers,
and so on.
>I learned this in grade school.
Wow! Good education. But yes, we learn in grade school how natural
numbers are stand-ins for isomorphism classes of finite sets. We don't
learn all the formalism lurking behind this process, but we do
learn how it works - or should: if we don't, numbers are meaningless.
>The reason you (OK, I admit it (but
>*I'm* anonymous): we) like to use the word "categorify" when talking
>about this is because it gives us a glamorous and mysterious question
>to ask about almost any mathematical thing: what does the thing
>categorify to? Glamorous because there is a richer theory of the
>categorified thing than of the original thing ("finite sets" is short
>for "finite sets and functions between them"; these are more
>interesting than natural numbers, which are interesting), and
>mysterious because the question is hard to answer.
Right! While decategorification is a systematic, turn-the-crank
affair, in practice we often start with some math that's been developed
using sets, and are trying to guess what it could be the decategorification
*of*. This is an *unsystematic* and therefore mysterious process.
And yes, it's very glamorous, because when one succeeds one taps into
a deeper world of meaning, full of its own interesting questions and
theorems.
>Like, the integers? Is a mathematical thing. What does it categorify
>to?
I have various tentative answers to this fascinating puzzle - only one
of which appears here:
http://math.ucr.edu/home/baez/counting/
but I'll give them in a separate post, because I have to go pick
up my new glasses from the optician. And then I'll also take a stab
at the more *precise* question at the end of your post - I need to
think about it a bit.
>There are probably all kinds of answers to this question, in varying
>degrees of half-bakedery.
Indeed!
Best,
jb
.
- References:
- the mysteries of counting
- From: John Baez
- Re: the mysteries of counting
- From: Apollonius de Tyane
- the mysteries of counting
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