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What's the deeper point?
* someone who likes the concept
(either b/c they think it intrinsically beautiful
or )
wishes to share the concept
* ...and gives an elementary example,
as this can easily be conveyed
* a skeptic hearing it says:
"that's useless: you're not telling me anything I don't already know,
and you're using gratuitous abstraction"
FIXME FIXME: this is key!
the enthused one may say:
"ok, so now that you know the concept, it helps you answer X"
skeptic: "I don't care about X".
What's the issue?
- some people just *enjoy math* (of some sort or every sort)
...and wish to share it
- others want to solve particular problems
...and aren't motivated by "this is intriguing/beautiful";
they're intrigued by "is this useful for this question?"
OIC:
- for the problem-solving mind,
what is motivating is:
"I have a question in which I'm interested.
Give me tools to solve it / ways to understand it."
...and for new material, what's motivating is:
"Here's an interesting / challenging question."
(An inquisitive problem-solver finds many questions interesting.)
- for the understanding mind,
what's motivating is:
"Here's an interesting structure / theory / phenomenon"
EG, "Poincare conjecture" is immediately interesting to a
problem-solving mind.
"Structure of 3-manifolds" is immediately interesting to an
understanding mind.
OIC:
- there is an attraction to theories, to phenomena
- there is a *different* attraction to problems
different traits / tastes
different ways of motivating
(some topics really are more interesting in one way or another:
eg of category theory
)
EG, for Galois theory, I'm more interested in the general philosophy
of solution than in the details:
- relate geometry to algebra
- to prove impossibility, consider *class* of all possible
- ...and find a property that distinguishes the possible from possible
(solveability of Galois group)
(these minds aren't so far, obviously:
a tool and a theory are essentially the same thing:
- a tool that works works for a reason (there's a theory)
- a theory that says anything interesting solves questions
Is a definition a tool or a theory?
)
OH!
For me, the question is *not* "can I solve this?"
(since having a solution to a math problem isn't interesting to me!)
but rather "is there something interesting here?"
(Now, I *do* sometimes want to solve something, when I have some
*other* motivation (as in work or programming: I want to get something
done *b/c I like the result*).
But in math I'm motivated by the intrinsic beauty/interest of the
matter, and if the matter is ugly, I've little interest in solving it!
[partly "process vs. product"]
)
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"Why abstraction seems useless"
FIXME: this is companion to "I hate elementary"
some people naturally find abstract concepts beautiful
(now, you may find particular abstractions silly.
sometimes this is correct (it's a bad concept),
other times it hasn't been introduced usefully)
EG:
* parallelograms have a 3-dimensional moduli
* rectangles and rhombi are both 2-dimensional subvarieties
(are each 1 linear condition: codimension 1)
* hence we'd expect a 1-dimensional (codimension 2) subvariety
of parallelograms that are both rectangles and rhombi,
and these exist, namely squares
Now, this is ridiculously heavy-duty:
every 5-year old knows that squares exist,
and anyone who's seen squares, rectangles, and rhombi
can see that squares are at once rectangles and rhombi.
However, it's an elementary exposition of the notion of intersecting
moduli.
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A similar EG is arthimetic and cohomology:
it's not that cohomology sheds light on carrying,
but rather that "adding with carrying"
is a very simple example of cohomology.
EG of adjunction: used in Grothendieck-Riemann-Roch
The point is, if you're of a problem-solving mind,
you'll come to category theory very *late*,
b/c it doesn't *solve* any questions you're interested in.
If you're of a theory-building mind,
you'll come to category theory very early
(indeed, in 1st year of college!)
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Hmmm...
Category theory is abstract nonsense
Combinatorics is just games
Well, category theory is abstract, and *applicable*
Combinatorics *is* clever.
Hmmm...does one see many *clever* things in category theory?
Category theory tries to make everything very *natural*.
Oh, it's Grothendieck's aesthetic:
everything "unfolds": no *obscure cleverness*.
BTW, matroids are a combinatorics object
with theoretical interest
(generating functions?)