[email alex hess once I'm done:
alhess@gmail.com]
proof & understanding
bottom line: proof? understanding?
don't feel need for elementary proof
(may be nice for exposition)
(combinatorics has depth; e.g. Gowers paper/talk)
math aesthetics:
terse, no scaffolding, textura
[not a cathedral! not magic!]
give EG of Terry Tao on the prime number theorem
(again, elementary proofs can be very clever
and have some value)
elementary proofs vs. intuitive
(isolate vs. understand what's going on)
Timothy Gowers’ essay The Two Cultures of Mathematics
http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
‘theory-builders’ and ‘problem-solvers’
[much more deeply:
question is "how did you come up with something?"
did you try stuff? what was your intuition?
This can be scaffolding.]
magic
...but like understanding how the trick works
goal is to train magicians, not leave people amazed
(if goal is to wow people, magic is ok)
"I hate elementary proofs!"
Not really, but they go against a POV of math that I like.
What are the goals of rigor?
Both *correctness* and *deep understanding*.
For Euclid, it was very much about *correctness*, unassailability.
Much subtler was rigor in the 19th century (don't know )
When you see a proof,
the big question is "Where did this come from?"
With the background, it's not so *magic*;
it can still be impressive and amazing.
(which indeed is one reason people omit background!)
Elementary proofs are easy to follow
(that is, you can follow the *steps* even if you don't understand what's going on),
but you have *no idea* where they came from.
High-tech proofs require lots of background
...but you can see exactly *why* they work.
(People who discover elementary proofs are stumbling onto
the deeper mysteries, using their existing tools.)
twin goals of rigor are
*correctness*/*evidence*
and
*understanding*
(rigor really does help understanding!)
people tend to have preferences (I prefer understanding)
[understanding often uses very *general* proof;
(simple proofs but lotta technology)
elementary proofs often use all the special features:
these aren't necessary, but make it shorter / simpler / clearer?]
is the content in the *definitions/concepts* or in the proofs?
intuitive more about building up suitable concepts;
proofs generally simple
How to convert between them:
intuitive -> elementary
find crucial computation
(may be able to simplify it by using special features)
("what's really necessary/meat?")
elementary -> intuitive
find appropriate setting / "what's really going on?"
intuitive proofs generally give more data,
hence it's easier to strip down an intuitive proof to an elementary one,
but there is the question of *which parts are really essential*
A proof sketch is just the idea, w/o the evidence/details;
a terse proof is just the evidence, w/o the idea.
EG:
self-adjoint real operators have an eigenvector
Noam gives terse, suffient proof.
I discuss critical set of Milnor-Bott function on projective space.
Noam doesn't say "derivative",
and uses special details of max, instead of general "critical set"
[I'm not saying mine is superior;
they are very *different* in effects,
but *be able to do both and choose the appropriate one*]
"Gauss usually declined to present the intuition behind his often very
elegant proofs--he preferred them to appear "out of thin air" and
erased all traces of how he discovered them."