What should we teach? Do you need to know proofs?
that is, can you prove everything you rely on?
[should we teach proofs of results?]
- yes, obviously it's nice to know,
but takes time
- proof can be for rigor / certainty,
...or for understanding underlying or related concepts
[proofs are about the *structure* of a theory,
and are often *irrelevant* for applications]
For applied people,
it's not worth proving it to them:
better to give context and applications
so they understand the statement and meaning.
EG, useless to teach proof of spectral theorem to economists
(Rayleigh quotient? complete flag?)
...but showing that you can interpret covariance
as a bilinear on random variables
(and variance as the associated quadratic form)
(...and interpret it geometrically!)
...and that the spectral theorem lets you diagonalize
a covariance matrix (by orthogonal axes)
*is* useful.
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As an example:
everyone learns Galois theory (or ought to).
Do you remember the statement of the Sylow theorems?
Their proofs?
Probably not, but you probably remember:
$A_5$ is simple, a fortiori it is not solvable (nor is $S_5$),
and the general quintic polynomial has Galois group $S_5$,
hence cannot be solved by radicals (as radicals correspond
to cyclic extensions).
You don't need to know \emph{why} $A_5$ is simple:
it is a key ingredient, but its proof isn't necessary
for the rest of the theory.
This doesn't mean that we shouldn't teach the Sylow theorems,
(Abel's impossibility theorem is one of the great glories of math:
if anything deserves a complete proof, it does)
but if we choose to do so, we should understand that only
finite group theorists are likely to remember them,
as they aren't used much elsewhere.
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More generally, how do you prioritize topics?
This isn't just about teaching linear algebra to economists:
how do you teach point-set topology?
Analysis? (Weak^* topology)
Algebraic Geometry? (Sheaves and Spec?)
Manifold Topology? (not much: all your spaces are nice)
Algebraic Topology? (categoricalness)
Logic / Topos theory? (locales, models of constructive logic)
[problem is that deep study of one area doesn't prepare you much for another,
and teaching is often narrow]
Yes, your students will develop some theory,
but they often don't need to know why their tools work.
A driver doesn't need to know how to change their oil;
even a racecar driver.
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This is a problem of diversity:
each professor teaches their point of view,
which often don't connect up.
This is in some ways a strength:
everyone gets a different set of perspectives,
but it undermines *coherence*
The opposite is to have a common bottom line, a common point of view,
(e.g., arithmetic algebraic geometry; microlocal analysis; whatever),
and teach everything from that point of view.
[A focused student can do this themselves:
``Ok, what's the relevance of this to (my favorite field)?'']
An intermediate way is teach many points of view *and the connections*.
(takes a long time!)
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Indeed, even pro mathematicians
need to use results that they can't prove
(else you'd never get anything done).
The deeper reason to teach proofs in math is
*that's what math is about*
...or rather, that's what a field is about.
If you just want to *use* group theory,
(say, in Galois theory)
you don't need to prove the Sylow theorems,
just know how to apply them.
...but if you want to be a group theorist
(or learn how they think and work),
you *do* need to prove it.
Are you a user or a participant?
[but *you can be a good mathematician
w/o knowing other approaches!*
Fields are focused!]
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To understand a result, you need to *use* it
(apply it, see where it works and doesn't).
Proving a result is good, but it's *deeper*
and it's *NOT* necessary for applications
...esp. b/c you won't use the methods of proof again in the applications.
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Proofs as "morally obligated" or "keep you honest"
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EG of algebraic topology:
- from Joe Harris
(algebraic geometer, user: didn't care about structure of theory)
- from Peter May
algebraic topologist: cared about structure, not applications
I loved both classes, though in different ways.
[and when Shmuel Weinberger, my advisor, taught the class,
I think he emphasized the geometric topology point of view.]