Re: Fundamental theorem

Re: Fundamental theorem

[Vaughan Pratt]

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The illustrator of Hal Abelson's 1970 "Calculus of Elementary
Functions", Ellis Cooper, pointed me at it as an earlier reference for a
proof in print (or out of print in this case) of the growing-circles
argument for FTAlg than Pontrjagin=92s 1982 article.  But then Mike Barr
just now doubled that with Courant and Robbins!  (So how far back *does*
that very nice argument go?)

Mike concluded with exactly the right question, which is what I wish I'd
asked in the first instance: =93what does it mean to give an intuitively
appealing non-proof and call it a proof?"

Apropos of both my original assertion and that question, Peter Freyd and
I went back and forth a bit, and he suggested I forward the ensuing
correspondence to the list, following.  My answer to Mike's question I
think would be in my fourth last sentence, "By all means tell the
audience..."

Vaughan Pratt



=3D=3D=3D=3D=3D
PJF:
Vaughan, surely you're not saying that the standard homotopy
argument for the FToA is new, are you"

It's the first one I ever heard -- and that was over 50 years ago.

    Peter

=3D=3D=3D=3D=3D
VRP:
Hi, Peter,

As usual you're in the vanguard in these things.  But then why didn't
you or whoever told you the proof pass it on to one of Garrett Birkhoff,
Marshall Hall, Pierre Samuel, Peter Hilton (who *surely* should have
known, yes?), or Paul Cohn?  They're the ones taking responsibility for
the algebra article for the Brittanica (1987 ed.).  On p.260a they say
"No elementary algebraic proof of [the FTAlg] exists, and the result is
not proved here."

I was being told the same thing in the 1960s - if Gauss couldn't find a
simple proof in half a dozen tries, there isn't one.  The basic message
was, if you don't possess the necessary higher maths or the stamina for
an intricate argument, we can't help you with that result, ask us about
solvability of z^n =3D a.

My cohort may well have run into you in the Edgeworth David building
then, you would appear to have missed the opportunity to disabuse the
rest of us of this notion.

If you know of an elementary exposition of any proof of FTAlg appearing
in the 1970's or earlier I'd love to see it.

Vaughan



=3D=3D=3D=3D=3D
PJF:
The proof was shown to me over 50 years ago as the standard argument
of why one should believe the FToA .It's an easily understandable
proof. But it is not elementary.

To complete the proof there's a lot of work to be done. Just try
proving -- from scratch -- that the circle is not contractible (if it
were, there's no way you could use it to prove the FToA).

There is a pretty elementary way that starts with the proof that for
any continuous self-map on the unit circle  C --> C  there's a
continuous  R --> R  such that

            h
         R --> R

       g |     | g
         v     v

         C --> C
            f

where  g(x) =3D exp(xi2\pi). (A proof of this doesn't require a lot
of background but it's a bit tedious.)

Then the "winding number' of  f  is the constant value (after you
prove it's constant) of  h(x+2\pi) - h(x)..And one can then
construct a tedious proof that the winding number is a homotopy
invariant.

But even with this there's still a lot of work to do.

There's another easy proof of the FToA (but also not elementary) that
rests on the fact that a polynomial mapping on the complex plane is
open and proper.

    Peter

Do you want to forward our correspondence (when done) to the cat net?


=3D=3D=3D=3D=3D
VRP:
No problem, just let me know when.

My position is that this is a perfect example of a nonelementary proof
that *is* fit for general consumption, in contrast to one that can't be
grasped without first understanding the definition of some nonelementary
concept such as holomorphic function [better example: local compactness
=96vp].  Any child knows intuitively that a rubber band wrapped one or
more times around a pencil can only be removed from the pencil by
pulling it off one end or the other or by magic.  By all means tell the
audience that we're not going to prove that bit because a proper proof
is surprisingly hard for such an intuitively obvious result.  But
otherwise I would say that, at least morally, this is an elementary proof.

If "the rest of us" have been deprived of this argument all this time
(up to the 1980s or whenever) on the ground that this property of loops
is not an elementary one, I for one am very sorry not to have been shown
the argument long ago, in place of the ones I *was* shown.

It's a good thing sailors and scouts aren't taught by mathematicians or
they wouldn't be allowed to study knots until they were officers or
Eagle scouts.

Vaughan


=3D=3D=3D=3D=3D
PJF:
Damned good point.

Fundamental theorem

[Michael Barr]

The proof that Vaughan outlined is found in almost exactly the same form
in Courant & Robbins, published around 60 years ago.

Here is a proof I learned in grad school, based on three facts, one
analytic and two algebraic.  The analytic fact, which is an irreducible
minimum, given that the reals cannot be defined algebraically (except as a
real closed field of continuum transcendence degree, which misses the
point) is the intermediate value theorem.  In other words, order
completeness.  From this follows the fact that every odd order polynomial
has a (real) root and that every real number--and with a bit of
manipulation, every complex number--has a complex square root.  The
algebraic facts are the existence of a splitting field and the theorem on
elementary symmetric functions, neither of which is quite trivial, nor
very deep.

The way you do it is by proving that every real polynomial of degree n =
2^k*m with m odd has a complex root, by induction on k.  The case k = 0 is
quite trivial, of course.  So suppose that f is a real polynomial of even
degree n and, in some splitting field has roots r_1,...,r_n.  For each
integer s form the polynomial f_s = \prod{i<j}(x - r_i - r_j - sr_ir_j)
which has degree n(n-1)/2, which is less 2 divisible than n.  The theorem
on symmetric functions implies it is real and hence for some i and j
dependent on s, r_i + r_j + sr_ir_j is in C.  Since there are only
finitely many pairs i and j and infinitely many integers, there are
distinct s and t for which both r_i + r_j + sr_ir_j and r_i + r_j +
tr_ir_j belong to C.  Given that C has square roots, one easily discovers
that r_i and r_j are both complex numbers.  To go from real to complex
polynomials, just multiply a complex polynomial by its conjugate; for a
root r of the product, one of r and its conjugate is a root of the
original.  And the process of factoring completely is well-known.

Incidentally, it is the case that if K is an algebraic extension of L and
every polynomial with coefficients in L has at least one root in K, then K
is an algebraic closure of L.  The argument above pretty much does the
characteristic 0; the prime case is trickier.

This is not entirely elementary, but then neither is the winding number
argument.  Comparing them is difficult because the algebraic argument I
have given could all be taught in a lower division course, while the
winding number argument, intuitively appealing, really requires some
sophisticated stuff to deal with rigorously.

Which raises an interesting question.  If we agree, as we seem to, that
proof is the essence of what we mean by mathematics, then what does it
mean to give an intuitively appealing non-proof and call it a proof?

Michael