Re: fundamental theorem of algebra

[Vaughan Pratt <pratt@cs.stanford.edu>]

John Baez wrote:

> I really doubt those authors were unaware of the topological proof
> of the fundamental theorem of calculus in 1987.  After all, it's

Right, both my claim and its premises needed a fair bit of tuning (as
with my recent question about the quasivariety "groups+free monoids" --
this is a good list to get corrective feedback from).  (But a neat piece
of historical research there, John.)

The issue seems to be coming down to Mike Barr's question, which if I
can paraphrase it without changing its intent, was, what is the proper
status of an appeal to the very plausible in a proof?   My suggestion in
my last message to Peter Freyd was that the prover should point out the
gap, its cause (lack of a simple proof), and its plausibility
notwithstanding.

This suggestion raises more questions than it answers.

1.  Is a proof with a gap more acceptable for expository purposes when
the bridgability of the gap is more plausible?  (The case in point being
an extreme example.)

2.  How is plausibility to be judged?  By consensus, or are there
objective criteria?

3.  It is certainly not necessary to prove A before B merely because B
depends on A; indeed one common-sense practice when proving a two-lemma
proof is to get the easier lemma out of the way first, even if it
depends on the harder one.  Is it kosher to truncate such a proof after
the first lemma (or in this case the final result), call it an
exposition, and point to the literature for the second lemma?

Regarding 3, the authors of the Britannica article seemed not to think
so, but perhaps this just reflects Garrett Birkhoff's attitude that "I
don't consider this algebra, but this doesn't mean that algebraists
can't use it" cited by Michael Artin when proving FTAlg in his 1991 book
"Algebra".  Who on this list considers the fundamental theorem of
algebra "not algebra"?

These questions are probably more appropriate for a philosophy of
mathematics list than this one.  What makes FTAlg such an interesting
case study for those with something at stake in such questions is that
the tensions here are so extreme.  The final result (FTAlg) is not at
all obvious, whereas the lemma it rests on, whether it be that |P(z)|
attains its minimum, or that circles around a hole don't retract, or the
intermediate value theorem, or the existence of a root for a real
polynomial of odd degree, seems self-evident.  Yet the one that is hard
to see is easy to prove, while the one that is easy to see is hard to prove.

If seeing is believing, what is proof?  In the real world, when
something is easy to see it is up to the opposition to demonstrate that
it is nonetheless false.  How did mathematics evolve to play by a
different rule book?

Vaughan Pratt