Amateur takes on Wiles's work

From: James Harris (jstevh_at_msn.com)
Date: 08/29/04 

Date: 29 Aug 2004 08:13:49 -0700

Now I'm not a professional mathematician. I do post about math on
Usenet, but that's not an indication of expertise!

I'm not beholding to any mathematical interests though, so I feel no
compulsion to protect a favored Golden Calf of the modern math world,
which is an argument that supposedly proves something by one Andrew
Wiles, which I fear doesn't, and I'll say exactly why I say it
doesn't.

It'll be up to others to answer the charge, dismiss it, or consider
that I might be right.

First off despite the assertions of great complexity to the area what
mathematicians initially noticed isn't that complicated:

They had these things they called modular forms, and these things they
called elliptic curves, which didn't seem at ALL related.

But there are these 4 numbers that you can get from elliptic curves,
and find modular forms with the same 4 numbers. Those numbers are
kind of like a description.

So there's some way that modular forms and elliptic curves could have
the same description!

Mathematicians would check various elliptic curves and find they could
always find some modular form to associate with it.

Taniyama and Shimura conjectured that there was a pattern here that
held, as in fact modular forms and elliptic curves WERE related in
some deep way, and that what mathematicians were noticing wasn't just
one of those intriguing coincidences.

But you have the setup for a logical fallacy called Cum Hoc, Ergo
Propter Hoc, where people see what looks like a pattern, and leap to a
conclusion, though at this point mathematicians were ok, as it was
only a conjecture.

It took Andrew Wiles coming in, with an attempt at proof by
association for the logical fallacy to fully take hold.

The problem for many of you with such a charge is that it can seem
esoteric. I've had two posters on sci.math where I've discussed this
for a while actually come back to claim that Cum Hoc, Ergo Propter Hoc
is about time, so it can't appy to mathematics!!!

But notice, it's actually about false implication, where you see a
pattern, and your mind plays a trick on you and tells you that the
pattern is proof of itself!!!

To date, while mathematicians now apparently mostly believe the
Taniyama-Shimura Conjecture, they can't give you a reason why, or can
they?

It turns out that if the charge of Cum Hoc, Ergo Propter Hoc is itself
challenged, the next proper step is to ask for a null test.

What is a null test?

A null test is to go through the argument under challenge with the
assumption that its conclusion is false, and find a contradiction with
that assumption!

You see, math proofs begin with a truth and proceed by logical steps
to a conclusion which then MUST BE TRUE.

But the conclusion follows from the previous steps in the proof, so
any challenge to the conclusion must contradict a previous logical
step, or the truth with which the proof begins.

Math proofs are perfectly logical.

There is no way for a math proof to fail a null test.

It is just not logically possible.

Therefore, any math proof can be challenged by assuming the opposite
of its conclusion, and tracing through it until you reach the logical
step where you end up with a contradiction.

The resolution to the contradiction, if you have a proof, is that your
assumption is false and the conclusion IS true!

It's neat. It's beautiful. It's just cool.

Notice also that the null test, which can be requested whenever, and
not just when you have a case of Cum Hoc, Ergo Propter Hoc, is a great
way for someone who is not an expert in a particular feel to find a
limited area to check.

For instance, with my challenge to Wiles's work, someone should find a
single logical step where the assumption of a non-modular elliptic
curve will cause a contradiction, and be able to give the exact
section in his work where it occurs!!!

Then they can explain why it occurs and despite the entire work being
hundreds of pages you have the ability to look at the crucial link
without going through the entire thing.

You could call that logical step the keystone.

I'm asking for someone to produce the keystone in Wiles's work, which
will ring out loud and clear if you assume the existence of a
non-modular elliptic curve.

Let the full challenge--with witnesses now from alt.math.recreational
and others throughout the world through the Internet--begin.

James Harris