Re: Amateur takes on Wiles's work

From: *** T. Winter (***.Winter_at_cwi.nl)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 01:13:43 GMT

In article <3c65f87.0408290713.3a80d92c@posting.google.com> jstevh@msn.com (James Harris) writes:
> Now I'm not a professional mathematician. I do post about math on
> Usenet, but that's not an indication of expertise!

That shows.

> 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!

Yup, but as I have noted before, it is *not* that description by 4 integer
numbers that describes the relation.

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

Ah, yes. Given an elliptic curve defined by the four integers "a, b, c
and d", and a modular form defined by the same four integers. Off-hand
you have here an easy association between elliptic curves and modular
forms. But that is *not* the association that makes elliptic curves
modular. Try to read about the stuff and comprehend this.

> 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.

Eh?

> 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.

You are leaping to conclusions to assume that it is those four numbers
that establish the relation. In fact the relation through the four
numbers is trivial, but does *not* make elliptic curves modular.

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

Nope. Wiles did it for a subset of the elliptic curves. Your lack of
knowledge is showing.

> 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!!!

Eh? It is *you* that sees a pattern and assume that the association
is proven using that pattern. How wrong can you get.

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

Can you give a good reason why some of the equations:
   x^5 + y^5 = n.z^5
have solutions with integer x, y and z, and some do not have solutions?
Nevertheless, it is a fact that has been proven.

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

Except that some math proofs begin with something that is false, and by
logical steps show that the assumption is false...

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

Yup. In almost all null tests it is the final conclusion that will fail
the assumption. 

-- 
*** t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~***/
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