Sunday, August 29, 2004


Amateur takes on Wiles's work

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.

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