## Monday, June 12, 2006

### The proof of Fermat's theorem is a "Rube Goldberg contraption"

Doron Zeilberger does not pull his punches:
All that Andrew Wiles did, in his FLT proof, was solve one problem, using ad hoc arguments, that depend on historical contingencies of what was proved before. It is a huge Rube Goldberg contraption, that is unlikely to lead to anything further of any significance, just more of the same human drivel.
In context, he is talking about the importance of computer-generated proofs. Read the whole article.

Categories:

1. By the end of the first paragraph, I guessed correctly that Zielberger is reacting to one of his own papers getting rejected.

As a sanity check, I looked at the paper, and I can see exactly why it was rejected. Never mind that he's "merely" implementing known enumeration techniques. Never mind the snarky remarks about his program producing "infinitely many...PhD theses" and other researchers being "extremely lucky", No, the answer is right on the first page, where he writes "...so I had to spend a few weeks writing such a program myself." A few WEEKS?! So he *admits* that the paper describes trivial work! And then he has the gall to complain when his paper is rejected?

In any case, his basic point is absolutely wrong. The point of mathematics is not to explain things to computers. (That's not even the point of computer science!) The point of mathematics (and computer science) is to extend the boundaries of HUMAN understanding. Computer-generated proofs are certainly a useful tool in that pursuit, but it is patently ridiculous to assume (as Zielberger advocates) that they are the only such tool, even within the realm of mathematics.

2. Re: "Computer-generated proofs are certainly a useful tool in that pursuit, but it is patently ridiculous to assume (as Zielberger advocates) that they are the only such tool, even within the realm of mathematics."

I think Z's point is that strong AI will eventually make such advances in mathematics that any human progress we might make will be rendered trivial.

But what I fail to see is how programming efforts that will similarly be rendered trivial are any more of an advance than the human proofs he derides. If he wants to advance the cause of strong AI, he should work on automated reasoning more generally, not ad hoc programs for specialized combinatorial enumeration problems.

3. Zeilberger makes many entertaining (if not altogether well-founded) points in his notes: I admit though that he appears to have a very large bee in his bonnet about the value of computer-generated proofs. As a computer scientist, and one who is fully aware of the bugs that creep into code, I can't quite understand what he is so enamored of.

Posted by Suresh

4. Ernie Writes:

A few WEEKS?! So he *admits* that the paper describes trivial work! And then he has the gall to complain when his paper is rejected?

To be fair, Jeff, does his paper have any redeeming merit? I've written a paper or two where I made some trivial implementation and discovered a structure or disproved a conjecture. I once spent a WHOLE WEEKEND writing some code to disprove a conjecture about Ramsey numbers on a 6-dimensional hypercube or something (can't remember exactly) after my buddy told me about the problem while we were drinking on a ski lift, only to discover that another guy just published the results in Discrete and Computational Geometry that same month!!! It looks like Zoran did more work than that.

But if it's crap, it's crap.

I've had too much beer at the moment to verify this, but it seems he gives a method for generating a random domino tiling of a square. That's pretty cool. It would be even cooler if it were faster than the method in this paper:

Markov chain algorithms for planar lattice structures
Michael Luby, Dana Randall, and Alistair Sinclair
SIAM Journal on Computing, 31: 167-192 (2001)

http://www.math.gatech.edu/~randall/r-lrs2001.pdf

The Pig

Posted by Anonymous

5. Zeilberger makes many entertaining (if not altogether well-founded) points in his notes:

After reading a few, I think calling them "notes" is too kind.

Posted by Anonymous

>By the end of the first paragraph, I guessed correctly that Zielberger is reacting to one of his own papers getting rejected.

If you call me, Zielberger, I'll call you Eirckson. The point of my Opinion 72 was not to react to the stupid rejection by a stupid journal, but to make a general point. The rejection was just a symptom of the stupidity of some humans, and your
feedback is another symptom. I don't have time to defend my rejected paper, but it is probably more significant than
any of your accepted papers (that you wrote yourself without a million coauthors)

Re:
>The point of mathematics is not to explain things to computers. That's not even the point of computer science!

Says Who? A HCP (Human Chauvinist Pig) like you!

Posted by Doron Zeilberger

7. Feel free to call me whatever you like, Dr. Zeilberger, but I think Ireckson would be a better response to my misspelling, don't you?

8. Opinion 72: The Next Term in the Sequence: [Dog, Human, Mathematician, ...] is Computer-Programmer for Computer-Generated Mathematics''

I've carefully gone through the article by Dr. Doron Zeilberger.

It's really an interesting thought made with remarkable insights.

I agree with his observations -
"...one can do potentially much bigger and better things, and the computer's dA is much larger, so we can (potentially) reach a mountain-top much faster, and conquer new mountain-tops where no humans will ever tread with their naked brains.

So this activity of computer-generated mathematics is the future. Unfortunately, many human mathematicians still don't realize the importance of this activity, and dismiss it as "just a computer program" and "no new mathematics"."

I would request Dr. Jeff Erickson to refrain from making casual remarks about spelling mistakes of his name and evaluate this article with an open and true spirit of a scientific mind.

Regards,
Avijit Lahiri