[FOM] Brief Reply to Comments on my Opinion 57

Doron Zeilberger zeilberg at math.rutgers.edu
Tue Nov 11 20:48:45 EST 2003

Dear FOMers,
  Thanks for the interesting remarks about my Opinion 57.
Here are brief reactions to some of the comments.

1) Reply to: Alasdair Urquhart <urquhart at cs.toronto.edu> wrote:

>Hardy did not say that "chess is trivial."
>What he did say was that chess problems of 
>the kind "White to move and mate in 3"
>are trivial.  On the whole, this seems correct,
>since such problems are easily solved by 
>exhaustive search.  

Alrealy mate in 3 moves is not so easy by brute "brute force", and
one needs `clever brute force' + backtrack. Anyway,
Chess was just a metaphor for something extremely hard yet finite,
and my point was that human mathematics is also finite, but humans
pretend that it is infinite, and that's a fiction, and the activity
of math is a generalized Chess game, but of very limited depth

>Hardy goes on to say that chess problems are
>unimportant, but that the best mathematics is
>serious in the sense of connecting together 
>significant mathematical ideas.  In general,
>I don't find too much to quarrel with in his
>opinions, and I believe most mathematicians
>would agree with him.

Perhaps you are right about yourself and `most mathematicians'
but you and the `mathematicians' you speak of are humans,
so they go with  anthropocentric manifestos of people like
Hardy, Dieodonee, and Atiyah, who talk about `ideas' and
`mainstream' and `core' mathematics, thereby dictating to us
what's `important'. It is only important for 
maintainig the current party line and theological doctrines.
Diodonee didn't like combinatorics becuase its problems are
`without issue' and not connected to the `core' of mathematics.
But I'll trade any day a nice combinatorial solution to all
the volumes of Bourbaki.

2) Reply to: Anatoly Vorobey <mellon at pobox.com>

>I think you're completely missing the point on why Hardy thinks
>chess is trivial (thereby rendering most of your trashing of Hardy 
>irrelevant). It's not because chess is finite, unlike number theory 
>or real analysis or what have you; it's because the rules of chess are 
>incidental. A chess problem does not give rise to nontrivial mathematics 
>because it is governed by incidental (and historically accidental) rules
>which are very unlikely to connect in any meaningful way to the existing
>body of mathematics, 

The existing body of mathematics is every bit as contingent as
the rules of chess, with the difference that human math is less deep
since humans pick problems that they can do.

3) Reply to Ms. Kathy Garber

>I would very much like to see discussion on Mr. Zeilberger's core ideas
>beyond response to the introductory paragraphs of his article.

>For example, are there any thoughts on the notion of number crunching as a
>special case of symbol crunching? The implications?

>Has anyone worked seriously with an automated theorem prover in
>conjunction with a computer algebra system?  What *are* the philosophical
>implications; i.e., if these systems can do Heine-Borel, what more can
>they do and should we all keep doing Heine-Borel over and over? (much less
>decorating web pages over and over..)

>I have tinkered around with Isabelle recreationally, but don't really want
>to invest the time required to gain proficiency if there are sound
>mathematical grounds for using a different prover.

>Specifically, for Mr. Zeilberger - have you investigated the tools of the
>automated theorem proving (ATP) community yourself and found them lacking?

Interesting questions, Ms. Gerber. I have a great admiration to
ATP, and read with great interest Donald MacKenzie's book
`Automating Proofs', but I think that ATP have limited scope in
creating new non-trivial math, since it is logic-based.
I prefer what I call ansatz-based math, which would emulate
(much faster and better) the way math is actually created,
as opposed to written up, in the stultifying Euclidean-Fregean
style. So ultimately CAS would be more promising, but they are
still in a very primitive stage, and perhaps a synthesis of
ATP and CAS would be fruitful.

Finally I'd like to thank Tim Chow and Olivier Gerard for their
kind interprations and comments.

-Doron Zeilberger

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