ANALYSIS of ALGORITHMS, Bulletin Board
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Important message from Doron Zeilberger
- To: AofA@pommard.inria.fr
- Subject: Important message from Doron Zeilberger
- From: "Helmut Prodinger" <firstname.lastname@example.org>
- Date: Wed, 7 Apr 1999 09:48:31 GMT +2
- Organization: MS, University of the Witwatersrand
- Priority: normal
------- Forwarded Message Follows -------
Date: Tue, 6 Apr 1999 18:09:06 -0400 (EDT)
From: Doron Zeilberger <email@example.com>
Subject: Non-Junky Junk Mail
Hi My Direct-E-Mail (a.k.a. junk mail) recipients:
As you alredy know, Knapp&Friedlander turned down
my brilliant opinion 36, but I believe that
it deserves the largest diffusion. So, I will
soon ask you for a favor. But, FIRST:
Let me take this opportunity to announce that the
classic book A=B (by Petkovsek, W and Z)
is now available for FREE download
(just go to my homepage, http://www.math.temple.edu/~zeilberg/).
It is still very much in print, but unlike the `non-profit' AMS, that
charges for viewing the lousy papers of the Transactions,
our `commercial' (but far less greedy than the AMS directors)
publisher, Klaus Peters, agreed to have it downloadable.
But, the main point of this message is to ask you for a favor.
The Notices of the American Mathematical Society has
just stupidly (or perhaps, cunningly) turned down the
opinion below. Of course, it is available on the web for ever after
(go to my homepage and click on opinions, then on Opinion 36),
but some of my best friends still are non-surfers, and even those
who do, may miss it.
So, please, even if you don't agree completely with the content
of this opinion, or even if you totally disagree,
but believe in free speech,
PLEASE save it, and then send it as `junk mail' to all
your department, (in my system one would do:
mail allfaculty<Op36 )
You may also want to post it on the Department's
(physical) Bulletin Board, for the benefit of the
non-users of E-mail.
--Begin Opinion 36 submitted to, and rejected by FORUM editor S. Friedlander
Don't Ask: What Can The Computer do for ME?, But Rather: What CAN I
do for the COMPUTER?
By Doron Zeilberger,
Dept. of Mathematics, Temple University, Philadelphia,
PA 19122, firstname.lastname@example.org, http://www.math.temple.edu/~zeilberg/
Written: March 5, 1999
Revised: March 25, 1999
Rabbi Levi Ben Gerson, in his pre-algebra text (1321), Sefer Ma'asei
Khosev, had about fifty theorems, complete with rigorous proofs.
Nowadays, we no longer call them theorems, but rather (routine)
algebraic identities. For example, proving (a+b)c=ab+bc took him about
half a page, while proving (a+b)*(a+b)=a*a+2*a*b+b*b took a page and a
The reason that it took him so long is that while he already had the
algebraic concepts, he still was too hung-up on words, and while he
used symbols, (denoted by dotted Hebrew letters), he did not quite
utilize, systematically, the calculus of algebraic identities. The
reason was that he was still in a pre-algebra frame of mind, and it
was more than three hundred years later (even after Cardano), that
probably Vieta started the modern `high-school' algebra.
So Levi Ben Gerson had an inkling of the algebraic revolution to come,
but still did not go all the way, because we humans are creatures of
habit, and he liked proving these deep theorems so much that it did
not occur to him to streamline them, and hence kept repeating the same
old arguments again and again in long-winded natural language.
Believe it not, our current proofs are just as long-winded and
repetitive, since we use an informal language, a minor variation on
our everyday speech.
We are now on the brink of a much more significant revolution in
mathematics, not of algebra, but of COMPUTER ALGEBRA. All our current
theorems, FLT included, will soon be considered trivial, in the same
way that Levi Ben Gerson's theorems and `mophetim' (he used the word
MOPHET to designate proof, the literal meaning of mophet is `perfect',
`divine lesson', and sometimes even miracle), are considered trivial
today. I have a meta-proof that FLT is trivial. After all, a mere
human (even though a very talented as far as humans go), with a tiny
RAM, disk-space, and very unreliable circuitry, did it). So any theorem
that a human can prove is, ipso facto, utterly trivial.
(Of course, this was already known to Richard Feynman, who stated
the theorem (Surely You're Joking Mr. Feynman, p. 70)-
`mathematicians can prove only trivial theorems, because every theorem
that is proved is trivial'.)
Theorems that only computers can prove, like the Four Color Theorem,
Kepler's Conjecture, and Conway's Lost Cosmological Theorem, are also
not very deep, but not quite as trivial, since, after all, computers are
few order-of-magnitudes better and faster than humans. In fact,
if something is provable by computer, it is at best semi-trivial
(on complexity-theory grounds). So Erdos's BOOK may exist, but all the
proofs there, though elegant, are really trivial (since they are
short). So for non-trivial stuff we can only have, at best,
semi-rigorous proofs , and sometimes just empirical evidence.
Since Everything that we can prove today will soon be provable, faster
and better, by computers, it is a waste time to keep proving, in the
same old-way, either by only pencil and paper, and even doing
`computer-assisted' proofs, regarding the computer,
as George Andrews put it, as a `pencil with power-stirring'.
Very soon all our awkwardly phrased
proofs, in semi-natural language, with their endless redundancy, will
seem just as ludicrous as Levi's half-page statement of (a+b)c=ac+bc,
and his subsequent half-page proof.
We could be much more useful than we are now, if, instead of proving
yet another theorem, we would start teaching the computer everything
we know, so that it would have a headstart. Of course, eventually
computers will be able to prove everything humans did (and much more!)
ab initio, but if we want to reap the fruits of the computer
revolution as soon as possible, and see the proofs of the Riemann
Hypothesis and the Goldbach conjecture in OUR lifetime, we better get
to work, and transcribe our human mathematical heritage into Maple,
Mathematica, or whatever. Hopefully we will soon have super-symbolic
programming languages, of higher and higher levels, continuing the
sequence: Machine, Assembly, C, Maple, ... further and further up.
This will make the transcription task much easier. So another
worthwhile project is to develop these higher and higher math systems.
So we can serve our time much better by programming rather than
proving. If you still don't know how to program, you better get going!
And don't worry. If you were smart enough to earn a Ph.D. in math, you
should be able to learn how to program, once you overcome a possible
psychological block. More important, let's make sure that our grad
students are top-notch programmers, since very soon, being a good
programmer will be a prerequisite to being a good mathematician.
Once you learned to PROGRAM (rather than just use) Maple (or
Mathematica, etc.), you should immediately get to the business
of transcribing your math-knowledge into Maple. You can get a (very
crude) prototype by looking at my own Maple
in particular RENE (http://www.math.temple.edu/~zeilberg/tokhniot/RENE),
my modest effort in stating (and hence proving!) theorems in
Plane Geometry. Other noteworthy efforts are by Frederic Chyzak (Mgfun
and Holonomic), Christian Krattenthaler (HYP and qHYP), John
Stembridge (SF and Coxeter), the INRIA gang (Salvy and Zimmermann's
gfun and Automatic Average Case Analysis), Peter Paule and his RISC
gang (WZ-stuff and Omega), and many others (but still a tiny fraction
of all mathematicians).
What, if like me, you are addicted to proving? Don't worry, you can
still do it. I go jogging every day for an hour, even though I own a
car, since jogging is fun, and it keeps my body in shape. So proving
can still be pursued as a very worthy recreation (it beats watching
TV!), and as mental calisthenics, BUT, PLEASE, not instead of working!
The real work of us mathematicians, from now until, roughly, fifty
years from now, when computers won't need us anymore, is to make the
transition from homocentric math to machine-centric math as smooth and
efficient as possible.
If we will dawdle, and keep loafing, pretending that `proving' is real
work, we would be doomed to never see non-utterly-trivial results. Our
only hope at seeing the proofs of RH, P!=NP, Goldbach etc., is to try
to teach our much more reliable, more competent, smarter. and of
course faster, but inexperienced, silicon-colleagues, what we know, in
a language that they can understand! Once enough edges will be
established, we will very soon see a PERCOLATING phase-transition of
mathematics from the UTTERLY TRIVIAL state to the SEMI-TRIVIAL state.
------end Opinion 36 of DZ------
Professor Helmut Prodinger
University of the Witwatersrand
Tel. +27-11-716 2919
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