[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

**To**:**AofA@pommard.inria.fr****Subject**:**Important message from Doron Zeilberger****From**:**"Helmut Prodinger" <helmut@cam.wits.ac.za>**- 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 <zeilberg@euclid.math.temple.edu> To: zeilberg@euclid.math.temple.edu Subject: Non-Junky Junk Mail Cc: notices@math.sunysb.edu 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, zeilberg@math.temple.edu, 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 half. 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 packages (http://www.math.temple.edu/~zeilberg/programs.html)., 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 Mathematics Department University of the Witwatersrand P.O. Wits 2050 Johannesburg South Africa Tel. +27-11-716 2919 Fax. +27-11-4032017 Email. helmut@gauss.cam.wits.ac.za Homepage. http://www.wits.ac.za/helmut/index.htm

- Prev by Date:
**complexite pseudo-diadique** - Next by Date:
**Set symmetric difference** - Prev by thread:
**complexite pseudo-diadique** - Next by thread:
**Set symmetric difference** - Index(es):