Last Update: June 1, 1999.
Feedback from Rabbi Professor Dror Bar Natan (Oops, I meant to say Senior Lecturer Dr. Dror Bar-Natan (added June 1, 1999: Dror just became (Assoc.) Professor, which means that in Israel he should be addressed as Professor Dror, hence Brendan McKay was prophetic when he addressed him as Prof.. Judging from this astounding prediction, we should expect Dror to become a Rabbi pretty soon, and perhaps he would even drop his hyphen)):
Shalom Doron!
What a disaster it was that the French (Cauchy and his generation, and then Bourbaki) found that practically all of mathematics can be formalized! This formalization procedure seemed so powerful, that we have substituted "formal mathematics" to "mathematics", and for many of us math is ain't math until it is reduced to a sequence of theorems and proofs. Hence for example, as a group we were, and largely still are, blind to the discovery of path integrals, and we left this whole arena to the physicists, whose motivations are entirely different. Who knows how much more have we missed by requiring that the whole process, from imagination to formalization, not only be fully carried out within each mathematical context, not only be carried out in full by each generation of thinkers on each mathematical context, not only be fully carried out by each individual throughout her lifetime, but even be carried out in full within each individual submission to a journal!
Mathematics is not about proofs, it is about ideas. It is not about theorems, but about interpretations. Thus the main thing I dislike about your opinion piece is your very narrow view of what should be regarded as a part of "the BOOK". I counted 29 occurrences of the word "proof" and its derivatives in your piece, but not a single "idea".
Lehitraot,
Dror. (An avid programmer, as you probably know).
---------end of Dror's interesting feedback---------
Response from a distinguished mathematical statistician, who prefers to remain anonymous:
This is just a short comment (not to be made public, because I don't have the inclination to battle with the many people who will object to my opinion). My comment is that it's no surprise that many people will object to your viewpoints, as expressed in your opinion piece. The fact of the matter is that mathematicians, like so many other scientists, are extremely conservative in their own way. In part, the conservatism is, and can be, justified by the high level of training and rigor required to become a serious mathematician. (And frankly, after reading parts of Alan Sokal's superb hoax on the postmodern people, I am kind of happy that the barriers to becoming a serious mathematical scientist!)
However, there are aspects of the conservative nature of mathematicians which, in my view, cannot be substantiated by concerns about training, rigor, style, or whatever. This part of the conservative outlook stems purely from obstinacy, biases ingrained by mathematical upbringing, etc. This attitude is very hard to crack, and I think this is the root cause of the objections to the comments in your opinion piece.
The only way to crack this hard-core bias is to keep chipping away. I wish you lots of luck (and whatever little support I have to offer). Just keep hammering away ...
Take care,
xxxxxx
P.S. Have you ever heard the story about the really smart young Hungarian mathematician who was making his "coming-out" tour of the English universities (this took place nearer to the turn of the 20th century). The guy gave a talk at Cambridge (or Oxford?) and among the audience was the great G. H. Hardy. After the lecture was over, someone asked Hardy for his opinion of the lecturer. Hardy's response was "Obviously a very bright young man, but was that mathematics?"
As you can guess, the lecturer was John von Neumann. And imagine that as good a mathematician Hardy was unable to see that von Neumann's approach to mathematics was about to take the world by storm.
Of course, by bringing Ramanujan to England, Hardy did enough for mathematics. So I don't want to be too hard on him.
Bottom line: You and v-N are in the same situation, sort of.
-----end of response by friend who would rather stay anonymous-----------
From plouffe@math.uqam.ca Mon Apr 5 01:56:28 1999
I endorse your paper(!), fully.
I always thought that. I am very poor in proving something in mathematics (of course I know basic proofs but it never helped my life of mathematician to 'know' how to prove something), I always found that knowing to prove was a waste of brain energy and of course a waste of time.
I never understood what the other mathematicians are doing all day long (true!), doing numbers , sequences and constructive hypothesis on computers and programming it took all of my time in the past 25 years. I will continue doing the same, I had complexes for a long time when I was considering the way I work with numbers and computers, only recently I realized that I was not wrong at all : My method of research seems completely foreign to others I met in my life of mathematician. (apart from a few exceptions from which many are listed in your paper).
I think the same : we should concentrate on ways to use the computer and our brains to work out something usefull and new instead of using the papers and power-steering pencils!
Simon Plouffe
-------end of Simon's response-------
Reponse from Marko Petkovsek:
From marko.petkovsek@fmf.uni-lj.si Wed Apr 7 14:00:36 1999
Marko Petkovsek, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia , E-mail: marko.petkovsek@fmf.uni-lj.si,
***********************************************
From ira@cs.brandeis.edu Wed Apr 7 12:57:03 1999
Dear Doron,
You write
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'.)
You say that any theorem that a human can prove is trivial because humans have a small amount of memory and an unreliable processor. I find this argument neither convincing nor (what is worse!) interesting.
You also write
Everything that we can prove today will soon be provable, faster and better, by computers.
However, you give no justification for this statement, and I believe that it is false.
Best regards,
Ira
----------end response from Ira Gessel---------------
From: "Douglas E. Iannucci"
I forwarded your opinion #36 to the entire
science & math division at UVI, and one of
our biologists had a reply for me, which I
am forwarding to you at the end of this
message (in case you are interested). I
believe Wiles did not use a computer at all
in proving FLT. When you say "a human with a
tiny RAM, disk space," etc., are you referring
to Wiles himself? I believe Rich Hall (our
biologist) interpreted your statement as
such, hence his reply.
Also, I do not believe you
were comparing the computer to the human brain
per se, but rather you compared merely our respective
theorem-proving capabilities. I think that's a big
difference.
Doug Iannucci
From: Richard Hall
Newer computers easily exceed the ability of the human brain in some types
of operations as measured in cycles per seconds or bits handled per second,
however, the human brain is also described as massively parallel. For
example:
The human eye transmits information to the thalamus at approximately 1 KHz
but over 1 million parallel lines (bits). That works out to one billion
bits per second. There are a total of 10 million parallel sensory lines
into the cns, so in theory the maximum would be 10 billion bits per second.
The maximum is never obtained because peripheral and central filters
prioritize information at multiple levels... Thus the human nervous system
can collect and analyze huge amounts of information.
A 300 MHz computer typically sends information in 32 bit packets or 0.9
billion bits per second. Newer computers can easily outperform one
component of the system. However, the human brain is not one computer, it
is many, many computers linked serially and in parallel. Consider
Shakespeare's writing versus 10 monkeys on keyboards. Who would be
considered the more productive? Substitute one thousand 400 MHz PC's and
Shakespeare still gets all the royalities.
It gets better. A computer program tends to freeze when inappropriate
information or missing information bolexes operations. The human brain can
fill in the gaps and keep right on calculating using on average only 600
Kcal/day, rain or shine. Trivial problems like getting out of bed,
feeding oneself, finding social companionship are computational nightmares
for mere computers.
rlh
---------end of responses from Iannucci and Hall-----
----interesting remark from Roger Howe----------
From howe@math.nus.edu.sg Thu Apr 8 22:11:14 1999
Dear Doron,
I wanted to add a comment about your opinion #36. You have a very
good point if you take as goal the proving of deep theorems, a goal
which many mathematicians would grant. But I would say, it
is the trivial theorems which are in many ways the best.
Cheers,
---------------end remark from R. Howe-----
From larsen@iu-math.math.indiana.edu Wed Apr 7 10:43:39 1999
Dear Doron,
I enjoyed reading your "Opinion 36" very much. As usual, with your
opinions, I found it thought provoking and at times infuriating. But I
want to take issue only with one rather specific issue, namely that of
"depth". Obviously, we don't mean the same thing by depth as by "length"
or we wouldn't bother with using a different term.
It is known that there is an effectively computable constant C such that
every odd number less than C is a sum of three primes. Unfortunately, C
is out of reach of any computer we can imagine today. But we could
imagine a more powerful computer (possibly in a different universe)
checking all the cases up to C and providing a proof that every odd
integer 7 or greater is the sum of three primes. This proof would
certainly be much longer than the proof of the weaker statement known
today, but it would not be any deeper.
A deep proof should have *explanatory* power. It should force you to
rearrange your thinking about a large block of conceptual material. It is
likely to involve unexpected connections. It is also likely to seem
inevitable once it has been discovered.
The proof of Fermat's Last Theorem is very deep and fairly long.
Falting's proof of the Mordell conjecture and Deligne's proof of the
Riemann hypothesis in characteristic p do not require many pages but are
also very deep.
I am not bothered by the proofs of Haken-Appel or Hales. But the day may
come when an AI announces a proof of the Riemann hypothesis and explains
to the expectant human mathematical community that although it would like
to at least sketch the approach, it's probably too difficult conceptually
for a human being to grasp. That would bother me.
All the best,
--------end of Larsen's comments-----------
------Fascinating reaction by Ron Bruck that appeared in sci.math.research
I'm rather curious WHY the letter was rejected. Doron Zeilberger has
invented one of the most astounding and far-reaching algorithms in
computer algebra (of course, there were predecessors; it didn't come from
NOWHERE; that's why it's usually called the "Gosper-Zeilberge" or
"Wilf-Zeilberger" Algorithm, and the "A=B" book discusses earlier
contributions).
Given that standing, his opinions SHOULD be of great interest to the
mathematical community. The letter SHOULD have stimulated discussion, and
I'm surprised it was rejected. Are we hearing the whole story? Did the
editors perhaps ask for a redaction, and Doron refused? Did the editors
suggest an alternative publication as being more appropriate? Are they
perhaps concerned about his example of Rabbi Levi Ben Gerson (which I find
the most fascinating part of the whole letter, incidentally)? The Notices
have had problems with anti-semitic abstracts in the past, and perhaps
they're hypersensitive about a "pro-semitic" letter? Farfetched, surely!
Oh, maybe Zeilberger overstates his views; read his other essays on his
web page and you'll understand this isn't unusual for him. (I've been
enjoying them for years.) It may take longer than he predicts; and the
eventual evolution of man/machine collaboration is likely to be very
different from what anybody today will imagine. E.g., What happens if man
and machine MERGE? If machine becomes an integral part of us? Not the
klutzy chips of today, of course, but our neural circuitry revised and
optimized and supplemented with biological or quantum-computing implants.
The machine may then have human aesthetic sensibilities, because it WILL
be (partly) human; and who can say what sort of mathematics would evolve?
My first grandchild was born on Friday, and I watched as the doctors put
silver nitrate (or whatever they use today) into her eyes. Perhaps one
day the doctor will routinely insert her neural implant; more likely
that'll be done prenatally, and it will be genetically engineered (to HER
gene pattern) to "grow in". Phooey--why bother?--perhaps the whole thing
will be genetically engineered, at least the optimization part, from the
moment of conception. Who knows?
Yes, "suppression" is probably too harsh a word. (I don't read Doron
Zeilberger using it, incidentally--only his student.) Some explanation of
the editorial decision would be appropriate.
--Ron Bruck
bruck@math.usc.edu preferred e-mail
-------end of Bruck's comments _______________________________
---- Annoying reaction from Greg Kuperberg (in sci.math.research)----
Author: Greg Kuperberg
>I don't see any reason to have a hostile reaction to Doron Zeilberger's
This was not the BASIC thesis (this part is obvious), the basic
thesis was that we MUST change our working habits!
>I suspect
That's very math-centric! Math is but a small part of
human endeavor, just slightly more gereral than Chess. Just because
Deep Blue can beat us all does not mean that we have to commit suicide.
>I don't know how far
If you will shoot yourself, that's your business, but frankly I
hope that you won't (even though you sometimes are annoying).
>Doron's argument for his thesis is a little crazy, but it's not
Of course, you should not interpret it literally, but neither should
you interpret anybody's text literally. As Derrida, Rorty and
several others have shown, we are slaves to our own final vocabularies
and we always have hidden agendas, and our `objective views'
are just an instrument to bolster our ego, and to justify to
ourselves our miserable existence. Now that plain Racism and
Sexism is out of style, we cling to Human Chauvinism.
>Although it is not very original or strongly argued,
I have not yet seen any mathematician who said that
paper-and-pencil proving
is a waste of time and we should all be programming computers
instead.
So, even though I never claimed to be original, I only wanted to state
my opinion, it is in fact original, for what's it is worth.
Perhaps it is not as strongly argued as possible, but
its rhetoric beats your underhanded `defense' of my views from the
humanist mob.
>it obviously is provocative.
It was not meant to be provocative, neither did Galileo
or Darwin mean to be, it just had to be.
>It's like shock art.
I am glad that I was able to shock you, since more than once
you shocked me!
>When I'm in the mood for it, I like it.
>The only part of "opinion 36" that I object to is the accusations
Of course, in spite of your pioneering efforts in the XXX archives
and in electronic publishing, when it comes to the substance
of doing mathematics, you are yet another HCP (Human Chauvinist Pig).
>In fact, sci.math.research is about as
So according to you I should thank Susan&Anthony
for sparing my neck!, thanks!
------end Greg Kuperberg's annoying reaction and my (reaction)^2---
-----fascinating feedback from Andrej Bauer
Author: Andrej Bauer
What is the point of the above statement? Of course nobody expects
that planar geometry will help solve Goldbach Conjecture! I fail
to see what you were going after there. Did Zeilberger suggest that?
I think there is a more constructive way to understand the above
remark. Maybe you mean to say that providing a computer with a large
knowledge base of theorems in a given branch of mathematics will NOT
give the computer a head start in that branch. (I do not intend to put
words in your mouth, so please accept my apology if you did not mean
to say that.)
I would like to argue that it is extremely useful to have a large base
of mathematical knowledge organized in a way that can be manipulated
by computers, even if computers can do only the most trivial sort of
theorem proving. In this sense Zeilberger is correct when he says that
we should be "entering mathematical facts into computers".
At some level we already have a kind of Math-Internet knowledge database.
Mathematicians correspond with each other via e-mail and put their papers
on web pages. What is missing is the kind of knowledge that a computer
could manipulate *semantically*. For example, there ought to be a
Math-AltaVista where you could ask "Has anyone proved this theorem yet?",
and the computer would go off searching the planet.
I believe that a large body of knowledge would help a theorem prover
enormously, provided the searching mechanisms were efficient enough. I
base this opinion on an analogy with how mathematicians operate (they
know a whole lot of theorems, tricks and techniques), and on my
experience with the 'Analytica' theorem prover, developed by Ed Clarke
at CMU. Analytica uses Mathematica to do algebraic manipulations, and
it has a large knowledge base of the basic properties of real numbers.
In other words, it does not try to prove everything from the axioms
all the time. It knows about useful definitions, and it does not
automatically eliminate them (because that causes an exponential
blow-up). To see what sort of things it can do, see the paper in
Journal of Automated Reasoning, vol. 23, no. 3, December 1998, pp.
295--325. The point is that you could never prove certain kinds of
theorems without *extensive* knowledge of properties of reals, and a
lot of algebraic manipulation that Mathematica does.
>I don't think time spent on
>entering such facts into Maple or Mathematica will advance mathematical
>knowledge, now or ever, by one iota.
I agree that entering mathematical knowledge into existing computer
algebra systems and theorem provers is a waste of time. We have not
yet developed the infrastructure that is needed for a global
mathematical database that could be manipulated on the semantic level.
Mathematica and Maple are nowhere near being satisfactory systems for
such an endeavor. I think Zeilberger is wrong when he thinks that we
can do it today. We do not even have a good language to do it in.
However, I have no doubts that the required tools, mathematical and
computer theoretic, are going to be developed in two or three decades,
if not sooner. They *are* being developed by various groups, mostly by
theoretical computer scientists (one that comes to mind is the Cornell
Nuprl group). And once we have those you will be proven wrong---there
will come a day, not too far from now, when computers will
significantly advance mathematical knowledge. I do not want to make a
prediction as to whether this will be just because of a global
mathematical database, or also because of very smart theorem provers.
But I do predict it will be in my lifetime.
>In the end no human mathematicians will be useful or needed, right? In
>fact, "in the end" the universe will die of heat exhaustion and all of
>the elementary particles will decay. But we don't have to make
>decisions about what to do right now based on what will happen "in the
>end"; we can decide based on the circumstances that prevail at the
>moment.
You know very well that I was not thinking of the end of the universe
and I was not talking about mathematician's angst of being useless.
So, silly remarks and intentional misunderstandings aside, the point
I was trying to make was that either our generation or the one coming
after us is going to make the big leap from doing math "by hand" to
doing math "*with* computers". In that sense, we do have to make
decisions now. I think the correct decision is to wait until those
pesky theoretical computer scientists come up with decent languages
and knowledge manipulation techniques that will bless the happy
matrimony of math and computers. I suspect Zeilberger is afraid
that the arrogant mathematicians will refuse to listen to the pesky
theoretical computer scientists.
>I don't read the article as a modest exhortation to mathematicians to
>become more familiar with what computers can do for them, and to use
>them more in their research, which I would wholly support. I read it as
>either lunacy or satire. Perhaps the editors read it the same way, and
>that's why it was not accepted.
Yes, the article is not written in the the most diplomatic and
convincing way. It is too enthusiastic, and that is why many people
will probably dismiss it easily (which is just as well).
If you attempt to understand Zeilberger in a less dismissive way,
though, then you could still find something positive in what he is
saying. He's drawing our attention to a new technology which will, in
his opinion, revolutionize mathematics. Zeilberger thinks this is
happening now, I think it will happen in my lifetime, and if
understand what you are saying, according to you it will never happen
(even though you heavily use computers?). Opinions, opinions.
********
Let me also respond to Phil Diamond.
Phil Diamond, pmd@maths.uq.edu.au writes:
>OK, I accept that the machines will be able to do these nontrivial
>problems. But after that? Who will provide more nontrivial (or even
>trivial) problems? Who will invent the concepts that are used?
Humans will invent new concepts, of course. May I ask why you are
asking these questions? I do not understand what you are getting at.
If I understand you correctly, you are making the point that even if
machines could prove theorems much more efficiently than humans, they
still would not know *what* to prove. So what? We are going to tell
them what to prove. Machines are *tools*! They will replace those
mathematicians who spend their days devising formulas for compound
interest rates, and solving differential equations for the design of
new cars. That's good!
I cannot help but to view your opinion as a form of technophobia. Is it?
Maybe something needs to be said about where mathematical problems
come from. I think the best mathematical problems are the ones that
originate from real-world problems, and I mean this in a very general
sense. For example, I would claim that classical analysis was invented
because of the needs of physics to understand the macroscopic world
(Newtonian mechanics). A more recent example would be the way computer
science is driving certain branches of mathematics (discrete math,
type theory, constructive logic). I can't imagine we'd ever run out of
problems to solve.
>This is a question that goes far beyond computer algebra and enters
>the AI area. And after 40 years and zillions of $, the Holy Grail
>of machine intelligence (whatever that is?) seems as unattainable
>as ever.
Knowing the kind of stuff Zeilberger does, I do not think he has
AI-ish inclinations. My understanding is that he is suggesting that
mathematicians should be finding *algorithms* for solving problems (he
talks about *programming*, not about automated theorem proving). The
kinds of algorithms that he might have in mind are Schur's algorithm
for finding the closed form of an indefinite integral, or
Gosper-Zeilberger-Wilf algorithms for finding the closed form of
summations, or Buchberger's algorithm for finding a Groebner basis. Of
course, having blown his vision out of proportion, he paints a future
which resembles a sci-fi movie.
Is Prof. Zeilberger reading this discussion? It would be great
to hear his opinion. Maybe his student can provoke him into
replying by showing him a printout of this thread.
>It is the difference between developing chess playing
>systems that can beat any human being at the game, and **inventing**
>the game.
Yes, yes. But don't you think that even if computers can be "just"
very good at proving theorems, that would still have a huge impact on
math? And that we should pay attention to such a possibility, even
if computers will always lack the "human spirit and creativity"?
I'll borrow your analogy. We all know that computers have become very
good at chess. It is perhaps less known that they are very bad at go,
the Japanese game. Today's computers are as bad at math as they are at
go, but thirty years from now they will be as good at math as they are
at chess today.
-------end Andrej Bauer's reaction
--begin fascinating, profound, and humor-sparked feedback by Olivier Gerard--
About Doron Zeilberger's Opinion #36
While I am very sympathetic with Doron Zeilberger's frank and direct
way of expressing his ideas and while most of the mathematics he does
with his companion Shalosh B. Ekhad has a strong appeal to me, I am far
to agree with it on many minor aspects of his opinions.
-> No, a Ph. D. in math does not warrant at all the ability to program,
even when one has removed those very common psychological blocks. There is
a lot to unlearn, there is the whole social implication of a researcher in
the academic system. Moreover, we all now math Ph. D. who aren't smart
*at all* and especially about mathematics. Introducing this kind of public
to even mathematically-minded computer macro-languages like TeX is worth
Hercules' works. Part of the trouble with any academic system is that there
are areas, time and circonstances were you can be dubbed "researcher" or
"professor" by only being a good schoolboy, no need to be smart about the
subject.
-> The typical attitude in areas of mathematics where designing programs for
calculations could be useful for one's research, was precisely to have
top-notch programmers as graduate students to do the ground (and despised)
work. Many of them, investing their energy in almost confidential project
like this left the mathematical field disgusted, especially by the poor
credit given to their efforts, often refered to as "non-mathematical".
-> The center of the mathematical discovery and of short (and you would say
trivial) proofs, is not proving in itself (we agree), but not programming
either. The common intellectual process is *finding a suitable
representation of the problem*.
-> What do we do for computers, most of the time? We translate our
mathematical folklore into a recordable or computable discrete model and we
discharge ourselves of the burden of computing, veryfying or applying the
said models.
-> Short or elegant proofs are similarly based on efficient models, even if
they were not conceived to be implemented on a computer system.
-> In fact the real work of a mathematician is often to translate his ideas
on various media: another part of his own brain, blackboard, paper, a given
sub-language or sub-area of mathematics, another mathematician's brain and
naturally Doron, computers. Each translation gives a chance to trivialize a
step further what is passed through. But the triviality resides only in the
final statement, whereas the mathematics is the sum of the modelisation(s)
plus the statement, not the theorem alone.
-> If we come back to Rabbi Levi ben Gerson, I agree with Doron that probably,
like many of us, he was fascinated by the plays of these charming entities
that satisfied as by miracle identities and wanted to collect as many as
possible in the same kind. Moreover, the use of a page-length word
explanations made the results even more fascinating, like gems worked out of
convoluted reasoning.
-> What is not present in Doron's text is that this attitude is probably a
necessary step in the development of any mathematical knowledge by humans
being. This is not only a matter of habit, it is also part of the relation
of knowledge, power and attitude towards life.
-> I strongly support your view that the sequence:
Machine code, assembly language, C (or other general purpose programming
language), Mathematica (or other general purpose symbolic computation
program), ...
is just in its infancy and should be expanded with all our might. My own
prejudice is to enhance the graphical interaction between us and the computer
to exchange more intensively with it. This is the purpose of at least two of
my projects: "Tasteful algebra(tm)" and "Suukan".
-> Another direction that systematically enhance our mathematical knowledge
is the challenge of conveying our most recent mathematics to the youngest
possible children or the most general public.
-> To mention another direction than strict computer theorem proving I
recommand you to look to a recent article by Simon Colton published in the
JIS (The Journal of Integer Sequences) at
http://www.research.att.com/~njas/sequences/JIS/,
the article can be reached directly at
http://www.research.att.com/~njas/sequences/JIS/colton/joisol.html
were you will find the reference of Colton's web site on his project.
I plan to write later more in detail about this kind of endeavour which
matches very well Doron's view of the future of mathematics and certain
enterprises of mine.
-> However powerful, experienced and smart computers may become, I am still
of the opinion that
a) we need mathematics more badly than they do.
b) we will still be a source of mathematical questions or concepts
they wouldn't care about.
Of course, I expect Shalosh B. Ekhad to say exactly the same on its side.
I only hope that, if computers do replace us on the surface of the Earth,
they'll have a little thought of our destiny when they are replaced by
another living form I can't even dream of they will inevitably create someday
to do even less trivial mathematics.
-> I sometimes suspect Doron Zeilberger of giving due credits to
Shalosh B. Ekhad (when credit is due) in the hope that Shalosh will consent
to keep him as a pet in his old days. It's a feeling I have sometimes about
my own personified computer network and program library,
Charles O. Gely which is systematically credited for all my computer
aided mathematical discoveries.
-> A interesting side effect of putting more and more mathematical knowledge
into computer systems will lead us to reconsider most of the hierarchy of
mathematical subareas. I am convinced for instance, that a lot of algebraic
geometry and algebraic topology will look rather easy and under complete
dependence of combinatorics when this process is complete, but this is a
view Doron has already forcefully expressed.
-> Doron Zeilberger adds to the end of his opinion that he proposed his text
to the AMS Notices. I would have loved to see it in print in this venerable
organ of the american mathematical community. But if I had been the Editor of
the Forum section, I would have rejected it. I don't know what was the
reasoning of Susan Friedlander, but here is what would have been mine:
<< Since the AMS Notices Forum Column is quite small and that we don't have
the resource to manage real, open, and democratic debates in it, we better
not start polemics when they don't stem from articles already published in
former issues. If I reject his column, he will have a better case to make
his opinions known and rally people under his banner, and mathematicians
that do not minimally use online technologies won't understood Zeilberger
points anyway. >>
It may be that Susan Friedlander has rejected DZ column because she had very
different motivations, bad or good. But in effect her decision serves you
quite well, and if she was trying to block you, the result is not in that
direction.
****
Added May 17, 1999: Jaak Peetre (of Lions-Peetre Interpolation of
Operators fame) wrote a very interesting
defence of Rabbi Levi Ben Gerson
Added Jul 14, 1999:
Ursula Martin wrote
a very interesting article
available from
Ursula Martin's On-Line publications
, that is highly relevant to the present issues.
Dear Doron:
Subject: Re: Free Speech! (non-junky junkmail)
Roger
-------begin interesting response from Michael Larsen----
Michael Larsen
----(with my (reaction)^2-----
>opinion 36. The basic thesis --- that computers are replacing
>mathematicians --- is true in some respects and not in others.
>that when computers make mathematicians obselete, all
>human activity
>will be superfluous and human history will end.
>in the future this eventuality might lay, but even if
>it were soon,
>nothing we do now would matter much afterwards.
>completely crazy. You shouldn't interpret it literally,
>even though Doron himself might.
Enjoy!
>against the editors of the Notices. The Notices is not the
> be-all of
>mathematical soapboxes, to be protected at all costs from
> the fallible judgement of its editors.
>widely read as the Notices, although more by amateur mathematicians
>and less by professionals. I take the two forums equally seriously.
>One difference is that the moderators of sci.math.research give people
>more rope to hang themselves than the Notices does.
Date: 1999/04/15
Forum: sci.math.research
_________________________________________________________________
David desJardins
>I don't think that a computer that can prove the Goldbach Conjecture
>will get any kind of "head start" from reading a list of definitions
>and facts used in proving theorems in plane geometry.
From ogerard@ext.jussieu.fr Tue Apr 20 16:08:09 1999
(for DZ text see: http://sites.math.rutgers.edu/~zeilberg/Opinion36.html)
(This is Olivier Gerard's Opinion #6, the previous ones will be translated in
english before being released on my web page.)
------end Olivier Gerard's reaction (much better than the cause!)------
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Opinion 36 of Doron Zeilberger