Last Update: Aug. 21 , 2001

Snail:

Dr.
Doron Zeilberger,

Board of Governors Professor of Mathematics,

Department of Mathematics

Rutgers University , Busch Campus-Hill Center

110 Frelinghuysen Rd

Pistacaway, NJ 08854-8019, USA

FAX:(732) 445-5530

Mon. : Hill 704 (732) 445-1326.

Tue., Wed., Thu., Fri. : Home (609)921-7873.

Office Hours: By appointment (during summer)

Math 085 (Calculus, Fall 2000), Barton Hall BB309, Mon. and Wed. 9:40-11:30.

Math 517, Combinatorics, Wachman 617, Mon. and Wed. 11:40-1:00.
Math 684 (Combinatorics, Spring 1999),
MW 11:40-1:00, Wachman Hall 527.

Math 683 (Fall 1998),MW 11:40-1:00, Wachman Hall 527.

Math 502
:A class I taught in 1997-1998.

PROJECT: Prove that sum(mobius(i),i=1..n) < (A_eps)*n^(1/2+eps)

"What we know about for sure is bad enough already. Figuring out the answers to the questions that remain open might make things a little better but will probably make them a lot worse. And what's really frustrating is the uncertainty that comes from not really knowing."---- David Harel, "Computers Ltd.'', Oxford Univ. Press, 2000, p. 155.

`The progress of mathematics can be viewed as progress from the infinite to the finite.'

--Gian-Carlo Rota (1983) [quoted by Xavier G. Viennot. Opening plenary talk, LACIM 2000, Sept. 7, 2000]

`The many mathematical theorems, that have finitary proofs, are certain, hence there is absolute certainty in mathematics. What is still problematic, like in Pythagoras's time, is the use of infinite objects.'

--Arnon Avron [Goedel's Theorems and the Foundations of Mathematics Problem (in Hebrew, Ministry of Defence, Israel, 1998, p. 167]

`Like musicians who can read and write complicated scores in a world without sounds, for us mathematics is a source of delight, excitement, and even controversy which are hard to share with non mathematicians. In our small micro-cosmos we should ever seek the right balance between competition and solidarity, criticism and empathy, exclusion and inclusion.'

--Gil Kalai, "Combinatorics with a Geometric Flavor: Some Examples", to appear

`I'm frightened stiff by the Internet, billions of people all over the world have access to it.'

- Gabriel Bach (retired Israel Supreme Court Judge)

"In high school, I wanted to study logic, which I thought would be useful in political debate or in the legal battles against evil once I fulfilled my dream of becoming a solicitor. Unfortunately, I became neither a lawyer nor a politician,and I have since come to understand that logic is not a very useful tool in these areas in any case''.

---Ariel Rubinstein ("Economics and Language", p. 3)

`Sometimes a good idea comes to you when you are not looking for it. Through an improbable combination of coincidence, naiveté and lucky mistakes ...'

--Kary B. Mullis (`The Unusual Origin of the Polymerase Chain Reaction', Sci. Amer., April 1990) p. 445)

`A mathematician experiments,amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so - and proves a theorem.'

--Steven Krantz (`Conformal mappings', Amer. Sci. Sept.-Oct. 1999, 436-445, p. 445)

`So too is mathematics, the proof of the refined alternating sign matrix conjecture is the ground from which we can begin to seek its true significance. At the next stage, we seek the theories that can explain what we have seen and predict the directions that should be most fruitful.'

--Dave Bressoud (`Proof and Confirmation', 1999, p.259

Philosophers and psychiatrists should explain why it is that we mathematicians are in the habit of systematically erasing our footsteps. Scientists have always looked askance at this strange habit of mathematicians, which has changed little from Pythagoras to our day.

--Gian-Carlo Rota ( April 27 1932- April 18 1999), `(Two Turning Points in Invariant Theory', Math. Intell. 21(1) (Winter 1999), p. 26)

``Sometimes attaining the deepest familiarity with a question is our best substitute for actually having the answer''

--Brian Greene (`The Elegant Universe', p. 271).

...`Two lines must meet at a point. Therefore there are only two surprises here'.

...`This is a rather unusual situation in physics. We perform approximate calculations which are valid only in some regime and this gives us the exact answer. This is a theorist's heaven- exact results with approximate methods'.

...`We do not know how to formulate string theory nor do we know its underlying principles. Surprisingly, this fact does not stop us from making progress'.

--Nathan Seiberg (`The Superworld', hep-th/9802144, 20 Feb 1998)

`They had proved a significant outstanding conjecture, but not the one they had set out to prove'...

`The study of alternating sign matrices should continue to bear fruit for many years to come-and to tantalize us with fruit that is just beyond our reach.'

--Dave Bressoud and Jim Propp (`How the Alternating Sign Matrix was Solved', Notices of the Amer. Math. Soc. 46(6) (June/July 1999), 637-646, p. 641, p.646)

`The extensive use of computers in the discovery and proofs of the ASM formula may be a harbinger of future trends both within and beyond the theory of determinant evaluations. Zeilberger envisions an age in which the work that was done by his troupe of hand-picked referees will be done entirely by computers; computers will take over nearly all the tedious parts of mathematics, freeing human mathematicians, and perhaps a few artificially intelligent electronic colleagues, to spend their time in the more creative side of the enterprise.'

--Dave Bressoud and Jim Propp (Previous version of `How the Alternating Sign Matrix was Solved', Notices of the Amer. Math. Soc. 46(6) (June/July 1999), 637-646, deleted from the published (final) version, but restored in a letter to the editor by Bressoud and Propp, Notices of the AMS 46(9), 1030-1031,Oct. 1999).

``Over the last few years, in a rush to exploit a society driven by buzzwords and Microsoft press releases, computer book publishers have spewed out a mountain of meaningless books, at best worthy of being used as toilet paper and in some cases not even. This constant onslaught of garbage has left the task of finding good books about as difficult as getting Internet Explorer uninstalled. The good books are out there, but they are swamped in a sea of "Build your own ...", "... in C++", and "Advanced ..." trash.''

--Noam Zeilberger

It is more important that a proposition be interesting than that it be true. This statement is almost a tautology. For the energy of operation of a proposition in an occasion of experience is its interest and is its importance. But of course a true proposition is more apt to be interesting than a false one.

---Alfred North Whitehead, quoted in: W. Auden and L. Kronenberger, The Viking Book of Aphorisms, New York, Viking Press, 1966.

[Contributed by Olivier Gerard.]

``I thought about this mathematical need to reduce fractions to their minimum. I thought about all the energy that students put out on reducing thirty four divided by sixty eight to one half. And I was reminded of the feeling of relief in the discovery that some fraction is actually one fifth.

Now I thought that this was all baloney [bilbul moakh]. As though that one fifth is clearer. They gave a semblance of clarity to semething that is not necessarily clearer and they idiotized people. They idiotized me. What is clear in the fraction `one fifth'?''

Same thing with logarithms....''

--Orly Castel Bloom, `The New Book of Orly Castel-Bloom' [Taking the Trend] (in Hebrew), Keter, 1998,p. 185.

``Perhaps you have seen me. I know well, my purpose was merely that of a symbol, `equals', `times'... ; but what is said, for all that, was identity-less: a kind of live geometry''.

--Albert Goldbarth, ``Dark Waves and Light Matter'', The University of Georgia Press, 1999, p. 12.

``In my opinion, it is not just the serious accomplishments of distinguished men which are worthy of being recorded but also what they did for fun.''

--Xenophon, "The Symposium." [contributed by Ilan Vardi]

``More than any other time in history, mankind faces a crossroads. One path leads to despair and utter hopelessness. The other, to total extinction. Let us pray that we have the wisdom to choose correctly''.

--Woody Allen, My Speech to the Graduates, Side Effects, p. 81 (1981)

``We decided that `trivial' means `proved'. So we joked with the mathematicians: `We have a new theorem- that mathematicians can prove only trivial theorems, because every theorem that's proved is trivial.''

--Richard Feynman, `Surely You're Joking, Mr. Feynman!', p. 70 (1985)

``Here is something Category-Theorists like: it is trivial, but not trivially trivial.''

--W. Timothy Gowers, Grosswald Lecture, Temple Univ., March 29, 2001.

``The research reported on in our book "A=B" (A K Peters, Ltd., Wellesley, MA), has moved a whole active field of mathematics from the province of human thought to the realm of computer-fodder. It is quite exciting to think about what other fields of pure mathematics, hitherto thought to be reserved to human intelligence, might be moved to that realm next. The goal is to put ourselves out of business completely, and the work is well underway.''

--Herb Wilf, Chronicles of Higher Education (c. 5/1998)

It is worth noting that despite this work, which apparently shows that there is a class of theorems for which computers can find proofs, Prof. Wilf is still working as a research mathematician :-)

--Michael Albert , sci.physics Forum, message dated 5/14/2000 .

``This is like dreaming, but now you have to wake up, and the minus sign is what is waking you up''.

--Alain Connes, Grosswald Lecture, Temple Univ., April 27, 1998.

``Every human activity, EXCEPT Mathematics, must come to an end.''

--Paul Erdos, quoted by Bela Bollobas, Amer. Math. Monthly 105(1998), p. 209

`In teaching, the greatest sin is to bore.''

--Jet Wimp Amer. Math. Monthly 105(1998), p. 292.

`A beautiful problem is like a funny joke.'

-- Daniel Ullman, Amer. Math. Monthly 105(1998), p. 292.

``If you had done something twice, you are likely to do it again.''

-- Brian Kernighan and Bob Pike, `The Unix Programming Environment', Prentice Hall, p. 97.

``What promotes math progress even more than new ideas (and is there such a thing as a truly new idea?) are new technical tools and habits of thought that encapsulate existing ideas, so that insights of one generation become the instincts of the next.''

``MacPherson told me that my theorem can be viewed as blah blah blah Grothendick blah blah blah, which makes it much more respectable''

``I think some intuition leaks out in every step of an induction proof''

-- Jim Propp, talk at AMS special session, 1/22/00

``Induction makes you feel guilty for getting something out of nothing, and it is artificial, but it is one of the greatest ideas of civilization. It is very hard to teach, but WE are TEACHERS.''

--Herbert S. Wilf, invited MAA address, Baltimore, Jan. 10, 1998.

``A scientist can not be measured quantitavely by the number of degrees or the accumulation of information. A true scientist should have a measure of courage to correct error and seek truth- no matter how painful. The alternative is more painful. To build error upon error is to drift into dogmas, metaphysics, science fiction, and mythology.''

``...I regard as impertinent any science which purports to explain and instruct me or my behavior based on the antics of a hungry rat in a cage.''

--Alexander Wilf, `Origin and Destinity of the Moral Species', 1969, p. 9

``The cubic root of 2 is not constructible by ruler and compass, but the cubic root of 2+sqrt(5), which looks more complicated, is, (since it equals the golden ratio). Things like this make it fun to be a mathematican.''

--Tom Osler, Temple talk, 3/25/98.

``Everybody knows that mathematics is about Miracles, only mathematicians have a name for them: Theorems.''

--Roger Howe, invited MAA address, Baltimore, Jan. 9, 1998.

``1/r^2 has a nasty singularity at r=0, but it did not bother Newton-the Moon is far enough.''

--Edward Witten, AMS Gibbs Lecture, Baltimore, Jan. 7, 1998.

"We haven't the money, so we have to think."

--Ernst Rutherford [contributed by Edoardo Milotti]

``Dans les sciences mathématiques, une bonne notation a la m&ehat;me importance philosophique qu'une bonne classification dans les sciences naturelles''

---Henri Poincaré (Preface to Laguerre's CW)

``Nein! Wir Sind Dichter''

--Kronecker (quoted by Sylvester, CW IV, 625)

`...Mathematics... is a bit like discovering oil. ... But mathematics has one great advantage over oil, in that no one has yet ... found a way that you can keep using the same oil forever.'

--- Andrew Wiles, Notices of the AMS, May 1997, p. 588.

``I never use a computer''

--- Andrew Wiles, Nova program on FLT and `Fermat's Enigma' by Simon Singh, p. 211, line 4

``He [Taniyama] was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this and tried to imitate him, but found it quite difficult to make good mistakes''

--- Goro Shimura, Nova program on FLT and `Fermat's Enigma' by Simon Singh, p. 174

`Nature might be somehow more powerful than a digital computer'--

Aviezri S. Fraenkel [quoted in NY Times, March 25, 1997, p. C5, col. 6]

``The recent development of combinatorics is somewhat like a cinderella story. It used to be looked down on by ``mainstream'' mathematicians as being somehow less respectable than other areas.... Then along came the prince of computer science with its many mathematical problems and needs-and it was combinatorics that best fitted the glass slipper held out.

--- --- Anders Bjorner and Richard Stanley, `A combinatorial Miscellany', preprint, p.2

``Certain functions appear so often that it is convenient to give them names. These are collectively called special functions. There are many examples and no single way of looking at them can illuminate all examples or even all the important properties of a single example of a special function.''

---Richard A. Askey [preface to `Special Functions: Group Theoretical Aspects and Applications', edited by T.H.Koornwinder and W. Schempp, Reidel, 1984]

``If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do.''

--- Richard Askey [Ramanujan and Important Formulas, p. 32, in ``Srinivasa Ramanujan (1887-1920), a Tribute'', K.R. Nagarajan and T. Soundarajan, eds., Madurai Kamaraj University, 1987.]

``When I entered graduate school I had carried out the instructions given to me by my father and had knocked on both Murray Gell-Mann's and Feynman's doors and asked them what they were currently doing. Murray wrote down the partition function for the three-dimensional Ising model and said it would be nice if I could solve it (at least that is how I remember the conversation). Feynman's answer was `nothing'.''

--Ken G. Wilson [quoted in J.M. Yeomans, `Staistical Mechanics of phase transitions', p. 35]

``One's fingers are frequently smarter than one's mind''

-- Richard J. Duffin (quoted by Clarence Zener in: `The influence of Dick Duffin on an Engineer', that appeared in `Constructive approachs to mathematical models', C.V. Coffman and G.J. Fix, editors, a volume in honor of R.J. Duffin (1909-1996)).

``Dear Sirs: Nash is a mathematical genius''

---Richard J. Duffin, ibid.

``My dear friend and at that time research director Dick Duffin rehearsed me for the talk. After my first presentation he said: `Very good, Raul, but cut it in half'. When I had done so I tried again: `Excellent', he said, `but cut it in half'. And I must say this principle of cutting in half twice stood me well ever since. Would that all my professional brethern had learned it also.''

--Raul Bott, Notices of the AMS 37(7)[Sept. 1990], 806.

``A mathematician is a conjurer who gives away his secrets''

-- (John Horton Conway quoting)**(INFINITY)) [in: Open problems in communication and computation, T.M. Cover and B. Gopinath, eds., Springer, 1987; p. 9].

``What good your beautiful proof on [the transcendence of] Pi: Why investigate such problems, given that irrational numbers do not even exist?

--Leonard Kronecker (to Ferdinand Lindemann) [quoted in: `Fractals, Chaos, Power Laws', by M. Schroeder, W.H. Freeman, 1991]

``Surely with as good reason as had Archimedes to have the cylinder, cone and sphere engraved on his tombstone might our distinguished countrymen [Arthur Cayley and George Salmon] leave testamentary directions for the cubic eikosiheptagram to be engraved on theirs. Spirit of the Universe! wither are we drifting, and when, where, and how is all this to end?''

---J.J. Sylvester, Proceedings of the London Math. Soc. #2 (1867), P. 155. Found in Coxeter's ``Regular Complex Polytopes,''

``The work of Bourbaki is like a beautiful symphony with too many French horns.''

--Ilan Vardi

``He is a very humble person who has a lot to be humble about.''

--Ilan Vardi

``1 1 10 10 20'' (To be read in French)

--Ilan Vardi (his homepage) [ This was Vardi's attempt at outdoing the "pi" movie. ]

``The popular press often seems to be a modern form of the Inquisition.

--Ilan Vardi (in `A Classical Reeducation', a forthcoming book).

``Mathematical study and research are very suggestive of mountaineering. Whymper made several efforts before he climbed the Matterhorn in the 1860's and even then it cost the life of four of his party. Now, however, any tourist can be hauled up for a small cost, and perhaps does not appreciate the difficulty of the original ascent. So in mathematics, it may be found hard to realise the great initial difficulty of making a little step which now seems so natural and obvious, and it may not be suprising if such a step has been found and lost again.''

--L.J. Mordell, Three Lectures on Fermat's Last Theorem, p.4

`Imaginons par example qu'on nommé Dupont, ayant fait le premier l'ascension du Bréevet, en rapporte un croquis de la chaîne du Mont-Blanc de ce sommet... Imaginons que Durand, faisant \à son tour la même ascension, écrive: "J'ai fait une découverte qui avait échappé à Dupont"... Dupont n'est-il pas en droit de répondre: "Cela va sans dire. Il n'y a qu'a regarder mon dessin." ? Mais Durand peut répliquer: "Cela va encore mieux en le disant, et c'est moi qui l'ai dit'

---Paul Levy, Quelques aspects de la pensée d'un mathématicien, (quoted by Ilan Vardi in `A Classical Reeducation', a forthcoming book). Ilan Vardi's translationn of the above plus.

``Everyone else would climb a peak by looking for a path somewhere in the mountain. Nash would climb another mountain altogether and from that distant peak would shine a searchlight back onto the first peak.'' ''

-- Donald Newman (quoted in `A Beautiful Mind' (Simon and Schuster, 1998) by S. Nasar, p. 12)

``For us who grew up thinking that all there is to learn about sl(N) is already in sl(2), this is not a big surprise''

---Dror Bar-Natan (Comobinatorica 17(1997), 43-52. quoted by R. Thomas, Notices of the Amer. Math. SOc. Aug. 1998, p. 851)

``Science is what we understand well enough to explain to a computer, Art is all the rest.''

-- Donald E. Knuth (foreword to ``A=B'' by Petkovsek, W and Z)

``Premature optimization is the root of all evil''

--Donald E. Knuth (quoted in `The Unix Programming Environment' by Kernighan and Pine, p. 91)

"If you think you're a really good programmer... read [Knuth's] Art of Computer Programming...You should definitely send me a resume if you can read the whole thing.''

--Bill Gates [ quoted on the jacket of `ACP I', third ed.]

"If you can enjoy at least a few pages of ACP, you have in your mind something even Bill Gates cannot buy for himself."

--Olivier Gerard

"Most of the programers in ten years will be us, and we won't get much smarter''

--Bjarne Strustrup [lecture at Temple U., 11/25/97]

"If you think it's simple, then you have misunderstood the problem''

--Bjarne Strustrup [lecture at Temple U., 11/25/97]

"I think that I am among the few lucky ones who are exploiting complexity. Most people are unhappy with the emergernce of complexity, they would prefer it if the world were very simple, but then it would be a doom for a cryptographer like myself.''

--Adi Shamir, in: `THe Emergence of Complexity', B. Pullman, ed., p. 93.

``We should give up the attempt to derive results and answers with complete certainty''

-- Michael O. Rabin [in `Out of their minds' by D. Sasha and C. Lazare, p. 68]

`If you need more than five lines to prove something, then you are on the wrong track'

-- Edgser W. Dijkstra's mother [ibid, p. 55]

`If after two weeks of computing, I quit the program with an error message: `System Error-ran out of memory' it does not mean that you are on the wrong track, all it means is that you need a bigger and/or faster computer.'

-- Shalosh B. Ekhad

`Uri Zwick, at Tel Aviv Univ, proved a 4n lower bound, and I am almost sure he is using the full basis. I really don't care too much if it is 3 or 4...'

--Avi Wigderson, E-mail message to D. Zeilberger dated Thu Apr 11 09:29:47 1996.

[By the way Zwick only used the AND-type functions basis, and the best known lower bound for the Boolean Circit Complexity, over the full basis, due to N. Blum (extending ideas of Paul's 1977 2.5n bound) is still 3n, and was not improved since 1984]

`When other computer scientists would be satisfied to say that a certain algorithm takes time proportional to the square of its input, Knuth would prove that it takes exactly 3.65 times the square of the input'

--D. Sasha and C. Lazare [`Out of their minds', p. 96].

``The axioms of set theory are inconsistent, but the proof of inconsistency is too long for our physical universe''

--Pierre Cartier (quoted by Ruelle, `Chance and Chaos'.)

``Counting pairs is the oldest trick in combinatorics... Every time we count pairs, we learn something from it''

-- Gil Kalai (``Combinatorics and Convexity'', lecture)

``Physicists are more like avant-garde composers, willing to bend traditional rules ... Mathematicians are more like classical composers... ''

-- Brian Greene (`The Elegant Universe', p. 271)

``When I give this talk to a physics audience, I remove the quotes from my "Theorem". ''

-- Brian Greene, invited talk at Joint Math Soc. meeting, Washington, DC, Jan. 19, 2000.

``Mathematicians tend to prefer a worst-case analysis, a kind of paranoia that is especially understandable if you live in Israel!''

-- Noga Alon ( quoted in SIAM News, 31(9)[Nov. 1998], p. 8)

``This isn't an offer for the Tome. This is a gift for Ethan deigning to even discuss trading the Tome with me.''

-- Richard Garfield (Magic:The Gathering, PPG, first edition, p.15)

``I used to be surprised and even depressed when I met someone I had admired and discovered him or her to be a jerk or, perhaps worse, rather common. Now I half expect this reaction and even find it a little reassuring, maybe even uplifting. Isn't it amazing that somebody like that could produce such and such''

-- John Allen Paulos (A Mathematician Reads the Newspaper, p.111)

`` The only bit of logic-based public bathroom humor I know is: the difference between men and women is that between the statement [P and not Q] and the statement [Q and not P]. ''

-- John Allen Paulos (Once Upon A Number, Fall 1998)

``We ourselves are co-called non-linear dynamical systems... I don't feel quite so pathetic when I interrupt a project to check on some obscure web site or newsgroup or derive an iota of cheer by getting rid of pocketful of change.'' ''

-- John Allen Paulos (Once Upon A Number, Fall 1998)

``We might be the holographic image of a two-dimensional structre''

-Brian Greene, invited talk at Math meeting, 1/19/2000

``Algebra With Personality=Combinatorics''

-- Dave Bayer (Lecture MSRI, 10/14/98)

``Combinatorics without Algebra and Topology is like Sex without Love''

-- Anthony Joseph

Omar Foda's corollary of Joseph's maxim: ``Algebra without Combinatorics is like Love without Sex''.

``Mathematicians are like lovers...; Grant a mathematician the least principle, he will draw you a consequence that you must grant him as well, and from that consequence yet another...; and in spite of yourself, he will carry you to out-of-sight places that you would never believe existed. These two kinds of people, the mathematicians and lovers, always take more than what one has given them.''

--- Fontenelle, quoted by Lucas, cover of Recreations Mathematiques IV (new printing: A. Blanchard, Paris, 1979)

``There is something in statistics that makes it very similar to astrology''

-- Gian Carlo Rota

``There is something inhuman and vaguely pornographic about statistics.... Pornography, on the other hand, with its loosely bound sequences of storyless sexual couplings (or triplings) often has the feel of a statistical survery.'' ''

-- John Allen Paulos (Once Upon A Number, Fall 1998)

``ha-ekhad sofer atsmo ve-ein akher sofro, ve-hu kol mispar, hu shoresh ve-yesod u-meruba u-meukav, ve-hu dome le-etsem ha-davar ha-nose kol hamikrim, ve-kol mispar be-kokho, ve-hu be-kol mispar be-ma'ase, ve-hu ha-hoveh, ve-kol mispar hoveh be-avuru, ve-hu kadmon, ve-kol mispar mitkhadesh, ve-hu sibat kol mispar, zug ve-she-eino zug, hu eino mispar, ve-lo yarbe ve-lo yekhalek''.

-- Abraham Ibn Ezra, Sefer HaEkhad. translation

``..ve-yadanu ki ha-mishgal nekhlak le-gimel khalakim ha-ekhad le-peria ve-revia be-lo taava ve-hasheni le-hakel me-leut ha-guf ve-ha-shlishi le-taaava ha-nimshelet le-taavat ha-behemah''

--Abraham Ibn Ezra, Commentary to Va-Yikra (Leviticus) 18:20. translation

``A lot of mathematicians are a little bit strange in one way or another. It goes with creativity''.

-- Peter L. Duren (NYT,5/26/96 , p.23)

There would be this algebraic equation with an equals sign in the middle, and all the components would have different letters of the alphabet. It would come out right with x+z^2+t/q=y+co, and the co would be clothes off!

--Tom Stoppard, The Pennsylvania Gazette, April 1996, p.25

On part d'un detail quelconque, parfois mesquin, et on en arrive a decouvrir sans le vouloir de grands principes.

--Georges Simenon, L'homme qui regardait passer les trains, p. 135

Il ne s'agit toutefois pas d'une revolution mathematique... mais bien d'une revolte. Revolte contre les mathematiques structurelles et la methode axiomatique (bourbakisme).

---Michel Mendes France, `Roger Apery et l'irrationel', La Recherche, No. 97, Fevrier 1979, 170-172.

One good thing about teaching calculus is that you develop a hardened attitude towards repeating yourself.

--Phil Hanlon

Are we going to do any thinking today, or is it going to be all math?

--Thelma M. (quoted by Phil Hanlon)

The worst thing you can do to a problem is solve it completely

--Daniel Kleitman (quoted by W. Tom Trotter)

This was NOT a proper math talk. Many in the audience could almost understand it

--[Univ. of Stockholm official to Kimmo Eriksson]

The great advances in science usually result from new tools rather than from new doctrines.

--Freeman Dyson (AMM 103(1996), p. 805).

I am firmly on the side of nature. But nature, I suspect, is on the side of the machines.

--George Dyson (Darwin among the machines (1997), p. ix).

"Mathematics is a collection of cheap tricks and dirty jokes."

--Lipman Bers (quoted by Boris Datskovsky).

"There are two kind of mathematicians, smart ones, and dumb ones. I am one of the dumb ones."

--Lipman Bers (quoted by Raymond O Wells).

"The total amount of information that humanity can claim to know currently doubles every five years; by the year 2020, when today's elementary schoolchildren are in their 20s and 30s, it will double Every 17 days."

--Mary Flynn, publisher of Creative Publications, Mountain View, CA, in an opinion piece in "American Teacher", February, 1997. (brought to my attention by Richard Askey)

"To sum up: I am the man who when the concern pressed him and his way was straitened and he could find no other device by which to teach a demonstrable truth other than by giving satisfaction to a single virtuous man while displeasing ten thousand ignoramuses--I am he who prefers to address that single man by himself, and I do not heed the blame of those many creatures."

-Moses Maimonedes, "Guide for the Perplexed." [contributed by Ilan Vardi]

``The paper obviously addresses the physics community and also uses its style. The author does not care if the things he manipulates are defined or make sense. This by itself can be accepted. What is more serious in the reviewer's eyes is the fact that he does not pay attention to the existing mathematical literature on the same circle of problems...''

--Dieter H. Mayer, Mathematical Reviews, review of E.B. Bogolmony, "Semiclassical quantization of multidimensional system,." Nonlinearity #5 (1992), 805-866. (Contributed by Ilan Vardi)

`THE COMPUTER IS JUST AN INSTRUMENT for doing faster what we already know how to do slower. All pretensions to computer intelligence and paradise-tomorrow promises should be toned down before the public turns away in disgust. And if that should happen, our civilization might not survive.

--Gian Carlo Rota, Discrete Thoughts, p. 263

`THE HUMAN IS JUST A CREATURE for doing slower (and unreliably) (a small part of) what we already know (or soon will know) to do faster. All pretensions of human superiority should be withdrawn if humans want to survive in the future.

--Shalosh B. Ekhad

It is a common public relations gimmick to give the entire credit for the solution of famous problems to the one mathematician who is responsible for the last step.

It would probably be counterproductive to let it be known that behind every "genius" there lurks a beehive of research mathematicians who gradually built up to the "final" step in seemingly pointless research papers. And it would be fatal to let it be known that the showcase problems of mathematics are of little or nonterest for the progress of mathematics. We all know that they are dead ends, curiosities, good only as confirmation of the effectiveness of theory. What mathematicians privately celebrate when one of their showcase problems is solved is Polya's adage : "no problem is ever solved directly".

--Gian-Carlo Rota (Foreword to `Species' by Bergeron, Labelle and Leroux)

``Is my understanding only blidness to my own lack of understanding? It often seems so to me.''

--Ludwig Wittgenstein, (On Certainty, #418)

``(I Once wrote: ``In mathematics process and result are equivalent.'')''

--Ludwig Wittgenstein, (`Remarks on the Foundations of Mathematics', MIT press, p. 68)

``Proof, one might say, does not merely shew that it is like this, but: how it is like this. It shows how 13+14 yield 27.

--Ludwig Wittgenstein, (ibid, p. 159)

``Suppose someone follows the series ``1,3,5,7, ..'', and in writing the series 2x+1; and he asked himself ``But am I always doing the same thing, or something different every time?'

If from one day to the next someone promises: ``Tomorrow I will give up smoking', does he say the same thing every day, or every day something differet?''

---Ludwig Wittgenstein, (ibid, p. 415)

``Wovon man nich sprechen kann, darueber muss man schweigen''

---Ludwig Wittgenstein, (Logisch-Philosophische Abhandlung 7)

``Combinatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum and ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late 18th century.''

--E. T. Bell, The Development of Mathematics', p. 290 (quoted by Volker Strehl, in his invited lecture at CRM, Montreal, May 1997)

``The simple equations that generate the convoluted Mandelbrot fractal have been called the wittiest remarks ever made''

--John Allen Paulos, Once Upon a Number, p. 130-131

``The once-surprising existence of non-Euclidean models of Euclid's first four axioms can be seen as a sort of mathematical joke.''

--John Allen Paulos, Once upon a number, p. 132.

``So in the end it wasn't G\:odel, it wasn't Turing, and it wasn't my results that are making mathematics go into an experimental mathematics direction, in a quasi-emirical direction. The reason why mathematicians are changing their working habits is the computer. I think that this is an excellent joke!''

--Gregory J. Chaitin, `Randomness in arithmetic and the decline and fall of reductionism', in: `Nature's Imagination', J. Cornwell, ed., Oxford, 1995. (p. 44)

``But if we play it safe, the problem is that we may be losing out, and I believe we are.''

--Gregory J. Chaitin, ibid (p. 42)

"I think logicians hate my work, they detest it! And I'm like pornography, I'm sort of an unmentionable subject in the world of logic, because my results are so disgusting!"

--Gregory J. Chaitin, "Exploring Randomness", Springer, p. 26

``We now show that the proposition [R(q);q] is undecidable in PM. For supposing the proposition [R(q);q] were provable, it would also be correct; but that means, as has been said, that q would belong to K, i.e. \neg Bew[R(q);q] would also hold good, in contradiction to our initial assumption. If, in the contrary,the negation of [R(q);q] were provable, then \neg {q \in K}, i.e. Bew[R(q);q] would hold good. [R(q);q] would then be provable at the same time as its negation, which again is impossible.''

--Kurt G\:odel, Monat. f. Math. u. Phy. v.38(1931), 173-198, p.175.

``All sentences of the type `deconstruction is X' or `deconstruction is not X', a priori miss the point, which is to say that they are at least false. As you know, one of the principal things at stake in what is called in my texts `deconstruction', is precisely the delimiting of ontology and above all of the third-person present indicative: S is P.''

--Jacques Derrida, `Letter to a Japanese Friend', in: D. Wood (ed.), Derrida and Diff\'erance, 1-8.

``It depends on what the meaning of the word `is' means. If `is' means `is and never has been', that's one thing. If it means, `there is none', that was a completely true statement.''

--Bill Clinton

``Mathematics is much less formally complete and precise than computer programs.''

-- William P. Thurston, BAMS (2)30(199), 161-177, p. 170.

``Probability of human error is considerably higher than that of machine error''.

--Ken Appel and Wolfgang Haken, `The four-color problem', in Mathematics Today, L. Steen (ed.), Springer, 1978, 151-180, p. 179.

``Building intellignet machines can teach us about our minds - about who we are - and those lessons will make our world a better place. To win that knowledge, though, our species will have to trade in another piece of its vanity.''

--Astro Teller, NYT, c. 3/20/98.

``As for explaining mathematical phenomena it opens the question: explaining to whom? humans?, other computers? ''

--Gil Kalai, "Combinatorics with a Geometric Flavor: Some Examples", to appear

``With randomness it is very unlikely to be embarassed, but even if you get embarassed, you can't replicate it''

-- Carl Pomerance [Plenary Lecture, Aug. 1, 1997, Topics in Number Theory, Penn State.]

``Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems.''

-- Carl Pomerance [ MAA invited talk, 1/21/2000]

``logloglog n has been proved to go to infinity, but has never been observed to do so''

--Anon., quoted by Carl Pomerance [ MAA invited talk, 1/21/2000]

``The very term `combinatorial methods' has an oxymoronic character''

--Joel Spencer, Handbook of Combinatorics, p. 1807.

``Any time your are stuck on a problem, introduce more notation''

-- Chris Skinner, [Plenary Lecture Aug. 3, 1997, Topics in Number Theory, Penn State.]

``What cannot be known is more revealing than what can.''

-- John D. Barrow [Impossibility, (p. 252), Oxford Univ. Press, 1998]

Why is it that Serge Lange'sLinear Algebra, published by no less a Verlag than Springer, ostentatiously displays the sale of a few thousand copies over a period of fifteen years, while the same title by Seymour Lipschutz in theThe Schaum's Outlineswill be considered a failure unless it brings in a steady annual income from the sale of a few hundred thousand copies in twenty-six languages?

-- Gian-Carlo Rota [``Indiscrete Thoughts'', Birkhauser, p. 238.]

Stan Ulam was lazy, .... He talked too much .... He was self-centered ... . He had an overpowering personality...

--- Gian-Carlo Rota [``Indiscrete Thoughts'',Birkhauser, p. 85]

Rota's personality is compatible with mine

---Stan Ulam [``Adventures of a Mathematician'', University of California Press, p. 264]

``One day, when I was doing well in class and had finished my lessons, I was sitting there trying to analyze the game of tic-tac-toe... The teacher came along and snatched my papers on which I had been doodling... She did not realize that analyzing tic-tac-toe can lead into dozens of non-trivial mathematical questions.''

-- Martin Gardner [Math. Intell. 19(4), (Fall 1997), p. 40]

``...You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS math. ''

-- John H. Conway, Public Lecture, Princeton, Oct. 27, 1999.

``Most popular mathematics puzzles and games, such as Rubik's cube and jigsaw puzzles, are essentially problems in combinatorics''

--- Anders Bjorner and Richard Stanley, `A combinatorial Miscellany', preprint, p.2

``I am not in the business of making money, I am a professor of mathematics''

George Papanicolau, invited SIAM-AMS talk, Washington, DC, 1/21/2000.

``My occupation is an open question. I was once an assistant professor of mathematics. Since then, I have spent time living in the woods of Montana.''

--Theodore J. Kaczynski, New York Times Jan. 23, 1998, p. A18

Why is Zeilberger so willing to give up absolute truth? The most reasonable answer is that he is pursuing deeper truths.

-- J. Borwein, P. Borwein, R. Girgensohn and S. Parnes, Math. Intell. 18(4)(Fall 1996) p.15.

"If you don't like your analyst, see your local algebraist!".

--- Gert Almkvist, founder and director of The Institute for Algebraic Meditation

``Are you an Analyst?, I am an Oralist''

-- George Bergmann

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