Last Update: April 21, 2015

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

``The circle is solved."

--Emmanuel Xagorarakis, The Groundlessness of Infinity

``In 2005, I was sitting at a lunch table in the Institute [for Advanced Study], and someone mentioned a certain product (the Chas-Sullivan product), and I got really excited, and I exclaimedI know that product!, where is the other product?''

``I have checked it so many times, but I still don't understand it!''.

---Nancy Hingston, Invited Hour Address, American Mathematical Society's Spring Eastern Sectional Metting, Washington, D.C., March 8, 2015

``Only Numbers. Pure math. You have to accustom yourself to thinking that way"

---Anthoby Doerr, ("All the light we cannot see", Scribner, 2014, p. 388)

"School reformers must turn their attention from structural changes, such as curricular alteration and modification in the physical environment, and address themselves to the nettlesome issue of teacher quality. Reform efforts must break the traditional patterns of limited professional stature, low wage structures, and inadequate professional commitment."

---Richard B. Fishbane (Pennsylvania Magazine of History and Biography(1979):66-84, available here)

"Real numbers are good if you add the word `random'"

---Peter Sarnak, (last sentence of Rutgers math colloquium, Dec. 5, 2014)

"Without mathematical thinking it would be hard to function in the modern world. (Just to remove any possible doubt, the above is written in irony.)" ...

"The truth is,there is no such thing as mathematical thinking."

---Zvi Artstein, Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics , pp. 383, and 384 (resp.)

"It seems to me there's this grand mathematical world out there, and I am wandering through it and discovering fascinating phenomena that often totally suprise me. I do not think of mathemaatics as invented but rather discovered".

---George E. Andrews, quoted by the staff of Powell's bookstore.

"There is nosignificantdifference between human activities and those by amoebas and even bacteria, well, on the GRAND SCALE" (p.2)

"A mathematician would hardly call a correspondence between the set of 64 triples of four units and a set of twenty other units, "universal", while such correspondence is, probably, the most fundamental general feature of life on Earth" (p.8)

"Understanding what and why did not work may be more instructive than celebrating our successes." (p. 28)

"But anything that can be called "rigor" is lost exactly where the things become interesting and non trivial". (p. 31)

"But `the physical level of rigor' is higher on certainty than the logical one, since reproducible experiments are more reliable than anybody's, be it Hilbert's, Einstein's or Gödel's intuition"

---Misha Gromov, Ergostructures, Ergologic, and the Universal Learning Problem, Chapters 1,2,3.

"Reason is a powerful-but nevertheless limited-tool."

---Noson S. Yanofsky, "The Outer Limits of Reason", p. 352.

"André Weil once told to me (in private) that rigor is like a parking lot in Chicago, and axiomatics is the big light at the very center. As long as you stand below the light, you have a small chance of getting robbed, but as you venture to the dark edges, it becomes less and less safe, but much more interesting."

---Pierre Cartier in: minutes 4:00-4:55 of this excellent lecture delivered, Nov. 14, 2013, at the Rutgers Mathematical Physics seminar.

"Thers is this wonderful iconoclast at Rutgers, Doron Zeilberger, who says that our mathematics is the result of a random walk, by which he means whatWEcall mathematics. Likewise, I think, for the sciences."

---Ian Hacking, in: Ian Hacking interviewed by Andrew Lakoff (requires (private or institutional) subscription),Public Culture, v. 24(2012)(1), 217-232, p. 227.

"I have Doron Zeilberger for my number theory class this semester! .... He hates proofs, and thinks that if some conjecture is true for the first 1000 cases that's proof enough"

"...he is not a complete nut ..."

"...That's pretty much verbatim what he's told our class. If he's more subtle he hasn't let on..."

---Reddit User DJBJ, in this thread

"In highschool I was very excited that

a^{log(b)}=b^{log(a)},

and still find it useful today."

---Noga Alon, Rutgers Theory of Computing Seminar, Oct. 9, 2013

"I would rather be right than rigorous"[Contributed by Drew Sills]

---Stephen Hawking to Kip Thorne,as quoted by Kitty Ferguson, *Stephen Hawking: An Unfettered Mind*, Palgrave Macmillan, 2012, p. 96.

"This is not very important what I'm doing. I'm just proving something"[Contributed by Drew Sills]

---Richard Feynman, *The Feynman Lectures on Physics*, Sophomore year Lecture 2: Differential Calculus of Vector Fields, October 1, 1962, 7:55, into section 3.

[Incidentally, this remark was edited out (censored?!) from the printed Lectures, so you willnotfind it on page 2-5 of volume II, where it should have appeared!]

"...In the audience was a "goblin" with gleaming brilliant eyes: Michael Kiessling, he told me with contagious enthusisam his "amours de jeunesse" (young loves) for plasma physics, ...""... -What, Vladimir Scheffer works here!!

-Yes, he has been here for ages. Why, Cédric, you know his work?

-But of course, I gave a Bourbaki seminar on his famous theorem about the existence of paradoxical solutions of Euler's equation ... I must meet him!"[...One page later] "After two or three general phrases, he stopped abruptly, saying that he must go home, since he uses public transportation, and what with the snow it is very slippery ... and `my journey is rather long, and' .... The end of the meeting [with Scheffer] passed by him stating all the good reasons why he must take his leave. The mathematical discussion lasted five minutes, during which I learned nothing. To say that this is the man behind the most surprising theorem in the whole theory of fluid mechanics! This is a living proof that that one can be a superior mind yet a poor communicator.

When I returned to Joel [Lebowitz], I told him about my meeting [with Scheffer] and regretted that it only lasted five minutes.

---You know, Cédric, [replied Joel] five minutes with Vlad is the absolute longest that any one can talk with him during the last five years."---Cédric Villani, (free translation from French of excerpts from pp. 95-97 in: "Théorème vivant", Grasset 2012)

"Tout mathématicien digne de ce nom a ressenti, même si ce n'est que quelquefois, l'état d'exaltation lucide dans lequel une pensée succède à une autre comme par miracle... Contrairement au plaisir sexuel, ce sentiment peut durer pendant plusieurs heures, voire plusieurs jours."

---André Weil (quoted by Cédric Villani, p. 135 in: "Théorème vivant", Grasset 2012)

"Give math a chance

before you condemn it-

you may be surprised.

I liken it to poetry-

keep reading it and

eventually you find

something you like."

--- Tom ? to his nephew Steven (or Stephen) ? [found in the dedication to a used copy of "The most beautiful mathematical formulas" (by L. Salem, F. Testard, and C. Salem) bought at the Princeton Public Library used bookstore, May 14, 2013. (signed: "Your Uncle and Friend, Tom")]

"The computer was, to the best of my feelings about the subject, not thinking like a mathematician, and it was much more successful, because it was thinking not like a mathematician"

"Without computers we will be stuck only proving theorems that have short proofs".

---Kenneth I. Appel(1932-2013), quoted in the New York Times, April 29, 2013, p. A19.

"Data-driven statistics has the danger of isolating statistics from the rest of the scientific and mathematical communities by not allowing valuable cross-pollination of ideas from other fields."

"I feel that one should employ methods that reflect the physics of the problem at hand rather than the methods one happens to know."

"Of course it was always a friendly debate though sometimes I got too loud."

---Larry Shepp(1936-2013), Statistical thinking: From Tukey to Vardi and beyond,

"It is a bit challenging to talk about cluster algebras. It is still a young theory, depending on how you define the beginning of life. It was conceived nine years ago, appeared in preprint form eight years ago, and the first official publication was seven years ago. It is still young, but it grew faster than our wildest expectations."

"...It is a very simple dynamical system, ..., and what was surprising is that it appears in so many places, including mirror symmetry in string theory."

"Q. (from audience): Can you elaborate on the connection [of cluster algebras] to mirror symmetry?

A. (from speaker AZ): No [laugh]. I hope to learn it one day, but it is still mysterious to me."

---Andrei Zelevinsky(1953-2013), "Cluster Algebras via Quivers with Potentials", Worldwide Center of Mathematics Lecture Series, Sept. 26, 2011

"There are two types of types (Church and Curry) coming from two different communities (computation and logic). One can be dogmatic, and say that only one of them is the right one, but they may be unified".

"Nicolaas de Bruijn (of proof assistants fame) was apparently influenced by early discussions with Don Knuth (of computation fame)."

"This is usually called `Refinement-Type', but to be rebellious I will call it `Type Refinement'."

--- Noam Zeilberger, Substructural Type Theory, "Univalent Foundations" seminar, Institute for Advanced Study, March 22, 2013

"Pi is NOT a number, but rather something much more exciting, a game!"

--- Joey Reichert, during Dr. Z.'s Experimental Math Class 3.14 of 2013

"If you have never heard of Betti numbers, then this is not the right time to teach you what they are"

--- Gerard Ben-Arous, Invited AMS talk, San Diego, Jan. 9, 2013 (10:05-10:55)

"[Don] Zagier speaks English, German, and French faster than any one I know"

--- Richard Guy, MAA invited paper session, San Diego, Jan. 9, 2013 (10:15-10:45)

"Cedric Villani illustrated to me Optimal Transport theory, by showing that it is not possible to go from 'ALICE GUIONNET' to 'OPTIMAL GIRL'"

--- Alice Guionnet, First AMS Colloquium talk, San Diego, Jan. 9, 2013 (13:00-14:00)

"I will end this talk with a riddle. What is common between this beautiful rug and this nice car with a boat on its roof"?

--- Alice Guionnet, Second AMS Colloquium talk, San Diego, Jan. 10, 2013 (13:00-14:00)

"Adding and subtracting one is one of the most important tricks in mathematics"

--- Alice Guionnet, Third AMS Colloquium talk, San Diego, Jan. 11, 2013 (13:00-14:00)

"Here the answer is more important than the question"

"Copernicus was not the first to propose the Heliocentric system, but he was at the right place in the right time"

--- Efim Zelmanov, AMS-SIAM Special Session, San Diego, Jan. 9, 2013 (15:15-16:05)

"On Sept. 5, 2006, it was exactly one hundred years after the tragic death of my great hero, Ludwig Boltzmann, so I decided to go on a pilgrimage to the Vienna cemetery where he is buried. I got lost, so I asked some strangers, "is there a map somewhere?". Then they asked me, "Whose grave are you looking for?". So I said "Boltzmann", and then they replied "Ah, S=klog(W)", it is that way". I realized that S=klog(W) is like a secret password.

--- Cedric Villani, AMS Josiah Willard Gibbs Lecture, Jan. 9, 2013 (20:30-21:30)

"There are also numbers that are not prime" --- Jean Bourgain, Invited AMS talk, San Diego, Jan. 10, 2013 (15:20-16:10)

"According to Jon Borwein, high precision computations is the electron microscope of math"

"I once told Helaman Ferguson, the inventor of the PSLQ algorithm, that very soon he would be more famous for his sculptors than for his algorithm, and he replied 'I passed that point long ago'" --- David H. Bailey, AMS-SIAM special session, San Diego, Jan. 11, 2013 (8:00-8:30)

"Defining math precisely and non-recursively is a moving target"

"Barbie had a point when she said `Math is hard', it is hard for everyone."

"The villain and the hero of math communication is the special language of math".

"The word `obvious' is yet another loaded and overused word in math"

"Don't answer questions students are incapable of asking"

--- Paul Zorn, MAA retiring address San Diego, Jan. 11, 2013 (9:00-9:50)

"André Weil always looked like that [pointing to a non-smiling stern photo], not only when he talked about the devil"

"This is a collaboration between a complex analyst, a dynamical system expert, and an arithmetical algebraic geometer (myself). It sounds like a joke, a complex analyst, a dynamical system expert, and an arithmetical algebraic geometer walk into a bar ..."

--- Jordan Ellenberg, Invited AMS talk, San Diego, Jan. 11, 2013 (10:05-10:55)

"Gröbner bases are the new Gaussian elimination, like sixty is the new forty. Very soon every student will learn Gröbner bases instead of Gaussian elimination"

"To my embarrassment, my most cited paper only asked questions, it did not give any answers"

"When I was a teenager I was fascinated by the early video game RATMAZ. Later I found out that it is based on Broder's algorithm to generate a random spanning tree on a lattice"

--- Robin Pemantle, Invited Joint MAA-AMS talk, San Diego, Jan. 11, 2013 (11:10-12:00)

"Basil Gordon told Krishna Alladi that whatever works for 5 should work for 8"

--- Alexander Berkovich, AMS Special Session, San Diego, Jan. 11, 2013 (14:30-15:50)

"Please pardon my foul language: 'Locally Compact Groups'"

--- Persi Diaconis, AMS Special Session, San Diego, Jan. 11, 2013 (17:00-17:20)

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding"

---William Paul Thurston (1946-2012), p. 76 in "Mathematicians: an outer view of the inner world", photographs by Mariana Cook, Princeton University Press.

"These disturbing phenomena [Extra Sensory Perception] seem to deny all our scientific ideas. How we should like to discredit them! Unfortunately the statistical evidence, at least for telepathy, is overwhelming."

---Alan Turing, p. 453 in "Computing Machinery and Intelligence", Mind v. 59 (1950), No. 236, 433-460.

"In my view, using technology too soon is definitely detrimental to education. I have often used the analogy 'it's like wine-tasting for first-graders'. One can be both a strong advocate offirst-gradersandwine-tasting, but strongly opposed towine-tasting for first-graders".

---George Andrews, 2:07-2:30 of this video (interview before the Amer. Math. Soc. 2012 Presidential Retiring Address)

"However the machine would permit us to test the hypothesis for any special value of n. We could carry out such tests for a sequence of consecutive values n=2,3,.. up to, say, n=100. If the result of at least one test were negative, the hypothesis would prove to be false; otherwise our confidence in the hypothesis would increase, and we should feel encouraged to attempt establishing the hypothesis (by means of a normal mathematical proof), instead of trying to construct a counterexample."

---Alfred Tarski (1948) [Contributed by David John Wilson]

Remark by DZ: There are many cases where checking it for one hundred values (or one thousand, or whatever, it depends on the problem, and sometimes ONE case suffices) constitutes a perfectly rigorous proof. If the finite number of cases that needs to be checked is too large, then checking it for a random large number of values would be what I call a

"Don't Fear LARGE Constants!"

---Daniel Spielman, Computer Science Distinguished Talk, Rutgers University, Dec. 6, 2011.

"According to the Platonic philosophy, Reality is a perturbation of an ideal system, just like in Singularity Perturbed Control Systems"

"Young measures are not Juvenile measures, but named after L.C. Young."

"Recently I gave a talk to a general audience, using postage stamps to illustrate science concepts and scientific personalities, and they really appreciated it. I then gave the same talk to high school students, and they were not as appreciative, since they have never heard of postage stamps".

---Zvi Artstein, invited talk at Sontagfest, DIMACS, Rutgers Univ., May 23, 2011.

"When I was in graduate school in Princeton (during the early sixties), I was told to take three courses. One of them to work on really hard, another to work on moderately hard, and the third one just to absorb. In my case, I never showed up to the latter class, taught by Robert Gunning, on Several Complex Variables. Several Complex Variables (C^{n}) was starting to get vary fashionable then, but I decided to specialize in n=1/2"

---Richard Askey, talk entitled "TBA" (sic!), March 15, 2011, Conference in honor of Mourad Ismail and Dennis Stanton (organized by Drew Sills), Georgia Southern University

"Computer assisted proofs are getting better and better and computers will play a bigger and bigger role in the future"

--- Enrico Bombieri, public lecture, Institute for Advanced Study, Oct. 29, 2010.

"Bombieri's Law: of Finance: Profits are on paper, losses are in cash"

--- Enrico Bombieri's father, quoted in Enrico Bombieri's public lecture, Institute for Advanced Study, Oct. 29, 2010.

"The usual conversational example of the second proposition [that the explanatory gap Brain->Mind cannot be understood because our minds are not up to the task] is to compare our inability to understand the mind-body problem to a monkey who cannot do calculus or quantum mechanics."

---Jacob I. Sage, MD Mind, Brain and Consciousness, p. 166 .

"A theorem is not a child.

No a theorem is not a child."

--- Marion Deutsche Cohen, in: "Crossing the Equal Sign", Plain View Press, 2006, p. 105.

"Mais il n'y a même pas besoin de cours, l'algèbre ce n'est pas la natation ni même l'anglais, cela s'apprend aussi bien dans un livre "

--Marcel Proust, La Prisonnière, p. 152

"Comme un géomètre qui dépouillant les choses des leur qualités sensibles ne voit que leur substratum linéaire, ..."

--Marcel Proust, Le Temps Retrouvé, p. 24

"Mathematics is the summit of human thinking. It has all the creativity and imagination that you can find in all kinds of art, but unlike art-charlatans and all kinds of quacks will not succeed there."

--- Meir Shalev, Yediot Aharonot, July 30, 2010 (p. 23 of musaf shabbat)

"Math is dirty, if it is done right"

---Günter Ziegler, Darf Ich Zahlen, Piper, 2010.

"No two thingsthe same, the equals sign a scandal."

---John Banville, "The infinities", Alfred Knopf, 2010, (p. 195)

"I now have easy proofs for the following identities

1+&Sigma^{∞}_{0}(-1)^{n}q^{n(n-1)/2}/((1-q)(1-q^{2}) ...(1-q^{n})=1

1+&Sigma^{∞}_{0}(-1)^{n}q^{n2}/((1-q)(1-q^{2}) ...(1-q^{(2n+1)})=2

1+&Sigma^{∞}_{0}q^{n(n-1)/2}/((1+q)(1+q^{2}) ...(1+q^{n})=3

These are as easy as "one, two, three". I had to stick a "+1" on the left sides, since otherwise I would have had to say "as easy as zero, one, two", and no one would have known what I meant."

---George E. Andrews, invited talk, Tom Osler Fest (April 17, 2010)

"We will evaluate these integrals rigorously if we can, and non-rigorously if we must".

---Victor Moll, invited talk, Tom Osler Fest (April 17, 2010)

"No Victor, you got it backwards, you should evaluate these integrals non-rigorously if you can, and rigorously if you must".

---Doron Zeilberger (comment made at the end of Victor Moll's talk). Tom Osler Fest (April 17, 2010)

"You can keep counting forever. The answer is infinity. But, quite frankly, I don't think I ever liked it. I always found something repulsive about it.

I prefer finite mathematics much more than infinite mathematics. I think that it is much more natural, much more appealing and the theory is much more beautiful. It is very concrete. It is something that you can touch and something you can feel and something to relate to.

Infinity mathematics, to me, is something that is meaningless, because it is abstract nonsense."

----Doron Zeilberger, "To infinity and Beyond", BBC2 Horizon, 9:00pm GMT Feb. 10, 2010.

(quoted here)

"What Idon'tlike about measure theory is that you have to say "almost everywhere" almost everywhere.

-----Kurt Friedrichs (quoted by Brian White during an invited talk at the 2010 AMS meeting [San Francisco]).

"Someone has remarked that `An ideal math talk should have one proof and one joke and they should not be the same'"

"AB=(1/4)((A+B)^{2}-(A-B)^{2}) is anamazing identity, and unfortunately I have to remind mycurrentstudents (in an undergrad class that I am teaching) how to prove it."

"Someone has said that all the great jugglers are dead (that means that I am not a great juggler)"

"I was reminded of the Sydney Harris cartoon that said 'adding two numbers that have not been added before does not constitute a mathematical breakthrough' "

--Ron Graham, Key-note talk, DIMACS 20th Birthday conference, Nov. 20, 2009

"We proved a negative result, but not as negative as one could have hoped for. But let's be optimistic, perhaps our negative-but-not- too-negative statement hints at possible cracks in the negativity arguments, that may lead to something positive."

--Russel Impagliazzo, DIMACS Theoretical Computer Science seminar, March 11, 2009. (note: this is a slight paraphrase, I don't remember the exact wording).

"The standard model gives us an accuracy of ten decimal digits, this is an amazing success that has never been achieved before in science."

"Newton's theory is not `not right', it just does not cover all distances. Contrary to popular belief, theories in science are not proven wrong, they are just replaced by more complete and convenient theories. To sound provocative, even the geocentric theory was never "proven" wrong, it is just not as convenient as the heliocentric theory, since it requires endless epicycles."

"What will the [greatest] experiment discover? I have no idea!, maybe nothing!, If we won't check, we will never know. In science there is never a guarantee of success. If you understood this last sentence, then you understood the most important message of this lecture."

--Nati Seiberg, The World's Greatest Experiment (public lecture [in Hebrew]), Hebrew University of Jerusalem, Jan 4, 2009.

"There are two schools in Statistics. The first school says that it is a reasonable question to ask `what does a typical matrix, with non-negative entries and given row- and column- sums, look like?'. The other school says, `no, it is not!', and the first school replies, `yes, it is!'"

--Alexander Barvinok, Invited talk, 4th ICC conference, Auckland, New Zealand, Dec. 19, 2008.

Christos Papadimitriou once said that the theory of NP-hardness is the largest intellectual export of CS to Math and Science. And indeed when I checked google scholar, I found 2000 titles mentioning "NP" inphysics and chemistry alone. This means that science has adopted this notion and use it to demonstrate structural nastiness.

The fact that a problem is NP-complete does not make it go away.

The Theory of complexity has given new insight to basic concepts that engaged philosophers for millenia: Learning, Proof, Randomness, Knowledge, and Secret.

--Avi Wigderson, Rutgers Univ. Computer Science Distinguished Lecture, Dec. 4, 2008.

"A computer would to whatever it is programmed to do"

--Andrew Appel (CNN show)

"The difference between physicists and mathematicians is that physicists study things that exist, while mathematicians often try torule outthings that obviously do not exist." [This was accompanied by a cartoon of a dinosaur (as a metaphor for the research interests of physicists), and a unicorn (that mathematicians try very hard to prove does not exist).]

--Andrei Okounkov, lecture in honor of Israel Gelfand's 95^{th}birthday , Rutgers University, Sept. 19, 2008.

"I am a professor at the computer science department, but I don't know how to use a computer, not even for Email."

"You must all know about Bourgain, so I don't have to write his name on the board-for an obvious reason [not being sure how to spell it]."

"Too much knowledge could be a bad thing. I was lead to the Szemerédi theorem by proving a result, about squares, that Euler had already proven, and I relied on an "obvious" fact, about arithmetical progressions, that was unproved at the time. But that lead me to try and prove that formerly unproved statement- about arithmetical progressions-and that ultimately lead to the Szemerédi Theorem."

----Endre Szemerédi, Rutgers Univ. Mathematics Colloquium, February 29, 2008.

" `You might think of combinatorics as a machine too', the major [MacMahon] says. 'A different sort of machine, though. Have you heard of Babbage's analytic engine? He never built it. .... I have an analytic machine of my own-right here.' He taps his own skull."

in: David Leavitt'sThe Indian Clerk, p.289.

"You are not supposed to get it. It's a paradox. All of mathematics is built on paradoxes. That's the biggest paradox of all-all this orderliness, and at the heart, impossibility. Contradiction. Heaven built on the foundations of hell."

in: David Leavitt'sThe Indian Clerk, p.366.

"It is cheaper [and better for the world] to pay mathematicians and computer scientists to design algorithms that will eliminate webspamming, rather than to pay lawyers to do lawsuits."

"I described to my patent lawyer our new algorithm-that I was hoping to patent- about detecting clustering, that involved three probabilities α , β, γ that add-up to 1, and mentioned that it is like "a three-sided coin". A few days later he came up with a patent application for a "three-sided-coin"."

"I don't know whether guys are more promiscuous or just bigger liars."

"Always try to innovate. If you lose your old fitness, you lose out to Bose-Einstein condensation. I am sure that companies that go under would feel better if they knew that they were victims of Bose-Einstein condensation."

----Jennifer Tour Chayes (MAA 2007 Hedrick Lecture (I), MathFest, Aug. 3., 2007.)

"My work on prime gaps lead to lots of media coverage, some good, some bad, some ugly, and some merely ridiculous. For example, a reporter of our university newspaper, who admitted that he is still learning English, wrote that "Prof. Goldston solved one of the most controversial problems in the prime number theory last month with support from his Turkish partner."

----Dan Goldston (MAA invited talk, MathFest, Aug. 4, 2007).

"The largest known prime number is 2^{32582657}-1. I am proud to say that I memorized all its digits-in binary."

----Carl Pomerance (MAA special talk, Aug. 4, 2007).

"It would be nice if we could design a virtual reality in Hyperbolic Space, and meet each other there".

-----Donald Knuth (Π Μ &Epsilon J. S. Frame Talk, MathFest, Aug. 4, 2007).

"Theoretical Computer Science is just as useless as everything we mathematicians do."

"Combinatorialists and analysts always have different names for everything, in order to keep themselves from interacting."

"With a metric you can really go to town, otherwise it is just abstract nonsense."

----Jennifer Tour Chayes (MAA 2007 Hedrick Lecture (III), MathFest, Aug. 5., 2007.)

"When Daniel Gorenstein was chair (1975-1981), he did mathematics from 5am to 12noon, spending the second half of his working day on administration. When I was chair, I also spent half of my time on research: every other minute."

"When I told my son that I had to give a talk about my work to non-mathematicians, he warned me that regular people don't think like mathematicians."

"When I told my mother that I have to give a talk, and was debating what could I possibly say to non-mathematicians, she said: "[more important] you got what to wear?"."

"The Method of Bisection is a sophisticated version of a tool used in fifth grade called "Guess and Check". "

"I once got a call from a bank, asking me to compute a mortgage, since their computers were down. This was a very depressing moment [since the formula is very simple]."

"I don't like to end my talk with a 700 million dollar loss, even if it shows the importance of Numerical Analysis"

-----Richard Falk, 14th Daniel Gorenstein Award Lecture, Rutgers University, April 24, 2007.

"The Torus is my enemy!"

-----Peter Sarnak, Rutgers Mathematics Colloquium talk, April 6, 2007.

"Cauchy ruined mathematics. Let's throw out all that epsilon-delta nonsense!"

-----Adriano Garsia, talk at the Journees Leroux, Montreal, Sept. 8, 2006.

Purblind specialization, of which such ignorance [of the WZ method] is the inevitable issue, is the constant enemy of mathematical progress.

-----Jet J. Foncannon, Math. Intell. v. 28, no. 3 (Summer 2006), p. 68 .

"Math is just a tool. Used wisely, math can indeed help .... . But use it unwisely-as seems to be the case here-and your approval ratings might just hit a new all-time low.

-----Jonathan David Farley, New York Times, May 16, 2006, Op-Ed page.

"There is a place in mathematics for another kind of mind"

----Brian Hayes ("Gauss's Day of Reckoning", Computing Science Column in American Scientist, May-June 2006, 200-205, p. 205)

"To be is to be a value of a variable"

----W.V. Quine

What's the most difficult aspect of your life as a mathematician, Diane Maclagan, an assistant professor at Rutgers, was asked.

"Trying to prove theorems," she said.

And the most fun?

"Trying to prove theorems."

--quoted in the NY Times, June 19, 2005, Section 14 (New Jersey), p.1 (article by Fran Schumer).

"(...(...(will there be infinitely many parentheses? Is it love when there are infinitely many parentheses?)...)...)

--Aner Shalev, "Dark Matter" [Zmora-Bitan, 2004](in Hebrew), p. 115.

We worked hard, applying Permutation Group Theory, Character theory, and Classification of Finite Simple Groups, to show that a certain problem cannot be solved using a computer which doesn't exist.

--Aner Shalev, "Permutation Groups and Quantum Computing", Rutgers Univ. Colloq. talk, Feb. 24, 2006.

"Most current mathematical research, since the 60s, is devoted to fancy situations: it brings solutions which nobody understands to questions nobody asked. Nevertheless, those who bring these solutions are called distinguished by the academic community. This word by itself gives a measure of the social distance: real life mathematics do not require distinguished mathematicians. On the contrary, it requires barbarians: people willing to fight, to conquer, to build, to understand, with no predetermined idea about which tool should be used. We, at SCM, are not yet barbarians, but we are working on this. Of course, it takes some time, due to our earlier education. In academic life, people have specific ambitions, and customer satisfaction is not one of these ambitions. They want to be recognized, to obtain some notoriety, to be published and quoted, to get prizes and honors. We, at SCM, live on the service we give to our customers. And we remember what Thomas Grey wrote in 1732 (Elegy in a churchyard): The paths of glory lead but to the grave."

--Bernard Beauzamy, "Real life Mathematics", Irish Math. Soc. Bull. 48 (2002), 43-46.

"...All the wonders of our universe can in effect be captured by simple rules, yet [...] there can be no way to know all the consequences of these rules, except in effect just to watch and see how they unfold."

--Stephen Wolfram ("A New Kind of Science", last sentence (p. 846))

"The evidence peculiar to this defective way of knowing - an evidence on the strength of which mathematics plumes itself and proudly struts before philosophy - rests solely on the poverty of its purpose and the defectiveness of its material, and is on that account of a kind that philosophy must scorn to have anything to do with."

--Georg Wilhelm Friedrich Hegel ("The Phenomenology of Mind", 0.45)

``It would be fun some day, Turing, to listen to a discussion, say on the Fourth Programme, between two machines on why human beings think that they think! ''

--Sir Geoffrey Jefferson (in: `Can Automatic Calculating Machines Be Said to Think?', by M.H.A. Newman et al., a BBC broadcast recorded 10 Jan. 1952, Turing Archives, reprinted in ch. 7 of `The Turing Test', S. Shieber, ed., MIT Press, 2004.)``The result of the mathematician's creative work is demonstrative reasoning, a proof, but the proof is discovered by plausible reasoning, by GUESSING''

--George Polya, (`Mathematics and Plausible Reasoning', 1953.)``Truth is no more a true notion than reality is real''

--Jean-Yves Girard ("Locus Solum", in Mathematical Structures in Computer Science, 11(3):301-506, 2001.)"The mind is an evolved computer program."

``Thought, then, is the execution of this computer code.''

``The self is the resultant of the interest of the genes.''

``Meaning comes from the correspondence between the code and its execution, and the compact underlying structure of the world and its dynamics.''

--Eric B. Baum (`What is thought?', MIT Press, 2004 (pp. 438, 439))"We live in a world with huge repositories of logic and even greater such of information-but, alas, so little wisdom."

--Apostolos Doxiadis, ("Writing Incompleteness-the play")``Mathematics is an experimental science, and definitions do not come first, but later on.''

---Oliver Heaviside , `On operators in physical mathematics', part II, Proceedings of the Royal Society of London, Vol. 54, 1893, p. 121. [contributed by Stephen J. Greenfield.]``It is shocking that young people should be addling their brains over mere logical subtleties [in Euclid's Elements], trying to understand the proof of one obvious fact in terms of something equally .. obvious.''

---Oliver Heaviside , [contributed by Tom Robinson]"University politics make me long for the simplicity of the Middle East" --

--Henry Kissinger [contributed By Herb Wilf]``Everything that is not symmetric, can be just made so.''

--Michael Weingart, Rutgers Grad Students Pizza seminar, March 7, 2003.``Computers are here to stay. It is a major challenge for the future to use computers efficiently in combinatorics without losing its special appeal.''

``I was interviewed in the Israeli Radio for five minutes and I said that more than 2000 years ago, Euclid proved that there are infinitely many primes. Immediately the host interrupted me and asked: `Are there still infinitely many primes?'

--Noga Alon, (Distinguished Lecture series, Princeton Applied and Computational Mathematics Program, Feb. 13, 2003).``As for mathematicians themselves: don't expect too much help. Most of them are too far removed in their ivory towers to take up such challenges. And anyway, they are not competent. After all, they are just mathematicians-what we need is paramathematicians, like you... It is you who can be the welding force, between mathematicians and stories, in order to achieve the synthesis."

--Apostolos Doxiadis, (Opening address to the Third Mediterranean Conference on mathematics education, Jan. 3, 2003).``Nature is a good approximation of Mathematics''

--Zvi Artstein (Rutgers Univ. Colloquium, Dec. 6, 2002).``I had made an empirical discovery and it carried all the weight of a mathematical proof.''

--Paul Auster (p.9-10, "The Book of Illusions", 2002)``It's like a mathematical law, Grace.''

--Paul Auster (p. 171, "Oracle Night", 2003)``In the old physics, three times two equals six and two times three equals 6 are reversible propositions. Not in quantum physics. Three times two and two times three are two different matters, distinct and separate propositions."

--Paul Auster (p. 284, "Sunset Park", 2010)``Let's not try to define knowledge, but try to define zero-knowledge''

--Shafi Goldwasser (ICM 2002 plenary talk, Aug. 23, 2002)``There is an explicit way to define what explicit is''

--Noga Alon (ICM 2002 plenary talk, Aug. 23, 2002)``Risk used to be viewed as uncertainty about the future. Now it is quantified and abstracted.''

``Mathematics is abused by `consultants' to lend speculations an aura of objectivity and precision that they do not posses.''

--Mary Poovey (ICM 2002 public lecture, Aug. 22, 2002).``I am sorry that it is esoteric from the start, but this is K-theory...''

--Vladimir Voevodsky (ICM 2002 special seminar, Aug. 22, 2002).``Nowadays we can do computer experiments using Mathematica, and even solve a system of 42 equations. This offers another route to knowledge, rather than mere ideas.''

--John F. Nash, Jr. (ICM 2002 public lecture, evening, Aug. 21, 2002).``Logic has virtually nothing to do with the way we think.''

``The world is a very complicated place, as babies know.''

``The world is continuous, but the mind is discrete.''

``The extreme possibilities are the most illuminating.''

``Images of the world are Renormalization Group fixed points.''

--David Mumford (ICM 2002 plenary talk, Aug. 21, 2002).``First I thought that I had to desingularize, but I made a mistake. Then I realized that I didn't have to do it.''

--Laurent Lafforgue (ICM 2002 plenary talk, Aug. 21, 2002).``In Mathematics it is always best to cheat.''

--Herb Wilf (E-mail message, 5/10/02).``One of the reasons we don't do as well as we should is that we are all over-taught.''

``My life may be encapsulated by one of Graham Greene's "entertainments'" titles: `Loser Takes All'. Since I was thrown out of highschool for political reasons, I was free to study on my own and develop my own ways of thinking.''

--Israel Gelfand (Gelfand Workshop, Rutgers Univ., May 6, 2002)``A mark of a good theory is that it proves even the most trivial results''.

--Hector Sussmann (Gelfand Workshop, Rutgers Univ., Feb. 11, 2002)``Music is part of Number Theory. Nowadays when a number-theorist applies for a grant, he says that it is good for security, but in those days, way before America, he would say that it's good for music. I will not comment whether we have progressed...''

``Recreational Number Theory is that part of Number Theory that is too difficult to study seriously.''

``A math talk without a proof is like a movie without a love scene''

---H. W. Lenstra, Jr., AMS-MAA Invited talk, 2002 Annual meeting, San Diego, Jan. 8, 2002.``In The Renormalization Group method you take a structure you don't understand and convert it to another structure you don't understand. You keep doing it until you finally understand.''

---Sir Michael Berry (2002 Gibbs Lecture, San Diego, Jan. 6, 2002)

``Perhaps we should go back to Pythagoras's time and make math forbidden knowledge''

---Brian Hayes (2002 AMS meeting, San Diego, Panel discussion, Jan. 6, 2002)

``Cardinal Arithmetics is much older than Number Theory. People used to exchange things (in one-to-one-correspondence) way before there were numbers. Expressing numbers like 762 is already a sign of a very advanced civilization.''

``Given a conjecture, the best thing is to prove it. The second best thing is to disprove it. The third best thing is to prove that it is not possible to disprove it, since it will tell you not to waste your time trying to disprove it. That's what Gödel did for the Continuum Hypothesis.''

``Number theorists say that number theory is too complicated, so let's pretend that there is only one prime number (at a time), and then let's combine all these results. Surprisingly, sometimes it works.''

``There is an old maxim that says that two empires that are too large will collapse. The analog in set theory is that two different theories that are too powerful must necessarily contradict each other.''

----Saharon Shelah, Rutgers Univ. Colloqium, Oct. 26, 2001.

``If you understand something, you understand that it is obvious.''

``...The approach of von Neumann and Connes to the use of non-commutative algebra in physics is naive, the situation is much more complicated...''

They have the idea that non-commutative algebra should remind one of commutative algebra, but the former is more sophisticated. I believe that non-commutative algebra is just as simple, but it isdifferent''.

``I used to say: "Everything is Representation Theory".NowI say: "Nothing is Representation Theory".''

In the Middle Age, in Germany, if you wanted to learn addition and multiplication, you could go to any university. But if you wanted to learn division, you could only do it in one place, Heidelberg. This makes sense, since in my theory with Vladimir Retakh and Robert Wilson, addition and multiplications are cheap, but division is expensive.

---Israel Gelfand (Gelfand seminar and workshop, Rutgers University, Sept. 10, and Nov. 19, 2001)``The continuous evolution of objects that we see is just an illusion. Our brains are actually digital and what we see us actually a fast discrete measurement''

---Saber Elaydi, ``Is the world evolving discretely?'', [to appear] ."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)``I was just sitting on the train, just staring out the window at some cows. It was not the most inspiring subject. When all of a sudden the idea of Harry just appeared in my mind's eye.''

--J. K. Rowling (interview)`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"Maybe Natural Deduction is not so natural"

"Pattern matching is not just syntactical sugar"

"There are many ways to answer this question, one of them being `it is a stupid question'"

--Noam Zeilbergerז ה ל א ק ר , ז ה נ ע י ם

נ ו ע ם צ י י ל ב ר ג רIt 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 something 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....What is a generalization? Who said that one should not generalize?... To say that one should not generalize is not a generalization? If it is forbidden, how come that there are generalizations?...

--Orly Castel Bloom, `Human Parts', Kineret, 2002, p. 150. ``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 [quoted in: `Constructive approachs to mathematical models', C.V. Coffman and G.J. Fix, editors, a volume in honor of R.J. Duffin (1909-1996), p. xviii, line N]``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?

--Leopold 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,''"...le lycéen qui découvre simultanément Rimbaud et la rigeur des mathématiques..."

----Roger Vailland, "La Fête" (1960), p. 243."Duc se plut vivement à la conversation du professeur de mathématiques; ce que le Bourbakistes se proposent de faire à propos de la géométrie n'est sans rapport avec l'expression commune qu'il essaie d'élaborer des diverses actions dans lesquelles s'engage un homme"

----Roger Vailland, "La Fête" (1960), p. 192.``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 surprising 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 translation 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 programmers 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 emergence 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 structure''

-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 JosephOmar 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 survey.'' ''

-- 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.25Il existe sans doute des vérités mathématiques.[...] Mais il était encore bien plus sûr cette fois-ci!

--Georges Simenon, Les Pitard (Gallimard, 1935), p.214On 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. 135Et puis, un seul petit fait, s'il est bien choisi, ne suffit-il pas à l'expérimentateur pour décider d'une loi générale qui fera connaître la vérité sur des milliers de faits analogues?

-- Marcel Proust, Albertine Disparue, pp. 95-96Il 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 HanlonAre 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. EkhadIt 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 no interest 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 blindness 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 different?''

---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 nauseum 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-empirical 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.``... the atlasisa manifold. This is a typical mathematician's use of the word "is'', and should not be confused with the normal use."

--- Timothy Gowers, p. 45, col. 2, lines 5-8 in "The Princeton Companion to Mathematics", [W.]T.Gowers, ed., Princeton Univ. Press, 2008.``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 intelligent 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 embarrassed, but even if you get embarrassed, 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."In anger, my hostility is directed toward another's action and can be extinguished by getting even - an action that reestablishes the equilibrium..."

--Jon Elster ("Alchemists and the Mind: Rationality and the Emotions", p.65)``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``As for my advice to you, it was just that, advice."

--Capt. Nick Sutter (Nicholas Sutter, at the time Lt.), Princeton Police Department, Email message to DZ, Nov. 2009.``Amanda [Folsom] is now a tenure-track [assistant] professor at Yale" (loud applause)

--Ken Ono, Adding and Counting (videotaped lecture), 1:08-1:15.``In a mathematical proposition, for example, the objectivity is given, but therefore its truth is also an indifferent truth''.

-Soren Kierkegaarde, Concluding unscientific postscript to Philosophical fragments, VII 170.``Nothing is a waste, everything adds up to your experience"

--Alizah Spivak"-Mais c'est terriblement grave d'accuser cette femme de vendre tout le monde-sans preuve, sur un calcul de probabilites, sans une seule preuve formelle..."

Roger Vailland, "Drole de Jeu", (p. 292)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

[Added Feb. 28, 2015]: Thanks to David Brown for numerous corrections.

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