Selected Excogitations and General Exegesis

Go to Newest Quotes.

``a certain impression I had of mathematicians was ... that they spent immoderate amounts of time declaring each other's work trivial.''
(Richard Preston, New Yorker, 1992)

``It's about as interesting as going to the beach and counting sand. I wouldn't be caught dead doing that kind of work.''
( Professor Take Your Pick)

``The universe contains at most `two to the power fifty' grains of sand.''

``Americans are broad-minded people. They'll accept the fact that a person can be alcoholic, a dope fiend or a wife-beater, but if a man doesn't drive a car, everybody thinks that something is wrong with him.''
(Art Buchwald, local newspaper, March 1996)

``Caution, skepticism, scorn, distrust and entitlement seem to be intrinsic to many of us because of our training as scientists... . These qualities hinder your job search and career change.''

Former astrophysicist Stephen Rosen, now director, Scientific Career Transitions Program, New York City, giving job-hunting advice in an on-line career counseling session.

(Quoted in Science, 4 August 1995, page 637)

``Consider a precise number that is well known to generations of parents and doctors: the normal human body temperature of 98.6 Farenheit. Recent investigations involving millions of measurements reveal that this number is wrong; normal human body temperature is actually 98.2 Farenheit. The fault, however, lies not iwth Dr. Wunderlich's original measurements - they were averaged and sensibly rounded to the nearest degree: 37 Celsius. When this temperature was converted to Farenheit, however, the rounding was forgotten and 98.6 was taken to be accurate to the nearest tenth of a degree. Had the original interval between 36.5 and 37.5 Celsius been translated, the equivalent Farenheit temperatures would have ranged from 97.7 to 99.5. Apparently, discalculia can even cause fevers.''

John Allen Paulus, in `A Mathematician Reads the Newspaper' (Basic Books)

(Quoted in Science, August 18, 1995, page 992)

``When Gladstone was British Prime Minister he visited Michael Faraday's laboratory and asked if some esoteric substance called `Electricity' would ever have practical significance.

was the answer.''

(Quoted in Science, 1994). As Michael Saunders points out, this can not be correct because Faraday died in 1867 and Gladstone became PM in 1868. A more plausible PM would be Peel as electricity was discovered in 1831. Equally well it may be an `urbane legend'.

`` "the proof is left as an exercise" occurred in `De Triangulis Omnimodis' by Regiomontanus, written 1464 and published 1533. He is quoted as saying "This is seen to be the converse of the preceding. Moreover, it has a straightforward proof, as did the preceding. Whereupon I leave it to you for homework." ''
(Quoted in Science, 1994)

``As the fading light of a dying day filtered through the window blinds, Roger stood over his victim with a smoking .45, surprised at the serenity that filled him after pumping six slugs into the bloodless tyrant that had mocked him day after day, and then he shuffled out of the office with one last look back at the shattered computer terminal lying there like a silicon armidillo left to rot on the information highway.''

  • From the winner of the 1994 Bulwer-Lytton Fiction contest for lousy literature

    Named for the author of `It was a dark and stormy night.' in the novel `Paul Clifford', 1830. (Later winners are quoted below.)

  • ``I imagine most of that stuff on the information highway is roadkill anyway.''
    (John Updike, 1994)

    ``It's going to be about bad news. It's going to be about the future of this country, about foreign policy, about defense policy. There are a lot of issues left. I'm certain something will pop up in November. So we'll be able to put it together.''

    Robert Dole on `what is the key issue?' in the '96 Presidential election.

    (Quoted in the Economist, March 16, 1996, page 23)

    ``My dearest Miss Dorothea Sankey

    My affectionate & most excellent wife is as you are aware still living - and I am proud to say her health is good. Nevertheless it is always well to take time by the forelock and be prepared for all events. Should anything happen to her, will you supply in her place - as soon as the proper period for decent mourning is over.

    Till then I am your devoted servant
    Anthony Trollope.''

    Anthony Trollope taking precautions in 1861.

  • Sotheby's at auction in 1942 described this letter as "one of the most extraordinary letters ever offered for sale".
  • (Quoted from The Oxford Book of Letters in the Economist, March 23, 1996, page 90)

    ``I believe that the motion picture is destined to revolutionize our educational system and that in a few years it will supplant largely, if not entirely, the use of textbooks.''

    (Thomas Alva Edison, 1922)

    ``Keynes distrusted intellectual rigour of the Ricardian type as likely to get in the way of original thinking and saw that it was not uncommon to hit on a valid conclusion before finding a logical path to it.

    `I don't really start', he said, `until I get my proofs back from the printer. Then I can begin serious writing.' ''

    two excerpts from Keynes the man written on the 50th Anniverary of Keynes' death.

    (Sir Alec Cairncross, in the Economist, April 20, 1996)

    ``One major barrier to entry into new markets is the requirement to see the future with clarity. It has been said that to so fortell the future, one has to invent it. To be able to invent the future is the dividend that basic research pays.''

    An econonomic case for basic research, by Eugen Wong, Hong Kong University of Science and Technology

    (In Nature, May 16, 1996, pages 178-9)

    `` 'Ace, watch your head!' hissed Wanda urgently, yet somehow provocatively, through red, full, sensuous lips, but he couldn't, you know, since nobody can actually watch more than part of his nose or a little cheek or lips if he really tries, but he appreciated her warning.''

  • Janice Estey of Aspen 1996 Bulwer-Lytton Grand Prize Winner

  • ``Because the Indians of the high Andes were believed to have little sense of humor, Professor Juan Lyner was amazed to hear this knee-slapper that apparently had been around for centuries at all of the Inca spots: `Llama ask you this. Guanaco on a picnic? Alpaca lunch.' ''

  • John Ashman of Houston 1995 Bulwer-Lytton Grand Prize Winner

  • ``We know [smoking is] not good for kids. But a lot of other things aren't good. Drinking's not good. Some would say milk's not good.''

    Robert Dole echoing the tobacco companies on smoking?

    ( Page 27 in the Economist, July 6, 1996)

    ``I feel so strongly about the wrongness of reading a lecture that my language may seem immoderate .... The spoken word and the written word are quite different arts .... I feel that to collect an audience and then read one's material is like inviting a friend to go for a walk and asking him not to mind if you go alongside him in your car.''

    Sir Lawrence Bragg. What would he say about overheads?

    ( Page 76 in Science, July 5, 1996)

    ``I know, it's hard to believe that Microsoft would release a product before it was ready, but there you have it. A Seattle cyberwag says, "At Microsoft, quality is job 1.1." We had him killed. ''

    from Welcome to Stale

    A take-off of Microsoft's ``webzine'', Slate, Stale, August 1996.

    ``No presidential candidate in the future will be so inept that four of his major speeches can be boiled down to these four historic sentences: Agriculture is important. Our rivers are full of fish. You cannot have freedom with out liberty. The future lies ahead.''

    the (no doubt partisan) Louisville Courier-Journal on Thomas Dewey in 1948, quoted in Jack Beatty's review of James Patterson's Grand Expectations, The United States, 1945-1974. Beatty goes on to say:

    `Tom Dewey, make room for Bob (``like everyone else in this room I was born'') Dole.`

    and lists many other fine quotes from Patterson's book.

    (From pp. 107-112 in the Atlantic Monthly, September 1996)

    ``Writers often thank their colleagues for their help. Mine have given none. .. Writers often thank their typists. I thank mine. Mrs George Cook is not a particularly good typist, but her spelling and grammar are good. The responsibility for any mistakes is mine, but the fault is hers. Finally, writers too often thank their wives. I have no wife.''

    Acknowledgement by Edward Ingram in The Beginning of the Great Game in Asia, 1828-1834.

    (From p. 83 in the Economist, September 7th 1996)

    ``I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style.''

    Freeman Dyson's review of Nature's Numbers by Ian Stewart (Basic Books, 1995).

    (From p. 612 in the American Mathematical Monthly, August-September 1996)

    ``I was sitting by Dr. Franklin, who perceived that I was not insensible to these mutilations. I have made a rule, said he, whenever in my power, to avoid becoming the draughtsman of papers to be reviewed by a public body.''

    Jefferson writing in 1818 of the drafting of the Declaration of Independence.

    (From p. 74 of Conor Cruise O'Brien's disturbing article Thomas Jefferson: Radical and Racist, in the Atlantic Monthly, October 1996)

    ["The tree of liberty must be refreshed from time to time with the blood of patriots and tyrants."
    (Jefferson quoted on Oklahoma bomb suspect McVeigh's T-shirt.)]

    ``My morale has never been higher than since I stopped asking for grants to keep my lab going.''

    Robert Pollack, Columbia Professor of biology. Speaking on "the crisis in scientific morale", September 19, 1996 at GWU's symposium Science in Crisis at the Millenium.

    (Quoted from p. 1805 in the September 27, 1996 Science)

    ``smugness, brutality, unctuous rectitude and tact"

    Cecil Rhodes own sardonic paraphrase of the criteria for a Rhodes Scholarship:

    (Quoted from Cecil Rhodes Traduced, pp. 80-81 in the October 5, 1996 Economist)

    ``The dictum that everything that people do is 'cultural' ... licenses the idea that every cultural critic can meaningfully analyze even the most intricate accomplishments of art and science. ... It is distinctly weird to listen to pronouncements on the nature of mathematics from the lips of someone who cannot tell you what a complex number is!''

    Norman Levitt, from "The flight From Science and Reason," New York Academy of Science.

    (Quoted from p. 183 in the October 11, 1996 Science)

    ``Church discipline is also somewhat of a remove from the time when the Emperor Henry IV was made to stand in the snow for three days outside the Pope's castle at Canossa, awaiting forgiveness. A French Bishop, Jacques Gaillot, because of his ultra-liberal views was recently transferred from his position at Evreux, in Normandy, and given charge instead of the defunct dioscese of Partenia, in Southern Algeria, which has been covered by sand since the Middle Ages. Gaillot has retaliated by creating a virtual dioscese on the Internet, which can be reached at ''

    Cullen Murphy, "Broken Covenant?"

    (Quoted from p. 24 in the November, 1996 Atlantic Monthly)

    ``We were a polite society and I expected to lead a quiet life teaching mechanics and listening to my senior colleagues gently but obliquely poking fun at one another. This dream of somnolent peace vanished very quickly when Rutherford came to Cambridge. Rutherford was the only person I have met who immediately impressed me as a great man. He was a big man and made a big noise and he seemed to enjoy every minute of his life. I remember that when transatlantic broadcasting first came in, Rutherford told us at a dinner in Hall how he had spoken into a microphone to America and had been heard all over the continent. One of the bolder of our Fellows said "Surely you did not need to use apparatus for that." ''

    Geoffrey Fellows, 1952, as quoted by George Batchelor in The Life and Legacy of G.I. Taylor (Cambridge University Press).

    (Quoted in "Vignettes: Yesteryear in Oxbridge" p. 733 in Science November 1, 1996)

    ``Then, owls and bats,
    Cowls and twats,
    Monks and nuns, in a cloister's moods,
    Adjourn to the oak-stump pantry.''

    From Robert Browning's (1841) Pippa Passes, which also contains "God's in his Heaven, all's right with the world."

    (Quoted on page 56-66 of Bill Byerson, Mother Tongue: The English Language, Penguin, 1990.)

    He goes on to say about "this disconcerting quote" that

    ``Browning had apparently somewhere come across the word twat - which meant precisely the same as it does now - but somehow took it to mean a piece of head gear for nuns. The verse became a source of twittering amusement for generations of schoolboys and a perennial embarrassment to their elders, but the word was never altered and Browning was allowed to live out his life in wholesome ignorance because no one could think of a suitably delicate way of explaining his mistake to him.''

    ``Two major advances are responsible for both the recent progress and current optimism. First, recombinant DNA technology has made it possible to identify every gene and protein in an organism and to manipulate them in order to explore their functions. Second, it has been discovered that the molecular mechanisms of development have been conserved during animal evolution to a far greater extent than had been imagined. This conservation means that discoveries about the development of worms and files, which come from the kind of powerful genetic studies that are not possible in mammals, greatly accelerate the rate at which we can discover the mechanisms and molecules that operate during our own development.
    ... ...
    It is tempting to think that the main principles of neural development will have been discovered by the end of the century and that the cellular and molecular basis of the mind will be the main challenge for the next. An alternative view is that this feeling that understanding is just a few steps away is a recurring and necessary delusion that keeps scientists from dwelling on the complexity the face and how much more remains to be discovered.''

    (Editorial on page 1063 of Science November 15, 1996)

    Neural Development: Mysterious No More? written by Martin Raff (University College, London).

    [It is hard to imagine a better case for ``basic science'' than that afforded by this conservation principle -- if worms were good enough for Darwin ... !]


    Vice President Gore, who was clearly on top of the technical issues, met on Wednesday with a group of tough-minded scientists, clergy and fuzzy romantics to discuss the questions raised by evidence of extraterrestrial life. For physicist/astronomer John Bahcall, the remarkable thing was not that such questions were being asked, but that we have the tools to answer them.''

    (From WHAT'S NEW by Robert L. Park -- Friday, 13 Dec 96)

    WHAT'S NEW is published every Friday by the AMERICAN PHYSICAL SOCIETY.

    ``As the test beds begin to prove WDM (`wavelength division multiplexing') networks feasible, telephone company executives will have to judge whether they are wise. If a single glass fiber can carry all the voice, fax, video and data traffic for a large corporation yet costs little more than today's high-speed Internet connections, how much will they be able to charge for telephone service? Peter Cochrane of BT Laboratories in Ipswich, England, predicts that "photonics will transform the telecoms industry by effectively making bandwidth free and distance irrelevant." Joel Birnbaum, director of Hewlett-Packard Laboratories, expects that this will relegate telephone companies to the role of digital utilities. "You will buy computing like you now buy water or power," he says.

    Others, such as industry analyst Francis McInerney, believe the double-time march of technology has already doomed them to fall behind. AT&T and its ilk, he claims, "are already dead. When individuals have [megabits per second of bandwidth], telephone service should cost about three cents a month." Having discovered how to offer high-bandwidth service, telephone companies may now need to invent useful things to do with it, just to stay in business. ''

    (From BANDWIDTH, UNLIMITED by W. Wayt Gibbs)

    In the January 1997 on-line Scientific American.

    ``Before Canada jeopardizes its scientific future and compromises its scientific community to achieve short-term budgetary solutions, it must recognize that the funding of university sicence is both a government responsibility and a long-range investment. Without government support, Canada's university science infrastructure will erode, and along with it, the country's competitiveness in a world economy increasingly based on knowledge.''
    (Editorial on page 139 of Science January 10, 1997)

    Canada's Crisis: Can Business Rescue Science? written by Albert Aguyo and Richard A. Murphy (McGill, Montreal Neurological).


    178 new bills were introduced in the Senate on Tuesday -- one, S.124, is a thing of beauty: "The National Research Investment Act of 1997." It calls for doubling the federal investment in basic science and medical research over a 10-year period (WN 17 Jan 97). Funds must be allocated by a peer review system and can not be used for the commercialization of technologies. A dozen non-defense agencies and programs are covered by the bill, which is the work of Phil Gramm (R-TX). Gramm pointed out that in 1965, 5.7% of the federal budget went for non-defense R&D -- 32 years later, that has dropped to only 1.9%, and real spending on research has declined for four straight years. Ten-year doubling requires an annual increase of 7% -- just what leaders of the science community have been calling for (WN 10 Jan 97)."

    (From WHAT'S NEW by Robert L. Park -- Friday, 24 January, 1997)

    WHAT'S NEW is published every Friday by the AMERICAN PHYSICAL SOCIETY. It is interesting to contrast a conservative US senator (an ex-academic) from a liberal Canadian government.

    ``a British officer told a sergeant to post four lookouts to watch for the German army which was advancing through Belgium. Later, the officer discovered that the sergeant had posted only three. Asked to explain his lapse, the soldier said he had judged the fourth guard unnecessary. 'The enemy would hardly come from that direction,' he explained, 'it's private property.' ''

    (Quoted from Back to the Front by Stephen O'Shea)

    From page 59 in MACLEANS Magazine of February 10, 1997.

    ``Admirers of Thomas Jefferson have long quoted his statement about black men and women that is inscribed on the Jefferson Memorial: 'Nothing is more certainly written in the book of fate than that these people are to be free.' But they and the inscription, as Conor Cruise O'Brien pointed out in 'Thomas Jefferson: Radical and Racist' (October, 1996, Atlantic), omit Jefferson's subsequent clause: 'Nor is it less certain that the two races, equally free, cannot live in the same government.'"

    (Quoted from What Jefferson Helps to Explain by Benjamin Schwarz)

    From page 60 in the Atlantic Monthly of March, 1997. [There are well established copyright notions of "paternity" and "integrity" in the use of material -- the later which this clearly violates!]

    ``A centre of excellence is, by definition, a place where second class people may perform first class work.''

    ``A truly popular lecture cannot teach, and a lecture that truly teaches cannot be popular.''

    ``The most prominent requisite to a lecturer, though perhaps not really the most important, is a good delivery; for though to all true philosophers science and nature will have charms innumerably in every dress, yet I am sorry to say that the generality of mankind cannot accompany us one short hour unless the path is strewed with flowers.''

    (Three quotes from Michael Faraday)

    Excerpted from "Michael Faraday -- and the Royal Institution, the genius of man and place", by J.M. Thomas, Adam Hilger, Bristol, 1991.

    ``The body of mathematics to which the calculus gives rise embodies a certain swashbuckling style of thinking, at once bold and dramatic, given over to large intellectual gestures and indifferent, in large measure, to any very detailed description of the world. It is a style that has shaped the physical but not the biological sciences, and its success in Newtonian mechanics, general relativity and quantum mechanics is among the miracles of mankind. But the era in thought that the calculus made possible is coming to an end. Everyone feels this is so and everyone is right.''
    (From Vignettes: Changing Times in Science, 28 February 1997, page 1276)

    From David Berlinski's "A Tour of the Calculus" (Pantheon Books, 1995)

    ``[8] 94m:94015 Beutelspacher, Albrecht Cryptology. An introduction to the art and science of enciphering, encrypting, concealing, hiding and safeguarding described without any arcane skullduggery but not without cunning waggery for the delectation and instruction of the general public. Transformation from German into English succored and abetted by J. Chris Fisher. MAA Spectrum. Mathematical Association of America, Washington, DC, 1994. xvi+156 pp. ISBN: 0-88385-504-6 94A60 (94-01)''
    (From Math Reviews)

    A serious "best title" candidate...

    ``It's generally the way with progress that it looks much greater than it really is.''
    (From The Wittgenstein Controversy, by Evelyn Toynton in the Atlantic Monthly June 1997, pp. 28-41.)

    The epigraph that Ludwig Wittgenstein (1889-1951) ("whereof one cannot speak, thereof one must be silent") had wished for an unrealized joint publication of Tractatus Logico-Philosophicus (1922) and Philosophical Investigations (1953): suggesting the two volumes are not irreconcilable.

    Compare the following for which I have no good source:

    "The world will change. It will probably change for the better. It won't seem better to me."

    (J. B. Priestly)

    ``In 1965 the Russian mathematician Alexander Konrod said "Chess is the Drosophila of artificial intelligence." However, computer chess has developed as genetics might have if the geneticists had concentrated their efforts starting in 1910 on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies.''
    (From John McCarthy's review of Kasparov versus Deep Blue by Monty Newborn (Springer, 1996) in Science, 6 June 1997, page 1518)

    He goes on to point out that of three features of human chess play two were used in early programs (forward pruning, identifying parallel moves, and partitioning (never used)). None survives in present programs. Material on Making computer chess scientific is available from John McCarthy's web site

    ``A research policy does not consist of programs, but of hiring high-quality scientists. When you hire someone good, you've made your research policy for the next 20 years.''
    Chief CNRS advisor Vincent Courtillot quoted in New CNRS Chief Gets Marching Orders, Science, 18 July, 1997, page 308)

    ``Mathematicians are like pilots who maneuver their great lumbering planes into the sky without ever asking how the damn things stay aloft.
    The computer has in turn changed the very nature of mathematical experience, suggesting for the first time that mathematics, like physics, may yet become an empirical discipline, a place where things are discovered because they are seen.
    The existence and nature of mathematics is a more compelling and far deeper problem than any of the problems raised by mathematics itself.''
    (From David Berlinski's somewhat negative review of The Pleasures of Counting by T. W. Korner (Cambridge, 1996)
    in The Sciences, July/August 1997, pages 37-41)

    Korner is a careful and stimulating writer/teacher.

    ``If I can give an abstract proof of something, I'm reasonably happy. But if I can get a concrete, computational proof and actually produce numbers I'm much happier. I'm rather an addict of doing things on the computer, because that gives you an explicit criterion of what's going on. I have a visual way of thinking, and I'm happy if I can see a picture of what I'm working with.''
    (John Milnor)

    Page 78 of Who got Einstein's Office? by Ed Regis, Addison-Wesley, 1986. A history of the Institute for Advanced Study. The answer is Arnie Beurling.

    ``The term "reviewed publication" has an appealing ring for the naive rather than the realistic... Let's face it: (1) in this day and age of specialization, you may not find competent reviewers for certain contributions; (2) older scientists may agree that over the past two decades, the relative decline in research funds has been accompanied by an increasing number of meaningless, often unfair reviews; (3) some people are so desperate to get published that they will comply with the demands of reviewers, no matter how asinine they are.''
    (August Epple)

    From Organizing Scientific Meetings quoted on page 400 of Science October 17, 1997.

    ``The NYT also has a stunning revelation about the way the Ivy League used to do business. Last Friday, the President of Darmouth used the occasion of dedicating a campus Jewish student center to haul out a 1934 letter between an alumnus of the school and the director of admissions. The alum complained that "the campus seems more Jewish each time I arrive in Hanover. And unfortunately many of them (on quick judgment) seem to be the 'kike' type." And the Dartmouth admissions man wrote back, "I am glad to have your comments on the Jewish problem, and I shall appreciate your help along this line in the future. If we go beyond the 5 percent or 6 percent in the Class of 1938, I shall be grieved beyond words." In reacting to the revelation, Elie Wiesel summons a simple fact that suggests how much times have changed: the current presidents of Harvard, Yale, and Princeton are Jewish.''

    (SLATE, Tuesday November 11, 1997)

    ``This is the essence of science. Even though I do not understand quantum mechanics or the nerve cell membrane, I trust those who do. Most scientists are quite ignorant about most sciences but all use a shared grammar that allows them to recognize their craft when they see it. The motto of the Royal Society of London is 'Nullius in verba' : trust not in words. Observation and experiment are what count, not opinion and introspection. Few working scientists have much respect for those who try to interpret nature in metaphysical terms. For most wearers of white coats, philosophy is to science as pornography is to sex: it is cheaper, easier, and some people seem, bafflingly, to prefer it. Outside of psychology it plays almost no part in the functions of the research machine.''
    (Steve Jones, University College, London)

    From his review of How the Mind Works (by Steve Pinker) in The New York Review of Books (pages 13-14) November 6, 1997. [Two solitudes indeed!]

    ``If you have a great idea, solid science, and earthshaking discoveries, you are still only 10% of the way there,''
    (David Tomei, LXR Biotechnology Inc.)

    Quoted in Science page 1039, November 7, 1997. [On the vicissitudes of startup companies.]

    ``There he received his hardest job of the war - a rush request to convert typewriters to twenty-one different languages of Asia and the South Pacific.
    The implications of the work and its difficulty brought him to near collapse, but he completed it with only one mistake: on the Burmese typewriter he put a letter upside down. Years later, after he had discovered his error, he told the language professor he had worked with that he would fix that letter on the professor's Burmese typewriter. The professor said not to bother; in the intervening years, as a result of typewriters copied from Martin's original, that upside-down letter had been accepted in Burma as proper typewriter style.''
    (Ian Frazier)

    Page 88 in Typewriter Man, the Atlantic Monthly, November 1997: "For Martin Tytell, the machinery of writing has been a life's work." [A fine example of convergence.]

    The T-bone terror proves that ministers have no grasp of science or maths - let alone our liberties

    ``The giant finger whooshes out of the night sky and points at the dumbstruck face in the window. "It could be you," says a voice. This week the Agriculture Minister Jack Cunningham impersonated the National Lottery advertiser. As the nation's fork was poised with a T-bone steak on its way to the nation's mouth, Dr Cunningham screamed: "Don't touch it." According to the great god science, new variant Creutzfeldt-Jakob disease (nvCJD) could be lurking in that mouthful. There is a small risk, and where there is risk, a government must ban.

    Perhaps only mathematicians are aware of the enormity of what the Government did this week. It took a risk that is statistically negligible and exploited it as an act of insufferable nannying. Beef ribs, T-bones and oxtails present a public health risk publicised as "very small" and "a chance of one case per year" (though none of Britain's 22 nvCJD cases has been positively linked to beef). Most newspapers cluelessly converted "a chance" into a certainty, and ridiculed the risk as a tiny one in 56 million. But that is not what the scientists said. They suggested the chance was "5 per cent", so the risk is nearer to one in 1.1 billion, or one in 560 million among the half of the population that eats beef. There can have been no more tenuous basis for an infringement of personal liberty.''

    (Simon Jenkins)

    Simon Jenkins on Boneless Wonders in the Times of London, Dec 6, 1997

    ``The common situation is this: An experimentalist performs a resolution analysis and finds a limited-range power law with a value of D smaller than the embedding dimension. Without necessarily resorting to special underlying mechanistic arguments, the experimentalist then often chooses to label the object for which she or he finds this power law a ''fractal''. This is the fractal geometry of nature.''

    (David Avnir et al, Hebrew University)

    From Is the geometry of nature fractal? in Science January 2, 1998, 39-40. Their review of all articles from 1990 to 1996 in Physical Reviews suggests very little substance for claims of fractility.

    ``Most nonscientists who like to think of themselves as knowledgeable about modern science really know only about technologies - and specifically those technologies likely to bring economic profits in the short term.''

    (Takashi Tachibana, Japanese Journalist)

    From Closing the Knowledge Gap Between Scientist and Nonscientist in Science August 7, 1998, 778-779.

    ``Another thing I must point out is that you cannot prove a vague theory wrong. ... Also, if the process of computing the consequences is indefinite, then with a little skill any experimental result can be made to look like the expected consequences.''

    (Richard Feynman, 1964)

    Quoted by Gary Taubes in The (Political) Science of Salt, Science August 14, 1998, 898-907.

    ``Renyi would become one of Erdos's most important collaborators. ... Their long collaborative sessions were often fueled by endless cups of strong coffee. Caffeine is the drug of choice for most of the world's mathematicians and coffee is the preferred delivery system. Renyi, undoubtedly wired on espresso, summed this up in a famous remark almost always attributed to Erdos: "A mathematician is a machine for turning coffee into theorems." ... Turan, after scornfully drinking a cup of American coffee, invented the corollary: "Weak coffee is only fit for lemmas." ''

    (Bruce Schechter, 1998)

    On page 155 of My Brain is Open, Schechter's 1998 Simon and Schuster biography of Erdos. Schechter's Erdos is recognisable. The book contains interesting material on the Erdos-Selberg controversy (pp. 144-151).

    ``Once the opening ceremonies were over, the real meat of the Congress was then served up in the form of about 1400 individual talks and posters. I estimated that with luck I might be able to comprehend 2% of them. For two successive weeks in the halls of a single University, ICM'98 perpetuated the myth of the unity of mathematics; which myth is supposedly validated by the repetition of that most weaselly of rhetorical phrases: "Well, in principle, you could understand all the talks." ''

    (Philip Davis, 1998)

    Describing the Berlin International Congress of Mathematicians in the October 1998 SIAM News.

    ``Looking over the past 150 years -- at the tiny garden at Brno, the filthy fly room at Columbia, the labs of the New York Botanical Garden, the basement lab at Stanford, and the sun-drenched early gatherings at Cold Spring Harbor -- it seems that the fringes, not the mainstream, are the most promising places to discover revolutionary advances.''

    (Paul Berg and Maxine Singer, 1998)

    In Inspired Choices, Science October 30, 1998, 873-874s, on the past 150 years of biological research.

    ``Should we teach mathematical proofs in the high school? In my opinion, the answer is yes...Rigorous proofs are the hallmark of mathematics, they are an essential part of mathematics' contribution to general culture.''
    George Polya (1981). Mathematical discovery: On understanding, learning, and teaching problem solving (Combined Edition), New York, Wiley & Sons (p. 2-126)
    ``A mathematical deduction appears to Descartes as a chain of conclusions, a sequence of successive steps. What is needed for the validity of deduction is intuitive insight at each step which shows that the conclusion attained by that step evidently flows and necessarily follows from formerly acquired knowledge (acquired directly by intuition or indirectly by previous steps of I think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning.'' (ibid, p. 2-128)
    This "quasi-experimental" approach to proof can help to de-emphasis a focus on rigor and formality for its own sake, and to instead support the view expressed by Hadamard when he stated "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it" (J. Hadamard, in E. Borel, Lecons sur la theorie des fonctions, 3rd ed. 1928, quoted in Polya, (1981), (p. 2/127).
    ``intuition comes to us much earlier and with much less outside influence than formal arguments which we cannot really understand unless we have reached a relatively high level of logical experience and sophistication. Therefore, 1 think that in teaching high school age youngsters we should emphasize intuitive insight more than, and long before, deductive reasoning.'' (ibid, p. 2-128)

    ``In the first place, the beginner must be convinced that proofs deserve to be studied, that they have a purpose, that they are interesting.'' (ibid, p. 2-128)

    ``The purpose of a legal proof is to remove a doubt, but this is also the most obvious and natural purpose of a mathematical proof. We are in doubt about a clearly stated mathematical assertion, we do not know whether it is true or false. Then we have a problem: to remove the doubt, we should either prove that assertion or disprove it.'' (ibid, p. 2-129)

    (Polya quotes are thanks to Laurie Edwards)

    ``The basic difference between playing a human and playing a supermatch against Deep Blue is the eerie and almost empty sensation of not having a human sitting opposite you. With humans, you automatically know a lot about their nationality, gender, mannerisms, and such minor things as a persistent cough or bad breath. Years ago we had to endure chain-smokers who blew smoke our way. But Deep Blue wasn't obnoxious, it was simply nothing at all, an empty chair not an opponent but something empty and relentless.''
    (Garry Kasparov, 1998)

    Kasparov writing on TechMate in Forbes (22/2/98) - a collection on super computing.

    ``All professions look bad in the movies ... why should scientists expect to be treated differently?''
    (Michael Crichton, 1999)

    Addressing the 1999 AAAS Meetings, and quoted in Science February 19, 1999, page 1111.

    ``the academy was a sort of club for retired Parisian scientists, happy to be able to come together once a week to talk about science for 2 hours after lunch and a little nap.''
    (Guy Ourison, January 1999)

    Inaugural speech as President to the French Academy of Science quoted in Science April 23, 1999, page 580.

    ``User-interface criticism is a genre to watch. It will probably be more influential and beneficial to the next century than film criticism was to the twentieth century. The twenty-first century will be filled with surprises, but one can safely count on it to bring more complexity to almost everything. Bearing the full brunt of that complexity, the great user-interface designers of the future will provide people with the means to understand and enrich their own humanity, and to stay human.''
    (Jaron Lanier, June 1999)

    From page 43 of Interface-off in The Sciences May/June 1999, pages 38-43.

    ``A real number complexity model appropriate for this context is given in the recent landmark work of Blum, Cucker, Shub and Smale*. In discussing their motivation for seeking a suitable theoretical foundation for modern scientific computing, where most of the algorithms are `real number algorithms' the authors of this work quote the following illuminating remarks of John von Neumann, made in 1948: ``There exists today a very elaborate system of formal logic, and specifically, of logic applied to mathematics. This is a discipline with many good sides but also serious weaknesses.... Everybody who has worked in formal logic will confirm that it is one of the technically most refactory parts of mathematics. The reason for this is that it deals with rigid, all-or-none concepts, and has very little contact with the continuous concept of the real or the complex number, that is with mathematical analysis. Yet analysis is the technically most successful and best-elaborated part of mathematics. Thus formal logic, by the nature of its approach, is cut off from the best cultivated portions of mathematics, and forced onto the most difficult mathematical terrain, into combinatorics.

    The theory of automata, of the digital, all-or-none type as discussed up to now, is certainly a chapter in formal logic. It would, therefore, seem that it will have to share this unattractive property of formal logic. It will have to be, from the mathematical point of view, combinatorial rather than analytical.''

    (*L. Blum, P. Cucker, M. Shub and S. Smale (1998), Complexity and Real Computation, Springer-Verlag, New York)

    Commentary thanks to Larry Nazareth

    ``Considerable obstacles generally present themselves to the beginner, in studying the elements of Solid Geometry, from the practice which has hitherto uniformly prevailed in this country, of never submitting to the eye of the student, the figures on whose properties he is reasoning, but of drawing perspective representations of them upon a plane. ...I hope that I shall never be obliged to have recourse to a perspective drawing of any figure whose parts are not in the same plane.''
    (Augustus De Morgan)

    Adrian Rice (What Makes a Great Mathematics Teacher?) from page 540 of The American Mathematical Monthly, June-July 1999

    ``In 1831, Fourier's posthumous work on equations showed 33 figures of solution, got with enormous labour. Thinking this is a good opportunity to illustrate the superiority of the method of W. G. Horner, not yet known in France, and not much known in England, I proposed to one of my classes, in 1841, to beat Fourier on this point, as a Christmas exercise. I received several answers, agreeing with each other, to 50 places of decimals. In 1848, I repeated the proposal, requesting that 50 places might be exceeded: I obtained answers of 75, 65, 63, 58, 57, and 52 places.''
    (Augustus De Morgan)

    Adrian Rice from page 542 of The American Mathematical Monthly, June-July 1999

    ``I think we need more institutes, but then you run into the question, Is it better to spend $2 million and have another institute or to fund another twenty-five or so researchers each year? It's a question of trying to keep the discipline alive and thriving. There's no doubt the really big ideas in mathematics come from maybe 5 percent of the people, but you need a broad base to nourish the 5 percent and to work out all the details as they move on to more adventuresome things. Look at, say, mathematicians at Group III universities. It's a rarity when they get funding. How do you keep them in the system? ... We're under terrific pressure to increase the size of our grants. If we did what the [National Science] board wants us to do, we would fund 800 people instead of 1,400. It's a question of whether DMS did the right thing when they pulled so many people down to one month of summer support. This took some of the pressure off the Foundation to put more money in mathematics. Suppose we funded 800 people. How much noise would it create? Would there be a march on Washington? I often think that's the way to go. See whether mathematicians would stand up for themselves or whether they'd just meekly accept. In chemistry, people get declined, and in two months they turn around with another proposal. Mathematicians --- they get declined twice, and they fold. I think mathematicians have such a personal investment in their problems that if you turn down their proposals, they take it as if you're judging them as mathematicians. They're not as flexible and often don't seem to be able to move to another class of problems. We fund proposals, not individuals.''
    (D. J. Lewis)

    Interview with Allyn Jackson from page 669 of The Notices of The AMS, June-July 1999

    ``Notices: After your time at the NSF, do you have any advice for the math community about what they should be doing to try to improve the funding for mathematics?

    Lewis: I don't think that up to this date they've made a very good case for why they should be funded. The bottom line is, What are you doing for the citizens of the country?

    Notices: When you say ``make the case,'' what do you mean concretely? Do groups of mathematicians have to descend on Capitol Hill?

    Lewis: They've got to do some demonstrations of what mathematics has accomplished for the good of society. One of the things mathematicians have done is education. For example, if mathematicians took seriously the job of training elementary and middle school teachers, they could make some claim that they really improve things. Also, science is getting so complicated, it can be done only with the help of mathematics. Is the math community willing to step up and participate?

    If so, they will have nonmathematicians making the case for greater funding of mathematics. It is always best to have outsiders make your case for you. Once upon a time I thought going to Capitol Hill would be effective. I don't think it will get very far if mathematicians go to Capitol Hill without the support of others. These days information technology and biology and medicine are the themes that echo well with the president and Congress.''

    (D. J. Lewis)

    Interview with Allyn Jackson from page 672 of The Notices of The AMS, June-July 1999

    ``The work then proceeds in a manner unique to science. Because practitioners publish their work electronically, through the e-print archives at the Los Alamos National Laboratory in New Mexico, the entire community can read a paper hours after its authors finish typing the last footnote. As a result, no one theorist or even a collaboration does definitive work. Instead, the field progresses like a jazz performance: A few theorists develop a theme, which others quickly take up and elaborate. By the time it's fully developed, a few dozen physicists, working anywhere from Princeton to Bombay to the beaches of Santa Barbara, may have played important parts.''

    (Gary Taubes)

    From String Theorists Find a Rosetta Stone on page 513 of Science, 23rd July, 1999

    'where almost one quarter hour was spent, each beholding the other with admiration before one word was spoken: at last Mr. Briggs began "My Lord, I have undertaken this long journey purposely to see your person, and to know by what wit or ingenuity you first came to think of this most excellent help unto Astronomy, viz. the Logarithms: but my Lord, being by you found out, I wonder nobody else found it out before, when now being known it appears so easy." '
    (Henry Briggs, 1617)

    Briggs, later the first Savelian Professor of Geometry in Oxford, is describing his first meeting with Napier whom he had traveled from London to Edinburgh to meet. From H.W. Turnbull's The Great Mathematicians, Methuen, 1929.

    ``Far better an approximate answer to the right question, which is often vague, than the exact answer to the wrong question, which can always be made precise.''
    (J. W. Tuckey, 1962)

    From the Annals of Mathematical Statistics , Volume 33. Compare the 1964 Feynman quote above!

    `` One of the beauties of learning is that it admits its provisionality, its imperfections. This scholarly scrupulousness, this willingness to admit that even the best-supported of theories is still a theory, is now being exploited by the unscrupulous. But that we do not know everything does not mean we know nothing. Not all theories are of equal weight. The moon, even the moon over Kansas, is not made of green cheese. Genesis, as a theory, is bunk.

    If the overabundant new knowledge of the modern age is, let's say, a tornado, then Oz is the extraordinary, Technicolored new world in which it has landed us, the world from which --- life not being a movie --- there is no way home. In the immortal words of Dorothy Gale, `Toto, something tells me we're not in Kansas any more.' To which one can only add: Thank goodness, baby, and amen.''

    (Salman Rushdie)

    From his article "Locking out that disruptive Darwin fellow" in the Globe and Mail , September 2, 1999

    ``The mental maps, gave rise to industries that could not have been predicted, and created a new class of technological workers whom wise societies took pains to nurture. Are we about to go through this process again? A renowned social analyst and management philosopher looks to history for insights.''
    (Peter Drucker)

    Beyond the Information Revolution in The Atlantic Monthly Online November 3, 1999

    ``When the facts change, I change my mind. What do you do, sir?''
    (John Maynard Keynes)

    Quoted in The Economist, December 18 1999, page 47

    ``Look miss, if I disagree with Darwin, he's not going to send me to hell.''
    (An anonymous student's "Pascal wager-style" rationale)

    Quoted in The Globe and Mail, January 1, 2000, page D22 by Laura Penny describing a first year University class in Buffalo in which one third of the students were creationists.

    `` Most working scientists may be naive about the history of their discipline and therefore overly susceptible to the lure of objectivist mythology. But I have never met a pure scientific realist who views social context as entirely irrelevant, or only as an enemy to be expunged by the twin lights of universal reason and incontrovertible observation. And surely, no working scientist can espouse pure relativism at the other pole of the dichotomy. (The public, I suspect, misunderstands the basic reason for such exceptionless denial. In numerous letters and queries, sympathetic and interested nonprofessionals have told me that scientists cannot be relativists because their commitment to such a grand and glorious goal as the explanation of our vast and mysterious universe must presuppose a genuine reality "out there" to discover. In fact, as all working scientists know in their bones, the incoherence of relativism arises from virtually opposite and much more quotidian motives. Most daily activity in science can only be described as tedious and boring, not to mention expensive and frustrating. Thomas Edison was just about right in his famous formula for invention as 1% inspiration mixed with 99% perspiration. How could scientists ever muster the energy and stamina to clean cages, run gels, calibrate instruments, and replicate experiments, if they did not believe that such exacting, mindless, and repetitious activities can reveal truthful information about a real world? If all science arises as pure social construction, one might as well reside in an armchair and think great thoughts.)

    Similarly, and ignoring some self-promoting and cynical rhetoricians, I have never met a serious social critic or historian of science who espoused anything close to a doctrine of pure relativism. The true, insightful, and fundamental statement that science, as a quintessentially human activity, must reflect a surrounding social context does not imply either that no accessible external reality exists, or that science, as a socially embedded and constructed institution, cannot achieve progressively more adequate understanding of nature's facts and mechanisms. ''

    (Stephen J. Gould)

    From the article: 'Deconstructing the "Science Wars" by Reconstructing an Old Mold' in Science, Jan 14, 2000: 253-261.

    `` caused Thorstein Veblen to comment acerbically in 1908 that "business principles" were transforming higher education into "a merchantable commodity, to be produced on a piece-rate plan, rated, bought, and sold by standard units, measured, counted and reduced to staple equivalence by impersonal, mechanical tests. ''


    ``New products and new processes do not appear full-grown," Vannevar Bush, President Franklin Roosevelt's chief science adviser, declared in 1944. "They are founded on new principles and new conceptions, which in turn are painstakingly developed by research in the purest realms of science." ''

    (Eyal Press and Jennifer Washburn)

    From The Kept University in The Atlantic Monthly Online, March 2000. Which quote better reflects Science in 2001?

    ``Most important to Fox was a young instructor who had arrived at Cornell two years before from Williams and Mary. William Lloyd Garrison Williams had written his Ph.D thesis under Leonard Dickson at Chicago in 1920. Born in Friendship, Kansas, Williams, who was named for the famous abolitionist William LLoyd Garrison, attended a small Quaker school in Indiana, taught school briefly in North Dakota and then attended Haverford College where he earned a B.A. degree. From 1910-13 he attended Oxford University as a Rhodes Scholar, and after receving a B.A. and M.A., he took a faculty position at Miami University of Ohio. His Ph.D. work at Chicago was done during the summers. He also taught briefly at Gettysburg College and William and Mary before coming to Cornell.

    In 1924, Williams moved to McGill University, where he helped develop the graduate program. He was the founder and organizer of the Canadian Mathematical Congress, the first meeting of which was in Montreal in 1945. Nearly all of the support the Congress was able to acquire was due to William's efforts (see W.L.G. Williams, 1888-1976, G. De B. Robinson, Proc. Royal Society of Canada, 1976). A man of remarkable ability and compassion, Williams took a strong personal interest in his fellowman. A lifelong member of the Society of Friends, he was a tireless worker for Quaker causes.''

    (James A. Donaldson and Richard J. Fleming)

    From "Elbert F. Fox: An Early Pioneer", American Math Monthly 107 (2000) 105-128.

    Gravity Turntable Sets New Record

    `LONG BEACH, CALIFORNIA--Scientists have been scrutinizing gravity since the time of Newton, but they've had difficulty measuring the power of its pull. Now, thanks to a clever device, physicists have the most precise measurement yet.


    "[It] should have been obvious" that previous measures of big G were off, says physicist Randy Newman of the University of California, Irvine. The new result, announced this week at the American Physical Society meeting, sets big G tentatively at 6.67423 0.00009 x 10-11 m3/(kg s2). "It's one of the fundamental constants," Gundlach says. "Mankind should just know it. It's a philosophical thing."'

    From ScienceNow May 5, 2000.

    ``Imagine Dostoyevsky. There are some incidents like this, two boys killing other children, in his famous diary. Imagine what Dostoyevsky would do with that. He would deal with the transcendentally important question of evil in the child. Today the editor would say, "Fyodor, tomorrow, please, your piece. Don't tell me you need ten months for thinking. Fyodor, tomorrow!" "

    (George Steiner)

    Quoted in James Gleick's Faster (Pantheon 1999), pages 97-88, on instant opinion -- sound bites and 'hurry sickness'.

    ``So my reaction surprises me. I tell Natalie that math is important and relevant and that I wished I'd made the effort to understand. I wish somebody had found a way of making sense of it all. This revelation comes from reading a stack of magazines about the future, about computers and artificial intelligence, cars and planes, food production and global warming. And I have come to the conclusion that Mr. Kool was right.

    Math has something to do with calculations, formulas, theories and right angles. And everything to do with real life. Mathematicians not only have the language of the future (they didn't send Taming of the Shrew into space, just binary blips) but they can use it to predict when Andromeda will perform a cosmic dance with the Milky Way. It's mathematicians who are designing the intelligent car that knows when you're falling asleep at the wheel or brakes to avoid an accident. It can predict social chaos and the probability of feeding billions. It even explains the stock market and oil prices.''


    Paulette Bourgeois lives in Toronto where she is calculating the probability of ever balancing her chequebook. She is the author of the Franklin the Turtle books for children.

    (Paulette Bourgeois)

    Quoted from "The Numbers Game," The Globe and Mail July 13, 2000, page A14.

    `` Mathematics is the language of high technology. Indeed it is, but I think it is also becoming the eyes of science.''

    (Tom Brzustowski, NSERC President)

    Addressing the MITACS NCE annual general meeting June 6, 2000.

    ``This is fundamentally wrong. We are not entering a time when copyright is more threatened than it is in real space. We are instead entering a time when copyright is more effectively protected than at any time since Gutenberg. The power to regulate access to and use of copyrighted material is about to be perfected. Whatever the mavens of the mid-1990s may have thought, cyberspace is about to give the holders of copyrighted property the biggest gift of protection they have ever known.

    In such an age -- in a time when the protections are being perfected -- the real question for law is not, how can law aid in that protection? but rather, is that protection too great? The mavens were right when they predicted that cyberspace will teach us that everything we thought about copyright was wrong. But the lesson in the future will be that copyright is protected far too well. The problem will center not on copy-right but on copy-duty -- the duty of owners of protected property to make that property available.''

    (Lawrence Lessig)

    Quoted from page 127 of his book: "Code and other laws of Cyberspace", Basic Books, 1999.

    ``An informed list of the most profound scientific developments of the 20th century is likely to include general relativity, quantum mechanics, big bang cosmology, the unraveling of the genetic code, evolutionary biology, and perhaps a few other topics of the reader's choice. Among these, quantum mechanics is unique because of its profoundly radical quality. Quantum mechanics forced physicists to reshape their ideas of reality, to rethink the nature of things at the deepest level, and to revise their concepts of position and speed, as well as their notions of cause and effect. ''

    (Daniel Kleppner and Roman Jackiw)

    Quoted from the article "One Hundred Years of Quantum Physics" in Science August 11, pages 893-898.

    ``A wealthy (15th Century) German merchant, seeking to provide his son with a good business education, consulted a learned man as to which European institution offered the best training. "If you only want him to be able to cope with addition and subtraction," the expert replied, "then any French or German university will do. But if you are intent on your son going on to multiplication and division -- assuming that he has sufficient gifts -- then you will have to send him to Italy.''

    (Georges Ifrah)

    From page 577 of "The Universal History of Numbers: From Prehistory to the Invention of the Computer", translated from French, John Wiley, 2000. (Emphasizing quite how great an advance positional notation was!)

    ``2000 was a banner year for scientists deciphering the "book of life"; this year saw the completion of the genome sequences of complex organisms ranging from the fruit fly to the human.

    Genomes carry the torch of life from one generation to the next for every organism on Earth. Each genome--physically just molecules of DNA--is a script written in a four-letter alphabet. Not too long ago, determining the precise sequence of those letters was such a slow, tedious process that only the most dedicated geneticist would attempt to read any one "paragraph"--a single gene. But today, genome sequencing is a billion-dollar, worldwide enterprise. Terabytes of sequence data generated through a melding of biology, chemistry, physics, mathematics, computer science, and engineering are changing the way biologists work and think. Science marks the production of this torrent of genome data as the Breakthrough of 2000; it might well be the breakthrough of the decade, perhaps even the century, for all its potential to alter our view of the world we live in.''

    (Elizabeth Pennisi)

    From ''BREAKTHROUGH OF THE YEAR: Genomics Comes of Age.'' Cover story in Science of December 22, 2000.

    "Not until the creation and maintenance of decent conditions of life for all people are recognized and accepted as a common obligation of all people and all countries - not until then shall we, with a certain degree of justification, be able to speak of humankind as civilized."

    (Albert Einstein, 1945)

    "Capitalism is the extraordinary belief that the nastiest of men, for the nastiest of reasons, will somehow work for the benefit of us all."

    (John Maynard Keynes)

    "When we have before us a fine map, in which the line of the coast, now rocky, now sandy, is clearly indicated, together with the winding of the rivers, the elevations of the land, and the distribution of the population, we have the simultaneous suggestion of so many facts, the sense of mastery over so much reality, that we gaze at it with delight, and need no practical motive to keep us studying it, perhaps for hours altogether. A map is not naturally thought of as an aesthetic object... And yet, let the tints of it be a little subtle, let the lines be a little delicate, and the masses of the land and sea somewhat balanced, and we really have a beautiful thing; a thing the charm of which consists almost entirely in its meaning, but which nevertheless pleases us in the same way as a picture or a graphic symbol might please. Give the symbol a little intrinsic worth of form, line and color, and it attracts like a magnet all the values of things it is known to symbolize. It becomes beautiful in its expressiveness."

    (George Santayana}

    From "The Sense of Beauty", 1896.

    "If my teachers had begun by telling me that mathematics was pure play with presuppositions, and wholly in the air, I might have become a good mathematician, because I am happy enough in the realm of essence. But they were overworked drudges, and I was largely inattentive, and inclined lazily to attribute to incapacity in myself or to a literary temperament that dullness which perhaps was due simply to lack of initiation."

    (George Santayana}

    From pp. 238-9 "Persons and Places", 1945.

    "He designed and built chess-playing, maze-solving, juggling and mind-reading machines. These activities bear out Shannon's claim that he was more motivated by curiosity than usefulness. In his words `I just wondered how things were put together.' "

    (Claude Shannon)

    From Claude Shannon's (1916-2001) obituary.

    "The price of metaphor is eternal vigilance"

    (Arturo Rosenblueth and Norbert Wiener)

    Quoted by R. C. Leowontin, in Science page 1264, Feb 16, 2001 (The Human Genome Issue).

    "What is particularly ironic about this is that it follows from the empirical study of numbers as a product of mind that it is natural for people to believe that numbers are not a product of mind!"

    (George Lakoff and Rafael E. Nunez)

    On page 81 of Where Mathematics Comes From, Basic Books, 2000.

    Recent Discoveries about the Nature of Mind. In recent years, there have been revolutionary advances in cognitive science ---- advances that have a profound bearing on our understanding of mathematics. Perhaps the most profound of these new insights are the following:
    1. The embodiment of mind. The detailed nature of our bodies, our brains and our everyday functioning in the world structures human concepts and human reason. This includes mathematical concepts and mathematical reason.
    2. The cognitive unconscious. Most thought is unconscious --- not repressed in the Freudian sense but simply inaccessible to direct conscious introspection. We cannot look directly at our conceptual systems and at our low-level thought processes. This includes most mathematical thought.
    3. Metaphorical thought. For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in sensory-motor systems. The mechanism by which the abstract is comprehended in terms of the concept is called conceptual metaphor. Mathematical thought also makes use of line."

    (George Lakoff and Rafael E. Nunez)

    On page 5 of Where Mathematics Comes From, Basic Books, 2000.

    "The early study of Euclid made me a hater of geometry."

    (James Joseph Sylvester, 1814-97, Second LMS President)

    quoted in D. MacHale, "Comic Sections" (Dublin 1993).

    "a thrill which is indistinguishable from the thrill I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me the austere beauty of the four statues representing 'Day', 'Night', 'Evening', and 'Dawn' which Michelangelo has set over the tomb of Guiliano de'Medici and Lorenzo de'Medici."

    (G. N. Watson, 1886-1965)

    "All physicists and a good many quite respectable mathematicians are contemptuous about proof."

    (G. H. Hardy, 1877-1947)

    A century after biology started to think physically:

    "The idea that we could make biology mathematical, I think, perhaps is not working, but what is happening, strangely enough, is that maybe mathematics will become biological,!"

    (Greg Chaitin, Interview, 2000.)
    "The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology, and all of them the physicist can more or less easily read and adequately solve."

    (D'Arcy Thompson, ``On Growth and Form'' 1917)

    In Philip Ball's "The Self-Made Tapestry: Pattern Formation in Nature,''

    "A doctorate compels most of us to be detailed and narrow, and to carve out our own specialities, and tenure commitees rarely like boldness. Later, when our jobs are safe we can be synthetic, and generalize."

    (Paul Kennedy)

    Writing critically about A.J.P. Taylor (`The Nonconformist') in the Atlantic Monthly April 2001, page 114.

    ``... it is no doubt important to attend to the eternally beautiful and true. But it is more important not to be eaten."

    (Jerry Fodor)

    Distinguishing effortless early learning of language and social customs from later labourious general purpose concept acquisition, Egan writes:
    "The bad news is that our evolution equipped us to live in small, stable, hunter-gatherer societies. We are Pleistocene people, but our languaged brains have created massive, multicultural, technologically sophisticated and rapidly changing societies for us to live in."
    "The cement like learning of our early years can accomodate almost anything, then it fixes and becomes almost unmovable."
    "we can, as a result, change our earlier beliefs and commitments. We also know this is difficult for most people."

    (Kieran Egan)

    In Kieran Egan's, Getting it Wrong from the Beginning -- Major Mistakes in the Project to Educate Everybody (in press).

    This is what Albert Einstein said quoting Max Planck
    "...a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents die and a new generation grows up that's familiar with it."
    or ...
    "A new scientific truth usually does not make its way in the sense that its opponents are persuaded and declare themselves enlightened, but rather that the opponents become extinct and the rising generation was made familiar with the truth from the very beginning".

    Max Planck, in THE QUANTUM BEAT by F.G.Major, Springer (1998).

    "And Max Planck, surveying his own career in his Scientific Autobiography, sadly remarked that 'a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.'"
    (Thomas Kuhn)

    On page 151 of T.S. Kuhn, The Structure of Scientific Revolutions, 3rd ed., Univ. of Chicago Press, 1996. (Quoting: Max Planck, Scientific Autobiography and Other Papers, trans. F. Gaynor (New York, 1949), pp. 33-34. See also "Conversations with a Mathematician" by Greg Chaitin.)

    ``the idea that we could make biology mathematical, I think, perhaps is not working, but what is happening, strangely enough, is that maybe mathematics will become biological, not that biology will become mathematical, mathematics may go in that direction!"

    (Interview with Gregory Chaitin by Hans-Ulrich Obrist (Musee d'Art Moderne de la Ville de Paris), Paris/CDG Airport, October 2000.)


    "The message is that mathematics is quasi-empirical, that mathematics is not the same as physics, not an empirical science, but I think it's more akin to an empirical science than mathematicians would like to admit."

    "Mathematicians normally think that they possess absolute truth. They read God's thoughts. They have absolute certainty and all the rest of us have doubts. Even the best physics is uncertain, it is tentative. Newtonian science was replaced by relativity theory, and then---wrong!---quantum mechanics showed that relativity theory is incorrect. But mathematicians like to think that mathematics is forever, that it is eternal. Well, there is an element of that. Certainly a mathematical proof gives more certainty than an argument in physics or than experimental evidence, but mathematics is not certain. This is the real message of Godel's famous incompleteness theorem and of Turing's work on uncomputability."

    "You see, with Godel and Turing the notion that mathematics has limitations seems very shocking and surprising. But my theory just measures mathematical information. Once you measure mathematical information you see that any mathematical theory can only have a finite amount of information. But the world of mathematics has an infinite amount of information. Therefore it is natural that any given mathematical theory is limited, the same way that as physics progresses you need new laws of physics."

    "Mathematicians like to think that they know all the laws. My work suggests that mathematicians also have to add new axioms, simply because there is an infinite amount of mathematical information. This is very controversial. I think mathematicians, in general, hate my ideas. Physicists love my ideas because I am saying that mathematics has some of the uncertainties and some of the characteristics of physics. Another aspect of my work is that I found randomness in the foundations of mathematics. Mathematicians either don't understand that assertion or else it is a nightmare for them... ":

    "This skyhook-skyscraper construction of science from the roof down to the yet unconstructed foundations was possible because the behaviour of the system at each level depended only on a very approximate, simplified, abstracted characterization at the level beneath1. This is lucky, else the safety of bridges and airplanes might depend on the correctness of the "Eightfold Way" of looking at elementary particles.

    1 ... More than fifty years ago Bertrand Russell made the same point about the architecture of mathematics. See the "Preface" to Principia Mathematica "... the chief reason in favour of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question allows us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reason rather for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." Contemporary preferences for deductive formalisms frequently blind us to this important fact, which is no less true today than it was in 1910."

    (Herbert A. Simon)

    On page 16 of ``The Sciences of the Artificial," MIT Press, 1996.

    " Hardy `asked `What's your father doing these days. How about that esthetic measure of his?' I replied that my father's book was out. He said, 'Good, now he can get back to real mathematics'."

    (Garret Birkoff)

    Quoted in Towering Figures, 1890-1950, by David E. Zitarelli on page 618 of MAA Monthly Aug-Sept, Vol 108, (2001), 606-635 : regarding G. D. Birkhoff's Aesthetic Measures (1933).

    "I DO CONSIDER it appropriate to pay one's tribute to Prof. Subramanyan Chandrasekhar at the outset, before taking a plunge into the aesthetics of macro-causality, based on his book Truth and Beauty: Aesthetics and Motivations in Science. Brought up on the refined diet of music, mathematics and aesthetics, Chandrasekhar's own writing is probably the most appropriate mirror of his personality. I quote: "When Michelson was asked towards the end of his life, why he had devoted such a large fraction of his time to the measurement of the velocity of light, he is said to have replied 'It was so much fun'." Prof. Chandrasekhar goes on to some length to explain the term quoting even the Oxford Dictionary -- "fun" means "drollery", what Michelson really meant, Chandrasekhar asserts is "pleasure" and "enjoyment" - evidently "fun" in the colloquial sense, a concept, so familiar in our so called ordinary life has no place in Chandrasekhar's dictionary..."

    (Bikash Sinha)


    " `His peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. I fancy his preeminence is due to his muscles of intuition being the strongest and most enduring with which a man has ever been gifted. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one's mind and apply all one's powers of concentration to piercing through it, and how it will dissolve and escape and you find that what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret. Then being a supreme mathematical technician he could dress it up, how you will, for purposes of exposition, but it was his intuition which was pre-eminently extraordinary---"so happy in his conjectures", said de Morgan, "as to seem to know more than he could possibly have any means of proving."'-- J. M. Keynes 1956


    If Edison, Fineman, Gauss, and Newton had all been intensely tutored from the age of three by brilliant parents, as J.S. Mill was, then I might at least consider the possibility that my own mental muscles might have been stronger if my own parents had been more demanding. But they were not and I will not. `When you see [Edison's] mind at play in his notebooks, the sheer multitude and richness of his ideas makes you recognize that there is something that can't be understood easily---that we may never be able to understand.' (historian Paul Israel, quoted in McAuliffe 1995). I think what lies at the heart of these mysteries is genetic, probably emergenic. The configuration of traits of intellect, mental energy, and temperament with which, during the plague years of 1665--6, Isaac Newton revolutionized the world of science were, I believe, the consequence of a genetic lottery that occurred about nine months prior to his birth, on Christmas day, in 1642.


    Gauss's second son, Eugene, emigrated to the United States in 1830, enlisted in the army, and later went into business in Missouri. Eugene is said to have had some of his father's gift for languages and the ability to perform prodigious arithmetic calculations, which he did for recreation after his sight failed him in old age. "

    (David T. Lykken)


    `For Poincaré, ignoring the emotional sensibility, even in mathematical demonstrations "would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility" (p. 2047).'

    (Nathalie Sinclair)

    Quoting Henri Poincaré's "Mathematical creation" (1956). In J. Newman (Ed.), The World of Mathematics ( pp. 2041-2050). Simon and Schuster.

    "The controversy between those who think mathematics is discovered and those who think it is invented may run and run, like many perennial problems of philosophy. Controversies such as those between idealists and realists, and between dogmatists and sceptics, have already lasted more than two and a half thousand years. I do not expect to be able to convert those committed to the discovery view of mathematics to the inventionist view. However what I have shown is that a better case can be put for mathematics being invented than our critics sometimes allow. Just as realists often caricature the relativist views of social constructivists in science, so too the strengths of the fallibilist views are not given enough credit. For although fallibilists believe that mathematics has a contingent, fallible and historically shifting character, they also argue that mathematical knowledge is to a large extent necessary, stable and autonomous. Once humans have invented something by laying down the rules for its existence, like chess, the theory of numbers, or the Mandelbrot set, the implications and patterns that emerge from the underlying constellation of rules may continue to surprise us. But this does not change the fact that we invented the game in the first place. It just shows what a rich invention it was. As the great eighteenth century philosopher Giambattista Vico said, the only truths we can know for certain are those we have invented ourselves. Mathematics is surely the greatest of such inventions."

    (Paul Ernst)

    From Is Mathematics Discovered or Invented? (THES, 1996 and after).

    " Who owns the Internet? Until recently, nobody. That's because, although the Internet was "Made in the U.S.A.," its unique design transformed it into a resource for innovation that anyone in the world could use. Today, however, courts and corporations are attempting to wall off portions of cyberspace. In so doing, they are destroying the Internet's potential to foster democracy and economic growth worldwide. "

    (Lawrence Lessig)

    From Who Owns The Internet? Foreign Policy, November-December 2001.

    "Predicting the future is an activity fraught with error. Wilbur Wright, co-inventor of the motorized airplane that successfully completed the first manned flight in 1903, seems to have learned this lesson when he noted: "In 1901, I said to my brother Orville that man would not fly for 50 years. Ever since I have ... avoided predictions." Despite the admonition of Wright, faulty future forecasting seems a favored human pastime, especially among those who would presumably avoid opportunities to so easily put their feet in their mouths.

    What follows are some of the more striking exemplars of expert error in forecasting the future of technological innovations.

    "Louis Pasteur's theory of germs is ridiculous fiction." -- Piem Pachet, Professor of Physiology, 1872

    "The abdomen, the chest, and the brain will forever be shut from the intrusion of the wise and humane surgeon." -- Sir John Eric Erickren, British surgeon to Queen Victoria, 1873

    "Radio has no future. Heavier than air flying machines are impossible. X-rays will prove to be a hoax." -- William Thomson (Lord Kelvin), English physicist and inventor, 1899

    "There is not the slightest indication that nuclear energy will ever be obtainable. It would mean that the atom would have to be shattered at will." -- Albert Einstein, 1932

    "Man will never reach the moon, regardless of all future scientific advances." -- Lee De Forest, Radio pioneer, 1957

    Computers and information technologies seem to hold a special place in the forecasters' hall of humiliation, be they predictions from the media, business, politicians, scientists, or technologists. Here are some examples:

    "This 'telephone' has too many shortcomings to be seriously considered as a means of communication. The device is inherently of no value to us." -- Western Union internal memo, 1876

    "I think there is a world market for maybe five computers." -- Thomas Watson, chair of IBM, 1943

    "The problem with television is that people must sit and keep their eyes glued on a screen; the average American family hasn't time for it." -- New York Times, 1949

    "Where ... the ENIAC is equipped with 18,000 vacuum tubes and weights 30 tons, computers in the future may have only 1,000 vacuum tubes and weigh only 1.5 tons." -- Popular Mechanics, 1949

    "Folks, the Mac platform is through -- totally." -- John C. Dvorak, PC Magazine, 1998

    "There is no reason anyone would want a computer in their home." -- Ken Olson, president, chairman and founder, Digital Equipment Corp, 1977

    "640K ought to be enough for anybody." -- Attributed to Bill Gates, Microsoft chair, 1981

    "By the turn of this century, we will live in a paperless society." -- Roger Smith, chair of General Motors, 1986

    "I predict the Internet ... will go spectacularly supernova and in 1996 catastrophically collapse." -- Bob Metcalfe, 3Com founder and inventor, 1995

    "Credit reports are particularly vulnerable ... [as] are billing, payroll, accounting, pension and profit-sharing programs." -- Leon A Kappelman [author of this article] on likely Y2K problems, 1999 "

    (Leon A Kappelman)

    From "The Future is Ours," Communications of the ACM, March 2001, pg. 46.

    " Computation with Roman numerals is certainly algorithmic - it's just that the algorithms are complicated.

    In 1953, I had a summer job at Bell Labs in New Jersey (now Lucent), and my supervisor was Claude Shannon (who has died only very recently). On his desk was a mechanical calculator that worked with Roman numerals. Shannon had designed it and had it built in the little shop Bell Labs had put at his disposal. On a name plate, one could read that the machine was to be called: Throback I.

    Martin from a foggy morning in Berkeley"

    (Martin Davis)

    Martin Davis, Visiting Scholar UC Berkeley, Professor Emeritus, NYU. Following up on queries on the Historia Mathematica list, Jan 12, 2002.

    " If nature has made any one thing less susceptible than all others of exclusive property, it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of everyone, and the receiver cannot dispossess himself of it. [2] Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it. He who receives an idea from me, receives instruction himself without lessening mine; as he who lites his taper at mine, receives light without darkening me. [3] That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, when she made them, like fire, expansible over all space, without lessening their density at any point, and like the air in which we breathe, move, and have our physical being, incapable of confinement, or exclusive appropriation. [4] Inventions then cannot, in nature, be a subject of property. "

    (Thomas Jefferson)

    Letter from Thomas Jefferson to Issac McPherson (August 13, 1813), in The Writings of Thomas Jefferson 6 quoted from page 94 of the future of ideas by Lawrence Lessig, Random House, 2001.

    "The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. 'Mathematizing' may well be a creative activity of man, like language or music, of primary originality, whose Historical decisions defy complete objective rationalisation."

    (Hermann Weyl)

    In "Obituary: David Hilbert 1862 - 1943", RSBIOS, 4, 1944, pp. 547 - 553; and American Philosophical Society Year Book, 1944, pp. 387 - 395, p. 392.

    "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate."

    (Bertrand Russell)

    From "Recent Work on the Principles of Mathematics in International Monthly, 4 (July, 1901), 83-101. (Collected Papers, v3, p.366; revised version in Newman's World of Mathematics, v3, p. 1577.)

    "The problems of mathematics are not problems in a vacuum. There pulses in them the life of ideas which realize themselves in concreto through our [or throught] human endeavors in our historical existence, but forming an indissoluble whole transcending any particular science."

    (Hermann Weyl)

    In "David Hilbert and his mathematical work," Bull. Am. Math. Soc., 50 (1944), p. 615.

    THE FUTURE OF E-PUBLISHING. Although e-publishing has suffered a series of setbacks this year, Wired magazine still found plenty of optimism about the future of e-books. Michael S. Hart of Project Guttenberg, which offers books in electronic form, says: "The number of e-books available for free download on the Net will pass 20,000. The number of Net users will start heading towards 1 billion." Librarian Cynthia Orr, a co-founder of, thinks e-publishers should pay more attention to libraries, and says that if the major publishers worked with librarians or distributors "to figure out how to let libraries purchase or license their e-books, and let readers 'check them out' for free," that would help build "a market that otherwise threatens to just collapse for lack of interest. Librarians have been careful defenders of copyright over the years ... and our budgets are far higher than they realize." And Mark Gross, president of Data Conversion Laboratory, thinks that the e-publishing has already won a stealth war: ""What people forget is e-books were going strong before they were called e-books and they went on to sweep into many aspects of business and publishing. Most of this has gone unnoticed by the media. Probably because it has been a kind of backdoor revolution. To cite one example: Print law books are just about gone. People don't use them in law firms anymore. It's all electronic books or online. A revolution has occurred, but no one's noticed."
    (Wired Magazine)
    Wired, December 25, 2001.

    "Dear brother;

    I have often been surprised that Mathematics, the quintessence of Truth, should have found admirers so few and so languid. Frequent consideration and minute scrutiny have at length unravelled the cause; viz. that though Reason is feasted, Imagination is starved; while Reason is luxuriating in its proper Paradise, Imagination is wearily travelling on a dreary desert. To assist Reason by the stimulus of Imagination is the design of the following production."

    Samuel Taylor Coleridge then launches into an ode on mathematics, the first verses of which are as follows:

    " On a given finite line
    Which must no way incline;
    To describe an equi -
    - lateral tri -
    A -N -G -L -E.
    Now let AB
    Be the given line
    Which must no way incline;
    The great Mathematician
    Makes this requisition,
    That we describe an Equi -
    - lateral Tri -
    - angle on it;
    Aid us, Reason - aid us, Wit!

    From the centre A at the distance AB,
    Describe the circle BCD
    At the distance BA from B the centre
    The round ACE to describe boldly venture.
    (Third postulate see)
    And from the point C
    In which the circles make a pother
    cutting and slashing one another
    Bid the straight lines a journeying go.
    CACB those lines will show
    To the points, which by AB are reckoned
    And postulate the second
    For authority you know
    ABC Triumphant shall be
    An equilateral Triangle
    No Peter Pindar carp, nor Zoilus can wrangle."

    (Samuel Coleridge)

    In a letter to his brother the Reverend George Coleridge.

    "There is a story, no doubt exaggerated, that the Pope once remarked that two types of proposals exist for peace in the Middle East: The realistic and the miraculous. The realistic solution is divine intervention. The miraculous involves a voluntary agreement between the two sides."

    (Paul Adams)

    From his article "Israel, Palestinians now further apart than two years ago" in the The Globe and Mail, Monday, April 15,2002

    "Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts. It should be to us a guidepost on the mazy path to hidden truths, and ultimately a reminder of our pleasure in the successful solution.


    Besides it is an error to believe that rigor in the proof is the enemy of simplicity."

    (David Hilbert)

    In his `23' Mathematische Probleme lecture to the Paris International Congress, 1900 (see Yandell's, fine account in The Honors Class, A.K. Peters, 2002).

    ``... waved his manuscript and confessed his publishing woes. ... "I said, 'I'm afraid no one's going to get to read these words. And I love these words.'"

    Ann Sparanese, a librarian in the audience, sent an SOS over the Internet to fellow librarians. Within hours, they inundated HarperCollins with angry e-mails - and orders for Stupid White Men. Some also threatened a boycott.

    "Those librarians," says Moore, ... "That's one terrorist group you don't want to mess with."

    HarperCollins caved.


    (Jan Wong)

    Quoted from "Lunch with Michael Moore - A smart white guy with attitude," The Globe and Mail May 18, 2002, page F2.

    `` Old ideas give way slowly; for they are more than abstract logical forms and categories. They are habits, predispositions, deeply engrained attitudes of aversion and preference. Moreover, the conviction persists-though history shows it to be a hallucination that all the questions that the human mind has asked are questions that can be answered in terms of the alternatives that the questions themselves present. But in fact intellectual progress usually occurs through sheer abandonment of questions together with both of the alternatives they assume an abandonment that results from their decreasing vitality and a change of urgent interest. We do not solve them: we get over them.

    Old questions are solved by disappearing, evaporating, while new questions corresponding to the changed attitude of endeavor and preference take their place. Doubtless the greatest dissolvent in contemporary thought of old questions, the greatest precipitant of new methods, new intentions, new problems, is the one effected by the scientific revolution that found its climax in the " Origin of Species." ``

    (John Dewey)

    Quoted from The Influence of Darwin on Philosophy, 1910.

    ``The first [axiom] said that when one wrote to the other (they often preferred to exchange thoughts in writing instead of orally), it was completely indifferent whether what they said was right or wrong. As Hardy put it, otherwise they could not write completely as they pleased, but would have to feel a certain responsibility thereby. The second axiom was to the effect that, when one received a letter from the other, he was under no obligation whatsoever to read it, let alone answer it, - because, as they said, it might be that the recipient of the letter would prefer not to work at that particular time, or perhaps that he was just then interested in other problems.... The third axiom was to the effect that, although it did not really matter if they both thought about the same detail, still, it was preferable that they should not do so. And, finally, the fourth, and perhaps most important axiom, stated that it was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name; otherwise there would constantly arise quarrels and difficulties in that now one, and now the other, would oppose being named co-author.''
    (Harald Bohr)

    Hardy and Littlewood's Four Axioms for Collaboration quoted from the preface of Bella Bollobas' 1988 edition of Littlewood's Miscellany. (Other quotes from the Miscellany.)

    "I got into a research project which can be very simply described as concerned with the realization of the "Nash program" (making use of words made conventional by others that refer to suggestions I had originally made in my early works in game theory).

    In this project a considerable quantity of work in the form of calculations has been done up to now. Much of the value of this work is in developing the methods by which tools like Mathematica can be used with suitable special programs for the solution of problems by successive approximation methods."

    (John Nash)

    On page 241 of "The Essential John Nash", edited by Harold W. Kuhn and Sylvia Nasar, Princeton Univ. Press, 2001.

    "A proof is a proof. What kind of a proof? It's a proof. A proof is a proof. And when you have a good proof, it's because it's proven."
    (Jean Chretien)

    The Canadian Prime Minister explaining Canada's conditions for determining if Iraq has complied, September 5, 2002. Sounds a lot like Bertrand Russell!

    "No man can worthely praise Ptolemye ... yet muste ye and all men take heed, that both in him and in all mennes workes, you be not abused by their autoritye, but evermore attend to their reasons, and examine them well, ever regarding more what is saide, and how it is proved, than who saieth it, for autorite often times deceaveth many menne."
    (Robert Record)

    The great textbook writer in his cosmology text `The castle of knowledge' (1556) quoted on page 47 of Oxford Figures, Oxford University Press, 2000.

    "The future has arrived; it's just not evenly distributed."
    (Douglas Gibson)

    On his Vancouver home page.

    "The plural of 'anecdote' is not 'evidence'."
    (Alan L. Leshner)

    Science's publisher speaking at the Federal S&T Forum, Oct 2, 2002.

    " ... Several years ago I was invited to contemplate being marooned on the proverbial desert island. What book would I most wish to have there, in addition to the Bible and the complete works of Shakespeare? My immediate answer was: Abramowitz and Stegun's Handbook of Mathematical Functions. If I could substitute for the Bible, I would choose Gradsteyn and Ryzhik's Table of Integrals, Series and Products. Compounding the impiety, I would give up Shakespeare in favor of Prudnikov, Brychkov And Marichev's of Integrals and Series ... On the island, there would be much time to think about waves on the water that carve ridges on the sand beneath and focus sunlight there; shapes of clouds; subtle tints in the sky... With the arrogance that keeps us theorists going, I harbor the delusion that it would be not too difficult to guess the underlying physics and formulate the governing equations. It is when contemplating how to solve these equations - to convert formulations into explanations - that humility sets in. Then, compendia of formulas become indispensable."
    (Michael Berry)

    "Why are special functions special?" Physics Today, April 2001.

    "I will be glad if I have succeeded in impressing the idea that it is not only pleasant to read at times the works of the old mathematical authors , but this may occasionally be of use for the actual advancement of science."
    (Constantin Caratheodory)

    Speaking to an MAA meeting in 1936.

    "I have myself always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations. His object is simply to distinguish clearly and notify to others as many different peaks as he can. There are some peaks which he can distinguish easily, while others are less clear. He sees A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B. B is now fixed in his vision, and from this point he can proceed to further discoveries. In other cases perhaps he can distinguish a ridge which vanishes in the distance, and conjectures that it leads to a peak in the clouds or below the horizon. But when he sees a peak he believes that it is there simply because he sees it. If he wishes someone else to see it, he points to it, either directly or through the chain of summits which led him to recognize it himself. When his pupil also sees it, the research, the argument, the proof is finished.

    The analogy is a rough one, but I am sure that it is not altogether misleading. If we were to push it to its extreme we should be led to a rather paradoxical conclusion; that we can, in the last analysis, do nothing but point; that proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils. This is plainly not the whole truth, but there is a good deal in it. The image gives us a genuine approximation to the processes of mathematical pedagogy on the one hand and of mathematical discovery on the other; it is only the very unsophisticated outsider who imagines that mathematicians make discoveries by turning the handle of some miraculous machine. Finally the image gives us at any rate a crude picture of Hilbert's metamathematical proof, the sort of proof which is a ground for its conclusion and whose object is to convince ."

    (G.H. Hardy)

    From the Preface to David Broussoud's recent book "Proofs and Confirmation: The Story of the Alternating Sign Matrix Conjecture," MAA, 1999. Broussoud cites Hardy's "Rouse Ball Lecture of 1928".

    "[T]o suggest that the normal processes of scholarship work well on the whole and in the long run is in no way contradictory to the view that the processes of selection and sifting which are essential to the scholarly process are filled with error and sometimes prejudice."
    (Kenneth Arrow)

    From E. Roy Weintraub and Ted Gayer, "Equilibrium Proofmaking," Journal of the History of Economic Thought, 23 (Dec. 2001), 421-442.

    "Mathematical proofs like diamonds should be hard and clear, and will be touched with nothing but strict reasoning."

    (John Locke)

    From The Mathematical Universe by William Dunham, John Wiley, 1994.

    ``In his review of Winchester's previous book, The Map That Changed the World (3), Stephen Jay Gould wrote:

    I don't mean to sound like an academic sourpuss, but I just don't understand the priorities of publishers who spare no expense to produce an elegantly illustrated and beautifully designed book and then permit the text to wallow in simple, straight-out factual errors, all easily corrected for the minimal cost of one scrutiny of the galleys by a reader with professional expertise... (4)

    With Krakatoa, the publisher clearly spared considerable expense, and this new book also wallows in errors. Perhaps, given our popular culture's appetite for sensationalized disasters, a modern publisher would rather not see all those pesky details corrected.''
    (Tom Simkin and Richard S. Fiske)

    Review entitled "Clouded Picture of a Big Bang" from Science, July 4, 2003, page 50-51)

    "Again, I have to repeat the dictum of Harvard's president, Larry Summers: "In the history of the world, no one has ever washed a rented car." Most Iraqis still feel they are renting their own country --- first from Saddam and now from us. They have to be given ownership. If the Bush team is ready to put in the time, energy and money to make that happen --- great. But if not, it's going to have to make the necessary compromises to bring in the U.N. and the international community to help. ''
    (Thomas Freedman)

    New York Times August 26, 2003.

    "The paomnnehil pweor of the hmuan mnid. Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae. The rset can be a total mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe."

    Psased on by Kevin Hare, Septemeber 2003.

    " "The great tragedy of science," the biologist Thomas Henry Huxley lamented, is "the slaying of a beautiful hypothesis by an ugly fact." By that standard, political science is going through a homely phase. It's not even three weeks since the Iowa caucuses, and voters have wiped out several decades' worth of conventional wisdom about presidential primaries."

    Some columnist in February 2004.

    "By 1948, the Marxist-Leninist ideas about the proletariat and its political capacity seemed more and more to me to disagree with reality ... I pondered my doubts, and for several years the study of mathematics was all that allowed me to preserve my inner equilibrium. Bolshevik ideology was, for me, in ruins. I had to build another life."

    (Jean Van Heijenoort, 1913-1986)

    From his autobiography With Trotsky in Exile, quoted in Anita Feferman's From Trotsky to G\"odel

    "Numbers are not the only thing that computers are good at processing. Indeed, only a cursory familiarity with fractal geometry is needed to see that computers are good at creating and manipulating visual representations of data. There is a story told of the mathematician Claude Chevalley, who, as a true Bourbaki, was extremely opposed to the use of images in geometric reasoning. He is said to have been giving a very abstract and algebraic lecture when he got stuck. After a moment of pondering, he turned to the blackboard, and, trying to hide what he was doing, drew a little diagram, looked at it for a moment, then quickly erased it, and turned back to the audience and proceeded with the lecture. It is perhaps an apocryphal story, but it illustrates the necessary role of images and diagrams in mathematical reasoning-even for the most diehard anti-imagers. The computer offers those less expert, and less stubborn than Chevalley, access to the kinds of images that could only be imagined in the heads of the most gifted mathematicians, images that can be coloured, moved and otherwise manipulated in all sorts of ways.

    (Nathalie Sinclair, 2004)

    From Making the Connection: Research and Practice in Undergraduate Mathematics, M. Carlson and C. Rasmussen (Eds), MAA Notes, in press.

    Try the following large collections of Mathematical Quotes from: Platonic Realms and from Furman.

    We start at the end where the most recent quotes lie.

    And, since many came from the Economist, here is a link to the Economist.

    Revised 6/6/2004 by Jonathan Borwein. SFU / CECM / ~jborwein /