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Abel, Niels H. (1802 - 1829)
If you disregard the very simplest cases, there is in all of
mathematics not a single infinite series whose sum has been
rigorously determined. In other words,the most important parts of
mathematics stand without a foundation.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992,
p. 188.
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Abel, Niels H. (1802 - 1829)
[A reply to a question about how he got his expertise:]
By studying the masters and not their pupils.
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Abel, Niels H. (1802 - 1829)
[About Gauss' mathematical writing style]
He is like the fox, who effaces his tracks in the sand with his tail.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992,
p. 177.
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Adams, Douglas (1952 - )
Bistromathics itself is simply a revolutionary new way of
understanding the behavior of numbers. Just as Einstein observed that
space was not an absolute but depended on the observer's movement in
space, and that time was not an absolute, but depended on the
observer's movement in time, so it is now realized that numbers are
not absolute, but depend on the observer's movement in restaurants.
Life, the Universe and Everything. New York: Harmony Books, 1982.
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Adams, Douglas (1952 - )
The first nonabsolute number is the number of people for whom the
table is reserved. This will vary during the course of the first
three telephone calls to the restaurant, and then bear no apparent
relation to the number of people who actually turn up, or to the
number of people who subsequently join them after the
show/match/party/gig, or to the number of people who leave when they
see who else has turned up.
The second nonabsolute number is the given time of arrival, which is
now known to be one of the most bizarre of mathematical concepts, a
recipriversexcluson, a number whose existence can only be defined as
being anything other than itself. In other words, the given time of
arrival is the one moment of time at which it is impossible that any
member of the party will arrive. Recipriversexclusons now play a
vital part in many branches of math, including statistics and
accountancy and also form the basic equations used to engineer the
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Adams, Douglas (1952 - )
Numbers written on restaurant bills within the confines of
restaurants do not follow the same mathematical laws as numbers
written on any other pieces of paper in any other parts of the
Universe.
This single statement took the scientific world by storm. It
completely revolutionized it. So many mathematical conferences got
held in such good restaurants that many of the finest minds of a
generation died of obesity and heart failure and the science of math
was put back by years.
Life, the Universe and Everything. New York: Harmony Books, 1982.
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Adams, John (1735 - 1826)
I must study politics and war that my sons may have liberty to study
mathematics and philosophy. My sons ought to study mathematics and
philosophy, geography, natural history, naval architecture,
navigation, commerce and agriculture in order to give their children
a right to study painting, poetry, music, architecture, statuary,
tapestry, and porcelain.
Letter to Abigail Adams, May 12, 1780.
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Adler, Alfred
Each generation has its few great mathematicians, and mathematics
would not even notice the absence of the others. They are useful as
teachers, and their research harms no one, but it is of no importance
at all. A mathematician is great or he is nothing.
"Mathematics and Creativity." The New Yorker Magazine, February 19,
1972.
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Adler, Alfred
The mathematical life of a mathematician is short. Work rarely
improves after the age of twenty-five or thirty. If little has been
accomplished by then, little will ever be accomplished.
"Mathematics and Creativity." The New Yorker Magazine, February 19,
1972.
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Adler, Alfred
In the company of friends, writers can discuss their books,
economists the state of the economy, lawyers their latest cases, and
businessmen their latest acquisitions, but mathematicians cannot
discuss their mathematics at all. And the more profound their work,
the less understandable it is.
Reflections: mathematics and creativity, New Yorker, 47(1972), no.
53, 39 - 45.
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Allen, Woody
Standard mathematics has recently been rendered obsolete by the
discovery that for years we have been writing the numeral five
backward. This has led to reevaluation of counting as a method of
getting from one to ten. Students are taught advanced concepts of
Boolean algebra, and formerly unsolvable equations are dealt with by
threats of reprisals.
In Howard Eves' Return to Mathematical Circles, Boston: Prindle,
Weber, and Schmidt, 1988.
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Anglin, W.S.
Mathematics is not a careful march down a well-cleared highway, but
a journey into a strange wilderness, where the explorers often get
lost. Rigour should be a signal to the historian that the maps have
been made, and the real explorers have gone elsewhere.
"Mathematics and History", Mathematical Intelligencer, v. 4, no. 4.
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If thou art able, O stranger, to find out all these things and
gather them together in your mind, giving all the relations, thou
shalt depart crowned with glory and knowing that thou hast been
adjudged perfect in this species of wisdom.
In Ivor Thomas "Greek Mathematics" in J. R. Newman (ed.) The World
of Mathematics, New York: Simon and Schuster, 1956.
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Defendit numerus: There is safety in numbers.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956, p. 1452.
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Like the crest of a peacock so is mathematics at the head of all
knowledge.
[An old Indian saying. Also, "Like the Crest of a Peacock" is the
title of a book by G.G. Joseph]
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Referee's report: This paper contains much that is new and much that
is true. Unfortunately, that which is true is not new and that which
is new is not true.
In H.Eves Return to Mathematical Circles, Boston: Prindle, Weber,
and Schmidt, 1988.
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Aristophanes (ca 444 - 380 BC)
Meton: With the straight ruler I set to work
To make the circle four-cornered
[First(?) allusion to the problem of squaring the circle]
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Aristotle (ca 330 BC)
Now that practical skills have developed enough to provide
adequately for material needs, one of these sciences which are not
devoted to utilitarian ends [mathematics] has been able to arise in
Egypt, the priestly caste there having the leisure necessary for
disinterested research.
Metaphysica, 1-981b
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Aristotle (ca 330 BC)
The whole is more than the sum of its parts.
Metaphysica 10f-1045a
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Aristotle
The so-called Pythagoreans, who were the first to take up
mathematics, not only advanced this subject, but saturated with it,
they fancied that the principles of mathematics were the principles
of all things.
Metaphysica 1-5
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Aristotle
It is not once nor twice but times without number that the same
ideas make their appearance in the world.
"On The Heavens", in T. L. Heath Manual of Greek Mathematics,
Oxford: Oxford University Press, 1931.
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Aristotle
To Thales the primary question was not what do we know, but how do
we know it.
Mathematical Intelligencer v. 6, no. 3, 1984.
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Aristotle
The mathematical sciences particularly exhibit order, symmetry, and
limitation; and these are the greatest forms of the beautiful.
Metaphysica, 3-1078b.
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Ascham, Roger (1515-1568)
Mark all mathematical heads which be wholly and only bent on these
sciences, how solitary they be themselves, how unfit to live with
others, how unapt to serve the world.
In E G R Taylor, Mathematical Practitioners of Tudor and Stuart
England, Cambridge: Cambridge University Press, 1954.
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Aubrey, John (1626-1697)
[About Thomas Hobbes:]
He was 40 years old before he looked on geometry; which happened
accidentally. Being in a gentleman's library, Euclid's Elements lay
open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read
the proposition "By God", sayd he, "this is impossible:" So he reads
the demonstration of it, which referred him back to such a
proposition; which proposition he read. That referred him back to
another, which he also read. Et sic deinceps, that at last he was
demonstratively convinced of that trueth. This made him in love with
geometry.
In O. L. Dick (ed.) Brief Lives, Oxford: Oxford University Press,
1960, p. 604.
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Auden, W. H. (1907-1973)
How happy the lot of the mathematician. He is judged solely by his
peers, and the standard is so high that no colleague or rival can
ever win a reputation he does not deserve.
The Dyer's Hand, London: Faber and Faber, 1948.
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Auden, W. H. (1907-1973)
Thou shalt not answer questionnaires
Or quizzes upon world affairs,
Nor with compliance
Take any test. Thou shalt not sit
with statisticians nor commit
A social science.
"Under which lyre" in Collected Poems of W H Auden, London: Faber
and Faber.
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Augarten, Stan
Computers are composed of nothing more than logic gates stretched
out to the horizon in a vast numerical irrigation system.
State of the Art: A Photographic History of the Integrated Circuit.
New York: Ticknor and Fields.
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St. Augustine (354-430)
Six is a number perfect in itself, and not because God created the
world in six days; rather the contrary is true. God created the world
in six days because this number is perfect, and it would remain
perfect, even if the work of the six days did not exist.
The City of God.
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St. Augustine (354-430)
The good Christian should beware of mathematicians, and all those
who make empty prophecies. The danger already exists that the
mathematicians have made a covenant with the devil to darken the
spirit and to confine man in the bonds of Hell.
DeGenesi ad Litteram, Book II, xviii, 37 [Note: mathematician =
astrologer]
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St. Augustine (354-430)
If I am given a formula, and I am ignorant of its meaning, it cannot
teach me anything, but if I already know it what does the formula
teach me?
De Magistro ch X, 23.
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Babbage, Charles (1792-1871)
Errors using inadequate data are much less than those using no data
at all.
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Babbage, Charles (1792-1871)
On two occasions I have been asked [by members of Parliament],
'Pray, Mr. Babbage, if you put into the machine wrong figures, will
the right answers come out?' I am not able rightly to apprehend the
kind of confusion of ideas that could provoke such a question.
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Babbage, Charles (1792-1871)
I wish to God these calculations had been executed by steam.
In H. Eves In Mathematical Circles,, Boston: Prindle, Weber and
Schmidt, 1969.
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Bacon, Sir Francis (1561-1626)
And as for Mixed Mathematics, I may only make this prediction, that
there cannot fail to be more kinds of them, as nature grows further
disclosed.
Advancement of Learning book 2; De Augmentis book 3.
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Bacon, Roger
For the things of this world cannot be made known without a
knowledge of mathematics.
Opus Majus part 4 Distinctia Prima cap 1, 1267.
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Bacon, Roger
In the mathematics I can report no deficience, except that it be
that men do not sufficiently understand the excellent use of the pure
mathematics, in that they do remedy and cure many defects in the wit
and faculties intellectual. For if the wit be too dull, they sharpen
it; if too wandering, they fix it; if too inherent in the sense, they
abstract it. So that as tennis is a game of no use in itself, but of
great use in respect it maketh a quick eye and a body ready to put
itself into all postures; so in the mathematics, that use which is
collateral and intervenient is no less worthy than that which is
principal and intended.
John Fauvel and Jeremy Gray (eds.) A History of Mathematics: A
Reader, Sheridan House, 1987.
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Baker, H. F.
[On the concept of group:]
... what a wealth, what a grandeur of thought may spring from what
slightbeginnings.
Florian Cajori, A History of Mathematics, New York, 1919, p 283.
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Bagehot, Walter
Life is a school of probability.
Quoted in J. R. Newman (ed.) The World of Mathematics, Simon and
Schuster, New York,1956, p. 1360.
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Balzac, Honore de (1799 - 1850)
Numbers are intellectual witnesses that belong only to mankind.
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Banville, John
Throughout the 1960s and 1970s devoted Beckett readers greeted each
successively shorter volume from the master with a mixture of awe and
apprehensiveness; it was like watching a great mathematician wielding
an infinitesimal calculus, his equations approaching nearer and still
nearer to the null point.
Quoted in a review of Samuel Beckett's Nohow On: I11 Seen I11 Said,
Worstward Ho, in The New York Review of Books, August 13, 1992.
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Bell, Eric Temple (1883-1960)
Euclid taught me that without assumptions there is no proof.
Therefore, in any argument, examine the assumptions.
In H. Eves Return to Mathematical Circles., Boston: Prindle, Weber
and Schmidt, 1988.
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Bell, Eric Temple (1883-1960)
Wherever groups disclosed themselves, or could be introduced,
simplicity crystallized out of comparative chaos.
Mathematics, Queen and Servant of Science, New York, 1951, p 164.
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Bell, Eric Temple (1883-1960)
It is the perennial youthfulness of mathematics itself which marks
it off with a disconcerting immortality from the other sciences.
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Bell, Eric Temple (1883-1960)
The Handmaiden of the Sciences.
[Book by that title.]
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Bell, Eric Temple (1883-1960)
Abstractness, sometimes hurled as a reproach at mathematics, is its
chief glory and its surest title to practical usefulness. It is also
the source of such beauty as may spring from mathematics.
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Bell, Eric Temple (1883-1960)
Guided only by their feeling for symmetry, simplicity, and
generality, and an indefinable sense of the fitness of things,
creative mathematicians now, as in the past, are inspired by the art
of mathematics rather than by any prospect of ultimate usefulness.
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Bell, Eric Temple (1883-1960)
"Obvious" is the most dangerous word in mathematics.
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Bell, Eric Temple (1883-1960)
The pursuit of pretty formulas and neat theorems can no doubt
quickly degenerate into a silly vice, but so can the quest for
austere generalities which are so very general indeed that they are
incapable of application to any particular.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
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Bell, Eric Temple (1883-1960)
If a lunatic scribbles a jumble of mathematical symbols it does not
follow that the writing means anything merely because to the inexpert
eye it is indistinguishable from higher mathematics.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956, p. 308.
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Bell, Eric Temple (1883-1960)
The longer mathematics lives the more abstract -- and therefore,
possibly also the more practical -- it becomes.
In The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991.
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Bell, Eric Temple (1883-1960)
The cowboys have a way of trussing up a steer or a pugnacious bronco
which fixes the brute so that it can neither move nor think. This is
the hog-tie, and it is what Euclid did to geometry.
In R Crayshaw-Williams The Search For Truth, p. 191.
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Bell, Eric Temple (1883-1960)
If "Number rules the universe" as Pythagoras asserted, Number is
merely our delegate to the throne, for we rule Number.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber
and Schmidt, 1971.
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Bell, Eric Temple (1883-1960)
I have always hated machinery, and the only machine I ever
understood was a wheelbarrow, and that but imperfectly.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
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Belloc, Hillaire (1870-1953)
Statistics are the triumph of the quantitative method, and the
quantitative method is the victory of sterility and death.
The Silence of the Sea
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Bentham, Jeremy (1748-1832)
O Logic: born gatekeeper to the Temple of Science, victim of
capricious destiny: doomed hitherto to be the drudge of pedants: come
to the aid of thy master, Legislation.
In J. Browning (ed.) Works.
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Bernoulli, Daniel
...it would be better for the true physics if there were no
mathematicians on earth.
In The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
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Bernoulli, Jacques (Jakob?) (1654-1705)
I recognize the lion by his paw.
[After reading an anonymous solution to a problem that he realized
was Newton's solution.]
In G. Simmons, Calculus Gems, New York: McGraw Hill, 1992, p. 136.
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Bernoulli, Johann
But just as much as it is easy to find the differential of a given
quantity, so it is difficult to find the integral of a given
differential. Moreover, sometimes we cannot say with certainty
whether the integral of a given quantity can be found or not.
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Besicovitch, A.S.
A mathematician's reputation rests on the number of bad proofs he
has given.
In J. E. Littlewood A Mathematician's Miscellany, Methuen and Co.
Ltd., 1953.
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Blake
God forbid that Truth should be confined to Mathematical
Demonstration!
Notes on Reynold's Discourses, c. 1808.
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Blake
What is now proved was once only imagin'd.
The Marriage of Heaven and Hell, 1790-3.
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Bohr, Niels Henrik David (1885-1962)
An expert is a man who has made all the mistakes, which can be made,
in a very narrow field.
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The Bible
I returned and saw under the sun that the race is not to the swift,
nor the battle to the strong, neither yet bread to the wise, nor yet
riches to men of understanding, nor yet favour to men of skill; but
time and chance happeneth to them all.
Ecclesiastes.
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Bolyai, János (1802 - 1860)
Out of nothing I have created a strange new universe.
[A reference to the creation of a non-euclidean geometry.]
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Bolyai, Wolfgang (1775-1856)
[To son János:]
For God's sake, please give it up. Fear it no less than the sensual
passion, because it, too, may take up all your time and deprive you
of your health, peace of mind and happiness in life.
[Bolyai's father urging him to give up work on non-Euclidian
geometry.]
In P. Davis and R. Hersh The Mathematical Experience , Boston:
Houghton Mifflin Co., 1981, p. 220.
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Bourbaki
Structures are the weapons of the mathematician.
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Bridgman, P. W.
It is the merest truism, evident at once to unsophisticated
observation, that mathematics is a human invention.
The Logic of Modern Physics, New York, 1972.
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Brown, George Spencer (1923 - )
To arrive at the simplest truth, as Newton knew and practiced,
requires years of contemplation. Not activity Not reasoning. Not
calculating. Not busy behaviour of any kind. Not reading. Not
talking. Not making an effort. Not thinking. Simply bearing in mind
what it is one needs to know. And yet those with the courage to tread
this path to real discovery are not only offered practically no
guidance on how to do so, they are actively discouraged and have to
set abut it in secret, pretending meanwhile to be diligently engaged
in the frantic diversions and to conform with the deadening personal
opinions which are continually being thrust upon them.
The Laws of Form. 1969.
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Browne, Sir Thomas (1605-1682)
God is like a skilful Geometrician.
Religio Medici I, 16.
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Browne, Sir Thomas (1605-1682)
All things began in Order, so shall they end, and so shall they
begin again, according to the Ordainer of Order, and the mystical
mathematicks of the City of Heaven.
Hydriotaphia, Urn-burial and the Garden of Cyrus, 1896.
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Browne, Sir Thomas (1605-1682)
...indeed what reason may not go to Schoole to the wisdome of Bees,
Aunts, and Spiders? what wise hand teacheth them to doe what reason
cannot teach us? ruder heads stand amazed at those prodigious pieces
of nature, Whales, Elephants, Dromidaries and Camels; these I
confesse, are the Colossus and Majestick pieces of her hand; but in
these narrow Engines there is more curious Mathematicks, and the
civilitie of these little Citizens more neatly sets forth the
wisedome of their Maker.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956, p. 1001.
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Buck, Pearl S. (1892 - 1973)
No one really understood music unless he was a scientist, her father
had declared, and not just a scientist, either, oh, no, only the real
ones, the theoreticians, whose language mathematics. She had not
understood mathematics until he had explained to her that it was the
symbolic language of relationships. "And relationships," he had told
her, "contained the essential meaning of life."
The Goddess Abides, Pt. I, 1972.
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Burke, Edmund
The age of chivalry is gone. That of sophisters, economists and
calculators has succeeded.
Reflections on the Revolution in France.
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Butler, Bishop
To us probability is the very guide of life.
Preface to Analogy.
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Butler, Samuel (1612 - 1680)
... There can be no doubt about faith and not reason being the
ultima ratio. Even Euclid, who has laid himself as little open to the
charge of credulity as any writer who ever lived, cannot get beyond
this. He has no demonstrable first premise. He requires postulates
and axioms which transcend demonstration, and without which he can do
nothing. His superstructure indeed is demonstration, but his ground
his faith. Nor again can he get further than telling a man he is a
fool if he persists in differing from him. He says "which is absurd,"
and declines to discuss the matter further. Faith and authority,
therefore, prove to be as necessary for him as for anyone else.
The Way of All Flesh.
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Byron
When Newton saw an apple fall, he found ...
A mode of proving that the earth turnd round
In a most natural whirl, called gravitation;
And thus is the sole mortal who could grapple
Since Adam, with a fall or with an apple.
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Caballero, JamesI advise my students to listen carefully the moment
they decide to take no more mathematics courses. They might be able
to hear the sound of closing doors.
Everybody a mathematician?,CAIP Quarterly 2 (Fall, 1989).
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Cardano, Girolamo (1501 - 1576)
To throw in a fair game at Hazards only three-spots, when something
great is at stake, or some business is the hazard, is a natural
occurrence and deserves to be so deemed; and even when they come up
the same way for a second time if the throw be repeated. If the third
and fourth plays are the same, surely there is occasion for suspicion
on the part of a prudent man.
De Vita Propria Liber.
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Carlyle, Thomas (1795 - 1881)
It is a mathematical fact that the casting of this pebble from my
hand alters the centre of gravity of the universe.
Sartor Resartus III.
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Carlyle, Thomas (1795-1881)
Teaching school is but another word for sure and not very slow
destruction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
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Carlyle, Thomas (1795-1881)
A witty statesman said, you might prove anything by figures.
Chartism.
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Carroll, Lewis
What I tell you three times is true.
The Hunting of the Snark.
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Carroll, Lewis
The different branches of Arithmetic -- Ambition, Distraction,
Uglification, and Derision.
Alice in Wonderland.
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Carroll, Lewis
"Can you do addition?" the White Queen asked. "What's one and one
and one and one and one and one and one and one and one and one?" "I
don't know," said Alice. "I lost count."
Through the Looking Glass.
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Carroll, Lewis
"Alice laughed: "There's no use trying," she said; "one can't
believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I
was younger, I always did it for half an hour a day. Why, sometimes
I've believed as many as six impossible things before breakfast."
Alice in Wonderland.
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Carroll, Lewis
"Then you should say what you mean," the March Hare went on.
"I do, " Alice hastily replied; "at least I mean what I say, that's
the same thing, you know."
"Not the same thing a bit!" said the Hatter. "Why, you might just as
well say that "I see what I eat" is the same thing as "I eat what I
see!"
Alice in Wonderland.
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Carroll, Lewis
"It's very good jam," said the Queen.
"Well, I don't want any to-day, at any rate."
"You couldn't have it if you did want it," the Queen said. "The rule
is jam tomorrow and jam yesterday but never jam to-day."
"It must come sometimes to "jam to-day,""Alice objected.
"No it can't," said the Queen. "It's jam every other day; to-day
isn't any other day, you know."
"I don't understand you," said Alice. "It's dreadfully confusing."
Through the Looking Glass.
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Carroll, Lewis
"When I use a word," Humpty Dumpty said, in a rather scornful tone,
"it means just what I choose it to mean - neither more nor less."
"The question is," said Alice, "whether you can make words mean so
many different things."
"The question is," said Humpty Dumpty, "which is to be master -
that's all."
Through the Looking Glass.
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Carmichael, R. D.
A thing is obvious mathematically after you see it.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.
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Cauchy, Augustin-Louis (1789 - 1857)
Men pass away, but their deeds abide.
[His last words (?)]
In H. Eves Mathematical Circles Revisted, Boston: Prindle, Weber and
Schmidt, 1971.
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Cayley, Arthur
As for everything else, so for a mathematical theory: beauty can be
perceived but not explained.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
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Cayley, Arthur
Projective geometry is all geometry.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
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Cézanne, Paul (1839 - 1906)
...treat Nature by the sphere, the cylinder and the cone...
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Chebyshev
To isolate mathematics from the practical demands of the sciences is
to invite the sterility of a cow shut away from the bulls.
In G. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992,
page 198.
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Chekov, Anton (1860 - 1904)
There is no national science just as there is no national
multiplication table; what is national is no longer science.
In V. P. Ponomarev Mysli o nauke Kishinev, 1973.
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Chesterton, G. K. (1874 - 1936)
Poets do not go mad; but chess-players do. Mathematicians go mad,
and cashiers; but creative artists very seldom. I am not, as will be
seen, in any sense attacking logic: I only say that this danger does
lie in logic, not in imagination.
Orthodoxy ch. 2.
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Chesterton, G. K. (1874 - 1936)
You can only find truth with logic if you have already found truth
without it.
The Man who was Orthodox. 1963.
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Chesterton, G. K. (1874 - 1936)
It isn't that they can't see the solution. It is that they can't see
the problem.
The Point of a Pin in The Scandal of Father Brown.
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Christie, Agatha
"I think you're begging the question," said Haydock, "and I can see
looming ahead one of those terrible exercises in probability where
six men have white hats and six men have black hats and you have to
work it out by mathematics how likely it is that the hats will get
mixed up and in what proportion. If you start thinking about things
like that, you would go round the bend. Let me assure you of that!"
The Mirror Crack'd. Toronto: Bantam Books, 1962.
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Christie, Agatha
I continued to do arithmetic with my father, passing proudly through
fractions to decimals. I eventually arrived at the point where so
many cows ate so much grass, and tanks filled with water in so many
hours I found it quite enthralling.
An Autobiography.
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Churchill, [Sir] Winston Spencer (1874-1965)
It is a good thing from an uneducated man to read books of
quotations.
Roving Commission in My Early Life. 1930.
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Churchill, Sir Winston Spencer (1874-1965)
I had a feeling once about Mathematics - that I saw it all. Depth
beyond depth was revealed to me - the Byss and Abyss. I saw - as one
might see the transit of Venus or even the Lord Mayor's Show - a
quantity passing through infinity and changing its sign from plus to
minus. I saw exactly why it happened and why the tergiversation was
inevitable but it was after dinner and I let it go.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Churchman, C. W.
The measure of our intellectual capacity is the capacity to feel
less and less satisfied with our answers to better and better
problems.
In J.E. Littlewood A Mathematician's Miscellany. Methuen and Co.,
Ltd. 1953.
%
Cocteau
The composer opens the cage door for arithmetic, the draftsman gives
geometry its freedom.
%
Coleridge, Samuel Taylor (1772-1834)
...from the time of Kepler to that of Newton, and from Newton to
Hartley, not only all things in external nature, but the subtlest
mysteries of life and organization, and even of the intellect and
moral being, were conjured within the magic circle of mathematical
formulae.
The Theory of Life.
%
Conrad, Joseph
Don't talk to me of your Archimedes' lever. He was an absentminded
person with a mathematical imagination. Mathematics commands all my
respect, but I have no use for engines. Give me the right word and
the right accent and I will move the world.
Preface to A Personal Record.
%
Coolidge, Julian Lowell (1873 - 1954)
[Upon proving that the best betting strategy for "Gambler's Ruin"
was to bet all on the first trial.]
It is true that a man who does this is a fool. I have only proved
that a man who does anything else is an even bigger fool.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Copernicus, Nicholaus (1473-1543)
Mathematics is written for mathematicians.
De Revolutionibus.
%
Crick, Francis Harry Compton (1916 - )
In my experience most mathematicians are intellectually lazy and
especially dislike reading experimental papers. He (René Thom) seemed
to have very strong biological intuitions but unfortunately of
negative sign.
What Mad Pursuit. London: Weidenfeld and Nicolson, 1988.
%
Crowe, Michael
Revolutions never occur in mathematics.
Historia Mathematica. 1975.
%
D'Alembert, Jean Le Rond (1717-1783)
Just go on..and faith will soon return.
[To a friend hesitant with respect to infinitesimals.]
In P. J. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhauser, 1981.
%
D'Alembert, Jean Le Rond (1717-17830
Thus metaphysics and mathematics are, among all the sciences that
belong to reason, those in which imagination has the greatest role. I
beg pardon of those delicate spirits who are detractors of
mathematics for saying this .... The imagination in a mathematician
who creates makes no less difference than in a poet who invents....
Of all the great men of antiquity, Archimedes may be the one who most
deserves to be placed beside Homer.
Discours Preliminaire de L'Encyclopedie, Tome 1, 1967. pp 47 - 48.
%
Dantzig
The mathematician may be compared to a designer of garments, who is
utterly oblivious of the creatures whom his garments may fit. To be
sure, his art originated in the necessity for clothing such
creatures, but this was long ago; to this day a shape will
occasionally appear which will fit into the garment as if the garment
had been made for it. Then there is no end of surprise and delight.
%
Dantzig
Neither in the subjective nor in the objective world can we find a
criterion for the reality of the number concept, because the first
contains no such concept, and the second contains nothing that is
free from the concept. How then can we arrive at a criterion? Not by
evidence, for the dice of evidence are loaded. Not by logic, for
logic has no existence independent of mathematics: it is only one
phase of this multiplied necessity that we call mathematics.
How then shall mathematical concepts be judged? They shall not be
judged. Mathematics is the supreme arbiter. From its decisions there
is no appeal. We cannot change the rules of the game, we cannot
ascertain whether the game is fair. We can only study the player at
his game; not, however, with the detached attitude of a bystander,
for we are watching our own minds at play.
%
Darwin, Charles
Every new body of discovery is mathematical in form, because there
is no other guidance we can have.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.
%
Darwin, Charles
Mathematics seems to endow one with something like a new sense.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.
%
Davis, Philip J.
The numbers are a catalyst that can help turn raving madmen into
polite humans.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.
%
Davis, Philip J.
One of the endlessly alluring aspects of mathematics is that its
thorniest paradoxes have a way of blooming into beautiful theories.
Number, Scientific American, 211, (Sept. 1964), 51 - 59.
%
Davis, Philip J. and Hersh, Reuben
One began to hear it said that World War I was the chemists' war,
World War II was the physicists' war, World War III (may it never
come) will be the mathematicians' war.
The Mathematical Experience, Boston: Birkhauser, 1981.
%
Dehn, Max
Mathematics is the only instructional material that can be presented
in an entirely undogmatic way.
In The Mathematical Intelligencer, v. 5, no. 2, 1983.
%
De Morgan, Augustus (1806-1871)
[When asked about his age.] I was x years old in the year x^2.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
%
De Morgan, Augustus (1806-1871)
It is easier to square the circle than to get round a mathematician.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
%
De Morgan, Augustus (1806-1871)
Every science that has thriven has thriven upon its own symbols:
logic, the only science which is admitted to have made no
improvements in century after century, is the only one which has
grown no symbols.
Transactions Cambridge Philosophical Society, vol. X, 1864, p. 184.
%
Descartes, René (1596-1650)
Of all things, good sense is the most fairly distributed: everyone
thinks he is so well supplied with it that even those who are the
hardest to satisfy in every other respect never desire more of it
than they already have.
Discours de la Méthode. 1637.
%
Descartes, René (1596-1650)
Each problem that I solved became a rule which served afterwards to
solve other problems.
Discours de la Méthode. 1637.
%
Descartes, René (1596-1650)
If I found any new truths in the sciences, I can say that they
follow from, or depend on, five or six principal problems which I
succeeded in solving and which I regard as so many battles where the
fortunes of war were on my side.
Discours de la Méthode. 1637.
%
Descartes, René (1596-1650)
I concluded that I might take as a general rule the principle that
all things which we very clearly and obviously conceive are true:
only observing, however, that there is some difficulty in rightly
determining the objects which we distinctly conceive.
Discours de la Méthode. 1637.
%
Descartes, René (1596-1650)
I thought the following four [rules] would be enough, provided that
I made a firm and constant resolution not to fail even once in the
observance of them. The first was never to accept anything as true if
I had not evident knowledge of its being so; that is, carefully to
avoid precipitancy and prejudice, and to embrace in my judgment only
what presented itself to my mind so clearly and distinctly that I had
no occasion to doubt it. The second, to divide each problem I
examined into as many parts as was feasible, and as was requisite for
its better solution. The third, to direct my thoughts in an orderly
way; beginning with the simplest objects, those most apt to be known,
and ascending little by little, in steps as it were, to the knowledge
of the most complex; and establishing an order in thought even when
the objects had no natural priority one to another. And the last, to
make throughout such complete enumerations and such general
%
Descartes, René (1596-1650)
These long chains of perfectly simple and easy reasonings by means
of which geometers are accustomed to carry out their most difficult
demonstrations had led me to fancy that everything that can fall
under human knowledge forms a similar sequence; and that so long as
we avoid accepting as true what is not so, and always preserve the
right order of deduction of one thing from another, there can be
nothing too remote to be reached in the end, or to well hidden to be
discovered.
Discours de la Méthode. 1637.
%
Descartes, René (1596-1650)
When writing about transcendental issues, be transcendentally clear.
In G. Simmons Calculus Gems. New York: McGraw Hill Inc., 1992.
%
Descartes, René (1596-1650)
If we possessed a thorough knowledge of all the parts of the seed of
any animal (e.g. man), we could from that alone, be reasons entirely
mathematical and certain, deduce the whole conformation and figure of
each of its members, and, conversely if we knew several peculiarities
of this conformation, we would from those deduce the nature of its
seed.
%
Descartes, René (1596-1650)
Cogito Ergo Sum. "I think, therefore I am."
Discours de la Méthode. 1637.
%
Descartes, René (1596-1650)
I hope that posterity will judge me kindly, not only as to the
things which I have explained, but also to those which I have
intentionally omitted so as to leave to others the pleasure of
discovery.
La Geometrie.
%
Descartes, René (1596-1650)
Perfect numbers like perfect men are very rare.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Descartes, René (1596-1650)
omnia apud me mathematica fiunt.
With me everything turns into mathematics.
%
Descartes, René (1596-1650)
It is not enough to have a good mind. The main thing is to use it
well.
Discours de la Méthode. 1637.
%
Descartes, René (1596-1650)
If you would be a real seeker after truth, you must at least once in
your life doubt, as far as possible, all things.
Discours de la Méthode. 1637.
%
De Sua, F. (1956)
Suppose we loosely define a religion as any discipline whose
foundations rest on an element of faith, irrespective of any element
of reason which may be present. Quantum mechanics for example would
be a religion under this definition. But mathematics would hold the
unique position of being the only branch of theology possessing a
rigorous demonstration of the fact that it should be so classified.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
%
Diophantus
[His epitaph.]
This tomb hold Diophantus Ah, what a marvel! And the tomb tells
scientifically the measure of his life. God vouchsafed that he should
be a boy for the sixth part of his life; when a twelfth was added,
his cheeks acquired a beard; He kindled for him the light of marriage
after a seventh, and in the fifth year after his marriage He granted
him a son. Alas! late-begotten and miserable child, when he had
reached the measure of half his father's life, the chill grave took
him. After consoling his grief by this science of numbers for four
years, he reached the end of his life.
In Ivor Thomas Greek Mathematics, in J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956.
%
Dirac, Paul Adrien Maurice (1902- )
I think that there is a moral to this story, namely that it is more
important to have beauty in one's equations that to have them fit
experiment. If Schroedinger had been more confident of his work, he
could have published it some months earlier, and he could have
published a more accurate equation. It seems that if one is working
from the point of view of getting beauty in one's equations, and if
one has really a sound insight, one is on a sure line of progress. If
there is not complete agreement between the results of one's work and
experiment, one should not allow oneself to be too discouraged,
because the discrepancy may well be due to minor features that are
not properly taken into account and that will get cleared up with
further development of the theory.
Scientific American, May 1963.
%
Dirac, Paul Adrien Maurice (1902- )
Mathematics is the tool specially suited for dealing with abstract
concepts of any kind and there is no limit to its power in this field.
In P. J. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhauser, 1981.
%
Dirac, Paul Adrien Maurice (1902- )
In science one tries to tell people, in such a way as to be
understood by everyone, something that no one ever knew before. But
in poetry, it's the exact opposite.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Disraeli, Benjamin
There are three kinds of lies: lies, damned lies, and statistics.
Mark Twain. Autobiography.
%
Donatus, Aelius (4th Century)
Pereant qui ante nos nostra dixerunt.
"To the devil with those who published before us."
[Quoted by St. Jerome, his pupil]
%
Doyle, Sir Arthur Conan (1859-1930)
Detection is, or ought to be, an exact sciences and should be
treated in the same cold and unemotional manner. You have attempted
to tinge it with romanticism, which produces much the same effect as
if you worked a love story or an elopement into the fifth proposition
of Euclid.
The Sign of Four.
%
Doyle, Sir Arthur Conan (1859-1930)
When you have eliminated the impossible, what ever remains, however
improbable must be the truth.
The Sign of Four.
%
Doyle, Sir Arthur Conan (1859-1930)
From a drop of water a logician could predict an Atlantic or a
Niagara.
A study in Scarlet 1929.
%
Doyle, Sir Arthur Conan (1859-1930)
It is a capital mistake to theorize before one has data.
Scandal in Bohemia.
%
Dryden, John (1631-1700)
Mere poets are sottish as mere drunkards are, who live in a
continual mist, without seeing or judging anything clearly. A man
should be learned in several sciences, and should have a reasonable,
philosophical and in some measure a mathematical head, to be a
complete and excellent poet.
Notes and Observations on The Empress of Morocco. 1674.
%
Dubos, René J.
Gauss replied, when asked how soon he expected to reach certain
mathematical conclusions, that he had them long ago, all he was
worrying about was how to reach them!
In Mechanisms of Discovery in I. S. Gordon and S. Sorkin (eds.) The
Armchair Science Reader, New York: Simon and Schuster, 1959.
%
Dunsany, Lord
Logic, like whiskey, loses its beneficial effect when taken in too
large quantities.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Durer, Albrecht (1471-1528)
But when great and ingenious artists behold their so inept
performances, not undeservedly do they ridicule the blindness of such
men; since sane judgment abhors nothing so much as a picture
perpetrated with no technical knowledge, although with plenty of care
and diligence. Now the sole reason why painters of this sort are not
aware of their own error is that they have not learnt Geometry,
without which no one can either be or become an absolute artist; but
the blame for this should be laid upon their masters, who are
themselves ignorant of this art.
The Art of Measurement. 1525.
%
Durer, Albrecht (1471-1528)
Whoever ... proves his point and demonstrates the prime truth
geometrically should be believed by all the world, for there we are
captured.
J Heidrich (ed.) Albrecht Durer's schriftlicher Nachlass Berlin,
1920.
%
Durer, Albrecht (1471-1528)
And since geometry is the right foundation of all painting, I have
decided to teach its rudiments and principles to all youngsters eager
for art...
Course in the Art of Measurement
%
Dyson, Freeman
I am acutely aware of the fact that the marriage between mathematics
and physics, which was so enormously fruitful in past centuries, has
recently ended in divorce.
Missed Opportunities, 1972. (Gibbs Lecture?)
%
Dyson, Freeman
For a physicist mathematics is not just a tool by means of which
phenomena can be calculated, it is the main source of concepts and
principles by means of which new theories can be created.
Mathematics in the Physical Sciences.
%
Dyson, Freeman
The bottom line for mathematicians is that the architecture has to
be right. In all the mathematics that I did, the essential point was
to find the right architecture. It's like building a bridge. Once the
main lines of the structure are right, then the details miraculously
fit. The problem is the overall design.
"Freeman Dyson: Mathematician, Physicist, and Writer". Interview
with Donald J. Albers, The College Mathematics Journal, vol 25, no.
1, January 1994.
%
Eddington, Sir Arthur (1882-1944)
Proof is the idol before whom the pure mathematician tortures
himself.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Eddington, Sir Arthur (1882-1944)
We used to think that if we knew one, we knew two, because one and
one are two. We are finding that we must learn a great deal more
about `and'.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Eddington, Sir Arthur (1882-1944)
We have found a strange footprint on the shores of the unknown. We
have devised profound theories, one after another, to account for its
origins. At last, we have succeeded in reconstructing the creature
that made the footprint. And lo! It is our own.
Space, Time and Gravitation. 1920.
%
Eddington, Sir Arthur (1882-1944)
It is impossible to trap modern physics into predicting anything
with perfect determinism because it deals with probabilities from the
outset.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Eddington, Sir Arthur (1882-1944)
I believe there are
15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116
,709,366,231,425,076,185,631,031,296 protons in the universe and the
same number of electrons.
The Philosophy of Physical Science. Cambridge, 1939.
%
Eddington, Sir Arthur (1882-1944)
To the pure geometer the radius of curvature is an incidental
characteristic - like the grin of the Cheshire cat. To the physicist
it is an indispensable characteristic. It would be going too far to
say that to the physicist the cat is merely incidental to the grin.
Physics is concerned with interrelatedness such as the
interrelatedness of cats and grins. In this case the "cat without a
grin" and the "grin without a cat" are equally set aside as purely
mathematical phantasies.
The Expanding Universe..
%
Eddington, Sir Arthur (1882-1944)
Human life is proverbially uncertain; few things are more certain
than the solvency of a life-insurance company.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Edwards, Jonathon When I am violently beset with temptations, or
cannot rid myself of evil thoughts, [I resolve] to do some
Arithmetic, or Geometry, or some other study, which necessarily
engages all my thoughts, and unavoidably keeps them from wandering.
In T. Mallon A Book of One's Own. Ticknor and Fields, New York,
1984, p. 106-107.
%
Egrafov, M.
If you ask mathematicians what they do, yo always get the same
answer. They think. They think about difficult and unusual problems.
They do not think about ordinary problems: they just write down the
answers.
Mathematics Magazine, v. 65 no. 5, December 1992.
%
Eigen, Manfred (1927 - )
A theory has only the alternative of being right or wrong. A model
has a third possibility: it may be right, but irrelevant.
Jagdish Mehra (ed.) The Physicist's Conception of Nature, 1973.
%
Einstein, Albert (1879-1955)
[During a lecture:]This has been done elegantly by Minkowski; but
chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.]
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co. Ltd.,
1953.
%
Einstein, Albert (1879-1955)
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.
%
Einstein, Albert (1879-1955)
I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.
%
Einstein, Albert (1879-1955)
Imagination is more important than knowledge.
On Science.
%
Einstein, Albert (1879-1955)
The most beautiful thing we can experience is the mysterious. It is
the source of all true art and science.
What I Believe.
%
Einstein, Albert (1879-1955)
The bitter and the sweet come from the outside, the hard from
within, from one's own efforts.
Out of My Later Years.
%
Einstein, Albert (1879-1955)
Gott wurfelt nicht.
%
Einstein, Albert (1879-1955)
Common sense is the collection of prejudices acquired by age
eighteen.
In E. T. Bell Mathematics, Queen and Servant of the Sciences. 1952.
%
Einstein, Albert (1879-1955)
God does not care about our mathematical difficulties. He integrates
empirically.
L. Infeld Quest, 1942.
%
Einstein, Albert (1879-1955)
How can it be that mathematics, being after all a product of human
thought independent of experience, is so admirably adapted to the
objects of reality?
%
Einstein, Albert (1879-1955)
[About Newton]
Nature to him was an open book, whose letters he could read without
effort.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.
%
Einstein, Albert (1879-1955)
As far as the laws of mathematics refer to reality, they are not
certain; and as far as they are certain, they do not refer to reality.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Einstein, Albert (1879-1955)
What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert]
In H. Eves Mathematical Circles Squared Boston: Prindle, Weber and
Schmidt, 1972.
%
Einstein, Albert (1879-1955)
Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is
subtle, but he is not malicious.
Inscribed in Fine Hall, Princeton University.
%
Einstein, Albert (1879-1955)
Nature hides her secrets because of her essential loftiness, but not
by means of ruse.
%
Einstein, Albert (1879-1955)
The human mind has first to construct forms, independently, before
we can find them in things.
%
Einstein, Albert (1879-1955)
Since the mathematicians have invaded the theory of relativity, I do
not understand it myself anymore.
In A. Sommerfelt "To Albert Einstein's Seventieth Birthday" in Paul
A. Schilpp (ed.) Albert Einstein, Philosopher-Scientist, Evanston,
1949.
%
Einstein, Albert (1879-1955)
Do not worry about your difficulties in mathematics, I assure you
that mine are greater.
%
Einstein, Albert (1879-1955)
The truth of a theory is in your mind, not in your eyes.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Einstein, Albert (1879-1955)
These thoughts did not come in any verbal formulation. I rarely
think in words at all. A thought comes, and I may try to express it
in words afterward.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Einstein, Albert (1879-1955)
A human being is a part of the whole, called by us "Universe," a
part limited in time and space. He experiences himself, his thoughts
and feelings as something separated from the resta kind of optical
delusion of his consciousness. This delusion is a kind of prison for
us, restricting us to our personal desires and to affection for a few
persons nearest to us. Our task must be to free ourselves from this
prison by widening our circle of compassion to embrace all living
creatures and the whole of nature in its beauty. Nobody is able to
achieve this completely, but the striving for such achievement is in
itself a part of the liberation and a foundation for inner security.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Einstein, Albert (1879-1955)
The world needs heroes and it's better they be harmless men like me
than villains like Hitler.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Einstein, Albert (1879-1955)
It is nothing short of a miracle that modern methods of instruction
have not yet entirely strangled the holy curiousity of inquiry.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Einstein, Albert (1879-1955)
Everything that is really great and inspiring is created by the
individual who can labor in freedom.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Einstein, Albert (1879-1955)
The search for truth is more precious than its possession.
The American Mathematical Monthly v. 100 no. 3.
%
Einstein, Albert (1879-1955)
If my theory of relativity is proven successful, Germany will claim
me as a German and France will declare that I am a citizen of the
world. Should my theory prove untrue, France will say that I am a
German and Germany will declare that I am a Jew.
Address at the Sorbonne, Paris.
%
Einstein, Albert (1879-1955)
We come now to the question: what is a priori certain or necessary,
respectively in geometry (doctrine of space) or its foundations?
Formerly we thought everything; nowadays we think nothing. Already
the distance-concept is logically arbitrary; there need be no things
that correspond to it, even approximately.
"Space-Time." Encyclopaedia Britannica, 14th ed.
%
Einstein, Albert (1879-1955)
Most of the fundamental ideas of science are essentially simple, and
may, as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.
%
Einstein, Albert (1879-1955)
Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.
%
Ellis, Havelock
The mathematician has reached the highest rung on the ladder of
human thought.
The Dance of Life.
%
Ellis, Havelock
It is here [in mathematics] that the artist has the fullest scope of
his imagination.
The Dance of Life.
%
Erath, V.
God is a child; and when he began to play, he cultivated
mathematics. It is the most godly of man's games.
Das blinde Spiel. 1954.
%
Erdös, Paul
Mathematics is not yet ready for such problems.
[Attributed by Paul Halmos.]
The American Mathematical Monthly, Nov. 1992
%
Erdös, Paul
A Mathematician is a machine for turning coffee into theorems.
%
Euler, Leonhard (1707 - 1783)
If a nonnegative quantity was so small that it is smaller than any
given one, then it certainly could not be anything but zero. To those
who ask what the infinitely small quantity in mathematics is, we
answer that it is actually zero. Hence there are not so many
mysteries hidden in this concept as they are usually believed to be.
These supposed mysteries have rendered the calculus of the infinitely
small quite suspect to many people. Those doubts that remain we
shall thoroughly remove in the following pages, where we shall
explain this calculus.
%
Euler, Leonhard (1707-1783)
Mathematicians have tried in vain to this day to discover some order
in the sequence of prime numbers, and we have reason to believe that
it is a mystery into which the human mind will never penetrate.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Euler, Leonhard (1707-1783)
[upon losing the use of his right eye]
Now I will have less distraction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
%
Everett, Edward (1794-1865)
In the pure mathematics we contemplate absolute truths which existed
in the divine mind before the morning stars sang together, and which
will continue to exist there when the last of their radiant host
shall have fallen from heaven.
Quoted by E.T. Bell in The Queen of the Sciences, Baltimore, 1931.
%
Eves, Howard W.
A formal manipulator in mathematics often experiences the
discomforting feeling that his pencil surpasses him in intelligence.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
%
Eves, Howard W.
An expert problem solver must be endowed with two incompatible
qualities, a restless imagination and a patient pertinacity.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
%
Eves, Howard W.
Mathematics may be likened to a large rock whose interior
composition we wish to examine. The older mathematicians appear as
persevering stone cutters slowly attempting to demolish the rock from
the outside with hammer and chisel. The later mathematicians resemble
expert miners who seek vulnerable veins, drill into these strategic
places, and then blast the rock apart with well placed internal
charges.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
%
Eves, Howard W.
One is hard pressed to think of universal customs that man has
successfully established on earth. There is one, however, of which he
can boast the universal adoption of the Hindu-Arabic numerals to
record numbers. In this we perhaps have man's unique worldwide
victory of an idea.
Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt,
1972.
%
Ewing, John
If the entire Mandelbrot set were placed on an ordinary sheet of
paper, the tiny sections of boundary we examine would not fill the
width of a hydrogen atom. Physicists think about such tiny objects;
only mathematicians have microscopes fine enough to actually observe
them.
"Can We See the Mandelbrot Set?", The College Mathematics Journal,
v. 26, no. 2, March 1995.
%
Focus Newsletter (MAA)
Sample recommendation letter:
Dear Search Committee Chair,
I am writing this letter for Mr. John Smith who has applied for a
position in your department. I should start by saying that I cannot
recommend him too highly.
In fact, there is no other student with whom I can adequately
compare him, and I am sure that the amount of mathematics he knows
will surprise you.
His dissertation is the sort of work you don't expect to see these
days. It definitely demonstrates his complete capabilities.
In closing, let me say that you will be fortunate if you can get him
to work for you.
Sincerely,
A. D. Visor (Prof.)
%
de Fermat, Pierre (1601?-1665)
[In the margin of his copy of Diophantus' Arithmetica, Fermat wrote]
To divide a cube into two other cubes, a fourth power or in general
any power whatever into two powers of the same denomination above the
second is impossible, and I have assuredly found an admirable proof
of this, but the margin is too narrow to contain it.
%
de Fermat, Pierre (1601?-1665)
And perhaps, posterity will thank me for having shown it that the
ancients did not know everything.
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon,
Inc., 1976.
%
Feynman, Richard Philips (1918 - 1988)
We have a habit in writing articles published in scientific journals
to make the work as finished as possible, to cover up all the tracks,
to not worry about the blind alleys or describe how you had the wrong
idea first, and so on. So there isn't any place to publish, in a
dignified manner, what you actually did in order to get to do the
work.
Nobel Lecture, 1966.
%
Finkel, Benjamin Franklin
The solution of problems is one of the lowest forms of mathematical
research, ... yet its educational value cannot be overestimated. It
is the ladder by which the mind ascends into higher fields of
original research and investigation. Many dormant minds have been
aroused into activity through the mastery of a single problem.
The American Mathematical Monthly, no. 1.
%
Fisher, Irving
The effort of the economist is to "see," to picture the interplay of
economic elements. The more clearly cut these elements appear in his
vision, the better; the more elements he can grasp and hold in his
mind at once, the better. The economic world is a misty region. The
first explorers used unaided vision. Mathematics is the lantern by
which what before was dimly visible now looms up in firm, bold
outlines. The old phantasmagoria disappear. We see better. We also
see further.
Transactions of Conn. Academy, 1892.
%
Fisher, Ronald Aylmer (1890 - 1962)
Natural selection is a mechanism for generating an exceedingly high
degree of improbability.
%
Fisher, Ronald Aylmer (1890-1962)
To call in the statistician after the experiment is done may be no
more than asking hm to perform a postmortem examination: he may be
able to say what the experiment died of.
Indian Statistical Congress, Sankhya, ca 1938.
%
Flaubert, Gustave (1821-1880)
Poetry is as exact a science as geometry.
%
Flaubert, Gustave (1821-1880)
Since you are now studying geometry and trigonometry, I will give
you a problem. A ship sails the ocean. It left Boston with a cargo of
wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is
broken, the cabin boy is on deck, there are 12 passengers aboard, the
wind is blowing East-North-East, the clock points to a quarter past
three in the afternoon. It is the month of May. How old is the
captain?
%
Fontenelle, Bernard Le Bovier (1657-1757)
Mathematicians are like lovers. Grant a mathematician the least
principle, and he will draw from it a consequence which you must also
grant him, and from this consequence another.
Quoted in V. H. Larney Abstract Algebra: A First Course, Boston:
Prindle, Weber and Schmidt, 1975.
%
Fontenelle, Bernard Le Bovier (1657-1757)
A work of morality, politics, criticism will be more elegant, other
things being equal, if it is shaped by the hand of geometry.
Preface sur l'Utilité des Mathématiques et de la Physique, 1729.
%
Fontenelle, Bernard Le Bovier (1657-1757)
Leibniz never married; he had considered it at the age of fifty; but
the person he had in mind asked for time to reflect. This gave
Leibniz time to reflect, too, and so he never married.
Eloge de le Leibniz.
%
Frankland, W.B.
Whereas at the outset geometry is reported to have concerned herself
with the measurement of muddy land, she now handles celestial as well
as terrestrial problems: she has extended her domain to the furthest
bounds of space.
Hodder and Stoughton, The Story of Euclid. 1901.
%
Frayn, Michael
For hundreds of pages the closely-reasoned arguments unroll, axioms
and theorems interlock. And what remains with us in the end? A
general sense that the world can be expressed in closely-reasoned
arguments, in interlocking axioms and theorems.
Constructions. 1974.
%
Frederick the Great (1712-1786)
To your care and recommendation am I indebted for having replaced a
half-blind mathematician with a mathematician with both eyes, which
will especially please the anatomical members of my Academy.
[To D'Alembert about Lagrange. Euler had vacated the post.]
In D. M. Burton, Elementary Number Theory, Boston: Allyn and Bacon,
Inc., 1976.
%
Frege, Gottlob (1848 - 1925)
A scientist can hardly meet with anything more undesirable than to
have the foundations give way just as the work is finished. I was put
in this position by a letter from Mr. Bertrand Russell when the work
was nearly through the press.
In Scientific American, May 1984, p 77.
%
Galbraith, John Kenneth
There can be no question, however, that prolonged commitment to
mathematical exercises in economics can be damaging. It leads to the
atrophy of judgement and intuition...
Economics, Peace, and Laughter.
%
Galilei, Galileo (1564 - 1642)
[The universe] cannot be read until we have learnt the language and
become familiar with the characters in which it is written. It is
written in mathematical language, and the letters are triangles,
circles and other geometrical figures, without which means it is
humanly impossible to comprehend a single word.
Opere Il Saggiatore p. 171.
%
Galilei, Galileo (1564 - 1642)
Measure what is measurable, and make measurable what is not so.
Quoted in H. Weyl "Mathematics and the Laws of Nature" in I Gordon
and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and
Schuster, 1959.
%
Galilei, Galileo (1564 - 1642)
And who can doubt that it will lead to the worst disorders when
minds created free by God are compelled to submit slavishly to an
outside will? When we are told to deny our senses and subject them
to the whim of others? When people devoid of whatsoever competence
are made judges over experts and are granted authority to treat them
as they please? These are the novelties which are apt to bring about
the ruin of commonwealths and the subversion of the state.
[On the margin of his own copy of Dialogue on the Great World
Systems].
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956, p. 733.
%
Galois, Evariste
Unfortunately what is little recognized is that the most worthwhile
scientific books are those in which the author clearly indicates what
he does not know; for an author most hurts his readers by concealing
difficulties.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.
%
Galton, [Sir] Francis (1822-1911)
Whenever you can, count.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Galton, Sir Francis (1822-1911)
[Statistics are] the only tools by which an opening can be cut
through the formidable thicket of difficulties that bars the path of
those who pursue the Science of Man.
Pearson, The Life and Labours of Francis Galton, 1914.
%
Galton, Sir Francis (1822-1911)
I know of scarcely anything so apt to impress the imagination as the
wonderful form of cosmic order expressed by the "Law of Frequency of
Error." The law would have been personified by the Greeks and
deified, if they had known of it. It reigns with serenity and in
complete self-effacement, amidst the wildest confusion. The huger the
mob, and the greater the apparent anarchy, the more perfect is its
sway. It is the supreme law of Unreason. Whenever a large sample of
chaotic elements are taken in hand and marshaled in the order of
their magnitude, an unsuspected and most beautiful form of regularity
proves to have been latent all along.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956. p. 1482.
%
Gardner, Martin
Biographical history, as taught in our public schools, is still
largely a history of boneheads: ridiculous kings and queens, paranoid
political leaders, compulsive voyagers, ignorant generals -- the
flotsam and jetsam of historical currents. The men who radically
altered history, the great scientists and mathematicians, are seldom
mentioned, if at all.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.
%
Gardner, Martin
Mathematics is not only real, but it is the only reality. That is
that entire universe is made of matter, obviously. And matter is made
of particles. It's made of electrons and neutrons and protons. So
the entire universe is made out of particles. Now what are the
particles made out of? They're not made out of anything. The only
thing you can say about the reality of an electron is to cite its
mathematical properties. So there's a sense in which matter has
completely dissolved and what is left is just a mathematical
structure.
Gardner on Gardner: JPBM Communications Award Presentation.
Focus-The Newsletter of the Mathematical Association of America v.
14, no. 6, December 1994.
%
Gauss, Karl Friedrich (1777-1855) I confess that Fermat's Theorem as
an isolated proposition has very little interest for me, because I
could easily lay down a multitude of such propositions, which one
could neither prove nor dispose of.
[A reply to Olbers' attempt in 1816 to entice him to work on
Fermat's Theorem.] In J. R. Newman (ed.) The World of Mathematics,
New York: Simon and Schuster, 1956. p. 312.
%
Gauss, Karl Friedrich (1777-1855)
If others would but reflect on mathematical truths as deeply and as
continuously as I have, they would make my discoveries.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956. p. 326.
%
Gauss, Karl Friedrich (1777-1855)
There are problems to whose solution I would attach an infinitely
greater importance than to those of mathematics, for example touching
ethics, or our relation to God, or concerning our destiny and our
future; but their solution lies wholly beyond us and completely
outside the province of science.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956. p. 314.
%
Gauss, Karl Friedrich (1777-1855)
You know that I write slowly. This is chiefly because I am never
satisfied until I have said as much as possible in a few words, and
writing briefly takes far more time than writing at length.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.
%
Gauss, Karl Friedrich (1777-1855)
God does arithmetic.
%
Gauss, Karl Friedrich (1777-1855)
We must admit with humility that, while number is purely a product
of our minds, space has a reality outside our minds, so that we
cannot completely prescribe its properties a priori.
Letter to Bessel, 1830.
%
Gauss, Karl Friedrich (1777-1855)
I mean the word proof not in the sense of the lawyers, who set two
half proofs equal to a whole one, but in the sense of a
mathematician, where half proof = 0, and it is demanded for proof
that every doubt becomes impossible.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.
%
Gauss, Karl Friedrich (1777-1855)
I have had my results for a long time: but I do not yet know how I
am to arrive at them.
In A. Arber The Mind and the Eye 1954.
%
Gauss, Karl Friedrich (1777-1855)
[His motto:]
Few, but ripe.
%
Gauss, Karl Friedrich (1777-1855)
[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
W. Shakespeare King Lear.
%
Gauss, Karl Friedrich (1777-1855)
[attributed to him by H.B Lubsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lubsen's geometry textbook.
%
Gauss, Karl Friedrich (1777-1855)
It is not knowledge, but the act of learning, not possession but the
act of getting there, which grants the greatest enjoyment. When I
have clarified and exhausted a subject, then I turn away from it, in
order to go into darkness again; the never-satisfied man is so
strange if he has completed a structure, then it is not in order to
dwell in it peacefully, but in order to begin another. I imagine the
world conqueror must feel thus, who, after one kingdom is scarcely
conquered, stretches out his arms for others.
Letter to Bolyai, 1808.
%
Gauss, Karl Friedrich (1777-1855)
Finally, two days ago, I succeeded - not on account of my hard
efforts, but by the grace of the Lord. Like a sudden flash of
lightning, the riddle was solved. I am unable to say what was the
conducting thread that connected what I previously knew with what
made my success possible.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Gauss, Karl Friedrich (1777-1855)
A great part of its [higher arithmetic] theories derives an
additional charm from the peculiarity that important propositions,
with the impress of simplicity on them, are often easily discovered
by induction, and yet are of so profound a character that we cannot
find the demonstrations till after many vain attempts; and even then,
when we do succeed, it is often by some tedious and artificial
process, while the simple methods may long remain concealed.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Gauss, Karl Friedrich (1777-1855)
I am coming more and more to the conviction that the necessity of
our geometry cannot be demonstrated, at least neither by, nor for,
the human intellect...geometry should be ranked, not with arithmetic,
which is purely aprioristic, but with mechanics.
Quoted in J. Koenderink Solid Shape, Cambridge Mass.: MIT Press,
1990.
%
Gay, John
Lest men suspect your tale untrue,
Keep probability in view.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956. p. 1334.
%
Gibbs, Josiah Willard (1839-1903)
Mathematics is a language.
%
Gilbert, W. S. (1836 - 1911)
I'm very good at integral and differential calculus, I know the
scientific names of beings animalculous; In short, in matters
vegetable, animal, and mineral, I am the very model of a modern
Major-General.
The Pirates of Penzance. Act 1.
%
Glaisher, J.W.
The mathematician requires tact and good taste at every step of his
work, and he has to learn to trust to his own instinct to distinguish
between what is really worthy of his efforts and what is not.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Glanvill, Joseph
And for mathematical science, he that doubts their certainty hath
need of a dose of hellebore.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956, p. 548.
%
Goethe
It has been said that figures rule the world. Maybe. But I am sure
that figures show us whether it is being ruled well or badly.
In J. P. Eckermann, Conversations with Goethe.
%
Goethe
Mathematics has the completely false reputation of yielding
infallible conclusions. Its infallibility is nothing but identity.
Two times two is not four, but it is just two times two, and that is
what we call four for short. But four is nothing new at all. And thus
it goes on and on in its conclusions, except that in the higher
formulas the identity fades out of sight.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956, p. 1754.
%
Goodman, Nicholas P.
There are no deep theorems -- only theorems that we have not
understood very well.
The Mathematical Intelligencer, vol. 5, no. 3, 1983.
%
Gordon, P
This is not mathematics, it is theology.
[On being exposed to Hilbert's work in invariant theory.]
Quoted in P. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhauser, 1981.
%
Graham, Ronald
It wouild be very discouraging if somewhere down the line you could
ask a computer if the Riemann hypothesis is correct and it said,
`Yes, it is true, but you won't be able to understand the proof.'
John Horgan. Scientific American 269:4 (October 1993) 92-103.
%
Grunbaum, Branko (1926 - ), and Shephard, G. C. (?)
Mathematicians have long since regarded it as demeaning to work on
problems related to elementary geometry in two or three dimensions,
in spite of the fact that it it precisely this sort of mathematics
which is of practical value.
Handbook of Applicable Mathematics.
%
Hadamard, Jacques
The shortest path between two truths in the real domain passes
through the complex domain.
Quoted in The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
%
Hadmard, Jacques
Practical application is found by not looking for it, and one can
say that the whole progress of civilization rests on that principle.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Haldane, John Burdon Sanderson (1892-1964)
In scientific thought we adopt the simplest theory which will
explain all the facts under consideration and enable us to predict
new facts of the same kind. The catch in this criterion lies in the
world "simplest." It is really an aesthetic canon such as we find
implicit in our criticisms of poetry or painting. The layman finds
such a law as dx/dt = K(d^2x/dy^2) much less simple than "it oozes,"
of which it is the mathematical statement. The physicist reverses
this judgment, and his statement is certainly the more fruitful of
the two, so far as prediction is concerned. It is, however, a
statement about something very unfamiliar to the plainman, namely,
the rate of change of a rate of change.
Possible Worlds, 1927.
%
Haldane, John Burdon Sanderson (1892-1964)
A time will however come (as I believe) when physiology will invade
and destroy mathematical physics, as the latter has destroyed
geometry.
Daedalus, or Science and the Future, London: Kegan Paul, 1923.
%
Halmos, Paul R.
Mathematics is not a deductive science -- that's a cliche. When you
try to prove a theorem, you don't just list the hypotheses, and then
start to reason. What you do is trial and error, experimentation,
guesswork.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
%
Halmos, Paul R.
... the student skit at Christmas contained a plaintive line: "Give
us Master's exams that our faculty can pass, or give us a faculty
that can pass our Master's exams."
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
%
Halmos, Paul R.
I remember one occasion when I tried to add a little seasoning to a
review, but I wasn't allowed to. The paper was by Dorothy Maharam,
and it was a perfectly sound contribution to abstract measure theory.
The domains of the underlying measures were not sets but elements of
more general Boolean algebras, and their range consisted not of
positive numbers but of certain abstract equivalence classes. My
proposed first sentence was: "The author discusses valueless measures
in pointless spaces."
I want to be a Mathematician, Washington: MAA Spectrum, 1985, p.
120.
%
Halmos, Paul R.
...the source of all great mathematics is the special case, the
concrete example. It is frequent in mathematics that every instance
of a concept of seemingly great generality is in essence the same as
a small and concrete special case.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
%
Halmos, Paul R.
The joy of suddenly learning a former secret and the joy of suddenly
discovering a hitherto unknown truth are the same to me -- both have
the flash of enlightenment, the almost incredibly enhanced vision,
and the ecstasy and euphoria of released tension.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
%
Halmos, Paul R.
Don't just read it; fight it! Ask your own questions, look for your
own examples, discover your own proofs. Is the hypothesis necessary?
Is the converse true? What happens in the classical special case?
What about the degenerate cases? Where does the proof use the
hypothesis?
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
%
Halmos, Paul R.
To be a scholar of mathematics you must be born with talent,
insight, concentration, taste, luck, drive and the ability to
visualize and guess.
I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
%
Hamilton, [Sir] William Rowan (1805-1865)
Who would not rather have the fame of Archimedes than that of his
conqueror Marcellus?
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber
and Schmidt, 1971.
%
Hamilton, Sir William Rowan (1805-1865)
I regard it as an inelegance, or imperfection, in quaternions, or
rather in the state to which it has been hitherto unfolded, whenever
it becomes or seems to become necessary to have recourse to x, y, z,
etc..
In a letter from Tait to Cayley.
%
Hamilton, Sir William Rowan (1805-1865)
On earth there is nothing great but man; in man there is nothing
great but mind.
Lectures on Metaphysics.
%
Hamming, Richard W.
Does anyone believe that the difference between the Lebesgue and
Riemann integrals can have physical significance, and that whether
say, an airplane would or would not fly could depend on this
difference? If such were claimed, I should not care to fly in that
plane.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Hamming, Richard W.
Mathematics is an interesting intellectual sport but it should not
be allowed to stand in the way of obtaining sensible information
about physical processes.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Hardy, Godfrey H. (1877 - 1947)
[On Ramanujan]
I remember once going to see him when he was lying ill at Putney. I
had ridden in taxi cab number 1729 and remarked that the number
seemed to me rather a dull one, and that I hoped it was not an
unfavorable omen. "No," he replied, "it is a very interesting number;
it is the smallest number expressible as the sum of two cubes in two
different ways."
Ramanujan, London: Cambridge Univesity Press, 1940.
%
Hardy, Godfrey H. (1877 - 1947)
Reductio ad absurdum, which Euclid loved so much, is one of a
mathematician's finest weapons. It is a far finer gambit than any
chess play: a chess player may offer the sacrifice of a pawn or even
a piece, but a mathematician offers the game.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Godfrey H. (1877 - 1947)
I am interested in mathematics only as a creative art.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Godfrey H. (1877 - 1947)
Pure mathematics is on the whole distinctly more useful than
applied. For what is useful above all is technique, and mathematical
technique is taught mainly through pure mathematics.
%
Hardy, Godfrey H. (1877 - 1947)
In great mathematics there is a very high degree of unexpectedness,
combined with inevitability and economy.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Godfrey H. (1877 - 1947)
There is no scorn more profound, or on the whole more justifiable,
than that of the men who make for the men who explain. Exposition,
criticism, appreciation, is work for second-rate minds.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Godfrey H. (1877 - 1947)
Young Men should prove theorems, old men should write books.
Quoted by Freeman Dyson in Freeman Dyson: Mathematician, Physicist,
and Writer. Interview with Donald J. Albers, The College Mathematics
Journal, vol. 25, No. 1, January 1994.
%
Hardy, Godfrey H. (1877 - 1947)
A science is said to be useful of its development tends to
accentuate the existing inequalities in the distribution of wealth,
or more directly promotes the destruction of human life.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Godfrey H. (1877 - 1947)
The mathematician's patterns, like the painter's or the poet's must
be beautiful; the ideas, like the colors or the words must fit
together in a harmonious way. Beauty is the first test: there is no
permanent place in this world for ugly mathematics.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Godfrey H. (1877 - 1947)
I believe that mathematical reality lies outside us, that our
function is to discover or observe it, and that the theorems which we
prove, and which we describe grandiloquently as our "creations," are
simply the notes of our observations.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Godfrey H. (1877 - 1947)
Archimedes will be remembered when Aeschylus is forgotten, because
languages die and mathematical ideas do not. "Immortality" may be a
silly word, but probably a mathematician has the best chance of
whatever it may mean.
A Mathematician's Apology, London, Cambridge University Press,1941.
%
Hardy, Godfrey H. (1877 - 1947)
The fact is that there are few more "popular" subjects than
mathematics. Most people have some appreciation of mathematics, just
as most people can enjoy a pleasant tune; and there are probably more
people really interested in mathematics than in music. Appearances
may suggest the contrary, but there are easy explanations. Music can
be used to stimulate mass emotion, while mathematics cannot; and
musical incapacity is recognized (no doubt rightly) as mildly
discreditable, whereas most people are so frightened of the name of
mathematics that they are ready, quite unaffectedly, to exaggerate
their own mathematical stupidity.
A Mathematician's Apology, London, Cambridge University Press, 1941.
%
Hardy, Thomas
...he seemed to approach the grave as an hyperbolic curve approaches
a line, less directly as he got nearer, till it was doubtful if he
would ever reach it at all.
Far from the Madding Crowd.
%
Harish-Chandra
I have often pondered over the roles of knowledge or experience, on
the one hand, and imagination or intuition, on the other, in the
process of discovery. I believe that there is a certain fundamental
conflict between the two, and knowledge, by advocating caution, tends
to inhibit the flight of imagination. Therefore, a certain naivete,
unburdened by conventional wisdom, can sometimes be a positive asset.
R. Langlands, "Harish-Chandra," Biographical Memoirs of Fellows of
the Royal Society 31 (1985) 197 - 225.
%
Harris, Sydney J.
The real danger is not that computers will begin to think like men,
but that men will begin to think like computers.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Hawking, Stephen Williams (1942- ) God not only plays dice. He also
sometimes throws the dice where they cannot be seen.
[See related quotation from Albert Einstein.] Nature 1975 257.
%
Heath, Sir Thomas
[The works of Archimedes] are without exception, monuments of
mathematical exposition; the gradual revelation of the plan of
attack, the masterly ordering of the propositions, the stern
elimination of everything not immediately relevant to the purpose,
the finish of the whole, are so impressive in their perfection as to
create a feeling akin to awe in the mind of the reader.
A History of Greek Mathematics. 1921.
%
Heaviside, Oliver (1850-1925)
[Criticized for using formal mathematical manipulations, without
understanding how they worked:]
Should I refuse a good dinner simply because I do not understand the
process of digestion?
%
Heisenberg, Werner (1901-1976)
An expert is someone who knows some of the worst mistakes that can
be made in his subject, and how to avoid them.
Physics and Beyond. 1971.
%
Hempel, Carl G.
The propositions of mathematics have, therefore, the same
unquestionable certainty which is typical of such propositions as
"All bachelors are unmarried," but they also share the complete lack
of empirical content which is associated with that certainty: The
propositions of mathematics are devoid of all factual content; they
convey no information whatever on any empirical subject matter.
"On the Nature of Mathematical Truth" in J. R. Newman (ed.) The
World of Mathematics, New York: Simon and Schuster, 1956.
%
Hempel, Carl G.
The most distinctive characteristic which differentiates mathematics
from the various branches of empirical science, and which accounts
for its fame as the queen of the sciences, is no doubt the peculiar
certainty and necessity of its results.
"Geometry and Empirical Science" in J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956.
%
Hempel, Carl G.
...to characterize the import of pure geometry, we might use the
standard form of a movie-disclaimer: No portrayal of the
characteristics of geometrical figures or of the spatial properties
of relationships of actual bodies is intended, and any similarities
between the primitive concepts and their customary geometrical
connotations are purely coincidental.
"Geometry and Empirical Science" in J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956.
%
Henkin, Leon
One of the big misapprehensions about mathematics that we perpetrate
in our classrooms is that the teacher always seems to know the answer
to any problem that is discussed. This gives students the idea that
there is a book somewhere with all the right answers to all of the
interesting questions, and that teachers know those answers. And if
one could get hold of the book, one would have everything settled.
That's so unlike the true nature of mathematics.
L.A. Steen and D.J. Albers (eds.), Teaching Teachers, Teaching
Students, Boston: Birkhauser, 1981, p89.
%
Hermite, Charles (1822 - 1901)
There exists, if I am not mistaken, an entire world which is the
totality of mathematical truths, to which we have access only with
our mind, just as a world of physical reality exists, the one like
the other independent of ourselves, both of divine creation.
In The Mathematical Intelligencer, v. 5, no. 4.
%
Hermite, Charles (1822-1901)
Abel has left mathematicians enough to keep them busy for 500 years.
In G. F. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Hermite, Charles (1822-1901)
We are servants rather than masters in mathematics.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Hertz, Heinrich
One cannot escape the feeling that these mathematical formulas have
an independent existence and an intelligence of their own, that they
are wiser that we are, wiser even than their discoverers, that we get
more out of them than was originally put into them.
Quoted by ET Bell in Men of Mathematics, New York, 937.
%
Hesse, Hermann (1877-1962)
You treat world history as a mathematician does mathematics, in
which nothing but laws and formulae exist, no reality, no good and
evil, no time, no yesterday, no tomorrow, nothing but an eternal,
shallow, mathematical present.
The Glass Bead Game, 1943.
%
Hilbert, David (1862-1943)
Wir mussen wissen.
Wir werden wissen.
[Engraved on his tombstone in Göttingen.]
%
Hilbert, David (1862-1943)
Before beginning I should put in three years of intensive study, and
I haven't that much time to squander on a probable failure.
[On why he didn't try to solve Fermat's last theorem]
Quoted in E.T. Bell Mathematics, Queen and Servant of Science, New
York: McGraw Hill Inc., 1951.
%
Hilbert, David (1862-1943)
Galileo was no idiot. Only an idiot could believe that science
requires martyrdom - that may be necessary in religion, but in time a
scientific result will establish itself.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1971.
%
Hilbert, David (1862-1943)
Mathematics is a game played according to certain simple rules with
meaningless marks on paper.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Hilbert, David (1862-1943)
Physics is much too hard for physicists.
C. Reid Hilbert, London: Allen and Unwin, 1970.
%
Hilbert, David (1862-1943)
How thoroughly it is ingrained in mathematical science that every
real advance goes hand in hand with the invention of sharper tools
and simpler methods which, at the same time, assist in understanding
earlier theories and in casting aside some more complicated
developments.
%
Hilbert, David (1862-1943)
The art of doing mathematics consists in finding that special case
which contains all the germs of generality.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Hilbert, David (1862-1943)
The further a mathematical theory is developed, the more
harmoniously and uniformly does its construction proceed, and
unsuspected relations are disclosed between hitherto separated
branches of the science.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Hilbert, David (1862-1943)
I have tried to avoid long numerical computations, thereby following
Riemann's postulate that proofs should be given through ideas and not
voluminous computations.
Report on Number Theory, 1897.
%
Hilbert, David (1862-1943)
One can measure the importance of a scientific work by the number of
earlier publications rendered superfluous by it.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber and
Schmidt,1971.
%
Hilbert, David (1862-1943)
Mathematics knows no races or geographic boundaries; for
mathematics,the cultural world is one country.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Hilbert, David (1862-1943)
The infinite! No other question has ever moved so profoundly the
spirit of man.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Hirst, Thomas Archer
10th August 1851: On Tuesday evening at Museum, at a ball in the
gardens. The night was chill, I dropped too suddenly from
Differential Calculus into ladies' society, and could not give myself
freely to the change. After an hour's attempt so to do, I returned,
cursing the mode of life I was pursuing; next morning I had already
shaken hands, however, with Diff. Calculus, and forgot the ladies....
J. Helen Gardner and Robin J. Wilson, "Thomas Archer Hirst -
Mathematician Xtravagant II - Student Days in Germany", The American
Mathematical Monthly , v. 6, no. 100.
%
Hobbes, Thomas
There is more in Mersenne than in all the universities together.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Hobbes, Thomas
To understand this for sense it is not required that a man should be
a geometrician or a logician, but that he should be mad.
["This" is that the volume generated by revolving the region under
1/x from 1 to infinity has finite volume.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Hobbes, Thomas
Geometry, which is the only science that it hath pleased God
hitherto to bestow on mankind.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Hobbes, Thomas
The errors of definitions multiply themselves according as the
reckoning proceeds; and lead men into absurdities, which at last they
see but cannot avoid, without reckoning anew from the beginning.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Holmes, Oliver Wendell
Descartes commanded the future from his study more than Napoleon
from the throne.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Holmes, Oliver Wendell
Certitude is not the test of certainty. We have been cocksure of
many things that are not so.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Holmes, Oliver Wendell
I was just going to say, when I was interrupted, that one of the
many ways of classifying minds is under the heads of arithmetical and
algebraical intellects. All economical and practical wisdom is an
extension of the following arithmetical formula: 2 + 2 = 4. Every
philosophical proposition has the more general character of the
expression a + b = c. We are mere operatives, empirics, and egotists
until we learn to think in letters instead of figures.
The Autocrat of the Breakfast Table.
%
Holt, M. and Marjoram, D. T. E.
The truth of the matter is that, though mathematics truth may be
beauty, it can be only glimpsed after much hard thinking. Mathematics
is difficult for many human minds to grasp because of its
hierarchical structure: one thing builds on another and depends on it.
Mathematics in a Changing World Walker, New York 1973.
%
Hofstadter, Douglas R. (1945 - )
Hofstadter's Law: It always takes longer than you expect, even when
you take into account Hofstadter's Law.
Gödel, Escher, Bach 1979.
%
Hughes, Richard
Science, being human enquiry, can hear no answer except an answer
couched somehow in human tones. Primitive man stood in the mountains
and shouted against a cliff; the echo brought back his own voice, and
he believed in a disembodied spirit. The scientist of today stands
counting out loud in the face of the unknown. Numbers come back to
him - and he believes in the Great Mathematician.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Hume, David (1711 - 1776)
If we take in our hand any volume; of divinity or school
metaphysics, for instance; let us ask, `Does it contain any abstract
reasoning concerning quantity or number?' No. `Does it contain any
experimental reasoning concerning matter of fact and existence?' No.
Commit it then to the flames: for it can contain nothing but
sophistry and illusion.
Treatise Concerning Human Understanding.
%
Huxley, Aldous
I admit that mathematical science is a good thing. But excessive
devotion to it is a bad thing.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.
%
Huxley, Aldous
If we evolved a race of Isaac Newtons, that would not be progress.
For the price Newton had to pay for being a supreme intellect was
that he was incapable of friendship, love, fatherhood, and many other
desirable things. As a man he was a failure; as a monster he was
superb.
Interview with J. W. N. Sullivan, Contemporary Mind, London, 1934.
%
Huxley, Aldous
...[he] was as much enchanted by the rudiments of algebra as he
would have been if I had given him an engine worked by steam, with a
methylated spirit lamp to heat the boiler; more enchanted, perhapsfor
the engine would have got broken, and, remaining always itself, would
in any case have lost its charm, while the rudiments of algebra
continued to grow and blossom in his mind with an unfailing
luxuriance. Every day he made the discovery of something which seemed
to him exquisitely beautiful; the new toy was inexhaustible in its
potentialities.
Young Archimedes.
%
Huxley, Thomas Henry (1825-1895)
This seems to be one of the many cases in which the admitted
accuracy of mathematical processes is allowed to throw a wholly
inadmissible appearance of authority over the results obtained by
them. Mathematics may be compared to a mill of exquisite workmanship,
which grinds your stuff of any degree of fineness; but, nevertheless,
what you get out depends on what you put in; and as the grandest mill
in the world will not extract wheat flour from peascods, so pages of
formulae will not get a definite result out of loose data.
Quarterly Journal of the Geological Society, 25,1869.
%
Huxley, Thomas Henry (1825-1895)
The mathematician starts with a few propositions, the proof of which
is so obvious that they are called selfevident, and the rest of his
work consists of subtle deductions from them. The teaching of
languages, at any rate as ordinarily practised, is of the same
general nature authority and tradition furnish the data, and the
mental operations are deductive.
"Scientific Education -Notes of an After-dinner Speech." Macmillan's
Magazine Vol XX, 1869.
%
Huxley, Thomas Henry (1825-1895)
It is the first duty of a hypothesis to be intelligible.
%
Ibn Khaldun (1332-1406)
Geometry enlightens the intellect and sets one's mind right. All of
its proofs are very clear and orderly. It is hardly possible for
errors to enter into geometrical reasoning, because it is well
arranged and orderly. Thus, the mind that constantly applies itself
to geometry is not likely to fall into error. In this convenient way,
the person who knows geometry acquires intelligence.
The Muqaddimah. An Introduction to History.
%
Isidore of Seville (ca 600 ad)
Take from all things their number and all shall perish.
%
Jacobi, Carl
It is true that Fourier had the opinion that the principal aim of
mathematics was public utility and explanation of natural phenomena;
but a philosopher like him should have known that the sole end of
science is the honor of the human mind, and that under this title a
question about numbers is worth as much as a question about the
system of the world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Jacobi, Carl
God ever arithmetizes.
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber
and Schmidt, 1971.
%
Jacobi, Carl
One should always generalize.
(Man muss immer generalisieren)
In P. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhauser, 1981.
%
Jacobi, Carl
The real end of science is the honor of the human mind.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
%
Jacobi, Carl
It is often more convenient to possess the ashes of great men than
to possess the men themselves during their lifetime.
[Commenting on the return of Descartes' remains to France]
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Jacobi, Carl
Mathematics is the science of what is clear by itself.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
James, William (1842 - 1910)
The union of the mathematician with the poet, fervor with measure,
passion with correctness, this surely is the ideal.
Collected Essays.
%
Jeans, Sir James
The essential fact is that all the pictures which science now draws
of nature, and which alone seem capable of according with
observational facts, are mathematical pictures.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Jeans, Sir James
From the intrinsic evidence of his creation, the Great Architect of
the Universe now begins to appear as a pure mathematician.
Mysterious Universe.
%
Jefferson, Thomas
...the science of calculation also is indispensable as far as the
extraction of the square and cube roots: Algebra as far as the
quadratic equation and the use of logarithms are often of value in
ordinary cases: but all beyond these is but a luxury; a delicious
luxury indeed; but not be in indulged in by one who is to have a
profession to follow for his subsistence.
In J. Robert Oppenheimer "The Encouragement of Science" in I.
Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York:
Simon and Schuster, 1959.
%
Jevons, William Stanley
It is clear that Economics, if it is to be a science at all, must be
a mathematical science.
Theory of Political Economy.
%
Johnson, Samuel (1709-1784)
Sir, I have found you an argument. I am not obliged to find you an
understanding.
J. Boswell The Life of Samuel Johnson, 1784.
%
Jowett, Benjamin (1817 - 1893)
Logic is neither a science or an art, but a dodge.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Kant, Emmanual (1724 - 1804)
The science of mathematics presents the most brilliant example of
how pure reason may successfully enlarge its domain without the aid
of experience.
The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
%
Kant, Emmanual (1724 - 1804)
All human knowledge thus begins with intuitions, proceeds thence to
concepts, and ends with ideas.
Quoted in Hilbert's Foundations of Geometry.
%
Kaplan, Abraham
Mathematics is not yet capable of coping with the naivete of the
mathematician himself.
Sociology Learns the Language of Mathematics.
%
Kaplansky, Irving
We [he and Halmos] share a philosophy about linear algebra: we think
basis-free, we write basis-free , but when the chips are down we
close the office door and compute with matrices like fury.
Paul Halmos: Celebrating 50 Years of Mathematics.
%
Karlin, Samuel (1923 - )
The purpose of models is not to fit the data but to sharpen the
questions.
11th R A Fisher Memorial Lecture, Royal Society 20, April 1983.
%
Kasner, E. and Newman, J.
Mathematics is man's own handiwork, subject only to the limitations
imposed by the laws of thought.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
%
Kasner, E. and Newman, J.
...we have overcome the notion that mathematical truths have an
existence independent and apart from our own minds. It is even
strange to us that such a notion could ever have existed.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
%
Kasner, E. and Newman, J.
Mathematics is the science which uses easy words for hard ideas.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
%
Kasner, E. and Newman, J.
Mathematics is often erroneously referred to as the science of
common sense. Actually, it may transcend common sense and go beyond
either imagination or intuition. It has become a very strange and
perhaps frightening subject from the ordinary point of view, but
anyone who penetrates into it will find a veritable fairyland, a
fairyland which is strange, but makes sense, if not common sense.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
%
Kasner, E. and Newman, J.
Perhaps the greatest paradox of all is that there are paradoxes in
mathematics.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
%
Kasner, E. and Newman, J.
When the mathematician says that such and such a proposition is true
of one thing, it may be interesting, and it is surely safe. But when
he tries to extend his proposition to everything, though it is much
more interesting, it is also much more dangerous. In the transition
from one to all, from the specific to the general, mathematics has
made its greatest progress, and suffered its most serious setbacks,
of which the logical paradoxes constitute the most important part.
For, if mathematics is to advance securely and confidently it must
first set its affairs in order at home.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
%
Kasner, E. and Newman, J. R.
The testament of science is so continually in a flux that the heresy
of yesterday is the gospel of today and the fundamentalism of
tomorrow.
E. Kasner and J. R. Newman, Mathematics and the Imagination, Simon
and Schuster, 1940.
%
Keller, Helen (1880 - 1968)
Now I feel as if I should succeed in doing something in mathematics,
although I cannot see why it is so very important... The knowledge
doesn't make life any sweeter or happier, does it?
The Story of My Life. 1903.
%
Kelley, John
A topologist is one who doesn't know the difference between a
doughnut and a coffee cup.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Kepler, Johannes (1571-1630)
A mind is accustomed to mathematical deduction, when confronted with
the faulty foundations of astrology, resists a long, long time, like
an obstinate mule, until compelled by beating and curses to put its
foot into that dirty puddle.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Kepler, Johannes (1571-1630)
Where there is matter, there is geometry.
(Ubi materia, ibi geometria.)
J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990
%
Kepler, Johannes (1571-1630)
The chief aim of all investigations of the external world should be
to discover the rational order and harmony which has been imposed on
it by God and which He revealed to us in the language of mathematics.
%
Kepler, Johannes (1571-1630)
Nature uses as little as possible of anything.
%
Keynes, John Maynard
It has been pointed out already that no knowledge of probabilities,
less in degree than certainty, helps us to know what conclusions are
true, and that there is no direct relation between the truth of a
proposition and its probability. Probability begins and ends with
probability.
The Application of Probability to Conduct.
%
Kleinhenz, Robert J.
When asked what it was like to set about proving something, the
mathematician likened proving a theorem to seeing the peak of a
mountain and trying to climb to the top. One establishes a base camp
and begins scaling the mountain's sheer face, encountering obstacles
at every turn, often retracing one's steps and struggling every foot
of the journey. Finally when the top is reached, one stands examining
the peak, taking in the view of the surrounding countrysideand then
noting the automobile road up the other side!
%
Kline, Morris
A proof tells us where to concentrate our doubts.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Kline, Morris
Statistics: the mathematical theory of ignorance.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Kline, Morris
Logic is the art of going wrong with confidence.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Kline, Morris
Universities hire professors the way some men choose wives - they
want the ones the others will admire.
Why the Professor Can't Teach. St. Martin's Press, 1977. p 92.
%
Koestler, Arthur (1905- )
In the index to the six hundred odd pages of Arnold Toynbee's A
Study of History, abridged version, the names of Copernicus, Galileo,
Descartes and Newton do not occur yet their cosmic quest destroyed
the medieval vision of an immutable social order in a walled-in
universe and transformed the European landscape, society, culture,
habits and general outlook, as thoroughly as if a new species had
arisen on this planet.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Koestler, Arthur (1905- )
Nobody before the Pythagoreans had thought that mathematical
relations held the secret of the universe. Twenty-five centuries
later, Europe is still blessed and cursed with their heritage. To
non-European civilizations, the idea that numbers are the key to both
wisdom and power, seems never to have occurred.
The Sleepwalkers. 1959.
%
Kovalevsky, Sonja
Say what you know, do what you must, come what may.
[Motto on her paper "On the Problem of the Rotation of a Solid Body
about a Fixed Point."]
%
Kraft, Prinz zu Hohlenlohe-Ingelfingen (1827 - 1892)
Mathematics is indeed dangerous in that it absorbs students to such
a degree that it dulls their senses to everything else.
Attributed by Karl Schellbach. In H. Eves Mathematical Circles
Adieu, Boston: Prindle, Weber and Schmidt, 1977.
%
Kronecker, Leopold (1823 - 1891)
God made the integers, all else is the work of man.
Jahresberichte der Deutschen Mathematiker Vereinigung.
%
Kronecker, Leopold (1823-1891)
Number theorists are like lotus-eaters -- having once tasted of this
food they can never give it up.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
La Touche, Mrs.
I do hate sums. There is no greater mistake than to call arithmetic
an exact science. There are permutations and aberrations discernible
to minds entirely noble like mine; subtle variations which ordinary
accountants fail to discover; hidden laws of number which it requires
a mind like mine to perceive. For instance, if you add a sum from the
bottom up, and then from the top down, the result is always different.
Mathematical Gazette, v. 12.
%
LaGrange, Joseph-Louis
The reader will find no figures in this work. The methods which I
set forth do not require either constructions or geometrical or
mechanical reasonings: but only algebraic operations, subject to a
regular and uniform rule of procedure.
Preface to Mécanique Analytique.
%
LaGrange, Joseph-Louis
[said about the chemist Lavoisier:]
It took the mob only a moment to remove his head; a century will not
suffice to reproduce it.
H. Eves An Introduction to the History of Mathematics, 5th Ed.,
Saunders.
%
LaGrange, Joseph-Louis
When we ask advice, we are usually looking for an accomplice.
%
Lakatos, Imre
That sometimes clear ... and sometimes vague stuff ... which is ...
mathematics.
In P. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhauser, 1981.
%
Lanczos, Cornelius
Most of the arts, as painting, sculpture, and music, have emotional
appeal to the general public. This is because these arts can be
experienced by some one or more of our senses. Such is not true of
the art of mathematics; this art can be appreciated only by
mathematicians, and to become a mathematician requires a long period
of intensive training. The community of mathematicians is similar to
an imaginary community of musical composers whose only satisfaction
is obtained by the interchange among themselves of the musical scores
they compose.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Landau, E.
[Asked for a testimony to the effect that Emmy Noether was a great
woman mathematician, he said:]
I can testify that she is a great mathematician, but that she is a
woman, I cannot swear.
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co ltd.,
1953.
%
Landau, Susan
There's a touch of the priesthood in the academic world, a sense
that a scholar should not be distracted by the mundane tasks of
day-to-day living. I used to have great stretches of time to work.
Now I have research thoughts while making peanut butter and jelly
sandwiches. Sure it's impossible to write down ideas while reading
"curious George" to a two-year-old. On the other hand, as my husband
was leaving graduate school for his first job, his thesis advisor
told him, "You may wonder how a professor gets any research done when
one has to teach, advise students, serve on committees, referee
papers, write letters of recommendation, interview prospective
faculty. Well, I take long showers."
In Her Own Words: Six Mathematicians Comment on Their Lives and
Careers. Notices of the AMS, V. 38, no. 7 (September 1991), p. 704.
%
Lang, Andrew (1844-1912)
He uses statistics as a drunken man uses lamp posts -- for support
rather than illumination.
Treasury of Humorous Quotations.
%
Langer, Rudoph E.
[about Fourier] It was, no doubt, partially because of his very
disregard for rigor that he was able to take conceptual steps which
were inherently impossible to men of more critical genius.
In P. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhauser, 1981.
%
Lao Tze (604-531 B.C.)
A good calculator does not need artificial aids.
Tao Te Ching, ch 27.
%
de Laplace, Pierre-Simon (1749 - 1827)
What we know is not much. What we do not know is immense.
(Allegedly his last words.)
DeMorgan's Budget of Paradoxes.
%
de Laplace, Pierre-Simon (1749 - 1827)
[His last words, according to De Morgan:]
Man follows only phantoms.
DeMorgan's Budget of Paradoxes.
%
de Laplace, Pierre-Simon (1749 - 1827)
Nature laughs at the difficulties of integration.
In J. W. Krutch "The Colloid and the Crystal", in I. Gordon and S.
Sorkin (eds.) The Armchair Science Reader, New York: Simon and
Schuster, 1959.
%
de Laplace, Pierre-Simon (1749 - 1827)
Read Euler: he is our master in everything.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
de Laplace, Pierre-Simon (1749 - 1827)
Such is the advantage of a well constructed language that its
simplified notation often becomes the source of profound theories.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
de Laplace, Pierre-Simon (1749 - 1827)
Napoleon: You have written this huge book on the system of the world
without once mentioning the author of the universe.
Laplace: Sire, I had no need of that hypothesis.
Later when told by Napoleon about the incident, Lagrange commented:
Ah, but that is a fine hypothesis. It explains so many things.
DeMorgan's Budget of Paradoxes.
%
de Laplace, Pierre-Simon (1749 - 1827)
[said about Napier's logarithms:]
...by shortening the labors doubled the life of the astronomer.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
%
de Laplace, Pierre-Simon (1749 - 1827)
It is India that gave us the ingenious method of expressing all
numbers by means of ten symbols, each symbol receiving a value of
position as well as an absolute value; a profound and important idea
which appears so simple to us now that we ignore its true merit. But
its very simplicity and the great ease which it has lent to
computations put our arithmetic in the first rank of useful
inventions; and we shall appreciate the grandeur of the achievement
the more when we remember that it escaped the genius of Archimedes
and Apollonius, two of the greatest men produced by antiquity.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Leach, Edmund Ronald (1910 - 1989)
How can a modern anthropologist embark upon a generalization with
any hope of arriving at a satisfactory conclusion? By thinking of the
organizational ideas that are present in any society as a
mathematical pattern.
Rethinking Anthropology. 1961.
%
Leacock, Stephen
How can you shorten the subject? That stern struggle with the
multiplication table, for many people not yet ended in victory, how
can you make it less? Square root, as obdurate as a hardwood stump
in a pasturenothing but years of effort can extract it. You can't
hurry the process. Or pass from arithmetic to algebra; you can't
shoulder your way past quadratic equations or ripple through the
binomial theorem. Instead, the other way; your feet are impeded in
the tangled growth, your pace slackens, you sink and fall somewhere
near the binomial theorem with the calculus in sight on the horizon.
So died, for each of us, still bravely fighting, our mathematical
training; except for a set of people called "mathematicians" -- born
so, like crooks.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Lebesgue, Henri (1875 - 1941)
In my opinion, a mathematician, in so far as he is a mathematician,
need not preoccupy himself with philosophy -- an opinion, moreover,
which has been expressed by many philosophers.
Scientific American, 211, September 1964, p. 129.
%
Lehrer, Thomas Andrew (1928- )
In one word he told me the secret of success in mathematics:
plagiarize only be sure always to call it please research.
Lobachevski (A musical recording.)
%
Leibniz, Gottfried Whilhem (1646-1716)
[about him:]
It is rare to find learned men who are clean, do not stink and have
a sense of humour.
[attributed variously to Charles Louis de Secondat Montesquieu and
to the Duchess of Orléans]
%
Leibniz, Gottfried Whilhem (1646-1716)
Nothing is more important than to see the sources of invention which
are, in my opinion more interesting than the inventions themselves.
J. Koenderink, Solid Shape, Cambridge Mass.: MIT Press, 1990.
%
Leibniz, Gottfried Whilhem (1646-1716)
Music is the pleasure the human soul experiences from counting
without being aware that it is counting.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Leibniz, Gottfried Whilhem (1646-1716)
The imaginary number is a fine and wonderful recourse of the divine
spirit, almost an amphibian between being and not being.
%
Leibniz, Gottfried Whilhem (1646-1716)
He who understands Archimedes and Apollonius will admire less the
achievements of the foremost men of later times.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Leibniz, Gottfried Whilhem (1646-1716)
In symbols one observes an advantage in discovery which is greatest
when they express the exact nature of a thing briefly and, as it
were, picture it; then indeed the labor of thought is wonderfully
diminished.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Leibniz, Gottfried Whilhem (1646-1716)
The art of discovering the causes of phenomena, or true hypothesis,
is like the art of decyphering, in which an ingenious conjecture
greatly shortens the road.
New Essays Concerning Human Understanding, IV, XII.
%
Leibniz, Gottfried Whilhem (1646-1716)
Although the whole of this life were said to be nothing but a dream
and the physical world nothing but a phantasm, I should call this
dream or phantasm real enough, if, using reason well, we were never
deceived by it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Leibniz, Gottfried Whilhem (1646-1716)
The soul is the mirror of an indestructible universe.
The Monadology.
%
da Vinci, Leonardo (1452-1519)
Whoever despises the high wisdom of mathematics nourishes himself on
delusion and will never still the sophistic sciences whose only
product is an eternal uproar.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
da Vinci, Leonardo (1452 - 1519)
Mechanics is the paradise of the mathematical sciences, because by
means of it one comes to the fruits of mathematics.
Notebooks, v. 1, ch. 20.
%
da Vinci, Leonardo (1452-1519)
He who loves practice without theory is like the sailor who boards
ship without a rudder and compass and never knows where he may cast.
%
da Vinci, Leonardo (1452-1519)
No human investigation can be called real science if it cannot be
demonstrated mathematically.
%
da Vinci, Leonardo (1452-1519)
Inequality is the cause of all local movements.
%
Leybourn, William (1626-1700)
But leaving those of the Body, I shall proceed to such Recreation as
adorn the Mind; of which those of the Mathematicks are inferior to
none.
Pleasure with Profit, 1694.
%
Lichtenberg, Georg Christoph (1742 - 1799)
All mathematical laws which we find in Nature are always suspect to
me, in spite of their beauty. They give me no pleasure. They are
merely auxiliaries. At close range it is all not true.
In J P Stern Lichtenberg, 1959.
%
Lichtenberg, Georg Christoph (1742 - 1799)
The great trick of regarding small departures from the truth as the
truth itself -- on which is founded the entire integral calculus --
is also the basis of our witty speculations, where the whole thing
would often collapse if we considered the departures with
philosophical rigour.
Aphorisms.
%
Lichtenberg, Georg Christoph (1742 - 1799)
In mathematical analysis we call x the undetermined part of line a:
the rest we don't call y, as we do in common life, but a-x. Hence
mathematical language has great advantages over the common language.
%
Lichtenberg, Georg Christoph (1742 - 1799)
I have often noticed that when people come to understand a
mathematical proposition in some other way than that of the ordinary
demonstration, they promptly say, "Oh, I see. That's how it must be."
This is a sign that they explain it to themselves from within their
own system.
%
le Lionnais, Francois
Who has not be amazed to learn that the function y = e^x , like a
phoenix rising again from its own ashes, is its own derivative?
Great Currents of Mathematical Thought, vol. 1, New York: Dover
Publications.
%
Lippman, Gabriel (1845-1921)
[On the Gaussian curve, remarked to Poincaré:]
Experimentalists think that it is a mathematical theorem while the
mathematicians believe it to be an experimental fact.
In D'Arcy Thompson On Growth and Form, 1917.
%
Littlewood, J. E. (1885 -1977)
It is true that I should have been surprised in the past to learn
that Professor Hardy had joined the Oxford Group. But one could not
say the adverse chance was 1:10. Mathematics is a dangerous
profession; an appreciable proportion of us go mad, and then this
particular event would be quite likely.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
A good mathematical joke is better, and better mathematics, than a
dozen mediocre papers.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
I recall once saying that when I had given the same lecture several
times I couldn't help feeling that they really ought to know it by
now.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
In passing, I firmly believe that research should be offset by a
certain amount of teaching, if only as a change from the agony of
research. The trouble, however, I freely admit, is that in practice
you get either no teaching, or else far too much.
"The Mathematician's Art of Work" in Béla Bollobás (ed.)
Littlewood's Miscellany, Cambridge: Cambridge University Press, 1986.
%
Littlewood, J. E. (1885 -1977)
It is possible for a mathematician to be "too strong" for a given
occasion. He forces through, where another might be driven to a
different, and possible more fruitful, approach. (So a rock climber
might force a dreadful crack, instead of finding a subtle and
delicate route.)
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
I constantly meet people who are doubtful, generally without due
reason, about their potential capacity [as mathematicians]. The first
test is whether you got anything out of geometry. To have disliked or
failed to get on with other [mathematical] subjects need mean
nothing; much drill and drudgery is unavoidable before they can get
started, and bad teaching can make them unintelligible even to a born
mathematician.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
The infinitely competent can be uncreative.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Littlewood, J. E. (1885 -1977)
In presenting a mathematical argument the great thing is to give the
educated reader the chance to catch on at once to the momentary point
and take details for granted: his successive mouthfuls should be such
as can be swallowed at sight; in case of accidents, or in case he
wishes for once to check in detail, he should have only a clearly
circumscribed little problem to solve (e.g. to check an identity: two
trivialities omitted can add up to an impasse). The unpractised
writer, even after the dawn of a conscience, gives him no such
chance; before he can spot the point he has to tease his way through
a maze of symbols of which not the tiniest suffix can be skipped.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
A linguist would be shocked to learn that if a set is not closed
this does not mean that it is open, or again that "E is dense in E"
does not mean the same thing as "E is dense in itself".
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
The surprising thing about this paper is that a man who could write
it would.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
A precisian professor had the habit of saying: "... quartic
polynomial ax^4+bx^3+cx^2+dx+e , where e need not be the base of the
natural logarithms."
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
%
Littlewood, J. E. (1885 -1977)
I read in the proof sheets of Hardy on Ramanujan: "As someone said,
each of the positive integers was one of his personal friends." My
reaction was, "I wonder who said that; I wish I had." In the next
proof-sheets I read (what now stands), "It was Littlewood who said..."
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.
%
Littlewood, J. E. (1885 -1977)
We come finally, however, to the relation of the ideal theory to
real world, or "real" probability. If he is consistent a man of the
mathematical school washes his hands of applications. To someone who
wants them he would say that the ideal system runs parallel to the
usual theory: "If this is what you want, try it: it is not my
business to justify application of the system; that can only be done
by philosophizing; I am a mathematician". In practice he is apt to
say: "try this; if it works that will justify it". But now he is not
merely philosophizing; he is committing the characteristic fallacy.
Inductive experience that the system works is not evidence.
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.
%
Littlewood, J. E. (1885 -1977)
The theory of numbers is particularly liable to the accusation that
some of its problems are the wrong sort of questions to ask. I do not
myself think the danger is serious; either a reasonable amount of
concentration leads to new ideas or methods of obvious interest, or
else one just leaves the problem alone. "Perfect numbers" certainly
never did any good, but then they never did any particular harm.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
%
Lobatchevsky, Nikolai
There is no branch of mathematics, however abstract, which may not
some day be applied to phenomena of the real world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Locke, John
...mathematical proofs, like diamonds, are hard and clear, and will
be touched with nothing but strict reasoning.
D. Burton, Elementary Number Theory, Boston: Allyn and Bacon 1980.
%
Luther, Martin (1483-1546)
Medicine makes people ill, mathematics make them sad and theology
makes them sinful.
%
Mach, Ernst (1838 - 1916)
Archimedes constructing his circle pays with his life for his
defective biological adaptation to immediate circumstances.
%
Mach, Ernst (1838-1916)
The mathematician who pursues his studies without clear views of
this matter, must often have the uncomfortable feeling that his paper
and pencil surpass him in intelligence.
"The Economy of Science" in J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956.
%
Mackay, Alan Lindsay (1926- )
Like the ski resort full of girls hunting for husbands and husbands
hunting for girls, the situation is not as symmetrical as it might
seem.
A Dictionary of Scientific Quotations, Bristol: IOP Publishing, 1991.
%
Mackay, Charles (1814-1889)
Truth ... and if mine eyes
Can bear its blaze, and trace its symmetries,
Measure its distance, and its advent wait,
I am no prophet -- I but calculate.
The Poetical Works of Charles Mackay. 1876.
%
Maistre Joseph Marie de (1753 - 1821)
The concept of number is the obvious distinction between the beast
and man. Thanks to number, the cry becomes a song, noise acquires
rhythm, the spring is transformed into a dance, force becomes
dynamic, and outlines figures.
%
Mann, Thomas (1875-1955)
A great truth is a truth whose opposite is also a great truth.
Essay on Freud. 1937.
%
Mann, Thomas (1875-1955)
I tell them that if they will occupy themselves with the study of
mathematics they will find in it the best remedy against the lusts of
the flesh.
The Magic Mountain. 1927.
%
Mann, Thomas (1875-1955)
Some of the men stood talking in this room, and at the right of the
door a little knot had formed round a small table, the center of
which was the mathematics student, who ws eagerly talking. He had
made the assertion that one could draw through a given point more
than one parallel to a straight line; Frau Hagenström had cried out
that this was impossible, and he had gone on to prove it so
conclusively that his hearers were constrained to behave as though
they understood.
Little Herr Friedemann.
%
Mathesis, Adrian
If your new theorem can be stated with great simplicity, then there
will exist a pathological exception.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1988.
%
Mathesis, Adrian
All great theorems were discovered after midnight.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1988.
%
Mathesis, Adrian
The greatest unsolved theorem in mathematics is why some people are
better at it than others.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1988.
%
Matthias, Bernd T
If you see a formula in the Physical Review that extends over a
quarter of a page, forget it. It's wrong. Nature isn't that
complicated.
%
Maxwell, James Clerk (1813-1879)
... that, in a few years, all great physical constants will have
been approximately estimated, and that the only occupation which will
be left to men of science will be to carry these measurements to
another place of decimals.
Scientific Papers 2, 244, October 1871.
%
Mayer, Maria Goeppert (1906 -1972)
Mathematics began to seem too much like puzzle solving. Physics is
puzzle solving, too, but of puzzles created by nature, not by the
mind of man.
J. Dash, Maria Goeppert-Mayer, A Life of One's Own.
%
McDuff, Dusa
Gel'fand amazed me by talking of mathematics as though it were
poetry. He once said about a long paper bristling with formulas that
it contained the vague beginnings of an idea which could only hint at
and which he had never managed to bring out more clearly. I had
always thought of mathematics as being much more straightforward: a
formula is a formula, and an algebra is an algebra, but Gel'fand
found hedgehogs lurking in the rows of his spectral sequences!
Mathematical Notices v. 38, no. 3, March 1991, pp. 185-7.
%
McShane, E. J.
There are in this world optimists who feel that any symbol that
starts off with an integral sign must necessarily denote something
that will have every property that they should like an integral to
possess. This of course is quite annoying to us rigorous
mathematicians; what is even more annoying is that by doing so they
often come up with the right answer.
Bulletin of the American Mathematical Society, v. 69, p. 611, 1963.
%
Mencken, H. L. (1880 - 1956)
It is now quite lawful for a Catholic woman to avoid pregnancy by a
resort to mathematics, though she is still forbidden to resort to
physics and chemistry.
Notebooks, "Minority Report".
%
Millay, Edna St. Vincent (1892 - 1950)
Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.
%
Milton, John (1608 - 1674)
From Man or Angel the great Architect
Did wisely to conceal, and not divulge,
His secrets, to be scanned by them who ought
Rather admire. Or, if they list to try
Conjecture, he his fabric of the Heavens
Hath left to their disputes -- perhaps to move
His laughter at their quaint opinions wide
Hereafter, when they come to model Heaven
And calculate the stars: how they will wield
The mighty frame: how build, unbuild, contrive
To save appearances; how gird the Sphere
With Centric and Eccentric scribbled o'er,
Cycle and Epicycle, Orb in Orb.
Paradise Lost.
%
Milton, John (1608-1674)
Chaos umpire sits
And by decision more
embroils the fray
by which he reigns: next
him high arbiter
Chance governs all.
%
Minkowski, Herman
From henceforth, space by itself, and time by itself, have vanished
into the merest shadows and only a kind of blend of the two exists in
its own right.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Minsky, Marvin Lee (1927 -)
Logic doesn't apply to the real world.
D. R. Hofstadter and D. C. Dennett (eds.) The Mind's I, 1981.
%
Mitchell, Margaret
...She knew only that if she did or said thus-and-so, men would
unerringly respond with the complimentary thus-and-so. It was like a
mathematical formula and no more difficult, for mathematics was the
one subject that had come easy to Scarlett in her schooldays.
Gone With the Wind.
%
Mittag-Leffler, Gösta
The mathematician's best work is art, a high perfect art, as daring
as the most secret dreams of imagination, clear and limpid.
Mathematical genius and artistic genius touch one another.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Mordell, L.J.
Neither you nor I nor anybody else knows what makes a mathematician
tick. It is not a question of cleverness. I know many mathematicians
who are far abler than I am, but they have not been so lucky. An
illustration may be given by considering two miners. One may be an
expert geologist, but he does not find the golden nuggets that the
ignorant miner does.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Moore, E.H. (1862 - 1932)
We lay down a fundamental principle of generalization by abstraction:
"The existence of analogies between central features of various
theories implies the existence of a general theory which underlies
the particular theories and unifies them with respect to those
central features...."
In H. Eves Mathematical Circles Revisited, Boston: Prindle, Weber
and Schmidt, 1971.
%
Moroney, M.J.
The words figure and fictitious both derive from the same Latin
root, fingere. Beware!
Facts from Figures.
%
Mueller, Ian
[about Hypatia:]
In an era in which the domain of intellect and politics were almost
exclusively male, Theon [her father] was an unusually liberated
person who taught an unusually gifted daughter and encouraged her to
achieve things that, as far as we know, no woman before her did or
perhaps even dreamed of doing.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Napoleon (1769-1821)
A mathematician of the first rank, Laplace quickly revealed himself
as only a mediocre administrator; from his first work we saw that we
had been deceived. Laplace saw no question from its true point of
view; he sought subtleties everywhere; had only doubtful ideas, and
finally carried the spirit of the infinitely small into
administration.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc.,1988.
%
Nebeuts, E. Kim
Teach to the the problems, not to the text.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Nebeuts, E. Kim
To state a theorem and then to show examples of it is literally to
teach backwards.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Nebeuts, E. Kim
A good preparation takes longer than the delivery.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Neumann, Franz Ernst (1798 - 1895)
The greatest reward lies in making the discovery; recognition can
add little or nothing to that.
%
von Neumann, Johann (1903 - 1957)
In mathematics you don't understand things. You just get used to
them.
In G. Zukav The Dancing Wu Li Masters.
%
Newman, James R.
The most painful thing about mathematics is how far away you are
from being able to use it after you have learned it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Newman, James, R.
The discovery in 1846 of the planet Neptune was a dramatic and
spectacular achievement of mathematical astronomy. The very existence
of this new member of the solar system, and its exact location, were
demonstrated with pencil and paper; there was left to observers only
the routine task of pointing their telescopes at the spot the
mathematicians had marked.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Newman, James R.
It is hard to know what you are talking about in mathematics, yet no
one questions the validity of what you say. There is no other realm
of discourse half so queer.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Newman, James R.
Mathematical economics is old enough to be respectable, but not all
economists respect it. It has powerful supporters and impressive
testimonials, yet many capable economists deny that mathematics,
except as a shorthand or expository device, can be applied to
economic reasoning. There have even been rumors that mathematics is
used in economics (and in other social sciences) either for the
deliberate purpose of mystification or to confer dignity upon common
places as French was once used in diplomatic communications.
In J. R. Newman (ed.) The World of Mathematics, New Yorl: Simon and
Schuster, 1956.
%
Newman, James R.
To be sure, mathematics can be extended to any branch of knowledge,
including economics, provided the concepts are so clearly defined as
to permit accurate symbolic representation. That is only another way
of saying that in some branches of discourse it is desirable to know
what you are talking about.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Newman, James R.
The Theory of Groups is a branch of mathematics in which one does
something to something and then compares the result with the result
obtained from doing the same thing to something else, or something
else to the same thing.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Newman, James R.
Games are among the most interesting creations of the human mind,
and the analysis of their structure is full of adventure and
surprises. Unfortunately there is never a lack of mathematicians for
the job of transforming delectable ingredients into a dish that
tastes like a damp blanket.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Newton, Isaac (1642-1727)
...from the same principles, I now demonstrate the frame of the
System of the World.
Principia Mathematica.
%
Newton, Isaac (1642-1727)
Hypotheses non fingo.
I feign no hypotheses.
Principia Mathematica.
%
Newton, Isaac (1642-1727)
To explain all nature is too difficult a task for any one man or
even for any one age. `Tis much better to do a little with certainty,
and leave the rest for others hat come after you, than to explain all
things.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Newton, Isaac (1642-1727)
The description of right lines and circles, upon which geometry is
founded, belongs to mechanics. Geometry does not teach us to draw
these lines, but requires them to be drawn.
Principia Mathematica.
%
Newton, Isaac (1642-1727)
The latest authors, like the most ancient, strove to subordinate the
phenomena of nature to the laws of mathematics.
%
Newton, Isaac (1642-1727)
[His epitaph:]
Who, by vigor of mind almost divine, the motions and figures of the
planets, the paths of comets, and the tides of the seas first
demonstrated.
%
Thomas R. Nicely
Usually mathematicians have to shoot somebody to get this much
publicity.
[On the attention he received after finding the flaw in Intel's
Pentium chip in 1994]
Cincinnati Enquirer, December 18, 1994, Section A, page 19.
%
Nightingale, Florence (1820-1910)
[Of her:]
Her statistics were more than a study, they were indeed her
religion. For her Quetelet was the hero as scientist, and the
presentation copy of his Physique sociale is annotated by her on
every page. Florence Nightingale believed -- and in all the actions
of her life acted upon that belief -- that the administrator could
only be successful if he were guided by statistical knowledge. The
legislator -- to say nothing of the politician -- too often failed
for want of this knowledge. Nay, she went further; she held that the
universe -- including human communities -- was evolving in accordance
with a divine plan; that it was man's business to endeavor to
understand this plan and guide his actions in sympathy with it. But
to understand God's thoughts, she held we must study statistics, for
these are the measure of His purpose. Thus the study of statistics
was for her a religious duty.
K. Pearson The Life, Letters and Labours for
%
Oakley, C.O.
The study of mathematics cannot be replaced by any other activity
that will train and develop man's purely logical faculties to the
same level of rationality.
The American Mathematical Monthly, 56, 1949, p19.
%
Ogyu, Sorai (1666 - 1729)
Mathematicians boast of their exacting achievements, but in reality
they are absorbed in mental acrobatics and contribute nothing to
society.
Complete Works on Japan's Philosophical Thought. 1956.
%
Oppenheimer, Julius Robert (1904 - 1967)
Today, it is not only that our kings do not know mathematics, but
our philosophers do not know mathematics and -- to go a step further
-- our mathematicians do not know mathematics.
"The Tree of Knowledge" in Harper's, 217, 1958.
%
Osgood, W. F.
The calculus is the greatest aid we have to the application of
physical truth in the broadest sense of the word.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Pascal, Blaise (1623-1662)
We are usually convinced more easily by reasons we have found
ourselves than by those which have occurred to others.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
It is the heart which perceives God and not the reason.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Man is equally incapable of seeing the nothingness from which he
emerges and the infinity in which he is engulfed.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Our nature consists in movement; absolute rest is death.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Man is full of desires: he loves only those who can satisfy them
all. "This man is a good mathematician," someone will say. But I
have no concern for mathematics; he would take me for a proposition.
"That one is a good soldier." He would take me for a besieged town. I
need, that is to say, a decent man who can accommodate himself to all
my desires in a general sort of way.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
We run carelessly to the precipice, after we have put something
before us to prevent us from seeing it.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
We do not worry about being respected in towns through which we
pass. But if we are going to remain in one for a certain time, we do
worry. How long does this time have to be?
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Few men speak humbly of humility, chastely of chastity, skeptically
of skepticism.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Those who write against vanity want the glory of having written
well, and their readers the glory of reading well, and I who write
this have the same desire, as perhaps those who read this have also.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Our notion of symmetry is derived form the human face. Hence, we
demand symmetry horizontally and in breadth only, not vertically nor
in depth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
When we encounter a natural style we are always surprised and
delighted, for we thought to see an author and found a man.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Everything that is written merely to please the author is worthless.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
I cannot judge my work while I am doing it. I have to do as painters
do, stand back and view it from a distance, but not too great a
distance. How great? Guess.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Contradiction is not a sign of falsity, nor the lack of
contradiction a sign of truth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
All err the more dangerously because each follows a truth. Their
mistake lies not in following a falsehood but in not following
another truth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Perfect clarity would profit the intellect but damage the will.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Those who are accustomed to judge by feeling do not understand the
process of reasoning, because they want to comprehend at a glance and
are not used to seeking for first principles. Those, on the other
hand, who are accustomed to reason from first principles do not
understand matters of feeling at all, because they look for first
principles and are unable to comprehend at a glance.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
To deny, to believe, and to doubt well are to a man as the race is
to a horse.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Words differently arranged have a different meaning and meanings
differently arranged have a different effect.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Pascal, Blaise (1623-1662)
Nature is an infinite sphere of which the center is everywhere and
the circumference nowhere.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
We arrive at truth, not by reason only, but also by the heart.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
When the passions become masters, they are vices.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Men despise religion; they hate it, and they fear it is true.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Religion is so great a thing that it is right that those who will
not take the trouble to seek it if it be obscure, should be deprived
of it.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
It is not certain that everything is uncertain.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
We are so presumptuous that we should like to be known all over the
world, even by people who will only come when we are no more. Such is
our vanity that the good opinion of half a dozen of the people around
us gives us pleasure and satisfaction.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
The sole cause of man's unhappiness is that he does not know how to
stay quietly in his room.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Reason's last step is the recognition that there are an infinite
number of things which are beyond it.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Through space the universe grasps me and swallows me up like a
speck; through thought I grasp it.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Let no one say that I have said nothing new... the arrangement of
the subject is new. When we play tennis, we both play with the same
ball, but one of us places it better.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
The excitement that a gambler feels when making a bet is equal to
the amount he might win times the probability of winning it.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Pascal, Blaise (1623-1662)
Reason is the slow and tortuous method by which these who do not
know the truth discover it. The heart has its own reason which reason
does not know.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Reverend Fathers, my letters did not usually follow each other at
such close intervals, nor were they so long.... This one would not be
so long had I but the leisure to make it shorter.
Lettres provinciales.
%
Pascal, Blaise (1623-1662)
The last thing one knows when writing a book is what to put first.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
What is man in nature? Nothing in relation to the infinite, all in
relation to nothing, a mean between nothing and everything.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
[I feel] engulfed in the infinite immensity of spaces whereof I know
nothing, and which know nothing of me, I am terrified The eternal
silence of these infinite spaces alarms me.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Let us weigh the gain and the loss in wagering that God is. Let us
consider the two possibilities. If you gain, you gain all; if you
lose, you lose nothing. Hesitate not, then, to wager that He is.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
Look somewhere else for someone who can follow you in your
researches about numbers. For my part, I confess that they are far
beyond me, and I am competent only to admire them.
[Written to Fermat]
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Pascal, Blaise (1623-1662)
The more I see of men, the better I like my dog.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Pascal, Blaise (1623-1662)
The more intelligent one is, the more men of originality one finds.
Ordinary people find no difference between men.
Pensees. 1670.
%
Pascal, Blaise (1623-1662)
However vast a man's spiritual resources, he is capable of but one
great passion.
Discours sur les passions de l'amour. 1653.
%
Pascal, Blaise (1623-1662)
There are two types of mind ... the mathematical, and what might be
called the intuitive. The former arrives at its views slowly, but
they are firm and rigid; the latter is endowed with greater
flexibility and applies itself simultaneously to the diverse lovable
parts of that which it loves.
Discours sur les passions de l'amour. 1653.
%
Passano, L.M.
This trend [emphasizing applied mathematics over pure mathematics]
will make the queen of the sciences into the quean of the sciences.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Pasteur, Louis Chance favors only the prepared mind.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Pearson, Karl
The mathematician, carried along on his flood of symbols, dealing
apparently with purely formal truths, may still reach results of
endless importance for our description of the physical universe.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Peirce, Benjamin (1809-1880)
Mathematics is the science which draws necessary conclusions.
Memoir read before the National Academy of Sciences in Washington,
1870.
%
Peirce, Charles Sanders (1839-1914)
The one [the logician] studies the science of drawing conclusions,
the other [the mathematician] the science which draws necessary
conclusions.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956.
%
Peirce, Charles Sanders (1839-1914)
...mathematics is distinguished from all other sciences except only
ethics, in standing in no need of ethics. Every other science, even
logic, especially in its early stages, is in danger of evaporating
into airy nothingness, degenerating, as the Germans say, into an
arachnoid film, spun from the stuff that dreams are made of. There is
no such danger for pure mathematics; for that is precisely what
mathematics ought to be.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956.
%
Peirce, Charles Sanders (1839-1914)
Among the minor, yet striking characteristics of mathematics, may be
mentioned the fleshless and skeletal build of its propositions; the
peculiar difficulty, complication, and stress of its reasonings; the
perfect exactitude of its results; their broad universality; their
practical infallibility.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956.
%
Peirce, Charles Sanders (1839-1914)
The pragmatist knows that doubt is an art which hs to be acquired
with difficulty.
Collected Papers.
%
Pedersen, Jean
Geometry is a skill of the eyes and the hands as well as of the mind.
%
Plato (ca 429-347 BC)
He who can properly define and divide is to be considered a god.
%
Plato (ca 429-347 BC)
The ludicrous state of solid geometry made me pass over this branch.
Republic, VII, 528.
%
Plato (ca 429-347 BC)
He is unworthy of the name of man who is ignorant of the fact that
the diagonal of a square is incommensurable with its side.
%
Plato (ca 429-347 BC)
Mathematics is like checkers in being suitable for the young, not
too difficult, amusing, and without peril to the state.
%
Plato (ca 429-347 BC)
The knowledge of which geometry aims is the knowledge of the eternal.
Republic, VII, 52.
%
Plato (ca 429-347 BC)
I have hardly ever known a mathematician who was capable of
reasoning.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Plato (ca 429-347 BC)
There still remain three studies suitable for free man. Arithmetic
is one of them.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Plutarch (ca 46-127)
[about Archimedes:]
... being perpetually charmed by his familiar siren, that is, by his
geometry, he neglected to eat and drink and took no care of his
person; that he was often carried by force to the baths, and when
there he would trace geometrical figures in the ashes of the fire,
and with his finger draws lines upon his body when it was anointed
with oil, being in a state of great ecstasy and divinely possessed by
his science.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Poe, Edgar Allen
To speak algebraically, Mr. M. is execrable, but Mr. G. is (x + 1)-
ecrable.
[Discussing fellow writers Cornelius Mathews and William Ellery
Channing.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.
%
Poincaré, Jules Henri (1854-1912)
Mathematics is the art of giving the same name to different things.
[As opposed to the quotation: Poetry is the art of giving different
names to the same thing].
%
Poincaré, Jules Henri (1854-1912)
Later generations will regard Mengenlehre (set theory) as a disease
from which one has recovered.
[Whether or not he actually said this is a matter of debate amongst
historians of mathematics.]
The Mathematical Intelligencer, vol 13, no. 1, Winter 1991.
%
Poincaré, Jules Henri (1854-1912)
What is it indeed that gives us the feeling of elegance in a
solution, in a demonstration? It is the harmony of the diverse parts,
their symmetry, their happy balance; in a word it is all that
introduces order, all that gives unity, that permits us to see
clearly and to comprehend at once both the ensemble and the details.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Poincaré, Jules Henri (1854-1912)
Thus, be it understood, to demonstrate a theorem, it is neither
necessary nor even advantageous to know what it means. The geometer
might be replaced by the "logic piano" imagined by Stanley Jevons;
or, if you choose, a machine might be imagined where the assumptions
were put in at one end, while the theorems came out at the other,
like the legendary Chicago machine where the pigs go in alive and
come out transformed into hams and sausages. No more than these
machines need the mathematician know what he does.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Poincaré, Jules Henri (1854-1912)
Talk with M. Hermite. He never evokes a concrete image, yet you soon
perceive that the more abstract entities are to him like living
creatures.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Poincaré, Jules Henri (1854-1912)
Science is built up with facts, as a house is with stones. But a
collection of facts is no more a science than a heap of stones is a
house.
La Science et l'hypothèse.
%
Poincaré, Jules Henri (1854-1912)
A scientist worthy of his name, about all a mathematician,
experiences in his work the same impression as an artist; his
pleasure is as great and of the same nature.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Poincaré, Jules Henri (1854-1912)
The mathematical facts worthy of being studied are those which, by
their analogy with other facts, are capable of leading us to the
knowledge of a physical law. They reveal the kinship between other
facts, long known, but wrongly believed to be strangers to one
another.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Poincaré, Jules Henri (1854-1912)
Mathematicians do not study objects, but relations between objects.
Thus, they are free to replace some objects by others so long as the
relations remain unchanged. Content to them is irrelevant: they are
interested in form only.
%
Poincaré, Jules Henri (1854-1912)
Thought is only a flash between two long nights, but this flash is
everything.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Poincaré, Jules Henri (1854-1912)
The mind uses its faculty for creativity only when experience forces
it to do so.
%
Poincaré, Jules Henri (1854-1912)
Mathematical discoveries, small or greatare never born of
spontaneous generation They always presuppose a soil seeded with
preliminary knowledge and well prepared by labour, both conscious and
subconscious.
%
Poincaré, Jules Henri (1854-1912)
Absolute space, that is to say, the mark to which it would be
necessary to refer the earth to know whether it really moves, has no
objective existence.... The two propositions: "The earth turns
round" and "it is more convenient to suppose the earth turns round"
have the same meaning; there is nothing more in the one than in the
other.
La Science et l'hypothèse.
%
Poincaré, Jules Henri (1854-1912)
...by natural selection our mind has adapted itself to the
conditions of the external world. It has adopted the geometry most
advantageous to the species or, in other words, the most convenient.
Geometry is not true, it is advantageous.
Science and Method.
%
Poisson, Siméon (1781-1840)
Life is good for only two things, discovering mathematics and
teaching mathematics.
Mathematics Magazine, v. 64, no. 1, Feb. 1991.
%
Polyá, George (1887, 1985)
Mathematics consists of proving the most obvious thing in the least
obvious way.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Polyá, George (1887, 1985)
The traditional mathematics professor of the popular legend is
absentminded. He usually appears in public with a lost umbrella in
each hand. He prefers to face the blackboard and to turn his back to
the class. He writes a, he says b, he means c; but it should be d.
Some of his sayings are handed down from generation to generation.
"In order to solve this differential equation you look at it till a
solution occurs to you."
"This principle is so perfectly general that no particular
application of it is possible."
"Geometry is the science of correct reasoning on incorrect figures."
"My method to overcome a difficulty is to go round it."
"What is the difference between method and device? A method is a
device which you used twice."
How to Solve It. Princeton: Princeton University Press. 1945.
%
Polyá, George (1887, 1985)
Mathematics is the cheapest science. Unlike physics or chemistry, it
does not require any expensive equipment. All one needs for
mathematics is a pencil and paper.
D. J. Albers and G. L. Alexanderson, Mathematical People, Boston:
Birkhauser, 1985.
%
Polyá, George (1887, 1985)
There are many questions which fools can ask that wise men cannot
answer.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Polyá, George (1887, 1985)
When introduced at the wrong time or place, good logic may be the
worst enemy of good teaching.
The American Mathematical Monthly, v. 100, no. 3.
%
Polyá, George (1887, 1985)
Even fairly good students, when they have obtained the solution of
the problem and written down neatly the argument, shut their books
and look for something else. Doing so, they miss an important and
instructive phase of the work. ... A good teacher should understand
and impress on his students the view that no problem whatever is
completely exhausted.
One of the first and foremost duties of the teacher is not to give
his students the impression that mathematical problems have little
connection with each other, and no connection at all with anything
else. We have a natural opportunity to investigate the connections of
a problem when looking back at its solution.
How to Solve It. Princeton: Princeton University Press. 1945.
%
Polyá, George (1887, 1985)
In order to translate a sentence from English into French two things
are necessary. First, we must understand thoroughly the English
sentence. Second, we must be familiar with the forms of expression
peculiar to the French language. The situation is very similar when
we attempt to express in mathematical symbols a condition proposed in
words. First, we must understand thoroughly the condition. Second, we
must be familiar with the forms of mathematical expression.
How to Solve It. Princeton: Princeton University Press. 1945.
%
Pope, Alexander (1688-1744)
Epitaph on Newton:
Nature and Nature's law lay hid in night:
God said, "Let Newton be!," and all was light.
[added by Sir John Collings Squire:
It did not last: the Devil shouting "Ho.
Let Einstein be," restored the status quo]
[Aaron Hill's version:
O'er Nature's laws God cast the veil of night,
Out blaz'd a Newton's souland all was light.
%
Pope, Alexander (1688-1744)
Order is Heaven's first law.
An Essay on Man IV.
%
Pope, Alexander (1688-1744)
See skulking Truth to her old cavern fled,
Mountains of Casuistry heap'd o'er her head!
Philosophy, that lean'd on Heav'n before,
Shrinks to her second cause, and is no more.
Physic of Metaphysic begs defence,
And Metaphysic calls for aid on Sense!
See Mystery to Mathematics fly!
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Pordage, Matthew
One of the endearing things about mathematicians is the extent to
which they will go to avoid doing any real work.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Proclus Diadochus (412 - 485)
It is well known that the man who first made public the theory of
irrationals perished in a shipwreck in order that the inexpressible
and unimaginable should ever remain veiled. And so the guilty man,
who fortuitously touched on and revealed this aspect of living
things, was taken to the place where he began and there is for ever
beaten by the waves.
Scholium to Book X of Euclid V.
%
Purcell, E. and Varberg, D.
The Mean Value Theorem is the midwife of calculus -- not very
important or glamorous by itself, but often helping to delivery other
theorems that are of major significance.
Calculus with Analytic Geomety, fifth edition, Englewood Cliffs, NJ:
Prentice Hall, 1987.
%
Pushkin, Aleksandr Sergeyevich (1799 - 1837)
Inspiration is needed in geometry, just as much as in poetry.
Likhtenshtein
%
Quine, Willard Van Orman
Just as the introduction of the irrational numbers ... is a
convenient myth [which] simplifies the laws of arithmetic ... so
physical objects are postulated entities which round out and simplify
our account of the flux of existence... The conceptional scheme of
physical objects is [likewise] a convenient myth, simpler than the
literal truth and yet containing that literal truth as a scattered
part.
In J. Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990.
%
Raleigh, [Sir] Walter Alexander (1861-1922)
In an examination those who do not wish to know ask questions of
those who cannot tell.
Some Thoughts on Examinations.
%
Recorde, Robert (1557)
To avoide the tediouse repetition of these woordes: is equalle to:
I will settle as I doe often in woorke use, a paire of paralleles, or
gemowe [twin] lines of one lengthe: =, bicause noe .2. thynges, can
be moare equalle.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Reid, Thomas
It is the invaluable merit of the great Basle mathematician Leonard
Euler, to have freed the analytical calculus from all geometric
bounds, and thus to have established analysis as an independent
science, which from his time on has maintained an unchallenged
leadership in the field of mathematics.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Renan, Ernest
The simplest schoolboy is now familiar with facts for which
Archimedes would have sacrificed his life.
Souvenirs d'enfance et de jeunesse.
%
Rényi, Alfréd
If I feel unhappy, I do mathematics to become happy. If I am happy,
I do mathematics to keep happy.
P. Turán, "The Work of Alfréd Rényi", Matematikai Lapok 21, 1970, pp
199 - 210.
%
Richardson, Lewis Fry (1881 - 1953)
Another advantage of a mathematical statement is that it is so
definite that it might be definitely wrong; and if it is found to be
wrong, there is a plenteous choice of amendments ready in the
mathematicians' stock of formulae. Some verbal statements have not
this merit; they are so vague that they could hardly be wrong, and
are correspondingly useless.
Mathematics of War and Foreign Politics.
%
Riskin, Adrian
(after Edna St. Vincent Millay)
...Euclid alone
Has looked on Beauty bare.
He turned away at once;
Far too polite to stare.
The Mathematical Intelligencer, V. 16, no. 4 (Fall 1994), p. 20.
%
R. Rivest, A. Shamir, and L. Adleman
The magic words are squeamish ossifrage
[This sentence is the result when a coded message in Martin
Gardner's column about factoring the famous number RSA-129 is
decoded. See the article whose title is the above sentence by Barry
Cipra, SIAM News July 1994, 1, 12-13.]
%
Rohault, Jacques (17th century)
It was by just such a hazard, as if a man should let fall a handful
of sand upon a table and the particles of it should be so ranged that
we could read distinctly on it a whole page of Virgil's Aenead.
Traité de Physique, Paris, 1671.
%
Rosenblueth, A
[with Norbert Wiener]
The best material model of a cat is another, or preferably the same,
cat.
Philosophy of Science 1945.
%
Rosenlicht, Max (1949)
You know we all became mathematicians for the same reason: we were
lazy.
%
Hugo Rossi
In the fall of 1972 President Nixon announced that the rate of
increase of inflation was decreasing. This was the first time a
sitting president used the third derivative to advance his case for
reelection.
Mathematics Is an Edifice, Not a Toolbox, Notices of the AMS, v. 43,
no. 10, October 1996.
%
Rota, Gian-carlo
We often hear that mathematics consists mainly of "proving
theorems." Is a writer's job mainly that of "writing sentences?"
In preface to P. Davis and R. Hersh The Mathematical Experience,
Boston: Birkhauser, 1981.
%
Russell, Bertrand (1872-1970)
How dare we speak of the laws of chance? Is not chance the
antithesis of all law?
Calcul des probabilités.
%
Russell, Bertrand (1872-1970)
Mathematics takes us into the region of absolute necessity, to which
not only the actual word, but every possible word, must conform.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Russell, Bertrand (1872-1970)
Although this may seem a paradox, all exact science is dominated by
the idea of approximation.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Russell, Bertrand (1872-1970)
At the age of eleven, I began Euclid, with my brother as my tutor.
This was one of the great events of my life, as dazzling as first
love. I had not imagined there was anything so delicious in the
world. From that moment until I was thirty-eight, mathematics was my
chief interest and my chief source of happiness.
The Autobiography of Bertrand Russell .
%
Russell, Bertrand (1872-1970)
A good notation has a subtlety and suggestiveness which at times
make it almost seem like a live teacher.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Russell, Bertrand (1872-1970)
If I were a medical man, I should prescribe a holiday to any patient
who considered his work important.
The Autobiography of Bertrand Russell .
%
Russell, Bertrand (1872-1970)
Ordinary language is totally unsuited for expressing what physics
really asserts, since the words of everyday life are not sufficiently
abstract. Only mathematics and mathematical logic can say as little
as the physicist means to say.
The Scientific Outlook, 1931.
%
Russell, Bertrand (1872-1970)
With equal passion I have sought knowledge. I have wished to
understand the hearts of men. I have wished to know why the stars
shine. And I have tried to apprehend the Pythagorean power by which
number holds sway about the flux. A little of this, but not much, I
have achieved.
The Autobiography of Bertrand Russell .
%
Russell, Bertrand (1872-1970)
At first it seems obvious, but the more you think about it the
stranger the deductions from this axiom seem to become; in the end
you cease to understand what is meant by it.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Russell, Bertrand (1872-1970)
Calculus required continuity, and continuity was supposed to require
the infinitely little; but nobody could discover what the infinitely
little might be.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Russell, Bertrand (1872-1970)
The fact that all Mathematics is Symbolic Logic is one of the
greatest discoveries of our age; and when this fact has been
established, the remainder of the principles of mathematics consists
in the analysis of Symbolic Logic itself.
Principles of Mathematics. 1903.
%
Russell, Bertrand (1872-1970)
A habit of basing convictions upon evidence, and of giving to them
only that degree or certainty which the evidence warrants, would, if
it became general, cure most of the ills from which the world suffers.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Russell, Bertrand (1872-1970)
The method of "postulating" what we want has many advantages; they
are the same as the advantages of theft over honest toil.
Introduction to Mathematical Philosophy, New York and London, 1919,
p 71.
%
Russell, Bertrand (1872-1970)
Aristotle maintained that women have fewer teeth than men; although
he was twice married, it never occurred to him to verify this
statement by examining his wives' mouths.
The Impact of Science on Society, 1952.
%
Russell, Bertrand (1872-1970)
[Upon hearing via Littlewood an exposition on the theory of
relativity:]
To think I have spent my life on absolute muck.
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co. ltd.,
1953.
%
Russell, Bertrand (1872-1970)
"But," you might say, "none of this shakes my belief that 2 and 2
are 4." You are quite right, except in marginal cases -- and it is
only in marginal cases that you are doubtful whether a certain animal
is a dog or a certain length is less than a meter. Two must be two of
something, and the proposition "2 and 2 are 4" is useless unless it
can be applied. Two dogs and two dogs are certainly four dogs, but
cases arise in which you are doubtful whether two of them are dogs.
"Well, at any rate there are four animals," you may say. But there
are microorganisms concerning which it is doubtful whether they are
animals or plants. "Well, then living organisms," you say. But there
are things of which it is doubtful whether they are living organisms
or not. You will be driven into saying: "Two entities and two
entities are four entities." When you have told me what you mean by
%
Russell, Bertrand (1872-1970)
I wanted certainty in the kind of way in which people want religious
faith. I thought that certainty is more likely to be found in
mathematics than elsewhere. But I discovered that many mathematical
demonstrations, which my teachers expected me to accept, were full of
fallacies, and that, if certainty were indeed discoverable in
mathematics, it would be in a new field of mathematics, with more
solid foundations than those that had hitherto been thought secure.
But as the work proceeded, I was continually reminded of the fable
about the elephant and the tortoise. having constructed an elephant
upon which the mathematical world could rest, I found the elephant
tottering, and proceeded to construct a tortoise to keep the elephant
from falling. But the tortoise was no more secure than the elephant,
and after some twenty years of very arduous toil, I came to the
conclusion that there was nothing more that I could do in the way of
making
%
Russell, Bertrand (1872-1970)
Men who are unhappy, like men who sleep badly, are always proud of
the fact.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Russell, Bertrand (1872-1970)
Work is of two kinds: first, altering the position of matter at or
near the earth's surface relatively to other such matter; second,
telling other people to do so. The first kind is unpleasant and ill
paid; the second is pleasant and highly paid.
%
Russell, Bertrand (1872-1970)
A sense of duty is useful in work but offensive in personal
relations. Certain characteristics of the subject are clear. To
begin with, we do not, in this subject, deal with particular things
or particular properties: we deal formally with what can be said
about "any" thing or "any" property. We are prepared to say that one
and one are two, but not that Socrates and Plato are two, because, in
our capacity of logicians or pure mathematicians, we have never heard
of Socrates or Plato. A world in which there were no such individuals
would still be a world in which one and one are two. It is not open
to us, as pure mathematicians or logicians, to mention anything at
all, because, if we do so we introduce something irrelevant and not
formal.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Russell, Bertrand (1872-1970)
The desire to understand the world and the desire to reform it are
the two great engines of progress.
Marriage and Morals.
%
Russell, Bertrand (1872-1970)
It can be shown that a mathematical web of some kind can be woven
about any universe containing several objects. The fact that our
universe lends itself to mathematical treatment is not a fact of any
great philosophical significance.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
%
Rutherford, Ernest (1871-1937)
If your experiment needs statistics, you ought to have done a better
experiment.
In N. T. J. Bailey the Mathematical Approach to Biology and
Medicine, New York: Wiley, 1967.
%
Sanford, T. H.
The modern, and to my mind true, theory is that mathematics is the
abstract form of the natural sciences; and that it is valuable as a
training of the reasoning powers not because it is abstract, but
because it is a representation of actual things.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Santayana, George
It is a pleasant surprise to him (the pure mathematician) and an
added problem if he finds that the arts can use his calculations, or
that the senses can verify them, much as if a composer found that
sailors could heave better when singing his songs.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Sarton, G.
The main duty of the historian of mathematics, as well as his
fondest privilege, is to explain the humanity of mathematics, to
illustrate its greatness, beauty and dignity, and to describe how the
incessant efforts and accumulated genius of many generations have
built up that magnificent monument, the object of our most legitimate
pride as men, and of our wonder, humility and thankfulness, as
individuals. The study of the history of mathematics will not make
better mathematicians but gentler ones, it will enrich their minds,
mellow their hearts, and bring out their finer qualities.
%
Sayers, Dorothy L.
The biologist can push it back to the original protist, and the
chemist can push it back to the crystal, but none of them touch the
real question of why or how the thing began at all. The astronomer
goes back untold million of years and ends in gas and emptiness, and
then the mathematician sweeps the whole cosmos into unreality and
leaves one with mind as the only thing of which we have any immediate
apprehension. Cogito ergo sum, ergo omnia esse videntur. All this
bother, and we are no further than Descartes. Have you noticed that
the astronomers and mathematicians are much the most cheerful people
of the lot? I suppose that perpetually contemplating things on so
vast a scale makes them feel either that it doesn't matter a hoot
anyway, or that anything so large and elaborate must have some sense
in it somewhere.
With R. Eustace, The Documents in the Case, New York: Harper and
Row, 1930, p 54.
%
Schopenhauer
Of all the intellectual faculties, judgment is the last to mature.
A child under the age of fifteen should confine its attention either
to subjects like mathematics, in which errors of judgment are
impossible, or to subjects in which they are not very dangerous, like
languages, natural science, history, etc.
%
Seneca
If you would make a man happy, do not add to his possessions but
subtract from the sum of his desires.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1988.
%
Shakespeare, William (1564 - 1616)
I cannot do it without comp[u]ters.
The Winter's Tale.
%
Shakespeare, William (1564-1616)
Though this be madness, yet there is method in't.
%
Shakespeare, William (1564-1616)
O God! I could be bounded in a nutshell, and count myself king of
infinite space, were it not that I have bad dreams.
Hamlet.
%
Shakespeare, William (1564-1616)
I am ill at these numbers.
Hamlet.
%
Shaw, George Bernard (1856-1950)
Tyndall declared that he saw in Matter the promise and potency of
all forms of life, and with his Irish graphic lucidity made a picture
of a world of magnetic atoms, each atom with a positive and a
negative pole, arranging itself by attraction and repulsion in
orderly crystalline structure. Such a picture is dangerously
fascinating to thinkers oppressed by the bloody disorders of the
living world. Craving for purer subjects of thought, they find in the
contemplation of crystals and magnets a happiness more dramatic and
less childish than the happiness found by mathematicians in abstract
numbers, because they see in the crystals beauty and movement without
the corrupting appetites of fleshly vitality.
Preface to Back to Methuselah.
%
Shaw, J. B.
The mathematician is fascinated with the marvelous beauty of the
forms he constructs, and in their beauty he finds everlasting truth.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Simmons, G. F.
Mathematical rigor is like clothing; in its style it ought to suit
the occasion, and it diminishes comfort and restrains freedom of
movement if it is either too loose or too tight.
In The Mathematical Intelligencer, v. 13, no. 1, Winter 1991.
%
Slaught, H.E.
...[E.H.] Moore ws presenting a paper on a highly technical topic to
a large gathering of faculty and graduate students from all parts of
the country. When half way through he discovered what seemed to be an
error (though probably no one else in the room observed it). He
stopped and re-examined the doubtful step for several minutes and
then, convinced of the error, he abruptly dismissed the meeting -- to
the astonishment of most of the audience. It was an evidence of
intellectual courage as well as honesty and doubtless won for him the
supreme admiration of every person in the group -- an admiration
which was in no wise diminished, but rather increased, when at a
later meeting he announced that after all he had been able to prove
the step to be correct.
The American Mathematical Monthly, 40 (1933), 191-195.
%
Smith, Adam
I have no faith in political arithmetic.
%
Smith, David Eugene
One merit of mathematics few will deny: it says more in fewer words
than any other science. The formula, e^i pi = -1 expressed a world of
thought, of truth, of poetry, and of the religious spirit "God
eternally geometrizes."
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Smith, Henry John Stephen (1826 - 1883)
[His toast:]
Pure mathematics, may it never be of any use to anyone.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Smith, Henry John Stephen (1826-1883)
It is the peculiar beauty of this method, gentlemen, and one which
endears it to the really scientific mind, that under no circumstance
can it be of the smallest possible utility.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Soddy, Frederick (1877-1956)
Four circles to the kissing come,
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of squares of all four bends
Is half the square of their sum.
Nature, v. 137, 1936.
%
Somerville, Mary (1780-1872)
Nothing has afforded me so convincing a proof of the unity of the
Deity as these purely mental conceptions of numerical and
mathematical science which have been by slow degrees vouchsafed to
man, and are still granted in these latter times by the Differential
Calculus, now superseded by the Higher Algebra, all of which must
have existed in that sublimely omniscient Mind from eternity.
Martha Somerville (ed.) Personal Recollections of Mary Somerville,
Boston, 1874.
%
Spengler, Oswald (1880 -1936)
The mathematic, then, is an art. As such it has its styles and style
periods. It is not, as the layman and the philosopher (who is in this
matter a layman too) imagine, substantially unalterable, but subject
like every art to unnoticed changes form epoch to epoch. The
development of the great arts ought never to be treated without an
(assuredly not unprofitable) side-glance at contemporary mathematics.
The Decline of the West.
%
Steinmetz, Charles P.
Mathematics is the most exact science, and its conclusions are
capable of absolute proof. But this is so only because mathematics
does not attempt to draw absolute conclusions. All mathematical
truths are relative, conditional.
In E. T. Bell Men of Mathematics, New York: Simona and Schuster,
1937.
%
Sternberg, S.
Kepler's principal goal was to explain the relationship between the
existence of five planets (and their motions) and the five regular
solids. It is customary to sneer at Kepler for this. It is
instructive to compare this with the current attempts to "explain"
the zoology of elementary particles in terms of irreducible
representations of Lie groups.
%
Stewart, Ian
The successes of the differential equation paradigm were impressive
and extensive. Many problems, including basic and important ones, led
to equations that could be solved. A process of self-selection set
in, whereby equations that could not be solved were automatically of
less interest than those that could.
Does God Play Dice? The Mathematics of Chaos. Blackwell, Cambridge,
MA, 1989, p. 39.
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Sullivan, John William Navin (1886 - 1937)
The mathematician is entirely free, within the limits of his
imagination, to construct what worlds he pleases. What he is to
imagine is a matter for his own caprice; he is not thereby
discovering the fundamental principles of the universe nor becoming
acquainted with the ideas of God. If he can find, in experience, sets
of entities which obey the same logical scheme as his mathematical
entities, then he has applied his mathematics to the external world;
he has created a branch of science.
Aspects of Science, 1925.
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Sullivan, John William Navin (1886-1937)
Mathematics, as much as music or any other art, is one of the means
by which we rise to a complete self-consciousness. The significance
of mathematics resides precisely in the fact that it is an art; by
informing us of the nature of our own minds it informs us of much
that depends on our minds.
Aspects of Science, 1925.
%
Sun Tze (5th - 6th century)
The control of large numbers is possible, and like unto that of
small numbers, if we subdivide them.
Sun Tze Ping Fa.
%
Swift, Jonathan
If they would, for Example, praise the Beauty of a Woman, or any
other Animal, they describe it by Rhombs, Circles, Parallelograms,
Ellipses, and other geometrical terms ...
"A Voyage to Laputa" in Gulliver's Travels.
%
Jonathan Swift
What vexes me most is, that my female friends, who could bear me
very well a dozen years ago, have now forsaken me, although I am not
so old in proportion to them as I formerly was: which I can prove by
arithmetic, for then I was double their age, which now I am not.
Letter to Alexander Pope. 7 Feb. 1736.
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Sylvester, J.J. (1814 - 1897)
...there is no study in the world which brings into more harmonious
action all the faculties of the mind than [mathematics], ... or, like
this, seems to raise them, by successive steps of initiation, to
higher and higher states of conscious intellectual being....
Presidential Address to British Association, 1869.
%
Sylvester, J.J. (1814 - 1897)
So long as a man remains a gregarious and sociable being, he cannot
cut himself off from the gratification of the instinct of imparting
what he is learning, of propagating through others the ideas and
impressions seething in his own brain, without stunting and
atrophying his moral nature and drying up the surest sources of his
future intellectual replenishment.
%
Sylvester, J.J. (1814 - 1897)
[on graph theory...]
The theory of ramification is one of pure colligation, for it takes
no account of magnitude or position; geometrical lines are used, but
these have no more real bearing on the matter than those employed in
genealogical tables have in explaining the laws of procreation.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Sylvester, J.J. (1814 - 1897)
Time was when all the parts of the subject were dissevered, when
algebra, geometry, and arithmetic either lived apart or kept up cold
relations of acquaintance confined to occasional calls upon one
another; but that is now at an end; they are drawn together and are
constantly becoming more and more intimately related and connected by
a thousand fresh ties, and we may confidently look forward to a time
when they shall form but one body with one soul.
Presidential Address to British Association, 1869.
%
Sylvester, J.J. (1814 - 1897)
The world of ideas which it [mathematics] discloses or illuminates,
the contemplation of divine beauty and order which it induces, the
harmonious connexion of its parts, the infinite hierarchy and
absolute evidence of the truths with which it is concerned, these,
and such like, are the surest grounds of the title of mathematics to
human regard, and would remain unimpeached and unimpaired were the
plan of the universe unrolled like a map at our feet, and the mind of
man qualified to take in the whole scheme of creation at a glance.
Presidential Address to British Association, 1869.
%
Sylvester, J.J. (1814 - 1897)
I know, indeed, and can conceive of no pursuit so antagonistic to
the cultivation of the oratorical faculty ... as the study of
Mathematics. An eloquent mathematician must, from the nature of
things, ever remain as rare a phenomenon as a talking fish, and it is
certain that the more anyone gives himself up to the study of
oratorical effect the less will he find himself in a fit state to
mathematicize.
%
Thales (CA 600 BC)
I will be sufficiently rewarded if when telling it to others you
will not claim the discovery as your own, but will say it was mine.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.
%
Thompson, D'Arcy Wentworth (1860-1948)
Cell and tissue, shell and bone, leaf and flower, are so many
portions of matter, and it is in obedience to the laws of physics
that their particles have been moved, moulded and conformed. They are
no exceptions to the rule that God always geometrizes. Their problems
of form are in the first instance mathematical problems, their
problems of growth are essentially physical problems, and the
morphologist is, ipso facto, a student of physical science.
On Growth and Form, 1917.
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Thomson, [Lord Kelvin] William (1824-1907)
Fourier is a mathematical poem.
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Thoreau
He is not a true man of science who does not bring some sympathy to
his studies, and expect to learn something by behavior as well as by
application. It is childish to rest in the discovery of mere
coincidences, or of partial and extraneous laws. The study of
geometry is a petty and idle exercise of the mind, if it is applied
to no larger system than the starry one. Mathematics should be mixed
not only with physics but with ethics; that is mixed mathematics. The
fact which interests us most is the life of the naturalist. The
purest science is still biographical.
%
Tietze
The story was told that the young Dirichlet had as a constant
companion all his travels, like a devout man with his prayer book, an
old, worn copy of the Disquisitiones Arithmeticae of Gauss.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
%
Tillotson, Archbishop
How often might a man, after he had jumbled a set of letters in a
bag, fling them out upon the ground before they would fall into an
exact poem, yea, or so much as make a good discourse in prose. And
may not a little book be as easily made by chance as this great
volume of the world.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Titchmarsh, E. C.
Perhaps the most surprising thing about mathematics is that it is so
surprising. The rules which we make up at the beginning seem ordinary
and inevitable, but it is impossible to foresee their consequences.
These have only been found out by long study, extending over many
centuries. Much of our knowledge is due to a comparatively few great
mathematicians such as Newton, Euler, Gauss, or Riemann; few careers
can have been more satisfying than theirs. They have contributed
something to human thought even more lasting than great literature,
since it is independent of language.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Titchmarsh, E. C.
It can be of no practical use to know that Pi is irrational, but if
we can know, it surely would be intolerable not to know.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Todhunter, Isaac (1820 - 1910)
[Asked whether he would like to see an experimental demonstration of
conical refraction]
No. I have been teaching it all my life, and I do not want to have
my ideas upset.
%
Tolstoy, [Count] Lev Nikolgevich (1828-1920)
A modern branch of mathematics, having achieved the art of dealing
with the infinitely small, can now yield solutions in other more
complex problems of motion, which used to appear insoluble. This
modern branch of mathematics, unknown to the ancients, when dealing
with problems of motion, admits the conception of the infinitely
small, and so conforms to the chief condition of motion (absolute
continuity) and thereby corrects the inevitable error which the human
mind cannot avoid when dealing with separate elements of motion
instead of examining continuous motion. In seeking the laws of
historical movement just the same thing happens. The movement of
humanity, arising as it does from innumerable human wills, is
continuous. To understand the laws of this continuous movement is the
aim of history. Only by taking an infinitesimally small unit for
observation (the differential of history, that is, the individual
tendencies of
%
Tolstoy, Count Lev Nikolgevich (1828-1920)
A man is like a fraction whose numerator is what he is and whose
denominator is what he thinks of himself. The larger the denominator
the smaller the fraction.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber
and Schmidt, 1989.
%
Truesdell, Clifford
This paper gives wrong solutions to trivial problems. The basic
error,however, is not new.
Mathematical Reviews 12, p561.
%
Turgenev, Ivan Sergeievich (1818 - 1883)
Whatever a man prays for, he prays for a miracle. Every prayer
reduces itself to this: `Great God, grant that twice two be not four'.
%
Turnbull, H.W.
Attaching significance to invariants is an effort to recognize what,
because of its form or colour or meaning or otherwise, is important
or significant in what is only trivial or ephemeral. A simple
instance of failing in this is provided by the poll-man at Cambridge,
who learned perfectly how to factorize a^2 - b^2 but was floored
because the examiner unkindly asked for the factors of p^2 - q^2.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Ulam, Stanislaw
In many cases, mathematics is an escape from reality. The
mathematician finds his own monastic niche and happiness in pursuits
that are disconnected from external affairs. Some practice it as if
using a drug. Chess sometimes plays a similar role. In their
unhappiness over the events of this world, some immerse themselves in
a kind of self-sufficiency in mathematics. (Some have engaged in it
for this reason alone.)
Adventures of a Mathematician, Scribner's, New York, 1976.
%
Valéry, Paul (1871 - 1945)
In the physical world, one cannot increase the size or quantity of
anything without changing its quality. Similar figures exist only in
pure geometry.
%
van Vleck, E. B.
This new integral of Lebesque is proving itself a wonderful tool. I
might compare it with a modern Krupp gun, so easily does it penetrate
barriers which were impregnable.
Bulletin of the American Mathematical Society, vol. 23, 1916.
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Veblen, Thorstein (1857-1929)
The outcome of any serious research can only be to make two
questions grow where only one grew before.
The Place of Science in Modern Civilization and Other Essays.
%
Veblen, Thorstein (1857-1929)
Invention is the mother of necessity.
J. Gross, The Oxford Book of Aphorisms, Oxford: Oxford University
Press, 1983.
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Voltaire (1694-1778)
Vous avez trouve par de long ennuis
Ce que Newton trouva sans sortir de chez lui.
[Written to La Condamine after his measurement of the equator.]
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Voltaire (1694-1778)
He who has heard the same thing told by 12,000 eye-witnesses has
only 12,000 probabilities, which are equal to one strong probability,
which is far from certain.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Voltaire (1694-1778)
There are no sects in geometry.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1962.
%
Walton, Izaak
Angling may be said to be so like mathematics that it can never be
fully learned.
The Compleat Angler, 1653.
%
Warner, Sylvia Townsend
For twenty pages perhaps, he read slowly, carefully, dutifully, with
pauses for self-examination and working out examples. Then, just as
it was working up and the pauses should have been more scrupulous
than ever, a kind of swoon and ecstasy would fall on him, and he read
ravening on, sitting up till dawn to finish the book, as though it
were a novel. After that his passion was stayed; the book went back
to the Library and he was done with mathematics till the next bout.
Not much remained with him after these orgies, but something
remained: a sensation in the mind, a worshiping acknowledgment of
something isolated and unassailable, or a remembered mental joy at
the rightness of thoughts coming together to a conclusion, accurate
thoughts, thoughts in just intonation, coming together like
unaccompanied voices coming to a close.
Mr. Fortune's Maggot.
%
Warner, Sylvia Townsend
Theology, Mr. Fortune found, is a more accommodating subject than
mathematics; its technique of exposition allows greater latitude. For
instance when you are gravelled for matter there is always the moral
to fall back upon. Comparisons too may be drawn, leading cases
cited, types and antetypes analysed and anecdotes introduced. Except
for Archimedes mathematics is singularly naked of anecdotes.
Mr. Fortune's Maggot.
%
Warner, Sylvia Townsend
He resumed:
"In order to ascertain the height of the tree I must be in such a
position that the top of the tree is exactly in a line with the top
of a measuring stick or any straight object would do, such as an
umbrella which I shall secure in an upright position between my feet.
Knowing then that the ratio that the height of the tree bears to the
length of the measuring stick must equal the ratio that the distance
from my eye to the base of the tree bears to my height, and knowing
(or being able to find out) my height, the length of the measuring
stick and the distance from my eye to the base of the tree, I can,
therefore, calculate the height of the tree."
"What is an umbrella?"
Mr. Fortune's Maggot.
%
Warren, Robert Penn (1905-)
What if angry vectors veer
Round your sleeping head, and form.
There's never need to fear
Violence of the poor world's abstract storm.
Lullaby in Encounter, 1957.
%
Weil, Andre (1906 -1998)
Every mathematician worthy of the name has experienced ... the state
of lucid exaltation in which one thought succeeds another as if
miraculously... this feeling may last for hours at a time, even for
days. Once you have experienced it, you are eager to repeat it but
unable to do it at will, unless perhaps by dogged work...
The Apprenticeship of a Mathematician.
%
Weil, Andre (1906- 1998)
God exists since mathematics is consistent, and the Devil exists
since we cannot prove it.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
%
Weil, Simone (1909 - 1943)
Algebra and money are essentially levelers; the first
intellectually, the second effectively.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
West, Nathanael
Prayers for the condemned man will be offered on an adding machine.
Numbers constitute the only universal language.
Miss Lonelyhearts.
%
Weyl, Hermann (1885 - 1955)
Our federal income tax law defines the tax y to be paid in terms of
the income x; it does so in a clumsy enough way by pasting several
linear functions together, each valid in another interval or bracket
of income. An archeologist who, five thousand years from now, shall
unearth some of our income tax returns together with relics of
engineering works and mathematical books, will probably date them a
couple of centuries earlier, certainly before Galileo and Vieta.
The Mathematical Way of Thinking, an address given at the
Bicentennial Conference at the University of Pennsylvania, 1940.
%
Weyl, Hermann (1885 - 1955)
We are not very pleased when we are forced to accept a mathematical
truth by virtue of a complicated chain of formal conclusions and
computations, which we traverse blindly, link by link, feeling our
way by touch. We want first an overview of the aim and of the road;
we want to understand the idea of the proof, the deeper context.
Unterrichtsblatter fur Mathematik und Naturwissenschaften, 38,
177-188 (1932). Translation by Abe Shenitzer appeared in The American
Mathematical Monthly, v. 102, no. 7 (August-September 1995), p. 646.
%
Weyl, Hermann (1885 - 1955)
A modern mathematical proof is not very different from a modern
machine, or a modern test setup: the simple fundamental principles
are hidden and almost invisible under a mass of technical details.
Unterrichtsblatter fur Mathematik und Naturwissenschaften, 38,
177-188 (1932). Translation by Abe Shenitzer appeared in The American
Mathematical Monthly, v. 102, no. 7 (August-September 1995), p. 646.
%
Weyl, Hermann (1885-1955)
The constructs of the mathematical mind are at the same time free
and necessary. The individual mathematician feels free to define his
notions and set up his axioms as he pleases. But the question is will
he get his fellow mathematician interested in the constructs of his
imagination. We cannot help the feeling that certain mathematical
structures which have evolved through the combined efforts of the
mathematical community bear the stamp of a necessity not affected by
the accidents of their historical birth. Everybody who looks at the
spectacle of modern algebra will be struck by this complementarity of
freedom and necessity.
%
1951.
%
Weyl, Hermann (1885 - 1955)
My work has always tried to unite the true with the beautiful and
when I had to choose one or the other, I usually chose the beautiful.
In an obituary by Freeman J. Dyson in Nature, March 10, 1956.
%
Weyl, Hermann (1885 - 1955)
... numbers have neither substance, nor meaning, nor qualities. They
are nothing but marks, and all that is in them we have put into them
by the simple rule of straight succession.
"Mathematics and the Laws of Nature" in The Armchair Science Reader,
New York: Simon and Schuster, 1959.
%
Weyl, Hermann (1885 - 1955)
Without the concepts, methods and results found and developed by
previous generations right down to Greek antiquity one cannot
understand either the aims or achievements of mathematics in the last
50 years.
[Said in 1950]
The American Mathematical Monthly, v. 100. p. 93.
%
Weyl, Hermann (1885 - 1955)
Logic is the hygiene the mathematician practices to keep his ideas
healthy and strong.
The American Mathematical Monthly, November, 1992.
%
Whewell
Nobody since Newton has been able to use geometrical methods to the
same extent for the like purposes; and as we read the Principia we
feel as when we are in an ancient armoury where the weapons are of
gigantic size; and as we look at them we marvel what manner of man he
was who could use as a weapon what we can scarcely lift as a burden.
In E. N. Da C. Andrade "Isaac Newton" in J. R. Newman (ed.) The
World of Mathematics, New York: Simon and Schuster, 1956.
%
Whitehead, Alfred North (1861 - 1947)
The science of pure mathematics ... may claim to be the most
original creation of the human spirit.
Science and the Modern World.
%
Whitehead, Alfred North (1861 - 1947)
Mathematics as a science, commenced when first someone, probably a
Greek, proved propositions about "any" things or about "some" things,
without specifications of definite particular things.
%
Whitehead, Alfred North (1861 - 1947)
So far as the mere imparting of information is concerned, no
university has had any justification for existence since the
popularization of printing in the fifteenth century.
The Aims of Education.
%
Whitehead, Alfred North (1861 - 1947)
No Roman ever died in contemplation over a geometrical diagram.
[A reference to the death of Archimedes.]
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Whitehead, Alfred North (1861 - 1947)
Life is an offensive, directed against the repetitious mechanism of
the Universe.
Adventures of Ideas, 1933.
%
Whitehead, Alfred North (1861 - 1947)
There is no nature at an instant.
%
Whitehead, Alfred North (1861 - 1947)
Let us grant that the pursuit of mathematics is a divine madness of
the human spirit, a refuge from the goading urgency of contingent
happenings.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Whitehead, Alfred North (1861 - 1947)
There is a tradition of opposition between adherents of induction
and of deduction. In my view it would be just as sensible for the two
ends of a worm to quarrel.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Whitehead, Alfred North (1861 - 1947)
It is a profoundly erroneous truism, repeated by all copy books and
by eminent people when they are making speeches, that we should
cultivate the habit of thinking of what we are doing. The precise
opposite is the case. Civilization advances by extending the number
of important operations which we can perform without thinking about
them.
An Introduction to Mathematics.
%
Whitehead, Alfred North (1861 - 1947)
Our minds are finite, and yet even in these circumstances of
finitude we are surrounded by possibilities that are infinite, and
the purpose of life is to grasp as much as we can out of that
infinitude.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Whitehead, Alfred North (1861 - 1947)
In modern times the belief that the ultimate explanation of all
things was to be found in Newtonian mechanics was an adumbration of
the truth that all science, as it grows towards perfection, becomes
mathematical in its ideas.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.
%
Whitehead, Alfred North (1861 - 1947)
Algebra reverses the relative importance of the factors in ordinary
language. It is essentially a written language, and it endeavors to
exemplify in its written structures the patterns which it is its
purpose to convey. The pattern of the marks on paper is a particular
instance of the pattern to be conveyed to thought. The algebraic
method is our best approach to the expression of necessity, by reason
of its reduction of accident to the ghostlike character of the real
variable.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
Be relieving the brain of all unnecessary work, a good notation sets
it free to concentrate on more advanced problems, and, in effect,
increases the mental power of the race.
In P. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhauser, 1981.
%
Whitehead, Alfred North (1861 - 1947)
Everything of importance has been said before by somebody who did
not discover it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Whitehead, Alfred North (1861 - 1947)
Seek simplicity, and distrust it.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
Fundamental progress has to do with the reinterpretation of basic
ideas.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
We think in generalities, but we live in details.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
Apart from blunt truth, our lives sink decadently amid the perfume
of hints and suggestions.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
"Necessity is the mother of invention" is a silly proverb.
"Necessity is the mother of futile dodges" is much nearer the truth.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
It is more important that a proposition be interesting than that it
be true. This statement is almost a tautology. For the energy of
operation of a proposition in an occasion of experience is its
interest and is its importance. But of course a true proposition is
more apt to be interesting than a false one.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
War can protect; it cannot create.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
The progress of Science consists in observing interconnections and
in showing with a patient ingenuity that the events of this
ever-shifting world are but examples of a few general relations,
called laws. To see what is general in what is particular, and what
is permanent in what is transitory, is the aim of scientific thought.
An Introduction to Mathematics.
%
Whitehead, Alfred North (1861 - 1947)
Through and through the world is infested with quantity: To talk
sense is to talk quantities. It is not use saying the nation is large
.. How large? It is no use saying the radium is scarce ... How
scarce? You cannot evade quantity. You may fly to poetry and music,
and quantity and number will face you in your rhythms and your
octaves.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Whitehead, Alfred North (1861 - 1947)
"One and one make two" assumes that the changes in the shift of
circumstance are unimportant. But it is impossible for us to analyze
this notion of unimportant change.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
I will not go so far as to say that to construct a history of
thought without profound study of the mathematical ideas of
successive epochs is like omitting Hamlet from the play which is
named after him. That would be claiming too much. But it is certainly
analogous to cutting out the part of Ophelia. This simile is
singularly exact. For Ophelia is quite essential to the play, she is
very charming ... and a little mad.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
Whitehead, Alfred North (1861 - 1947)
The study of mathematics is apt to commence in disappointment....We
are told that by its aid the stars are weighed and the billions of
molecules in a drop of water are counted. Yet, like the ghost of
Hamlet's father, this greatest science eludes the efforts of our
mental weapons to grasp it.
An Introduction to Mathematics
%
Whitehead, Alfred North (1861 - 1947)
In the study of ideas, it is necessary to remember that insistence
on hard-headed clarity issues from sentimental feeling, as it were a
mist, cloaking the perplexities of fact. Insistence on clarity at all
costs is based on sheer superstition as to the mode in which human
intelligence functions. Our reasonings grasp at straws for premises
and float on gossamers for deductions.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Whitehead, Alfred North (1861 - 1947)
Familiar things happen, and mankind does not bother about them. It
requires a very unusual mind to undertake the analysis of the obvious.
Science and the Modern World.
%
Whitman, Walt (1819-1892)
Do I contradict myself? Very well then I contradict myself. (I am
large, I contains multitudes).
Song of Myself, 1939.
%
Whitman, Walt (1819-1892)
When I heard the learn'd astronomer,
When the proofs, the figure, were ranged in columns before me,
When I was shown the charts and diagrams, to add, divide, and
measure them,
When I sitting heard the astronomer where he lectured with much
applause in the lecture room,
How soon unaccountable I became tired and sick,
Till rising and gliding out I wander'd off by myself,
In the mystical moist night-air, and from time to time,
Look'd up in perfect silence at the stars.
%
Wiener, Norbert (1894 - 1964)
A professor is one who can speak on any subject -- for precisely
fifty minutes.
%
Wiener, Norbert (1894-1964)
The modern physicist is a quantum theorist on Monday, Wednesday, and
Friday and a student of gravitational relativity theory on Tuesday,
Thursday, and Saturday. On Sunday he is neither, but is praying to
his God that someone, preferably himself, will find the
reconciliation between the two views.
%
Wiener, Norbert (1894-1964)
Progress imposes not only new possibilities for the future but new
restrictions.
The Human Use of Human Beings.
%
Wiener, Norbert (1894-1964)
The Advantage is that mathematics is a field in which one's blunders
tend to show very clearly and can be corrected or erased with a
stroke of the pencil. It is a field which has often been compared
with chess, but differs from the latter in that it is only one's best
moments that count and not one's worst. A single inattention may lose
a chess game, whereas a single successful approach to a problem,
among many which have been relegated to the wastebasket, will make a
mathematician's reputation.
Ex-Prodigy: My Childhood and Youth.
%
Wilder, R. L.
There is nothing mysterious, as some have tried to maintain, about
the applicability of mathematics. What we get by abstraction from
something can be returned.
Introduction to the Foundations of Mathematics.
%
Wilder, R. L.
Mathematics was born and nurtured in a cultural environment. Without
the perspective which the cultural background affords, a proper
appreciation of the content and state of present-day mathematics is
hardly possible.
In The American Mathematical Monthly, March 1994.
%
William of Occam (1300-1439)
[Occam's Razor:]
Entities should not be multiplied unnecessarily.
Quodlibeta.
%
Wilson, John (1741 - 1793)
A monument to Newton! a monument to Shakespeare! Look up to Heaven
look into the Human Heart. Till the planets and the passionsthe
affections and the fixed stars are extinguishedtheir names cannot die.
%
Wittgenstein, Ludwig (1889-1951)
We could present spatially an atomic fact which contradicted the
laws of physics, but not one which contradicted the laws of geometry.
Tractatus Logico Philosophicus, New York, 1922.
%
Wittgenstein, Ludwig (1889-1951)
Mathematics is a logical method ... Mathematical propositions
express no thoughts. In life it is never a mathematical proposition
which we need, but we use mathematical propositions only in order to
infer from propositions which do not belong to mathematics to others
which equally do not belong to mathematics.
Tractatus Logico Philosophicus, New York, 1922, p. 169.
%
Wittgenstein, Ludwig (1889-1951)
There can never be surprises in logic.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
%
Wittgenstein, Ludwig (1889-1951)
The riddle does not exist. If a question can be put at all, then it
can also be answered.
Tractatus Logico Philosophicus, New York, 1922.
%
Wordsworth, William (1770 - 1850)
[Mathematics] is an independent world
Created out of pure intelligence.
%
Wren, Sir Christopoher
In things to be seen at once, much variety makes confusion, another
vice of beauty. In things that are not seen at once, and have no
respect one to another, great variety is commendable, provided this
variety transgress not the rules of optics and geometry.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New
York: Viking Press, 1966.
%
X, Malcom
I'm sorry to say that the subject I most disliked was mathematics.
I have thought about it. I think the reason was that mathematics
leaves no room for argument. If you made a mistake, that was all
there was to it.
Mascot.
%
Young, J. W. A.
Mathematics has beauties of its own -- a symmetry and proportion in
its results, a lack of superfluity, an exact adaptation of means to
ends, which is exceedingly remarkable and to be found only in the
works of the greatest beauty When this subject is properly ...
presented, the mental emotion should be that of enjoyment of beauty,
not that of repulsion from the ugly and the unpleasant.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.
%
Zeeman, E Christopher (1925 - )
Technical skill is mastery of complexity while creativity is mastery
of simplicity.
Catastrophe Theory, 1977.