Mathematical Quotes
Here are some math quotes. Many of these care from Professor Mark Woodard
at Furman:
http://math.furman.edu/~mwoodard/mquot.html,
many more were added by Professor L. C. Berselli ,
http://docenti.ing.unipi.it/~d9378/
and I've added a few. Enjoy!
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.
Abel, Niels H. (1802 - 1829)
[A reply to a question
about how he got his expertise:]
By studying the masters and not
their pupils.
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.
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.
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
Somebody Else's Problem field.
The third and most mysterious piece of
nonabsoluteness of all lies in the relationship between the number of items
on the bill, the cost of each item, the number of people at the table and what
they are each prepared to pay for. (The number of people who have actually
brought any money is only a subphenomenon of this field.)
Life, the
Universe and Everything. New York: Harmony Books, 1982.
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.
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.
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.
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.
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.
Aiken, Conrad
[At a musical concert:]
...the
music's pure algebra of enchantment.
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.
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.
Anonymous
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 w
isdom.
In Ivor Thomas "Greek Mathematics" in J. R. Newman
(ed.) The World of Mathematics, New York: Simon and Schuster,
1956.
Anonymous
Defendit numerus: There is safety in
numbers.
In J. R. Newman (ed.) The World of Mathematics, New
York: Simon and Schuster, 1956, p. 1452.
Anonymous
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]
Anonymous
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, Web
er, and Schmidt, 1988.
Arbuthnot, John
The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of
being reduc'd to a Mathematical Reasoning; and when they cannot it's a sign our knowledge of them is very small and confus'd; and when a Mathematical Reasoning can be had it's as great a folly to make use of any other, as to grope for a thing in the dark,
when you have a Candle standing by you.
Of the Laws of Chance.
(1692)
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]
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
Aristotle (ca 330 BC)
The whole is more than the sum of
its parts.
Metaphysica 10f-1045a
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
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.
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.
Aristotle
The mathematical sciences particularly exhibit
order, symmetry, and limitation; and these are the greatest forms of the
beautiful.
Metaphysica, 3-1078b.
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 Practit
ioners of Tudor and Stuart England, Cambridge: Cambridge University Press, 1954.
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" [Pythago
ras' 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.
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: Fabe
r & Faber, 1948.
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 lyr
e" in Collected Poems of W H Auden, London: Faber and Faber.
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.
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 i
f the work of the six days did not exist.
The City of God.
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 m
an in the bonds of Hell.
DeGenesi ad Litteram, Book II, xviii, 37
[Note: mathematician = astrologer]
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.
Babbage, Charles (1792-1871)
Errors using inadequate
data are much less than those using no data at all.
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 confu
sion of ideas that could provoke such a question.
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.
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.
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.
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 intellect
ual. 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 its
elf 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
.
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.
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.
Balzac, Honore de (1799 - 1850)
Numbers are intellectual
witnesses that belong only to mankind.
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.
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.
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.
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.
Bell, Eric Temple (1883-1960)
The Handmaiden of the
Sciences.
[Book by that title.]
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.
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 rath
er than by any prospect of ultimate usefulness.
Bell, Eric Temple (1883-1960)
"Obvious" is the
most dangerous word in mathematics.
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 appl
ication to any particular.
In H. Eves Mathematical Circles
Squared, Boston: Prindle, Weber and Schmidt, 1972.
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.
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.
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-Wi
lliams The Search For Truth, p. 191.
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.
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.
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
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.
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.
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.
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 give
n quantity can be found or not.
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 & Co. Ltd., 1953.
Blake
God forbid that Truth should be confined to
Mathematical Demonstration!
Notes on Reynold's Discourses, c.
1808.
Blake
What is now proved was once only imagin'd.
The Marriage of Heaven and Hell, 1790-3.
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.
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.
Bolyai, János (1802 - 1860)
Out of nothing I have
created a strange new universe.
[A reference to the creation of a non
euclidean geometry.]
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 l
ife.
[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.
Bourbaki
Structures are the weapons of the
mathematician.
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.
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 makin
g 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 se
t 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.
Browne, Sir Thomas (1605-1682)
God is like a skilful
Geometrician.
Religio Medici I, 16.
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 Gard
en of Cyrus, 1896.
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 n
ature, 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 wi
sedome of their Maker.
In J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956, p. 1001.
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 underst
ood 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.
Burke, Edmund
The age of chivalry is gone. That of
sophisters, economists and calculators has succeeded.
Reflections
on the Revolution in France.
Butler, Bishop
To us probability is the very guide of
life.
Preface to Analogy.
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 h
as 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 h
e 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.
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 appl
e.
Caballero, James
I 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./P>
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.
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.
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.
Carlyle, Thomas (1795-1881)
A witty statesman said, you
might prove anything by figures.
Chartism.
Carroll, Lewis
What I tell you three times is true.
The Hunting of the Snark.
Carroll, Lewis
The different branches of Arithmetic --
Ambition, Distraction, Uglification, and Derision.
Alice in
Wonderland.
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."
Th
rough the Looking Glass.
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 di
d it for half an hour a day. Why, sometimes I've believed as many as six
impossible things before breakfast."
Alice in
Wonderland.
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.
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 yeste
rday 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 un
derstand you," said Alice. "It's dreadfully confusing."
Through the Looking Glass.
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.
Céline, Louis-Ferdinand (1894 - 1961)
Entre le
pénis et les mathématiques... il n'existe rien. Rien! C'est le
vide.
Voyage au bout de la nuit. Paris: Gallimard.
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.
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.
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.
Cayley, Arthur
Projective geometry is all geometry.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.
Cézanne, Paul (1839 - 1906)
...treat Nature by
the sphere, the cylinder and the cone...
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.
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.
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 i
n logic, not in imagination.
Orthodoxy ch. 2.
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.
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.
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.
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 qui
te enthralling.
An Autobiography.
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.
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: P
rindle, 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.
Comte, Auguste (1798-1857)
C'este donc par
l'étude des mathématiques, et seulement par elle, que
l'on peut se faire une idée juste et approfondie de ce que c'est qu'une
science.
Quoted by T. H. Huxley in F
ortnightly Review, Vol. II, N.S. 5.
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: Birkhäuser, 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 mathemat
ics 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 Preli
minaire 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 mathema
tical 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: Birkhäuser, 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.
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 th
ey 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 fortune
s 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 determi
ning 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 eac
h 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
surveys that I might be sure of leaving nothing out.
Discours de la
Méthode. 1637.
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 hidde
n 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 Sch
midt, 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 cheek
s 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 grav
e 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 c
ould 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: Birkhäuser, 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 A
dieu, 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. Gordo
n 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.
Dürer, 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.
Dürer, 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
Dürer's schriftlicher Nachlas
s Berlin, 1920.
Dürer, 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 s
tructure 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 t
hat 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 tha
t 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 &
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 Mis
cellany, 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 würfelt
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: P
rindle, 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 e
mbrace 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 C
ircles 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 an
d 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 distanc
e-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 t
hat 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 w
e 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. S
immons 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 hav
e 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 ch
isel. 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, 19
69.
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 ha
ve 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 to
o 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 d
emonstrates 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 a
bove 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 researc
h 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 on
ce, 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 brok
en, 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.
Elog
e 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 a
xioms 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'A
lembert 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 n
early 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 an
d 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 Yor
k: 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.
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.
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 an
d 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 Y
ork: 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 his
torical 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 dissolv
ed 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 Sch
uster, 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 futu
re; 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. Simm
ons 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, 1
830.
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 Lübsen]
Theory attracts practice as the magnet attracts
iron.
Foreword of H.B Lübsen'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 or
der 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 threa
d 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 inductio
n, 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, whi
ch 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)
One of the principal
objects of theoretical research in my department of knowledge is to find the
point of view from which the subject appears in its greatest simplicity.
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 Mat
hematical 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.
Goedel, Kurt
I don't believe in natural science.
[Said to
physicist John Bahcall.]
Ed Regis, Who Got Einstein's Office?
Addison Wesley, 1987.
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:
Birkhäuser, 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. Scienti
fic American 269:4 (October 1993) 92-103.
Grünbaum, 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 s
ort 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 s
tatement. 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 ra
te 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: Ke
gan 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 W
ant 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 s a faculty that can pass our Master's exams."
I
Want to be a Mathematician, Washi
ngton: 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 i
n 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 sp
ecial 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 degene
rate 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 cl
aimed, 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 n
ot 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, bu
t 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 Mathemati
cian'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 Journa
l, 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," a
re 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) a
s 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, 194
1.
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 o
f 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 immedia
tely 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?
Heinlein, Robert A.
Anyone who cannot cope with
mathematics is not fully human. At best he is a tolerable subhuman who has
learned to wear shoes, bathe, and not make messes in the
house.
Time Enough for Love.
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.) T
he 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 necess
ity 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 an
d 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 some
where 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: Birkhäuser, 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, mathema
tical present.
The Glass Bead Game, 1943.
Hilbert, David (1862-1943)
Wir müssen 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 the
ories 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. Aft
er 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 - Mat
hematician 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 infi
nity 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.)
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 Autocr
at 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 st
ructure: 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 di
sembodied 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, 1
956.
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 exquis
itely 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 pa
ges 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 language
s, 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 th
e 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: Birkhäuser, 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 Mathema
tics, 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 be
yond 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, Winte
r 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 Im
agination, 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 frighteni
ng 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, 19
40.
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 int
eresting, 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 Li
fe. 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 i
nto 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 no
ting 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 civilizat
ions, 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 C
ircles 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: Birkhäuser, 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 Mathematici
an'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 Ow
n 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
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 Pr
ess 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 Napo
leon 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 wh
ich 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 Schm