**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,

**Abel, Niels H. (1802 - 1829) **

**Abel, Niels H. (1802 - 1829) **

In G. F. Simmons,

**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.

**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.)

**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.

**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."

**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."

**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 **

**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.)

**Anonymous **Like the crest of a peacock so is mathematics
at the head of all knowledge.

[An old Indian saying. Also, "

**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

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.

**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.

*
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

**Aristotle **To Thales the primary question was not what do
we know, but how do we know it.

**Aristotle **The mathematical sciences particularly exhibit
order, symmetry, and limitation; and these are the greatest forms of the
beautiful.

**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,

*
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.

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.

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.

D*eGenesi 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?

**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

**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.

book 3.

**Bacon, Roger **For the things of this world cannot be made
known without a knowledge of mathematics.

**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,

**Bagehot, Walter **Life is a school of probability.

Quoted in J. R. Newman (ed.)

**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

**Bell, Eric Temple (1883-1960) **Wherever groups disclosed
themselves, or could be introduced, simplicity crystallized out of
comparative chaos.

**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

**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

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

**Belloc, Hillaire (1870-1953) **Statistics are the triumph of
the quantitative method, and the quantitative method is the victory of
sterility and death.

**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.)

**Bernoulli, Daniel **...it would be better for the true physics
if there were no mathematicians on earth.

In

**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,

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

**Blake **God forbid that Truth should be confined to
Mathematical Demonstration!

**Blake **What is now proved was once only imagin'd.

**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.

**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.

**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.

*
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.

**Butler, Bishop **To us probability is the very guide of
life.

Preface to

**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**

Everybody a mathematician?,

/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.

**Carlyle, Thomas (1795-1881) **Teaching school is but
another word for sure and not very slow destruction.

In H. Eves

**Carlyle, Thomas (1795-1881) **A witty statesman said, you
might prove anything by figures.

**Carroll, Lewis **What I tell you three times is true.

**Carroll, Lewis **The different branches of Arithmetic --
Ambition, Distraction, Uglification, and Derision.

**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."

*
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) **

Voyage au bout de la nuit.

**Carmichael, R. D. **A thing is obvious mathematically after
you see it.

In N. Rose (ed.)

**Cauchy, Augustin-Louis (1789 - 1857) **Men pass away, but
their deeds abide.

[His last words (?)]

In H. Eves

**Cayley, Arthur **As for everything else, so for a
mathematical theory: beauty can be perceived but not explained.

In J. R. Newman (ed.)

**Cayley, Arthur **Projective geometry is all geometry.

In J. R. Newman (ed.)

**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,

**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

**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.

**Chesterton, G. K. (1874 - 1936) **It isn't that they can't
see the solution. It is that they can't see the problem.

**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

**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

.

**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,

Quoted by T. H. Huxley in

*
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.

**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.

**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

**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.)

**Darwin, Charles **Mathematics seems to endow one with
something like a new sense.

In N. Rose (ed.)

**Davis, Philip J. **The numbers are a catalyst that can help
turn raving madmen into polite humans.

In N. Rose (ed.)

**Davis, Philip J. **One of the endlessly alluring aspects of
mathematics is that its thorniest paradoxes have a way of blooming into
beautiful theories.

**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.

*
*, Boston: Birkhäuser, 1981.

**Dehn, Max **Mathematics is the only instructional material
that can be presented in an entirely undogmatic way.

In

**De Morgan, Augustus (1806-1871) **[When asked about his
age.] I was x years old in the year x^2.

In H. Eves

**De Morgan, Augustus (1806-1871) **It is easier to square
the circle than to get round a mathematician.

In H. Eves

**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.

**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

**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) **

**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

**Descartes, René (1596-1650) **

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.

**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.

**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

, 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

*
dieu*, Boston: Prindle, Weber and Schmidt, 1977.

**Disraeli, Benjamin **There are three kinds of lies: lies,
damned lies, and statistics.

Mark Twain.

**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.

**Doyle, Sir Arthur Conan (1859-1930) **From a drop of
water a logician could predict an Atlantic or a Niagara.

**Doyle, Sir Arthur Conan (1859-1930) **It is a capital
mistake to theorize before one has data.

**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

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.)

**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.)

*
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...

**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.

**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.

*
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

**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

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.)

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.

*
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.)

**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.

*
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.)

**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,

*
cellany*, Methuen and Co. Ltd., 1953.

**Einstein, Albert (1879-1955) **Everything should be made
as simple as possible, but not simpler.

**Einstein, Albert (1879-1955) **I don't believe in
mathematics.

Quoted by Carl Seelig.

**Einstein, Albert (1879-1955) **Imagination is more
important than knowledge.

**Einstein, Albert (1879-1955) **The most beautiful thing we
can experience is the mysterious. It is the source of all true art and
science.

**Einstein, Albert (1879-1955) **The bitter and the sweet
come from the outside, the hard from within, from one's own efforts.

**Einstein, Albert (1879-1955) **

**Einstein, Albert (1879-1955) **Common sense is the
collection of prejudices acquired by age eighteen.

In E. T. Bell

**Einstein, Albert (1879-1955) **God does not care about our
mathematical difficulties. He integrates empirically.

L. Infeld

**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

**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.)

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

**Einstein, Albert (1879-1955) **

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.)

*
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

**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

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

**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

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

**Einstein, Albert (1879-1955) **The search for truth is
more precious than its possession.

**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.

**Einstein, Albert (1879-1955) **Science without religion is
lame; religion without science is blind.

**Ellis, Havelock **The mathematician has reached the highest
rung on the ladder of human thought.

**Ellis, Havelock **It is here [in mathematics] that the artist
has the fullest scope of his imagination.

**Erath, V. **God is a child; and when he began to play, he
cultivated mathematics. It is the most godly of man's games.

**Erdös, Paul **Mathematics is not yet ready for such
problems.

[Attributed by Paul Halmos.]

**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

**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.

**Eves, Howard W. **An expert problem solver must be
endowed with two incompatible qualities, a restless imagination and a patient
pertinacity.

**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

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'

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,

**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

*
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.

*
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...

**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.)

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.

> [On the margin of his own copy of *Dialogue on the Great World
Systems*].

In J. R. Newman (ed.) *The World of Mathematics*,
New York: Simon and Schuster, 1956, p. 733.

**Galois, Evariste **Unfortunately what is little recognized is
that the most worthwhile scientific books are those in which the author
clearly indicates what he does not know; for an author most hurts his
readers by concealing difficulties.

> In N. Rose (ed.)

**Galton, [Sir] Francis (1822-1911) **Whenever you can,
count.

In J. R. Newman (ed.)

**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,

*
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.)

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

**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

**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.)

**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

**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

*
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,

**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.

**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

**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.

*
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

**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

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.

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.

*
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."

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.

**Hamilton, [Sir] William Rowan (1805-1865) **Who would not
rather have the fame of Archimedes than that of his conqueror
Marcellus?

In H. Eves

**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.

**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

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.

**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.

**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.

*
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.

*
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.

**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

**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.

**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.

**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.), T*eaching 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

**Hermite, Charles (1822-1901) **We are servants rather
than masters in mathematics.

In H. Eves

**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

*
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

**Hilbert, David (1862-1943) **Physics is much too hard for
physicists.

C. Reid

**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

**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

**Hilbert, David (1862-1943)**Mathematics knows no races
or geographic boundaries; for mathematics,the cultural world is one
country.

In H. Eves

**Hilbert, David (1862-1943) **The infinite! No other question
has ever moved so profoundly the spirit of man.

In J. R. Newman (ed.)

**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

**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.)

**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

**Holmes, Oliver Wendell **Certitude is not the test of
certainty. We have been cocksure of many things that are not so.

In G. Simmons

**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.

**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,

**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

**Jacobi, Carl **One should always generalize.

**Jacobi, Carl **The real end of science is the honor of the
human mind.

In H. Eves

**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

*
*, Boston: Prindle, Weber and Schmidt, 1977.

**Jacobi, Carl **Mathematics is the science of what is clear by
itself.

In J. R. Newman (ed.)

**James, William (1842 - 1910) **The union of the
mathematician with the poet, fervor with measure, passion with
correctness, this surely is the ideal.

**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.)

*
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.

**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.

**Johnson, Samuel (1709-1784) **Sir, I have found you an
argument. I am not obliged to find you an understanding.

J. Boswell

**Jowett, Benjamin (1817 - 1893) **Logic is neither a science
or an art, but a dodge.

In J. R. Newman (ed.)

**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.

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.

**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.

*
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.

**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.

*
agination*, New York: Simon and Schuster, 1940.

**Kasner, E. and Newman, J. **Mathematics is the science
which uses easy words for hard ideas.

**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.

**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,

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?

*
fe.* 1903.

**Kelley, John **A topologist is one who doesn't know the
difference between a doughnut and a coffee cup.

In N. Rose

**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.

**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

**Kline, Morris **Statistics: the mathematical theory of
ignorance.

In N. Rose

**Kline, Morris **Logic is the art of going wrong with
confidence.

In N. Rose

**Kline, Morris **Universities hire professors the way some
men choose wives - they want the ones the others will admire.

**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

*
ircles Adieu*, Boston: Prindle, Weber and Schmidt, 1977.

**Kronecker, Leopold (1823 - 1891) **God made the integers,
all else is the work of man.

**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

**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

.

**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

**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,

*
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.

**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.

**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

**de Laplace, Pierre-Simon (1749 - 1827) **[His last words,
according to De Morgan:]

Man follows only phantoms.

DeMorgan's

**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.)

Schuster, 1959.

**de Laplace, Pierre-Simon (1749 - 1827) **Read Euler: he is
our master in everything.

In G. Simmons

**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

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

**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 Schmidt,
1988.

**Leach, Edmund Ronald (1910 - 1989) **How can a modern
anthropologist embark upon a generalization with any hope of arriving at a
satisfactory conclusion? By thinking of the organizational ideas that are
present in any society as a mathematica

l pattern.

*Rethinking Anthropology*. 1961.

**Leacock, Stephen **How can you shorten the subject? That
stern struggle with the multiplication table, for many people not yet ended in
victory, how can you make it less? Square root, as obdurate as a hardwood
stump in a pasturenothing but y

ears of effort can extract it. You can't hurry the process. Or pass from arithmetic to algebra; you can't shoulder your way past quadratic equations or ripple through the binomial theorem. Instead, the other way; your feet are impeded in the tangled growt

h, your pace slackens, you sink and fall somewhere near the binomial theorem with the calculus in sight on the horizon. So died, for each of us, still bravely fighting, our mathematical training; except for a set of people called "mathematicians"

; -- born so, like crooks.

In H. Eves *Return to Mathematical
Circles*, Boston: Prindle, Weber and Schmidt, 1988.

**Lebesgue, Henri (1875 - 1941) **In my opinion, a
mathematician, in so far as he is a mathematician, need not preoccupy
himself with philosophy -- an opinion, moreover, which has been expressed by
many philosophers.

*
/I>, 211, September 1964, p. 129.*

*
*

* Lehrer, Thomas Andrew (1928- ) In one word he told me
the secret of success in mathematics: plagiarize only be sure always to call
it please research. Lobachevski (A musical recording.)*

*
*

* Leibniz, Gottfried Whilhem (1646-1716) [about him:]
It is rare to find learned men who are clean, do not stink and have a sense of
humour. [attributed variously to Charles Louis de Secondat Montesquieu
and to the Duchess of Orl&eacu*

*
te;ans]*

*
*

* Leibniz, Gottfried Whilhem (1646-1716) Nothing is more
important than to see the sources of invention which are, in my opinion more
interesting than the inventions themselves. J. Koenderink, Solid
Shape, Cambridge Mass.: MIT Pres*

*
s, 1990.*

*
*

* Leibniz, Gottfried Whilhem (1646-1716) Music is the
pleasure the human soul experiences from counting without being aware that
it is counting. In N. Rose Mathematical Maxims and Minims,
Raleigh NC:Rome Press Inc., 1988.*

*
*

* Leibniz, Gottfried Whilhem (1646-1716)*

The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.

*
*

* Leibniz, Gottfried Whilhem (1646-1716) He who
understands Archimedes and Apollonius will admire less the achievements of
the foremost men of later times. In G. Simmons Calculus Gems,
New York: McGraw Hill Inc., 1992.*

*
*

* Leibniz, Gottfried Whilhem (1646-1716)*

In symbols one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully di

*
minished. In G. Simmons Calculus Gems, New York: McGraw Hill
Inc., 1992.*

*
*

* Leibniz, Gottfried Whilhem (1646-1716) The art of
discovering the causes of phenomena, or true hypothesis, is like the art of
decyphering, in which an ingenious conjecture greatly shortens the road.
New Essays Concerning Human Unders*

*
tanding*, IV, XII.

*
*

* Leibniz, Gottfried Whilhem (1646-1716)*

Although the whole of this life were said to be nothing but a dream and the physical world nothing but a phantasm, I should call this dream or phantasm real enough, if, using reason well, we were never

*
deceived by it. In J. R. Newman (ed.) The World of Mathematics,
New York: Simon and Schuster, 1956.*

*
*

* Leibniz, Gottfried Whilhem (1646-1716) The soul is the
mirror of an indestructible universe. The Monadology.*

*
*

* da Vinci, Leonardo (1452-1519) Whoever despises the
high wisdom of mathematics nourishes himself on delusion and will never still
the sophistic sciences whose only product is an eternal uproar. In N.
Rose Mathematical Maxims and Mini*

*
ms*, Raleigh NC:Rome Press Inc., 1988.

*
*

* da Vinci, Leonardo (1452 - 1519) Mechanics is the
paradise of the mathematical sciences, because by means of it one comes to
the fruits of mathematics. Notebooks, v. 1, ch. 20.*

*
*

* da Vinci, Leonardo (1452-1519)*

He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast.

*
*

**da Vinci, Leonardo (1452-1519) **No human investigation
can be called real science if it cannot be demonstrated mathematically.

*
*

**da Vinci, Leonardo (1452-1519) **Inequality is the cause of
all local movements.

*
*

* Leybourn, William (1626-1700) But leaving those of the
Body, I shall proceed to such Recreation as adorn the Mind; of which those of
the Mathematicks are inferior to none. Pleasure with Profit,
1694.*

*
*

* Lichtenberg, Georg Christoph (1742 - 1799) All
mathematical laws which we find in Nature are always suspect to me, in spite
of their beauty. They give me no pleasure. They are merely auxiliaries. At
close range it is all not true. In J *

*
P Stern Lichtenberg, 1959.*

*
*

* Lichtenberg, Georg Christoph (1742 - 1799)*

The great trick of regarding small departures from the truth as the truth itself -- on which is founded the entire integral calculus -- is also the basis of our witty speculations, where the whole

*
thing would often collapse if we considered the departures with philosophical
rigour. Aphorisms.*

*
*

**Lichtenberg, Georg Christoph (1742 - 1799) **In
mathematical analysis we call

*
ntages over the common language.*

*
*

* Lichtenberg, Georg Christoph (1742 - 1799)*

I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, "Oh, I see. That's how it must

*
be." This is a sign that they explain it to themselves from within their
own system.*

*
*

* le Lionnais, Francois Who has not be amazed to learn that
the function y = e^x , like a phoenix rising again from its own ashes, is its
own derivative? Great Currents of Mathematical Thought, vol. 1,
New York: Dover Publications.*

*
*

*
*

* Lippman, Gabriel (1845-1921) [On the Gaussian curve,
remarked to Poincaré:] Experimentalists think that it is a
mathematical theorem while the mathematicians believe it to be an
experimental fact. In D'Arcy Thompson On Gro*

*
wth and Form*, 1917.

*
*

* Littlewood, J. E. (1885 -1977)*

It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group. But one could not say the adverse chance was 1:10. Mathematics is a dangerous profession; an

*
appreciable proportion of us go mad, and then this particular event would be
quite likely. A Mathematician's Miscellany, Methuen and Co. ltd.,
1953.*

*
*

* Littlewood, J. E. (1885 -1977) A good mathematical joke
is better, and better mathematics, than a dozen mediocre papers. A
Mathematician's Miscellany, Methuen and Co. ltd., 1953.*

*
*

* Littlewood, J. E. (1885 -1977) I recall once saying that
when I had given the same lecture several times I couldn't help feeling that
they really ought to know it by now. A Mathematician's
Miscellany, Methuen and Co. ltd., 1953.<*

*
/P>*

*
*

* Littlewood, J. E. (1885 -1977)*

In passing, I firmly believe that research should be offset by a certain amount of teaching, if only as a change from the agony of research. The trouble, however, I freely admit, is that in practice you get ei

*
ther no teaching, or else far too much. "The Mathematician's Art
of Work" in Béla Bollobás (ed.) Littlewood's
Miscellany, Cambridge: Cambridge University Press, 1986.*

*
*

* Littlewood, J. E. (1885 -1977)*

It is possible for a mathematician to be "too strong" for a given occasion. He forces through, where another might be driven to a different, and possible more fruitful, approach. (So a rock climber m

*
ight force a dreadful crack, instead of finding a subtle and delicate
route.) A Mathematician's Miscellany, Methuen and Co. ltd.,
1953.*

*
*

* Littlewood, J. E. (1885 -1977)*

I constantly meet people who are doubtful, generally without due reason, about their potential capacity [as mathematicians]. The first test is whether you got anything out of geometry. To have disliked or fail

*
ed to get on with other [mathematical] subjects need mean nothing; much
drill and drudgery is unavoidable before they can get started, and bad
teaching can make them unintelligible even to a born mathematician.
A Mathematician's Miscellany, Met*

*
huen and Co. ltd., 1953.*

*
*

* Littlewood, J. E. (1885 -1977) The infinitely competent
can be uncreative. In H. Eves Mathematical Circles Squared,
Boston: Prindle, Weber and Schmidt, 1972.*

*
*

* Littlewood, J. E. (1885 -1977)*

In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary point and take details for granted: his successive mouthfuls should be such

*
as can be swallowed at sight; in case of accidents, or in case he wishes for
once to check in detail, he should have only a clearly circumscribed little
problem to solve (e.g. to check an identity: two trivialities omitted can add
up to an impasse). The u*

*
npractised writer, even after the dawn of a conscience, gives him no such
chance; before he can spot the point he has to tease his way through a maze
of symbols of which not the tiniest suffix can be skipped. A
Mathematician's Miscellany, Methu*

*
en Co. Ltd., 1953.*

*
*

* Littlewood, J. E. (1885 -1977)*

A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that "E is dense in E" does not mean the same thing as "E is dense in itself".<

*
BR> A Mathematician's Miscellany, Methuen Co. Ltd., 1953.*

*
*

* Littlewood, J. E. (1885 -1977) The surprising thing about
this paper is that a man who could write it would. A Mathematician's
Miscellany, Methuen Co. Ltd., 1953.*

*
*

* Littlewood, J. E. (1885 -1977) A precisian professor had
the habit of saying: "... quartic polynomial ax^4+bx^3+cx^2+dx+e ,
where e need not be the base of the natural logarithms." A
Mathematician's Miscellany, Methuen *

*
Co. Ltd., 1953.*

*
*

* Littlewood, J. E. (1885 -1977)*

I read in the proof sheets of Hardy on Ramanujan: "As someone said, each of the positive integers was one of his personal friends." My reaction was, "I wonder who said that; I wish I had."

*
In the next proof-sheets I read (what now stands), "It was Littlewood
who said..." A Mathematician's Miscellany, Methuen Co. Ltd,
1953.*

*
*

* Littlewood, J. E. (1885 -1977)*

We come finally, however, to the relation of the ideal theory to real world, or "real" probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone w

*
ho wants them he would say that the ideal system runs parallel to the usual
theory: "If this is what you want, try it: it is not my business to justify
application of the system; that can only be done by philosophizing; I am a
mathematician". I*

*
n practice he is apt to say: "try this; if it works that will justify
it". But now he is not merely philosophizing; he is committing the
characteristic fallacy. Inductive experience that the system works is not
evidence. *

*
Miscellany,* Methuen Co. Ltd, 1953.

*
*

* Littlewood, J. E. (1885 -1977)*

The theory of numbers is particularly liable to the accusation that some of its problems are the wrong sort of questions to ask. I do not myself think the danger is serious; either a reasonable amount of conce

*
ntration leads to new ideas or methods of obvious interest, or else one just
leaves the problem alone. "Perfect numbers" certainly never did
any good, but then they never did any particular harm. A
Mathematician's Miscellany, Methuen *

*
Co. Ltd., 1953.*

*
*

* Lobatchevsky, Nikolai There is no branch of mathematics,
however abstract, which may not some day be applied to phenomena of the
real world. In N. Rose Mathematical Maxims and Minims, Raleigh
NC:Rome Press Inc., 1988.*

*
*

* Locke, John ...mathematical proofs, like diamonds, are
hard and clear, and will be touched with nothing but strict reasoning. D.
Burton, Elementary Number Theory, Boston: Allyn and Bacon
1980.*

*
*

**Luther, Martin (1483-1546) **Medicine makes people ill,
mathematics make them sad and theology makes them sinful.

*
*

* Mach, Ernst (1838 - 1916)*

Archimedes constructing his circle pays with his life for his defective biological adaptation to immediate circumstances.

*
*

* Mach, Ernst (1838-1916) The mathematician who pursues
his studies without clear views of this matter, must often have the
uncomfortable feeling that his paper and pencil surpass him in
intelligence. "The Economy of Science" in*

*
J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.*

*
*

* Mackay, Alan Lindsay (1926- ) Like the ski resort full of
girls hunting for husbands and husbands hunting for girls, the situation is not
as symmetrical as it might seem. A Dictionary of Scientific
Quotations, Bristol: IOP Publis*

*
hing, 1991.*

*
*

* Mackay, Charles (1814-1889) Truth ... and if mine
eyes Can bear its blaze, and trace its symmetries, Measure its
distance, and its advent wait, I am no prophet -- I but calculate.
The Poetical Works of Charles Mackay.*

*
1876.*

*
*

* Maistre Joseph Marie de (1753 - 1821)*

The concept of number is the obvious distinction between the beast and man. Thanks to number, the cry becomes a song, noise acquires rhythm, the spring is transformed into a dance, force becomes dynamic

*
, and outlines figures.*

*
*

* Mann, Thomas (1875-1955) A great truth is a truth
whose opposite is also a great truth. Essay on Freud.
1937.*

*
*

* Mann, Thomas (1875-1955) I tell them that if they will
occupy themselves with the study of mathematics they will find in it the best
remedy against the lusts of the flesh. The Magic Mountain.
1927.*

*
*

* Mann, Thomas (1875-1955)*

Some of the men stood talking in this room, and at the right of the door a little knot had formed round a small table, the center of which was the mathematics student, who ws eagerly talking. He had made the assertio

*
n that one could draw through a given point more than one parallel to a
straight line; Frau Hagenström had cried out that this was impossible,
and he had gone on to prove it so conclusively that his hearers were
constrained to behave as though they u*

*
nderstood. Little Herr Friedemann.*

*
*

* Mathesis, AdrianIf your new theorem can be stated with
great simplicity, then there will exist a pathological exception.In H. Eves
Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt,
1988.*

*
*

* Mathesis, AdrianAll great theorems were discovered after
midnight.In H. Eves Return to Mathematical Circles, Boston:
Prindle, Weber and Schmidt, 1988.*

*
*

* Mathesis, AdrianThe greatest unsolved theorem in
mathematics is why some people are better at it than others.In H. Eves
Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt,
1988.*

*
*

* Matthias, Bernd T*

If you see a formula in the Physical Review that extends over a quarter of a page, forget it. It's wrong. Nature isn't that complicated.

*
*

* Maxwell, James Clerk (1813-1879)*

... that, in a few years, all great physical constants will have been approximately estimated, and that the only occupation which will be left to men of science will be to carry these measurements to another

*
place of decimals. Scientific Papers 2, 244, October 1871.*

*
*

* Mayer, Maria Goeppert (1906 -1972) Mathematics began
to seem too much like puzzle solving. Physics is puzzle solving, too, but of
puzzles created by nature, not by the mind of man. J. Dash, Maria
Goeppert-Mayer, A Life of One's Own.<*

*
/I>*

*
*

* McDuff, Dusa*

Gel'fand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which could only hint at and which he had never m

*
anaged to bring out more clearly. I had always thought of mathematics as
being much more straightforward: a formula is a formula, and an algebra is
an algebra, but Gel'fand found hedgehogs lurking in the rows of his spectral
sequences! Mathematical*

*
Notices* v. 38, no. 3, March 1991, pp. 185-7.

*
*

* McShane, E. J.*

There are in this world optimists who feel that any symbol that starts off with an integral sign must necessarily denote something that will have every property that they should like an integral to possess. This of course is

*
quite annoying to us rigorous mathematicians; what is even more annoying is
that by doing so they often come up with the right answer. Bulletin
of the American Mathematical Society, v. 69, p. 611, 1963.*

*
*

* Mencken, H. L. (1880 - 1956) It is now quite lawful for a
Catholic woman to avoid pregnancy by a resort to mathematics, though she
is still forbidden to resort to physics and chemistry.
Notebooks, "Minority Report". *

*
*

* Mermin, Norman David (1935 -) Bridges would not be
safer
if only
people who knew the proper definition of a real number were allowed to
design them. "Topological Theory of Defects" in Review
of Modern Physics, v. 51 no. 3, Jul
y 1979.*

*
*

**Millay, Edna St. Vincent (1892 - 1950) **Euclid alone has
looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing

*
, intricately drawn nowhere In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek releaseFrom dusty bondage into
luminous air.O blinding hour, O holy, terrible day, When first the
shaft into his vision shone Of *

*
light anatomized! Euclid alone Has looked on Beauty bare. Fortunate
they Who, though once only and then but far away, Have heard her
massive sandal set on stone.*

*
*

**Milton, John (1608 - 1674) **From Man or Angel the great
Architect

Did wisely to conceal, and not divulge,

His secrets, to be scanned by them who ought

Rather admire. Or, if they list to try

Conjecture, he his fabric of the Hea

*
vens Hath left to their disputes -- perhaps to move His laughter at
their quaint opinions wide Hereafter, when they come to model
Heaven And calculate the stars: how they will wield The mighty
frame: how build, unbuild, contrive To*

*
save appearances; how gird the Sphere With Centric and Eccentric
scribbled o'er, Cycle and Epicycle, Orb in Orb. *

*
*

* Milton, John (1608-1674) Chaos umpire sits And by
decision more embroils the fray by which he reigns: next him
high arbiter Chance governs all.*

*
*

* Minkowski, Herman From henceforth, space by itself, and
time by itself, have vanished into the merest shadows and only a kind of
blend of the two exists in its own right. In J. R. Newman (ed.) The
World of Mathematics, New York: *

*
Simon and Schuster, 1956.*

*
*

* Minsky, Marvin Lee (1927 -) Logic doesn't apply to the real
world. D. R. Hofstadter and D. C. Dennett (eds.) The Mind's I,
1981.*

*
*

* Mitchell, Margaret*

...She knew only that if she did or said thus-and-so, men would unerringly respond with the complimentary thus-and so. It was like a mathematical formula and no more difficult, for mathematics was the one subject that had

*
come easy to Scarlett in her schooldays. Gone With the
Wind.*

*
*

* Mittag-Leffler, Gösta The mathematician's best
work is art, a high perfect art, as daring as the most secret dreams of
imagination, clear and limpid. Mathematical genius and artistic genius touch
one another. In N. Rose Mathemat*

*
ical Maxims and Minims*, Raleigh NC:Rome Press Inc., 1988.

*
*

* Mordell, L.J.*

Neither you nor I nor anybody else knows what makes a mathematician tick. It is not a question of cleverness. I know many mathematicians who are far abler than I am, but they have not been so lucky. An illustration may be give

*
n by considering two miners. One may be an expert geologist, but he does not
find the golden nuggets that the ignorant miner does. In H. Eves
Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt,
1977.*

*
*

* Moore, E.H. (1862 - 1932) We lay down a fundamental
principle of generalization by abstraction: "The existence of
analogies between central features of various theories implies the existence
of a general theory which underlies the *

*
particular theories and unifies them with respect to those central
features...." In H. Eves Mathematical Circles Revisited,
Boston: Prindle, Weber and Schmidt, 1971.*

*
*

* Moroney, M.J.*

The words figure and fictitious both derive from the same Latin root, fingere. Beware!

Facts from Figures.

*
*

* Mueller, Ian [about Hypatia:] In an era in which the
domain of intellect and politics were almost exclusively male, Theon [her
father] was an unusually liberated person who taught an unusually gifted
daughter and encouraged her to achie*

*
ve things that, as far as we know, no woman before her did or perhaps even
dreamed of doing. In G. Simmons Calculus Gems, New York:
McGraw Hill Inc., 1992.*

*
*

* Napoleon (1769-1821)*

A mathematician of the first rank, Laplace quickly revealed himself as only a mediocre administrator; from his first work we saw that we had been deceived. Laplace saw no question from its true point of view; he sought

*
subtleties everywhere; had only doubtful ideas, and finally carried the spirit
of the infinitely small into administration. In N. Rose Mathematical
Maxims and Minims, Raleigh NC:Rome Press Inc.,1988.*

*
*

* Nebeuts, E. Kim Teach to the the problems, not to the
text. In H. Eves Return to Mathematical Circles, Boston: Prindle,
Weber and Schmidt, 1988.*

*
*

* Nebeuts, E. Kim To state a theorem and then to show
examples of it is literally to teach backwards. In H. Eves Return to
Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.*

*
*

* Nebeuts, E. Kim A good preparation takes longer than the
delivery. In H. Eves Return to Mathematical Circles, Boston:
Prindle, Weber and Schmidt, 1988.*

*
*

* Neumann, Franz Ernst (1798 - 1895)*

The greatest reward lies in making the discovery; recognition can add little or nothing to that.

*
*

* von Neumann, Johann (1903 - 1957) In mathematics you
don't understand things. You just get used to them. In G. Zukav The
Dancing Wu Li Masters.*

*
*

* Newman, James R. The most painful thing about
mathematics is how far away you are from being able to use it after you
have learned it. In J. R. Newman (ed.) The World of Mathematics,
New York: Simon and Schuster, 1956.*

*
*

* Newman, James, R.*

The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pe

*
ncil and paper; there was left to observers only the routine task of pointing
their telescopes at the spot the mathematicians had marked. In J. R.
Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.*

*
*

* Newman, James R. It is hard to know what you are talking
about in mathematics, yet no one questions the validity of what you say.
There is no other realm of discourse half so queer. In J. R. Newman
(ed.) The World of Mathematics,*

*
New York: Simon and Schuster, 1956.*

*
*

* Newman, James R.*

Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or

*
expository device, can be applied to economic reasoning. There have even
been rumors that mathematics is used in economics (and in other social
sciences) either for the deliberate purpose of mystification or to confer
dignity upon common places as French *

*
was once used in diplomatic communications. In J. R. Newman (ed.)
The World of Mathematics, New Yorl: Simon and Schuster, 1956.*

*
*

* Newman, James R.*

To be sure, mathematics can be extended to any branch of knowledge, including economics, provided the concepts are so clearly defined as to permit accurate symbolic representation. That is only another way of saying that in

*
some branches of discourse it is desirable to know what you are talking
about. In J. R. Newman (ed.) The World of Mathematics, New
York: Simon and Schuster, 1956.*

*
*

* Newman, James R.*

The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing

*
. In J. R. Newman (ed.) The World of Mathematics, New York:
Simon and Schuster, 1956.*

*
*

* Newman, James R.*

Games are among the most interesting creations of the human mind, and the analysis of their structure is full of adventure and surprises. Unfortunately there is never a lack of mathematicians for the job of transforming del

*
ectable ingredients into a dish that tastes like a damp blanket. In J. R.
Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.*

*
*

* Newton, Isaac (1642-1727) ...from the same principles, I
now demonstrate the frame of the System of the World. Principia
Mathematica.*

*
*

* Newton, Isaac (1642-1727) Hypotheses non fingo. I
feign no hypotheses. Principia Mathematica.*

*
*

* Newton, Isaac (1642-1727)*

To explain all nature is too difficult a task for any one man or even for any one age. `Tis much better to do a little with certainty, and leave the rest for others hat come after you, than to explain all things. R> In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.

*
*

* Newton, Isaac (1642-1727) The description of right lines
and circles, upon which geometry is founded, belongs to mechanics.
Geometry does not teach us to draw these lines, but requires them to be
drawn. Principia Mathematica.*

*
*

* Newton, Isaac (1642-1727)*

The latest authors, like the most ancient, strove to subordinate the phenomena of nature to the laws of mathematics.

*
*

* Newton, Isaac (1642-1727) [His epitaph:] Who, by
vigor of mind almost divine, the motions and figures of the planets, the
paths of comets, and the tides of the seas first demonstrated.*

*
*

**Thomas R. Nicely**

Usually mathematicians have to shoot
somebody to get this much publicity.

[On the attention he received after
finding the flaw in Intel's Pentium chip in 1994]*Cincinnati
Enquirer,* December 18, 1994, Section A, p

*
age 19.*

*
*

* Nightingale, Florence (1820-1910) [Of her:] Her
statistics were more than a study, they were indeed her religion. For her
Quetelet was the hero as scientist, and the presentation copy of his
Physique sociale is annotated by her on every*

*
page. Florence Nightingale believed -- and in all the actions of her life acted
upon that belief -- that the administrator could only be successful if he were
guided by statistical knowledge. The legislator -- to say nothing of the
politician -- too ofte*

*
n failed for want of this knowledge. Nay, she went further; she held that the
universe -- including human communities -- was evolving in accordance with a
divine plan; that it was man's business to endeavor to understand this plan
and guide his actions in*

*
sympathy with it. But to understand God's thoughts, she held we must study
statistics, for these are the measure of His purpose. Thus the study of
statistics was for her a religious duty. K. Pearson The Life, Letters
and Labours for
Francis Galton, vol. 2, 1924.*

*
*

* Oakley, C.O. The study of mathematics cannot be replaced
by any other activity that will train and develop man's purely logical faculties
to the same level of rationality. The American Mathematical
Monthly, 56, 1949, p19.*

*
*

* Ogyu, Sorai (1666 - 1729) Mathematicians boast of their
exacting achievements, but in reality they are absorbed in mental acrobatics
and contribute nothing to society. Complete Works on Japan's
Philosophical Thought. 1956.*

*
*

* Oppenheimer, Julius Robert (1904 - 1967) Today, it is not
only that our kings do not know mathematics, but our philosophers do not
know mathematics and -- to go a step further -- our mathematicians do not
know mathematics. "The Tre*

*
e of Knowledge" in Harper's, 217, 1958.*

*
*

* Osgood, W. F. The calculus is the greatest aid we have to
the application of physical truth in the broadest sense of the word. In N.
Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc.,
1988.*

*
*

* Pascal, Blaise (1623-1662) We are usually convinced
more easily by reasons we have found ourselves than by those which have
occurred to others. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) It is the heart which perceives
God and not the reason. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Man is equally incapable of
seeing the nothingness from which he emerges and the infinity in which he is
engulfed. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Our nature consists in
movement; absolute rest is death. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662)*

Man is full of desires: he loves only those who can satisfy them all. "This man is a good mathematician," someone will say. But I have no concern for mathematics; he would take me for a proposition. &qu

*
ot;That one is a good soldier." He would take me for a besieged town. I
need, that is to say, a decent man who can accommodate himself to all my
desires in a general sort of way. W. H. Auden and L. Kronenberger (eds.)
The Viking Book of Aphori*

*
sms*, New York: Viking Press, 1966.

*
*

* Pascal, Blaise (1623-1662) We run carelessly to the
precipice, after we have put something before us to prevent us from seeing
it. W. H. Auden and L. Kronenberger (eds.) The Viking Book of
Aphorisms, New York: Viking Press, 1966.*

*
*

*
*

* Pascal, Blaise (1623-1662) We do not worry about being
respected in towns through which we pass. But if we are going to remain in
one for a certain time, we do worry. How long does this time have to
be? W. H. Auden and L. Kronenberger *

*
(eds.) The Viking Book of Aphorisms, New York: Viking Press,
1966.*

*
*

* Pascal, Blaise (1623-1662) Few men speak humbly of
humility, chastely of chastity, skeptically of skepticism. W. H. Auden
and L. Kronenberger (eds.) The Viking Book of Aphorisms, New York:
Viking Press, 1966.*

*
*

* Pascal, Blaise (1623-1662) Those who write against
vanity want the glory of having written well, and their readers the glory of
reading well, and I who write this have the same desire, as perhaps those who
read this have also. W. H. Aud*

*
en and L. Kronenberger (eds.) The Viking Book of Aphorisms, New
York: Viking Press, 1966.*

*
*

* Pascal, Blaise (1623-1662) Our notion of symmetry is
derived form the human face. Hence, we demand symmetry horizontally and
in breadth only, not vertically nor in depth. W. H. Auden and L.
Kronenberger (eds.) The Viking Book of Apho*

*
risms*, New York: Viking Press, 1966.

*
*

* Pascal, Blaise (1623-1662) When we encounter a natural
style we are always surprised and delighted, for we thought to see an author
and found a man. W. H. Auden and L. Kronenberger (eds.) The Viking
Book of Aphorisms, New York: V*

*
iking Press, 1966.*

*
*

* Pascal, Blaise (1623-1662) Everything that is written
merely to please the author is worthless. W. H. Auden and L.
Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking
Press, 1966.*

*
*

* Pascal, Blaise (1623-1662) I cannot judge my work while I
am doing it. I have to do as painters do, stand back and view it from a
distance, but not too great a distance. How great? Guess. W. H. Auden
and L. Kronenberger (eds.) The V*

*
iking Book of Aphorisms*, New York: Viking Press, 1966.

*
*

* Pascal, Blaise (1623-1662) Contradiction is not a sign of
falsity, nor the lack of contradiction a sign of truth. W. H. Auden and L.
Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking
Press, 1966.*

*
*

* Pascal, Blaise (1623-1662) All err the more dangerously
because each follows a truth. Their mistake lies not in following a falsehood
but in not following another truth. W. H. Auden and L. Kronenberger
(eds.) The Viking Book of Aphor*

*
isms*, New York: Viking Press, 1966.

*
*

* Pascal, Blaise (1623-1662) Perfect clarity would profit
the intellect but damage the will. W. H. Auden and L. Kronenberger
(eds.) The Viking Book of Aphorisms, New York: Viking Press,
1966.*

*
*

* Pascal, Blaise (1623-1662)*

Those who are accustomed to judge by feeling do not understand the process of reasoning, because they want to comprehend at a glance and are not used to seeking for first principles. Those, on the other hand, who

*
are accustomed to reason from first principles do not understand matters
of feeling at all, because they look for first principles and are unable to
comprehend at a glance. W. H. Auden and L. Kronenberger (eds.) The
Viking Book of Aphorisms, Ne*

*
w York: Viking Press, 1966.*

*
*

* Pascal, Blaise (1623-1662) To deny, to believe, and to
doubt well are to a man as the race is to a horse. W. H. Auden and L.
Kronenberger (eds.) The Viking Book of Aphorisms, New York: Viking
Press, 1966.*

*
*

* Pascal, Blaise (1623-1662) Words differently arranged
have a different meaning and meanings differently arranged have a different
effect. W. H. Auden and L. Kronenberger (eds.) The Viking Book of
Aphorisms, New York: Viking Press*

*
, 1966.*

*
*

* Pascal, Blaise (1623-1662) Nature is an infinite sphere of
which the center is everywhere and the circumference nowhere.
Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) We arrive at truth, not by
reason only, but also by the heart. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) When the passions become
masters, they are vices. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Men despise religion; they
hate it, and they fear it is true. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Religion is so great a thing
that it is right that those who will not take the trouble to seek it if it be
obscure, should be deprived of it. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) It is not certain that
everything is uncertain. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662)*

We are so presumptuous that we should like to be known all over the world, even by people who will only come when we are no more. Such is our vanity that the good opinion of half a dozen of the people around us gi

*
ves us pleasure and satisfaction. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) The sole cause of man's
unhappiness is that he does not know how to stay quietly in his room.
Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Reason's last step is the
recognition that there are an infinite number of things which are beyond
it. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Through space the universe
grasps me and swallows me up like a speck; through thought I grasp it.
Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Let no one say that I have
said nothing new... the arrangement of the subject is new. When we play
tennis, we both play with the same ball, but one of us places it better.
Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) The excitement that a
gambler feels when making a bet is equal to the amount he might win times
the probability of winning it. In N. Rose Mathematical Maxims and
Minims, Raleigh NC:Rome Press Inc., 1988*

*
.*

*
*

* Pascal, Blaise (1623-1662) Reason is the slow and
tortuous method by which these who do not know the truth discover it. The
heart has its own reason which reason does not know. Pensees.
1670.*

*
*

**Pascal, Blaise (1623-1662) **Reverend Fathers, my letters
did not usually follow each other at such close intervals, nor were they so
long.... This one would not be so long had I but the leisure to make it
shorter.

*
*

*
*

* Pascal, Blaise (1623-1662) The last thing one knows when
writing a book is what to put first. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) What is man in nature?
Nothing in relation to the infinite, all in relation to nothing, a mean between
nothing and everything. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) [I feel] engulfed in the infinite
immensity of spaces whereof I know nothing, and which know nothing of me, I
am terrified The eternal silence of these infinite spaces alarms me.
Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) Let us weigh the gain and the
loss in wagering that God is. Let us consider the two possibilities. If you gain,
you gain all; if you lose, you lose nothing. Hesitate not, then, to wager that
He is. Pensees.*

*
1670.*

*
*

* Pascal, Blaise (1623-1662) Look somewhere else for
someone who can follow you in your researches about numbers. For my part,
I confess that they are far beyond me, and I am competent only to admire
them. [Written to Fermat] In G. Si*

*
mmons Calculus Gems, New York: McGraw Hill Inc., 1992.*

*
*

* Pascal, Blaise (1623-1662) The more I see of men, the
better I like my dog. In H. Eves Return to Mathematical Circles,
Boston: Prindle, Weber and Schmidt, 1988.*

*
*

* Pascal, Blaise (1623-1662) The more intelligent one is,
the more men of originality one finds. Ordinary people find no difference
between men. Pensees. 1670.*

*
*

* Pascal, Blaise (1623-1662) However vast a man's
spiritual resources, he is capable of but one great passion. Discours
sur les passions de l'amour. 1653.*

*
*

* Pascal, Blaise (1623-1662)*

There are two types of mind ... the mathematical, and what might be called the intuitive. The former arrives at its views slowly, but they are firm and rigid; the latter is endowed with greater flexibility and app

*
lies itself simultaneously to the diverse lovable parts of that which it
loves. Discours sur les passions de l'amour. 1653.*

*
*

* Passano, L.M. This trend [emphasizing applied
mathematics over pure mathematics] will make the queen of the sciences
into the quean of the sciences. In H. Eves Mathematical Circles
Squared, Boston: Prindle, Weber and Schmidt, 197*

*
2.*

*
*

* Pasteur, Louis Chance favors only the prepared mind. In
H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1988.*

*
*

* Pearson, Karl The mathematician, carried along on his
flood of symbols, dealing apparently with purely formal truths, may still
reach results of endless importance for our description of the physical
universe. In N. Rose Mathematical*

*
Maxims and Minims*, Raleigh NC:Rome Press Inc., 1988.

*
*

* Peirce, Benjamin (1809-1880) Mathematics is the science
which draws necessary conclusions. Memoir read before the National
Academy of Sciences in Washington, 1870.*

*
*

* Peirce, Charles Sanders (1839-1914) The one [the
logician] studies the science of drawing conclusions, the other [the
mathematician] the science which draws necessary conclusions.
"The Essence of Mathematics" in J. R. Newman (*

*
ed.) The World of Mathematics, New York: Simon and Schuster,
1956.*

*
*

* Peirce, Charles Sanders (1839-1914)*

...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating i

*
nto airy nothingness, degenerating, as the Germans say, into an arachnoid
film, spun from the stuff that dreams are made of. There is no such danger
for pure mathematics; for that is precisely what mathematics ought to
be. "The Essence of Mathema*

*
tics" in J. R. Newman (ed.) The World of Mathematics, New York:
Simon and Schuster, 1956.*

*
*

* Peirce, Charles Sanders (1839-1914)*

Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; th

*
e perfect exactitude of its results; their broad universality; their practical
infallibility. "The Essence of Mathematics" in J. R. Newman
(ed.) The World of Mathematics, New York: Simon and Schuster,
1956.*

*
*

* Peirce, Charles Sanders (1839-1914) The pragmatist
knows that doubt is an art which hs to be acquired with difficulty.
Collected Papers.*

*
*

* Pedersen, Jean*

Geometry is a skill of the eyes and the hands as well as of the mind.

*
*

* Plato (ca 429-347 BC)*

He who can properly define and divide is to be considered a god.

*
*

**Plato (ca 429-347 BC) **The ludicrous state of solid
geometry made me pass over this branch.

*
*

* Plato (ca 429-347 BC)*

He is unworthy of the name of man who is ignorant of the fact that the diagonal of a square is incommensurable with its side.

*
*

* Plato (ca 429-347 BC)*

Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.

*
*

* Plato (ca 429-347 BC) The knowledge of which geometry
aims is the knowledge of the eternal. Republic, VII, 52.*

*
*

* Plato (ca 429-347 BC) I have hardly ever known a
mathematician who was capable of reasoning. In N. Rose
Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc.,
1988.*

*
*

* Plato (ca 429-347 BC) There still remain three studies
suitable for free man. Arithmetic is one of them. In J. R. Newman (ed.)
The World of Mathematics, New York: Simon and Schuster,
1956.*

*
*

* Plutarch (ca 46-127) [about Archimedes:]... being
perpetually charmed by his familiar siren, that is, by his geometry, he
neglected to eat and drink and took no care of his person; that he was often
carried by force to the baths, and wh*

*
en there he would trace geometrical figures in the ashes of the fire, and with
his finger draws lines upon his body when it was anointed with oil, being in a
state of great ecstasy and divinely possessed by his science. In G.
Simmons Calculus Gems<*

*
/I>, New York: McGraw Hill Inc., 1992.*

*
*

* Poe, Edgar Allen To speak algebraically, Mr. M. is
execrable, but Mr. G. is (x + 1)- ecrable. [Discussing fellow writers
Cornelius Mathews and William Ellery Channing.] In N. Rose
Mathematical Maxims and Minims, Raleigh NC: R*

*
ome Press Inc., 1988.*

*
*

* Poincaré, Jules Henri (1854-1912) Mathematics is
the art of giving the same name to different things. [As opposed to the
quotation: Poetry is the art of giving different names to the same
thing].*

*
*

* Poincaré, Jules Henri (1854-1912) Later
generations will regard Mengenlehre (set theory) as a disease from which one
has recovered. [Whether or not he actually said this is a matter of
debate amongst historians of mathematics.]
R> The Mathematical Intelligencer, vol 13, no. 1, Winter 1991.*

*
*

* Poincaré, Jules Henri (1854-1912)*

What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introd

*
uces order, all that gives unity, that permits us to see clearly and to
comprehend at once both the ensemble and the details. In N. Rose
Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc.,
1988.*

*
*

* Poincaré, Jules Henri (1854-1912)*

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanle

*
y Jevons; or, if you choose, a machine might be imagined where the
assumptions were put in at one end, while the theorems came out at the
other, like the legendary Chicago machine where the pigs go in alive and come
out transformed into hams and sausages.*

*
No more than these machines need the mathematician know what he
does. In J. R. Newman (ed.) The World of Mathematics, New York:
Simon and Schuster, 1956.*

*
*

* Poincaré, Jules Henri (1854-1912) Talk with M.
Hermite. He never evokes a concrete image, yet you soon perceive that the
more abstract entities are to him like living creatures. In G. Simmons
Calculus Gems, New York: McGra*

*
w Hill Inc., 1992.*

*
*

* Poincaré, Jules Henri (1854-1912) Science is built
up with facts, as a house is with stones. But a collection of facts is no more
a science than a heap of stones is a house. La Science et
l'hypothèse.*

*
*

* Poincaré, Jules Henri (1854-1912) A scientist
worthy of his name, about all a mathematician, experiences in his work the
same impression as an artist; his pleasure is as great and of the same
nature. In N. Rose Mathematical Ma*

*
xims and Minims*, Raleigh NC:Rome Press Inc., 1988.

*
*

* Poincaré, Jules Henri (1854-1912)*

The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other

*
facts, long known, but wrongly believed to be strangers to one another.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.*

*
*

* Poincaré, Jules Henri (1854-1912)*

Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: th

*
ey are interested in form only.*

*
*

**Poincaré, Jules Henri (1854-1912) **Thought is only
a flash between two long nights, but this flash is everything.

In J. R. Newman (ed.)

*
*

* Poincaré, Jules Henri (1854-1912)*

The mind uses its faculty for creativity only when experience forces it to do so.

*
*

* Poincaré, Jules Henri (1854-1912)*

Mathematical discoveries, small or greatare never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subcon

*
scious.*

*
*

* Poincaré, Jules Henri (1854-1912)*

Absolute space, that is to say, the mark to which it would be necessary to refer the earth to know whether it really moves, has no objective existence.... The two propositions: "The earth turns

*
round" and "it is more convenient to suppose the earth turns
round" have the same meaning; there is nothing more in the one than in
the other. La Science et l'hypothèse.*

*
*

* Poincaré, Jules Henri (1854-1912)*

...by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geom

*
etry is not true, it is advantageous. Science and Method.*

*
*

* Poisson, Siméon (1781-1840) Life is good for only
two things, discovering mathematics and teaching mathematics.
Mathematics Magazine, v. 64, no. 1, Feb. 1991.*

*
*

* Polyá, George (1887, 1985) Mathematics consists
of proving the most obvious thing in the least obvious way. In N. Rose
Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc.,
1988.*

*
*

* Polyá, George (1887, 1985)*

The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back to the cl

*
ass. He writes a, he says b, he means c; but it should be d. Some of his
sayings are handed down from generation to generation. "In order
to solve this differential equation you look at it till a solution occurs to
you." "This princ*

*
iple is so perfectly general that no particular application of it is
possible." "Geometry is the science of correct reasoning on
incorrect figures." "My method to overcome a difficulty is
to go round it." "What is*

*
the difference between method and device? A method is a device which you
used twice." How to Solve It. Princeton: Princeton
University Press. 1945.*

*
*

* Polyá, George (1887, 1985) Mathematics is the
cheapest science. Unlike physics or chemistry, it does not require any
expensive equipment. All one needs for mathematics is a pencil and
paper. D. J. Albers and G. L. Alexanderson, <*

*
I>Mathematical People*, Boston: Birkhäuser, 1985.

*
*

* Polyá, George (1887, 1985) There are many
questions which fools can ask that wise men cannot answer. In H. Eves
Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt,
1988.*

*
*

* Polyá, George (1887, 1985) When introduced at
the wrong time or place, good logic may be the worst enemy of good
teaching. The American Mathematical Monthly, v. 100, no.
3.*

*
*

* Polyá, George (1887, 1985)*

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instru

*
ctive phase of the work. ... A good teacher should understand and impress on
his students the view that no problem whatever is completely
exhausted. One of the first and foremost duties of the teacher is not
to give his students the impression that ma*

*
thematical problems have little connection with each other, and no
connection at all with anything else. We have a natural opportunity to
investigate the connections of a problem when looking back at its
solution. How to Solve It. Princeton: Pr*

*
inceton University Press. 1945.*

*
*

* Polyá, George (1887, 1985)*

In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression pecul

*
iar to the French language. The situation is very similar when we attempt to
express in mathematical symbols a condition proposed in words. First, we
must understand thoroughly the condition. Second, we must be familiar with
the forms of mathematical expr*

*
ession. How to Solve It. Princeton: Princeton University Press.
1945.*

*
*

* Pope, Alexander (1688-1744) Epitaph on Newton:
Nature and Nature's law lay hid in night: God said, "Let Newton
be!," and all was light. [added by Sir John Collings Squire: It
did not last: the Devil shouting "*

*
Ho. Let Einstein be," restored the status quo] [Aaron Hill's
version: O'er Nature's laws God cast the veil of night, Out blaz'd a
Newton's souland all was light.*

*
*

* Pope, Alexander (1688-1744) Order is Heaven's first
law. An Essay on Man IV.*

*
*

* Pope, Alexander (1688-1744) See skulking Truth to her
old cavern fled, Mountains of Casuistry heap'd o'er her head!
Philosophy, that lean'd on Heav'n before, Shrinks to her second cause,
and is no more. Physic of Metaphysic *

*
begs defence, And Metaphysic calls for aid on Sense! See Mystery
to Mathematics fly! In J. R. Newman (ed.) *

*
*

* Pordage, Matthew One of the endearing things about
mathematicians is the extent to which they will go to avoid doing any real
work. In H. Eves Return to Mathematical Circles, Boston: Prindle,
Weber and Schmidt, 1988.*

*
*

* Proclus Diadochus (412 - 485)*

It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortu

*
itously touched on and revealed this aspect of living things, was taken to the
place where he began and there is for ever beaten by the waves.
Scholium to Book X of Euclid V.*

*
*

* Purcell, E. and Varberg, D. The Mean Value Theorem is the
midwife of calculus -- not very important or glamorous by itself, but often
helping to delivery other theorems that are of major significance.
Calculus with Analytic Geomety, *

*
fifth edition,* Englewood Cliffs, NJ: Prentice Hall, 1987.

*
*

* Pushkin, Aleksandr Sergeyevich (1799 - 1837) Inspiration
is needed in geometry, just as much as in poetry.
Likhtenshtein*

*
*

**Quine, Willard Van Orman **Just as the introduction of the
irrational numbers ... is a convenient myth [which] simplifies the laws of
arithmetic ... so physical objects are postulated entities which round out and
simplify our account of the fl

*
ux of existence... The conceptional scheme of physical objects is [likewise] a
convenient myth, simpler than the literal truth and yet containing that literal
truth as a scattered part. In J. Koenderink Solid Shape,
Cambridge Mass.: MIT Press, *

*
1990.*

*
*

* Raleigh, [Sir] Walter Alexander (1861-1922) In an
examination those who do not wish to know ask questions of those who
cannot tell. Some Thoughts on Examinations.*

*
*

* Recorde, Robert (1557)*

To avoide the tediouse repetition of these woordes: is equalle to: I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: =, bicause noe .2. thynges, can be moare eq

*
ualle. In G. Simmons Calculus Gems, New York: McGraw Hill Inc.,
1992.*

*
*

* Reid, Thomas*

It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on

*
has maintained an unchallenged leadership in the field of mathematics.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press
Inc., 1988.*

*
*

* Renan, Ernest The simplest schoolboy is now familiar with
facts for which Archimedes would have sacrificed his life. Souvenirs
d'enfance et de jeunesse.*

*
*

* Rényi, Alfréd If I feel unhappy, I do
mathematics to become happy. If I am happy, I do mathematics to keep
happy. P. Turán, "The Work of Alfréd
Rényi", Matematikai Lapok 21, 1970, pp 1*

*
99 - 210.*

*
*

* Richardson, Lewis Fry (1881 - 1953)*

Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong; and if it is found to be wrong, there is a plenteous choice of amendments ready in the mathematic

*
ians' stock of formulae. Some verbal statements have not this merit; they
are so vague that they could hardly be wrong, and are correspondingly
useless. Mathematics of War and Foreign Politics.*

*
*

* Richardson, Lewis Fry (1881 - 1953)*

Quotations from the book Lewis F. Richardson, Weather prediction by numerical process, Dover, New York, 1965. (originally published in 1922)

"The extensive researches of V. Bjerknes and his school are pervaded by the idea of using differential equations for all they're worth." page X I I.

"Finite arithmetical differences have proved remarkably successful in dealing with differential equations, ... in this book it is shown that similar methods can be extended to the very complicated system of differential equations which express the changes in the weather." page 1.

"The upper limit to the size of an eddy is, like the length of the piece of string, a matter of human convenience." page 65.

"... the aim has been to lay down theoretically only so much as can be determined with strictness, leaving all uncertainties to be decided by observations." page 65.

"Draw a sphere in a fluid. Let the radius be as large as is necessary to include a considerable number of eddie's but no larger. Let the sphere move so that the whole momentum of fluid inside it is equal to the mass of the same fluid multiplied by the velocity vector of the center of the sphere. The center it may then be said to be a point moving with the mean motion." page 66-67.

"The measure of diversity which fits best of our mathematical habits is half the smoothed square of the deviation." page 100.

"The road to full knowledge of the variations of viscosity appears to lie in the study of diffusion of eddies." page 79.

* Riskin, Adrian (after Edna St. Vincent Millay)...Euclid
aloneHas looked on Beauty bare. He turned away at once; Far
too polite to stare. The Mathematical Intelligencer, V. 16, no. 4
(Fall 1994), p. 20.*

*
*

* R. Rivest, A. Shamir, and L. Adleman The magic words are
squeamish ossifrage [This sentence is the result when a coded message
in Martin Gardner's column about factoring the famous number RSA-129 is
decoded. See the article whose title *

*
is the above sentence by Barry Cipra, SIAM News July 1994, 1, 12
13.]*

*
*

* Rohault, Jacques (17th century) It was by just such a
hazard, as if a man should let fall a handful of sand upon a table and the
particles of it should be so ranged that we could read distinctly on it a whole
page of Virgil's Aenead.
>Traité de Physique*, Paris, 1671.

**Rosenblueth, A **[with Norbert Wiener]

The best material model of a cat is another, or preferably the same, cat.

**Rosenlicht, Max (1949) **You know we all became
mathematicians for the same reason: we were lazy.

**Hugo Rossi**

In the fall of 1972 President Nixon
announced that the rate of increase of inflation was decreasing. This was
the first time a sitting president used the third derivative to advance his
case for reelection.

*Mathem*

*
atics Is an Edifice, Not a Toolbox*, **Notices of the AMS**, v. 43, no.
10, October 1996.

**Rota, Gian-carlo **We often hear that mathematics consists
mainly of "proving theorems." Is a writer's job mainly that of
"writing sentences?"

In preface to P. Davis and R. Hersh

on: Birkhäuser, 1981.

**Russell, Bertrand (1872-1970) **How dare we speak of the
laws of chance? Is not chance the antithesis of all law?

**Russell, Bertrand (1872-1970) **Mathematics takes us into
the region of absolute necessity, to which not only the actual word, but
every possible word, must conform.

In N. Rose

Inc., 1988.

**Russell, Bertrand (1872-1970) **Although this may seem a
paradox, all exact science is dominated by the idea of approximation.

W. H. Auden and L. Kronenberger (eds.)

**Russell, Bertrand (1872-1970) **At the age of eleven, I
began Euclid, with my brother as my tutor. This was one of the great events
of my life, as dazzling as first love. I had not imagined there was anything so
delicious in the world. From

that moment until I was thirty-eight, mathematics was my chief interest
and my chief source of happiness.

*The Autobiography of Bertrand
Russell *.

**Russell, Bertrand (1872-1970) **A good notation has a
subtlety and suggestiveness which at times make it almost seem like a live
teacher.

In J. R. Newman (ed.)

**Russell, Bertrand (1872-1970) **If I were a medical man, I
should prescribe a holiday to any patient who considered his work
important.

**Russell, Bertrand (1872-1970) **Ordinary language is
totally unsuited for expressing what physics really asserts, since the words
of everyday life are not sufficiently abstract. Only mathematics and
mathematical logic can say as little as th

e physicist means to say.

*The Scientific Outlook*, 1931.

**Russell, Bertrand (1872-1970) **With equal passion I have
sought knowledge. I have wished to understand the hearts of men. I have
wished to know why the stars shine. And I have tried to apprehend the
Pythagorean power by which number holds sw

ay about the flux. A little of this, but not much, I have achieved.

*The
Autobiography of Bertrand Russell *.

**Russell, Bertrand (1872-1970) **At first it seems obvious,
but the more you think about it the stranger the deductions from this axiom
seem to become; in the end you cease to understand what is meant by
it.

In N. Rose

*
ms and Minims*, Raleigh NC:Rome Press Inc., 1988.

**Russell, Bertrand (1872-1970) **Calculus required
continuity, and continuity was supposed to require the infinitely little; but
nobody could discover what the infinitely little might be.

In N. Rose

aleigh NC:Rome Press Inc., 1988.

**Russell, Bertrand (1872-1970) **The fact that all
Mathematics is Symbolic Logic is one of the greatest discoveries of our age;
and when this fact has been established, the remainder of the principles of
mathematics consists in the analysis of

Symbolic Logic itself.

*Principles of Mathematics*. 1903.

**Russell, Bertrand (1872-1970) **A habit of basing
convictions upon evidence, and of giving to them only that degree or
certainty which the evidence warrants, would, if it became general, cure
most of the ills from which the world suffers.

In G. Simmons *Calculus Gems*, New York: McGraw Hill Inc., 1992.

**Russell, Bertrand (1872-1970) **The method of
"postulating" what we want has many advantages; they are the
same as the advantages of theft over honest toil.

19, p 71.

**Russell, Bertrand (1872-1970) **Aristotle maintained that
women have fewer teeth than men; although he was twice married, it never
occurred to him to verify this statement by examining his wives'
mouths.

*
y*, 1952.

**Russell, Bertrand (1872-1970) **[Upon hearing via
Littlewood an exposition on the theory of relativity:]

To think I have spent my life on absolute muck.

J.E. Littlewood,

**Russell, Bertrand (1872-1970) **"But," you
might say, "none of this shakes my belief that 2 and 2 are 4." You
are quite right, except in marginal cases -- and it is only in marginal cases
that you are doubtful whether a ce

rtain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you ar

e doubtful whether two of them are dogs. "Well, at any rate there are four animals," you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. "Well, then living organisms," you say.

But there are things of which it is doubtful whether they are living
organisms or not. You will be driven into saying: "Two entities and two
entities are four entities." When you have told me what you mean by

"entity," we will resume the argument.

In N. Rose
*Mathematical Maxims and Minims*, Raleigh NC:Rome Press Inc.,
1988.

**Russell, Bertrand (1872-1970) **I wanted certainty in the
kind of way in which people want religious faith. I thought that certainty is
more likely to be found in mathematics than elsewhere. But I discovered that
many mathematical demonstrati

ons, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure.

But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to k

eep the elephant from falling. But the tortoise was no more secure than the
elephant, and after some twenty years of very arduous toil, I came to the
conclusion that there was nothing more that I could do in the way of making

mathematical knowledge indubitable.

*Portraits from
Memory.*

**Russell, Bertrand (1872-1970) **Men who are unhappy, like
men who sleep badly, are always proud of the fact.

W. H. Auden and L. Kronenberger (eds.)

**Russell, Bertrand (1872-1970) **Work is of two kinds:
first, altering the position of matter at or near the earth's surface relatively
to other such matter; second, telling other people to do so. The first kind is
unpleasant and ill paid; the

second is pleasant and highly paid.

**Russell, Bertrand (1872-1970) **A sense of duty is useful
in work but offensive in personal relations. Certain characteristics of the
subject are clear. To begin with, we do not, in this subject, deal with
particular things or particular pro

perties: we deal formally with what can be said about "any" thing or "any" property. We are prepared to say that one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematician

s, we have never heard of Socrates or Plato. A world in which there were no such individuals would still be a world in which one and one are two. It is not open to us, as pure mathematicians or logicians, to mention anything at all, because, if we do so w

e introduce something irrelevant and not formal.

In J. R. Newman (ed.)
*The World of Mathematics*, New York: Simon and Schuster,
1956.

**Russell, Bertrand (1872-1970) **The desire to understand
the world and the desire to reform it are the two great engines of
progress.

**Russell, Bertrand (1872-1970) **It can be shown that a
mathematical web of some kind can be woven about any universe containing
several objects. The fact that our universe lends itself to mathematical
treatment is not a fact of any great phil

osophical significance.

W. H. Auden and L. Kronenberger (eds.) *The
Viking Book of Aphorisms*, New York: Viking Press, 1966.

**Rutherford, Ernest (1871-1937) **If your experiment needs
statistics, you ought to have done a better experiment.

In N. T. J. Bailey

**Sanford, T. H. **The modern, and to my mind true, theory is
that mathematics is the abstract form of the natural sciences; and that it is
valuable as a training of the reasoning powers not because it is abstract, but
because it is a represent

ation of actual things.

In N. Rose *Mathematical Maxims and
Minims*, Raleigh NC:Rome Press Inc., 1988.

**Santayana, George **It is a pleasant surprise to him (the
pure mathematician) and an added problem if he finds that the arts can use
his calculations, or that the senses can verify them, much as if a composer
found that sailors could heave be

tter when singing his songs.

In J. R. Newman (ed.) *The World of
Mathematics*, New York: Simon and Schuster, 1956.

**Sarton, G. **The main duty of the historian of mathematics,
as well as his fondest privilege, is to explain the humanity of mathematics, to
illustrate its greatness, beauty and dignity, and to describe how the
incessant efforts and accumulate

d genius of many generations have built up that magnificent monument, the object of our most legitimate pride as men, and of our wonder, humility and thankfulness, as individuals. The study of the history of mathematics will not make better mathematicians

but gentler ones, it will enrich their minds, mellow their hearts, and bring out their finer qualities.

**Sayers, Dorothy L. **The biologist can push it back to the
original protist, and the chemist can push it back to the crystal, but none of
them touch the real question of why or how the thing began at all. The
astronomer goes back untold milli

on of years and ends in gas and emptiness, and then the mathematician sweeps the whole cosmos into unreality and leaves one with mind as the only thing of which we have any immediate apprehension. Cogito ergo sum, ergo omnia esse videntur. All this bother

, and we are no further than Descartes. Have you noticed that the astronomers and mathematicians are much the most cheerful people of the lot? I suppose that perpetually contemplating things on so vast a scale makes them feel either that it doesn't matter

a hoot anyway, or that anything so large and elaborate must have some
sense in it somewhere.

With R. Eustace, *The Documents in the
Case*, New York: Harper and Row, 1930, p 54.

**Schopenhauer **Of all the intellectual faculties, judgment is
the last to mature. A child under the age of fifteen should confine its
attention either to subjects like mathematics, in which errors of judgment
are impossible, or to subjects i

n which they are not very dangerous, like languages, natural science, history, etc.

**Seneca **If you would make a man happy, do not add to his
possessions but subtract from the sum of his desires.

In H. Eves

**Shakespeare, William (1564 - 1616) **I cannot do it without
comp[u]ters.

**Shakespeare, William (1564-1616) **Though this be
madness, yet there is method in't.

**Shakespeare, William (1564-1616) **O God! I could be
bounded in a nutshell, and count myself king of infinite space, were it not
that I have bad dreams.

**Shakespeare, William (1564-1616) **I am ill at these
numbers.

**Shaw, George Bernard (1856-1950) **Tyndall declared that
he saw in Matter the promise and potency of all forms of life, and with his
Irish graphic lucidity made a picture of a world of magnetic atoms, each atom
with a positive and a negative

pole, arranging itself by attraction and repulsion in orderly crystalline structure. Such a picture is dangerously fascinating to thinkers oppressed by the bloody disorders of the living world. Craving for purer subjects of thought, they find in the cont

emplation of crystals and magnets a happiness more dramatic and less
childish than the happiness found by mathematicians in abstract numbers,
because they see in the crystals beauty and movement without the
corrupting appetites of fleshly vitality.

Pr

eface to *Back to Methuselah*.

**Shaw, J. B. **The mathematician is fascinated with the
marvelous beauty of the forms he constructs, and in their beauty he finds
everlasting truth.

In N. Rose

**Simmons, G. F. **Mathematical rigor is like clothing; in its
style it ought to suit the occasion, and it diminishes comfort and restrains
freedom of movement if it is either too loose or too tight.

In

, v. 13, no. 1, Winter 1991.

**Slaught, H.E. **...[E.H.] Moore ws presenting a paper on a
highly technical topic to a large gathering of faculty and graduate students
from all parts of the country. When half way through he discovered what
seemed to be an error (though prob

ably no one else in the room observed it). He stopped and re-examined the doubtful step for several minutes and then, convinced of the error, he abruptly dismissed the meeting -- to the astonishment of most of the audience. It was an evidence of intellect

ual *courage* as well as *honesty* and doubtless won for him the
supreme admiration of every person in the group -- an admiration which was
in no wise diminished, but rather increased, when at a later meeting he
announced that after all he had b

een able to prove the step to be correct.

*The American
Mathematical Monthly*, 40 (1933), 191-195.

**Smith, Adam **I have no faith in political arithmetic.

**Smith, David Eugene **One merit of mathematics few will
deny: it says more in fewer words than any other science. The formula,
e^iπ = -1 expressed a world of thought, of truth, of poetry, and of
the religious spirit "God eternally

geometrizes."

In N. Rose *Mathematical Maxims and Minims*,
Raleigh NC:Rome Press Inc., 1988.

**Smith, Henry John Stephen (1826 - 1883) **[His
toast:]

Pure mathematics, may it never be of any use to anyone.

In H. Eves

**Smith, Henry John Stephen (1826-1883) **It is the peculiar
beauty of this method, gentlemen, and one which endears it to the really
scientific mind, that under no circumstance can it be of the smallest possible
utility.

In H. Eves

*
ematical Circles Squared*, Boston: Prindle, Weber and Schmidt,
1972.

**Soddy, Frederick (1877-1956) **Four circles to the kissing
come,

The smaller are the benter.

The bend is just the inverse of

The distance from the centre.

Though their intrigue left Euclid dumb

There's now no need for rule

of thumb.

Since zero bend's a dead straight line

And concave
bends have minus sign,

The sum of squares of all four bends

Is
half the square of their sum.

*Nature*, v. 137, 1936.

**Somerville, Mary (1780-1872) **Nothing has afforded me so
convincing a proof of the unity of the Deity as these purely mental
conceptions of numerical and mathematical science which have been by slow
degrees vouchsafed to man, and are still g

ranted in these latter times by the Differential Calculus, now superseded by
the Higher Algebra, all of which must have existed in that sublimely
omniscient Mind from eternity.

Martha Somerville (ed.) *Personal
Recollections of Mary Somerville*,

Boston, 1874.

**Spengler, Oswald (1880 -1936) **The mathematic, then, is
an art. As such it has its styles and style periods. It is not, as the layman
and the philosopher (who is in this matter a layman too) imagine,
substantially unalterable, but subject li

ke every art to unnoticed changes form epoch to epoch. The development of
the great arts ought never to be treated without an (assuredly not
unprofitable) side-glance at contemporary mathematics.

*The Decline
of the West.*

**Steiner, G.**For all their wealth of content, for all the sum of
history and social institution invested in them, music, mathematics, and
chess are resplendently useless (applied mathematics is a higher plumbing, a
kind of music for the police

band). They are metaphysically trivial, irresponsible. They refuse to relate
outward, to take reality for arbiter. This is the source of their
witchery.*The American Mathematical Monthly*, v. 101, no. 9,
November, 1994.

**Steinmetz, Charles P. **Mathematics is the most exact
science, and its conclusions are capable of absolute proof. But this is so only
because mathematics does not attempt to draw absolute conclusions. All
mathematical truths are relative, con

ditional.

In E. T. Bell *Men of Mathematics,* New York: Simona and
Schuster, 1937.

**Sternberg, S. **Kepler's principal goal was to explain the
relationship between the existence of five planets (and their motions) and
the five regular solids. It is customary to sneer at Kepler for this. It is
instructive to compare this wit

h the current attempts to "explain" the zoology of elementary particles in terms of irreducible representations of Lie groups.

**Stewart, Ian **The successes of the differential equation
paradigm were impressive and extensive. Many problems, including basic and
important ones, led to equations that could be solved. A process of self
selection set in, whereby equations t

hat could not be solved were automatically of less interest than those that
could.

*Does God Play Dice? The Mathematics of Chaos.* Blackwell,
Cambridge, MA, 1989, p. 39.

**Sullivan, John William Navin (1886 - 1937) **The
mathematician is entirely free, within the limits of his imagination, to
construct what worlds he pleases. What he is to imagine is a matter for his
own caprice; he is not thereby discovering t

he fundamental principles of the universe nor becoming acquainted with the
ideas of God. If he can find, in experience, sets of entities which obey the
same logical scheme as his mathematical entities, then he has applied his
mathematics to the external w

orld; he has created a branch of science.

*Aspects of Science,*
1925.

**Sullivan, John William Navin (1886-1937) **Mathematics, as
much as music or any other art, is one of the means by which we rise to a
complete self-consciousness. The significance of mathematics resides
precisely in the fact that it is an art;

by informing us of the nature of our own minds it informs us of much that
depends on our minds.

*Aspects of Science*, 1925.

**Sun Tze (5th - 6th century) **The control of large numbers
is possible, and like unto that of small numbers, if we subdivide them.

**Swift, Jonathan **If they would, for Example, praise the
Beauty of a Woman, or any other Animal, they describe it by Rhombs,
Circles, Parallelograms, Ellipses, and other geometrical terms ...

"A Voyage to Laputa" in

*
s Travels*.

**Jonathan Swift **What vexes me most is, that my female
friends, who could bear me very well a dozen years ago, have now forsaken
me, although I am not so old in proportion to them as I formerly was: which I
can prove by arithmetic, for then I

was double their age, which now I am not.

*Letter to Alexander
Pope.* 7 Feb. 1736.

**Sylvester, J.J. (1814 - 1897) **...there is no study in the
world which brings into more harmonious action all the faculties of the mind
than [mathematics], ... or, like this, seems to raise them, by successive
steps of initiation, to higher

and higher states of conscious intellectual being....

*Presidential
Address to British Association*, 1869.

**Sylvester, J.J. (1814 - 1897) **So long as a man remains a
gregarious and sociable being, he cannot cut himself off from the
gratification of the instinct of imparting what he is learning, of propagating
through others the ideas and impression

s seething in his own brain, without stunting and atrophying his moral nature
and drying up the surest sources of his future intellectual
replenishment.

**Sylvester, J.J. (1814 - 1897) **[on graph theory...]

The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter th

an those employed in genealogical tables have in explaining the laws of
procreation.

In H. Eves *Mathematical Circles Adieu*, Boston:
Prindle, Weber and Schmidt, 1977.

**Sylvester, J.J. (1814 - 1897) **Time was when all the parts
of the subject were dissevered, when algebra, geometry, and arithmetic
either lived apart or kept up cold relations of acquaintance confined to
occasional calls upon one another; but

that is now at an end; they are drawn together and are constantly becoming
more and more intimately related and connected by a thousand fresh ties,
and we may confidently look forward to a time when they shall form but one
body with one soul.

*Pres*

*
idential Address to British Association*, 1869.

**Sylvester, J.J. (1814 - 1897) **The world of ideas which it
[mathematics] discloses or illuminates, the contemplation of divine beauty
and order which it induces, the harmonious connexion of its parts, the
infinite hierarchy and absolute evide

nce of the truths with which it is concerned, these, and such like, are the
surest grounds of the title of mathematics to human regard, and would
remain unimpeached and unimpaired were the plan of the universe unrolled
like a map at our feet, and the mind

of man qualified to take in the whole scheme of creation at a glance.

*Presidential Address to British Association*, 1869.

**Sylvester, J.J. (1814 - 1897) **I know, indeed, and can
conceive of no pursuit so antagonistic to the cultivation of the oratorical
faculty ... as the study of Mathematics. An eloquent mathematician must,
from the nature of things, ever remai

n as rare a phenomenon as a talking fish, and it is certain that the more anyone gives himself up to the study of oratorical effect the less will he find himself in a fit state to mathematicize.

**Thales (CA 600 BC) **I will be sufficiently rewarded if when
telling it to others you will not claim the discovery as your own, but will say it
was mine.

In H. Eves

**Thompson, D'Arcy Wentworth (1860-1948) **Cell and
tissue, shell and bone, leaf and flower, are so many portions of matter, and
it is in obedience to the laws of physics that their particles have been moved,
moulded and conformed. They are no

exceptions to the rule that God always geometrizes. Their problems of form
are in the first instance mathematical problems, their problems of growth
are essentially physical problems, and the morphologist is, *ipso facto*,
a student of physical scien

ce.

*On Growth and Form*, 1917.

**Thomson, [Lord Kelvin] William (1824-1907) **Fourier is a
mathematical poem.

**Thoreau **He is not a true man of science who does not
bring some sympathy to his studies, and expect to learn something by
behavior as well as by application. It is childish to rest in the discovery of
mere coincidences, or of partial and e

xtraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interes

ts us most is the life of the naturalist. The purest science is still biographical.

**Tietze **The story was told that the young Dirichlet had as
a constant companion all his travels, like a devout man with his prayer book,
an old, worn copy of the

In G. Simmons

*
*, New York: McGraw Hill Inc., 1992.

**Tillotson, Archbishop **How often might a man, after he had
jumbled a set of letters in a bag, fling them out upon the ground before they
would fall into an exact poem, yea, or so much as make a good discourse in
prose. And may not a little b

ook be as easily made by chance as this great volume of the world.

In J.
R. Newman (ed.) *The World of Mathematics*, New York: Simon and
Schuster, 1956.

**Titchmarsh, E. C. **Perhaps the most surprising thing about
mathematics is that it is so surprising. The rules which we make up at the
beginning seem ordinary and inevitable, but it is impossible to foresee their
consequences. These have only

been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed s

omething to human thought even more lasting than great literature, since it
is independent of language.

In N. Rose *Mathematical Maxims and
Minims*, Raleigh NC:Rome Press Inc., 1988.

**Titchmarsh, E. C. **It can be of no practical use to know
that Pi is irrational, but if we can know, it surely would be intolerable not to
know.

In N. Rose

**Todhunter, Isaac (1820 - 1910) **[Asked whether he would
like to see an experimental demonstration of conical refraction]

No. I have been teaching it all my life, and I do not want to have my ideas upset.

**Tolstoy, [Count] Lev Nikolgevich (1828-1920) **A modern
branch of mathematics, having achieved the art of dealing with the infinitely
small, can now yield solutions in other more complex problems of motion,
which used to appear insoluble. Thi

s modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error

which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable h

uman wills, is continuous. To understand the laws of this continuous
movement is the aim of history. Only by taking an infinitesimally small unit
for observation (the differential of history, that is, the individual tendencies
of

man) and attaining to the art of integrating them (that is, finding the sum of
these infinitesimals) can we hope to arrive at the laws of history.

*War and Peace.*

**Tolstoy, Count Lev Nikolgevich (1828-1920) **A man is like
a fraction whose numerator is what he is and whose denominator is what he
thinks of himself. The larger the denominator the smaller the fraction.

In H. Eves

*
cal Circles*, Boston: Prindle, Weber and Schmidt, 1989.

**Truesdell, Clifford **This paper gives wrong solutions to
trivial problems. The basic error,however, is not new.

**Turgenev, Ivan Sergeievich (1818 - 1883) **Whatever a
man prays for, he prays for a miracle. Every prayer reduces itself to this:
`Great God, grant that twice two be not four'.

**Turnbull, H.W. **Attaching significance to invariants is an
effort to recognize what, because of its form or colour or meaning or
otherwise, is important or significant in what is only trivial or ephemeral. A
simple instance of failing in thi

s is provided by the poll-man at Cambridge, who learned perfectly how to
factorize a^2 - b^2 but was floored because the examiner unkindly asked for
the factors of p^2 - q^2.

In J. R. Newman (ed.) *The World of
Mathematics*, New York: Simon and S

chuster, 1956.

**Ulam, Stanislaw **In many cases, mathematics is an escape
from reality. The mathematician finds his own monastic niche and happiness
in pursuits that are disconnected from external affairs. Some practice it as
if using a drug. Chess sometimes

plays a similar role. In their unhappiness over the events of this world, some
immerse themselves in a kind of self-sufficiency in mathematics. (Some
have engaged in it for this reason alone.)

*Adventures of a
Mathematician*, Scribner's, New Yor

k, 1976.

**Valéry, Paul (1871 - 1945) **In the physical world,
one cannot increase the size or quantity of anything without changing its
quality. Similar figures exist only in pure geometry.

**van Vleck, E. B. **This new integral of Lebesque is proving
itself a wonderful tool. I might compare it with a modern Krupp gun, so easily
does it penetrate barriers which were impregnable.

*
*, vol. 23, 1916.

**Veblen, Thorstein (1857-1929) **The outcome of any
serious research can only be to make two questions grow where only one
grew before.

**Veblen, Thorstein (1857-1929) **Invention is the mother of
necessity.

J. Gross,

**Voltaire (1694-1778) **Vous avez trouve par de long
ennuis

Ce que Newton trouva sans sortir de chez lui.

[Written to La Condamine after his measurement of the equator.]

In J. R. Newman (ed.)

rk: Simon and Schuster, 1956.

**Voltaire (1694-1778) **He who has heard the same thing
told by 12,000 eye-witnesses has only 12,000 probabilities, which are equal
to one strong probability, which is far from certain.

In J. R. Newman (ed.)

, New York: Simon and Schuster, 1956.

**Voltaire (1694-1778) **There are no sects in
geometry.

W. H. Auden and L. Kronenberger (eds.)

**Walton, Izaak **Angling may be said to be so like
mathematics that it can never be fully learned.

**Warner, Sylvia Townsend **For twenty pages perhaps, he
read slowly, carefully, dutifully, with pauses for self-examination and working
out examples. Then, just as it was working up and the pauses should have
been more scrupulous than ever, a

kind of swoon and ecstasy would fall on him, and he read ravening on, sitting up till dawn to finish the book, as though it were a novel. After that his passion was stayed; the book went back to the Library and he was done with mathematics till the next b

out. Not much remained with him after these orgies, but something remained: a sensation in the mind, a worshiping acknowledgment of something isolated and unassailable, or a remembered mental joy at the rightness of thoughts coming together to a conclusio

n, accurate thoughts, thoughts in just intonation, coming together like
unaccompanied voices coming to a close.

*Mr. Fortune's
Maggot*.

**Warner, Sylvia Townsend **Theology, Mr. Fortune found, is a
more accommodating subject than mathematics; its technique of exposition
allows greater latitude. For instance when you are gravelled for matter there
is always the moral to fall bac

k upon. Comparisons too may be drawn, leading cases cited, types and
antetypes analysed and anecdotes introduced. Except for Archimedes
mathematics is singularly naked of anecdotes.

*Mr. Fortune's
Maggot.*

**Warner, Sylvia Townsend **He resumed:

"In order to ascertain the height of the tree I must be in such a position that the top of the tree is exactly in a line with the top of a measuring stick or any straight object would do, such as

an umbrella which I shall secure in an upright position between my feet. Knowing then that the ratio that the height of the tree bears to the length of the measuring stick must equal the ratio that the distance from my eye to the base of the tree bears t

o my height, and knowing (or being able to find out) my height, the length of
the measuring stick and the distance from my eye to the base of the tree, I
can, therefore, calculate the height of the tree."

"What is
an umbrella?"

*M*

*
r. Fortune's Maggot.*

**Warren, Robert Penn (1905-) **What if angry vectors
veer

Round your sleeping head, and form.

There's never need to fear

Violence of the poor world's abstract storm.

**Weil, Andre (1906 -?) **Every mathematician worthy of the
name has experienced ... the state of lucid exaltation in which one thought
succeeds another as if miraculously... this feeling may last for hours at a
time, even for days. Once you h

ave experienced it, you are eager to repeat it but unable to do it at will,
unless perhaps by dogged work...

*The Apprenticeship of a
Mathematician.*

**Weil, Andre (1906-????) **God exists since mathematics is
consistent, and the Devil exists since we cannot prove it.

In H. Eves

**Weil, Simone (1909 - 1943) **Algebra and money are
essentially levelers; the first intellectually, the second effectively.

W.H. Auden and L. Kronenberger

**West, Nathanael **Prayers for the condemned man will be
offered on an adding machine. Numbers constitute the only universal
language.

**Weyl, Hermann (1885 - 1955) **Our federal income tax law
defines the tax y to be paid in terms of the income x; it does so in a clumsy
enough way by pasting several linear functions together, each valid in
another interval or bracket of incom

e. An archeologist who, five thousand years from now, shall unearth some of our income tax returns together with relics of engineering works and mathematical books, will probably date them a couple of centuries earlier, certainly before Galileo and Vieta

.

*The Mathematical Way of Thinking*, an address given at the
Bicentennial Conference at the University of Pennsylvania, 1940.

**Weyl, Hermann (1885 - 1955) **We are not very pleased
when we are forced to accept a mathematical truth by virtue of a
complicated chain of formal conclusions and computations, which we
traverse blindly, link by link, feeling our way by touch

. We want first an overview of the aim and of the road; we want to
understand the *idea* of the proof, the deeper context.

*Unterrichtsblätter für Mathematik und
Naturwissenschaften*, 38, 177-188 (1932). Translation by Abe Shenitze

r appeared in *The American Mathematical Monthly*, v. 102, no. 7
(August-September 1995), p. 646.

**Weyl, Hermann (1885 - 1955)**A modern mathematical proof
is not very different from a modern machine, or a modern test setup: the
simple fundamental principles are hidden and almost invisible under a mass of
technical details.

*
richtsblätter für Mathematik und Naturwissenschaften*, 38,
177-188 (1932). Translation by Abe Shenitzer appeared in *The American
Mathematical Monthly*, v. 102, no. 7 (August-September 1995), p.
646.

**Weyl, Hermann (1885-1955) **The constructs of the
mathematical mind are at the same time free and necessary. The individual
mathematician feels free to define his notions and set up his axioms as he
pleases. But the question is will he get hi

s fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affecte

d by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity.

1951.

**Weyl, Hermann (1885 - 1955) **My work has always tried to
unite the true with the beautiful and when I had to choose one or the other, I
usually chose the beautiful.

In an obituary by Freeman J. Dyson in

**Weyl, Hermann (1885 - 1955) **... numbers have neither
substance, nor meaning, nor qualities. They are nothing but marks, and all
that is in them we have put into them by the simple rule of straight
succession.

"Mathematics and the L

aws of Nature" in *The Armchair Science Reader*, New York:
Simon and Schuster, 1959.

**Weyl, Hermann (1885 - 1955) **Without the concepts,
methods and results found and developed by previous generations right down
to Greek antiquity one cannot understand either the aims or achievements of
mathematics in the last 50 years.

[

Said in 1950]

*The American Mathematical Monthly*, v. 100. p.
93.

**Weyl, Hermann (1885 - 1955) **Logic is the hygiene the
mathematician practices to keep his ideas healthy and strong.

**Whewell **Nobody since Newton has been able to use
geometrical methods to the same extent for the like purposes; and as we
read the Principia we feel as when we are in an ancient armoury where the
weapons are of gigantic size; and as we look

at them we marvel what manner of man he was who could use as a weapon
what we can scarcely lift as a burden.

In E. N. Da C. Andrade
"Isaac Newton" in J. R. Newman (ed.) *The World of
Mathematics*, New York: Simon and Schuster, 1956.

**Whitehead, Alfred North (1861 - 1947) **The science of
pure mathematics ... may claim to be the most original creation of the
human spirit.

**Whitehead, Alfred North (1861 - 1947) **Mathematics as a
science, commenced when first someone, probably a Greek, proved
propositions about "any" things or about "some"
things, without specifications of definite particular

things.

**Whitehead, Alfred North (1861 - 1947) **So far as the mere
imparting of information is concerned, no university has had any justification
for existence since the popularization of printing in the fifteenth
century.

*
*.

**Whitehead, Alfred North (1861 - 1947) **No Roman ever
died in contemplation over a geometrical diagram.

[A reference to the death of Archimedes.]

In H. Eves

.

**Whitehead, Alfred North (1861 - 1947) **Life is an
offensive, directed against the repetitious mechanism of the Universe.

**Whitehead, Alfred North (1861 - 1947) **There is no nature
at an instant.

**Whitehead, Alfred North (1861 - 1947) **Let us grant that
the pursuit of mathematics is a divine madness of the human spirit, a refuge
from the goading urgency of contingent happenings.

In N. Rose

leigh NC:Rome Press Inc., 1988.

**Whitehead, Alfred North (1861 - 1947) **There is a
tradition of opposition between adherents of induction and of deduction. In
my view it would be just as sensible for the two ends of a worm to
quarrel.

In N. Rose

*
nd Minims*, Raleigh NC:Rome Press Inc., 1988.

**Whitehead, Alfred North (1861 - 1947) **It is a profoundly
erroneous truism, repeated by all copy books and by eminent people when
they are making speeches, that we should cultivate the habit of thinking of
what we are doing. The precise oppo

site is the case. Civilization advances by extending the number of important
operations which we can perform without thinking about them.

*An
Introduction to Mathematics.*

**Whitehead, Alfred North (1861 - 1947) **Our minds are
finite, and yet even in these circumstances of finitude we are surrounded by
possibilities that are infinite, and the purpose of life is to grasp as much as
we can out of that infinitude.<

BR> In N. Rose *Mathematical Maxims and Minims*, Raleigh NC:Rome
Press Inc., 1988.

**Whitehead, Alfred North (1861 - 1947) **In modern times
the belief that the ultimate explanation of all things was to be found in
Newtonian mechanics was an adumbration of the truth that all science, as it
grows towards perfection, becomes ma

thematical in its ideas.

In N. Rose *Mathematical Maxims and
Minims*, Raleigh NC:Rome Press Inc., 1988.

**Whitehead, Alfred North (1861 - 1947) **Algebra reverses
the relative importance of the factors in ordinary language. It is essentially a
written language, and it endeavors to exemplify in its written structures the
patterns which it is its p

urpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghostlike chara

cter of the real variable.

W.H. Auden and L. Kronenberger *The Viking
Book of Aphorisms*, New York: Viking Press, 1966.

**Whitehead, Alfred North (1861 - 1947) **Be relieving the
brain of all unnecessary work, a good notation sets it free to concentrate on
more advanced problems, and, in effect, increases the mental power of the
race.

In P. Davis and R. Hers

h *The Mathematical Experience*, Boston: Birkhäuser, 1981.

**Whitehead, Alfred North (1861 - 1947) **Everything of
importance has been said before by somebody who did not discover it.

In J. R. Newman (ed.)

**Whitehead, Alfred North (1861 - 1947) **Seek simplicity,
and distrust it.

W.H. Auden and L. Kronenberger

**Whitehead, Alfred North (1861 - 1947) **Fundamental
progress has to do with the reinterpretation of basic ideas.

W.H. Auden and L. Kronenberger

**Whitehead, Alfred North (1861 - 1947) **We think in
generalities, but we live in details.

W.H. Auden and L. Kronenberger

**Whitehead, Alfred North (1861 - 1947) **Apart from blunt
truth, our lives sink decadently amid the perfume of hints and
suggestions.

W.H. Auden and L. Kronenberger

**Whitehead, Alfred North (1861 - 1947) **"Necessity
is the mother of invention" is a silly proverb. "Necessity is the
mother of futile dodges" is much nearer the truth.

W.H. Auden and L. Kronenberger

*
of Aphorisms*, New York: Viking Press, 1966.

**Whitehead, Alfred North (1861 - 1947) **It is more
important that a proposition be interesting than that it be true. This
statement is almost a tautology. For the energy of operation of a
proposition in an occasion of experience is its intere

st and is its importance. But of course a true proposition is more apt to be
interesting than a false one.

W.H. Auden and L. Kronenberger *The
Viking Book of Aphorisms*, New York: Viking Press, 1966.

**Whitehead, Alfred North (1861 - 1947) **War can protect;
it cannot create.

W.H. Auden and L. Kronenberger

**Whitehead, Alfred North (1861 - 1947) **The progress of
Science consists in observing interconnections and in showing with a patient
ingenuity that the events of this ever-shifting world are but examples of a
few general relations, called law

s. To see what is general in what is particular, and what is permanent in what
is transitory, is the aim of scientific thought.

*An Introduction to
Mathematics.*

**Whitehead, Alfred North (1861 - 1947) **Through and
through the world is infested with quantity: To talk sense is to talk
quantities. It is not use saying the nation is large .. How large? It is no use
saying the radium is scarce ... How scar

ce? You cannot evade quantity. You may fly to poetry and music, and
quantity and number will face you in your rhythms and your octaves.

In
J. R. Newman (ed.) *The World of Mathematics*, New York: Simon and
Schuster, 1956.

**Whitehead, Alfred North (1861 - 1947) **"One and one
make two" assumes that the changes in the shift of circumstance are
unimportant. But it is impossible for us to analyze this notion of unimportant
change.

W.H. Auden and L. Kr

onenberger *The Viking Book of Aphorisms*, New York: Viking Press,
1966.

**Whitehead, Alfred North (1861 - 1947) **I will not go so far
as to say that to construct a history of thought without profound study of
the mathematical ideas of successive epochs is like omitting Hamlet from
the play which is named after him

. That would be claiming too much. But it is certainly analogous to cutting out
the part of Ophelia. This simile is singularly exact. For Ophelia is quite
essential to the play, she is very charming ... and a little mad.

W.H.
Auden and L. Kronenberger

*The Viking Book of Aphorisms*, New York: Viking Press, 1966.

**Whitehead, Alfred North (1861 - 1947) **The study of
mathematics is apt to commence in disappointment....We are told that by its
aid the stars are weighed and the billions of molecules in a drop of water are
counted. Yet, like the ghost of Ha

mlet's father, this greatest science eludes the efforts of our mental
weapons to grasp it.

*An Introduction to Mathematics*

**Whitehead, Alfred North (1861 - 1947) **In the study of
ideas, it is necessary to remember that insistence on hard-headed clarity
issues from sentimental feeling, as it were a mist, cloaking the perplexities
of fact. Insistence on clarity at

all costs is based on sheer superstition as to the mode in which human
intelligence functions. Our reasonings grasp at straws for premises and float
on gossamers for deductions.

In J. R. Newman (ed.) *The World of
Mathematics*, New York: Simon an

d Schuster, 1956.

**Whitehead, Alfred North (1861 - 1947) **Familiar things
happen, and mankind does not bother about them. It requires a very unusual
mind to undertake the analysis of the obvious.

**Whitman, Walt (1819-1892) **Do I contradict myself? Very
well then I contradict myself. (I am large, I contains multitudes).

**Whitman, Walt (1819-1892) **When I heard the learn'd
astronomer,

When the proofs, the figure, were ranged in columns before me,

When I was shown the charts and diagrams, to add, divide, and measure them,

When I sitting heard the a

stronomer where he lectured with much applause in the lecture room,

How soon unaccountable I became tired and sick,

Till rising and gliding
out I wander'd off by myself,

In the mystical moist night-air, and from
time to time,

Look'd up in

perfect silence at the stars.

**Wiener, Norbert (1894 - 1964) **A professor is one who
can speak on any subject -- for precisely fifty minutes.

**Wiener, Norbert (1894-1964) **The modern physicist is a
quantum theorist on Monday, Wednesday, and Friday and a student of
gravitational relativity theory on Tuesday, Thursday, and Saturday. On
Sunday he is neither, but is praying to his God

that someone, preferably himself, will find the reconciliation between the two views.

**Wiener, Norbert (1894-1964) **Progress imposes not only
new possibilities for the future but new restrictions.

**Wiener, Norbert (1894-1964) **The Advantage is that
mathematics is a field in which one's blunders tend to show very clearly and
can be corrected or erased with a stroke of the pencil. It is a field which has
often been compared with chess, b

ut differs from the latter in that it is only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will m

ake a mathematician's reputation.

*Ex-Prodigy: My Childhood and
Youth.*

**Wilder, R. L. **There is nothing mysterious, as some have
tried to maintain, about the applicability of mathematics. What we get by
abstraction from something can be returned.

**Wilder, R. L. **Mathematics was born and nurtured in a
cultural environment. Without the perspective which the cultural background
affords, a proper appreciation of the content and state of present-day
mathematics is hardly possible.

In

**William of Occam (1300-1439) **[Occam's Razor:]

Entities should not be multiplied unnecessarily.

**Wilson, John (1741 - 1793) **A monument to Newton! a
monument to Shakespeare! Look up to Heaven look into the Human Heart. Till
the planets and the passionsthe affections and the fixed stars are
extinguishedtheir names cannot die.

**Wittgenstein, Ludwig (1889-1951) **We could present
spatially an atomic fact which contradicted the laws of physics, but not one
which contradicted the laws of geometry.

**Wittgenstein, Ludwig (1889-1951) **Mathematics is a
logical method ... Mathematical propositions express no thoughts. In life it is
never a mathematical proposition which we need, but we use mathematical
propositions only in order to infer fr

om propositions which do not belong to mathematics to others which equally
do not belong to mathematics.

*Tractatus Logico Philosophicus,*
New York, 1922, p. 169.

**Wittgenstein, Ludwig (1889-1951) **There can never be
surprises in logic.

In J. R. Newman (ed.)

**Wittgenstein, Ludwig (1889-1951) **The riddle does not
exist. If a question can be put at all, then it can also be answered.

**Wordsworth, William (1770 - 1850) **[Mathematics] is an
independent world

Created out of pure intelligence.

**Wren, Sir Christopoher **In things to be seen at once, much
variety makes confusion, another vice of beauty. In things that are not seen
at once, and have no respect one to another, great variety is commendable,
provided this variety transgre

ss not the rules of optics and geometry.

W.H. Auden and L.
Kronenberger *The Viking Book of Aphorisms*, New York: Viking Press,
1966.

**X, Malcom **I'm sorry to say that the subject I most disliked
was mathematics. I have thought about it. I think the reason was that
mathematics leaves no room for argument. If you made a mistake, that was
all there was to it.

*
/I>*

*
*

* Young, J. W. A.*

Mathematics has beauties of its own -- a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found only in the works of the greatest

*
beauty When this subject is properly ... presented, the mental emotion should
be that of enjoyment of beauty, not that of repulsion from the ugly and the
unpleasant. In H. Eves Mathematical Circles Squared, Boston:
Prindle, Weber and Schmidt, 1*

*
972.*

*
*

* Zeeman, E Christopher (1925 - ) Technical skill is mastery
of complexity while creativity is mastery of simplicity. Catastrophe
Theory, 1977.*

*
*