We are usually convinced more easily by reasons we have found ourselves
than by those which have occurred to others.
Pensees. 1670.
It is the heart which perceives God and not the reason.
Pensees. 1670.
Man is equally incapable of seeing the nothingness from which he emerges
and the infinity in which he is engulfed.
Pensees. 1670.
Our nature consists in movement; absolute rest is death.
Pensees. 1670.
Man is full of desires: he loves only those who can satisfy them all.
"This man is a good mathematician," someone will say. But I have no concern
for mathematics; he would take me for a proposition. "That one is a good
soldier." He would take me for a besieged town. I need, that is to say,
a decent man who can accommodate himself to all my desires in a general
sort of way.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
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.
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.
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.
Those who write against vanity want the glory of having written well,
and their readers the glory of reading well, and I who write this have
the same desire, as perhaps those who read this have also.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
Our notion of symmetry is derived form the human face. Hence, we demand
symmetry horizontally and in breadth only, not vertically nor in depth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
When we encounter a natural style we are always surprised and delighted,
for we thought to see an author and found a man.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
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.
I cannot judge my work while I am doing it. I have to do as painters
do, stand back and view it from a distance, but not too great a distance.
How great? Guess.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
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.
All err the more dangerously because each follows a truth. Their mistake
lies not in following a falsehood but in not following another truth.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
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.
Those who are accustomed to judge by feeling do not understand the process
of reasoning, because they want to comprehend at a glance and are not
used to seeking for first principles. Those, on the other hand, who are
accustomed to reason from first principles do not understand matters of
feeling at all, because they look for first principles and are unable
to comprehend at a glance.
W. H. Auden and L. Kronenberger (eds.) The Viking Book of Aphorisms,
New York: Viking Press, 1966.
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.
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.
Nature is an infinite sphere of which the center is everywhere and the
circumference nowhere.
Pensees. 1670.
We arrive at truth, not by reason only, but also by the heart.
Pensees. 1670.
When the passions become masters, they are vices.
Pensees. 1670.
Men despise religion; they hate it, and they fear it is true.
Pensees. 1670.
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.
It is not certain that everything is uncertain.
Pensees. 1670.
We are so presumptuous that we should like to be known all over the world,
even by people who will only come when we are no more. Such is our vanity
that the good opinion of half a dozen of the people around us gives us
pleasure and satisfaction.
Pensees. 1670.
The sole cause of man's unhappiness is that he does not know how to stay
quietly in his room.
Pensees. 1670.
Reason's last step is the recognition that there are an infinite number
of things which are beyond it.
Pensees. 1670.
Through space the universe grasps me and swallows me up like a speck;
through thought I grasp it.
Pensees. 1670.
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.
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.
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.
Reverend Fathers, my letters did not usually follow each other at such
close intervals, nor were they so long.... This one would not be so long
had I but the leisure to make it shorter.
Lettres provinciales.
The last thing one knows when writing a book is what to put first.
Pensees. 1670.
What is man in nature? Nothing in relation to the infinite, all in relation
to nothing, a mean between nothing and everything.
Pensees. 1670.
[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.
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.
Look somewhere else for someone who can follow you in your researches
about numbers. For my part, I confess that they are far beyond me, and
I am competent only to admire them.
[Written to Fermat]
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
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.
The more intelligent one is, the more men of originality one finds. Ordinary
people find no difference between men.
Pensees. 1670.
However vast a man's spiritual resources, he is capable of but one great
passion.
Discours sur les passions de l'amour. 1653.
There are two types of mind ... the mathematical, and what might be called
the intuitive. The former arrives at its views slowly, but they are firm
and rigid; the latter is endowed with greater flexibility and applies
itself simultaneously to the diverse lovable parts of that which it loves.
Discours sur les passions de l'amour. 1653.
This trend [emphasizing applied mathematics over pure mathematics] will
make the queen of the sciences into the quean of the sciences.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Chance favors only the prepared mind.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988
.
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.
Mathematics is the science which draws necessary conclusions.
Memoir read before the National Academy of Sciences in Washington, 1870.
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.
...mathematics is distinguished from all other sciences except only ethics,
in standing in no need of ethics. Every other science, even logiclogic,
especiallyis in its early stages in danger of evaporating into airy nothingness,
degenerating, as the Germans say, into an arachnoid film, spun from the
stuff that dreams are made of. There is no such danger for pure mathematics;
for that is precisely what mathematics ought to be.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Among the minor, yet striking characteristics of mathematics, may be mentioned
the fleshless and skeletal build of its propositions; the peculiar difficulty,
complication, and stress of its reasonings; the perfect exactitude of
its results; their broad universality; their practical infallibility.
"The Essence of Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The pragmatist knows that doubt is an art which hs to be acquired with
difficulty.
Collected Papers.
Geometry is a skill of the eyes and the hands as well as of the mind.
He who can properly define and divide is to be considered a god.
The ludicrous state of solid geometry made me pass over this branch. Republic, VII, 528.
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.
Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.
The knowledge of which geometry aims is the knowledge of the eternal.
Republic, VII, 52.
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.
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.
[about Archimedes:]
... being perpetually charmed by his familiar siren,
that is, by his geometry, he neglected to eat and drink and took no care
of his person; that he was often carried by force to the baths, and when
there he would trace geometrical figures in the ashes of the fire, and
with his finger draws lines upon his body when it was anointed with oil,
being in a state of great ecstasy and divinely possessed by his science.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
To speak algebraically, Mr. M. is execrable, but Mr. G. is (x + 1)- ecrable.
[Discussing fellow writers Cornelius Mathews and William Ellery Channing.]
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
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].
Later generations will regard Mengenlehre (set theory) as a disease from
which one has recovered.
[Whether or not he actually said this is a matter of debate amongst historians of
mathematics.]
The Mathematical Intelligencer, vol 13, no. 1, Winter 1991.
What is it indeed that gives us the feeling of elegance in a solution,
in a demonstration? It is the harmony of the diverse parts, their symmetry,
their happy balance; in a word it is all that introduces order, all that
gives unity, that permits us to see clearly and to comprehend at once
both the ensemble and the details.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Thus, be it understood, to demonstrate a theorem, it is neither necessary
nor even advantageous to know what it means. The geometer might be replaced
by the "logic piano" imagined by Stanley Jevons; or, if you choose, a
machine might be imagined where the assumptions were put in at one end,
while the theorems came out at the other, like the legendary Chicago machine
where the pigs go in alive and come out transformed into hams and sausages.
No more than these machines need the mathematician know what he does.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Talk with M. Hermite. He never evokes a concrete image, yet you soon
perceive that the more abstract entities are to him like living creatures.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
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.
A scientist worthy of his name, about all a mathematician, experiences
in his work the same impression as an artist; his pleasure is as great
and of the same nature.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
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.
Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.
Thought is only a flash between two long nights, but this flash is everything.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The mind uses its faculty for creativity only when experience forces it to do so.
Mathematical discoveries, small or greatare never born of spontaneous generation They always presuppose a soil seeded with preliminary knowledge and well prepared by labour, both conscious and subconscious.
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.
...by natural selection our mind has adapted itself to the conditions
of the external world. It has adopted the geometry most advantageous
to the species or, in other words, the most convenient. Geometry is not
true, it is advantageous.
Science and Method.
Life is good for only two things, discovering mathematics and teaching
mathematics.
Mathematics Magazine, v. 64, no. 1, Feb. 1991.
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.
The traditional mathematics professor of the popular legend is absentminded.
He usually appears in public with a lost umbrella in each hand. He prefers
to face the blackboard and to turn his back to the class. He writes a,
he says b, he means c; but it should be d. Some of his sayings are handed
down from generation to generation.
"In order to solve this differential equation you look at it till a
solution occurs to you."
"This principle is so perfectly general that no particular application
of it is possible."
"Geometry is the science of correct reasoning on incorrect figures."
"My method to overcome a difficulty is to go round it."
"What is the difference between method and device? A method is a device
which you used twice."
How to Solve It. Princeton: Princeton University Press. 1945.
Mathematics is the cheapest science. Unlike physics or chemistry, it
does not require any expensive equipment. All one needs for mathematics
is a pencil and paper.
D. J. Albers and G. L. Alexanderson, Mathematical People, Boston: Birkhäuser, 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.
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.
Even fairly good students, when they have obtained the solution of the
problem and written down neatly the argument, shut their books and look
for something else. Doing so, they miss an important and instructive
phase of the work. ... A good teacher should understand and impress on
his students the view that no problem whatever is completely exhausted.
One of the first and foremost duties of the teacher is not to give his
students the impression that mathematical problems have little connection
with each other, and no connection at all with anything else. We have
a natural opportunity to investigate the connections of a problem when
looking back at its solution.
How to Solve It. Princeton: Princeton University Press. 1945.
In order to translate a sentence from English into French two things are
necessary. First, we must understand thoroughly the English sentence.
Second, we must be familiar with the forms of expression peculiar to the
French language. The situation is very similar when we attempt to express
in mathematical symbols a condition proposed in words. First, we must
understand thoroughly the condition. Second, we must be familiar with
the forms of mathematical expression.
How to Solve It. Princeton: Princeton University Press. 1945.
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.
Order is Heaven's first law.
An Essay on Man IV.
See skulking Truth to her old cavern fled,
Mountains of Casuistry heap'd o'er her head!
Philosophy, that lean'd on Heav'n before,
Shrinks to her second cause, and is no more.
Physic of Metaphysic begs defence,
And Metaphysic calls for aid on Sense!
See Mystery to Mathematics fly!
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.
It is well known that the man who first made public the theory of irrationals
perished in a shipwreck in order that the inexpressible and unimaginable
should ever remain veiled. And so the guilty man, who fortuitously touched
on and revealed this aspect of living things, was taken to the place where
he began and there is for ever beaten by the waves.
Scholium to Book X of Euclid V.
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.
Inspiration is needed in geometry, just as much as in poetry.
Likhtenshtein