Just go on..and faith will soon return.
[To a friend hesitant with respect to infinitesimals.]
In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Thus metaphysics and mathematics are, among all the sciences that belong
to reason, those in which imagination has the greatest role. I beg pardon
of those delicate spirits who are detractors of mathematics for saying
this .... The imagination in a mathematician who creates makes no less
difference than in a poet who invents.... Of all the great men of antiquity,
Archimedes may be the one who most deserves to be placed beside Homer.
Discours Preliminaire de L'Encyclopedie, Tome 1, 1967. pp 47 - 48.
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.
Neither in the subjective nor in the objective world can we find a criterion
for the reality of the number concept, because the first contains no such
concept, and the second contains nothing that is free from the concept.
How then can we arrive at a criterion? Not by evidence, for the dice
of evidence are loaded. Not by logic, for logic has no existence independent
of mathematics: it is only one phase of this multiplied necessity that
we call mathematics.
How then shall mathematical concepts be judged? They shall not be judged.
Mathematics is the supreme arbiter. From its decisions there is no appeal.
We cannot change the rules of the game, we cannot ascertain whether the
game is fair. We can only study the player at his game; not, however,
with the detached attitude of a bystander, for we are watching our own
minds at play.
Every new body of discovery is mathematical in form, because there is
no other guidance we can have.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
Mathematics seems to endow one with something like a new sense.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
The numbers are a catalyst that can help turn raving madmen into polite
humans.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.
Number, Scientific American, 211, (Sept. 1964), 51 - 59.
One began to hear it said that World War I was the chemists' war, World
War II was the physicists' war, World War III (may it never come) will
be the mathematicians' war.
The Mathematical Experience, Boston: Birkhäuser, 1981.
Mathematics is the only instructional material that can be presented in
an entirely undogmatic way.
In The Mathematical Intelligencer, v. 5, no. 2, 1983.
[When asked about his age.] I was x years old in the year x^2.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
It is easier to square the circle than to get round a mathematician.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Every science that has thriven has thriven upon its own symbols: logic,
the only science which is admitted to have made no improvements in century
after century, is the only one which has grown no symbols.
Transactions Cambridge Philosophical Society, vol. X, 1864, p. 184.
Of all things, good sense is the most fairly distributed: everyone thinks
he is so well supplied with it that even those who are the hardest to
satisfy in every other respect never desire more of it than they already
have.
Discours de la Méthode. 1637.
Each problem that I solved became a rule which served afterwards to solve
other problems.
Discours de la Méthode. 1637.
If I found any new truths in the sciences, I can say that they follow
from, or depend on, five or six principal problems which I succeeded in
solving and which I regard as so many battles where the fortunes of war
were on my side.
Discours de la Méthode. 1637.
I concluded that I might take as a general rule the principle that all
things which we very clearly and obviously conceive are true: only observing,
however, that there is some difficulty in rightly determining the objects
which we distinctly conceive.
Discours de la Méthode. 1637.
I thought the following four [rules] would be enough, provided that
I made a firm and constant resolution not to fail even once in the observance
of them. The first was never to accept anything as true if I had not
evident knowledge of its being so; that is, carefully to avoid precipitancy
and prejudice, and to embrace in my judgment only what presented itself
to my mind so clearly and distinctly that I had no occasion to doubt it.
The second, to divide each problem I examined into as many parts as was
feasible, and as was requisite for its better solution. The third, to
direct my thoughts in an orderly way; beginning with the simplest objects,
those most apt to be known, and ascending little by little, in steps as
it were, to the knowledge of the most complex; and establishing an order
in thought even when the objects had no natural priority one to another.
And the last, to make throughout such complete enumerations and such general
surveys that I might be sure of leaving nothing out. These long chains
of perfectly simple and easy reasonings by means of which geometers are
accustomed to carry out their most difficult demonstrations had led me
to fancy that everything that can fall under human knowledge forms a similar
sequence; and that so long as we avoid accepting as true what is not so,
and always preserve the right order of deduction of one thing from another,
there can be nothing too remote to be reached in the end, or to well hidden
to be discovered.
Discours de la Méthode. 1637.
When writing about transcendental issues, be transcendentally clear.
In G. Simmons Calculus Gems. New York: McGraw Hill Inc., 1992.
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.
Cogito Ergo Sum. "I think, therefore I am."
Discours de la Méthode. 1637.
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.
Perfect numbers like perfect men are very rare.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
omnia apud me mathematica fiunt.
With me everything turns into mathematics.
It is not enough to have a good mind. The main thing is to use it well.
Discours de la Méthode. 1637.
If you would be a real seeker after truth, you must at least once in your
life doubt, as far as possible, all things.
Discours de la Méthode. 1637.
Suppose we loosely define a religion as any discipline whose foundations
rest on an element of faith, irrespective of any element of reason which
may be present. Quantum mechanics for example would be a religion under
this definition. But mathematics would hold the unique position of being
the only branch of theology possessing a rigorous demonstration of the
fact that it should be so classified.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
[His epitaph.]
This tomb hold Diophantus Ah, what a marvel! And the tomb tells scientifically
the measure of his life. God vouchsafed that he should be a boy for the
sixth part of his life; when a twelfth was added, his cheeks acquired
a beard; He kindled for him the light of marriage after a seventh, and
in the fifth year after his marriage He granted him a son. Alas! late-begotten
and miserable child, when he had reached the measure of half his father's
life, the chill grave took him. After consoling his grief by this science
of numbers for four years, he reached the end of his life.
In Ivor Thomas Greek Mathematics, in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
I think that there is a moral to this story, namely that it is more important
to have beauty in one's equations that to have them fit experiment. If
Schroedinger had been more confident of his work, he could have published
it some months earlier, and he could have published a more accurate equation. It seems that if one is working from the point of view of getting beauty
in one's equations, and if one has really a sound insight, one is on a
sure line of progress. If there is not complete agreement between the
results of one's work and experiment, one should not allow oneself to
be too discouraged, because the discrepancy may well be due to minor features
that are not properly taken into account and that will get cleared up
with further development of the theory.
Scientific American, May 1963.
Mathematics is the tool specially suited for dealing with abstract concepts
of any kind and there is no limit to its power in this field.
In P. J. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
In science one tries to tell people, in such a way as to be understood
by everyone, something that no one ever knew before. But in poetry, it's
the exact opposite.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
There are three kinds of lies: lies, damned lies, and statistics.
Mark Twain. Autobiography.
.
Pereant qui ante nos nostra dixerunt.
"To the devil with those who published
before us."
[Quoted by St. Jerome, his pupil]
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.
When you have eliminated the impossible, what ever remains, however improbable
must be the truth.
The Sign of Four.
From a drop of water a logician could predict an Atlantic or a Niagara.
A study in Scarlet 1929.
It is a capital mistake to theorize before one has data.
Scandal in Bohemia.
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.
Gauss replied, when asked how soon he expected to reach certain mathematical
conclusions, that he had them long ago, all he was worrying about was
how to reach them!
In Mechanisms of Discovery in I. S. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.
Logic, like whiskey, loses its beneficial effect when taken in too large
quantities.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.
Whoever ... proves his point and demonstrates the prime truth geometrically
should be believed by all the world, for there we are captured.
J Heidrich (ed.) Albrecht Dürer's schriftlicher Nachlass Berlin, 1920.
And since geometry is the right foundation of all painting, I have decided
to teach its rudiments and principles to all youngsters eager for art...
Course in the Art of Measurement
I am acutely aware of the fact that the marriage between mathematics and
physics, which was so enormously fruitful in past centuries, has recently
ended in divorce.
Missed Opportunities, 1972. (Gibbs Lecture?)
For a physicist mathematics is not just a tool by means of which phenomena
can be calculated, it is the main source of concepts and principles by
means of which new theories can be created.
Mathematics in the Physical
Sciences.
The bottom line for mathematicians is that the architecture has to be
right. In all the mathematics that I did, the essential point was to
find the right architecture. It's like building a bridge. Once the main
lines of the structure are right, then the details miraculously fit.
The problem is the overall design.
"Freeman Dyson: Mathematician, Physicist, and Writer". Interview with Donald
J. Albers, The College Mathematics Journal, vol 25, no. 1, January 1994.