Proof is the idol before whom the pure mathematician tortures himself.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
We used to think that if we knew one, we knew two, because one and one
are two. We are finding that we must learn a great deal more about `and'.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.
We have found a strange footprint on the shores of the unknown. We have
devised profound theories, one after another, to account for its origins.
At last, we have succeeded in reconstructing the creature that made the
footprint. And lo! It is our own.
Space, Time and Gravitation. 1920.
It is impossible to trap modern physics into predicting anything with
perfect determinism because it deals with probabilities from the outset.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
I believe there are 15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,
425,076,185,631,031,296 protons in the universe and the same number of
electrons.
The Philosophy of Physical Science. Cambridge, 1939.
To the pure geometer the radius of curvature is an incidental characteristic - like
the grin of the Cheshire cat. To the physicist it is an indispensable
characteristic. It would be going too far to say that to the physicist
the cat is merely incidental to the grin. Physics is concerned with interrelatedness
such as the interrelatedness of cats and grins. In this case the "cat
without a grin" and the "grin without a cat" are equally set aside as
purely mathematical phantasies.
The Expanding Universe..
Human life is proverbially uncertain; few things are more certain than
the solvency of a life-insurance company.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
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.
If you ask mathematicians what they do, yo always get the same answer.
They think. They think about difficult and unusual problems. They do
not think about ordinary problems: they just write down the answers.
Mathematics Magazine, v. 65 no. 5, December 1992.
A theory has only the alternative of being right or wrong. A model has
a third possibility: it may be right, but irrelevant.
Jagdish Mehra (ed.) The Physicist's Conception of Nature, 1973.
[During a lecture:]
This has been done elegantly by Minkowski; but chalk is cheaper than grey
matter, and we will do it as it comes.
[Attributed by Pólya.]
J.E. Littlewood, A Mathematician's Miscellany, Methuen
and Co. Ltd., 1953.
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.
I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.
Imagination is more important than knowledge.
On Science.
The most beautiful thing we can experience is the mysterious. It is the
source of all true art and science.
What I Believe.
The bitter and the sweet come from the outside, the hard from within,
from one's own efforts.
Out of My Later Years.
Gott würfelt nicht.
Common sense is the collection of prejudices acquired by age eighteen.
In E. T. Bell Mathematics, Queen and Servant of the Sciences. 1952.
God does not care about our mathematical difficulties. He integrates empirically.
L. Infeld Quest, 1942.
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?
[About Newton]
Nature to him was an open book, whose letters he could
read without effort.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.
As far as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert]
In H. Eves Mathematical Circles Squared Boston: Prindle, Weber and Schmidt, 1972.
Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is subtle,
but he is not malicious.
Inscribed in Fine Hall, Princeton University.
Nature hides her secrets because of her essential loftiness, but not by means of ruse.
The human mind has first to construct forms, independently, before we can find them in things.
Since the mathematicians have invaded the theory of relativity, I do not
understand it myself anymore.
In A. Sommerfelt "To Albert Einstein's Seventieth Birthday" in Paul A. Schilpp (ed.) Albert Einstein, Philosopher-Scientist, Evanston, 1949.
Do not worry about your difficulties in mathematics, I assure you that mine are greater.
The truth of a theory is in your mind, not in your eyes.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
These thoughts did not come in any verbal formulation. I rarely think
in words at all. A thought comes, and I may try to express it in words
afterward.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
A human being is a part of the whole, called by us "Universe," a part
limited in time and space. He experiences himself, his thoughts and feelings
as something separated from the resta kind of optical delusion of his
consciousness. This delusion is a kind of prison for us, restricting
us to our personal desires and to affection for a few persons nearest
to us. Our task must be to free ourselves from this prison by widening
our circle of compassion to embrace all living creatures and the whole
of nature in its beauty. Nobody is able to achieve this completely, but
the striving for such achievement is in itself a part of the liberation
and a foundation for inner security.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
The world needs heroes and it's better they be harmless men like me than
villains like Hitler.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
It is nothing short of a miracle that modern methods of instruction have
not yet entirely strangled the holy curiousity of inquiry.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
Everything that is really great and inspiring is created by the individual
who can labor in freedom.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
The search for truth is more precious than its possession.
The American Mathematical Monthly v. 100 no. 3.
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.
We come now to the question: what is a priori certain or necessary, respectively
in geometry (doctrine of space) or its foundations? Formerly we thought
everything; nowadays we think nothing. Already the distance-concept is
logically arbitrary; there need be no things that correspond to it, even
approximately.
"Space-Time." Encyclopaedia Britannica, 14th ed.
Most of the fundamental ideas of science are essentially simple, and may,
as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.
Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.
The mathematician has reached the highest rung on the ladder of human thought.
The Dance of Life.
It is here [in mathematics] that the artist has the fullest scope of his
imagination.
The Dance of Life.
God is a child; and when he began to play, he cultivated mathematics.
It is the most godly of man's games.
Das blinde Spiel. 1954.
Mathematics is not yet ready for such problems.
[Attributed by Paul Halmos.]
The American Mathematical Monthly, Nov. 1992
A Mathematician is a machine for turning coffee into theorems.
If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero. To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people. Those doubts that remain we shall thoroughly remove in the following pages, where we shall explain this calculus.
Mathematicians have tried in vain to this day to discover some order in
the sequence of prime numbers, and we have reason to believe that it is
a mystery into which the human mind will never penetrate.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
[upon losing the use of his right eye]
Now I will have less distraction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
In the pure mathematics we contemplate absolute truths which existed in
the divine mind before the morning stars sang together, and which will
continue to exist there when the last of their radiant host shall have
fallen from heaven.
Quoted by E.T. Bell in The Queen of the Sciences, Baltimore, 1931.
A formal manipulator in mathematics often experiences the discomforting
feeling that his pencil surpasses him in intelligence.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
An expert problem solver must be endowed with two incompatible qualities,
a restless imagination and a patient pertinacity.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
Mathematics may be likened to a large rock whose interior composition
we wish to examine. The older mathematicians appear as persevering stone
cutters slowly attempting to demolish the rock from the outside with hammer
and chisel. The later mathematicians resemble expert miners who seek vulnerable
veins, drill into these strategic places, and then blast the rock apart
with well placed internal charges.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
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.
If the entire Mandelbrot set were placed on an ordinary sheet of paper, the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine enough to actually observe them.
"Can We See the Mandelbrot Set?", The College Mathematics Journal, v. 26, no. 2, March 1995.