Quotes from the Mathematical Quotations Server 2/5

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Dürer, Albrecht (1471-1528)
Whoever ... proves his point and demonstrates the prime truth
geometrically should be believed by all the world, for there we are
captured. J Heidrich (ed.) Albrecht Dürer's schriftlicher Nachlass
Berlin, 1920.

Dürer, Albrecht (1471-1528)
And since geometry is the right foundation of all painting, I have
decided to teach its rudiments and principles to all youngsters eager
for art... Course in the Art of Measurement

Dyson, Freeman
I am acutely aware of the fact that the marriage between mathematics
and physics, which was so enormously fruitful in past centuries, has
recently ended in divorce. Missed Opportunities, 1972. (Gibbs
Lecture?)

Dyson, Freeman
For a physicist mathematics is not just a tool by means of which
phenomena can be calculated, it is the main source of concepts and
principles by means of which new theories can be created. Mathematics
in the Physical Sciences.

Dyson, Freeman
The bottom line for mathematicians is that the architecture has to be
right. In all the mathematics that I did, the essential point was to
find the right architecture. It's like building a bridge. Once the
main lines of the 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.

Eddington, Sir Arthur (1882-1944)
Proof is the idol before whom the pure mathematician tortures himself.
In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome Press
Inc., 1988.

Eddington, Sir Arthur (1882-1944)
We used to think that if we knew one, we knew two, because one and one
are two. We are finding that we must learn a great deal more about
`and'. In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.

Eddington, Sir Arthur (1882-1944)
We have found a strange footprint on the shores of the unknown. We
have devised profound theories, one after another, to account for its
origins. At last, we have succeeded in reconstructing the creature
that made the footprint. And lo! It is our own. Space, Time and
Gravitation. 1920.

Eddington, Sir Arthur (1882-1944)
It is impossible to trap modern physics into predicting anything with
perfect determinism because it deals with probabilities from the
outset. In J. R. Newman (ed.) The World of Mathematics, New York:
Simon and Schuster, 1956.

Eddington, Sir Arthur (1882-1944)
I believe there are
15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,527,
116,709,366,231,425,076,185,631,031,296
protons in the universe and the same number of electrons.
The Philosophy of Physical Science. Cambridge, 1939.

Eddington, Sir Arthur (1882-1944)
To the pure geometer the radius of curvature is an incidental
characteristic - like the grin of the Cheshire cat. To the physicist
it is an indispensable characteristic. It would be going too far to
say 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..

Eddington, Sir Arthur (1882-1944)
Human life is proverbially uncertain; few things are more certain than
the solvency of a life-insurance company. In J. R. Newman (ed.) The
World of Mathematics, New York: Simon and Schuster, 1956.

Edwards, Jonathon
When I am violently beset with temptations, or cannot rid myself of
evil thoughts, [I resolve] to do some Arithmetic, or Geometry, or some
other study, which necessarily engages all my thoughts, and
unavoidably keeps them from wandering. In T. Mallon A Book of One's
Own. Ticknor & Fields, New York, 1984, p. 106-107.

Egrafov, M.
If you ask mathematicians what they do, yo always get the same answer.
They think. They think about difficult and unusual problems. They do
not think about ordinary problems: they just write down the answers.
Mathematics Magazine, v. 65 no. 5, December 1992.

Eigen, Manfred (1927 - )
A theory has only the alternative of being right or wrong. A model has
a third possibility: it may be right, but irrelevant. Jagdish Mehra
(ed.) The Physicist's Conception of Nature, 1973.

Einstein, Albert (1879-1955)
[During a lecture:]This has been done elegantly by Minkowski; but
chalk is cheaper than grey matter, and we will do it as it comes.
[Attributed by Pólya.] J.E. Littlewood, A Mathematician's Miscellany,
Methuen and Co. Ltd., 1953.

Einstein, Albert (1879-1955)
Everything should be made as simple as possible, but not simpler.
Reader's Digest. Oct. 1977.

Einstein, Albert (1879-1955)
I don't believe in mathematics.
Quoted by Carl Seelig. Albert Einstein.

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

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

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

Einstein, Albert (1879-1955)
Gott würfelt nicht.

Einstein, Albert (1879-1955)
Common sense is the collection of prejudices acquired by age eighteen.
In E. T. Bell Mathematics, Queen and Servant of the Sciences. 1952.

Einstein, Albert (1879-1955)
God does not care about our mathematical difficulties. He integrates
empirically.
L. Infeld Quest, 1942.

Einstein, Albert (1879-1955)
How can it be that mathematics, being after all a product of human
thought independent of experience, is so admirably adapted to the
objects of reality?

Einstein, Albert (1879-1955)
[About Newton]
Nature to him was an open book, whose letters he could read without
effort. In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.

Einstein, Albert (1879-1955)
As far as the laws of mathematics refer to reality, they are not
certain; and as far as they are certain, they do not refer to reality.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.

Einstein, Albert (1879-1955)
What is this frog and mouse battle among the mathematicians?
[i.e. Brouwer vs. Hilbert] In H. Eves Mathematical Circles Squared
Boston: Prindle, Weber and Schmidt, 1972.

Einstein, Albert (1879-1955)
Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is
subtle, but he is not malicious. Inscribed in Fine Hall, Princeton
University.

Einstein, Albert (1879-1955)
Nature hides her secrets because of her essential loftiness, but not
by means of ruse.

Einstein, Albert (1879-1955)
The human mind has first to construct forms, independently, before we
can find them in things.

Einstein, Albert (1879-1955)
Since the mathematicians have invaded the theory of relativity, I do
not understand it myself anymore. In A. Sommerfelt "To Albert
Einstein's Seventieth Birthday" in Paul A. Schilpp (ed.) Albert
Einstein, Philosopher-Scientist, Evanston, 1949.

Einstein, Albert (1879-1955)
Do not worry about your difficulties in mathematics, I assure you that
mine are greater.

Einstein, Albert (1879-1955)
The truth of a theory is in your mind, not in your eyes. In H. Eves
Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt,
1972.

Einstein, Albert (1879-1955)
These thoughts did not come in any verbal formulation. I rarely think
in words at all. A thought comes, and I may try to express it in words
afterward. In H. Eves Mathematical Circles Adieu, Boston: Prindle,
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 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.

Einstein, Albert (1879-1955)
The world needs heroes and it's better they be harmless men like me
than villains like Hitler. In H. Eves Return to Mathematical Circles,
Boston: Prindle, Weber and Schmidt, 1988.

Einstein, Albert (1879-1955)
It is nothing short of a miracle that modern methods of instruction
have not yet entirely strangled the holy curiousity of inquiry. In H.
Eves Return to Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1988.

Einstein, Albert (1879-1955)
Everything that is really great and inspiring is created by the
individual who can labor in freedom. In H. Eves Return to Mathematical
Circles, Boston: Prindle, Weber and Schmidt, 1988.

Einstein, Albert (1879-1955)
The search for truth is more precious than its possession.
The American Mathematical Monthly v. 100 no. 3.

Einstein, Albert (1879-1955)
If my theory of relativity is proven successful, Germany will claim me
as a German and France will declare that I am a citizen of the world.
Should my theory prove untrue, France will say that I am a German and
Germany will declare that I am a Jew. Address at the Sorbonne, Paris.

Einstein, Albert (1879-1955)
We come now to the question: what is a priori certain or necessary,
respectively in geometry (doctrine of space) or its foundations?
Formerly we thought everything; nowadays we think nothing. Already the
distance-concept is logically arbitrary; there need be no things that
correspond to it, even approximately. "Space-Time." Encyclopaedia
Britannica, 14th ed.

Einstein, Albert (1879-1955)
Most of the fundamental ideas of science are essentially simple, and
may, as a rule, be expressed in a language comprehensible to everyone.
The Evolution of Physics.

Einstein, Albert (1879-1955)
Science without religion is lame; religion without science is blind.
Reader's Digest, Nov. 1973.

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

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

Erath, V.
God is a child; and when he began to play, he cultivated mathematics.
It is the most godly of man's games. Das blinde Spiel. 1954.

Erdös, Paul
Mathematics is not yet ready for such problems.
[Attributed by Paul Halmos.]
The American Mathematical Monthly, Nov. 1992

Erdös, Paul
A Mathematician is a machine for turning coffee into theorems.

Euler, Leonhard (1707 - 1783)
If a nonnegative quantity was so small that it is smaller than any
given one, then it certainly could not be anything but zero. To those
who ask what the infinitely small quantity in mathematics is, we
answer 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.

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

Euler, Leonhard (1707-1783)
[upon losing the use of his right eye]
Now I will have less distraction.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and
Schmidt, 1969.

Everett, Edward (1794-1865)
In the pure mathematics we contemplate absolute truths which existed
in the divine mind before the morning stars sang together, and which
will continue to exist there when the last of their radiant host shall
have fallen from heaven. Quoted by E.T. Bell in The Queen of the
Sciences, Baltimore, 1931.

Eves, Howard W.
A formal manipulator in mathematics often experiences the
discomforting feeling that his pencil surpasses him in intelligence.
In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Eves, Howard W.
An expert problem solver must be endowed with two incompatible
qualities, a restless imagination and a patient pertinacity. In
Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.

Eves, Howard W.
Mathematics may be likened to a large rock whose interior composition
we wish to examine. The older mathematicians appear as persevering
stone cutters slowly attempting to demolish the rock from the outside
with hammer and 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.

Eves, Howard W.
One is hard pressed to think of universal customs that man has
successfully established on earth. There is one, however, of which he
can boast the universal adoption of the Hindu-Arabic numerals to
record numbers. In this we perhaps have man's unique worldwide victory
of an idea. Mathematical Circles Squared, Boston: Prindle, Weber and
Schmidt, 1972.

Ewing, John
If the entire Mandelbrot set were placed on an ordinary sheet of
paper, the tiny sections of boundary we examine would not fill the
width of a hydrogen atom. Physicists think about such tiny objects;
only mathematicians have 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 too 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 demonstrates his complete capabilities.
In closing, let me say that you will be fortunate if you can get him
to work for you.
Sincerely,
A. D. Visor (Prof.)

de Fermat, Pierre (1601?-1665)
[In the margin of his copy of Diophantus' Arithmetica, Fermat wrote]
To divide a cube into two other cubes, a fourth power or in general
any power whatever into two powers of the same denomination above the
second is impossible, and I have assuredly found an admirable proof of
this, but the margin is too narrow to contain it.

de Fermat, Pierre (1601?-1665)
And perhaps, posterity will thank me for having shown it that the
ancients did not know everything. In D. M. Burton, Elementary Number
Theory, Boston: Allyn and Bacon, Inc., 1976.

Feynman, Richard Philips (1918 - 1988)
We have a habit in writing articles published in scientific journals
to make the work as finished as possible, to cover up all the tracks,
to not worry about the blind alleys or describe how you had the wrong
idea first, and so on. So there isn't any place to publish, in a
dignified manner, what you actually did in order to get to do the
work. Nobel Lecture, 1966.

Finkel, Benjamin Franklin
The solution of problems is one of the lowest forms of mathematical
research, ... yet its educational value cannot be overestimated. It is
the ladder by which the mind ascends into higher fields of original
research 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 once, 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
broken, the cabin boy is on deck, there are 12 passengers aboard, the
wind is blowing East-North-East, the clock points to a quarter past
three in the afternoon. It is the month of May. How old is the
captain?

Fontenelle, Bernard Le Bovier (1657-1757)
Mathematicians are like lovers. Grant a mathematician the least
principle, and he will draw from it a consequence which you must also
grant him, and from this consequence another. Quoted in V. H. Larney
Abstract Algebra: A First Course, Boston: Prindle, Weber and Schmidt,
1975.

Fontenelle, Bernard Le Bovier (1657-1757)
A work of morality, politics, criticism will be more elegant, other
things being equal, if it is shaped by the hand of geometry. Preface
sur l'Utilité des Mathématiques et de la Physique, 1729.

Fontenelle, Bernard Le Bovier (1657-1757)
Leibniz never married; he had considered it at the age of fifty; but
the person he had in mind asked for time to reflect. This gave Leibniz
time to reflect, too, and so he never married. Eloge 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 axioms 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'Alembert 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 nearly through the press. In Scientific American, May 1984, p 77.

Galbraith, John Kenneth
There can be no question, however, that prolonged commitment to
mathematical exercises in economics can be damaging. It leads to the
atrophy of judgement and intuition... Economics, Peace, and Laughter.

Galilei, Galileo (1564 - 1642)
[The universe] cannot be read until we have learnt the language and
become familiar with the characters in which it is written. It is
written in mathematical language, and the letters are triangles,
circles and other geometrical figures, without which means it is
humanly impossible to comprehend a single word. Opere Il Saggiatore p.
171.

Galilei, Galileo (1564 - 1642)
Measure what is measurable, and make measurable what is not so. Quoted
in H. Weyl "Mathematics and the Laws of Nature" in I Gordon and S.
Sorkin (eds.) The Armchair Science Reader, New York: 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.) Mathematical Maxims and Minims, Raleigh
NC: Rome Press Inc., 1988.

Galton, [Sir] Francis (1822-1911)
Whenever you can, count.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956.

Galton, Sir Francis (1822-1911)
[Statistics are] the only tools by which an opening can be cut through
the formidable thicket of difficulties that bars the path of those who
pursue the Science of Man. Pearson, The Life and Labours of Francis
Galton, 1914.

Galton, Sir Francis (1822-1911)
I know of scarcely anything so apt to impress the imagination as the
wonderful form of cosmic order expressed by the "Law of Frequency of
Error." The law would have been personified by the Greeks and 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 York: 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 historical 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
dissolved and what is left is just a mathematical structure. Gardner
on Gardner: JPBM Communications Award Presentation. Focus-The
Newsletter of the Mathematical Association of America v. 14, no. 6,
December 1994.

Gauss, Karl Friedrich (1777-1855)
I confess that Fermat's Theorem as an isolated proposition has very
little interest for me, because I could easily lay down a multitude of
such propositions, which one could neither prove nor dispose of. [A
reply to Olbers' attempt in 1816 to entice him to work on Fermat's
Theorem.] In J. R. Newman (ed.) The World of Mathematics, New York:
Simon and Schuster, 1956. p. 312.

Gauss, Karl Friedrich (1777-1855)
If others would but reflect on mathematical truths as deeply and as
continuously as I have, they would make my discoveries. In J. R.
Newman (ed.) The World of Mathematics, New York: Simon and Schuster,
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
future; 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.
Simmons 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, 1830.

Gauss, Karl Friedrich (1777-1855)
I mean the word proof not in the sense of the lawyers, who set two
half proofs equal to a whole one, but in the sense of a mathematician,
where half proof = 0, and it is demanded for proof that every doubt
becomes impossible. In G. Simmons Calculus Gems, New York: McGraw Hill
inc., 1992.

Gauss, Karl Friedrich (1777-1855)
I have had my results for a long time: but I do not yet know how I am
to arrive at them. In A. Arber The Mind and the Eye 1954.

Gauss, Karl Friedrich (1777-1855)
[His motto:]
Few, but ripe.

Gauss, Karl Friedrich (1777-1855)
[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
W. Shakespeare King Lear.

Gauss, Karl Friedrich (1777-1855)
[attributed to him by H.B Lübsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lübsen's geometry textbook.

Gauss, Karl Friedrich (1777-1855)
It is not knowledge, but the act of learning, not possession but the
act of getting there, which grants the greatest enjoyment. When I have
clarified and exhausted a subject, then I turn away from it, in order
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 thread 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
induction, 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,
which is purely aprioristic, but with mechanics. Quoted in J.
Koenderink Solid Shape, Cambridge Mass.: MIT Press, 1990.

Gay, John
Lest men suspect your tale untrue,
Keep probability in view.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and
Schuster, 1956. p. 1334.

Gibbs, Josiah Willard (1839 - 1903)
One of the principal objects of theoretical research in my department
of knowledge is to find the point of view from which the subject
appears in its greatest simplicity.

Gibbs, Josiah Willard (1839-1903)
Mathematics is a language.

Gilbert, W. S. (1836 - 1911)
I'm very good at integral and differential calculus, I know the
scientific names of beings animalculous; In short, in matters
vegetable, animal, and mineral, I am the very model of a modern
Major-General. The Pirates of Penzance. Act 1.

Glaisher, J.W.
The mathematician requires tact and good taste at every step of his
work, and he has to learn to trust to his own instinct to distinguish
between what is really worthy of his efforts and what is not. In H.
Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt,
1972.

Glanvill, Joseph
And for mathematical science, he that doubts their certainty hath need
of a dose of hellebore. In J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956, p. 548.

Goedel, Kurt
I don't believe in natural science.
[Said to physicist John Bahcall.]
Ed Regis, Who Got Einstein's Office? Addison Wesley, 1987.

Goethe
It has been said that figures rule the world. Maybe. But I am sure
that figures show us whether it is being ruled well or badly. In J. P.
Eckermann, Conversations with Goethe.

Goethe
Mathematics has the completely false reputation of yielding infallible
conclusions. Its infallibility is nothing but identity. Two times two
is not four, but it is just two times two, and that is what we call
four for short. But four is nothing new at all. And thus it goes on
and on in its conclusions, except that in the higher formulas the
identity fades out of sight. In J. R. Newman (ed.) The World of
Mathematics, New York: Simon and Schuster, 1956, p. 1754.

Goodman, Nicholas P.
There are no deep theorems -- only theorems that we have not
understood very well. The Mathematical Intelligencer, vol. 5, no. 3,
1983.

Gordon, P
This is not mathematics, it is theology.
[On being exposed to Hilbert's work in invariant theory.]
Quoted in P. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhäuser, 1981.

Graham, Ronald
It wouild be very discouraging if somewhere down the line you could
ask a computer if the Riemann hypothesis is correct and it said, `Yes,
it is true, but you won't be able to understand the proof.' John
Horgan. Scientific 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 sort of mathematics which
is of practical value. Handbook of Applicable Mathematics.

Hadamard, Jacques
The shortest path between two truths in the real domain passes through
the complex domain. Quoted in The Mathematical Intelligencer, v. 13,
no. 1, Winter 1991.

Hadmard, Jacques
Practical application is found by not looking for it, and one can say
that the whole progress of civilization rests on that principle. In H.
Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt,
1972.

Haldane, John Burdon Sanderson (1892-1964)
In scientific thought we adopt the simplest theory which will explain
all the facts under consideration and enable us to predict new facts
of the same kind. The catch in this criterion lies in the world
"simplest." It is really an aesthetic canon such as we find implicit
in our criticisms of poetry or painting. The layman finds such a law
as dx/dt = K(d^2x/dy^2) much less simple than "it oozes," of which it
is the mathematical statement. 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 rate
of change. Possible Worlds, 1927.

Haldane, John Burdon Sanderson (1892-1964)
A time will however come (as I believe) when physiology will invade
and destroy mathematical physics, as the latter has destroyed
geometry. Daedalus, or Science and the Future, London: Kegan Paul,
1923.

Halmos, Paul R.
Mathematics is not a deductive science -- that's a cliche. When you
try to prove a theorem, you don't just list the hypotheses, and then
start to reason. What you do is trial and error, experimentation,
guesswork. I Want 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 us a
faculty that can pass our Master's exams." I Want to be a
Mathematician, Washington: 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 in 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 special 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 degenerate cases? Where does the proof use the hypothesis? I
Want to be a Mathematician, Washington: MAA Spectrum, 1985.

Halmos, Paul R.
To be a scholar of mathematics you must be born with talent, insight,
concentration, taste, luck, drive and the ability to visualize and
guess. I Want to be a Mathematician, Washington: MAA Spectrum, 1985.

Hamilton, [Sir] William Rowan (1805-1865)
Who would not rather have the fame of Archimedes than that of his
conqueror Marcellus? In H. Eves Mathematical Circles Revisited,
Boston: Prindle, Weber and Schmidt, 1971.

Hamilton, Sir William Rowan (1805-1865)
I regard it as an inelegance, or imperfection, in quaternions, or
rather in the state to which it has been hitherto unfolded, whenever
it becomes or seems to become necessary to have recourse to x, y, z,
etc.. In a letter from Tait to Cayley.

Hamilton, Sir William Rowan (1805-1865)
On earth there is nothing great but man; in man there is nothing great
but mind. Lectures on Metaphysics.

Hamming, Richard W.
Does anyone believe that the difference between the Lebesgue and
Riemann integrals can have physical significance, and that whether
say, an airplane would or would not fly could depend on this
difference? If such were claimed, I should not care to fly in that
plane. In N. Rose Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.

Hamming, Richard W.
Mathematics is an interesting intellectual sport but it should not be
allowed to stand in the way of obtaining sensible information about
physical processes. In N. Rose Mathematical Maxims and Minims, Raleigh
NC: Rome Press Inc., 1988.

Hardy, Godfrey H. (1877 - 1947)
[On Ramanujan]
I remember once going to see him when he was lying ill at Putney. I
had ridden in taxi cab number 1729 and remarked that the number seemed
to me rather a dull one, and that I hoped it was not 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, but a mathematician offers the game. A Mathematician's Apology,
London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
I am interested in mathematics only as a creative art.
A Mathematician's Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
Pure mathematics is on the whole distinctly more useful than applied.
For what is useful above all is technique, and mathematical technique
is taught mainly through pure mathematics.

Hardy, Godfrey H. (1877 - 1947)
In great mathematics there is a very high degree of unexpectedness,
combined with inevitability and economy. A Mathematician's Apology,
London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
There is no scorn more profound, or on the whole more justifiable,
than that of the men who make for the men who explain. Exposition,
criticism, appreciation, is work for second-rate minds. A
Mathematician'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 Journal, vol.
25, No. 1, January 1994.

Hardy, Godfrey H. (1877 - 1947)
A science is said to be useful of its development tends to accentuate
the existing inequalities in the distribution of wealth, or more
directly promotes the destruction of human life. A Mathematician's
Apology, London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
The mathematician's patterns, like the painter's or the poet's must be
beautiful; the ideas, like the colors or the words must fit together
in a harmonious way. Beauty is the first test: there is no permanent
place in this world for ugly mathematics. A Mathematician's Apology,
London, Cambridge University Press, 1941.

Hardy, Godfrey H. (1877 - 1947)
I believe that mathematical reality lies outside us, that our function
is to discover or observe it, and that the theorems which we prove,
and which we describe grandiloquently as our "creations," are 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) as 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, 1941.

Hardy, Thomas
...he seemed to approach the grave as an hyperbolic curve approaches
a line, less directly as he got nearer, till it was doubtful if he
would ever reach it at all. Far from the Madding Crowd.

Harish-Chandra
I have often pondered over the roles of knowledge or experience, on
the one hand, and imagination or intuition, on the other, in the
process of discovery. I believe that there is a certain fundamental
conflict between the two, and knowledge, by advocating caution, tends
to inhibit the flight of imagination. Therefore, a certain naivete,
unburdened by conventional wisdom, can sometimes be a positive asset.
R. Langlands, "Harish-Chandra," Biographical Memoirs of Fellows of the
Royal Society 31 (1985) 197 - 225.

Harris, Sydney J.
The real danger is not that computers will begin to think like men,
but that men will begin to think like computers. In H. Eves Return to
Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.

Hawking, Stephen Williams (1942- )
God not only plays dice. He also sometimes throws the dice where they
cannot be seen. [See related quotation from Albert Einstein.] Nature
1975 257.

Heath, Sir Thomas
[The works of Archimedes] are without exception, monuments of
mathematical exposition; the gradual revelation of the plan of attack,
the masterly ordering of the propositions, the stern elimination of
everything not immediately relevant to the purpose, the finish of the
whole, are so impressive in their perfection as to create a feeling
akin to awe in the mind of the reader. A History of Greek Mathematics.
1921.

Heaviside, Oliver (1850-1925)
[Criticized for using formal mathematical manipulations, without
understanding how they worked:] Should I refuse a good dinner simply
because I do not understand the process of digestion?

Heinlein, Robert A.
Anyone who cannot cope with mathematics is not fully human. At best he
is a tolerable subhuman who has learned to wear shoes, bathe, and not
make messes in the house. Time Enough for Love.

Heisenberg, Werner (1901-1976)
An expert is someone who knows some of the worst mistakes that can be
made in his subject, and how to avoid them. Physics and Beyond. 1971.

Hempel, Carl G.
The propositions of mathematics have, therefore, the same
unquestionable certainty which is typical of such propositions as "All
bachelors are unmarried," but they also share the complete lack of
empirical content which is associated with that certainty: The
propositions of mathematics are devoid of all factual content; they
convey no information whatever on any empirical subject matter. "On
the Nature of Mathematical Truth" in J. R. Newman (ed.) The 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 necessity 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 and
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 somewhere with all the right answers to all of the
interesting questions, and that teachers know those answers. And if
one could get hold of the book, one would have everything settled.
That's so unlike the true nature of mathematics. L.A. Steen and D.J.
Albers (eds.), Teaching Teachers, Teaching Students, Boston:
Birkhäuser, 1981, p89.

( cont'd )

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