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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.

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.

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 plain
man, namely the rate of change of a rate of change.

*Possible Worlds*, 1927.

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.

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.

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.

... 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.

...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.

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.

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.

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.

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.

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.

On earth there is nothing great but man; in man there is nothing great
but mind.

*Lectures on Metaphysics.*

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.

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.

[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.

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.

I am interested in mathematics only as a creative art.

*A Mathematician's Apology*, London, Cambridge University Press, 1941.

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.

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.

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.

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.
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.

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.

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.

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.

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.

...he seemed to approach the grave as an hyperbolic curve approaches a
lineless directly as he got nearer, till it was doubtful if he would ever
reach it at all.

*Far from the Madding Crowd*.

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.

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.

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.

[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.

[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?

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.*

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.

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.

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.

...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.

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.), T*eaching Teachers, Teaching Students*,
Boston: Birkhäuser, 1981, p89.

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.

Abel has left mathematicians enough to keep them busy for 500 years.

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

We are servants rather than masters in mathematics.

In H. Eves *Mathematical Circles Squared*, Boston: Prindle, Weber and Schmidt, 1972.

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.

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, mathematical
present.

*The Glass Bead Game*, 1943.

Wir müssen wissen.

Wir werden wissen.

[Engraved on his tombstone in Göttingen.]

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.

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 *Mathematical Circles Squared*, Boston: Prindle, Weber and Schmidt,
1971.

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.

Mathematics is a game played according to certain simple rules with meaningless
marks on paper.

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

Physics is much too hard for physicists.

C. Reid *Hilbert*, London: Allen and Unwin, 1970.

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 theories and in casting aside some more complicated developments.

The art of doing mathematics consists in finding that special case which
contains all the germs of generality.

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

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.

One can measure the importance of a scientific work by the number of earlier
publications rendered superfluous by it.

In H. Eves *Mathematical Circles Revisited*, Boston: Prindle, Weber and Schmidt,
1971.

Mathematics knows no races or geographic boundaries; for mathematics,
the cultural world is one country.

In H. Eves *Mathematical Circles Squared*, Boston: Prindle, Weber and Schmidt,
1972.

The infinite! No other question has ever moved so profoundly the spirit
of man.

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

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.
After 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 - Mathematician
Xtravagant II - Student Days in Germany", *The American Mathematical Monthly *,
v. 6, no. 100.

There is more in Mersenne than in all the universities together.

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

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 infinity has finite volume.]

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

Geometry, which is the only science that it hath pleased God hitherto
to bestow on mankind.

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

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

Descartes commanded the future from his study more than Napoleon from
the throne.

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

Certitude is not the test of certainty. We have been cocksure of many
things that are not so.

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

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.

T*he Autocrat of the Breakfast Table*.

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 structure: one thing builds on another and depends on it.*Mathematics in a Changing World* Walker, New York 1973.

Hofstadter's Law: It always takes longer than you expect, even when you
take into account Hofstadter's Law.

*Gödel, Escher, Bach* 1979.

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 disembodied 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, 1956.

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*.

I admit that mathematical science is a good thing. But excessive devotion
to it is a bad thing.

Interview with J. W. N. Sullivan, *Contemporary Mind*, London, 1934.

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.

...[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 exquisitely beautiful; the new toy was
inexhaustible in its potentialities.

*Young Archimedes.*

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 pages of formulae will not get a definite result out
of loose data.

*Quarterly Journal of the Geological Society*, 25,1869.

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 languages,
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.

It is the first duty of a hypothesis to be intelligible.