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The science of mathematics presents the most brilliant example of how
pure reason may successfully enlarge its domain without the aid of experience.

*The Mathematical Intelligencer*, v. 13, no. 1, Winter 1991.

All human knowledge thus begins with intuitions, proceeds thence to concepts, and ends with ideas.

Quoted in Hilbert's *Foundations of Geometry*.

Mathematics is not yet capable of coping with the naivete of the mathematician
himself.

*Sociology Learns the Language of Mathematics*.

We (he and Halmos) share a philosophy about linear algebra: we think basis-free,
we write basis-free , but when the chips are down we close the office
door and compute with matrices like fury.

*Paul Halmos: Celebrating 50 Years of Mathematics.*

The purpose of models is not to fit the data but to sharpen the questions.

11th R A Fisher Memorial Lecture, Royal Society 20, April 1983.

Mathematics is man's own handiwork, subject only to the limitations imposed
by the laws of thought.*Mathematics and the Imagination*, New York: Simon and Schuster, 1940.

The testament of science is so continually in a flux that the heresy of yesterday is the gospel of today and the fundamentalism of tomorrow.

*Mathematics and the Imagination*, Simon and Schuster, 1940.

...we have overcome the notion that mathematical truths have an existence
independent and apart from our own minds. It is even strange to us that
such a notion could ever have existed.

*Mathematics and the Imagination*, New York: Simon and Schuster, 1940.

Mathematics is the science which uses easy words for hard ideas.

*Mathematics and the Imagination*, New York: Simon and Schuster, 1940.

Mathematics is often erroneously referred to as the science of common
sense. Actually, it may transcend common sense and go beyond either imagination
or intuition. It has become a very strange and perhaps frightening subject
from the ordinary point of view, but anyone who penetrates into it will
find a veritable fairyland, a fairyland which is strange, but makes sense,
if not common sense.

*Mathematics and the Imagination*, New York: Simon and Schuster, 1940.

Perhaps the greatest paradox of all is that there are paradoxes in mathematics.*Mathematics and the Imagination*, New York: Simon and Schuster, 1940.

When the mathematician says that such and such a proposition is true of
one thing, it may be interesting, and it is surely safe. But when he
tries to extend his proposition to everything, though it is much more
interesting, it is also much more dangerous. In the transition from one
to all, from the specific to the general, mathematics has made its greatest
progress, and suffered its most serious setbacks, of which the logical
paradoxes constitute the most important part. For, if mathematics is
to advance securely and confidently it must first set its affairs in order
at home.

*Mathematics and the Imagination*, New York: Simon and Schuster, 1940.

Now I feel as if I should succeed in doing something in mathematics, although
I cannot see why it is so very important... The knowledge doesn't make
life any sweeter or happier, does it?

*The Story of My Life.* 1903.

A topologist is one who doesn't know the difference between a doughnut
and a coffee cup.

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

A mind is accustomed to mathematical deduction, when confronted with the
faulty foundations of astrology, resists a long, long time, like an obstinate
mule, until compelled by beating and curses to put its foot into that
dirty puddle.

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

Where there is matter, there is geometry.

*(Ubi materia, ibi geometria.)*

J. Koenderink *Solid Shape*, Cambridge Mass.: MIT Press, 1990

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.

Nature uses as little as possible of anything.

It has been pointed out already that no knowledge of probabilities, less
in degree than certainty, helps us to know what conclusions are true,
and that there is no direct relation between the truth of a proposition
and its probability. Probability begins and ends with probability.

*The Application of Probability to Conduct*.

When asked what it was like to set about proving something, the mathematician likened proving a theorem to seeing the peak of a mountain and trying to climb to the top. One establishes a base camp and begins scaling the mountain's sheer face, encountering obstacles at every turn, often retracing one's steps and struggling every foot of the journey. Finally when the top is reached, one stands examining the peak, taking in the view of the surrounding countrysideand then noting the automobile road up the other side!

A proof tells us where to concentrate our doubts.

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

Statistics: the mathematical theory of ignorance.

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

Logic is the art of going wrong with confidence.

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

Universities hire professors the way some men choose wives - they want the
ones the others will admire.

*Why the Professor Can't Teach.* St. Martin's
Press, 1977. p 92.

In the index to the six hundred odd pages of Arnold Toynbee's A Study
of History, abridged version, the names of Copernicus, Galileo, Descartes
and Newton do not occur yet their cosmic quest destroyed the medieval
vision of an immutable social order in a walled-in universe and transformed
the European landscape, society, culture, habits and general outlook,
as thoroughly as if a new species had arisen on this planet.

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

Nobody before the Pythagoreans had thought that mathematical relations
held the secret of the universe. Twenty-five centuries later, Europe
is still blessed and cursed with their heritage. To non-European civilizations,
the idea that numbers are the key to both wisdom and power, seems never
to have occurred.

*The Sleepwalkers.* 1959.

Say what you know, do what you must, come what may.

[Motto on her paper "On the Problem of the Rotation of a Solid Body about
a Fixed Point."]

Mathematics is indeed dangerous in that it absorbs students to such a
degree that it dulls their senses to everything else.

Attributed by Karl Schellbach. In H. Eves *Mathematical Circles Adieu*, Boston: Prindle, Weber and Schmidt, 1977.

Number theorists are like lotus-eaters -- having once tasted of this food
they can never give it up.

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

God made the integers, all else is the work of man.

(*Die Ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.*)

*Jahresberichte der Deutschen Mathematiker Vereinigung.*