Aristotle

Number proceeds from unity.

Roger Bacon

Mathematics is the gate and key to the sciences.

George Berkeley

And what are these same evanescent increments? They are neither
finite quantities, nor quantities infinitely small, nor yet nothing.
May we not call them the ghosts of departed quantities?

Jakob Bernoulli

Even as the finite encloses an infinite series,
And in the unlimited limits appear,
So the soul of immensity dwells in minuta
And in the narrowest limits, no limits inhere.
What joy to discern the minute in infinity!
The vast to perceive in the small, what Divinity!

Pierre Boutroux

Logic is invincible, because in order to
combat logic it is necessary to use logic.

Claude Bragdon

Mathematics is the handwriting on the human consciousness
of the very Spirit of Life itself.

T. Briffault

In Samoa, when elementary schools were first established,
the natives developed an absolute craze for arithmetical calculations.
They laid aside their weapons and were to be seen going about armed
with slate and pencil, setting sums and problems to one another
and to European visitors. The Honourable Frederick Walpole
declares that his visit to the beautiful island was positively
embittered by ceaseless multiplication and division.

Luitzen Brouwer

The construction itself is an art,
its application to the world an evil parasite.

Scott Buchanan

The structures with which mathematics deals are more like lace, the
leaves of trees and the play of the light and shadow on a human face
than they are like buildings and machines,
the least of their representatives.

Georg Cantor

I am so in favor of the actual infinite that instead of admitting that
Nature abhors it, as is commonly said, I hold that Nature makes
frequent use of it everywhere, in order to show more effectively
the perfections of its Author.

The essence of mathematics resides in its freedom.

G.K. Chesterton

You can only find truth with logic if you have
already found truth without it.

Cosgrove

The Groups of Wrath.

D'Alembert,

Algebra is generous; she often gives more than is asked of her.

Tobias Dantzig

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!

Charles Darwin

A mathematician is a blind man in a dark room
looking for a black cat which isn't there.

Richard Dedekind

Numbers are the free creation of the human mind.

Edmond Duranty

It takes immense genius to represent, simply and sincerely,

Albert Einstein

Pure mathematics is, in its way, the poetry of logical ideas.

Imagination is more important that knowledge.

Do not worry too much about your difficulties in mathematics,
I can assure you that mine are still greater.

Euripides

Mighty is geometry; joined with art, resistless.

Fontenelle

Mathematicians are like lovers . . . Grant a mathematician the least
principle, and he will draw from it a consequence which you
must grant him also, and from this consequence another.

Galileo

In questions of science the authority of a thousand
is not worth the humble reasoning of an individual.

The Universe is a grand book which cannot be read until one first
learns to comprehend the language and become familiar with the
characters in which it is composed.
It is written in the language of mathematics.

Gauss

Mathematics is the queen of the sciences,
and number theory the queen of mathematics.

Goethe

Mathematicians are a species of Frenchmen: if you say something
to them they translate it into their own language and presto!
It is something entirely different.

S. Gudder

The essence of mathematics is not to make simple things complicated,
but to make complicated things simple.

Paul Halmos

A good stack of examles, as large as possible, is indispensable
for a thorough understanding of any concept, and when I want
to learn something new, I make it my first job to build one.

Ross Hansberger

Paul Erdos has a theory that God has a book containing all the
theorems of mathematics with their absolutely most beautiful proofs,
and when he wants to express particular appreciation
of a proof he exclaims, "This is from the book!"

G.H. Hardy

I believe that mathematical reality lies outside of us, and 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 notes on our observations.

Werner Heisenberg

I think that modern physics has definitely decided in favor of Plato.
In fact the smallest units of matter are not physical objects in
the ordinary sense; they are forms, ideas which can be expressed
unambiguously only in mathematical language.

Herman Henkel

In most sciences one generation tears down what another has built
and what one has established another undoes.
In mathematics alone each generation adds a new story
to the old structure.

I.N. Herstein

Very often in mathematics the crucial problem is to recognize
and discover what are the relevant concepts;
once this is accomplished the job may be more than half done.

The value of a problem is not so much coming up with the answer
as in the ideas and attempted ideas it forces on the would be solver.

Heinrich Hertz

One cannot escape the feeling that these mathematical formulae
have an independent existence and an intelligence
of their own, that they are wiser than we are,
wiser even than their discoverers, that we get more out of them
than we originally put in to them.

David Hilbert

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

No one will expel us from the paradise
that Cantor has created for us.

(On Cantor's set theory:)
The finest product of mathematical genius
and one of the supreme achievements
of purely intellectual human activity.

Mathematical science is in my opinion an indivisible whole, an organism
whose vitality is conditioned upon the connection of its parts.

Harvey Jackins

If you don't read poetry how the hell can you solve equations?

William James

The union of the mathematician with the poet,
fervor with measure, passion with correctness,
this surely is the ideal.

James H. Jeans

All the pictures which science now draws of nature and which alone
seem capable of according with observational fact are mathematical
pictures . . . From the intrinsic evidence of his creation, the
Great Architect of the Universe now begins to appear
as a pure mathematician.

Bruce Joyce

We have to reinvent the wheel every once in a while, not because we
need a lot of wheels; but because we need a lot of inventors.

Lord Kelvin

Mathematics is the only good metaphysics.

Johannes Kepler

Who carved the nucleus, before it fell, into six horns of ice?

I believe the geometric proportion served the creator as an idea
when He introduced the continuous generation of similar
objects from similar objects.

J.P. King

One's intellectual and aesthetic life cannot be complete unless it
includes an appreciation of the power and the beauty of mathematics.
Simply put, aesthetic and intellectual fullfillment
requires that you know about mathematics.

Felix Klein

Thus, in a sense, mathematics has been most advanced by those who
distinguished themselves by intuition rather than by rigorous proofs.

Morris Kline

God exists since mathematics is consistent,
and the devil exists since we cannot prove the consistency.

The tantalizing and compelling pursuit of mathematical problems
offers mental absorption, peace of mind amid endless challenges,
repose in activity, battle without conflict,
refuge from the goading urgency of contingent happenings,
and the sort of beauty changeless mountains present
to senses tried by the present-day kaleidoscope of events.

Sophia Kovalevskaya

It is impossible to be a mathematician without being a poet in soul.

Francois Lasserre

Ask a philosopher "What is philosophy?" or a historian "What is
history?", and they will have no difficulty in giving an answer.
Neither of them, in fact, can pursue his own discipline without
knowing what he is searching for. But ask a mathematician "What
is mathematics?" and he may justifiably reply that he does not know
the answer but that this does not stop him from doing mathematics.

Gottfried Leibniz

Music is the pleasure the human mind experiences from counting
without being aware that it is counting.

Taking mathematics from the beginning of the world to the time of Newton,
what he has done is much the better half.

Nicolai Lobachevsky

There is no branch of mathematics, however abstract, which may not
someday be applied to the phenomena of the real world.

John Locke

Logic is the anatomy of thought.

George Mackey

Nowadays, group theoretical methods � expecially those involving
characters and representations, pervade all branches of quantum mechanics.

Thomas Mann

I tell them if they will occupy themselves with the study of mathematics
they will find in it the best remedy against the lusts of the flesh.

James Clerk Maxwell

Every existence above a certain rank has its singular points; the higher
the rank the more of them. At these points, influences whose physical
magnitude is too small to be taken account of by a finite being
may produce results of the greatest importance.

Thomas McCormack

That flower of modern mathematical thought, the notion of a function.

Norman David Mermin

Bridges would be safer if only people who knew the proper definition
of a real number were allowed to design them.

Walter Meyer

In a time when much of the world's geography has been explored,
and space exploration is restricted to astronauts,
mathematics offers fertile ground for exploring the unknown.

John Michel

The mathematical rules of the universe are visible to men
in the form of beauty.

Augustus de Morgan

The moving power of mathematical invention
in not reasoning but imagination.

Napoleon

The advancement and perfection of mathematics
are intimately connected with the prosperity of the State.

James Newman

The theory of groups is a branch of mathematics in which one does
something to something and then compares the results
with the result of doing the same thing to something else,
or something else to the same thing.

Isaac Newton

I do not know what I may appear to the world, but to myself I seem
to have been only a boy playing on the sea shore, and diverting
myself now and then finding a smoother pebble
or a prettier sea shell than ordinary
whilst the great ocean of truth lay all undiscovered before me.

I keep the subject constantly before me and wait
till the first dawnings open little by little into the full light.

If I have seen farther than others, it is because
I have stood on the shoulders of giants.

W.F. Osgood

The calculus is the greatest aid we have to the appreciation
of physical truth in the broadest sense of the word.

Rozso Peter

I love mathematics . . . principally because it is beautiful,
because man has breathed his spirit of play into it,
and because it has given him his greatest game
� the encompassing of the infinite.

Ivars Peterson

To most outsiders, modern mathematics is unknown territory. Its borders
are protected by dense thickets of technical terms; its landscapes
are a mass of indecipherable equations and incomprehensible
concepts. Few realize that the world of modern mathematics
is rich with vivid images and provocative ideas.

J.D. Phillips

Students must learn that mathematics is the most human of endeavors.
Flesh and blood representatives of their own species engaged in
a centuries long creative struggle to uncover and to erect
this magnificent edifice. And the struggle goes on today.
On the very campuses where mathematics is presented and received
as an inhuman discipline, cold and dead, new mathematics
is created. As sure as the tides.

Jean Piaget

A child['s] . . . first geometrical discoveries are topological . . .
If you ask him to copy a square or a triangle, he draws a closed circle.

C.S. Pierce

Mathematics is purely hypothetical: it produces
nothing but conditional propositions.

James Pierpont

The notion of infinity is our greatest friend;
it is also the greatest enemy of our peace of mind.

Charles Pinter

Mathematicians study structure independent of context, and their science
is a voyage of exploration through all the kinds of structure
and order which the human mind is capable of discerning.

Plato

Numbers are the highest degree of knowledge. It is knowledge itself.

Geometry existed before the creation.

God ever geometrizes.

He is unworthy of the name of man who is ignorant that the diagonal
of a square is incommensurate with its side.

[Inscription above Plato's Academy:]
Let no one ignorant of geometry enter here.

Plutarch

The mind is not a vessel to be filled, it is a fire to be kindled.

H. Poincare

Mathematicians do not study objects, but relations among objects;
they are indifferent to the replacement of objects by others
as long as the relations don't change.
Matter is not important, only form interests them.

A first fact should surprise us, or rather would surprise us if we were
not used to it. How does it happen there are people who do not
understand mathematics? If mathematics invokes only the rules
of logic, such as are accepted by all normal minds . . .
how does it come about that so many persons are here refractory?

Proculus

This therefore is Mathematics:
she reminds you of the invisible forms of the soul;
she gives life to her own discoveries;
she awakens the mind and purifies the intellect;
she brings light to our intrinsic ideas;
she abolishes oblivion and ignorance which are ours by birth.

Marcel Proust

The real voyage of discovery consists not in seeking
new landscapes, but in having new eyes.

Pythagoras

Number is the within of all things.

Geometry is knowledge of the eternally existent.

There is geometry in the humming of the strings.

Rudy Rucker

The study of infinity is much more than a dry academic game.
The intellectual pursuit of the absolute infinity is,
as Georg Cantor realized, a form of the soul's quest for God.
Whether or not the goal is ever reached,
an awareness of the process brings enlightenment.

Bertrand Russell

Mathematics takes us still further from what is human into the
region of absolute necessity, to which not only the
actual world, but every possible world, must conform.

The solution of the difficulties which formerly surrounded
the mathematical infinite is probably
the greatest achievement of which our age has to boast.

Mathematics, rightly viewed, possesses
not only truth, but supreme beauty
� a beauty cold and austere, like that of sculpture,
without appeal to any part of our weaker nature,
without the gorgeous trappings of paintings or music,
yet sublimely pure and capable of a stern perfection
such as only the greatest art can show.

Zeno was concerned with three problems . . . These are the problem
of the infinitesimal, the infinite, and continuity . . .
From his to our own day, the finest intellects of each
generation in turn attacked these problems, but achieved
broadly speaking nothing . . . Weierstrass, Dedekind, and Cantor,
. . . have completely solved them. Their . . . solutions are
so clear as to leave no longer the slightest doubt or difficulty.
This achievement is probably the greatest of which our age can boast.

George Sarton

On the basis of my historical experience, I fully believe
that mathematics of the twenty-fifth century will be as different from
that of today as the latter is from that of the sixteenth century.

W.W. Sawyer

In mathematics, if a pattern occurs, we can go on to ask,
Why does it occur? What does it signify? And we can find answers
to these questions. In fact, for every pattern that appears,
a mathematician feels he ought to know why it appears.

Erwin Schrodinger

The idea of the continuum seems simple to us. We have somehow lost sight
of the difficulties it implies . . . We are told such a number as
the square root of 2 worried Pythagoras and his school
almost to exhaustion. Being used to such queer numbers from
early childhood, we must be careful not to for a low idea of the
mathematical intuition of these ancient sages;
their worry was highly credible.

Shakespeare

I could be bounded in a nutshell and count myself a king of infinite space.

It was a sometime paradox, but now time has given it proof.

H.J.S. Smith

Poor teaching leads to the inevitable idea that the subject (mathematics)
is only adapted to peculiar minds, when it is the one universal science
and the one whose . . . ground rules are taught us almost
in infancy and reappear in the motions of the universe.

Joseph J. Sylvester

May not music be described as the mathematics of sense, mathematics
as music of the reason? The musician feels mathematics,
the mathematician thinks music �
music the dream, mathematics the working life.

Alan Turing

Mathematical reasoning may be regarded rather schematically
as the exercise of a combination of two facilities,
which we may call intuition and ingenuity.

Voltaire

There is an astonishing imagination even in the science of mathematics
. . . We repeat, there is far more imagination
in the head of Archimedes than in that of Homer.

Karl Weierstrass

A mathematician who is not also something of a poet
will never be a complete mathematician.

Steven Weinberg

The universe is an enormous direct product
of representations of symmetry groups.

Sir D'Arcy Wentworth

The harmony of the world is made manifest in Form and Number,
and the heart and soul and all the poetry of Natural Philosophy
are embodied in the concept of mathematical beauty.

Hermann Weyl

Besides language and music, [mathematics] is one of the primary
manifestations of the free creative powers of the human mind,
and it is the universal organ for world-understanding
through theoretical construction. Mathematics must therefore
remain an essential element of the knowledge and abilities which
we have to teach, of the culture we have to transmit,
to the next generation.

W.J. White

Mathematics is the science of definiteness,
the necessary vocabulary of those who know.

Alfred North Whitehead

We think of the number "five" as applying to appropriate groups of any
entities whatsoever � to five fishes, five children, five apples,
five days . . . We are merely thinking of those relationships
between those two groups which are entirely independent of the
individual essences of any of the members of either group. This is
is a very remarkable feat of abstraction; and it must have taken
ages for the human race to rise to it.

The pursuit of mathematics is a divine madness of the human spirit.

Algebra is the intellectual instrument which has been created for
rendering clear the quantitative aspects of the world.

A. Zygmund

Mathematics and art are quite different. We could not publish so many
papers that used, repeatedly, the same idea
and still command the respect of our colleagues.

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