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Standard mathematics has recently been rendered obsolete by the discovery
that for years we have been writing the numeral five backward. This has led to
reevaluation of counting as a method of getting from one to ten. Students are
taught advanced concepts of Boolean algebra, and formerly unsolvable equations
are dealt with by threats of reprisals.

Woody Allen, quoted in Howard Eves,

*Return to Mathematical Circles * (1988)

It is quite possible that mathematics was invented in the ancient Middle East
to keep track of tax receipts and grain stores. How odd that out of this should
come a subtle scientific language that can effectively describe and predict the
most arcane aspects of the Universe.

Isaac Asimov

One of the finest creations of the human mind is mathematics, for not only is
it the apotheosis of rational thought but it is also the spine that renders
scientific speculation sufficiently rigid to confront experience. Scientific
hypotheses themselves are like jelly; they need the rigidity of mathematical
formulation if they are to stand up to experimental verification and fit into
the network of concepts that constitute physical science.

Peter Atkins, *Galileo’s Finger: The Ten Great Ideas of Science* (2003)

“Arithmetic: The Limits of Reason”

For he who knows not mathematics cannot know any other science; what is more,
he cannot discover his own ignorance, or find its proper remedy.

Roger Bacon

Mathematics is the door and the key to the sciences.

Roger Bacon, * Opus Majur*, transl Robert Belle Burke

The greatest shortcoming of the human race is our inability to understand the
exponential function.

Albert A. Bartlett

Aftermath: The horrible headache you have when you've finished the algebra
test.

Tom Batiuk, * Funky Winkerbean * (comic strip, date unknown)

"Obvious" is the most dangerous word in mathematics.

Eric Temple Bell

The straightest line between a short distance is two points.

George Carlin, * Brain Droppings * (1997)

Recent polls reveal that some people have never been polled. Until recently.

George Carlin, * Brain Droppings * (1997)

The Nobel Prize in mathematics was awarded yesterday to a California
professor who has discovered a new number. The number is "bleen,"
which he says belongs between six and seven.

George Carlin, * Napalm & Silly Putty * (2001)

'I couldn't afford to learn it.' said the Mock Turtle with a sigh. 'I only took
the regular course.'

'What was that?' inquired Alice.

'Reeling and Writhing, of course, to begin with,' the Mock Turtle replied; 'and
then the different branches of Arithmetic — Ambition, Distraction, Uglification,
and Derision.'

'I never heard of "Uglification,"' Alice ventured to say. 'What is
it?'

The Gryphon lifted up both its paws in surprise. 'What! Never heard of
uglifying!' it exclaimed. 'You know what to beautify is, I suppose?'

'Yes,' said Alice doubtfully: 'it means — to — make — anything — prettier.'

'Well, then,' the Gryphon went on, 'if you don't know what to uglify is, you
* are*
a simpleton.'

Lewis Carroll, * Alice's Adventures in Wonderland * (1865)

You can be moved to tears by numbers — provided they are encoded and decoded
fast enough.

Richard Dawkins, * River Out of Eden:
*

As far as the properties of mathematics refer to reality, they are not
certain: and as far as they are certain, they do not refer to reality.

Albert Einstein

Statistics: the science of producing unreliable facts from reliable figures.

Evan Esar

If you took a sample of one million children and counted their digits at
birth, you would find that the vast majority of them had twenty in all, while
some might have more and some less. If you displayed this digital variation
graphically, you would come up with what statisticians call a normal curve. . .
. A practicing doctor would call all of the ten-toe-ten-finger kids normal. In
fact, they are a statistical mean.

Anne Fausto-Sterling, in * How Things Are: A Science Tool-Kit
*

A handle is topologically equivalent to a doughnut. (Hence the campus joke,
"A topologist is someone who doesn't know the difference between a coffee
cup and a doughnut.")

Timothy Ferris, * The Whole Shebang:
*

There are 10^{11} stars in the galaxy. That used to be a huge number. But it's only
a hundred billion. It's less than the national deficit! We used to call them
astronomical numbers. Now we should call them economical numbers.

Richard Feynman

If you are interested in the ultimate character of the physical world, or the
complete world, and at the present time our only way to understand that is
through a mathematical type of reasoning, then I don't think a person can fully
appreciate, or in fact can appreciate much of, these particular aspects of the
world, the great depth of character of the universality of the laws, the
relationships of things, without an understanding of mathematics. I don't know
any other way to do it, we don't know any other way to describe it accurately .
. . or to see the interrelationships without it. So I don't think a person who
hasn't developed some mathematical sense is capable of fully appreciating this
aspect of the world — don't misunderstand me, there are many, many aspects of
the world that mathematics is unnecessary for, such as love, which are very
delightful and wonderful to appreciate and to feel awed and mysterious about;
and I don't mean to say that the only thing in the world is physics, but you
were talking about physics and if that's what you're talking about, then to not
know mathematics is a severe limitation in understanding the world.

Richard Feynman, "The Pleasure of Finding Things Out"

(interview, BBC, *Horizon*, 1981; shown in US on *Nova*);

reprinted in *The Pleasure of Finding Things Out: The Best
Short
*

The Universe is a grand book of philosophy. The book lies continually open to
man's gaze, yet none can hope to comprehend it who has not first mastered the
language and the characters in which it has been written. This language is
mathematics; these characters are triangles, circles, and other geometric
figures.

Galileo Galilei

If the objects of mathematics are in some sense "out there,"
waiting to be discovered by human minds, permanently embedded in the structure
of the cosmos in the way the "form" of a vase is essential to the
material vase, then what is surprising about the fact that having extracted mathematics from the universe, of which our brain are a part, we are able to put
it back in?

Martin Gardner, * Weird Water & Fuzzy Logic:
*

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."

Paul R. Halmos, in * I want to be a Mathematician *(1985)

317 is a prime, not because we think so, or because our minds are shaped in
one way rather than another but because it *is* so. Mathematical reality is
built that way.

Godfrey Harold Hardy

Beauty is the first test; there is no permanent place in the world for ugly
mathematics.

Godfrey Harold Hardy, * A Mathematician's Apology * (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 our notes of
our observations.

Godfrey Harold Hardy, * A Mathematician's Apology * (1941)

... 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.

Carl G. Hempel, "Geometry and Empirical Science"

in J. R. Newman (ed.), *
The World of Mathematics * (1956)

Nobody knows why, but the only theories which work are the mathematical ones.

Michael Holt, * Mathematics in Art*

Mathematics: A tentative agreement that two and two make four.

Elbert Hubbard

A table of random numbers, once printed, requires no errata.

Mark Kak

I often say that when you can measure what you are speaking about, and
express it in numbers, you know something about it; but when you cannot express
it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be
the beginning of knowledge, but you have scarcely, in your thoughts, advanced to
the stage of Science, whatever the matter may be.

William Thomson, Lord Kelvin, Lecture at the

Institution of Civil Engineers (3
May 1883)

Do not imagine that mathematics is hard and crabbed, and repulsive to common
sense. It is merely the etherialization of common sense.

William Thomson, Lord Kelvin, in S. P.

Thompson, * Life of Lord Kelvin * (1910)

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 countryside and
then noting the automobile road up the other side!

Robert J. Kleinhenz

In real life, I assure you, there is no such thing as algebra.

Fran Lebowitz

... I find that abstract symbols on the blackboard automatically stimulate
the organ that secretes eye-glaze juice. If, for instance, I write * x = vt * (read:
ex equals vee times tee), a gasp arises in the lecture hall. It isn't that these
brilliant children of parents paying $20,000 tuition per year cannot deal with * x
= vt*. Give them numbers for * x * and * t * and ask them to solve for
*v*, and 48 percent
would get it right, 15 percent would refuse to answer on advice of counsel, and
5 percent would vote present. (Yes, I know doesn't add up to 100. But I'm an
experimenter not a theorist. Besides, dumb mistakes give my class confidence.)

Leon Lederman, * The God Particle: If the Universe is the
*

A good mathematical joke is better, and better mathematics, than a dozen
mediocre papers.

J. E. Littlewood, * A Mathematician's Miscellany * (1953)

Medicine makes people ill, mathematics make them sad and theology makes them
sinful.

Martin Luther

Strange as it may sound, the power of mathematics rests on its evasion of all
unnecessary thought and on its wonderful saving of mental operations.

Ernst Mach

A talk in mathematics should be one of four things: beautiful, deep,
surprising... or short.

Michel Mendès France, remark, c. 1986

A tendency to drastically underestimate the frequency of coincidence is a
prime characteristic of innumerates, who generally accord great significance to
correspondences of all sorts while attributing too little significance to quite
conclusive but less flashy statistical evidence.

John Allen Paulos, * Innumeracy: Mathematical
*

Consider a precise number that is well known to generations of parents and
doctors: the normal human body temperature of 98.6° Fahrenheit. Recent
investigations involving millions of measurements have revealed that this number
is wrong; normal human body temperature is actually 98.2° Fahrenheit. The
fault, however, lies not with Dr. Wunderlich's original measurements — they were
averaged and sensibly rounded to the nearest degree: 37° Celsius. When this
temperature was converted to Fahrenheit, however, the rounding was forgotten,
and 98.6 was taken to be accurate to the nearest tenth of a degree. Had the
original interval between 36.5° Celsius and 37.5° Celsius been translated, the
equivalent Fahrenheit temperatures would have ranged from 97.7° to 99.5°.
Apparently, dyscalculia can even cause fevers.

John Allen Paulos, * A Mathematician Reads the Newspaper
* (1995)

In a math course you learn whether or not there's a solution; in a physics
course you find the solution.

P. A. Piroue

Let no one ignorant of geometry enter my door.

Plato

Mathematics consists of proving the most obvious thing in the least obvious
way.

George Polyá, in N. Rose, * Mathematical
*

One of the endearing things about mathematicians is the extent to which they
will go to avoid doing any real work.

Matthew Pordage, in H. Eves, * Return
*

Pure mathematics consists entirely of such asservations as that, if such and
such a proposition is true of anything, then such and such another proposition
is true of that thing. It is essential not to discuss whether the first
proposition is really true, and not to mention what the anything is of which it
is supposed to be true. . . . If our hypothesis is about anything and not about
some one or more particular things, then our deductions constitute mathematics.
Thus mathematics may be defined as the subject in which we never know what we
are talking about, nor whether what we are saying is true.

Bertrand Russell, "Recent Work on the Principles of

Mathematics" in *
International Monthly*, vol. 4, p. 84 (1901)

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

Bertrand Russell, * The Study of Mathematics * (1902)

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.

Bertrand Russell, * Principles of Mathematics * (1903)

The true spirit of delight, the exaltation, the sense of being more than an,
which is the touchstone of the highest excellence, is to be found in mathematics
as surely as in poetry.

Bertrand Russell, * Mysticism and Logic * (1917)

That arithmetic is the basest of all mental activities is proved by the fact
that it is the only one that can be accomplished by a machine.

Arthur Schopenhauer

Numerology is the easiest — and consequently the most dangerous — method for
finding patterns. It is easy because anybody can do it, and dangerous for the
same reason. The difficulty lies in distinguishing significant numerical
patterns from accidental ones. ... The big problem with numerological
pattern-seeking is that it generates millions of accidentals for each universal.
Nor is it always obvious which is which.

Ian Stewart, * Nature's Numbers: The Unreal
*

Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics
is brilliant at helping us to solve puzzles. It is a more or less systematic way
of digging out the rules and structures that lie behind some observed pattern or
regularity, and then using those rules and structures to explain what's going
on. Indeed, mathematics has developed alongside our understanding of nature,
each reinforcing the other.

Ian Stewart, * Nature's Numbers: The Unreal
*

... mathematics is the science of patterns, and nature exploits just about
every pattern there is.

Ian Stewart, * Nature's Numbers: The Unreal
*

Chaos is overturning our comfortable assumptions about now the world works.
It tells us that the universe is far stranger than we think. It casts doubt on
many traditional methods of science: merely knowing the laws of nature is no
longer enough. On the other hand, it tells us that some things that we thought
were just random may actually be consequences of simple laws. Nature's chaos is
bound by rules. In the past, science tended to ignore events or phenomena that
seemed random, on the grounds that since they had no obvious patterns they could
not be governed by simple laws. Not so. There are simple laws right under our
noses — laws governing disease epidemics, or heart attacks, or plagues of
locusts. If we learn those laws, we may be able to prevent the disasters that
follow in their wake.

Ian Stewart, * Nature's Numbers: The Unreal
*

A circle is the longest distance to the same point.

Tom Stoppard

Math was always my bad subject. I couldn't convince my teachers that many of
my answers were meant ironically.

Calvin Trillin

Sometimes, half a dozen figures will reveal, as with a lightning-flash, the
importance of a subject which ten thousand labored words, with the same purpose
in view, had left at last but dim and uncertain.

Mark Twain, * Life on the Mississippi * (1883)

There are no sects in geometry.

Voltaire

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

Voltaire, * Dictionnaire Philosophique * (1764)

Anyone who considers arithmetical methods of producing random digits is, of
course, in a state of sin.

John von Neumann

In Mathematics, you don't understand things; you just get used to them.

John von Neumann

All science as it grows toward perfection becomes mathematical in its ideas.

Alfred North Whitehead

I will not go so far as to say that to construct a history of thought without
profound study of the mathematical ideas of successive epochs is like omitting
Hamlet from the play which is named after him. . . But it is certainly analogous
to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia
is quite essential to the play, she is very charming — and a little mad.

Alfred North Whitehead

The study of mathematics is apt to commence in disappointment. ... We are
told that by its aid the stars are weighed and the billions of molecules in a
drop of water are counted. Yet, like the ghost of Hamlet's father, this great
science eludes the efforts of our mental weapons to grasp it.

Alfred North Whitehead, * An Introduction to Mathematics* (1911)

The science of pure mathematics, in its modern developments, may claim to be
the most original creation of the human spirit.

Alfred North Whitehead, * Science and the Modern World* (1925)

Factorials were someone's attempt to make math * look* exciting.

Steven Wright

I'm kinda tired. I was up all night trying to round off infinity.

Steven Wright

In Vegas, I got into a long argument with the man at the roulette wheel over
what * I* considered to be an odd number.

Steven Wright

There are only two kinds of modern mathematics books — those you cannot read
beyond the first page and those you cannot read beyond the first sentence.

C. N. Yang