Angling may be said to be so like mathematics that it can never be fully
learned.
The Compleat Angler, 1653.
For twenty pages perhaps, he read slowly, carefully, dutifully, with pauses
for self-examination and working out examples. Then, just as it was working
up and the pauses should have been more scrupulous than ever, a kind of
swoon and ecstasy would fall on him, and he read ravening on, sitting
up till dawn to finish the book, as though it were a novel. After that
his passion was stayed; the book went back to the Library and he was done
with mathematics till the next bout. Not much remained with him after
these orgies, but something remained: a sensation in the mind, a worshiping
acknowledgment of something isolated and unassailable, or a remembered
mental joy at the rightness of thoughts coming together to a conclusion,
accurate thoughts, thoughts in just intonation, coming together like unaccompanied
voices coming to a close.
Mr. Fortune's Maggot.
Theology, Mr. Fortune found, is a more accommodating subject than mathematics;
its technique of exposition allows greater latitude. For instance when
you are gravelled for matter there is always the moral to fall back upon.
Comparisons too may be drawn, leading cases cited, types and antetypes
analysed and anecdotes introduced. Except for Archimedes mathematics
is singularly naked of anecdotes.
Mr. Fortune's Maggot.
He resumed:
"In order to ascertain the height of the tree I must be in such a position
that the top of the tree is exactly in a line with the top of a measuring
stickor any straight object would do, such as an umbrellawhich I shall
secure in an upright position between my feet. Knowing then that the
ratio that the height of the tree bears to the length of the measuring
stick must equal the ratio that the distance from my eye to the base of
the tree bears to my height, and knowing (or being able to find out) my
height, the length of the measuring stick and the distance from my eye
to the base of the tree, I can, therefore, calculate the height of the
tree."
"What is an umbrella?"
Mr. Fortune's Maggot.
What if angry vectors veer
Round your sleeping head, and form.
There's never need to fear
Violence of the poor world's abstract storm.
Lullaby in Encounter, 1957.
Every mathematician worthy of the name has experienced ... the state of
lucid exaltation in which one thought succeeds another as if miraculously...
this feeling may last for hours at a time, even for days. Once you have
experienced it, you are eager to repeat it but unable to do it at will,
unless perhaps by dogged work...
The Apprenticeship of a Mathematician.
God exists since mathematics is consistent, and the Devil exists since
we cannot prove it.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and Schmidt, 1977.
Algebra and money are essentially levelers; the first intellectually,
the second effectively.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Prayers for the condemned man will be offered on an adding machine. Numbers
constitute the only universal language.
Miss Lonelyhearts.
Our federal income tax law defines the tax y to be paid in terms of the
income x; it does so in a clumsy enough way by pasting several linear
functions together, each valid in another interval or bracket of income.
An archeologist who, five thousand years from now, shall unearth some
of our income tax returns together with relics of engineering works and
mathematical books, will probably date them a couple of centuries earlier,
certainly before Galileo and Vieta.
The Mathematical Way of Thinking,
an address given at the Bicentennial Conference at the University of Pennsylvania,
1940.
We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
Unterrichtsblätter für Mathematik und Naturwissenschaften, 38, 177-188 (1932). Translation by Abe Shenitzer appeared in The American Mathematical Monthly, v. 102, no. 7 (August-September 1995), p. 646.
A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.
The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity.
1951.
My work has always tried to unite the true with the beautiful and when
I had to choose one or the other, I usually chose the beautiful.
In an obituary by Freeman J. Dyson in Nature, March 10, 1956.
... numbers have neither substance, nor meaning, nor qualities. They are
nothing but marks, and all that is in them we have put into them by the
simple rule of straight succession.
"Mathematics and the Laws of Nature" in The Armchair Science Reader,
New York: Simon and Schuster, 1959.
Without the concepts, methods and results found and developed by previous
generations right down to Greek antiquity one cannot understand either
the aims or achievements of mathematics in the last 50 years.
[Said in 1950]
The American Mathematical Monthly, v. 100. p. 93.
Logic is the hygiene the mathematician practices to keep his ideas healthy
and strong.
The American Mathematical Monthly, November, 1992.
Nobody since Newton has been able to use geometrical methods to the same
extent for the like purposes; and as we read the Principia we feel as
when we are in an ancient armoury where the weapons are of gigantic size;
and as we look at them we marvel what manner of man he was who could use
as a weapon what we can scarcely lift as a burden.
In E. N. Da C. Andrade "Isaac Newton" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The science of pure mathematics ... may claim to be the most original
creation of the human spirit.
Science and the Modern World.
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 greatest science eludes the efforts of our mental weapons to grasp it.
An Introduction to Mathematics
Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about "any" things or about "some" things, without specifications of definite particular things.
So far as the mere imparting of information is concerned, no university
has had any justification for existence since the popularization of printing
in the fifteenth century.
The Aims of Education.
No Roman ever died in contemplation over a geometrical diagram.
[A reference to the death of Archimedes.]
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Life is an offensive, directed against the repetitious mechanism of the
Universe.
Adventures of Ideas, 1933.
There is no nature at an instant.
Let us grant that the pursuit of mathematics is a divine madness of the
human spirit, a refuge from the goading urgency of contingent happenings.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
There is a tradition of opposition between adherents of induction and
of deduction. In my view it would be just as sensible for the two ends
of a worm to quarrel.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
It is a profoundly erroneous truism, repeated by all copy books and by
eminent people when they are making speeches, that we should cultivate
the habit of thinking of what we are doing. The precise opposite is the
case. Civilization advances by extending the number of important operations
which we can perform without thinking about them.
An Introduction to
Mathematics.
Our minds are finite, and yet even in these circumstances of finitude
we are surrounded by possibilities that are infinite, and the purpose
of life is to grasp as much as we can out of that infinitude.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
In modern times the belief that the ultimate explanation of all things
was to be found in Newtonian mechanics was an adumbration of the truth
that all science, as it grows towards perfection, becomes mathematical
in its ideas.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Algebra reverses the relative importance of the factors in ordinary language.
It is essentially a written language, and it endeavors to exemplify in
its written structures the patterns which it is its purpose to convey.
The pattern of the marks on paper is a particular instance of the pattern
to be conveyed to thought. The algebraic method is our best approach
to the expression of necessity, by reason of its reduction of accident
to the ghostlike character of the real variable.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Be relieving the brain of all unnecessary work, a good notation sets it
free to concentrate on more advanced problems, and, in effect, increases
the mental power of the race.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Everything of importance has been said before by somebody who did not
discover it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Seek simplicity, and distrust it.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Fundamental progress has to do with the reinterpretation of basic ideas.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
We think in generalities, but we live in details.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
Apart from blunt truth, our lives sink decadently amid the perfume of
hints and suggestions.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
"Necessity is the mother of invention" is a silly proverb. "Necessity
is the mother of futile dodges" is much nearer the truth.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
It is more important that a proposition be interesting than that it be
true. This statement is almost a tautology. For the energy of operation
of a proposition in an occasion of experience is its interest and is its
importance. But of course a true proposition is more apt to be interesting
than a false one.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
War can protect; it cannot create.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
The progress of Science consists in observing interconnections and in
showing with a patient ingenuity that the events of this ever-shifting
world are but examples of a few general relations, called laws. To see
what is general in what is particular, and what is permanent in what is
transitory, is the aim of scientific thought.
An Introduction to Mathematics.
Through and through the world is infested with quantity: To talk sense
is to talk quantities. It is not use saying the nation is large .. How large?
It is no use saying the radium is scarce ... How scarce? You cannot evade
quantity. You may fly to poetry and music, and quantity and number will
face you in your rhythms and your octaves.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
"One and one make two" assumes that the changes in the shift of circumstance
are unimportant. But it is impossible for us to analyze this notion of
unimportant change.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
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. That
would be claiming too much. 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.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.
In the study of ideas, it is necessary to remember that insistence on
hard-headed clarity issues from sentimental feeling, as it were a mist,
cloaking the perplexities of fact. Insistence on clarity at all costs
is based on sheer superstition as to the mode in which human intelligence
functions. Our reasonings grasp at straws for premises and float on gossamers
for deductions.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Familiar things happen, and mankind does not bother about them. It requires
a very unusual mind to undertake the analysis of the obvious.
Science and the Modern World.
Do I contradict myself? Very well then I contradict myself. (I am large,
I contains multitudes).
Song of Myself, 1939.
When I heard the learn'd astronomer,
When the proofs, the figure, were ranged in columns before me,
When I was shown the charts and diagrams, to add, divide, and measure
them,
When I sitting heard the astronomer where he lectured with much applause
in the
lecture room,
How soon unaccountable I became tired and sick,
Till rising and gliding out I wander'd off by myself,
In the mystical moist night-air, and from time to time,
Look'd up in perfect silence at the stars.
A professor is one who can speak on any subject -- for precisely fifty minutes.
The modern physicist is a quantum theorist on Monday, Wednesday, and Friday and a student of gravitational relativity theory on Tuesday, Thursday, and Saturday. On Sunday he is neither, but is praying to his God that someone, preferably himself, will find the reconciliation between the two views.
Progress imposes not only new possibilities for the future but new restrictions.
The Human Use of Human Beings.
The Advantage is that mathematics is a field in which one's blunders tend
to show very clearly and can be corrected or erased with a stroke of the
pencil. It is a field which has often been compared with chess, but differs
from the latter in that it is only one's best moments that count and not
one's worst. A single inattention may lose a chess game, whereas a single
successful approach to a problem, among many which have been relegated
to the wastebasket, will make a mathematician's reputation.
Ex-Prodigy: My Childhood and Youth.
There is nothing mysterious, as some have tried to maintain, about the
applicability of mathematics. What we get by abstraction from something
can be returned.
Introduction to the Foundations of Mathematics.
Mathematics was born and nurtured in a cultural environment. Without
the perspective which the cultural background affords, a proper appreciation
of the content and state of present-day mathematics is hardly possible.
In The American Mathematical Monthly, March 1994.
[Occam's Razor:]
Entities should not be multiplied unnecessarily.
Quodlibeta.
A monument to Newton! a monument to Shakespeare! Look up to Heaven look into the Human Heart. Till the planets and the passionsthe affections and the fixed stars are extinguishedtheir names cannot die.
We could present spatially an atomic fact which contradicted the laws
of physics, but not one which contradicted the laws of geometry.
Tractatus Logico Philosophicus, New York, 1922.
Mathematics is a logical method ... Mathematical propositions express
no thoughts. In life it is never a mathematical proposition which we
need, but we use mathematical propositions only in order to infer from
propositions which do not belong to mathematics to others which equally
do not belong to mathematics.
Tractatus Logico Philosophicus, New York, 1922, p. 169.
There can never be surprises in logic.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The riddle does not exist. If a question can be put at all, then it can
also be answered.
Tractatus Logico Philosophicus, New York, 1922.
[Mathematics] is an independent world
Created out of pure intelligence.
In things to be seen at once, much variety makes confusion, another vice
of beauty. In things that are not seen at once, and have no respect one
to another, great variety is commendable, provided this variety transgress
not the rules of optics and geometry.
W.H. Auden and L. Kronenberger The Viking Book of Aphorisms, New York: Viking Press, 1966.