There can be no question, however, that prolonged commitment to mathematical
exercises in economics can be damaging. It leads to the atrophy of judgement
and intuition...
Economics, Peace, and Laughter.
[The universe] cannot be read until we have learnt the language and
become familiar with the characters in which it is written. It is written
in mathematical language, and the letters are triangles, circles and other
geometrical figures, without which means it is humanly impossible to comprehend
a single word.
Opere Il Saggiatore p. 171.
Measure what is measurable, and make measurable what is not so.
Quoted in H. Weyl "Mathematics and the Laws of Nature" in I Gordon
and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon
and Schuster, 1959.
And who can doubt that it will lead to the worst disorders when minds
created free by God are compelled to submit slavishly to an outside will?
When we are told to deny our senses and subject them to the whim of others?
When people devoid of whatsoever competence are made judges over experts
and are granted authority to treat them as they please? These are the novelties
which are apt to bring about the ruin of commonwealths and the subversion
of the state.
[On the margin of his own copy of Dialogue on the Great World Systems].
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956, p. 733.
Unfortunately what is little recognized is that the most worthwhile
scientific books are those in which the author clearly indicates what he
does not know; for an author most hurts his readers by concealing difficulties.
In N. Rose (ed.) Mathematical Maxims and Minims, Raleigh NC: Rome
Press Inc., 1988.
Whenever you can, count.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956.
[Statistics are] the only tools by which an opening can be cut through
the formidable thicket of difficulties that bars the path of those who
pursue the Science of Man.
Pearson, The Life and Labours of Francis Galton, 1914.
I know of scarcely anything so apt to impress the imagination as the
wonderful form of cosmic order expressed by the "Law of Frequency
of Error." The law would have been personified by the Greeks and deified,
if they had known of it. It reigns with serenity and in complete self-effacement,
amidst the wildest confusion. The huger the mob, and the greater the apparent
anarchy, the more perfect is its sway. It is the supreme law of Unreason.
Whenever a large sample of chaotic elements are taken in hand and marshaled
in the order of their magnitude, an unsuspected and most beautiful form
of regularity proves to have been latent all along.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956. p. 1482.
Biographical history, as taught in our public schools, is still largely
a history of boneheads: ridiculous kings and queens, paranoid political
leaders, compulsive voyagers, ignorant generals -- the flotsam and jetsam
of historical currents. The men who radically altered history, the great
scientists and mathematicians, are seldom mentioned, if at all.
In G. Simmons Calculus Gems, New York: McGraw Hill, 1992.
Mathematics is not only real, but it is the only reality. That is that
entire universe is made of matter, obviously. And matter is made of particles.
It's made of electrons and neutrons and protons. So the entire universe
is made out of particles. Now what are the particles made out of? They're
not made out of anything. The only thing you can say about the reality
of an electron is to cite its mathematical properties. So there's a sense
in which matter has completely dissolved and what is left is just a mathematical
structure.
Gardner on Gardner: JPBM Communications Award Presentation. Focus-The
Newsletter of the Mathematical Association of America v. 14, no. 6,
December 1994.
I confess that Fermat's Theorem as an isolated proposition has very
little interest for me, because I could easily lay down a multitude of
such propositions, which one could neither prove nor dispose of.
[A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem.]
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956. p. 312.
If others would but reflect on mathematical truths as deeply and as
continuously as I have, they would make my discoveries.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956. p. 326.
There are problems to whose solution I would attach an infinitely greater
importance than to those of mathematics, for example touching ethics, or
our relation to God, or concerning our destiny and our future; but their
solution lies wholly beyond us and completely outside the province of science.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956. p. 314.
You know that I write slowly. This is chiefly because I am never satisfied
until I have said as much as possible in a few words, and writing briefly
takes far more time than writing at length.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.
God does arithmetic.
We must admit with humility that, while number is purely a product of
our minds, space has a reality outside our minds, so that we cannot completely
prescribe its properties a priori.
Letter to Bessel, 1830.
I mean the word proof not in the sense of the lawyers, who set two
half proofs equal to a whole one, but in the sense of a mathematician,
where half proof = 0, and it is demanded for proof that every doubt becomes
impossible.
In G. Simmons Calculus Gems, New York: McGraw Hill inc., 1992.
I have had my results for a long time: but I do not yet know how I
am to arrive at them.
In A. Arber The Mind and the Eye 1954.
[His motto:]
Few, but ripe.
[His second motto:]
Thou, nature, art my goddess; to thy laws my services are bound...
W. Shakespeare King Lear.
[attributed to him by H.B Lübsen]
Theory attracts practice as the magnet attracts iron.
Foreword of H.B Lübsen's geometry textbook.
It is not knowledge, but the act of learning, not possession but the
act of getting there, which grants the greatest enjoyment. When I have
clarified and exhausted a subject, then I turn away from it, in order to
go into darkness again; the never-satisfied man is so strange if he has
completed a structure, then it is not in order to dwell in it peacefully,
but in order to begin another. I imagine the world conqueror must feel
thus, who, after one kingdom is scarcely conquered, stretches out his arms
for others.
Letter to Bolyai, 1808.
Finally, two days ago, I succeeded - not on account of my hard efforts,
but by the grace of the Lord. Like a sudden flash of lightning, the riddle
was solved. I am unable to say what was the conducting thread that connected
what I previously knew with what made my success possible.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber
and Schmidt, 1972.
A great part of its [higher arithmetic] theories derives an additional
charm from the peculiarity that important propositions, with the impress
of simplicity on them, are often easily discovered by induction, and yet
are of so profound a character that we cannot find the demonstrations till
after many vain attempts; and even then, when we do succeed, it is often
by some tedious and artificial process, while the simple methods may long
remain concealed.
In H. Eves Mathematical Circles Adieu, Boston: Prindle, Weber and
Schmidt, 1977.
I am coming more and more to the conviction that the necessity of our
geometry cannot be demonstrated, at least neither by, nor for, the human
intellect...geometry should be ranked, not with arithmetic, which is purely
aprioristic, but with mechanics.
Quoted in J. Koenderink Solid Shape, Cambridge Mass.: MIT Press,
1990.
Lest men suspect your tale untrue,
Keep probability in view.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956. p. 1334.
One of the principal objects of theoretical research in my department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.
Mathematics is a language.
Gilbert, W. S. (1836 - 1911)
I'm very good at integral and differential calculus, I know the scientific
names of beings animalculous; In short, in matters vegetable, animal, and
mineral, I am the very model of a modern Major-General.
The Pirates of Penzance. Act 1.
The mathematician requires tact and good taste at every step of his
work, and he has to learn to trust to his own instinct to distinguish between
what is really worthy of his efforts and what is not.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber
and Schmidt, 1972.
And for mathematical science, he that doubts their certainty hath need
of a dose of hellebore.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956, p. 548.
I don't believe in natural science.
[Said to physicist John Bahcall.]
Ed Regis, Who Got Einstein's Office? Addison Wesley, 1987.
It has been said that figures rule the world. Maybe. But I am sure that
figures show us whether it is being ruled well or badly.
In J. P. Eckermann, Conversations with Goethe.
Mathematics has the completely false reputation of yielding infallible
conclusions. Its infallibility is nothing but identity. Two times two is
not four, but it is just two times two, and that is what we call four for
short. But four is nothing new at all. And thus it goes on and on in its
conclusions, except that in the higher formulas the identity fades out
of sight.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon
and Schuster, 1956, p. 1754.
There are no deep theorems -- only theorems that we have not understood
very well.
The Mathematical Intelligencer, vol. 5, no. 3, 1983.
This is not mathematics, it is theology.
[On being exposed to Hilbert's work in invariant theory.]
Quoted in P. Davis and R. Hersh The Mathematical Experience, Boston:
Birkhäuser, 1981.
Graham, Ronald
It wouild be very discouraging if somewhere down the line you could
ask a computer if the Riemann hypothesis is correct and it said, `Yes,
it is true, but you won't be able to understand the proof.'
John Horgan. Scientific American 269:4 (October 1993) 92-103.
Mathematicians have long since regarded it as demeaning to work on problems
related to elementary geometry in two or three dimensions, in spite of
the fact that it it precisely this sort of mathematics which is of practical
value.
Handbook of Applicable Mathematics.