I do hate sums. There is no greater mistake than to call arithmetic an
exact science. There are permutations and aberrations discernible to
minds entirely noble like mine; subtle variations which ordinary accountants
fail to discover; hidden laws of number which it requires a mind like
mine to perceive. For instance, if you add a sum from the bottom up,
and then from the top down, the result is always different.
Mathematical Gazette, v. 12.
The reader will find no figures in this work. The methods which I set
forth do not require either constructions or geometrical or mechanical
reasonings: but only algebraic operations, subject to a regular and uniform
rule of procedure.
Preface to Mécanique Analytique.
[said about the chemist Lavoisier:]
It took the mob only a moment to remove his head; a century will not suffice to reproduce it.
H. Eves An Introduction to the History of Mathematics, 5th Ed., Saunders.
When we ask advice, we are usually looking for an accomplice.
That sometimes clear ... and sometimes vague stuff ... which is ... mathematics.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
Most of the arts, as painting, sculpture, and music, have emotional appeal
to the general public. This is because these arts can be experienced
by some one or more of our senses. Such is not true of the art of mathematics;
this art can be appreciated only by mathematicians, and to become a mathematician
requires a long period of intensive training. The community of mathematicians
is similar to an imaginary community of musical composers whose only satisfaction
is obtained by the interchange among themselves of the musical scores
they compose.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
[Asked for a testimony to the effect that Emmy Noether was a great woman
mathematician, he said:]
I can testify that she is a great mathematician, but that she is a woman,
I cannot swear.
J.E. Littlewood, A Mathematician's Miscellany, Methuen and Co ltd., 1953.
There's a touch of the priesthood in the academic world, a sense that a scholar should not be distracted by the mundane tasks of day-to-day living. I used to have great stretches of time to work. Now I have research thoughts while making peanut butter and jelly sandwiches. Sure it's impossible to write down ideas while reading "curious George" to a two-year-old. On the other hand, as my husband was leaving graduate school for his first job, his thesis advisor told him, "You may wonder how a professor gets any research done when one has to teach, advise students, serve on committees, referee papers, write letters of recommendation, interview prospective faculty. Well, I take long showers."
In Her Own Words: Six Mathematicians Comment on Their Lives and Careers. Notices of the AMS, V. 38, no. 7 (September 1991), p. 704.
He uses statistics as a drunken man uses lamp posts -- for support rather
than illumination.
Treasury of Humorous Quotations.
[about Fourier]
It was, no doubt, partially because of his very disregard for rigor that
he was able to take conceptual steps which were inherently impossible
to men of more critical genius.
In P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.
A good calculator does not need artificial aids.
Tao Te Ching, ch 27.
What we know is not much. What we do not know is immense.
(Allegedly his last words.)
DeMorgan's Budget of Paradoxes.
[His last words, according to De Morgan:]
Man follows only phantoms.
DeMorgan's Budget of Paradoxes.
Nature laughs at the difficulties of integration.
In J. W. Krutch "The Colloid and the Crystal", in I. Gordon and S. Sorkin (eds.) The Armchair Science Reader, New York: Simon and Schuster, 1959.
Read Euler: he is our master in everything.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Such is the advantage of a well constructed language that its simplified
notation often becomes the source of profound theories.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Napoleon: You have written this huge book on the system of the world
without once mentioning the author of the universe.
Laplace: Sire, I
had no need of that hypothesis.
Later when told by Napoleon about the
incident, Lagrange commented: Ah, but that is a fine hypothesis. It
explains so many things.
DeMorgan's Budget of Paradoxes.
[said about Napier's logarithms:]
...by shortening the labors doubled the
life of the astronomer.
In H. Eves In Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
It is India that gave us the ingenious method of expressing all numbers
by means of ten symbols, each symbol receiving a value of position as
well as an absolute value; a profound and important idea which appears
so simple to us now that we ignore its true merit. But its very simplicity
and the great ease which it has lent to computations put our arithmetic
in the first rank of useful inventions; and we shall appreciate the grandeur
of the achievement the more when we remember that it escaped the genius
of Archimedes and Apollonius, two of the greatest men produced by antiquity.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
How can a modern anthropologist embark upon a generalization with any
hope of arriving at a satisfactory conclusion? By thinking of the organizational
ideas that are present in any society as a mathematical pattern.
Rethinking Anthropology. 1961.
How can you shorten the subject? That stern struggle with the multiplication
table, for many people not yet ended in victory, how can you make it less?
Square root, as obdurate as a hardwood stump in a pasturenothing but years
of effort can extract it. You can't hurry the process.
Or pass from arithmetic to algebra; you can't shoulder your way past quadratic
equations or ripple through the binomial theorem. Instead, the other
way; your feet are impeded in the tangled growth, your pace slackens,
you sink and fall somewhere near the binomial theorem with the calculus
in sight on the horizon. So died, for each of us, still bravely fighting,
our mathematical training; except for a set of people called "mathematicians" -- born
so, like crooks.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1988.
In my opinion, a mathematician, in so far as he is a mathematician, need
not preoccupy himself with philosophy -- an opinion, moreover, which has
been expressed by many philosophers.
Scientific American, 211, September 1964, p. 129.
In one word he told me the secret of success in mathematics: plagiarize
only be sure always to call it please research.
Lobachevski (A musical recording.)
[about him:]
It is rare to find learned men who are clean, do not stink
and have a sense of humour.
[attributed variously to Charles Louis de Secondat Montesquieu and to
the Duchess of Orléans]
Nothing is more important than to see the sources of invention which are,
in my opinion more interesting than the inventions themselves.
J. Koenderink, Solid Shape, Cambridge Mass.: MIT Press, 1990.
Music is the pleasure the human soul experiences from counting without
being aware that it is counting.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.
He who understands Archimedes and Apollonius will admire less the achievements
of the foremost men of later times.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
In symbols one observes an advantage in discovery which is greatest when
they express the exact nature of a thing briefly and, as it were, picture
it; then indeed the labor of thought is wonderfully diminished.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
The art of discovering the causes of phenomena, or true hypothesis, is
like the art of decyphering, in which an ingenious conjecture greatly
shortens the road.
New Essays Concerning Human Understanding, IV, XII.
Although the whole of this life were said to be nothing but a dream and
the physical world nothing but a phantasm, I should call this dream or
phantasm real enough, if, using reason well, we were never deceived by
it.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
The soul is the mirror of an indestructible universe.
The Monadology.
Whoever despises the high wisdom of mathematics nourishes himself on delusion
and will never still the sophistic sciences whose only product is an eternal
uproar.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics.
Notebooks, v. 1, ch. 20.
He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast.
No human investigation can be called real science if it cannot be demonstrated mathematically.
Inequality is the cause of all local movements.
But leaving those of the Body, I shall proceed to such Recreation as adorn
the Mind; of which those of the Mathematicks are inferior to none.
Pleasure with Profit, 1694.
All mathematical laws which we find in Nature are always suspect to me,
in spite of their beauty. They give me no pleasure. They are merely
auxiliaries. At close range it is all not true.
In J P Stern Lichtenberg, 1959.
The great trick of regarding small departures from the truth as the truth
itself -- on which is founded the entire integral calculus -- is also the basis
of our witty speculations, where the whole thing would often collapse
if we considered the departures with philosophical rigour.
Aphorisms.
In mathematical analysis we call x the undetermined part of line a: the rest we don't call y, as we do in common life, but a-x. Hence mathematical language has great advantages over the common language.
I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, "Oh, I see. That's how it must be." This is a sign that they explain it to themselves from within their own system.
Who has not be amazed to learn that the function y = e^x , like a phoenix
rising again from its own ashes, is its own derivative?
Great Currents of Mathematical Thought, vol. 1, New York: Dover Publications.
[On the Gaussian curve, remarked to Poincaré:]
Experimentalists think
that it is a mathematical theorem while the mathematicians believe it
to be an experimental fact.
In D'Arcy Thompson On Growth and Form, 1917.
It is true that I should have been surprised in the past to learn that
Professor Hardy had joined the Oxford Group. But one could not say the
adverse chance was 1:10. Mathematics is a dangerous profession; an
appreciable proportion of us go mad, and then this particular event would
be quite likely.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
A good mathematical joke is better, and better mathematics, than a dozen
mediocre papers.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
I recall once saying that when I had given the same lecture several times
I couldn't help feeling that they really ought to know it by now.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
In passing, I firmly believe that research should be offset by a certain
amount of teaching, if only as a change from the agony of research. The
trouble, however, I freely admit, is that in practice you get either no
teaching, or else far too much.
"The Mathematician's Art of Work" in Béla Bollobás (ed.) Littlewood's Miscellany, Cambridge: Cambridge University Press, 1986.
It is possible for a mathematician to be "too strong" for a given occasion.
he forces through, where another might be driven to a different, and possible
more fruitful, approach. (So a rock climber might force a dreadful crack,
instead of finding a subtle and delicate route.)
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
I constantly meet people who are doubtful, generally without due reason,
about their potential capacity [as mathematicians]. The first test is
whether you got anything out of geometry. To have disliked or failed
to get on with other [mathematical] subjects need mean nothing; much drill
and drudgery is unavoidable before they can get started, and bad teaching
can make them unintelligible even to a born mathematician.
A Mathematician's Miscellany, Methuen and Co. ltd., 1953.
The infinitely competent can be uncreative.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
In presenting a mathematical argument the great thing is to give the educated
reader the chance to catch on at once to the momentary point and take
details for granted: his successive mouthfuls should be such as can be
swallowed at sight; in case of accidents, or in case he wishes for once
to check in detail, he should have only a clearly circumscribed little
problem to solve (e.g. to check an identity: two trivialities omitted
can add up to an impasse). The unpractised writer, even after the dawn
of a conscience, gives him no such chance; before he can spot the point
he has to tease his way through a maze of symbols of which not the tiniest
suffix can be skipped.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
A linguist would be shocked to learn that if a set is not closed this
does not mean that it is open, or again that "E is dense in E" does not
mean the same thing as "E is dense in itself".
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
The surprising thing about this paper is that a man who could write it
would.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
A precisian professor had the habit of saying: "... quartic polynomial
ax^4+bx^3+cx^2+dx+e , where e need not be the base of the natural
logarithms."
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
I read in the proof sheets of Hardy on Ramanujan: "As someone said, each
of the positive integers was one of his personal friends." My reaction
was, "I wonder who said that; I wish I had." In the next proof-sheets
I read (what now stands), "It was Littlewood who said..."
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.
We come finally, however, to the relation of the ideal theory to real
world, or "real" probability. If he is consistent a man of the mathematical
school washes his hands of applications. To someone who wants them he
would say that the ideal system runs parallel to the usual theory: "If
this is what you want, try it: it is not my business to justify application
of the system; that can only be done by philosophizing; I am a mathematician".
In practice he is apt to say: "try this; if it works that will justify
it". But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
A Mathematician's Miscellany, Methuen Co. Ltd, 1953.
The theory of numbers is particularly liable to the accusation that some
of its problems are the wrong sort of questions to ask. I do not myself
think the danger is serious; either a reasonable amount of concentration
leads to new ideas or methods of obvious interest, or else one just leaves
the problem alone. "Perfect numbers" certainly never did any good, but
then they never did any particular harm.
A Mathematician's Miscellany, Methuen Co. Ltd., 1953.
There is no branch of mathematics, however abstract, which may not some
day be applied to phenomena of the real world.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc., 1988.
...mathematical proofs, like diamonds, are hard and clear, and will be touched
with nothing but strict reasoning.
D. Burton, Elementary Number Theory, Boston: Allyn and Bacon 1980.
Medicine makes people ill, mathematics make them sad and theology makes them sinful.