If you disregard the very simplest cases, there is in all of mathematics
not a single infinite series whose sum has been rigorously determined.
In other words,the most important parts of mathematics stand without a
foundation.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 188.
[A reply to a question about how he got his expertise:]
By studying the masters and not their pupils.
[About Gauss' mathematical writing style]
He is like the fox, who effaces his tracks in the sand
with his tail.
In G. F. Simmons, Calculus Gems, New York: Mcgraw Hill, Inc., 1992, p. 177.
The first nonabsolute number is the number of people for whom the table is reserved. This will vary during the course of the first three telephone calls to the restaurant, and then bear no apparent relation to the number of people who actually turn up, or to the number of people who subsequently join them after the show/match/party/gig, or to the number of people who leave when they see who else has turned up.
The second nonabsolute number is the given time of arrival, which is now known to be one of the most bizarre of mathematical concepts, a recipriversexcluson, a number whose existence can only be defined as being anything other than itself. In other words, the given time of arrival is the one moment of time at which it is impossible that any member of the party will arrive. Recipriversexclusons now play a vital part in many branches of math, including statistics and accountancy and also form the basic equations used to engineer the Somebody Else's Problem field.
The third and most mysterious piece of nonabsoluteness of all lies in the relationship between the number of items on the bill, the cost of each item, the number of people at the table and what they are each prepared to pay for. (The number of people who have actually brought any money is only a subphenomenon of this field.)
Numbers written on restaurant bills within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe.
This single statement took the scientific world by storm. It completely revolutionized it. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years.
Life, the Universe and Everything. New York: Harmony Books, 1982.
I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry, and porcelain.
Letter to Abigail Adams, May 12, 1780.
Each generation has its few great mathematicians, and mathematics would
not even notice the absence of the others. They are useful as teachers,
and their research harms no one, but it is of no importance at all. A
mathematician is great or he is nothing.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.
In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.
Reflections: mathematics and creativity, New Yorker, 47(1972), no. 53, 39 - 45.
The mathematical life of a mathematician is short. Work rarely improves
after the age of twenty-five or thirty. If little has been accomplished
by then, little will ever be accomplished.
"Mathematics and Creativity." The New Yorker Magazine, February 19, 1972.
[At a musical concert:]
...the music's pure algebra of enchantment.
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.
In Howard Eves' Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.
Defendit numerus: There is safety in numbers.
In J. R. Newman (ed.) The World of Mathematics,
New York: Simon and Schuster, 1956, p. 1452.
Like the crest of a peacock so is mathematics at the head of all knowledge.
[An old Indian saying. Also, "Like the Crest of a Peacock" is the title of a book by G.G. Joseph]
Referee's report: This paper contains much that is new and much that
is true. Unfortunately, that which is true is not new and that which is new is not true.
In H.Eves Return to Mathematical Circles, Boston: Prindle, Weber, and Schmidt, 1988.
The whole is more than the sum of its parts.
Metaphysica 10f-1045a
The so-called Pythagoreans, who were the first to take up mathematics,
not only advanced this subject, but saturated with it, they fancied that
the principles of mathematics were the principles of all things.
Metaphysica 1-5
It is not once nor twice but times without number that the same ideas
make their appearance in the world.
"On The Heavens", in T.
L. Heath Manual of Greek Mathematics, Oxford: Oxford University Press, 1931.
To Thales the primary question was not what do we know, but how do we
know it.
Mathematical Intelligencer v. 6, no. 3, 1984.
The mathematical sciences particularly exhibit order, symmetry, and limitation;
and these are the greatest forms of the beautiful.
Metaphysica, 3-1078b.
Ascham, Roger (1515-1568)
Mark all mathematical heads which be wholly and only bent on these sciences,
how solitary they be themselves, how unfit to live with others, how unapt
to serve the world.
In E G R Taylor, Mathematical Practitioners
of Tudor and Stuart England, Cambridge: Cambridge University Press, 1954.
Thou shalt not answer questionnaires
Or quizzes upon world affairs,
Nor with compliance
Take any test.
Thou shalt not sit
with statisticians nor commit
A social science.
"Under which lyre" in Collected Poems of W H Auden, London: Faber and Faber.
The good Christian should beware of mathematicians, and all those who
make empty prophecies. The danger already exists that the mathematicians
have made a covenant with the devil to darken the spirit and to confine
man in the bonds of Hell.
DeGenesi ad Litteram, Book II, xviii,
37 [Note: mathematician = astrologer]
If I am given a formula, and I am ignorant of its meaning, it cannot teach
me anything, but if I already know it what does the formula teach me?
De Magistro ch X, 23.