Mathematical Quotations -- A

Mathematical Quotations -- A

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Abel, Niels H. (1802 - 1829)

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.

Adams, Douglas (1952 - 2001)

Bistromathics itself is simply a revolutionary new way of understanding the behavior of numbers. Just as Einstein observed that space was not an absolute but depended on the observer's movement in space, and that time was not an absolute, but depended on the observer's movement in time, so it is now realized that numbers are not absolute, but depend on the observer's movement in restaurants.
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.)
Life, the Universe and Everything. New York: Harmony Books, 1982.

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.

Adams, John (1735 - 1826)

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.

Adler, Alfred

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.

Aiken, Conrad

[At a musical concert:]
...the music's pure algebra of enchantment.

Allen, Woody

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.

Anglin, W.S.

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
"Mathematics and History", Mathematical Intelligencer, v. 4, no. 4.


If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.
In Ivor Thomas "Greek Mathematics" in J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

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.

Arbuthnot, John

The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning; and when they cannot it's a sign our knowledge of them is very small and confus'd; and when a Mathematical Reasoning can be had it's as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you.
Of the Laws of Chance. (1692)

Aristophanes (ca 444 - 380 BC)

Meton: With the straight ruler I set to work
To make the circle four-cornered
[First(?) allusion to the problem of squaring the circle]

Aristotle (ca 330 BC)

Now that practical skills have developed enough to provide adequately for material needs, one of these sciences which are not devoted to utilitarian ends [mathematics] has been able to arise in Egypt, the priestly caste there having the leisure necessary for disinterested research.
Metaphysica, 1-981b

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.

Aubrey, John (1626-1697)

[About Thomas Hobbes:]
He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition . "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.
In O. L. Dick (ed.) Brief Lives, Oxford: Oxford University Press, 1960, p. 604.

Auden, W. H. (1907-1973)

How happy the lot of the mathematician. He is judged solely by his peers, and the standard is so high that no colleague or rival can ever win a reputation he does not deserve.
The Dyer's Hand, London: Faber & Faber, 1948.

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.

Augarten, Stan

Computers are composed of nothing more than logic gates stretched out to the horizon in a vast numerical irrigation system.
State of the Art: A Photographic History of the Integrated Circuit. New York: Ticknor and Fields.

St. Augustine (354-430)

Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist.
The City of God.

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.

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