Mathematical Proof Quotes:

Mathematics is not a deductive science, that's a cliché ... What you do is trial and error, experimentation, guesswork.

- Paul Halmos

source: "I Want to be a Mathematician: An Automathography". Book by Paul Halmos, 1985.

topic: Errors, Trials, Trial And Error, Mathematical Proof, Mathematics And Science

We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?"

- Gian-Carlo Rota

topic: Jobs, Writing, Mathematics, Mathematical Proof, Mathematics And Science

I think some intuition leaks out in every step of an induction proof.

- Jim Propp

topic: Thinking, Intuition, Steps, Mathematical Proof

Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.

- Ibn Khaldun

topic: Fall, Errors, Mind, Mathematical Proof, Orderly

... mathematical knowledge ... is, in fact, merely verbal knowledge. "3" means "2+1", and "4" means "3+1". Hence it follows (though the proof is long) that "4" means the same as "2+2". Thus mathematical knowledge ceases to be mysterious.

- Bertrand Russell

topic: Mean, Numbers, Long, Mathematical Proof

Mathematics consists in proving the most obvious thing in the least obvious way.

- George Polya

topic: Way, Mathematics, Obvious, Mathematical Logic, Mathematical Proof

I have had my results for a long time: but I do not yet know how I am to arrive at them.

- Carl Friedrich Gauss

source: "The Mind and the Eye: A Study of the Biologist's Standpoint". Book by A. Arber, 1954.

topic: Time, Math, Science, Mathematical Proof, Funny Science

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.

- Hermann Weyl

source: Hermann Weyl (2013). “Levels of Infinity: Selected Writings on Mathematics and Philosophy”, p.33, Courier Corporation

topic: Simple, Simplicity, Principles, Mathematical Proof, Setups

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.

- Carl Friedrich Gauss

source: Letter to Heinrich Wilhelm Matthias Olbers on May 14, 1826. "Calculus Gems". Book by George F. Simmons, 1992.

topic: Mean, Math, Two, Mathematical Proof

Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional. In E. T. Bell Men of Mathematics, New York: Simona and Schuster, 1937.

- Charles Proteus Steinmetz

topic: New York, Men, Doe, Mathematical Proof

Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.

- David Hilbert

topic: Believe, Errors, Simplicity, Mathematical Proof, Rigour

To divide a cube into two other cubes, a fourth power, or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.

- Pierre de Fermat

source: Scribbled note in the margins of his copy of Diophantus's Arithmetica. He did not live to provide the promised proof, and the conjecture became famous as Fermat's Last Theorem. In 1993 Andrew Wiles, a British mathematician, claimed to have discovered the proof.

topic: Power, Two, Cubes, Mathematical Proof, Dividing

A mathematical proof is beautiful, but when you're finished, it's really only about one thing. A story can be about many things.

- Bonnie Jo Campbell

topic: Beautiful, Stories, Proof, Mathematical Proof

A mathematical proof must be perspicuous.

- Ludwig Wittgenstein

source: Ludwig Wittgenstein, Gertrude Elizabeth Margaret Anscombe, Rush Rhees, Georg Henrik Wright (1967). “Remarks on the foundations of mathematics”

topic: Mathematics, Proof, Mathematical, Mathematical Proof

A proof tells us where to concentrate our doubts.

- Morris Kline

source: Morris Kline (1964). “Mathematics in Western Culture”, p.478, Oxford University Press

topic: Doubt, Proof, Mathematical Proof

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.

- Eric Temple Bell

source: Mathematics Magazine, Vol. 23-24, (p. 161), 1949.

topic: Intelligence, Taught, Assumption, Mathematical Proof, Euclid

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.

- Hermann Weyl

source: Hermann Weyl (2013). “Levels of Infinity: Selected Writings on Mathematics and Philosophy”, p.33, Courier Corporation

topic: Ideas, Feelings, Links, Mathematical Proof, Overview

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? Where does the proof use the hypothesis?

- Paul Halmos

source: "I Want to be a Mathematician: An Automathography". Book by Paul Halmos, 1985.

topic: Fighting, Example, Doe, Mathematical Proof

A mathematician's reputation rests on the number of bad proofs he has given.

- Abram Samoilovitch Besicovitch

topic: Numbers, Reputation, Proof, Mathematical Proof, Numbers And Math

Nothing has afforded me so convincing a proof of the unity of the Deity as these purely mental conceptions of numerical and mathematical science which have been by slow degrees vouchsafed to man, and are still granted in these latter times by the Differential Calculus, now superseded by the Higher Algebra, all of which must have existed in that sublimely omniscient Mind from eternity.

- Mary Somerville

source: Kathryn A. Neeley, Mary Somerville (2001). “Mary Somerville: Science, Illumination, and the Female Mind”, p.180, Cambridge University Press

topic: Men, Unity, Mind, Mathematical Proof, Differential Calculus

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 truth. This made him in love with geometry.

- John Aubrey

topic: Love, Years, Gentleman, Mathematical Proof, Hobbes

Absence of proof isn't proof of absence.

- Robert Buettner

source: Robert Buettner (2005). “Orphan's Destiny”

topic: Atheist, Absence, Proof, Mathematical Proof

Absence of proof is not proof of absence.

- William Cowper

topic: Inspirational, Motivational, Suicide, Mathematical Proof

A witty statesman said, you might prove anything by figures.

- Thomas Carlyle

source: Chartism ch. 2 (1839) See Disraeli 38

topic: Witty, Math, Science, Mathematical Proof

It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out.

- Emil Artin

source: Emil Artin (2016). “Geometric Algebra”, p.14, Courier Dover Publications

topic: Proof, Mathematical, Mathematical Proof

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..."

- John Edensor Littlewood

source: John Edensor Littlewood, Béla Bollobás (1986). “Littlewood's Miscellany”, p.61, Cambridge University Press

topic: Friendship, Wish, Ramanujan, Mathematical Proof, Integers

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind.

- Lord Kelvin

source: Popular Lectures and Addresses "Electrical Units of Measurement" (1889).

topic: Science, Numbers, Kind, Mathematical Proof, Mathematics And Science

No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof.

- Leonardo da Vinci

topic: Ontology, Tests, Logic, Mathematical Proof

In a way, composing on the melodic level is an expression of a melodic truth, almost like a geometric truth. If it has clarity, other people will recognize it. There's no way of isolating it in a gallery on a white wall and saying, "This is a work of art. This is a mathematical proof."

- Eyvind Kang

source: Source: pitchfork.com

topic: Art, Wall, Expression, Mathematical Proof, White Walls

Poverty is a mathematical proof of the fact that mankind is a big failure!

- Mehmet Murat Ildan

topic: Poverty, Facts, Proof, Mathematical Proof

Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning.

- John Locke

topic: Math, Proof, Diamond, Math Is Like, Mathematical Proof

I had made an empirical discovery and it carried all the weight of a mathematical proof.

- Paul Auster

topic: Science, Discovery, Weight, Mathematical Proof

What a mathematical proof actually does is show that certain conclusions, such as the irrationality of , follow from certain premises, such as the principle of mathematical induction. The validity of these premises is an entirely independent matter which can safely be left to philosophers.

- Timothy Gowers

source: Timothy Gowers (2010). “Mathematics”, p.50, Sterling Publishing Company, Inc.

topic: Independent, Doe, Matter, Mathematical Proof

As far as I know, Clifford Pickover is the first mathematician to write a book about areas where math and theology overlap. Are there mathematical proofs of God? Who are the great mathematicians who believed in a deity? Does numerology lead anywhere when applied to sacred literature? Pickover covers these and many other off-trail topics with his usual verve, humor, and clarity. And along the way the reader will learn a great deal of serious mathematics.

- Martin Gardner

topic: Book, Writing, Math, Mathematical Proof, Proof Of God

There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was "style" and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a "style" that is different from that of experiments made with giant accelerators.

- J. A. Ratcliffe

topic: Science, Simple, Thinking, Mathematical Proof, Involving