Theorems Quotes:

One cannot really argue with a mathematical theorem.

- Stephen Hawking

topic: Arguing, Mathematical, Theorems

How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!

- Archimedes

source: Archimedes (2007). “The Works of Archimedes”, p.151, Cosimo, Inc.

topic: Firsts, Geometry, Theorems

If we knew what it was we were doing, it would not be called research, would it?

- Albert Einstein

topic: Love, Inspirational, Life, Science Research, Scientific Research

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

If only I had the Theorems! Then I should find the proofs easily enough.

- Bernhard Riemann

topic: Enough, Proof, Should, Mathematics By Mathematicians, Theorems

A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories.

- Stefan Banach

source: "Banach Spaces and Their Applications in Analysis: Proceedings of the International Conference at Miami University, May 22-27, 2006, in Honor of Nigel Kalton's 60th Birthday" by Beata Randrianantoanina, Narcisse Randrianantoanina, Walter de Gruyter, (p. 5), 2007.

topic: Analogies, Proof, Theory, Theorems

Young men should prove theorems, old men should write books.

- G. H. Hardy

topic: Book, Writing, Men, Theorems, Youth And Old Age

I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions - they are not just repetitions of each other.

- Michael Atiyah

source: "Interview with Michael Atiyah and Isadore Singer". Interview with Martin Raussen, Christian Skau, www.researchgate.net. February 2005.

topic: Thinking, Law, Two, Quadratics, Theorems

Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or "mathematics envy.

- Marvin Minsky

topic: Believe, Simple, Thinking, Simple Truths, Theorems

A mathematician is a device for turning coffee into theorems.

- Paul Erdos

topic: Witty, Coffee, Humorous, Math And Love, Math Love

There are three signs of senility. The first sign is that a man forgets his theorems. The second sign is that he forgets to zip up. The third sign is that he forgets to zip down.

- Paul Erdos

topic: Men, Firsts, Zips, Theorems, Senility

Economists often like startling theorems, results which seem to run counter to conventional wisdom.

- Joseph Stiglitz

source: Autobiographical Essay for the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, www.nobelprize.org. 2001.

topic: Running, Wisdom, Results, Conventional Wisdom, Theorems

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.

- George Polya

topic: Independent, Ideas, Numbers, Theorems

God has the Big Book, the beautiful proofs of mathematical theorems are listed here.

- Paul Erdos

topic: Beautiful, Book, Math, Big Book, Theorems

One geometry cannot be more true than another; it can only be more convenient.

- Henri Poincare

source: Henri Poincare (2012). “The Value of Science: Essential Writings of Henri Poincare”, p.85, Modern Library

topic: Mathematics, Geometry, Theorems

The principle is so perfectly general that no particular application of it is possible.

- George Polya

topic: Knowledge, Principles, Application, Culmination, Differential Equations

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

It gives me the same pleasure when someone else proves a good theorem as when I do it myself.

- Edmund Landau

topic: Giving, Pleasure, Prove, Theorems

You can learn to find unknowns in equations, draw equidistant lines and demonstrate theorems, but in real life there's nothing to position, calculate, or guess.

- Delphine de Vigan

source: Delphine de Vigan (2010). “No and Me”, p.67, A&C Black

topic: Real, Lines, Equations, Theorems

The "seriousness" of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects.

- G. H. Hardy

topic: Lying, Ideas, Mathematics, Theorems

In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.

- Martin Gardner

topic: Beautiful, Simple, Dull, Theorems, Analogue

A felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem.

- Atle Selberg

topic: May, Mathematics, Proof, Conjecture, Theorems

The missing piece in his stomach hurt so much-and eventually he stopped thinking about the Theorem and wondered only how something that isn't there can hurt you.

- John Green

source: John Green (2008). “An Abundance of Katherines”, p.98, Penguin

topic: Hurt, Thinking, Missing, Hurt So Much, Missing Piece

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples...

- John B. Conway

topic: Example, Purpose, Mathematics, Theorems

If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.

- George Polya

topic: Truth, Mean, Trying, Verify, Theorems

Combinatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum and ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late 18th century.

- Richard Askey

topic: Science, Germany, Analysis, Theorems

Share prices follow the theorem: hope divided by fear minus greed.

- Dominic Lawson

topic: Greed, Share, Minus, Theorems, Share Price

The great poem and the deep theorem are new to every reader, and yet are his own experiences, because he himself recreates them.

- Jacob Bronowski

topic: Justice, Social Justice, Reader, Theorems

There are infinitely many variations of the initial situation and therefore no doubt indefinitely many theorems of moral geometry.

- John Rawls

topic: Doubt, Variation, Moral, Theorems

Papers should include more side remarks, open questions, and such. Very often, these are more interesting than the theorems actually proved. Alas, most people are afraid to admit that they don't know the answer to some question, and as a consequence they refrain from mentioning the question, even if it is a very natural one. What a pity! As for myself, I enjoy saying 'I do not know'.

- Jean-Pierre Serre

source: Jean-Pierre Serre (2000). “Oeuvres, Collected Papers: 1985-1998”, Springer Verlag

topic: People, Interesting, Answers, Theorems

Heaven is angered by my arrogance; my proof [of the four-color theorem] is also defective.

- Hermann Minkowski

topic: Color, Heaven, Arrogance, Theorems

Toward the end of his life, Gödel feared that he was being poisoned, and he starved himself to death. His theorem is one of the most extraordinary results in mathematics, or in any intellectual field in this century. If ever potential mental instability is detectable by genetic analysis, an embryo of someone with Kurt Gödel's gifts might be aborted.

- Brian L. Silver

source: Brian L. Silver (2000). “The Ascent of Science”, p.506, Oxford University Press, USA

topic: Theorems

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.

- Stephen Leacock

topic: Struggle, Fall, Math, Quadratic Equation, Quadratics

I have found a very great number of exceedingly beautiful theorems.

- Pierre de Fermat

topic: Beautiful, Numbers, Found, Theorems

Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.

- Ivars Peterson

topic: Islands, Pieces, Unanswered Questions, Unanswered, Theorems

...contemporary physicists come in two varieties. Type 1 physicists are bothered by EPR and Bell's Theorem. Type 2 (the majority) are not, but one has to distinguish two subvarieties. Type 2a physicists explain why they are not bothered. Their explanations tend either to miss the point entirely (like Born's to Einstein) or to contain physical assertions that can be shown to be false. Type 2b are not bothered and refuse to explain why.

- David Mermin

topic: Science, Two, Missing, Not Bothered, Theorems

This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped.

- Emil Artin

source: Emil Artin (2016). “Geometric Algebra”, p.7, Courier Dover Publications

topic: Science, Important, Doe, Theorems

The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task.

- G. H. Hardy

topic: Science, Tasks, Raw Materials, Euclid, Theorems

An axiomatic system comprises axioms and theorems and requires a certain amount of hand-eye coordination before it works. A formal system comprises an explicit list of symbols, an explicit set of rules governing their cohabitation, an explicit list of axioms, and, above all, an explicit list of rules explicitly governing the steps that the mathematician may take in going from assumptions to conclusions. No appeal to meaning nor to intuition. Symbols lose their referential powers; inferences become mechanical.

- David Berlinski

topic: Eye, Hands, Intuition, Cohabitation, Theorems

The scientist has to take 95 per cent of his subject on trust. He has to because he can't possibly do all the experiments, therefore he has to take on trust the experiments all his colleagues and predecessors have done. Whereas a mathematician doesn't have to take anything on trust. Any theorem that's proved, he doesn't believe it, really, until he goes through the proof himself, and therefore he knows his whole subject from scratch. He's absolutely 100 per cent certain of it. And that gives him an extraordinary conviction of certainty, and an arrogance that scientists don't have.

- Christopher Zeeman

topic: Believe, Giving, Arrogance, Theorems

Humanism . . . is not a single hypothesis or theorem, and it dwells on no new facts. It is rather a slow shifting in the philosophic perspective, making things appear as from a new centre of interest or point of sight.

- William James

source: Dr. William James (2013). “The William James Reader”, p.102, Simon and Schuster

topic: Sight, Perspective, Shifting, Philosophic, Theorems

We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experiences, because he himself re-creates them. They are the marks of unity in variety; and in the instant when the mind seizes this for itself, in art or in science, the heart misses a beat.

- Jacob Bronowski

source: Jacob Bronowski (2011). “Science and Human Values”, p.18, Faber & Faber

topic: Art, Discovery, Missing, Theorems

That's the problem with false proofs of true theorems; it's not easy to produce a counterexample.

- Jeffrey Shallit

topic: Problem, Easy, Proof, Theorems

The story does what no theorem can quite do. It may not be "like real life" in the superficial sense: but it sets before us an image of what reality may well be like at some more central region.

- C. S. Lewis

source: C. S. Lewis (2002). “Of Other Worlds: Essays and Stories”, p.27, Houghton Mifflin Harcourt

topic: Real, Doe, Stories, Theorems

Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.

- Carl Friedrich Gauss

topic: Choices, Justified, Introduction, Theorems

When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached. It is this way that in mathematics speculative theorems and practical canons are reduced by analysis to definitions, axioms and postulates.

- Gottfried Leibniz

topic: Ideas, Words Of Wisdom, Analysis, Theorems

A theorem is a proposition which is a strict logical consequence of certain definitions and other propositions

- Anatol Rapoport

source: "Various meanings of 'theory'". American Political Science Review, 52.04, 1958.

topic: Definitions, Logical, Certain, Theorems

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.

- Henri Poincare

topic: Thinking, Law, Doubt, Theorems

The Mean Value Theorem is the midwife of calculus - not very important or glamorous by itself, but often helping to deliver other theorems that are of major significance.

- Edward Mills Purcell

topic: Mean, Important, Statistics, Theorems

Professor Eddington has recently remarked that 'The law that entropy always increases - the second law of thermodynamics - holds, I think, the supreme position among the laws of nature'. It is not a little instructive that so similar a law [the fundamental theorem of natural selection] should hold the supreme position among the biological sciences.

- Ronald Fisher

source: "The Genetical Theory of Natural Selection". Book by Ronald Fisher, 1930.

topic: Natural Selection, Theorems, Biological Science

I took a break from acting for four years to get a degree in mathematics at UCLA, and during that time I had the rare opportunity to actually do research as an undergraduate. And myself and two other people co-authored a new theorem: Percolation and Gibbs States Multiplicity for Ferromagnetic Ashkin-Teller Models on Two Dimensions, or Z2.

- Danica McKellar

topic: Opportunity, Two, Ucla, Undergraduate, Rare Opportunities