Theorem

A theorem in mathematics is a statement, that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.

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• It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a court of justice, without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel.
  • Isaac Barrow. Mathematical Lectures, London, 1734, p. 64; as cited in: Robert Edouard Moritz, Memorabilia mathematica or, The philomath's quotation-book, Published 1914. p. 18

• There is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master.
  • Eric Temple Bell, The Development of Mathematics (1940). Ch.3 Firmly established, pp.52

• Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.
  • 'I have discovered a truly remarkable proof of this theorem which this margin is too small to contain.
  • Pierre de Fermat; Note written on the margins of his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus, this was taken as an indication of what became known as Fermat's last theorem, a correct proof for which would be found only 357 years later; as quoted in Number Theory in Science and Communication (1997) by Manfred Robert Schroeder

• The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around.
  • Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly 87 (2), February 1980, pp. 81-90

• A “good theorem,” as Tate puts it, lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect.
  • Steve Nadis (1 November 2013). A History in Sum. Harvard University Press. p. 207. ISBN 978-0-674-72655-0. 

• Bell's theorem is the most profound discovery of science.
  • Henry P. Stapp, "Bell's Theorem and World Process", Nuovo Cimento, Vol. 29B, No. 2, p. 270 (1975).

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