“The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.”

— George Pólya

Induction and Analogy in Mathematics (1954)

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George Pólya 35

Hungarian mathematician 1887–1985

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“Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. …let us learn proving, but also let us learn guessing.”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)

“In plausible reasoning the principal thing is to distinguish… a more reasonable guess from a less reasonable guess.”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)

“The methods of mathematics are the main topic of the course, not a long list of finished mathematical results with such highly polished proofs that the poor student can only marvel at the results, with no hope of understanding how mathematics is actually created by practicing mathematicians.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)

“The efficient use of plausible reasoning is a practical skill and it is learned… by imitation and practice.”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)
Context: The efficient use of plausible reasoning is a practical skill and it is learned... by imitation and practice.... what I can offer are only examples for imitation and opportunity for practice.

“Anything new that we learn about the world involves plausible reasoning”

George Pólya (1887–1985) Hungarian mathematician

Induction and Analogy in Mathematics (1954)
Context: Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs.

“If we compare. e. g. the systems of classical mathematics and of intuitionistic mathematics, we find that the first is much simpler and technically more efficient, while the second is more safe from surprising occurences, e. g. contradictions. At the present time, any estimation of the degree of safety of the system of classical mathematics, in other words, the degree of plausibility of its principles, is rather subjective. The majority of mathematicians seem to regard this degree as sufficiently high for all practical purposes and therefore prefer the application of classical mathematics to that of intuitionistic mathematics. The latter has not, so far as I know, been seriously applied in physics by anybody.”

Rudolf Carnap (1891–1970) German philosopher

Rudolf Carnap (1939; 51), as cited in: Paul van Ulsen. Wetenschapsfilosofie http://www.illc.uva.nl/Research/Publications/Inaugurals/IV-10-Arend-Heyting.text.pdf, 6 november 2017.

“Empirical evidence can never establish mathematical existence--nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful.”

Richard Courant (1888–1972) German American mathematician (1888-1972)

Richard Courant in: The Parsimonious Universe, Stefan Hildebrandt & Anthony Tromba, Springer-Verlag, 1996, page 148

“The moving power of mathematical invention is not reasoning, but imagination.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Quoted in Robert Perceval Graves, The Life of Sir William Rowan Hamilton, Vol. 3 (1889), p. 219.

“However, in all honesty, I must say that one must essentially forget that all proofs are transcribed in this formal language. In order to think productively, one must use all the intuitive and informal methods of reasoning at one's disposal. At the very end one must check that no errors have been committed; but in practice set theory is treated as any other branch of mathematics. The reason that we can do this is that we will never speak about proofs but only about models.”

Paul Cohen (1934–2007) American mathematician

p. 1078 of "The discovery of forcing." http://www.logic.univie.ac.at/~ykhomski/ST2013/The%20Discovery%20of%20Forcing.pdf Rocky Mountain Journal of Mathematics 32, no. 4 (2002): 1071–1100.

“I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Principles of Mathematics (1903), Ch. II: Symbolic Logic, p. 11
1900s

“The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.”

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