Mathematics and Plausible Reasoning

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Mathematics and Plausible Reasoning
AuthorGeorge Pólya
GenreMathematics

Mathematics and Plausible Reasoning is a two-volume book by the mathematician George Pólya describing various methods for being a good guesser of new mathematical results.[1][2] In the Preface to Volume 1 of the book Pólya exhorts all interested students of mathematics thus: "Certainly, let us learn proving, but also let us learn guessing." P. R. Halmos reviewing the book summarised the central thesis of the book thus: ". . . a good guess is as important as a good proof."[3]

Outline

Volume I: Induction and analogy in mathematics

Polya begins Volume I with a discussion on induction, not mathematical induction, but as a way of guessing new results. He shows how the chance observations of a few results of the form 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, etc., may prompt a sharp mind to formulate the conjecture that every even number greater than 4 can be represented as the sum of two odd prime numbers. This is the well known Goldbach's conjecture. The first problem in the first chapter is to guess the rule according to which the successive terms of the following sequence are chosen: 11, 31, 41, 61, 71, 101, 131, . . . In the next chapter the techniques of generalization, specialization and analogy are presented as possible strategies for plausible reasoning. In the remaining chapters, these ideas are illustrated by discussing the discovery of several results in various fields of mathematics like number theory, geometry, etc. and also in physical sciences.

Volume II: Patterns of Plausible Inference

This volume attempts to formulate certain patterns of plausible reasoning. The relation of these patterns with the calculus of probability are also investigated. Their relation to mathematical invention and instruction are also discussed. The following are some of the patterns of plausible inference discussed by Polya.

Sl. No. Premise 1 Premise 2 Premise 3 plausible conclusion
1 A implies B B is true A is more credible.
2 A implies Bn+1 Bn+1 is very different from the
formerly verified consequences
B1, B2, . . . , Bn of A
Bn+1 true A much more credible
3 A implies Bn+1 Bn+1 is very similar to the
formerly verified consequences
B1, B2, . . . , Bn of A
Bn+1 true A just a little more credible
4 A implies B B is very improbable in itself B is true A very much more credible
5 A implies B B is quite probable in itself B is true A is just a little more credible
6 A analogous to B B is true A is more credible
7 A analogous to B B is more credible A is somewhat more credible
8 A is implied by B B is false A is less credible
9 A is incompatible with B B is false A is more credible

Reviews

  1. Bernhart, Arthur (1958-01-01). "Review of Mathematics and Plausible Reasoning". The American Mathematical Monthly 65 (6): 456–457. doi:10.2307/2310741. 
  2. Rado, Tibor (1956-01-01). "Review of Mathematics and Plausible Reasoning". Philosophy of Science 23 (2): 167–167. doi:10.1086/287478. 
  3. Van Dantzig, D. (1959-01-01). "Review of Mathematics and Plausible Reasoning, G. Pólya". Synthese 11 (4): 353–358. doi:10.1007/bf00486196. 
  4. Broadbent, T. A. A. (1956-01-01). "Review of Mathematics and Plausible Reasoning". The Mathematical Gazette 40 (333): 233–234. doi:10.2307/3608848. 
  5. Bush, Robert R. (1956-01-01). "Review of Mathematics and Plausible Reasoning". The American Journal of Psychology 69 (1): 166–167. doi:10.2307/1418146. 
  6. Johansson, I. (1955-01-01). "Review of Mathematics and plausible reasoning, I and II". Nordisk Matematisk Tidskrift 3 (1): 64–65. 
  7. Prager, W. (1955-01-01). "Review of Mathematics and plausible reasoning. Volume I: Induction and analogy. Volume II: Patterns of plausible inference". Quarterly of Applied Mathematics 13 (3): 344–345. 
  8. Meserve, Bruce E. (1955-01-01). "Review of Induction and Analogy in Mathematics, Vol. I, and Patterns of Plausible Inference, Vol. II, of Mathematics and Plausible Reasoning". The Mathematics Teacher 48 (4): 272–272. 
  9. Savage, Leonard J. (1955-01-01). "Review of Mathematics and Plausible Reasoning. Volume I, Induction and Analogy in Mathematics. Volume II, Patterns of Plausible Inference". Journal of the American Statistical Association 50 (272): 1352–1354. doi:10.2307/2281238. 
  10. פ., א. י. י. (1957-01-01). "Review of Mathematics and Plausible Reasoning. Volume I: Induction and Analogy in Mathematics; Volume II: Patterns of Plausible Reasoning". Iyyun: The Jerusalem Philosophical Quarterly / עיון: רבעון פילוסופי ח' (א'): 48–49. 
  11. Stein, Robert G. (1991-01-01). "Review of Patterns of Plausible Inference. Vol. 2 of Mathematics and Plausible Reasoning (R), George Pólya". The Mathematics Teacher 84 (7): 574–574. 
  12. Alexanderson, G. L. (1979-01-01). "Review of Mathematics and Plausible Reasoning: Vol. I: Induction and Analogy in Mathematics, ; Mathematics and Plausible Reasoning: Vol. II: Patterns of Plausible Inference, George Polya". The Two-Year College Mathematics Journal 10 (2): 119–122. doi:10.2307/3027025. 

References

  1. Polya, George (1954). Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics. Princeton University Press. 
  2. Polya, George (1954). Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Inference. Princeton University Press. 
  3. Halmos, Paul R. (1955). "Review: G. Pólya, Mathematics and plausible reasoning". Bulletin of the American Mathematical Society 61 (3 Part 1): 243–245. doi:10.1090/s0002-9904-1955-09904-x. http://projecteuclid.org/download/pdf_1/euclid.bams/1183519731. Retrieved 16 February 2015.