Mathematics and plausible reasoning
Mathematics and plausible reasoning is a two volume book by the mathematician George Polya describing various methods for being a good guesser of new mathematical results.[1][2] In the Preface to Volume 1 of the book Polya 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 the mathematical induction, 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 |
References
- ↑ Polya, George (1954). Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics. Princeton University Press.
- ↑ Polya, George (1954). Mathematics and Plausible Reasoning Volume II: Patterns of Plausible InferenceInduction and. Prnceton University Press.
- ↑ Halmos, Paul R. (1955). "Review: G. Pólya, Mathematics and plausible reasoning". Bulletin of American mathematical Society 61 (3 Part 1): 243–245. Retrieved 16 February 2015.