mathematics

Mathematics and Plausible Reasoning

Mathematics and Plausible Reasoning is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Mathematics and Plausible Reasoning rather than just read about it. In short: 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. 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.

Key takeaways

Mathematics and Plausible Reasoning belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mathematics and Plausible Reasoning to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mathematics and Plausible Reasoning from memory before moving on to harder problems.

Reference excerpt

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. 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."

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.

Reviews Bernhart, Arthur (1958-01-01). "Review of Mathematics and Plausible Reasoning". The American Mathematical Monthly. 65 (6): 456–457. doi:10.2307/2310741. hdl:2027/mdp.39015008206248. JSTOR 2310741. S2CID 121427033. Rado, Tibor (1956-01-01). "Review of Mathematics and Plausible Reasoning". Philosophy of Science. 23 (2): 167. doi:10.1086/287478. JSTOR 185607. Van Dantzig, D. (1959-01-01). "Review of Mathematics and Plausible Reasoning, G. Pólya". Synthese. 11 (4): 353–358. doi:10.1007/bf00486196. JSTOR 20114312. S2CID 46957889. Broadbent, T. A. A. (1956-01-01). "Review of Mathematics and Plausible Reasoning". The Mathematical Gazette. 40 (333): 233–234. doi:10.2307/3608848. hdl:2027/mdp.39015008206248. JSTOR 3608848. 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. hdl:2027/mdp.39015008206248. JSTOR 1418146. Johansson, I. (1955-01-01). "Review of Mathematics and plausible reasoning, I and II". Nordisk Matematisk Tidskrift. 3 (1): 64–65. JSTOR 24524537. 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. JSTOR 43634251. 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. JSTOR 27954884. 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. JSTOR 2281238. 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. JSTOR 23301574. 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. JSTOR 27967294. 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. JSTOR 3027025.

References

Worked examples

Example 1 — a first encounter with Mathematics and Plausible Reasoning

Start with the simplest possible case. Write down what Mathematics and Plausible Reasoning claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mathematics and Plausible Reasoning before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mathematics and Plausible Reasoning ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mathematics and Plausible Reasoning

In research: Mathematics and Plausible Reasoning appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mathematics and Plausible Reasoning in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mathematics and Plausible Reasoning is common in secondary-school and first-year university syllabi. It links to neighbouring topics Inference, Mathematics books, Reasoning, so understanding it makes those chapters shorter.
In everyday life: Look for Mathematics and Plausible Reasoning outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Mathematics and Plausible Reasoning in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mathematics and Plausible Reasoning means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mathematics and Plausible Reasoning out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mathematics and Plausible Reasoning in simple terms?

Why does Mathematics and Plausible Reasoning matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mathematics and Plausible Reasoning?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mathematics and Plausible Reasoning.