This is a combined review of *Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics* and *Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Inference*. These books deal with how mathematical facts are discovered (sometimes called heuristic). If we are math students, usually the facts are told to us and we discover the proofs; if we are math researchers we have to discover our own facts and hopefully prove them. This book focuses on both kinds of discovery.

The books are a sequel to Pólya’s much better known work *How to Solve It*. The present books have a different nature, and I think are aimed at a different audience. They deal with much more difficult mathematical problems (generally at the college level rather than high school) and delve much more deeply into the discovery process. The first volume is subtitled “Induction and Analogy in Mathematics.” Induction here is not the familiar mathematical induction, but induction as used more generally in the sciences: examining several or many instances and guessing a general law that describes them. There are many exercises, some asking for theorems or proofs and some reflective. The examples and

exercises are taken from many areas of mathematics, and most exercises will not be familiar. Most of the exercises are not difficult to work out, but they do require insight and experimentation, which is what the book is teaching.

The areas chosen are those that lend themselves to generating many examples, and include number theory, geometry, and calculus. For example, Chapter IV opens with a study of integer right triangles, that is, right triangles whose sides are all integers. We organize the study by the size of the hypotenuse. It turns out that integer right triangles are not that common; the only examples with a hypotenuse less than or equal 20 are 5, 10, 13, 15, and 17. The study then continues with attempting to discover what is special about these sizes, and developing some general hypotheses which we might then prove. Chapter III is a detailed investigation of polyhedra, leading to Euler’s Polyhedron Formula \( V − E + F = 2 \) and explorations of how far its validity goes. It’s not quite as in-depth as Lakatos’s study in *Proofs and Refutations*, but it goes farther than most books. A couple of the almost-hidden gems in this volume are evaluating definite integrals by differentiating a parameterized version under the integral sign (p. 23) and geometric reflection as a quick way of seeing a solution, such as the path of a billiard ball (p. 160) and geometrical optimization (p. 172 ff).

Volume II is subtitled “Patterns of Plausible Inference” and has a very different nature from the first volume: it is much less mathematical and much more philosophical, specifically epistemological. Much of the book deals with evaluating evidence and looking at considerations that make a conjecture more believable or less believable. Thus the first book focuses on how to come up with ideas for theorems and for proofs, while the second book focuses on how to evaluate the likelihood of these ideas being correct. There’s an especially long chapter XIV on probability, which includes a lot of technical material (not just philosophy in this one) and the cautionary title “Chance, the Ever-present Rival Conjecture”. One of the gems in this volume is Póya’s proof (p. 147) of Carleman’s inequality that for non-negative \( a_{1}, a_{2}, \ldots a_{n} \) we have \( \sum_{n=1}^{\infty} ( a_{1}a_{2} \cdots a_{n})^{1/n} < e \sum_{n=1}^{\infty} a_{n} \). It has a short and mysterious proof, but the book carefully leads you through how it was discovered and makes the proof much more motivated.