Theoremus

In Theoremus: A Student’s Guide to Mathematical Proofs, Lito Perez Cruz addresses a subject that has been covered many times elsewhere (that is the theoretical and practical aspects of proving mathematical statements), but he does so in a unique, energetic and playful style.

 

The book is divided into two parts. The first, which makes up about two-fifths of the work, consists of five chapters. In the first two very short chapters, the author offers some answers (with a touch of humour) to questions like: What is an argument? What is a proof? Why do mathematicians have this culture of proving? Why is this important to mathematics? What is a theorem? What distinguishes a lemma from a theorem or a corollary?

 

From the third chapter onwards, the tone becomes more serious and the author takes a more structured approach to introduce the reader to different forms of proofs, such as if–then proofs, if-and-only-if proofs, universal statements and existence results. An overview of the rudiments of mathematical logic is then given.

 

While it is true that practice makes perfect, it can also be very useful to spend some time observing a master at work before starting that solo practice. With this in mind, the author thought it wise to expose the reader to various examples of proofs (direct, by contraposition, by contradiction, by case and by induction) from elementary number theory and set theory.

 

In the second part – described, oddly enough, by the author as optional reading – the foundations of propositional logic and first-order logic are again reviewed, but in much greater depth. The author introduces the reader to the linear diagrammatic representation for natural deduction developed by the American logician Frederic Brenton Fitch and the tree structure proposed by the German mathematician Gerhard Gentzen. The text is embellished with a considerable number of figures intended to facilitate understanding. However, the figures suffer from two flaws: 1) low resolution makes some of the figures blurry; and 2) textual or symbolic inscriptions are sometimes superimposed on the line segments within the diagrams, which complicates reading.

 

All in all, Part I of this book is a bit overly simplistic for such a complex topic. In Part II, however, the author’s pace is too brisk and his explanations too terse to adequately serve as a reference book for undergraduate mathematics students. Moreover, despite thirty or so boxes entitled “Reflection” in which the reader’s attention is drawn to a particular point requiring attention, there are no practice exercises or problems to solve. Hence, the audience most likely to benefit from reading this compact book is undergraduate students in non-mathematics programs with a strong mathematical component, such as physics, engineering, bioinformatics, and computer science.

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Frederic Morneau-Guerin is a professor in the Department of Education at Universite TELUQ. He holds a Ph.D. in abstract harmonic analysis.