Mathematicians use variations of the word “formal” in many different, often inconsistent, ways. We talk of “formal proof” and emphasize the value of learning how such formal proofs are put together. But we also talk of “formal” manipulations when a calculation is done without any attempt at rigorous justification. There are the “formal systems” of logic. We also sometimes speak of “formalism” as one of three dominant philosophies of mathematics along with logicism and intuitionism. But what does formalism really mean? Does the question even make sense?

The author tries to set us straight. He begins with a definition:

*Formalism* is the practice of mathematics that is *formal* in the sense that it is

- precise; exact as to definition; or
- done according to established rules; or
- having the appearance if not the essence.

This definition (following the manner of Samuel Johnson in his *Dictionary of the English Language) *helped me not at all. Fortunately, Smoryński also has another characterization based on the motivations behind the formalism. He describes three types: In Type I Formalism, one does manipulations of objects or symbols heuristically, without concern for the validity of one’s manipulations; Type II Formalism arises when one replaces an intuitive notion by a by a more precisely defined concept; Type III Formalism is the modern axiomatic approach that replaces an intuitive notion with a precise delineation of the properties one is allowed to use in dealing with the notion.

Even this characterization is just a bare beginning. But the author goes on to develop and delineate the three types in considerable detail. He does this in four long chapters. The first chapter uses a single example — infinite series — to illustrate the three types of formalism. The following three chapters take up each type of formalism in turn using many examples and drawing on an abundance of historical background. In a short final chapter Smoryński considers criticisms of and dissatisfaction with formalism and the “crisis of intuition”.

I found the treatment of Type I formalism — heuristics and formal manipulation — to be the most interesting part of the book. Part of this derives from the charm of Euler and his methods, and part from a sense that the beginnings of new and creative ideas are strongly tied to Type I activities. Beside Euler’s work, this chapter also discusses Euclid (and his magnitudes), Galileo (and the infinite), Dirac’s delta function, and the treatment of Taylor series before Lagrange.

The treatment of Type II formalism was the most disappointing. Smoryński uses that chapter to develop and construct the various number systems from the natural numbers to the rational, real, complex, and hyperreal numbers. It’s perfectly fine for what it does, but it doesn’t give any perspective on the scope of Type II formalism.

By the time we get to the final chapter of this long book, we find ourselves hoping for a synthesis and a means of understanding how all the pieces of formalism fit together. But it’s not there.

It would also have been nice to see Smoryński’s take on formalism in the current practice of mathematics. Many mathematicians, I would argue, use a mixture of Types I, II, and III formalism, with Type II the default for formal presentation and Type I carefully hidden in the background. I also wonder about how the various formalisms play in the education of a mathematician. Learning mathematics must surely also involve all three types. Rigorous definitions and useful axioms don’t come out of thin air.

A prerequisite for this book is an undergraduate course in analysis. Who are the prospective readers? Anyone who teaches analysis would probably benefit from the perspective. Parts of it would be good supplemental reading for students of analysis. In many ways this is an eccentric book that is enlivened by the author’s enthusiasm and wide interests. It is good — even occasionally very good — locally, but rather flat and without direction on the whole.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.