Yuri Manin is a renowned mathematician with a broad spectrum of intellectual interests. He has made important contributions to algebraic geometry, number theory, diophantine geometry and mathematical physics, just to mention some. But Manin’s interests are not restricted to mathematics or physics, in particular he has had a lasting interest on *language*: From natural languages, e.g. Russian, to the formal languages of Physics and Mathematics. Thus, it shouldn’t come as a surprise that when Manin writes a book on Mathematical Logic and addresses this book to working mathematicians, we should pay attention to what he has to say and how he does it.

To begin with, since Manin is sharing his thoughts with a reader who is supposed to be mathematically sophisticated, some of his examples, motivations and digressions require some level of mathematical knowledge. Manin’s main concern, from the very beginning is the notion of *truth*, or to be more precise, the notion of *mathematical truth*. The first part of the book is devoted to the analysis of this concept, with emphasis on its meaning in several contexts or languages. The book starts with an introduction to formal languages, specifying the corresponding alphabets and rules for forming expressions with them, culminating with the uniqueness of the reading of any formula in formal languages with parentheses. Several well-chosen examples illustrate the concepts and results being discussed, and opportune remarks add the proper context of the discussion. For example, when the author observes that most natural or formal languages are linear, viewing a text as a long line of symbols, he notes that this is something that can be perceived as a drawback, since there are objects, such as graphs, where the natural description is nonlinear. Manin illustrates this remark with the so-called *snake lemma* of homological algebra, where given a commutative diagram of abelian groups and homomorphisms, the connecting morphism that it produces is better understood graphically, and its linear description, say in a text, is, at least, cumbersome. This remark and example highlight the unique approach taken by the author to introduce the reader to the realm of mathematical logic, from the perspective of a mathematician that has though long and deep about its subject and is at ease with the philosophical issues that come along.

The firsts two chapters, about 102 pages, set the tone for the whole book: It starts with a quick review of the syntax of first-order formal languages, its interpretations and the syntactic properties of truth, then moves to Boolean algebras, Gödel’s completeness theorem and the Löwenheim-Skolem theorem. These are staples of any book on logic, but the author has enriched them with asides, for example with a digression on the notion of *proof* that includes a discussion on the accepting of a proof as a *social act.* There is a discussion of the interesting and somehow controversial issue of *computer assisted proofs* and of when the *length of a proof* becomes so large that the possibility of errors is not negligible. This digression on proof ends with a delightful note regarding mechanical proofs, either by machine or by hand, quoting a paper by Mumford (*Inv. Math.* **3** (1967), p. 230) where Mumford writes that it took him several hours of “ghastly, but straightforward computations” to check that certain function was well defined, but in the end he was no *wiser* and decided to omit the details in the published paper. Manin then concludes that the moral of this is that a *good proof* is one that makes us *wiser*!

The first part of the book has also two chapters on chosen topics in set theory. Chapter III treats the continuum hypothesis and forcing, and Chapter IV Gödel’s proof of the consistency of the continuum hypothesis. As was observed before, Manin’s emphasis is on semantics, leaving to section 6 of the chapter a brief survey of the syntactic version of Gödel’s theory. The last section of Chapter IV enriches the previous systematic exposition with some remarks on the mathematical meaning of the question: What is the cardinality of the continuum?

The second part of the book, on computability, starts with a chapter on recursive functions, shifting our attention from the concept of mathematical truth to computational or algorithmic processes. This chapter begins with an intuitive discussion of the notion of algorithm and then makes this notion precise by introducing computable functions, partial recursive functions, and formulating Church’s thesis. The central result is a characterization of recursively enumerable sets. In the next chapter the important notion of Diophantine sets is introduced, and the rest of the chapter is devoted to proving Matiyasevic’s theorem: All enumerable sets are Diophantine. Many of the ideas and results obtained in this chapter are due to Manin, for example the use of Pell’s equation *x*^{2 }*– dy*^{2 }= 1 to construct a special Diophantine set that is crucial for the proof of the theorem.

The third part of the book, on provability and computability, has a chapter devoted to Gödel’s incompleteness theorem, formulated semantically: In a first order theory including Peano arithmetic, there is a true formula that is not provable. The next chapter treats some group-theoretical aspects related to the word problem, the main result being a theorem of G. Higman (*Proc. Royal Soc. Ser. A*, **262** (1961), 455-475) characterizing recursive groups that are finitely presented. All necessary preliminaries on free products and HNN extensions are included.

The first edition of this book (Springer, 1977) included a section in Chapter II on Quantum Logic, with a brisk summary of the relevant aspects of quantum mechanics, following Kochen and Specker (*J. Math. Mech.* **17 **(1967), 59-87), to prove a version of von Neumann’s theorem that there are no hidden variables in this formulation of quantum mechanics. Quantum logic in its modern guise, quantum computation, is retaken in the second part of Chapter IX, new to this second edition, including a discussion of the notion of entanglement and Peter Shor’s quantum algorithm that efficiently factors a given integer. The first part of Chapter IX introduces the basic ideas of the language of category theory, having in mind that this new (as compared to set theory) language could replace set theory as *the* language of mathematics. Again, the author does not take up only the now usual elementary aspects of category theory, but goes deeply into some recent trends in *homotopical algebra* that could be relevant for the foundations of mathematics.

For the second edition of this remarkable book, in addition to some minor corrections on the first eight chapters, a new section on Chapter IV, and the new chapter IX already mentioned, there is a new Part IV written by B. Zilber, devoted to model theory, something that was conspicuously missing in the first edition. In this chapter we find the standard topics: The compactness theorem, saturation, elimination of quantifiers in some theories, e.g., in the theory of algebraically closed fields (Tarski’s theorem). But again, this being the book that it is, there are some interesting mathematical applications. For example, as a corollary to Tarski’s theorem, a strong version of Lefschetz principle is obtained: An algebraic-geometry property proven in the context of complex algebraic geometry also holds for any algebraically closed field of characteristic zero. This opens the door to an introduction to a relatively new approach to some topics in Algebraic or Diophantine Geometry, allowing the author to mention, for example, Hrushovski’s proof of Lang’s conjecture. Moreover, here the author can introduce a topological component to some concepts originating in mathematical logic and give a quick introduction to the *Zariski geometries* of Hrushovski, Pillay and Zilber, where the closed sets on a structure are the subsets that are positively quantifier-free definable. Examples of these Zariski geometries are, of course, the class of smooth algebraic varieties over an algebraically closed field. Less obvious examples of Zariski geometries are the class of compact complex manifolds (which are essentially non algebraic) and the solution spaces of one-variable differential equations over differentiable closed fields, where in positive characteristic differentiation is in the sense of Hasse. This should give at least an idea of the new powerful methods that mathematical logic has provided to Algebraic and Diophantine Geometry.

It is clear that the authorial intention of presenting mathematical logic as mainstream mathematics is more than fulfilled. Manin’s book is a wonderful and original panorama of mathematics from the standpoint of logic, and not only a textbook on mathematical logic. It belongs on the bookshelf of any mathematician with even a slight interest in the meaning and unity of mathematics.

Lastly, I should mention that this new edition seems to have been completely retyped, and a few typos were introduced. For example, in page 37, line *–*7, where “t is not true” should be “it is not true”; in page 39 line 16, t_{1}, t_{2, }should be t_{1}= t_{2}; in page 52, line 11, a “prime” is missing just before the word “operation” (notation for the operation of rank one); in page 52 the footnote should be attached to the word *kenning*; in page 97 line 11, the decimal point is missing in the expansion of π; in pages 37, 40 and 41 the references to Mendelson’s book are to the first edition and the book is not listed in the bibliography or referenced in text.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is [email protected]