# The Foundations of Mathematics (2nd ed.)

The idea of this book is to facilitate the transition from "school mathematics" to a more formal rigorous, axiomatic approach of a professional mathematician, or at least of a more advanced study of mathematics. The first edition of the book dates back to 1977, and in that period so-called modern mathematics was stressing much more than today the axiomatic top-down approach to mathematics than is accepted today, and it might have been a reaction of the authors to this trend. They argue that this axiomaticism is neither how mathematics developed historically, nor is it the way how professional mathematicians think and develop new ideas, and hence it should not be the way how new mathematical concepts should be introduced in an educational context. A mathematical problem is approached on an intuitive basis pointing towards a solution, and once the solution has turned out to really work, only then is everything framed in a more formal approach. Precisely this process is what the authors illustrate in this book by describing the proper transition for example from an intuitive understanding of natural numbers to the Peano postulates and beyond.

To achieve this goal, the book has five parts (the first edition had only four). The first part starts by sketching the idea of how the learning process functions for mathematics and continues with a first approach to the real number system which consists in accepting numbers with infinitely many decimals. In Part II, just enough set theory and mathematical logic is introduced to see how one can give a formal definition and how from this a mathematical proof can be derived. The steps followed in such a proof are analyzed and may become after a while routine. From Part III on the more formal approach becomes central. A proof by induction leads to the Peano postulates of the natural numbers while a set theoretic approach leads to larger systems of integers, rationals, and reals. It turns out that these systems form some algebraic structures, and this viewpoint then triggers the introduction of complex numbers and more general structures. The idea of structure theorems is important in part IV because they may give a more imaginative, visual or intuitive interpretation of the axiomatic definition of a structure. This part has several new chapters that were not in the first edition like the one on structure theorems and the one on groups, where group elements may be interpreted as a permutation of the elements of the underlying set. In the first edition, there was a chapter on cardinals, but here, there is an extra chapter extending the reals with the infinites and their reciprocals, the infinitesimals, to form super ordered fields. The latter allow to fall back on ideas of Leibniz and Euler, to introduce concepts of calculus. The previous Part IV has now become Part V and discusses axiomatic set theory. Although the authors use the metaphor of foundations for a building in the text, the cover picture of the book has a stylized tree with crown and roots, which is a good metaphor for mathematics too. If the tree has many branches and leaves, then its root system should be large enough to support these, just like mathematics need strong foundations to support all the branches and applications where it is used.

Also new to this extended edition is an appendix on the `self-explanation' strategy for reading proofs. When reading the lines of a mathematical proof, the student should understand each element in the line, understand why these elements are used, be fully aware how it relates to the previous lines and how it fits in the whole concept of the proof. It has been shown that this strategy has improved the comprehension considerably.

I am not sure how the material of this book could be squeezed into a mathematical curriculum, given the level of mathematical training of students entering the university. There are so many other topics to be seen in a limited amount of time and the pressure of putting many things in the introductory courses coming from application courses or more advanced mathematical ones is often strong. Obviously the material in this book should come at the beginning of the curriculum, while it requires some knowledge from other courses like calculus or algebra. It is on the other hand providing the foundations of number systems, of formalizing, of proof techniques, but it also treats non-standard analysis and Gödel's incompleteness theorems. This makes me doubt that the whole book should be used an introductory course. However, I believe that making a careful selection of the topics can be used as the basis of a course in the first year. Every chapter ends with a number of exercises which are at the level of a student. So that emphasizes the course aspect of the book.

On the other hand, I am convinced that the book is even more useful and should be read by the instructors. They may have had the `wrong' introduction to the foundations and pass it on in the same way, or more dangerously, they had the `proper' introduction and now summarize their insights and provide their students with a direct axiomatic approach, which they think to be a shortcut. Reading this book may warn them to be careful and take the proper approach when they want to pass on their knowledge to fresh students. This is all the more so with this second edition. Where the first edition's main goal was to make the proper transition from intuitive undergraduate to formal graduate mathematics, in this second edition I have the impression that the goal is somewhat broadened and helps to understand and appreciate the mutual influence and the undeniable advantage of the interplay between intuition and axiomatic approaches in a broader mathematical realm.

Some small negative points: There is probably no book with formulas that has no typo at all, but even though I was not particularly scanning for them, I could spot several, while reading. Some a bit more annoying than others, but it is somewhat surprising that they do not only appear in the new chapters but also in the chapters that were also in the first edition. Some examples: On page 121: the name of the set $s=\{x\in\mathbf{R}|x<1\}$ should be a capital $S$; on page 126: $\forall x\in\mathbf{R}: x^2\ge0$ is false, but that is only false if $\ge$ is replaced by $>$; on page 266: $N\in N$ should be $N\in \mathbb{N}$ and it would not harm to have some blank space before the second quantifier, $\forall\varepsilon>0\exists N \in N$ doesn't look nice. On page 336: $x$ is an infinitesimal if $x\ne 0$ and $-r < x < r$ for all $x \in \mathbb{R}$ while it should be for all $r \in \mathbb{R}$; and on page 340: if $e$ is negative then $c < -r < 0$ while the latter should be $e < -r < 0$. Also because of different approaches, the number systems $\mathbf{R},\mathbf{Q},\mathbf{N}$ and $\mathbb{R},\mathbb{Q},\mathbb{N}$ may be considered as different, but the distinction is not made very explicit and could be confusing. Since this book is about formalization of logic and mathematics and there are some sections with historical background, it is a bit surprising that we do not find a reference to Whitehead and Russell with their *Principia Mathematica*, and the approach of the Bourbaki group.

**Submitted by Adhemar Bultheel |

**31 / Dec / 2015