Where do numbers come from

There is a modern trend in calculus courses to start from application examples and practical problem solving, and from there come to some abstraction and theorems. In this book, Körner presents some basic results in a way that is the opposite of this. It is a strict mathematical top-down approach, formulating definitions, properties and theorems with hard proofs starting from a minimal set of axioms. As the title suggests, his guiding idea is to introduce the number systems on a pure axiomatic basis, loosely following the historical evolution. In this sense it is not really a replacement for a calculus course, but rather a complement to it. It does have the structure of a course text, with a sequence of definitions, theorems and proofs that is interrupted with many exercises and challenges for the reader-student.

One would expect that the topics to be covered and the order in which they are introduced are more or less clear. Nevertheless it is somewhat surprising that rational numbers come first in part 1, and natural numbers are a "special case" in part 2, and finally the reals and complex numbers in part 3. The quaternions and polynomials over a field follow as some kind of "encores".

Rationals without integers sounds almost impossible, but here the historical context comes in as a motivation. Originally people counted quantities and that does not really involve numbers in the sense that one added 3 sheep ad 4 sheep to get 7 sheep, but adding 3 sheep to 4 apples was not something to consider. So, to come to abstract counting numbers, the first thing to do in a Greek tradition is to find an axiomatic system for what we call now the positive natural numbers $\mathbb{N}^+$. The Greek had rationals disguised as ratios of lengths or areas but the Indians and Chinese properly considered rationals as numbers. In this book these are introduced as equivalence classes of (numerator, denominator) pairs where numerator and denominator are elements from the previous set with their rules for adding and multiplication. This gives the strictly positive rationals $\mathbb{Q}^+$. Then zero is introduced first as a place holder, and eventually as a number, and this entails the definition of the negative rational numbers and thus besides inverses for multiplication now also inverses for addition will exist. So we now have the structure of $\mathbb{Q}$. This idea of introducing a new mathematical concept as a couple of items from a previous structure with their known composition rules (adding and multiplication) and then identifying them in equivalence classes, is a technique that is used at several places in this book.

In part 2 the natural numbers are derived via the introduction of 1 as the least positive rational whose (reduced) denominator is 1 and then using an induction to generate all the positive integers as part of the rationals, and eventually zero and the negative integers to give $\mathbb{Z}$. Along the way we learn about long division, Bezout's theorem, and prime numbers. Modular arithmetic is introduced via finite fields leading to Fermat's little theorem, coding theory, the Chinese remainder theorem and encryption. The Peano axioms to define the rational numbers $\mathbb{Q}$ as an ordered field is the ultimate conclusion of part 2. It includes some philosophical considerations about the existence of numbers and the idea of an axiomatic approach to mathematics in general. That includes the Russell paradox, Gödel's theorem, and the consistency problem of mathematics.

Historically, mathematics became a profession somewhere in the 17th century. First Körner introduces extensions like $\mathbb{Q}[\sqrt{2}]$ using again the technique of defining addition, multiplication and an order relation on couples of rational numbers. To come to analysis, it requires the fundamental axiom of an intermediate value and hence the existence of a limit for bounded sequences. Equivalence classes of converging sequences are introduced by identifying sequences with the same limit as equivalent. Pointwise operations can be defined for the sequences and the real numbers in the set $\mathbb{R}$ are then identified with these equivalence classes. The complex numbers are then easily introduced as couples of reals (using again the same trick of defining operations for couples of reals) and introducing limits and continuity in $\mathbb{C}$ is relatively easy. This paves the way to define polynomials over a field and to derive the fundamental theorem of algebra. The main reason for considering zeros of polynomials over a field, in particular with integer or rational coefficients, is that this allows to distinguish between the algebraic and the transcendental numbers. Integral domains and quaternions as generalizations of complex numbers are the remaining items with a short discussion.

Clearly this book is probing the fundamentals of mathematical analysis and will be useful as an extra reading for an introductory calculus course. It will certainly satisfy those readers who are looking for abstraction and who want to extract the maximal number of results from the minimal set of axioms. The historical elements on the side are entertaining but not essential. It is not exactly recreational mathematics, but the text is nicely written. It has many explicit proofs and many exercises and invitations to think about a statement. No solutions are provided though. It is an excellent way to get in touch with the foundations of mathematics at a relatively elementary level.

Adhemar Bultheel
Book details

This book gives an axiomatic approach at a first year university level to number systems and some elements from calculus. It could be a complement for a freshmen's calculus course.



978-1-1084-8806-8 (hbk), 978-1-1087-3838-5 (pbk)
£ 59.99 (hbk), £ 24.99 (pbk)

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