# Elements of Mathematics: From Euclid to Gödel

In this book Stillwell explores the boundary between elementary mathematics and advanced mathematics. This boundary is not strict and some elementary topics are close to the boundary and some advanced topics are just across, while others are way beyond. So it makes sense to look for what exactly makes a topic advanced or not, even though the classification can be the subject of discussion making it a bit fuzzy in some cases, Stillwell has a rather clear vision on it. There is clearly also an historical aspect to this. Mathematics evolved and it is interesting to see where, in hindsight, the line was crossed. When it comes to foundations of mathematics or to more subtle differences, there is also a philosophical component to this. So Stillwell gives a survey (or rather a limited selection) of elementary mathematics, proving basic theorems with elementary tools, but from time to time he has to cross the line and he also treats some topics just across to explore where the boundary lies and to explain why some theorems or concepts are more advanced than others. The sections dealing with advanced elements are clearly indicated with a * in their title.

In a first chapter he gives an introduction to the rest of the book, introducing the eight mathematical domains that form the successive subjects of the next chapters: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and finally logic. In a concluding chapter, some advanced topics in each of these subjects are discussed. Each chapter is concluded with historical and philosophical remarks. Although logic forms the foundation of mathematics, it does not come up first, but it is the last in the row because it was logic of the previous century that lifted mathematics in many different ways to a much more advanced level. On the other hand, computation comes up early in the list at the second place because it has become so important in modern society.

Looking at educational systems, there are three main parameters that help distinguish between elementary and advanced: infinity, abstraction, and proof. About the latter Stillwell advocates to include at least some proofs in the elementary curriculum. Mathematics, even elementary without proof is a delusion. Mathematics without abstraction is unthinkable. One should be able to work with symbols instead of numbers. Much of algebra is elementary, yet assumes some abstraction. Infinity is a bit dual. There is a soft form of infinity that cannot be avoided. There are infinite processes in Euclidean mathematics, but infinite is not an existing object, it is just representing something going on indefinitely. The infinity of analysis relying on set theory is much harder. This is Stillwell's main criterion to draw the line. Whenever a concept relies directly or indirectly on infinity (the hard version like in the concept of real numbers) it is considered to be advanced. This is what he does in his last chapter. Some of the topics already discussed in the domains of the eight previous chapters are reconsidered and pushed a but further usually by allowing some concept relying on infinity which lifts them to the advanced side of the boundary.

I will give some illustrations of where Stillwell draws the line in the eight chapters:

Elementary *arithmetic* contains topics such as prime numbers, finite arithmetic, Gaussian integers, and the Pell equation.

Under *computation* Stillwell starts with addition, multiplication and exponentiation, possibly in binary arithmetic. These are obviously very elementary. Also the P-NP problem, and Turing machines are still elementary but close to the boundary, while universal Turing machines and analysis of unsolvable problems cross the line.

Elementary *algebra* contains rings, fields (number fields and polynomial rings) and vector spaces. Groups (drops commutativity of the multiplication from the ring, hence is less natural) and the fundamental theorem of algebra (requires reals or complex numbers) are advanced.

In Euclid' *geometry*, it is about constructions with compass and straightedge, but proofs about irrational numbers could be given geometrically. Doubling an square (area) is possible, but not for the cube (volume), which means that a volume is 'more advanced' or 'less elementary' than area. Linear algebra arithmetizes geometry and it requires a vector space with inner product to do elementary geometry. Also constructible number fields are elementary. However non-Euclidean and projective geometry are not.*Calculus* obviously needs infinity, but only the `soft version' in its elementary part. Infinite series, and the concept of derivative and elementary integration but basically in their geometric meaning of tangent and area under a curve are elementary. Stillwell is carefully about integration which in its elementary version is only restricted to integrals of rational functions which give all the elementary functions and their inverses (logarithm, exponentials, trigonometric,...). The fundamental theorem of calculus introducing the primitive function is advanced, but a simple proof of the irrationality of e is elementary. Anything relying on the completeness of the reals (e.g. continuity) is advanced.

Under *combinatorics* we find topics usually described as discrete mathematics. Proving there are infinitely many primes, Fermat's little theorem, binomial coefficients, Fibonacci numbers and generating functions are all elementary, as are several elements from graph theory, including Euler's polyhedron formula. It becomes advanced when the graphs become infinite and one has to deal with Kőnig's infinity lemma, Bolzano-Weierstrass theorem, and Brouwer's fixed point theorem. All these need the `hard version' of infinity.*Probability* theory starts from combinatorics (the binomial coefficients gives approximations of the bell shape), random walk, mean, variance and standard deviation and the law of large numbers for coin tossing and random walks are all on the elementary side. You stay there as long as there is no concept of limit or measure.

Finally, *logic* can include propositions, quantifiers, Boolean algebra and induction. The latter basically relies on the definition of the natural numbers and can therefor be considered elementary, but most of the concepts introduced here are advanced like Peano arithmetic, the real numbers, countability, infinity, set theory. In fact reverse mathematics is an advanced topic that studies which axioms are needed for proving a theorem. This implies that it can somehow classify why some theorems are more advanced than others if they require more or more complicated axioms.

Clearly the subject to be discussed very broad, and there is definitely a limited selection of what can be said in just one book about defining a line bordering elementary mathematics. The line will probably remain fuzzy anyway. Since there is no comparable book, there is no list of references for reading on this boundary subject. There are few occasions where the reader is referred to for more information on a specific topic, but the long list of references consists mostly of original historical references. The subject index is also very complete, which is very much appreciated since there is obviously interference between the different chapters. Writing a mathematics book without a single typo is probably impossible. I found a blatant one on page 238 where it is said that $x^n$ goes to zero when $|x|<0$ where it should obviously be $|x|<1$. There probably are some more as in any first edition, but that hardly diminishes the overall quality of the text. It's a privilege and a pleasure to follow the pros and cons of Stillwell's arguments.

Most of what is described as elementary will be familiar if the reader has had a good mathematical secondary school course. Some of the advanced topics probably require some mathematics from a first year university course. It may give such a student an idea of what else there is to follow in his or her further mathematical education. However, I think the book is at least also (or perhaps even more so) of interest to professional mathematicians. The views and approaches that are presented by Stillwell are sometimes rather unconventional, and I am certain there will be some new unexpected connections or proofs to be discovered. The historical endnotes of the chapters are interesting but standard. In my opinion, the philosophical endnotes are often very interesting because this is where I recognize mostly Stillwell's idea that he borrows from Klein: "elementary mathematics from an advanced standpoint" and that he used as a *leitmotif* for this book.

**Submitted by Adhemar Bultheel |

**5 / Jul / 2016