Book Review: Mathematics a Very Long Introduction
I recently received my copy of the wonderful The Princeton Companion to Mathematics. In the title of this piece, I could not help making the obvious joke, as the editor Tim Gowers also wrote the brilliant Mathematics: A Very Short Introduction
. I first started looking for a book like this as an undergraduate. I wanted to get some overview of mathematics so I could have an idea how the different courses on offer might fit together. Over the years I have cobbled together some vague ideas of what the different areas of maths are, and more importantly to me, what questions they try to answer. Reading this book is like turning on the light (to steal and corrupt Andrew Wiles’ metaphor). There is a very serious attempt to explain what mathematics is and what the areas studied are. Also to give an impression of how they relate.
The book starts by identifying three methods of doing mathematics: Algebra, manipulating symbols; Geometry, using visualization; and Analysis, using limiting processes. Obviously these are very short description of longer passages of text aiming to sum up very subtle differences. Having explained these three methods the book starts to describe areas of mathematics, identifying the problem that the three methods are also areas of knowledge. The introduction then gets more specific defining some of the basic objects of mathematical study, from numbers through to manifolds, via groups and geometries. It concludes with a discussion of the goals of mathematics research. This final section is particularly valuable as it motivates mathematics as a way to answer questions, and talks about the general questions that we consider.
The second part of the book is historical, charting how modern mathematics came into being. This is followed by a more detailed discussion of 99 mathematical concepts. It is not every day that lies nestled between Phase Transitions and Probablility Distributions. The fourth part returns to the branches of mathematics and discusses them in more detail, showing how they consider, translate and transform the basic concepts. My apologies if this is starting to sound like a list! The book has many parts, each of which is worthy of at least a brief description as the structure has clearly been well thought through so that mathematics can be viewed from many different angles. This is especially true of the fifth part that considers 35 important theorems and open problems. This allows important single results to stand out, rather than considering the area in which they lie. Part six is similar in relation to the history chapter. It considers specific mathematicians (the cut off date was important work by the 1950s, so the list can be kept below 100!) rather the grand sweep of mathematical progress. The book concludes with a section on the influence of mathematics and some final thoughts. The section on the influence of mathematics is rather short, however this is in accordance with the books stated goals of describing mathematics, rather than its applications. At over 1000 pages there is hardly space to add a topic which could take up a possibly larger volume on its own.
Along with my excitement on opening the book, however, I felt a certain sense of trepidation. How would my own area be treated? Unfortunately this was justified, as it did not do so well. Tesselation is only mentioned in the briefest of ways and the Penrose tiling turns up only as a model for quasicrystals. Of course I accept that the book already covers a vast area, and mathematics is diverse enough that any book would have to miss something out. However I feel that what is missing is not just the particular topics on which I am working, but the whole area in which I work. I think that this is a more general problem than just this book, so it is a little unfair. However the book gives a clear target to attack, rather than relying on anecdotal evidence and insinuation. If asked what general area of mathematics I work in I would say geometry. However, although the book does describe geometry as the study of mathematics through visualisation, when it comes to describe geometry as a mathematical area the only topic considered is the study of manifolds (P 4-5). More tellingly even after describing geometry as visualisation the book states that:
If you look at a typical research paper in geometry, will it be full of pictures? Almost certainly not. (P 1)
Similarly the figures in the book, though reasonably plentiful are not considered worthy of their own contents list.
So what is left out by this? Here lies part of the problem. With the most natural term “Geometry” taken away to mean something else there are not even clearly defined terms for it. Discrete Geometry is certainly a large part of it as is the study of tilings, I would like to include sphere packings, space filling curves, fractals and nurbs. Maybe “visual mathematics” is a good name. This is meant to be a broad area, not a specific research topic, and manifolds especially in low dimensions would be both an object of study and an important tool, however I believe that the topic is a lot broader and contains many other concepts. Some books in the area might include bibles like Tilings and Patterns, The Symmetries of Things and Regular Polytopes, and many others worthy of mention such as Polyominoes and Indra’s Pearls.
Why is this area important? Firstly it shares with number theory the distinction of being able to ask difficult questions in very simple language. It is therefore an ideal way to explain mathematical concepts. In fact the book does exactly this, using the classification of the regular polytopes to explain classification (P 52-53) and sphere packings to talk about the discovery of mathematical patterns (P 58-59). As it also produces images, that are often strikingly beautiful, it might even go beyond number theory in some respects. People can be more drawn to images than to even the simple equations of number theory. It is true, however, that there is nothing to compete in grandeur to the quest and proof of Fermat’s last theorem. Though a natural extension of Hilbert’s 18th problem is a big open question. This asks if there is a single tile that will tile the plane but not periodically.
It is also of importance educationally, working with this sort of geometry and visualisation in 3 and 4 dimensions generates intuition. This intuition is a valuable skill for any area of work that creates objects whether in the real world (architecture and engineering) or in a computer (3d models for films and prototypes), as well as in mathematics.
Finally the that has been a consistent source of mathematics for thousands of years. The existence of irrational numbers for example was found by considering the diagonal of a square. Valuable contributions have also come from the study of perspective, leading to the projective plane and the work of Coxeter in group theory. The list could go on. Nor is it an area that will stop giving as it lies so close to undecidable problems. For example the domino problem, which asks if a set of shapes will tile the plane is undecidable. Though, as with all mathematics, there is a danger of simply wandering aimlessly down abstract paths, there are almost certainly beautiful and useful ideas waiting to be found. This is especially true as computers allow investigation of areas that are too complex to explore by hand. Klienian groups (as shown in Indra’s Pearls) and a wonderful example of this (though the wikipedia page contains no images!)
It is true that the influences section of the book is stronger on visual mathematics. It is here that you find the Penrose tiling for example. In particular I read with delight the wonderful section on Mathematics Art. It is great to see this acknowledged by mathematicians, especially citing Max Bill (P 950) and Constuctivism (P 948-9) along with the more traditional M C Escher (P 950-1) and Albrecht Dürer (P 945).
As I said before, though I see a lack of visual mathematics (as defined above) in the companion, this is perhaps more as a problem in the current priorities of mathematicians than the companion itself. In fact one of the great strengths of the book is that it helps clarify what mathematics currently is, so we can have a debate about how it might change and what it should become. Another example of this is Doron Zeilberger’s Opinion 92 pointing out that the importance of learning to program a computer to mathematicians.
To conclude I want to return to what the book does cover. It sets out with the admirable goal of explaining all of modern mathematics at a level comprehensible by people with a good level of high school mathematics. This goal, was dropped according to the preface (P xiii), as it was accepted that different topics needed different backgrounds. However much of the book is still accessible at the original level and even the harder parts should be accessible to maths graduates. The level of success achieved is remarkable and it is a great thing to see a stellar cast of mathematicians describing their topics in the simplest possible terms, letting motivation and ideas rule over technical detail. I certainly look forward to the many hours I will spend reading. As mathematics is getting larger having even a shallow understanding of the different areas being studied is hard. This book will become an essential tool. Hopefully it will inspire and challenge all mathematicians. It is possible to explain our research beyond our close colleagues!