Read Through: 'Mathematics: A Very Short Introduction' by Timothy Gowers
A review of Timothy Gowers's book 'Mathematics: A Very Short Introduction'
While I was looking for a specific book in one of the math bookshelfs in the Orell Füssli bookstore at Bellevue a few weeks ago, a rather small book managed to catch my attention because of the quite unusual but striking color of its cover (unusual for a math book, that is).
My first thought was simply that somebody deliberately put this book in the wrong shelf, but after picking it up and reading the title I was pleasantly surprised:
I already knew who Timothy Gowers is (he is a prominent British mathematician and recipient of the Fields Medal, the latter sometimes being referred to as the Nobel Prize in Mathematics), since I own another excellent book written by him, namely The Princeton Companion To Mathematics (I haven't read completely through it, though).
The next logical step for me after such a pleasant find was to simply purchase this conspicuously looking book, and I'm really glad I did it.
Small But Mighty
The book Mathematics: A Very Short Introduction, although being an introduction to mathematics (including some advanced topics) is a rather exciting excursion and learning experience in various mathematical topics. It starts with the explanation of how abstraction can be used to build mathematical models of existing systems, and then presents the reader with chapters covering numbers, proofs, limits and infinity, dimension, geometry, estimates and approximates, and ends with some interesting frequently asked questions.
Although it touches the one or other advanced mathematical topic, Mister Gowers definitely managed to do a very good job explaining the various mathematical topics as simple as possible. The various chapters are also spiked with a lot of examples, images and proofs, which help grasp the various presented concepts better.
Personally, I liked the Dimension chapter the most, since it was one of the more revealing chapters for me.
The following is a paraphrased passage from that chapter (I read the German edition of the book):
Imagine we want to draw a simple circle. Drawing a two-dimensional circle is a rather simple task. Add an additional dimension and we are able to draw the equivalent of a circle in the third dimension, namely a sphere. The interesting question now is: what happens if we move one dimension higher and start thinking about drawing a four-dimensional sphere? Is this even possible, and if so, what is this new form called and what does it look like?
This is where the author perfectly brings in the word abstraction. He just describes a circle or a sphere without ever mentioning the dimensions:
A circle or sphere is a set of points with a certain distance from a given point.
Having previously defined a distance function using the same abstraction (no mention of the dimension), we can now simply use the definition of a circle to construct spheres in the fourth or even twenty-seventh dimension.
This chapter was quite the learning experience for me, because a) I never delved into higher-dimensional geometry, and b) as soon as I began reading this chapter I started thinking about the mere impossibility to picture what a four-dimensional sphere may look like, let alone the vast number of things one could do with such a sphere. But right after I read the above-mentioned passage it just clicked. Why trying to do the impossible and imagine what such a figure may look like in higher dimensions? Just abstract its most important properties without ever mentioning the word dimension, and work with the new definition in higher dimensions.
I really enjoyed reading this book. Beside the various interesting topics the book covered, the author did a really good writing job; it is written in quite a captivating style, probably thanks to the fact that it reflects very well Mister Gowers's appreciation for the subject.
This book definitely earns 5 out of 5 stars!