The big questions: Mathematics

By Tony Crilly

With twenty skillfully written essays Tony Crilly paints a broad-stroke picture of modern mathematics, focusing on some of the most exciting topics. This book is intended for people whose acquaintance with mathematics is limited to their high school years, but who want to know "what all this fuss is about". It is ideal for those who have heard that mathematicians talk about imaginary numbers and unbreakable codes, and want to know how much of it, if any, is true.

This book may also be useful for experts in mathematics, albeit for a different reason. It shows how to make mathematics interesting and how to explain mathematical research to a lay person. The author has an unmistakable talent of presenting rather complex concepts in very plain words. For sure, he simplifies these concepts in the process, and certain ideas have to be twisted this way and that to make them easily comprehensible, but the distortions always stay within reasonable limits and the gist of the matter is delivered to the reader intact.

The Big Questions reminds us, professional mathematicians and amateurs alike, why people develop mathematics in the first place: because it is fun and because discovering absolute truths is one of the greatest intellectual pleasures that a human can experience.

Each chapter of this book is an elementary survey of a different mathematical field. Some of them are basic indeed: an overview of the development of numerical notation from marks on wooden sticks to decimal, binary and duodecimal systems, or a detailed explanation of the evolution of numbers in mathematics (from whole numbers to negative numbers, to rational ones, to irrational ones, to complex ones, and so on). Some are concerned with more "serious" mathematical subjects: set theory, group theory and chaos.

The chapters are seemingly disjoint. What connects them is the author's perspective: mathematics as a way to understand the world. Does the world really make sense to us, now that the mathematicians know so much more than the ancient Greeks did? Well, it depends. Mathematicians have realised that before we proceed to answering questions we need to agree about the precise meaning of those questions. For example, how many points are there in a continuous line? Does a three-dimensional space contain more points than a line? By themselves, these questions do not make sense. First, we need to agree how to count something so manifestly infinite as points in a line. The chapter How big is infinity? addresses this issue. It explains how mathematicians approached this problem, why their approach, although by no means the only one possible, made sense and what conclusions they have arrived at.

More books of this type should be written. It is true that mathematics has become such an enormous subject that people working in different areas can hardly understand each other's work, making it a hopeless task for the uninitiated. But what mathematicians can do is explain the general aims of their studies and why these aims are important. History has seen many examples of amateurs attempting to solve the great mathematical problems, such as squaring a circle, or to disprove mathematical truths, such as the impossibility of squaring a circle. Often this is the result of their misunderstanding of the problem, something that could be helped if the mathematical community did more towards keeping the general public informed about the nature of mathematical research. Tony Crilly's Big questions is a step in that direction, and let us hope for many more books like this one to come.

Book details:

The big questions: Mathematics

Tony Crilly

hardback — 208 pages (2010)

Quercus Publishing Plc

ISBN: 978-1849162401

About the author

Ilia Rushkin is a Postdoctoral Research Fellow at the University of Nottingham. His research is in theoretical physics.