'The big questions: Mathematics'

Review by 
Ilia Rushkin

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.

Comments

In his book The Big Questions Tony Crilly seems to have a high opinion of Isaac Newton's centripetal theory that centripetal forces convert a straight line orbit into an elliptical orbit. This is nonsense.

Mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.

In many cultures—under the stimulus of the needs of practical pursuits, such as commerce and agriculture—mathematics has developed far beyond basic counting. This growth has been greatest in societies complex enough to sustain these activities and to provide leisure for contemplation and the opportunity to build on the achievements of earlier mathematicians.

All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms. Inquiries into the logical and philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency. For full treatment of this aspect, see mathematics, foundations of.

This article offers a history of mathematics from ancient times to the present. As a consequence of the exponential growth of science, most mathematics has developed since the 15th century CE, and it is a historical fact that, from the 15th century to the late 20th century, new developments in mathematics were largely concentrated in Europe and North America. For these reasons, the bulk of this article is devoted to European developments since 1500.

This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe, it is necessary to know its history at least in ancient Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th century. The way in which these civilizations influenced one another and the important direct contributions Greece and Islam made to later developments are discussed in the first parts of this article.

India’s contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Islamic mathematics during its formative years. A separate article, South Asian mathematics, focuses on the early history of mathematics in the Indian subcontinent and the development there of the modern decimal place-value numeral system. The article East Asian mathematics covers the mostly independent development of mathematics in China, Japan, Korea, and Vietnam.

The substantive branches of mathematics are treated in several articles. See algebra; analysis; arithmetic; combinatorics; game theory; geometry; number theory; numerical analysis; optimization; probability theory; set theory; statistics; trigonometry.

During the last half of the twentieth century, striking applications of Mathematics appeared in all natural sciences, even in behavioural and social sciences. Mathematics is a universal tool to gain insight in highly complex systems. But mathematics is also a science of its own. It is highly alive, powered by its internal driving forces and by inspirations coming from new challenges in other fields.
The Mathematics Research Unit (Unité de Recherche en Mathématiques) RMATH carries out research in mathematics, both on its fundamental and its applied aspects