This book is the first of a trilogy, Secrets of Creation. Consequently, after reading it one is left in a situation similar to having read the first part of a detective story and not knowing how the story ends; but fortunately both the book and its website offer hints. Matthew Watkins' thesis is that numbers are more than how they are usually conceived, based on their use in counting. This counting use leads to numbers being visualised as entities stretched out in a line; but if instead we view a number as a product of its prime factors (for example, representing 60 as 2, 2, 3, 5), then different kinds of relationships appear.
Special interest attaches to those numbers (primes) that have no factors other than 1 and themselves, and the challenge is to characterise their apparently irregular distribution along the line of numbers determining their counting sequence.
Over time, mathematicians have found ways to approximate this distribution more and more accurately, culminating in a formula that represents this in terms of a collection of what Watkins calls "spiral waves". What is particularly remarkable about the latter is their mysterious irrational frequencies, such as 14.134725..., 21.022040...and 25.010856...
Time and time again, Watkins returns to the point that these facts about numbers, such as whether or not a number is prime, appear to be unalterable, the same for everyone. We will not find these facts in the outside world: they are in our minds alone; some inner machinery puts them there. Watkins claims that this is not a matter of culture because members of all cultures pick up numbers easily once they are introduced to them. And, according to the preview on his website, it seems that in the final book in the trilogy this will be taken further, arguing that patterns such as the spiral waves that specify the distribution of primes are present directly in nature at some level.
The author is at pains to make his exposition readily accessible to any intelligent reader, employing cartoons by illustrator Matt Tweed for this purpose. The way we think of numbers when they are used for counting is illustrated by placing them in sequence on a line, whereas to help readers picture them as products they are depicted as clusters of balloons labelled with the factors.
Every point in the argument is set out pictorially in a similar way, a process that works well except in the case of spiral waves, where pages of exposition are needed to explain what could otherwise be described by a simple formula (for the experts, some mathematical details are given in the appendices). In a more philosophical interlude, Watkins laments the fact that numbers seem to have taken over society, saying: "It's as if we're rapidly replacing the world around us with a crude replica, a world built out of numbers."
This is an unusual and fascinating book, which even experts on prime number theory are likely to find of interest. Even more interesting, perhaps, will be the second and third books of the trilogy, which Watkins promises will explain the origins of the mysterious world of mathematics.
Meanwhile in the present book, we can enjoy the sight of Watkins' wizard walking in a spiral, dropping a candle on each prime he passes, illustrating thus how these special numbers are distributed.
The Mystery of the Prime Numbers: Secrets of Creation, Volume One
By Matthew Watkins
Inamorata Press, 362pp, £15.00
Published 1 February 2010