The Mystery of the Prime Numbers uses an innovative visual approach to communicate some surprisingly advanced mathematical ideas without any need for formulas or equations. The issue of prime numbers acts as a gateway into some truly strange philosophical territory whose relevance extends well beyond mathematics.
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About the Author
Matthew Watkins was born in London in 1970. He trained to be a research mathematician, completing his PhD in 1994, but then left academia to travel and pursue other interests. He has stayed on the periphery of academic mathematics, having been an honorary research fellow at Exeter University since 2000.
Read an Excerpt
Secrets of Creation Volume 1
The Mystery of the Prime Numbers
By Matthew Watkins
John Hunt Publishing Ltd.Copyright © 2010 Matthew Watkins
All rights reserved.
numbers and counting
From the title of this volume you'll no doubt have guessed that numbers are somehow involved in the indisputable "something" which the introduction referred to. Indeed, the basic ideas of numbers and counting will act as our starting point. In the chapters that follow this one we'll be looking at them in a way which was inspired by my academic background in mathematics – this will lead us into the "indisputable" territory. Although presented in a gentle and accessible way, this approach to numbers may be unlike anything you've come across before. First, though, I think it's important that we take some time to have an informal look at numbers and counting from the perspective of ordinary human experience rather than from a strictly mathematical point of view. If you find yourself disputing some of what I have to say about this, don't worry, we haven't really started yet!
So, what exactly are numbers?
Because we all learnt about numbers and counting when we were very young, these ideas have come to be strongly associated with early childhood – they might even seem an unnecessarily and almost embarrassingly "childish" topic to be considering. But this brings us to the first thing that we should pay attention to: the fact that young children routinely and easily grasp the basic ideas of numbers and counting. Counting is one of the very first practical things a child learns to do and adults take it for granted that it's a sensible and appropriate thing to teach them. As children we all gradually learnt how to recite numbers in sequence, count things with them, recognise and draw the symbols that our culture uses to represent them, combine them by adding and multiplying, and so on. Some of us picked it up faster than others, but with very few exceptions (due to, for example, certain neurological conditions), children's minds absorb the basic ideas very easily. Some people end up very comfortable and capable working with numbers in adulthood. Some struggle. Most just get by. But it's almost unheard of for someone to remain completely baffled by the very idea of numbers and counting – everyone gets it.
And yet, if you think hard about numbers and what they really are, you'll probably get quite confused. You no doubt know how to work with them (at least to some extent), but if you spend enough time contemplating what they really are, you may well end up concluding that you don't know. The more you think about it, the more confusing it seems to become. Perhaps it all seems perfectly clear to you. But philosophers have been debating this matter for centuries and they're still far from providing us with a clear answer. You might not be able to imagine why, but it would be fair to say that there's still no straightforward consensus surrounding this issue at the deepest levels of philosophical discussion.
There are philosophical factions such as Platonists and social constructivists who continue to debate whether numbers and related concepts exist somehow independently of us and we "perceive" them with our minds or whether they are merely mental, social or cultural "construct". Much has been written about this question over the years. But these rigorous attempts to pin down exactly what numbers are would almost certainly confuse matters rather than clarify them, if presented to the "ordinary person in the street" who unproblematically deals with bus fares, football results, temperatures and shoe sizes.
Despite this puzzling situation, everyone should be able to agree that numbers are the common property of all. No one can be excluded from access to them. No one can take ownership of them. They're there for everyone equally. Wherever you find yourself in space or time, you'd expect the numbers to be there, accessible to you. But where is this "there"? They have this peculiar status of sort-of-existing (we're continually dealing with numbers of objects) but sort-of-not-existing (numbers don't exist in the way that actual objects do).
The fact that young children have no problem accepting numbers suggests to me that number concepts may be in some sense built into our minds. That is, a child learning about numbers is in fact recognising something which is already present within her or his mind. But even if I'm right in my vague suggestion that "it's in there" somewhere, there's still no agreed understanding of what "it" is, in what sense it's "in there" or even where "there" is.
Anthropologists have reported examples of cultures with extremely limited counting abilities which, once in contact with Western traders and the use of money, have suddenly switched into a highly competent number usage (without the introduction of any Western-style education). In The Emergence of Number, John Crossley suggests that "the idea of counting lies dormant until evoked". Having considered accounts of various indigenous peoples of Latin America, Polynesia, Australia and Malaya he concludes that "non-verbalized ideas of particular numbers appear to be present long before they may be needed in a verbal form and once counting is established there seems little difficulty in advancing rapidly".
The significant word here is "present". Present where?
I suggested that number concepts may be "built into our minds" but to some extent it now seems that they may be built into our brains. The relationship between the mind and the brain is another important matter which philosophers are unable to agree on. Certainly, the brain is the "physical part" with physically describable regions and components, while the mind is the "non-physical part" which is somehow related to the brain but in a way that no one is entirely sure about.
In recent years, neuropsychologists such as Stanislas Dehaene have been carrying out experimental work in the area of "numerical cognition" to explore the possibility that physical structures exist in the brain which relate to counting and basic operations with numbers, these having possibly evolved for survival-related reasons. Other researchers have carried out experimental work involving non-human animals, demonstrating the abilities of some to distinguish between various small numbers.
Despite the extremely widespread use of numbers in Western culture, the sense in which they "exist" and their relationships with the mind and the brain are rarely discussed – these are surprisingly marginal subjects. I find this situation strange, especially if we consider the incredible range of subjects which humans have explored in the most minute detail.
QUALITY- AND QUANTITY-BASED VIEWS OF NUMBER
Children first learning about numbers often describe feelings they have about each of the first few: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... Perhaps you have faint memories of something like this. I can still clearly remember sitting next to my friend Paul at school, aged six or seven, casually discussing our feelings about various numbers while we were working on our simple arithmetic problems. There were likes and dislikes, favourite numbers and numbers which seemed to have some sort of personality which we couldn't express clearly but we could somehow sense or feel. It felt as normal as discussing our feelings about various colours, songs or storybook characters.
In many cases, this kind of feeling might be linked to the shape of the numeral or the sound of the word associated with the number. Or it might be due to some association with an age, a birthdate, a house number or the shirt number of a favourite athlete. But I suspect that there may be something deeper going on with the overall phenomenon of these feelings, as suggested by the accounts of people with severe autism and related conditions, some of whom can perform baffling, almost superhuman feats of mental arithmetic and, at the same time, describe having a direct inner experience of numbers as having textures, colours and/or personalities. The combination of these people's extraordinary abilities with numbers and their claimed "inner perceptions" of them suggests that they might know something about number which the rest of us don't.
However, a "sensible grown-up" outlook dictates that there is no value in dwelling on these "childish" number-related feelings. In state-sanctioned systems of Western education, numbers are presented to children in a systematic, unemotional way. They are treated solely as quantities to be added, multiplied and so on. Their properties and interrelations are entirely unaffected by our feelings about them.
This brings into focus the distinction between two very different approaches to number. If I say "seven is between six and eight" or "seven is an odd number", those statements concern seven's properties as a quantity. But if I say, "seven is a lucky number", or "seven feels smooth, like a pebble", those are claims regarding supposed qualities of seven.
Prior to the emergence and expansion of science-based Western civilisation, many cultures had a kind of reverence for certain numbers or a belief that numbers have a qualitative aspect (a quality, personality or "meaning" of some kind) as well as the more obvious and mundane quantitative aspect (a quantity, an amount of something).
This distinction between the "qualitative" and "quantitative" approaches to number has hardly been discussed by academics outside a small fringe of thinkers. It seems that the unspoken, almost unconscious, belief among Western intellectuals is that because arithmetic and all higher mathematics involve the quantitative (and most definitely not the qualitative) approach, the qualitative obviously lacks any serious value and the quantitative is the "correct" view, so there's nothing to discuss. In this way, the quantitative view – the view used by mathematicians, scientists, stockbrokers, bookmakers and pocket calculators – triumphs.
Many Western children seem instinctively drawn to a qualitative approach to number but they are systematically directed away from this by their formal education. Wherever pre-Westernised cultures have gravitated to the qualitative approach, this tendency has been similarly countered by the nearly universal introduction of Western-style educational practices, part of that questionable ongoing global project sometimes called "progress".
Perhaps you're thinking "well, yes, this is progress, this is the correct way" – you may have no problem at all with completely dismissing the qualitative approach to number. After all, a dozen different people could "feel" a dozen different "qualities" associated with a number, so there's not much point trying to study this sort of thing, is there? Or perhaps you feel that there is something behind the qualitative approach to number worthy of more serious attention. In any case, as we proceed, try to keep in mind this distinction between the "qualitative" and "quantitative" approaches to number, ideally without judgement. Just remember that these two very different perspectives exist and try to avoid thinking about them in terms of true/false, right/wrong or valid/invalid.
THE QUALITATIVE VIEW
The qualitative view has no serious role in organised society. Still, remnants can still be seen at the level of individuals and their idiosyncrasies.
Any Western-style mathematics education is entirely based on the quantitative approach to number. In order to be considered "successful" on its own terms, it would have to involve any number-related feelings being "educated out" of children. But despite educators' best efforts, feelings of this type can persist into adulthood, and do, far more widely than some people would like to think. There are many curious remnants of "number mysticism" in our modern, scientific culture. Many people have lucky numbers, seven being the most notable for some reason. Fear of the number thirteen is still widespread in the Western world. Some major hotel operators routinely number their floors ..., 11, 12, 14, 15, ... for practical economic reasons – an economically significant proportion of their customers don't feel comfortable staying in a thirteenth floor room. Numerology books continue to proliferate. Telephone numerology consultations are commonly advertised in the back pages of popular newspapers. Websites and lucrative workshops abound. I've heard of a variety of eccentrically ritualistic and quasi-mystical ways in which people choose their lottery numbers – numbers which they see as the keys to a kind of salvation. And a significant number of people now struggle with variations of obsessive-compulsive disorder which involve an urgent need to repeat certain actions certain numbers of times.
There's a huge gulf between the dominant "scientific" (that is, quantitative) approach to number and the qualitative "folk beliefs" regarding numbers which can still be found throughout Western populations. This is similar to the gulf between the culturally dominant "scientific" view of numbers which now prevails and the views which were held throughout most of human history.
Although the social phenomena I've described could be worth examining for various reasons, they're still very marginal in the overall workings of the Western world. The powers-that-be (bankers, corporate leaders, politicians, economists, scientists, etc.), if they were to give the matter any thought, would certainly be of the opinion that such beliefs are nonsensical remnants of a pre-rational, pre-scientific age. Western science simply denies the validity of anything "numerological" and assumes the thinking behind it to be fundamentally misguided. Although relatively new in historical terms, this perspective is now firmly established as the dominant one.
THE QUANTITATIVE VIEW
Having mastered the basics of arithmetic as children, most people give very little thought to numbers beyond their immediate use in financial transactions and other such practical matters. There is a strong tendency to take them for granted. But if they were to stop and consider the extent to which numbers have become woven into their lives, many people would be quite surprised.
Suppose you were to switch on a radio and catch the end of the hourly news. You, together with possibly millions of other people, are listening to a publicly sanctioned source of information. You may well hear some new government statistics on crime, education or unemployment, the stock exchange index and the number of points it's gone up or down, some sports results (in the form of numbers), the time, a few temperatures, the identifying numbers of some major roads and junctions, the speeds of traffic in their vicinities and, finally, the frequency of the station you're listening to.
And it's not just actual numbers which you begin to notice everywhere once you've started looking, it's also the tendency for Westernised humans to measure and quantify the things they encounter. In almost every area of our lives, attempts are being made to reduce everything to measurements, which take the form of numerical data. We'll look at the main examples of this after a quick explanation of how I'm going to be using certain words.
By quantification, I mean the process of assigning a number to something. So quantification includes simple counting and all familiar forms of measurement (using a ruler, a stopwatch, a thermometer, etc.). But the word is more commonly applied to all of the other ways in which numbers get assigned to things which aren't obviously measurable – things like human intelligence, the "value" of a painting or the "performance" of a school or hospital. These things can be quantified when someone finds a way to measure them – an IQ test, an art auction, a governmental evaluation procedure.
By counting, I mean the application of number to the physical world, by means of agreed-upon categories of things-to-be-counted.
Excerpted from Secrets of Creation Volume 1 by Matthew Watkins. Copyright © 2010 Matthew Watkins. Excerpted by permission of John Hunt Publishing Ltd..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
An introduction 2
1 Numbers and counting and how they've taken over the world 5
2 How to build the number system five simple rules that take you out to infinity 35
3 Prime numbers something anyone could have noticed 51
4 Prime factors and the most important thing we know about numbers 81
5 A philosophical interlude and a journey into space 105
6 Addition versus multiplication they're surprisingly different things 117
7 An infinity of primes how we can be sure there isn't a biggest one 135
8 Patterns and formulas the questions everyone seems to ask 149
9 Spirals the central image in this story 167
10 The distribution the curiously random-looking arrangement of primes 187
11 Staircases a useful way to picture the distribution 203
12 The deviation isolating the deviant behaviour of the primes 231
13 Harmonic decomposition breaking everything down into waves 249
14 Spiral waves which no one's bothered to name properly 267
15 Mysterious frequencies the inevitable cliffhanger ending 287
Appendices 1-9 312