From 0 to Infinity in 26 Centuries

by Chris Waring

  First published in Great Britain in 2012 by

  Michael O’Mara Books Limited

  9 Lion Yard

  Tremadoc Road

  London SW4 7NQ

  Copyright © Michael O’Mara Books Limited 2012

  All rights reserved. You may not copy, store, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (electronic, digital, optical, mechanical, photocopying, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages.

  A CIP catalogue record for this book is available from the British Library.

  ISBN: 978-1-84317-873-6 in hardback print format

  ISBN: 978-1-84317-921-4 in EPub format

  ISBN: 978-1-84317-922-1 in Mobipocket format

  Designed and typeset by www.glensaville.com

  Illustrations by Aubrey Smith

  www.mombooks.com

  Contents

  Introduction

  Prehistoric Maths

  Early Civilized Maths

  The Ancient Greeks

  The Romans

  Eastern Mathematics

  The Middle Ages in Europe

  The Renaissance Onwards

  The Digital Age

  Modern Mathematics

  The Future of Mathematics

  Bibliography

  Index

  Introduction

  There’s no hiding from mathematics. It’s a subject so rich and diverse that we use it to explain everything from the Big Bang to how to improve your chances in a game show. Maths too plays an integral role in everyday life. You might work in a technical profession that demands frequent number-crunching, or you might only have to perform calculations when you’re working out your accounts or comparing special offers on the Internet.

  Maths is drummed into us from a young age. You have most likely received some degree of education in mathematics, probably up until the age of sixteen. You will have been taught arithmetic – how to perform calculations; geometry, which helps us to understand shape and space; and algebra, which allows us to solve problems without having to resort to trial and error.

  You may be one of an increasing number of people who has studied mathematics to degree level, or beyond. In which case you may be familiar with calculus, complex numbers, mechanics, statistics, decision mathematics or any myriad mathematical field that exists.

  Whatever level of mathematics you have so far reached, it is unlikely you have ever been told much of its back story. Who decided we should work in tens? Why are there 360 degrees in a circle? Who invented algebra? Every aspect of mathematics, from the numbers we use to the way modern mathematicians tackle the big unanswered problems, is the product of thousands of years of human endeavour that goes largely unmentioned in maths lessons and textbooks around the world.

  From 0 to Infinity in 26 Centuries sheds light on the fascinating history of mathematics, starting with the earliest people and working forwards to the modern day. It’s a chronicle of people and their cultures; their beliefs and aims. Why does the Mayan calendar end in 2012? Why were there no notable Roman mathematicians, and yet so many in Ancient Greece? When did scientists start using maths to develop theories?

  I hope that you will find the stories that follow interesting, touching and entertaining. I hope too that it will help you find a new respect for mathematics and for the people that helped to develop it into the wonderful subject that it is today.

  Prehistoric Maths

  BACK TO THE BEGINNING

  It has been estimated that the earliest humans arose in Africa approximately 250,000 years ago. These people left behind little evidence of their existence other than a few fossils, so we know very little about their culture, if indeed they had one.

  So, what can we say about their mathematical abilities?

  Early estimates

  A trait that all humans – and indeed primates and some other animals – have is the ability to subitize: to know at a glance how much a small number of things amounts to. Here is an example:

  lll

  If you have the ability to subitize, you will be able to look quickly at the lines above and spot that there are three of them, without having to count each line. Now try this one:

  lllllllllllllllllllllll

  There are twenty-three lines here, but I only know that because I typed them. At a glance, the best you would probably be able to do is to say that there are ‘around twenty’ or ‘two dozen’ lines. The instantly recognizable and countable pips on a die are a modern-day example of subitizing.

  We think that subitizing is a trait that has evolved in animals to allow them to make quick decisions with regard to fight-or-flight-type situations: one or two wild dogs and you might be happy to stand your ground (as long as there’s a stick nearby that you can use to fend them off); three or more dogs and you’re likely to run to the nearest tree.

  You and I are literate and numerate humans who can’t remember what it was like not to be able to count. We see three lines and we cannot help but think of the number three. Our first ancestors, however, would have had no word for three and, perhaps more significantly, possibly no concept of three as a number.

  THE STONE AGE

  Approximately 200,000 (notice the comma to help you subitize all those zeros!) years after they first walked this earth, humans gained what anthropologists call ‘behavioural modernity’: they started doing things that differentiated them from other animals. They developed language, tools, cooking, make-believe, painting, and had begun to ponder the nature of existence and all the other things that make us human. These were the Stone Age hunter-gatherers. We know a touch more about their mathematics because the remains of cavemen types were unearthed from the nineteenth century onwards and written about by their pith-helmeted discoverers.

  A counting controversy

  The Ishango bone is the thighbone of a baboon that was discovered in the Democratic Republic of the Congo, Africa in 1960. Dated at approximately 20,000 years old, the bone has caused much controversy among scientists. The bone has three sets of grooves carved deliberately into it, and if you count the grooves you find the following sequences: (9, 19, 21, 11), (19, 17, 13, 11) and (7, 5, 5, 10, 8, 4, 6, 3). Some scientists believe that this is evidence not only of the Stone Age peoples’ ability to count up to numbers much higher than the more recent Aboriginal tribes (well, higher than three anyway; see box below), but that the numbers in each set shows evidence of an understanding of counting in tens, odd numbers and prime numbers. This argument has been challenged by other scientists who suggest the grooves were either decorative or intended to make the smooth bone easier to grip, and therefore mathematically meaningless.

  Modern-Day Hunter-Gatherer Tribes

  The Pirahã tribe lives today in the Amazon rainforest. They are consummate experts at jungle survival. The tribe’s language is so simple that its hunters use a whistled version of it while out trailing game. Remarkably (at least to us), their language contains no numbers and, despite trading commodities such as T-shirts, metal knives and alcohol with other tribes and river traders, the Pirahã show no inclination to adopt a number system either. These people live in such a way that numbers have no function for them – they live hand to mouth in the equatorial rainforest, where food is available all year round.

  Australian Aboriginal tribes were living in a hunter-gatherer society when they were first encountered during the eighteenth century. The tribes that possessed a concept of numbers generally had words for one, two and sometimes three. Any
numbers larger than three they made by adding together a combination of the first three numbers. So a tribe with words for one, two and three would have been able to count to nine by saying: one, two, three, three-one, three-two, three-three, three-three-one, three-three-two, three-three-three. The fact that these people had no word for numbers larger than three suggests that they very rarely, if ever, needed to use them.

  The Ishango bone notwithstanding, it is fair to say that many Stone Age tribes would have had a fairly childlike grasp of numbers. And, like children today, we can be fairly sure that our early ancestors used their fingers for counting – a key development along the way to numeracy.

  COUNTING ON FINGERS (AND OTHER BITS TOO!)

  We humans have eight fingers and two thumbs, and if you watch any young child learning to count or add (counting is in fact just adding on one each time), you’ll see that these convenient ten counters are too tempting not to use. Consequently we instinctively like the idea of numbers coming to us in batches of ten.

  A means of communication

  We can also use our fingers to communicate numbers non-verbally, which is as useful now when you are in a foreign country as it was for Stone Age hunter-gatherers. They perhaps would have used their fingers to express numbers that they didn’t have words for, or to communicate an idea to other people across a language barrier.

  The upper limit for counting on your fingers is ten, which, as we have seen, would have been more than enough for many Stone Age people. As societies developed, larger numbers were required, yet counting in batches of ten continued. The modern-day words we use for numbers usually have their roots in this tradition – the English words ‘twenty’ and ‘thirty’ come from ‘two-tens’ and ‘three-tens’ respectively. Some ancient cultures such as the Mayans and the Celts used fingers and toes to count in batches of twenty rather than ten, the evidence of which still exists in some languages today. The French word for eighty, quatre-vingts, literally translates as ‘four-twenties’; the Welsh language uses a similar system: thirty-one in Welsh is un ar ddeg ar hugain, which is ‘one on ten on twenty’.

  When you count on your fingers, or, more importantly, use them to show that Greek barman how many shots of Metaxa you’d like to buy, it does not really matter which fingers you use – we tend to use what is anatomically more comfortable. For example, it is far easier to show the number four using the fingers of one hand rather than a thumb and three fingers on the same hand (try it!). It does mean, however, that there are ten different ways of showing one on your fingers, and that the finger system relies on a person’s ability to count up all the fingers that are shown.

  Some cultures have navigated these potential pitfalls by assigning a value to different parts of their body. On the Torres Strait Islands between Australia and Papua New Guinea, thirty-three different body parts are used for counting, and indeed as words to represent the numbers. For example, to a Torres Strait Islander the ring finger on your left hand means sixteen, your right shoulder means eight, your left knee twenty-four and your little toe on the right foot thirty-three. This system has evolved to allow for effective communication between islands that are home to several different languages.

  The Legacy Lives On

  With our highly evolved modern-day number systems, we have little need to communicate numbers with our hands, but we do still see it occasionally. Traders on the Stock Exchange floor have the ‘Open Outcry’ communication system, which involves shouting and hand signals, to buy and sell shares in a noisy trading pit.

  The Stone Age humans persisted in their ways, hunting and gathering without the need for many numbers at all. However, approximately 10,000 years ago, people in fertile areas around the world’s great rivers decided to settle down and get civilized. This led to the need for much bigger numbers, methods of recording them and every school kid’s favourite – arithmetic.

  Early Civilized Maths

  FROM HUNTER-GATHERERS TO HERDERS

  According to historians there are five main stages involved in a society becoming ‘civilized’. The first stage is the ability to make and control fire – Homo sapiens and their ancestors have been creating fires for approximately half a million years. The second stage is the cultivation of crops, which really requires the help of domesticated animals. The Neolithic Revolution, which occurred in independent locations across the globe approximately 10,000 years ago, saw humans begin to stay in one place, grow crops and domesticate and rear livestock.

  Counting sheep

  The first shepherds would have needed a method for counting their animals, so it seems obvious that they would have been good at counting. Or would they? Equally, the first farmers would have needed a method of working out what time of year it was so they knew when to plant their seeds – surely more numbers were involved here?

  The Neolithics used a system called pebble counting. In order to count their herds of sheep the shepherds placed one counter – perhaps a pebble or a fruit pip – in a bag for each member of the flock. To find out if all of the animals in the herd were safe the shepherds then removed a pip for each sheep counted. If by the time they got through all of the sheep they still had pips remaining they knew then that one or more of the sheep had been lost. Hopefully, if they were half-decent shepherds, this number would have been one of the low numbers (1–10), which we now know hunter-gatherers were equipped to deal with. The farmers would have used a similar system to count from a key event, such as the rains beginning, or the birds flying south, to let them know when to plant their crops.

  The economy takes shape

  Surplus is an inevitable consequence of agriculture and the domestication of animals. Once there was an excess of food, early civilizations could start to trade their surplus, which in turn promoted the idea of value. This is the third sign of civilization – the idea of an economy; of goods being bartered or sold. Naturally, they needed a way to cart around their surplus of goods, so the fourth sign of civilization is the wheel, which seemed to be widespread by c. 4000 BC.

  THE DAWN OF THE ACCOUNTANTS

  With a reliable surplus of food it then became possible for some people to fill their time doing things other than finding or growing food in order to survive. By c. 3000 BC towns and cities were filled with such people. These large urban societies needed organizing in order to function properly, and the ability to count large numbers was key to this. The need also arose for these new, large numbers to be recorded – and hence written – for the first time. Among the myriad flagship new professions that arose – craftsmen, soldiers, farmers, merchants – a new literate class – the scribe – was to be found; almost certainly many of these scribes were numerate accountants and – inevitably – tax collectors.

  The Bronze Age

  The fifth sign of a civilization is the use and working of metals, which started at approximately the same time as the first towns and cities arose. The easiest metal to work is copper, which is used to make bronze. Hence we call this period in history the Bronze Age.

  Most of the earliest civilizations developed around the fertile areas near rivers, where the land was suitable for farming and raising livestock. Three notable civilizations that arose during this period, and about which we know a great deal, were:

  1. The Mesopotamians: a collective name for the Sumerians, Akkadians, Babylonians and Assyrians. They lived in the Middle East from c. 3000 BC.

  2. The Ancient Egyptians: the baddies in the Old Testament. They were based along the River Nile from c. 3000 BC.

  3. The Mayans: from Central America. Their earliest stages of development began c. 1500 BC; they entered a stage of development similar to the Mesopotamians and Egyptians after AD 250.

  As some cities rose in importance and began to dominate the surrounding area, either economically or by strength of arms, certain regions began to adopt the most effective numbers system. Unfortunately the early records from many such cultures have not survived, perhaps because these people were sited next to large riv
ers that flooded often. But if we take a quick look at some of the highlights of the archaeological findings that have survived from each civilization, we gain some idea of an evolution of written numbers and number systems from our earliest urbanite ancestors.

  MESOPOTAMIAN MATHEMATICS

  Mesopotamia means ‘between rivers’ in Ancient Greek, and refers to the cultures that sprang up between the Tigris and Euphrates rivers in present-day Iraq – a very fertile area of land often called the Cradle of Civilization. Despite their early beginnings, we know a fair amount about the Mesopotamians because they performed all of their writing on clay tablets, which are hardy enough to withstand repeated soakings.

  Clay tablets, however, are not easy to write on. The Mesopotamians began with a pictographic language, when the written symbol for a word is a stylized picture of the thing being described. However, drawing decent images in thick wet clay is tricky, so they took to using the end of a wedge-shaped stylus (a rod-like implement with a pointed end) to make marks.

  Number system

  The Mesopotamian’s number system was base 60 (also referred to as sexagesimal), which means they counted in blocks of 60 rather than in blocks of 10, as we do today. Lots of numbers go into 60 (mathematicians would say 60 has many factors), which makes it a convenient number with which to do arithmetic. We still see a few reminders of it today – 60 seconds in a minute and 60 minutes in an hour hark back to the Babylonians.

  Cunningly, their number system contained only one symbol, made using the end of a stylus:

  They would use up to nine of these symbols. They would show a 10 by rotating the stylus by 90 degrees to get a slightly different symbol:

  So the number 47 would look like this: