From 0 to Infinity in 26 Centuries
Read From 0 to Infinity in 26 Centuries Online
Authors: 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
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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
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Contents
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
B
ACK TO THE
B
EGINNING
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.
T
HE
S
TONE
A
GE
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.
C
OUNTING ON
F
INGERS
(
AND
O
THER
B
ITS
T
OO
!)
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’.