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Origins of mathematics


Twelve thousand years ago Man was a hunter-gatherer, he had not yet learnt how to grow his own food or keep animals.

You have made a pitfall trap (a deep hole in the ground with steep sides and covered in branches to hide it so that an animal might fall into it and not be able to escape). You have caught a deer and are carrying it back to your camp. But you are keeping a good lookout for dangerous animals. You have walked a short distance and you see

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You are not unduly worried and you keep walking, but then you see

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You carry on walking but you look round for a tree you will be able to climb up if you need to, and you take your Big Stick out of your belt and keep it in your hand.

But now you see

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You start running towards the tree you have chosen, still carrying the deer. But now you see

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You drop the deer and sprint towards your tree. You get there just in time.

Being able to tell the difference between

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might just save your life; being able to tell the difference between

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is of absolutely no importance at all because anything more than

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is life-threatening.

If we are shown a picture with three animals in it we do not need to actually count the animals to know that there are three of them. We know that many other mammals and birds, and probably other animals, can do the same. This is called subitizing, from the Latin subito for suddenly, although instantly probably gives a better idea of what it means. The word subitize was invented by an American scientist, in 1949, which is why it is usually spelt with a z even in Britain. I have used pictures rather than words to give you some practice at subitizing. Most people cannot subitize more than four things , so few of us could say how many animals there are in the biggest of the above pictures without actually counting them.

It is important to understand that subitizing is about real things, which we can actually see, not about numbers. We subitize three animals or two coloured shapes, not three or two.

Note that we can instantly call out the number when we throw a dice (correctly a die, dice is the plural) not because we are counting or subitizing the number of spots but because we are recognising a pattern - we recognise different people by their faces in the same way, and so do octopuses and many other animals.

There is a crucial difference between three and three animals. Many people today, but particularly teenagers working for school exams, make a mistake which is related to this. You have been given all the dimensions of a house and need to work out the total external surface area. You do all the sums on your calculator. Your calculator says 435, but this is not the right answer: the right answer is 435 m². Not putting in the correct units is the second quickest way of losing marks in an exam. What’s the quickest? Not reading the question properly!

Hunter-gatherers, and of course many animals, must be able to subitize in order to survive, but they do not need to be able to count. The Australian Aborigines were still hunter-gatherers until the Europeans came, and even today there are still some tribes of hunter-gatherers in Amazonia and a few other places, and most of them either cannot count at all or cannot count beyond three.

There is much more about subitizing, and hunter-gatherers, and the whole history of mathematics, in a most wonderful book by a Yorkshire teacher called Chris Waring: “From zero to infinity in twenty six centuries.” Do try to read it if you get the opportunity.

Animal husbandry and agriculture

About 10 000 BCE, as the last glaciers of the last Ice Age were disappearing, Man began to take his first steps into animal husbandry and agriculture. He learned how to control animals such as sheep and goats and plant and harvest crops such as cereals.

Today we think of cereals as foods like cornflakes which we have for breakfast, but they are called cereals because they are made from cereal crops. Cereal crops are similar to wild grasses, a long stalk with a seed head on the top. The main cereal crops are wheat, barley, oats, maize, millet, rye, and (the World’s most important food crop but the one usually missed off the list of cereals by most Westerners) rice. Different cereals grow best in different places, and are planted and harvested at different times of the year, for example wheat grows well in England and the Great Plains of North America, oats grow well in Scotland, rye in Scandinavia, maize in the southern United States and Mexico, and rice in most of Asia. Until recently the word corn was used for the local cereal (that is, the main cereal grown in that area), for example until the 1950s farmers in England would talk about their cornfields rather than their wheatfields, and cereals are called cereals after Ceres the Greek Goddess of the Corn, but today corn is increasingly used to refer only to maize, for example “corn flakes” are made from maize.

Shepherds need to be able to count their sheep, but growing cereals needs more than just the ability to count, for example you must know when to plant the seeds, and how much seedcorn to set aside - the seedcorn is the grains you must save from one year’s harvest to plant later on in the year for next year’s harvest. Eating the seedcorn was the worst possible crime, far worse than murder. Once you have harvested and stored the grains (keeping the seedcorn quite separate of course) you need to know how many days it is before the next harvest so you can plan how to make your stores of food last until then. So you need to know how many days there are in a year, and how to count them, and you need some sort of calendar.

You also need containers for the stored grain, to keep it dry and protect it from rats and mice and fungi. For this fired clay pots were used. Fired clay pots were first made about 7000 BCE, and you can read about this, and also clay as a writing material, on the Clay Page.

Cuneiform Writing

Once you have made a clay pot you can easily put marks on the outside of the pot before you fire it to make it look nicer - remember that Man had been making the most beautiful paintings and designs on cave walls for at least thirty thousand years so making a pot look pretty was second nature to him.

The next step was to make the marks mean something, such as “This Pot For Seedcorn Only” - not in words of course but in marks which would be instantly recognisable even to little children: what little child today would not know was behind the door marked

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The final stage was to put marks onto clay objects which were not pots.

Simple baked clay tokens (counters) were being used to record the number of sheep before 4000 BCE. One sheep was represented by a token in the form of a small baked clay ball with special markings (not a picture of a sheep) on it, and ten sheep by a baked clay ball with different markings. The tokens were strung together on a necklace.

These tokens still did not represent numbers, you could not do anything with them except boast about how many sheep you had. By about 3300 BCE the inhabitants of a city called Eridu, in southern Mesopotamia, were writing on clay tablets using a wooden stylus. The stylus left a wedge-shaped mark in the clay, which is why this type of writing is called cuneiform (from the Greek for wedge-shaped). Cuneiform was the World’s first written language: in cuneiform you could write down how many sheep you owned and add your name and address as well!

You could bake it if you knew you would want to keep it for a long time, but otherwise you just let it dry in the Sun. Unbaked tablets are very fragile, but even so hundreds of thousands of them have survived for more than five thousand years! (Some unbaked tablets got baked later when the house caught fire, or invaders burnt the whole village!)

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Mesopotamia (from the Greek for between the rivers) is the once very fertile land between the River Tigris and the River Euphrates, which eventually join together and flow into the Persian Gulf. Most of Mesopotamia is in modern Iraq. Eridu was the earliest city in Mesopotamia, but over the next two thousand years many other cities were built, the best known being Babylon.

The Ancient Egyptians wrote on papyrus, made from the papyrus reeds which grow along the banks of the River Nile. I write my Web Pages on papyrus (only joking) and if you look very carefully you can see the strips of the papyrus reeds which were used to make the papyrus this Page is written on. Papyrus does not grow in Mesopotamia so the Mesopotamians continued to use clay tablets and cuneiform for more than two thousand years after they were first used in Eridu. Of course what they wrote and the way they used maths changed over the years, in Eridu only the simplest numbers and ideas, in Babylon highly sophisticated mathematics and some of the most wonderful poetry ever written. The Babylonian Empire eventually came to an end when Babylon was captured by Cyrus the Persian in 539 BCE.

Today almost all writing is two-dimensional: flat marks on a flat surface. You can easily make a hand-written copy, photocopy it, fax it to Australia or scan it to store in your computer or put it onto your Web Site. You can also do this with something written four thousand years ago on papyrus. But cuneiform writing on clay tablets is essentially three-dimensional: the surfaces of the tablet are not flat and the characters are not just marks on the surface but are actually cut into it. Not only that, you can read this Page only because the letters are a different colour to the papyrus, but on a clay tablet there is no difference in colour between the writing and the clay. It is very difficult indeed to copy the characters by hand onto a flat surface suitable for photocopying or faxing or scanning to store in a computer and put onto a web page without distorting them in some way, a way which might even change their meaning, in the same way that we cannot make a two-dimensional map of the Earth which does not distort the size and shape of the continents.

People who study Mesopotamian history are called Assyriologists even though the Assyrian Empire covered only a part of Mesopotamia. It is very difficult for Assyriologists to read the writing on a clay tablet from a photograph or scanned image or paper copy, they really need to be actually holding it in their hands, and at the moment this is not easy because unbaked clay tablets are so fragile that only a few people are allowed to handle them.

Things are beginning to change: a clay tablet can now be scanned at very high resolution in three dimensions and the image put on a web site, so anyone in the world can use their 3-D printer to make a perfect copy, in full colour but in unbreakable plastic.

Today school students, when they are not using their laptops or tablets (electronic not clay!), write on paper, but the paper must be bought and paid for out of the school budget so the school must be careful about the way paper is used.

No such restrictions on the use of clay in Babylonian schools! We know a lot about Babylonian maths, and the way it was taught, because hundreds of thousands of clay tablets have survived, covering every aspect of life in Babylon, allowing us to look at students’ maths homework exercises, even their very private teenage diaries!

The Babylonians used a mathematical system involving lots of sixties (this is described in the Page on Bases, but sadly it is beyond the scope of this Page. But there are lots of other web sites about it.

Numbers as tools

Before Eridu numbers were really only used for counting: if you said “Three” by itself someone would ask you “Three whats?” The story of mathematics is about the way Man learned to use numbers as tools to solve problems he could not solve without them. This is discussed on the Page on Natural Numbers.

For example aquaducts. Today when we think of aquaducts we think of the Romans and the huge stone arches still crossing rivers and valleys all over North Africa, the Near East and Europe. In fact the aquaduct was the whole water channel, from the spring where the water came out of the hillside to the city many kilometres away, it is just that the bridges (Latin pontes) are usually the only parts surviving today.

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The Roman aquaducts were built primarily to bring water to the great Bath Houses of their cities. Each aquaduct was fed from a spring and you cannot turn a spring off, so the wonderful fountains in Roman cities were just ways of using the surplus water.

Mesopotamia was a very fertile land but it also had a very low rainfall and, unlike Egypt, it did not have an annual flood. So crops could not be grown without irrigation, and that meant building aquaducts to bring the water to the fields. Mesopotamia was criss-crossed with aquaducts two thousand years before Rome was built.

An aquaduct has to be downhill all the way: if it is not downhill for even one centimetre the water will just overflow over the sides. So building an aquaduct involves surveying the route very carefully and checking the slope at every point. And you cannot do this without using maths for much, much, more than counting.

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© Barry Gray July 2018