Statue of Muhammad ibn Musa al-Khwarizmi (ca 780-850), Persian mathematician, Khiva, Uzbekistan. (Photo by DeAgostini/Getty Images)

Decolonise … Maths?

Algebra and arithmetic in the modern sense were not due to the Greeks, writes Mark Ronan

Artillery Row

“Decolonise mathematics” is a phrase I’ve heard recently. What does it mean? It occurs within the context of reassessing who contributed what, and where the basic ideas came from. Who were various races and creeds responsible, and do the Greeks deserve the credit they often receive?

These are reasonable questions. A hundred years ago our answers would have been different because we didn’t know as much as we do now. But one thing remains the same. Euclid’s great treatise on geometry — his Elements — is a massive contribution to mathematics. He defines everything he uses so there can be no uncertainty, and all assumptions are stated explicitly. For example between any two points there is a straight line. That’s an assumption. If for instance you puncture the plane with a small hole it’s no longer true because lines that go across the hole are broken in two. Reading Euclid’s work you find proofs that are exquisitely detailed. Nothing is left out.

I have heard claims that we unfairly credit the Greeks, but that is to ignore the work of historians in the past hundred years. Algebra and arithmetic in the modern sense were not due to the Greeks. What we do owe to them is clarity and methodology, contained within a beautifully structured whole. Little wonder Euclid’s Elements was translated into Arabic for scholars in the early Islamic world who wanted to learn from what had gone before. In fact the first Latin translation was made from Arabic in 1120 by Adelard of Bath. Euclid’s assumptions were so well judged that one in particular was thought by several Arabic and later European scholars to be unnecessary, a failure on his part to prove the result concerned and replace it by an axiom. It was not. It was truly necessary, and in the early nineteenth century a Hungarian mathematician proved that an alternative, consistent geometry was possible without it. Non-Euclidean geometry was born.

Algebra like many words beginning with al– has its roots in Arabic. It refers to rebalancing by adding or subtracting to two sides of an equation, and was pioneered by the Persian scholar Al-Khwarizmi who became a senior member of the House of Wisdom in Baghdad in the early ninth century. The word algorithm is taken from his name. The modern notation for algebra came much later, but the study of equations goes back far before Al-Khwarizmi. The Arithmetica written by Diophantus of Alexandria in the third century deals with solutions of algebraic equations, which involve the concept of adding lengths (x) and areas (x-squared). That idea is already present in the work of Heron of Alexandria in the first century, and goes back much further to the ‘Old Babylonian’ period from 2000 to 1600 BC in Mesopotamia. The word ‘Mesopotamia’ is Greek, meaning the land between the rivers (Tigris and Euphrates), the place from where Greeks of the Hellenistic period absorbed huge amounts of astronomy and mathematics. In local parlance from earlier days the area was called ‘Sumer and Akkad’.

Akkad was a city that rose to power in the second half of the third millennium BC and became the capital of the Akkadian Empire. Its founder Sargon the Great was a king who inspired a Moses-like story: as a baby his mother placed him in a basket, sealed with pitch, and floated him down the river. This was a man who later ruled an empire, and arranged for his native Akkadian language to be written in cuneiform for the first time.

Cuneiform was a writing system that emerged in Uruk (origin of the word Iraq) in the late fourth millennium BC, the largest city in the world at the time. It was used to express the Sumerian language. Uruk was home to king Gilgamesh, hero of a famous epic. The final tablets of this epic stunned the public when they were unearthed in Iraq during the late nineteenth century. Suddenly we had found a pre-Biblical version of the Flood story.

This number system is far and away superior to anything the Egyptians, Greeks or Romans had, but most people know nothing about it

This energised a whole new academic discipline called Assyriology, and by the middle of the twentieth century we had begun to understand their mathematical achievements. These were made possible by the greatest number system the world has ever seen. It worked to base-60, which is why we measure time in divisions of 60, sixty seconds in a minute, sixty minutes in an hour. But the really great innovation was that each number was written as a sequence of digits just as we do today. In our system the position of each digit determines its value up to a power of 10, in other words, units, tens, hundreds and so on — or in their case up to a power of 60. It was a fantastic advance and they used it equally for whole numbers and fractional quantities, just as we do with decimal fractions. In this beautifully flexible system you needed only to add extra digits to a number to get more accuracy, just as we do by adding extra places of decimals. That set them on the road to calculating square roots, and then solving equations involving squares. Something no other civilisation had done before. This number system, developed before 2000 BC by the Sumerians, is far and away superior to anything the Egyptians, Greeks or Romans had, but most people know nothing about it. It lasted for over two thousand years, until cuneiform writing disappeared completely. Its last hurrah was for astronomical texts in Mesopotamia.

In the ancient days when the number system was first created, Sumerians and Akkadians lived side by side, but by 2000 BC Sumerian had died out as a spoken language. It was still used for written documents, particularly technical ones, rather like Latin in mediaeval and Renaissance Europe, but who were the Sumerians? Unlike the Akkadians they were not a Semitic people, and DNA analysis is equivocal. They are distinguished by their language, which is unrelated to any other we know of. That hasn’t stopped people from trying to find its linguistic cousins. I recall one Sumerian expert telling me someone had written a book relating it to Japanese. She obviously found that amusingly foolish. And she was Japanese. We don’t know where the Sumerians came from, but they lived in the south of Iraq, not the north, and looked towards an earlier paradise on ‘Dilmun’ further south on the coast of Arabia. So a southern origin is plausible, and they referred to themselves as the ‘black-headed ones’.

By creating the greatest number system the world has ever seen, they made a greater contribution to mathematics than almost anyone else. In the present climate where accusations of racism are levelled across Western civilisation, it is rather convenient that we don’t know their skin colour or who their ethnic cousins were. Their pioneering number system allowed the Akkadians to do great mathematics and later on mathematical astronomy. Both were inherited by the Greeks, and later the Islamic world. A separate Greek tradition of philosophy imbued geometry with careful argument, definitions and axioms, and a treatise that lasted over two thousand years. You cannot take that away from them.

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