Ancient Mathematics

Mathematics is so integral to human existence that the development of mathematical thinking cannot be separated from the development of thinking itself.

So rather than look specifically for origins, we study archaeological and other artefacts for evidence of mathematical activity. However, attributing mathematical meaning to very ancient objects is no easy task.

It not only requires mathematical knowledge but it requires historical and cultural knowledge as well. And it often leads to controversy!

The Ishango Bone

A famous example, and one of the oldest objects believed to be of mathematical significance, is the Ishango bone which was dug up in the 1950s at a village called Ishango on the shores of Lake Edward in the Democratic Republic Of Congo.

ScienceMuseumOfBrussels via Wikimedia
The Ishango Bone [Image: Science Museum of Brussels]

The bone, which is engraved with a series of notches, has been carbon-dated to about 20,000 BCE. The bone’s discoverer suggested that the notches may represent an arithmetical game and that the patterning is strongly suggestive of a counting system based on 10 and knowledge of multiplication.

But other scholars have criticised this view, suggesting instead that the notches can be better explained by relating them to time-keeping and a count of periods of the moon.

Which of the two views is right? Or are they both wrong? We cannot know for sure. However, one thing we can be certain about is that historical sources do not speak for themselves. They require interpretation.

And that interpretation should take account not only of the content of the source but also of the context in which the source was produced. In the case of the Ishango bone, for example, it is its age combined with its mathematical content that makes it especially significant. Its mathematical content alone is not enough.

When considering ancient mathematical texts, such as those from Egypt and Mesopotamia, it is very easy to get seduced into considering only the numbers in the texts and to exclude everything else. This is because once you know the number system, the numbers themselves are easy to read—you do not need to be an Egyptologist or an Assyriologist to read them—and interpreting the ‘everything else’, including words in the text, is hard!

But if we allow ourselves to be seduced in this way, we not only run the risk of arriving at misleading or erroneous conclusions but we gain no understanding of the underlying culture.

Plimpton 322

A good example of the perils of the number only approach is the case of the Babylonian tablet known as Plimpton 322 (named after its first Western owner, the New York publisher George Plimpton, who bought it in 1922), arguably the most famous of all Babylonian mathematical tablets. The tablet contains elements of Pythagorean triples (sets of numbers that satisfy the equation x² = y² + z²) and in the past it was seen by some scholars (who considered only the numbers on the table) as a trigonometric table. But by considering the text in its entirety and placing it in its historical context, the leading Assyriologist and expert on Babylonian mathematics, Eleanor Robson, has shown this interpretation to be erroneous.

The fact that Robson knew from her detailed study of Babylonian culture that the Babylonians had no conceptual framework for angle measurement or trigonometry is, of course, no coincidence.