## Synopsis

## Excerpt

Early Greek mathematics was not one but many; there were various levels of practice, from calculations on the abacus to indirect proofs concerning incommensurable lines, and varying attitudes, from laughing off attempts to square the circle to using attempts to square the circle as examples in a second-order discussion about the nature of demonstration. In sum, different forms of mathematics were used for different purposes by different groups of people. Perhaps one common feature is clearly distinguishable: mathematics was a public activity, it was played out in front of an audience, and it fulfilled functions that were significant at a communal level, be they counting revenues, measuring out land or exploring the limits of persuasive speech. The first question addressed in this chapter is what I call the problem of political mathematics. I take 'political' in the literal Greek sense, as something that has to do with the *polis*, the city/community/state. When reading fifth-and fourth-century BC philosophical sources, I have always been struck by the frequency with which mathematical images or examples are used to make points which are not related to mathematics at all - often, points about politics. Moreover, Plato has some very interesting statements on the question of who mathematics should be for, and which mathematics ought to be done by whom: he established parallel hierarchies between forms of mathematics and categories of people. Once again, these were deeply political statements. So, having warned the reader in the introduction that I will ask questions rather than answering them, the first section will expand and muse on the theme, what were the political functions of early Greek mathematics?

The second section will tackle a historiographical issue: how later ancient sources depict early Greek mathematics, and what can be done with them. It will be, I am afraid, an exercise in scepticism.