### Claas Lattmann

Most of early Greek mathematics lies in darkness: though Thales at the beginning of the 6^{th} century BCE might have been the earliest Greek mathematician, as the ancient tradition has it, the first authentic first-hand testimonies date only from the first half of the 4^{th} century BCE. All that we know about the earlier stages of Greek mathematics comes from later authors who, however, were (re-)writing them in the terms of (post-) Euclidean mathematics.

Though not regarded as a mathematician proper himself, one of the first authors to give an undistorted glimpse of pre-Euclidean mathematics is Plato. His works contain the earliest direct references to Greek mathematics, and this in great number. Among them is a quite odd and often-discussed passage in the seventh book of the *Laws*. The speaker is the Athenian:

My dear Kleinias, I was utterly astounded myself having heard only quite lately of our condition in regard to this matter. It seemed to me to be the condition not of human beings, but of guzzling swine, and I was ashamed, not only of myself, but of all the Greeks.

What the Athenian is so ashamed about is his and the Greeks’ state of knowledge with regard to “incommensurability” – one of the most important concepts of ancient Greek mathematics, namely the insight that some mathematical magnitudes in principle do not have a common measure with each another. This concept may be regarded as equivalent to the modern concept of irrationality, if one of the magnitudes is the number 1. For example, the number 1 and the square root of 2 are incommensurable, insofar there is neither a number nor a part of a number that might serve as a common measure (or, in modern terms, factor) for these two numbers. One of the most well-known examples from ancient mathematics is the relation between the side and the diagonal of the square, as it is being discussed in Plato’s *Meno *(82a–85b).

What is most remarkable with regard to the *Laws* passage, however, is not that “the Athenian” thinks that the Greeks have an insufficient knowledge of mathematics with regard to some specific problem at some specific point in time – though, to be sure, this, too, is quite interesting from a history of science perspective. Rather, it is decidedly more interesting that this passage implies that the Greeks mathematicians acquired knowledge about the phenomenon of incommensurability only just recently – that is, shortly before the composition of the *Laws*, which took place around 350 BCE with an obvious *terminus ante quem* of 348/7 BCE (with the exception, of course, of revisions done by Philippos of Opus). Though the formulation ἀκούσας ὀψέ ποτε would leave it open that only “the Athenian” (Plato) has learned just lately of the phenomenon of incommensurability, in effect his being ashamed of all the Greeks does, in all probability, not, thus implying that the reference is to the general state of mathematical theory and not to some autobiographical circumstance. For all this, of course, the premiss has to be accepted that we may interpret (at least some of) the utterances made by the dialogue’s characters as referring to the time of composition and not to the time of its fictional setting. Though this in general seems to be admissible (and often advisable, for Plato was not interested in a “historical,” but in a philosophical argument), in this case this would not even be quite problematic at all, for the fictional date of the *Laws* seems to be some time in the (though early?) first half of the fourth century BCE.

### Session/Panel Title

Ancient Greek Philosophy (organized by the Society for Ancient Greek Philosophy)

### Session/Paper Number

61.1