Inventing Incommensurability. Traces of a Scientific Revolution in Early Greek Mathematics in the Time of Plato

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.