MATHEMATICS IN OLD GREECE FROM THE VIth TO THE IVth CENTURY Before Common Era (BCE) FROM PYTHAGORAS TO EUCLID History of mathematics, Bologna, October 2013 Part I From Arithmetic to Pythagoras’ theorem I. Introduction. Before beginning the course, we need to understand its interest, the main problems and difficulties, and the background. As very young people, something which happened more than 25 centuries ago, for most of you more than hundred times your age, could seem very, very, very boring. No blackboard, no pens, no cars and even no mobiles, what is it possible to expect from people living like this? Since mathematics has made huge progress from this time, why learning what was done then? Since we have no more any book from this time, how is it possible to say anything about it? 1 - What: Contrary to nowadays, at this time there were no clear borders between such different fields as Mathematics, Philosophy, Astronomy, Physics and even Music. As such it is a crucial period to try understanding what mathematics is. - Why: To understand what something is we need to know its origins. It is true for our present world, and it is not different for mathematics. To know what we are doing with mathematics, we need to know something about its origins, how people began to think such abstract objects like numbers, forms and space. - How: The first complete mathematical treaty we have is the Euclid’s Elements, from the 4th century BCE. Before, not only we have no treaty, even incomplete, but we have no mathematical text. Moreover, the most ancient mathematical passages are not found in mathematical books, there are no such books, but in texts from completely different fields, such as philosophy or rhetoric or even literature. We will have to study and interpret them with all the difficulties it implies. - Contrary to mathematics where every (solvable) problem seems to have one solution, we are never sure here of even the existence of a solution. At best we have more or less good hypotheses, so everyone has to make his own inquiry, to check and finally to decide which one is the most credible one. A claim, found from time to time, such the following: ‘This was proved by XXX in her dissertation YYY. Her results in this respect seem absolutely certain and have been universally accepted’1 is certainly wrong (‘universally accepted’), and shows rather a refusal to discuss the question (since it is solved once and for all), implying (almost) surely the weakness of the argumentation. 1 Excerpt of an article of Kurt von Fritz, The Discovery of Incommensurability by Hippasus of Metapontum, Annals of mathematics, 46, 2, 1945, otherwise an extremely interesting article. 2 II. Pythagoras and the Pythagorean School. The difficulties to study Pythagoras and his school are due, among other things, to the following hindrances: - We have no texts of him, not even indirectly, and he is said not to have ever written anything, though some authors as Diogenes Laertius disagree. The information we get are more and more precise as time is passing, a sure indication of a legend in building. - For instance, almost 1000 years after his death, we have a life of Pythagoras with a wealth of details written by Iamblichus. But till the 4th century BCE, we have only few sentences on him. Nevertheless, the absence of information on people especially mathematicians (including Euclid) is the norm in the Antiquity. - Since he was the chief of a school, his followers were keen to give him all the discoveries by members of the school. Later, when commentators wanted to consider some result of unknown author(s), it was convenient to put his name for discoveries made at his epoch. Because of the structure of mathematics, it is much more difficult to give a wrong period for the obtention of a result than to give a wrong name for the one who got it. So we have to carefully check everything attributed to him. Fortunately, we are more interested in the dating of mathematical results than in their authors. Another problem is his school lasts 2 centuries (roughly from the beginning of the 6th century to the beginning of the 4th century BCE). Even if we agree a result was known by the Pythagoreans, its dating is not obvious. So once again we need to compare the testimonies, to oppose them each other and to what was possibly known at this time in mathematics. We will proceed as the following: since even people who lived thousand years after Pythagoras were much closer from him and the sources on him than us, we have to take seriously their claims. But we have also to be very circumspect with these texts and to discuss their plausibility. The very existence of Pythagoras is sometimes put in doubt. Nevertheless many early authors are alluding directly or indirectly to him, among them: Herodotus, Heraclites and Empedocles (cf. Diogenes Laertius, Lives and Opinions…, book VIII). Plato spoke twice in the Republic about him or his school: ‘The second, I said, would seem relatively to the ears to be what the first is to the eyes; for I conceive that as the eyes are designed to look up at the stars, so are the ears to hear harmonious motions; and these are sister sciences - as the Pythagoreans say, and we, Glaucon, agree with them?’ (Republic,, VII, 530d) ‘But, if Homer never did any public service, was he privately a guide or teacher of any? Had he in his lifetime friends who loved to associate with him, and who handed down to posterity a Homeric way of life, such as was established by Pythagoras, who was so greatly beloved for his wisdom, and whose followers are to this day quite celebrated for the order which was named after him?’ (ib., X, 600b). 3 In another book, Plato refers probably to him or some of his followers, when he spoke about ‘some smart fellow, a Sicilian, I daresay, or Italian’ (Gorgias, 493a). There are also many texts of Aristotle on the ‘Pythagoreans’, but few on Pythagoras himself. Since his partisans as well as the commentators were eager to give the paternity of many discoveries to the chief of their school, it is preferable to understand the name ‘Pythagoras’ as indicating rather than the man himself the school of the ‘old Pythagoreans’ i.e. a group of men2 around the time of Pythagoras. Moreover in mathematics it is much more difficult to make wrong claims concerning the dating of results rather than its author. So without strong reasons running against them, we may accept at least the dating for the theorem attributed to Pythagoras or his school i.e. the 6th century BCE. Inversely some keep extremely sceptical views about Pythagoras (and the Pythagoreans) with respect to mathematics, as seen in the below quotation extracted from the Stranford Encyclopedia of Philosophy: ‘Pythagoras, one of the most famous and controversial ancient Greek philosophers, lived from ca. 570 to ca. 490 BCE. (…) Pythagoras wrote nothing, nor were there any detailed accounts of his thought written by contemporaries. By the first centuries BCE, moreover, it became fashionable to present Pythagoras in a largely unhistorical fashion as a semi-divine figure, who originated all that was true in the Greek philosophical tradition, including many of Plato's and Aristotle's mature ideas. A number of treatises were forged in the name of Pythagoras and other Pythagoreans in order to support this view. The Pythagorean question, then, is how to get behind this false glorification of Pythagoras (…) The early evidence shows, however, that, while Pythagoras was famous in his own day and even 150 years later in the time of Plato and Aristotle, it was not mathematics or science upon which his fame rested.’ 2 And of women, since if Iamblichus (Pythagoras’ life, 267) and (maybe) Diogenes Laertius (Lives and Opinions…, VIII, 41) are right, women were accepted in his school 4 III. About Pythagoras’ life. Everything about Pythagoras’ life is extremely controversial, more or less legendary. Nevertheless, keeping this in mind, we will give some indications on his life with which most scholars agree. - He lived in the 6th century (some authors give more precise date like 570-495 BCE). He was born at Samos (a Ionian Greek city) and died at Metapontum (near the actual Taranto) in so-called ‘Great Greece’ i.e. the south of actual Italy. - He traveled and stayed many years in Egypt, then in Babylon, according to Iamblichus, as a ‘prisoner of war’, but this is highly disputed. - Around 520 BCE he settled in ‘Great Greece’ firstly in Croton then in Metapontum where he died. In Croton, he founded a school with some strict rules of secrecy, at once religious, philosophical and political. These rules are often connected to the secrecy of religious ceremonies in Egypt (Herodotus, Histories, II, 81), but they could as easily be Greek, for instance to the so-called ‘Orphic mysteries’. To summarize, let us give once again a skeptical point of view, as the following quotation from Luc Brisson: 5 ‘From an historic point of view, we know very little on the origins, the formation and the activity of Pythagoras. He would be born at Samos at the beginning of the 6th century and would emigrate to Croton, where he would have taken up the power. A revolt would overturn the Pythagoreans at the end of the century, a little after 510, but the eclipse was short-lived 3 since it seems they control a solid bloc of territory between Metaponte and Locres till 450.’ 3 p.