Leonid Zhmud

This essay examines the quantitative aspects of Greco-Roman science, represented by a group of established disci¬plines, which since the fourth century BC were called mathēmata or mathē¬ma¬tikai epistē¬mai. In the group of mathēmata that in Antiquity normally comprised mathematics, mathematical astronomy, harmonics, mechanics and optics, we have also included geography. Using a dataset based on The Encyclopaedia of Ancient Natural Scientists, our essay considers a community of mathēmatikoi (as they called themselves), or ancient scientists (as they are defined for the purposes of the present paper) from a sociological point of view, focusing on the size of the scientific population known to us and its disciplinary, temporal and geographical distribution. A diachronic comparison of neighboring and partly overlapping communities, ancient scientists and philosophers, allows the pattern of their interrelationship to be traced. An examination of centers of science throughout ancient history reveals that there were five major centers – Athens, Alexandria, Rhodes, Rome and Byzantium/Constantinople – that appear and replace one another in succession as leaders. These conclusions serve to reopen the issue of the place of mathēmata and mathēmatikoi in ancient society.

This essay considers the origins of the arithmological genre, the first specimen of which was an anonymous Neopythagorean treatise of the first century BCE. Arithmology as a special genre of philosophical writings dealing with the properties of the first ten numbers should be distinguished from number symbolism, which is a universal cultural phenomenon related to individual significant numbers. As our analysis shows, the philosophical foundations of arithmology were laid down in the treatise of Plato’s successor Speusippus On Pythagorean Numbers, who relied on the Platonic doctrine of the ten ideal numbers, whereas in ancient Pythagoreanism arithmological notions, unlike number symbolism, are not attested. In the first century BCE, an epoch of revival of Platonism and Aristotelianism, Speusippus’ ideas received a second birth, thus marking the beginning of arithmology as a popular genre.

This chapter explores the kind of community founded by Pythagoras. It considers those types of association which actually existed in Greece in the Archaic and Classical periods. If the Pythagorean community was really a religious association, it should conform to the type of religious association of its time, and not to that of the Qumran community or a Christian monastery. To describe the nature of the society founded by Pythagoras, we may choose from a very small number of variants available for that period: a philosophical school, like those which appeared in the fourth century; a cultic community ; and a sociopolitical association. The chapter shows, first, that none of the features of a religious community is confirmed by reliable evidence. Second, that to apply to the Pythagoreans the term ‘sect’, as developed in the sociology of religion, is methodologically unsound. Third, that the tradition on Pythagorean ‘symbols’, reflected in Anaximander the Younger and Aristotle, mostly bears no relation to the realities of the Pythagorean way of life, while the picture which arose on the basis of that tradition, of acusmatici and mathematici dates from the Imperial period.

This chapter reviews references to Pythagoras by the authors of the pre-Platonic period. It shows that in the course of the fourth century, studies in mathematics, particularly geometry and arithmetic, became a constant element of the tradition of Pythagoras; astronomy and harmonics are less frequently mentioned. Mathematics did not displace metempsychosis and wonders, nor did the tradition of Pythagoras the politician which emerged concurrently with it, yet they did edge them aside, completing the ambivalent, contradictory image of Pythagoras, which was retained by his Neoplatonic biographers and passed from them into modern scholarship. It concludes that Pythagoras the mathematician is as little a product of the Academy as Pythagoras the philosopher.

This chapter first considers the estimates of how great the contribution of the Pythagoreans was to Plato's philosophy and how these views diverge substantially, and which vary across the range from ‘decisive’ to ‘insignificant’. It shows that Platonists were characterized by a benevolent attitude to Pythagoras and the Pythagoreans and an interest in their scientific, philosophical, and religious theories. Number doctrine is found in the testimonies of all three Platonists, but there is in them no picture of a Pythagorean philosophy even remotely reminiscent of number doctrine. The Platonists reacted, not to a common Pythagorean doctrine, but to various theories of Pythagoras and his successors: Philolaus, Archytas, Ecphantus and Hicetas, et al. The chapter then considers Aristotle's reports on the Pythagoreans in his surviving works.

This paper discusses continuity between ancient Pythagoreanism and the pseudo-Pythagorean writings, which began to appear after the end of the Pythagorean school ca. 350 BC. Relying on a combination of temporal, formal and substantial criteria, I divide Pseudopythagorica into three categories: 1) early Hellenistic writings ascribed to Pythagoras and his family members; 2) philosophical treatises written mostly, yet not exclusively, in pseudo-Doric from the turn of the first century BC under the names of real or fictional Pythagoreans; 3) writings attributed to Pythagoras and his relatives that continued to appear in the late Hellenistic and Imperial periods. I will argue that all three categories of pseudepigrapha contain astonishingly little that is authentically Pythagorean.

This chapter discusses how all the Pythagorean theories of the soul known to us are different. Only Simmias and Echecrates, the pupils of Philolaus, held identical views. The similarity among some of these theories can sometimes be explained by direct influence, but most often by the fact that many Pythagoreans shared the interpretation of the soul as the source of motion, which was the most widespread view amongst the Presocratics. It is also shown that there is neither a clear formulation of number doctrine, nor tangible examples of it, in ancient Pythagoreanism. It transpires that the theory which was taken to be the essence of Pythagorean philosophy, a theory backed by the authority of Aristotle as the source and as a historian of philosophy, is, like many other elements of the ancient tradition about the Pythagoreans, a retrospective projection.

This chapter begins by exploring notions of Shamanism in ancient Greece. It argues that there are no traces of shamanism or its most important component, ecstatic cult practice, either in Pythagoreanism or among the Scythians who supposedly influenced it, even if we assume that shamanism existed at that time. Furthermore, it seems unlikely that the Pythagoreans had any special cult at all. The chapter then considers the historical and religious context of metempsychosis. It addresses the following questions: Was metempsychosis borrowed by the Greeks, and if so from whom? Who in Greece was the first to preach it, the Orphics or Pythagoras, and did any Orphics live in Pythagoras' lifetime? How widespread was it among the Pythagoreans, and how consistent were they in observing the behavioural norms that sprang from it, such as the ban on eating meat? Can Pythagorean metempsychosis be considered an ethical doctrine? Behind each of these questions lies a long history of contradictory interpretations.

This introductory chapter focuses on the Pythagorean question, which remains one of the most intricate in the history of early Greek science, philosophy, and religion, and has every chance of being consigned to the category of insoluble problems. It suggests that like any other complex scientific problem, it can be broken down into a number of smaller, particular ones, which may prove amenable to solution. There are many facts on which agreement may be reached; there is also an undoubted hierarchy of interpretations, ranging from those admittedly impossible or unverifiable to those which are more probable and internally consistent.

This chapter begins with a discussion of Pythagoras and the science of music. The fundamental principles of Greek musical theory were taken up and developed by European musicologists. Three basic elements of that theory which the ancient tradition linked with Pythagoras continue to be associated with his name: the mathematical treatment of music; the doctrine of a musical ethos, or the psychagogic and educative effects of music; and the famous ‘harmony of the spheres’ generated by the movement of the heavenly bodies. The discussion then turns to the basic principles of correspondence between sounds and numbers adopted in Pythagorean harmonics, and acoustic experiments described in the ancient literature of musical theory.

This chapter turns from Pythagorean religion to Pythagorean science, primarily to mathematics. It shows that the similarities between Oriental calculations and Greek geometry are delusory, while others are perceived only by someone raised on the analytical geometry of Descartes and capable of translating Babylonian problems into the language of geometrical theorems. It then considers the systematic application of deductive proof, which was the most important factor in the formation in ancient Greece of theoretical mathematics on an axiomatic basis. This is followed by a discussion of Pythagoras' mathematics in fourth-century testimonies, Pythagoras as a mathematician, and Hippasus and Pythagorean mathematics of the first half of the fifth century.

This chapter discusses the Pythagorean ‘symbols’ and their custodians, the mathematici and the acusmatici. The ‘symbols’ are short sayings divided into three kinds according to the question they answer. The first kind answer the question, ‘What is...?’ The second kind answers the question, ‘What is most...?’ The third and most important kind contains precepts and prohibitions. In its application of the ‘symbols’, the following question is addressed: was there in the history of ancient Pythagoreanism a period in which the precepts and taboos they contain were observed to the letter, and, if there was, then what circle of persons was affected?

One of the epithets most frequently applied to Pythagoras in the majority of popular books, as well as many scholarly works, is ‘legendary’ or ‘semi-legendary’. In the tradition on Pythagoras it is true that from the very beginning facts have been interwoven with fantastic invention, but it is not too difficult to separate the two. Extracting the real events in his life from information which appears to be quite plausible is much more difficult. This is where we encounter the greatest number of controversies, though these occur, of course, in the biographies of any of the Presocratics, and Pythagoras is no exception. Moreover, we have far more reliable information about his life than about any other philosopher of the sixth century, be it Thales, Anaximander, Anaximenes, or Xenophanes. The reason is evident: Pythagoras was spoken of and written about much more often. His great fame had the twin effects of making his name the focus of legends, which multiplied over the centuries, and of preserving the memory of the historical events of his time. This chapter begins with a discussion of the main sources of biographical information on Pythagoras. It then details his life on Samos and his travels outside Greece.

This chapter begins with a discussion of Egyptian and Babylonian influences in Greek astronomy. It considers the development of Pythagorean astronomy before Philolaus. It then focuses on the difficulty of identifying an individual contribution to astronomy by Pythagoras or specific early Pythagoreans. It shows that Alexander relied on Aristotle, who connected with Philolaus neither the harmony of the spheres nor the geocentric model on which it is based. The surviving works of Aristotle actually contain no indication that he associated the harmony of the spheres with Philolaus' system.

The figure of the cordial host of the Academy, who invited the most gifted mathematicians and cultivated pure research, whose keen intellect was able if not to solve the particular problem then at least to show the method for its solution: this figure is quite familiar to students of Greek science. But was the Academy as such a center of scientific research, and did Plato really set for mathematicians and astronomers the problems they should study and methods they should use? Our sources tell about Plato's friendship or at least acquaintance with many brilliant mathematicians of his day (Theodorus, Archytas, Theaetetus), but they were never his pupils, rather vice versa -- he learned much from them and actively used this knowledge in developing his philosophy. There is no reliable evidence that Eudoxus, Menaechmus, Dinostratus, Theudius, and others, whom many scholars unite into the group of so-called "Academic mathematicians," ever were his pupils or close associates. Our analysis of the relevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' "Catalogue of geometers", and Philodemus' "History of the Academy", etc.) shows that the very tendency of portraying Plato as the architect of science goes back to the early Academy and is born out of interpretations of his dialogues

Apart from his teachings, wonders and scientific discoveries, Pythagoras was also known for his wide-ranging journeys. Ancient authors alleged that he visited many countries and nations from Egypt to India, stayed with the Phoenicians and the Ethiopians and talked to the Persian Magi and Gallic Druids. However, he never went to the North. If, nevertheless, he was eventually associated with the northern inhabitants, it is only because they themselves came into close contact with him. The first of them was Zalmoxis, a deity of a Thracian tribe, the Getae, who guaranteed them immortality after death. Having described a blood ritual that the Getae practised to become immortal, Herodotus relates a story he heard from the Hellespontine and Pontic Greeks. It goes that Zalmoxis was not adaimōnbut a former slave of Pythagoras on Samos and, having adopted the doctrine of immortality from him, he returned to Thrace and converted his tribesmen to it with a cunning trick. He invited the most prominent of them to a men's hall for entertainment and told them that neither he nor they or their descendants would die, but would live eternally. Then, having constructed a secret underground chamber, he suddenly disappeared from the eyes of the Getae and hid in his shelter for three full years, being lamented as dead. Then he showed himself again to the Getae, thus persuading them of the truth of his promises.