An examination of the emergence of the phenomenon of deductive argument in classical Greek mathematics.

The paper offers a running commentary on ps-Archytas’ On Intellect and Sense Perception (composed ca. 80 BCE), with the aim to provide a clear description of Hellenistic/post-Hellenistic Pythagorean epistemology. Through an analysis of the process of knowledge and of the faculties that this involves, ps-Archytas presents an original epistemological theory which, although grounded in Aristotelian and Platonic theories, results in a peculiar Pythagorean criteriology that accounts for the acquisition and production of knowledge, as well as for the specific competences of (...) each cognitive faculty. (shrink)

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. (shrink)

A fascinating portrait of the Pythagorean tradition, including a substantial account of the Neo-Pythagorean revival, and ending with Johannes Kepler on the threshold of modernism.

This is the first comprehensive study for nearly 200 years of what remains of the writings of the Presocratic philosopher Philolaus of Croton. These fragments are crucial to our understanding of one of the most influential schools of ancient philosophy, the Pythagoreans; they also show close ties with the main lines of development of Presocratic thought, and represent a significant response to thinkers such as Parmenides and Anaxagoras. Professor Huffman presents the fragments and testimonia with accompanying translations and introductory chapters (...) and interpretive commentary. He not only produces further argument for the authenticity of much that used to be neglected, but also undertakes a critique of Aristotle's testimony, opening the way for a quite new reading of fifth-century Pythagoreanism in general and of Philolaus in particular. Philolaus is revealed as a serious natural philosopher. (shrink)

Archytas of Tarentum is one of the three most important philosophers in the Pythagorean tradition, a prominent mathematician, who gave the first solution to the famous problem of doubling the cube, an important music theorist, and the leader of a powerful Greek city-state. He is famous for sending a trireme to rescue Plato from the clutches of the tyrant of Syracuse, Dionysius II, in 361 BC. This 2005 study was the first extensive enquiry into Archytas' work in any language. It (...) contains original texts, English translations and a commentary for all the fragments of his writings and for all testimonia concerning his life and work. In addition there are introductory essays on Archytas' life and writings, his philosophy, and the question of authenticity. Carl A. Huffman presents an interpretation of Archytas' significance both for the Pythagorean tradition and also for fourth-century Greek thought, including the philosophies of Plato and Aristotle. (shrink)

This is a comprehensive, authoritative and innovative account of Pythagoras and Pythagoreanism, one of the most enigmatic and influential philosophies in the West. In twenty-one chapters covering a timespan from the sixth century BC to the seventeenth century AD, leading scholars construct a number of different images of Pythagoras and his community, assessing current scholarship and offering new answers to central problems. Chapters are devoted to the early Pythagoreans, and the full breadth of Pythagorean thought is explored including politics, religion, (...) music theory, science, mathematics and magic. Separate chapters consider Pythagoreanism in Plato, Aristotle, the Peripatetics and the later Academic tradition, while others describe Pythagoreanism in the historical tradition, in Rome and in the pseudo-Pythagorean writings. The three great lives of Pythagoras by Diogenes Laertius, Porphyry and Iamblichus are also discussed in detail, as is the significance of Pythagoras for the Middle Ages and Renaissance. (shrink)

This long-awaited study of the most enigmatic figure of Greek philosophy reclaims Socrates' ground-breaking originality. Written by a leading historian of Greek thought, it argues for a Socrates who, though long overshadowed by his successors Plato and Aristotle, marked the true turning point in Greek philosophy, religion and ethics. The quest for the historical figure focuses on the Socrates of Plato's earlier dialogues, setting him in sharp contrast to that other Socrates of later dialogues, where he is used as a (...) mouthpiece for Plato's often anti-Socratic doctrine. At the heart of the book is the paradoxical nature of Socratic thought. But the paradoxes are explained, not explained away. The book highlights the tensions in the Socratic search for the answer to the question 'How should we live?' Conceived as a divine mandate, the search is carried out through elenctic argument, and dominated by an uncompromising rationalism. The magnetic quality of Socrates' personality is allowed to emerge throughout the book. Clearly and forcefully written, philosophically sophisticated but entirely accessible to non-specialists, this book will be of major importance and interest to all those studying ancient philosophy and the history of Western thought. (shrink)

The main source of Archytas, fr. 1 Diels-Kranz is Porphyr. in Ptol. harmon. p. 56,5–57,27 Düring; there is also an extensive quotation of its initial part in Nicomachus, Introd. Arithm. p. 6,16–7,5 Hoche. In recent years both the text and the interpretation of this fragment, whose authenticity was questioned by W. Burkert, have been re-examined, and a good deal of progress has been made especially by paying more attention to the nature of Nicomachus' quotation and its context.

Ancient Greek music was purely or predominantly melodic; and in such music subtleties of intonation count for much. If our sources of information about the intervals used in Greek music are not always easy to interpret, they are at any rate fairly voluminous. On the one hand we have Aristoxenus, by whom musical intervals were regarded spatially and combined and subdivided by the processes of addition and subtraction; for him the octave consisted of six tones, and the tone was exactly (...) divisible into fractions such as the half and quarter, so that the fourth was equal to two tones and a half, the fifth to three tones and a half, and so on. On the other hand we have preserved for us in Ptolemy's Harmonics the computations of a number of mathematicians, who realized correctly that intervals could only be expressed as ratios , that the octave was less than the sum of six whole tones and that this tone could not be divided into equal parts. These authorities are Archytas, the Pythagorean of the early fourth century, Eratosthenes , Didymus and Ptolemy himself . To these we must add the scale of Plato's Timaeus and, closely related to it, the computations of the pseudo-Philolaus and of Boethius himself . Aristoxenus is less easy to understand than the mathematicians because of the unscientific nature of his postulates. His importance, however, is very great, not only from his comparatively early date but because he claims to champion the direct musical consciousness against the scientific approach of some of his predecessors and contemporaries. (shrink)

In a long note in his epoch-making book on ancient Pythagoreanism Walter Burkert raised some grave doubts about the authenticity of Archytas Fr. 1 which have recently been challenged in an article by A. C. Bowen. In this paper I have two goals. First, I will evaluate Burkert's doubts and the success of some of Bowen's arguments against them. Second, I will present a further consideration that both clarifies the text of the fragment and also removes the most serious problem (...) raised by Burkert. The upshot of both these points is to increase the likelihood that the fragment is authentic. I reproduce the text of just the first part of the fragment as given in DK followed by the text as it appears in the two primary sources, Nicomachus and Porphyry. (shrink)

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. (shrink)

A review of Carl Huffman's new edition of the fragments of Archytas of Tarentum. Praises the extensive commentary on four fragments, but argues that at least two dubious works not included in the edition ("On Law and Justice" and "On Wisdom") deserve further consideration and contain important information for the interpretation of Archytas. Provides a complete translation for the fragments of those works.

This book is available either individually, or as part of the specially-priced Arguments of the Philosphers Collection.

Cicero may be best known as a politician, but he was also one of the few significant Roman writers of philosophy. This book presents a new and exciting selection of current scholarly work on this neglected side of him, establishing Cicero firmly on the agenda as a serious philosophical.

This volume presents the first Italian translation with commentary of the Doric Pseudo-Pythagorean texts, which are ascribed to Archytas and Brontinus and deal with metaphysical, logical, and epistemological questions. These texts probably date from the 1st century BCE and are the product of a re-emerging dogmatic interpretation of Plato's dialogues.

The aim -- The journey -- The goal -- The view -- The endless joy.

This article explores in greater depth the historiographical traditions concerning Herennius Pontius, a Samnite wisdom-practitioner who is said by the Peripatetic Aristoxenus of Tarentum to have been an interlocutor of the philosophers Archytas of Tarentum and Plato of Athens. Specifically, it argues that extant speeches attributed to Herennius Pontius in the writings of Cassius Dio and Appian preserve a philosophy of “extreme proportional benefaction” among unequals. Greek theories of ethics among unequals such as those of Aristotle and Archytas of Tarentum, (...) which aim for moderation, can be distinguished from that attributed to Herennius Pontius, which is circumstantial and stipulates extreme responses to extremes. Romans, in particular Appius Claudius Caecus and Sulla, celebrate and espouse proverbial wisdom strikingly similar to the theory of “extreme proportional benefaction” associated with Herennius Pontius, and in contrast to Greek ethics of moderation. Such comparisons suggest that starting in the late fourth century BCE, Romans and Samnites may have held shared ideological principles, as defined against Greek cultural paradigms. (shrink)

The Pythagorean idea that numbers are the key to understanding reality inspired philosophers in late Antiquity (4th and 5th centuries A.D.) to develop theories in physics and metaphysics based on mathematical models. This book draws on some newly discovered evidence, including fragments of Iamblichus's On Pythagoreanism, to examine these early theories and trace their influence on later Neoplatonists (particularly Proclus and Syrianus) and on medieval and early modern philosophy.

It has long been thought that the ancient Greeks did not take mechanics seriously as part of the workings of nature, and that therefore their natural philosophy was both primitive and marginal. In this book Sylvia Berryman challenges that assumption, arguing that the idea that the world works 'like a machine' can be found in ancient Greek thought, predating the early modern philosophy with which it is most closely associated. Her discussion ranges over topics including balancing and equilibrium, lifting water, (...) sphere-making and models of the heavens, and ancient Greek pneumatic theory, with detailed analysis of thinkers such as Aristotle, Archimedes, and Hero of Alexandria. Her book shows scholars of ancient Greek philosophy why it is necessary to pay attention to mechanics, and shows historians of science why the differences between ancient and modern reactions to mechanics are not as great as was generally thought. (shrink)

This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. An important first chapter looks at the mathematicians of the period and how mathematics was perceived by people at large. The central chapters of the book analyse sections of the Collection, identifying features typical of Pappus's mathematical practice. The final chapter draws (...) together the various threads and presents a fuller description of Pappus's mathematical 'agenda'. This is one of few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus's text as part of a wider context and relates it to other contemporary cultural practices and opens avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity. (shrink)

In ancient tradition, Pythagoras emerges as a wise teacher, an outstanding mathematician, an influential politician, and as a religious and ethical reformer. This volume offers a comprehensive study of Pythagoras, Pythagoreanism, and the early Pythagoreans through an analysis of the many representations of the individual and his followers.

For the first English edition of his distinguished study, Weisheit und Wissenschaft: Studien zu Pythagoras, Philoloas und Platon, Mr. Burkert has extensively ...

The ancient science of harmonics investigates the arrangements of pitched sounds which form the basis of musical melody, and the principles which govern them. It was the most important branch of Greek musical theory, studied by philosophers, mathematicians and astronomers as well as by musical specialists. This 2007 book examines its development during the period when its central ideas and rival schools of thought were established, laying the foundations for the speculations of later antiquity, the Middle Ages and the Renaissance. (...) It concentrates particularly on the theorists' methods and purposes and the controversies that their various approaches to the subject provoked. It also seeks to locate the discipline within the broader cultural environment of the period; and it investigates, sometimes with surprising results, the ways in which the theorists' work draws on and in some cases influences that of philosophers and other intellectuals. (shrink)

Conventional wisdom suggests that the Platonist philosophers of Late Antiquity, from Plotinus (third century) to the sixth-century schools in Athens and Alexandria, neglected the political dimension of their Platonic heritage in their concentration on an otherworldly life. Dominic O'Meara presents a revelatory reappraisal of these thinkers, arguing that their otherworldliness involved rather than excluded political ideas, and he reconstructs for the first time a coherent political philosophy of Late Platonism.

The nature of matter was as intriguing a question for ancient philosophers as it is for contemporary physicists, and Matter, Space, and Motion presents a fresh and illuminating account of the rich legacy of the physical theories of the Greeks from the fifth century B.C. to the late sixth century A.D.

The Seventh Platonic Letter describes Plato's attempts to turn the ruler of Sicily, Dionysius II, into a philosopher ruler along the lines of the Republic. It explains why Plato turned from politics to philosophy in his youth and how he then tried to apply his ideas to actual politics later on. It also sets out his views about language, writing and philosophy. But is it genuine? Scholars have debated the issue for centuries. The origin of this book was a seminar (...) given in Oxford in 2001 by Myles Burnyeat and Michael Frede, two of the most eminent scholars of ancient philosophy in recent decades. They question the authenticity of the letter head-on by showing how its philosophical content conflicts with what we find in the Platonic dialogues. They also reflect on the question of why the Letter was written, whether as an attempt to exculpate Plato from the charge of meddling in politics, or as an attempt to portray, through literary means, the ways in which human weakness and emotions can lead to disasters in political life. (shrink)