This is the first book to analyze systematically crucial aspects of ancient Greek philosophy in their original context of mystery, religion, and magic. The author brings to light recently uncovered evidence about ancient Pythagoreanism and its influence on Plato, and reconstructs the fascinating esoteric transmission of Pythagorean ideas from the Greek West down to the alchemists and magicians of Egypt, and from there into the world of Islam.
Reconstruction of the versions of Aristoxenos and Dikaiarchos.--The sources of Dikaiarchos and Aristoxenos and the reliability of their accounts. --Reconstruction of Timaios' version and the reliability of his accounts.--The chronological questions and the numismatic evidence.--The character of the "Pythagorean rule" in southern Italy.--Appendix.
Archytas of Tarentum was a central figure in fourth-century Greek life and thought and the last great philosopher in the early Pythagorean tradition. He solved a famous mathematical puzzle, saved Plato from the tyrant of Syracuse, led a powerful Greek city state, and was the subject of three books by Aristotle. This first extensive study of Archytas' work in any language presents a radically new interpretation of his significance for fourth-century Greek thought and his relationship to Plato, as well (...) as a full commentary on all the fragments and testimonia. (shrink)
: Some of the great physicists' belief in the existence of a connection between the aesthetical features of a theory (such as beauty and simplicity) and its truth is still one of the most intriguing issues in the aesthetics of science. In this paper I explore the philosophical credibility of a version of this thesis, focusing on the connection between the mathematical beauty and simplicity of a theory and its truth. I discuss a heuristic interpretation of this thesis, attempting to (...) clarify where the appeal of this Pythagorean view comes from and what are the arguments favoring its acceptance or rejection. Along the way, I sketch the historical context in which this heuristic interpretation gained credibility (the quantum crisis in physics in the 1920s), as well as the more general implications of this thesis for physicists' metaphysical outlook. (shrink)
The Pythagorean Life is the most extensive surviving source on Pythagoreanism, and has wider interest as an account of the religious aspirations of late antiquity. "...admirably clear translation and sensible introduction"--The Classical ...
A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic (...) velocities are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π. (shrink)
This book investigates the link Kant discerned between our experience of beauty and our experience of the moral law. By examining Kant's relation to Greek philosophy, to Plato and Pythagoras, as found in Kant's own writings, the author sheds new light on one the most intriguing and mysterious doctrines of Kant's third Critique.
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.
The origin of the Neoplatonist doctrine of the henads has been imputed to Iamblichus, mostly on indirect evidence found in later Neoplatonists, chiefly Proclus. Is there any trace of this concept to be found in the extant works or fragments of Iamblichus himself? The best candidates among his surviving texts are the excerpts in Psellus of his volume on Theological Arithmetic from his Pythagorean series, and the first book of de Mysteriis , where Iamblichus answers Porphyry's questions on the (...) nature of the gods. Such evidence as can be found there would most likely deal with the divine henads, given the subject matter of the text. Certain repeated items of vocabulary appear as technical usages that form the basis for arguing that Iamblichus already has in mind if not the explicit concept henad at least its functional equivalent: the term monoeides occurring in both the Psellan excerpts and de Mysteriis, and in the latter, mostly in Book I, the stated attributes of a high, divine principle uniting the gods which are also designated by Proclus as typical of the divine henads, particularly in the propositions of the Elements of Theology defining the henads. Iamblichus in Book I also ascribes to the gods the same role in the process of ellampsis as Proclus does for the divine henads. A theory is also advanced concerning the possible development of the concept of the henad by Iamblichus, based in part on the polemical nature of de Mysteriis and his relationship to Porphyry. (shrink)
The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...) dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists.. (shrink)
A new non-traditional quasi-classical description of the particle dynamics (QCDPD) is outlined. The “quasi-classical” attribute is suggested by the closeness—although not identity—to the description of a classical system, in the framework of classical dynamics. Founded on a suitable one-to-one mapping of the timelike 4-vectors of Minkowski's spacetime onto the real 4-dimension vector space, QCDPD is mathematically equivalent to the traditional description of special relativity. However, in QCDPD a new frequency fulfilling the same transformation law as the frequency of an oscillator (...) is employed. It turns out that the special-relativity particle is, in fact, modeled as an oscillator which is essentially classical. The associated wave is a natural consequence of such a model. (shrink)