Was Plato a Pythagorean? Plato's students and earliest critics thought so, but scholars since the nineteenth century have been more skeptical. With this probing study, Phillip Sidney Horky argues that a specific type of Pythagorean philosophy, called "mathematical" Pythagoreanism, exercised a decisive influence on fundamental aspects of Plato's philosophy. The progenitor of mathematical Pythagoreanism was the infamous Pythagorean heretic and political revolutionary Hippasus of Metapontum, a student of Pythagoras who is credited with experiments in harmonics that led to (...) innovations in mathematics. The innovations of Hippasus and other mathematical Pythagoreans, including Empedocles of Agrigentum, Epicharmus of Syracuse, Philolaus of Croton, and Archytas of Tarentum, presented philosophers like Plato with novel ways to reconcile empirical knowledge with abstract mathematical theories. Plato and Pythagoreanism demonstrates how mathematical Pythagoreanism established many of the fundamental philosophical questions Plato dealt with in his central dialogues, including Cratylus, Phaedo, Republic, Timaeus, and Philebus. In the process, it also illuminates the historical significance of the mathematical Pythagoreans, a group whose influence on the development of philosophical and scientific methods has been obscured since late antiquity. The picture that results is one in which Plato inherits mathematical Pythagorean method only to transform it into a powerful philosophical argument about the essential relationships between the cosmos and the human being. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Plato and Pythagoreanism by Phillip Sidney HorkyGabriele CornelliPhillip Sidney Horky. Plato and Pythagoreanism. Oxford University Press, 2013. xxi + 305 pp. Cloth, $74.Ceci n’est pas un livre sur Pythagore. With these clever and rather playful words, Horky’s book starts its literary journey through a wide range of Pythagorean sources, including Epicharmus, Empedocles, Philolaus, Eurytus and Arquitas, and Pythagorean themes, like numbers, immortality of the soul, limitness/unlimitness, (...) among others.From this explosive incipit, one would think that Horky agrees with Kahn’s almost forty-year-old observation about the fact that new theories of Pythagoreanism are not necessary in our present day and age (1974, 163, n. 6):It’s hard enough to satisfy minimal standards of historical rigor in discussing the Pythagoreans, without introducing arbitrary guesswork of this sort where no two students can come to the same conclusion on the basis of the same evidence. In fact, the direct testimony for Pythagorean doctrines is all too abundant. The task for a serious scholarship is not to enrich these data by inventing new theories or unattested stages of development but to sift the evidence so as to determine which items are most worthy (or least unworthy) of belief.On the contrary, however, Horky’s approach seems surprisingly to direct the reader towards a fundamentally historiographical rather than philological brand of work, that is, one devoted neither exclusively to the exegesis of Pythagorean sources nor even to the theoretical approach of one of the themes that received specific contributions from Pythagoreanism, such as mathematics, cosmology, politics, or the theory of the immortal soul. This is mainly true when one recognizes, from the title on, that the main goal of the book is to try to determine, once and for all, what is the extent of Pythagorean influence on Plato’s thought. Rather an historiographical problem, I would say. And a good one.In order to do so, Horky needs to trace down a sort of history of mathematical Pythagoreanism from a huge variety of informed ancient sources and perspectives. Horky’s approach to the sources is well-grounded, not only in Burkert’s fundamental works, but also in the recent revival in Pythagorean studies which took over ancient philosophy scholarship in the last two decades: Huffmann’s works on Philolaus (1993) and Arquitas (2005); Zhmud (1997) and Kahn (2001) on early Pythagoreans; Riedweg on Pythagoras (2002), among many others. After the publishing of Horky’s book, two sorts of Companions to Pythagoreanism confirm the trend of the revival: Cornelli, McKirahan, and Macris (2013), acts of the On Pythagoreanism seminar held in Brasilia by Archai [End Page 353] UNESCO Chair, and Huffman (2014). Luckily, Pythagorean scholarship has now gotten rid of that old-fashioned scepticism, starting from Zeller’s famous statement that the relevant sources are a “mixture of fables and poetry.” As Horky’s book seems to suggest, the job can be done. And he has definitely proved to be able to force his way through, with both clear philological arguments and refined historiographical hypotheses.Not surprisingly, a work on Plato and Pythagoreanism starts with a chapter on Aristotle: “Aristotle on Mathematical Pythagoreanism in the Fourth Century BCE” (chap. 1). The idea of the chapter is to rely on Aristotle’s distinction between the mathematical and acousmatic branches of Pythagoreanism. While acousmatic Pythagoreans supposedly did not engage in demonstration, the so-called mathematical Pythagoreans engaged in investigations based on mathemata in order to give an account (the reason why of the world they perceived). Horky seems to drive Aristotle’s account to a specific goal: mathematical Pythagoreans would have established some kind of correspondence between numbers and sensibles, and their ontology would have a direct relation to social and political organization.Chapter 2, “Hippasus of Metapontum and Mathematical Pythagoreanism,” is entirely dedicated to the account of the later doxographical tradition on Hippasus, the alleged progenitor of mathematical Pythagoreanism. Avoiding the risk of falling into the “bottomless pit” of research on the Pythagoreans—in Guthrie’s famous words (1962, 146, n. 1)—Horky manages his inquiry boldly and in a quite elegant way. Between the peripatetic systematization and the platonic derivation of Hippasus’ philosophy, Hippasus... (shrink)

HORKY, P. S. (2013). Plato and Pythagoreanism. Oxford University Press.

This chapter traces Plato's philosophical responses to the puzzle of Epicharmus' “Growing Argument” in the earlier and middle dialogues of Plato, especially Euthyphro and Cratylus. Plato's approach to this problem is rooted in his metaphysical propositions, including the correlative assumptions of participation of sensibles in Forms and imitation as a vehicle for names to obtain the properties of their governing Forms. By attacking a Sophistical version of the “Growing Argument” given by Cratylus, Plato simultaneously appropriates certain principles of ontological predication (...) given by Philolaus of Croton, thus pitting, in effect, the ideas of one mathematical Pythagorean against those of another. (shrink)

This book presents an up-to-date overview of the main new directions taken by ancient philosophy in the first century BC, a period in which the dominance exercised in the Hellenistic age by Stoicism, Epicureanism and Academic Scepticism gave way to a more diverse and experimental philosophical scene. Its development has been much less well understood, but here a strong international team of leading scholars of the subject reconstruct key features of the changed environment. They examine afresh the evidence for some (...) of the central Greek thinkers of the period, as well as illuminating Cicero's engagement with Plato both as translator and in his own philosophising. The intensity of renewed study of Aristotle's Categories and Plato's Timaeus is an especially striking outcome of their discussions. The volume will be indispensable for scholars and students interested in the history of Platonism and Aristotelianism. (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)

Recently, the Pythagoreans have received rather more attention, both in their own right and as part of the developing picture of Presocratic thought, than they received for much of the twentieth century. Thanks to these studies, a new and more complicated picture is emerging. This article refines this picture critically examining Aristotle's claims about Pythagorean influence on Plato, along with the related question of who among early Greek thinkers actually counts as a Pythagorean. It provides a reminder that Aristotle's account (...) of the history of earlier thought is always a history of just a part of the philosophy of his predecessors, and makes clear that the eagerness with which some present-day scholars find Pythagorean influence on later thought may be misplaced. (shrink)

This chapter describes the kinds of Pythagoreans who may have existed from the sixth through fourth centuries bce and their philosophical activities based on the evidence preserved by Aristotle. It identifies the characteristics that distinguished the mathematical Pythagorean pragmateia from the pragmateia of the rival acousmatic Pythagorean brotherhood in Magna Graecia. It argues that Aristotle establishes this distinction by appeal to the divergent philosophical methodologies of each group. The mathematical Pythagoreans, who are the same as the “so-called Pythagoreans” in Metaphysics (...) A, employ superordinate mathematical sciences in establishing something that approximates demonstrations that explain the “reason why” they hold their philosophical positions, whereas the acousmatic Pythagoreans, who are distinguished from the “so-called” Pythagoreans in Metaphysics A, appeal to basic, empirically derived “fact” in defense of their doctrines. (shrink)

Numenius is one of the most important authors who, in the Imperial Age, deal with the figure of Socrates. Socrates is important in the Platon­ic tradition, in particular in the sceptical tradition, when the Socratic dubitative “spirit” of the first Platonic dialogues became important to justify the “suspension of judgement.” Numenius criticises the whole Academic tradition by saying that the Academics (particularly the sceptics) betrayed the original doctrine of Plato and formulated a new image of Socrates. For Numenius, Socrates plays (...) a central role because Plato would have inherited his doctrine. What does Socrates’s doctrine consist in? According to Numenius, Socrates theorised a “doctrine of three Gods” (which can be likely found in the second Platonic epistle) which is strictly bound up with the main aspect of Plato’s thought. In fact, in Numenius’s view, Plato belongs to a genealogy which can be linked to Pythagoras himself. From this perspective, Numenius says that Socrates’s original thought is a theology which also belongs to the Pythagorean tradition and which Plato further developed. For Nume­nius, Socrates is not the philosopher of doubt, but a theologian who first theorised the existence of three levels of reality (Gods), which is also the kernel of Numenius’s metaphysical system. For this reason, Numenius puts Socrates within a theological genealogy that begins with Pythago­ras and continues with Socrates and Plato, and that the Academics and the Socratics failed to understand. (shrink)

This chapter examines the various ways the Peripatetics and the Academics treated the philosophy of Hippasus of Metapontum. It suggests that some figures in the Early Academy appear not only to have assimilated the philosophy they attributed to Hippasus into their own philosophical doctrines, but also have filled out the picture that makes Hippasus a Pythagorean.

Plato’s method of hypothesis is initiated in the Meno, is featured in the Phaedo and the Republic, and is further developed in the Theaetetus. His method of collection and division is mentioned in the Republic, is featured in the Phaedrus,and is elaborated with modifications in the Sophist and the Statesman. Both methods aim at definitions in terms of necessary and sufficient conditions. In the course of these developments, the former method is shown to be weak in its treatment of sufficient (...) conditions, and the latter is shown to be comparably weak in its treatment of necessary conditions. A third method, which avoids these difficulties, is introduced in the first part of the Parmenides and is applied in connection with the eight hypotheses that follow. This application yields a demonstration of serious shortcomings both in historical Eleaticism and in the Eleatically-inspired theory of Forms in the Phaedo and the Republic, along with a demonstration of comparative strengths in historical Pythagoreanism and in the Pythagorean- inspiried theory of Forms in the Statesman and the Philebus. (shrink)

In this paper I argue that we can better understand Plato’s Phaedo, if we don’t concentrate solely on the hints of Pythagoreanism among the characters and their doctrines, as though that were the principal key to the dialogue’s dialec- tical targets. I suggest that the dialogue is intended to make us think of the meta-physics of at least one other Presocratic predecessor, besides any Pythagorean influence (which may be much less than has been thought). Not least among the thinkers (...) of whom we are reminded is Heraclitus. In the Phaedo, Plato invites us to explore the limitations of these earlier metaphysical views, and to consider how far they can or cannot make space for the capacity of the soul to direct the body, to make decisions, to grasp eternal truths and potentially to survive in perpetuity beyond the decay of the body. (shrink)

This paper argues that Plato’s Meno does not offer evidence for a belief, commonly attributed to Plato, that we when learning something recollect what we learn from previous existences. This “theory of recollection” is a construct based on a reading of the relevant passages in the Meno which does not take into account the dialectical aspect of Socrates’ discussion with his interlocutor. And in one passage (81e3) it is based on a variant reading for which a better and better attested (...) reading is available. The reference to recollection is used to make Meno agree to an absurd position. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:From Death to Life: Key Themes in Plato’s Phaedo by Franco TRABATTONIAthanasia A. GiasoumiTRABATTONI, Franco. From Death to Life: Key Themes in Plato’s Phaedo. Boston: Brill, 2023. 190 pp. Cloth, $143.00In his comprehensive study of the Phaedo, Franco Trabattoni challenges the conventional interpretation of Plato’s thought by denying that Plato was ever a dogmatist or a skeptic. The opening chapter proposes that Plato employs a “third way” standing (...) on the Parmenidean distinction between dôxa (belief) and epistēme (knowledge). According to Trabattoni, the Phaedo (together with all other dialogues dealing with the distinction between sensible and intelligible reality) implies that knowledge of the forms can be obtained only after the separation of body and soul and only without the assistance of intermediaries. He contends that humans have no better instrument than lôgos to discover the intelligible realities (forms) that philosophy seeks. However, because lôgos serves as an intermediary, he believes that the mediation of lôgos introduces an unavoidable fault into the cognitive process, which is directly linked to the concept of dôxa. He maintains that this hypothesis is essential for the arguments presented in the Phaedo’s first part to make sense.Trabattoni delves into the Phaedo not just philosophically but also linguistically, particularly in passage 60c, where Cebes brings up the question of suicide. Notably, in chapter 2, Trabattoni provides a new translation of this passage that contrasts with the readings supplied by Dorter, Gallop, and Boter. Trabattoni claims that Plato responds to the problem of suicide by adhering to the eudaemonistic stance that no obligation that goes against the happiness of the individual subject to it is of any value.Chapter 3 features a counterargument to the predominant viewpoint positing an inconsistency between the Phaedo and the Republic. Trabattoni asserts that these dialogues agree in maintaining that true virtue can be ascribed only to those who believe in metaphysical realities and possess an exclusive desire to acquire knowledge of them. He posits that the disparities between the two dialogues can be attributed primarily to differing perspectives rather than a significant theoretical divide.In my opinion, chapter 4 is the book’s most interesting and original portion. Here, Trabattoni argues that Plato’s contrast in the Phaedo between genuine and false philosophers is directed at Antisthenes, specifically, that Plato opposes his Socratism to Antisthenes’ by depicting the Platonic Socrates as a genuine philosopher (philōsophos) in contrast to Antisthenes’ Socrates, who is portrayed as a false one (philōdoxos). According to Trabattoni, Plato endeavors to provide what Antisthenes’ portrayal of Socrates does not, namely, a metaphysical perspective that differentiates between opinion and truth. Though novel and charming, Trabattoni’s interpretation is not supported by the text since the Phaedo makes no direct allusion to Antisthenes’ Socrates, mentioning Antisthenes only at the beginning (58b) as someone present at Socrates’ last moments. Thus, the book’s strongest section could also be considered its weakest.Chapter 5 argues that the recollection theory is not intended to establish the possibility of a complete retrieval of the knowledge of the forms. [End Page 163] Trabattoni disagrees with Scott’s and others’ assertion that recollection is a method of inquiry on the grounds that Socrates never explains how the alleged “method” of recollection should be applied. Rather, Trabattoni contends, the function of recollection is to explain why human intellective knowledge cannot exceed the boundary of dôxa. Trabattoni argues further that the recollection theory indicates that only when the soul is separated from the body, that is, when a person dies, is it possible to gain knowledge of the forms.Chapters 6 and 7 examine in depth the philosophical, methodological, and linguistic aspects of the objections raised by Cebes and Simmias as well as Socrates’ ensuing arguments regarding the immortality of the soul. Additionally, Trabattoni explores the historical background of Plato’s Phaedo and the impact on it of Pythagoreanism, Orphism, and Eleatic philosophy.Chapter 8 is another particularly captivating section. It offers a fresh perspective on the concept of the “second voyage” (deũteros ploũs), challenging the prevailing interpretation, on which the second voyage signifies the triumph of the... (shrink)

The title of McClain’s book suggests that its theme is Plato’s seeming Pythagoreanism; yet it contains no commentary on the Philebus or the Phaedo. Its subtitle suggests that the theme is rather the five mathematical disciplines which may constitute a propaedeutic to dialectic ; yet it contains almost no account of the first four, and little account of the fifth as propaedeutic in this way. McClain’s book is rather an "adventure in musical imagination" whose proximate aim is a detailed (...) and coherent display of Plato’s musical imagination. (shrink)

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. (shrink)

In his Exhortation to Philosophy (Protrepticus), the Neoplatonic philosopher Iamblichus famously preserves material culled from lost works of ancient philosophy, including dialogues of Aristotle. He also preserves a work entitled On Wisdom and ascribed to the Pythagorean philosopher Archytas of Tarentum, who was a friend and challenger of Plato. The text On Wisdom is a later Hellenistic production, probably written in the 1st century BCE, but it presents an important piece in the puzzle of reconstructing Pythagoreanism for the Hellenistic (...) and Post-Hellenistic periods. On Wisdom provides ps-Archytas' views on the nature, pursuit, and purposes of philosophy, as well as the division of the parts of knowledge. This article presents the first complete modern translation into English and analysis of this work, whilst also explaining the precise ways in which its first critic, Iamblichus, appropriates its content to Neoplatonic theories. (shrink)

When did kosmos come to mean *the* kosmos, in the sense of ‘world-order’? I venture a new answer by examining later evidence often underutilised or dismissed by scholars. Two late doxographical accounts in which Pythagoras is said to be first to call the heavens kosmos (in the anonymous Life of Pythagoras and the fragments of Favorinus) exhibit heurematographical tendencies that place their claims in a dialectic with the early Peripatetics about the first discoverers of the mathematical structure of the universe. (...) Likewise, Xenophon and Plato refer to ‘wise men’ who nominate kosmos as the object of scientific inquiry into nature as a whole and the cosmic ‘communion’ (koinônia) between all living beings, respectively. Again, later testimonies help in identifying the anonymous ‘wise men’ by associating them with the Pythagoreans and, especially, Empedocles. As Horky argues, not only is Empedocles the earliest surviving source to use kosmos to refer to a harmonic ‘world-order’ and to illustrate cosmic ‘communities’ between oppositional pairs, but also his cosmology realises the mutual correspondence of these aspects in the cycle of love and strife. Thus, if later figures posited Pythagoras as the first to refer to the universal ‘world-order’ as the kosmos, they did so because they believed Empedocles to have been a Pythagorean natural scientist, whose combined focus on cosmology and ethics exemplified a distinctively Pythagorean approach to philosophy. (shrink)

Pythagoras and Pythagoreanism have a different position in the ancient philosophy tradition. The reason for this is the eclectical structure of Pythagoreanism which has syncretized from Orphism, Indian and Egyptian religions with philosophy. Orphism of these religions is especially important for affecting Pythagoreanism the most and giving to the ancient Greek religion a mystical content. Orphism which is a mystery cult is based on Orpheus, the poet, who sometimes is identified with Pythagoras in philosophy and the history (...) of religions. Orpheus, was attributed divine character by virtue of his beautiful voice and hymns in the ancient history, has brought many religious and philosophic elements such as reincarnation and unity with God to the Greek religion. The way of life and creeds of Orphism based on asceticism and wisdom, first affected Italian philosophy starting from Pythagoras and, later influenced both philosophy and the other religions such as Judaism and Christianity through Plato. The reason for this was the Orphism’s stronger theological structure than the Greek religion had. Thus, philosophy, the Miletus philosophers tried to purify from the Homeric religion, has regained some spirituality through Orphism and Pythagoras. Pythagoras, a philosopher and a religious leader, accepted students to his philosophy school by initiation in Crotone. In this way, Pythagoreanism gained the quality of both a school of philosophy and a community of mystery. In this school, Pythagoras constituted a strong natural philosophy based on the idea of arkhe consisting of numbers, and the system of mathematics and music. At the same time, for he was a religious leader, Pythagoras used his school both to teach philosophy and to celebrate the teletaic mystery rituals of Orphism. According to the Pythagoreans, as a master, Pythagoras has united with the spirit of Dionysos and has privileged to rise to the World-Soul of God. Due to this union, it has been believed that Pythagoras received revelations from Dionysos and knew the secrets of God’s spirit. Similarly, Pythagoras’ students of philosophy also joined these rituals. They, on the one hand, witnessed to the divine union of Pythagoras, on the other hand, experienced the communication with the subordinate gods, called daemons, allied to Dionysos. Through these mystical rituals in Pythagorean school, it was believed that the philosophical knowledge was not only a reasoning, but also a divine wisdom obtained from the divine word. Therefore, Pythagoreanism accepted that philosophy is from God and it is a way of life which rises the human to the divine level. According to Pythagoras philosophy, thinking about God is equal to dealing with natural philosophy. Because Pythagoreanism, that has a pantheistic belief in God, has acknowledged God and nature as the same. Additionally, the natural philosophy of the school was established on the Monad idea which has been accepted as the World-Soul. For Pythagoreanism, Monad is not a number, but it is the ONE as the source of all numbers and a theological principle. The ONE, which symbolizes God and the cosmic unity, brings into existence the life by limiting the chaos caused by the TWO. Similarly, the TWO, not a number but a principle, has been used in a positive sense in Anaximander and was called apeiron, meant “infiniteness”. But, for Pythagoras, infiniteness was a negative circumstance. For this reason, the TWO/Apeiron has been accepted as a female being which corresponds to limitless absence and evil. By Monad’s confinement the Apeiron had been passed into the cosmos, the cosmic harmony had been balanced and a spheric being had emerged. This is “unity” which corresponds with both the World Soul Monad and the nature in the Pythagorean philosophy. This study will discuss the notion of Monad which is known as the World-Soul and God in Pythagorean philosophy, from a perspective of the history of religions. In this context, the sources and knowledge of history of philosophy will be used by methods of the history of religions. Starting from Orphism, Monad will be discussed within similar mythical elements. Thus, the origin of the Pythagorean philosophy in ancient religions will be determined and the theological infrastructure will be shown. (shrink)

In the twenty-fourth aporia of Theophrastus' Metaphysics, there appears an important, if ‘bafflingly elliptical’, ascription to Plato and the ‘Pythagoreans’ of a theory of reduction to the first principles via ‘imitation’. Very little attention has been paid to the idea of Platonic and ‘Pythagorean’ reduction through the operation of ‘imitation’ as presented by Theophrastus in his Metaphysics. This article interrogates the concepts of ‘reduction’ and ‘imitation’ as described in the extant fragments of Theophrastus’ writings – with special attention to his (...) Metaphysics – in an attempt to make sense of how, precisely, Theophrastus characterizes the metaphysical systems of Plato and the ‘Pythagoreans’. A fuller examination of Aporia 24 produces important new understandings of the nature of the Early Academy in the mid fourth century B.C.E. Specifically, by ascribing the concept of reduction via ‘imitation’ to Plato and the ‘Pythagoreans’, Theophrastus is actually describing theories about the structure of the universe held by an important Platonist competitor and the contemporary head of the Academy in Athens, Xenocrates of Chalcedon (396/5–314/13 B.C.E.), who (a) took it upon himself to establish a Platonic doctrine that could be associated with his master’s writings and (b) coordinated that doctrine with the philosophical precepts of the Pythagoreans. As a consequence, we are prompted to question Aristotle’s famous ascription of a theory of ‘imitation’ to the metaphysics of the genuine contemporary or recently deceased Pythagoreans (such as Philolaus of Croton or Archytas of Tarentum) and consider how Xenocrates’ writings on Plato and the ‘Pythagoreans’ might be informing both Theophrastus’ and Aristotle’s descriptions of the Platonic and Pythagorean philosophical system(s). (shrink)

The treatise On the Nature of the Universe, attributed to the Pythagorean Ocellus, has frequently been the subject of scholarly attention due to its engagement with Aristotle’s theory of elemental transformation or its role in the late Hellenistic debate on the eternity of the universe. In this paper, I argue that its author endorses a peculiar conception of matter: prime matter is an unqualified body, only potentially perceptible. Ps.-Ocellus draws this doctrine from Stoicism but reworks it for his purposes outside (...) the framework of Stoic corporealism. The author presents it as an antecedent to both Plato’s receptacle and Aristotle’s prime matter. As it turns out, the doctrine was meant to address some of the challenges posed by an incorporeal material principle. (shrink)

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.

Mimesis can refer to imitation, emulation, representation, or reenactment - and it is a concept that links together many aspects of ancient Greek Culture. The Western Greek bell-krater on the cover, for example, is painted with a scene from a phlyax play with performers imitating mythical characters drawn from poetry, which also represent collective cultural beliefs and practices. One figure is shown playing a flute, the music from which might imitate nature, or represent deeper truths of the cosmos based upon (...) Pythagorean views (which were widespread in Western Greece at the time). The idea that mimesis should be restricted to ideals was made famous by Plato (whose connections to Pythagoreanism and Siracusa are well-known), and famously challenged by his student Aristotle (not to mention by the mimetic character of Plato’s own poetry). This volume gathers essays not only on the philosophical debate about mimesis, but also on its use in architecture, drama, poetry, history, music, ritual, and visual art. The emphasis is on examples from Hellenic cities in Southern Italy and Sicily, but the insights apply far beyond – even to modern times. Contributors include: Thomas Noble Howe, Francisco J. Gonzalez, Gene Fendt, Guilherme Domingues da Motta, Jeremy DeLong, Carolina Araújo, Marie-Élise Zovko, Lidia Palumbo, Sean Driscoll, Konstantinos Gkaleas, Anna Motta, Jure Zovko, Alexander H. Zistakis, Christos C. Evangeliou, Dorota Tymura, Iris Sulimani, Elliott Domagola, Jonah Radding, Giulia Corrente, Laura Tisi, Ewa Osek, Argyri G. Karanasiou, Rocío Manuela Cuadra Rubio, Jorge Tomás García, Aura Piccioni, and José Miguel Puebla Morón. (shrink)

This book is the scholarly & fully annotated edition of the award-winning _The Illustrated To Think Like God.__ _To Think Like God_ focuses on the emergence of philosophy as a speculative science, tracing its origins to the Greek colonies of Southern Italy, from the late 6th century to mid-5th century B.C. Special attention is paid to the sage Pythagoras and his movement, the poet Xenophanes of Colophon, and the lawmaker Parmenides of Elea. In their own ways, each thinker held that (...) true insight, whether as wisdom or certainty, belonged not to mortal human beings but to the gods. The Pythagoreans sought to approach this otherwordly knowledge by studying numerical relationships, believing them to govern the universe, and that those who know the number of a thing know its true nature. Yet their quest was a hopeless one, bogged down by cultism, numerology, political conspiracies, bloody uprisings, and exile. Above all, number did not turn out as the most reliable of mediums; it was certainly not a key to the realm of the divine. Thus, their contributions to philosophy's inception, while much better-publicized, was not the most significant. That particular role was reserved for an unusual challenge and the elaborate reaction it provoked. The challenge came from Xenophanes, who had argued that reliable truth was beyond mortal reach, because even if by accident a human being should state what is exactly the case, he had no way of knowing that he did, all things being susceptible to opinion. This dilemma is sure to have bothered a legislative mind like that of Parmenides, and we find him introducing techniques for testing the veracity of statements. These methods were meant to be carried out by reasoning and argument alone, without relying in physical evidence or mortal sense-perception, which was deemed untrustworthy. Reason was that one faculty shared by gods and humans alike. In time, Parmenides' ingenious arguments have earned him the titled of the first logician and metaphysician whose influence on subsequent thinkers was immeasurable. Parmenides taught us that philosophy was not about claims but about proof, which also makes him the father of theoretical science -- which, curiously, began as a quest into the mind of God. "Arnold Hermann makes a genuine contribution to Presocratics studies. This book, which is both an introduction to Pythagoras and Parmenides and a scholarly study, will interest novices and experts alike. Hermann's multi-leveled approach and his careful analyses of alternate views make his work a useful teaching tool, while his systematic inquiry into Pythagoreanism, the poem of Parmenides, and the development of early Greek thought will well repay the attention of scholars. — Patricia Curd_,_ _Purdue University_ "_To Think Like God_ is a highly ambitious book... Hermann's approach deserves to be taken seriously as an alternative to standard interpretations." — Richard D. McKirahan, Jr.,___ __Edwin Clarence Norton Professor of Classics and Professor of Philosophy, Pomona College_ "Arnold Hermann brings fresh life into the specialists' debates... a blow of wind that dissipates much fog." — Walter Burkert_, Professor Emeritus of Classical Philology, University of Zurich_ ARNOLD HERMANN_ is pursuing independent research on the origins of philosophy and methods of thinking. He specialices on subjects connected with Parmenides and Plato's _Parmenides._. (shrink)

29 o JOURNAL OF THE HISTORY OF PHILOSOPHY 34:2 APRIL t996 J. Burnet, Oxford, 19oz ) is excluded, as are influential works in foreign languages. Popper's The Open Society and Its Enemies, vol. I is included 077); it was later translated into German . The converse does not hold: P. Friedl~inder's Platon 049-43) is included, but its English translation is not. F. Solmsen's Plato's Theology is not included, nor is his "Plato and the Unity of Science,"s although it was reprinted (...) in Solmsen's Kleine Schriften4 and Soimen's German articles reprinted there are included. As the international Plato Forschungsberichte of Ritter5 demonstrate even for the Germany of the early decades of this century, the world of scholarship knows no national bound- aries. The present attempt to impose them on Platonic studies results in a highly artificial work of bibliography. 6 RICHARD McKIRAHAN Pomona College Carl A. Huffman. Philolaus of Croton, Pythagorean and Presocratic: A Commentary on the Fragments and Testimonia with Interpretive Essays. New York: Cambridge University Press, 1993. Pp. xix + 444- Cloth, $1oo.oo. Pythagoreanism used to play a major role in accounts of Presocratic philosophy. Follow- ing the testimony of ancient Platonists, the Pythagoreans were seen as Plato's precur- sors in cosmology, on the soul, and on Ideas and numbers; and attempts were made to reconstruct the systematic philosophy of the Pythagoreans, or even of Pythagoras himself. Now, however,.. (shrink)

Like his fellow first-generation Peripatetic Theophrastus, Aristoxenus wrote an extraordinary number of works. Many concerned music; one on Socrates contained evidence independent of Plato and Xenophon. At least five concerned Pythagoreanism: The Life of Pythagoras, On Pythagoras and His Associates, On the Pythagorean Way of Life, Life of Archytas, and the Pythagorean Precepts. This last one, as Carl Huffman...

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Cambridge Companion to Early Greek PhilosophyPatricia CurdA. A. Long, editor. The Cambridge Companion to Early Greek Philosophy. Cambridge: Cambridge University Press, 1999. Pp. xxxii + 427. Cloth, $54.95. Paper, $19.95.The Cambridge Companions are designed both to introduce and to survey, aims that anyone who teaches introductory courses knows are not fully compatible. The Cambridge Companion to Early Greek Philosophy is successful because its contributors have kept to (...) a uniform tone of mostly introduction, with some reference to recent developments and controversies in the field. There are sixteen essays, with a judicious mix of chapters on particular figures and on general topics.A. A. Long provides a helpful introduction, discussing the problems of dealing with the fragmentary evidence for these philosophers, and giving an account of the nature and scope of early Greek thought. (He also argues—not entirely persuasively—that we should give up the nineteenth century name "Presocratics" for these thinkers.) Jaap Mansfeld explains the problems of sources and doxography (the chapter is quite compressed and could be rather overwhelming for the newcomer to the area), while Keimpe Algra surveys Ionian cosmology. Carl Huffman provides a masterful discussion of early Pythagoreanism, attempting to sort out what we can and cannot be justified in claiming about Pre-Platonic Pythagoreanism. The chapters on individual thinkers mostly provide systematic overviews. Edward Hussey contributes an exemplary chapter on Heraclitus, and David Sedley one on Parmenides and Melissus (it is good to see Melissus treated seriously), with Richard McKirahan weighing in on Zeno (mostly on the paradoxes of motion but with some attention to the arguments against plurality). The discussions of individual Presocratics are rounded out with Daniel Graham's treatment of Empedocles' and Anaxagoras' responses to Parmenides, and C. C. W. Taylor on the Atomists. Sarah Broadie considers rational theology and James Lesher early Greek epistemology, while André Laks surveys ancient theories of soul, sensation, and thought. Paul Woodruff and Fernanda Decleva Caizzi provide chapters on the Sophists, the former discussing Protagoras and Gorgias, the latter, Protagoras and Antiphon. Mario Vegetti undertakes an examination of the notion of causation in early Greek thought, and Diskin Clay explores the poetics of the Presocratic philosophers. The book opens with short discussions—three paragraphs devoted to life, sources, and works for each of the thinkers who figure in the period (in alphabetical rather than even roughly chronological order), a chronological chart, and a map. There is a large and useful—though necessarily selective— bibliography (supplemented in the notes to some chapters by references to articles that are not included in the main bibliography), an index locorum, and an index of names and subjects that [End Page 429] includes modern scholars "only if their views are cited in the main text or discussed in footnotes" (which makes it difficult to use in tracking down references). Unfortunately, as in all the Cambridge Companions, notes are at the end of articles rather than at the foot of the page, making necessary a fair bit of flipping while reading.How well do the contributors succeed in meeting the needs of the intended readers of the series? For the most part, very well; the novice reader will find this an accessible guide to Early Greek philosophy. The various discussions are solid and well-informed, and the authors have indicated issues and interpretations that are controversial (for example, Taylor takes time to consider objections to his earlier criticism of Vlastos on the connection between atomistic metaphysics and ethics, and Hussey is careful to note the traditional flashpoints of disagreement in Heraclitus interpretation). It is also good that some of the Sophists are included (it is worth considering how viable the traditional distinction between the Sophists and the Philosophers really is), and inclusion of both the medical tradition and the issue of poetics is very welcome. Further chapters considering in more detail the cosmologies of post-Milesians (including Parmenides) and on the philosophical uses made of the early Greek thinkers by Plato and Aristotle would have rounded things out, but these would have made an already big book bigger. As perhaps befits an introductory survey, most of the interpretations lean to the traditional - someone looking for... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns by Arkady PlotnitskyNoam CohenPLOTNITSKY, Arkady. Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns. Cham: Springer, 2023. xvi + 294 pp. Cloth, $109.99The limits of thought in its relations to reality have defined Western philosophical inquiry from its very beginnings. The shocking discovery of the incommensurables in Greek mathematics (...) not only reintroduced these limits with greater force but also instigated two different forms of coping with the unthinkable in Western philosophy, mathematics, and science. As Arkady Plotnitsky sets forth, in response to the incommensurability crisis in the ancient Greek intellectual world, there appeared two different attitudes toward the alogon, that is, the unthinkable or irrational. Plato and his followers sought to overcome the unthinkable, of which the incommensurables in mathematics are only an example, through philosophy. In contrast, the Pythagoreans continued to hold on to mathematics as the primary means of knowing the world, but put a larger emphasis on geometry. Plotnitsky [End Page 359] claims that by not putting aside the primacy of mathematics, the Pythagorean shift from arithmetic to geometry established in Western thought a certain tolerance toward the unthinkable within the realm of the thinkable, “a possibility of the alogon in the logos.” Until modern times this remained no more than a possibility. However, in modern mathematics and mathematical physics Plotnitsky identifies a certain fulfillment of this potential, a “radical Pythagorean mathematics.” In his comprehensive and thought-provoking study, Plotnitsky recounts the development of such radical modern Pythagoreanism, not only to tell its history but also to suggest that the unthinkable is a condition of possibility for thought, and that a proper recognition of this role it plays is crucial for the advancement of mathematical thinking and thinking in general.Plotnitsky begins by portraying in broad strokes the two defining characteristics of radical Pythagorean mathematics: the presence of the unthinkable within mathematical thought itself, and the interplay of algebra and geometry. Relying on an extensive knowledge of the history of modern mathematics and physics, in the first two chapters of the book Plotnitsky gradually demonstrates the key role that the alogon plays in some theories of modern mathematical thought, such as Gödel’s incompleteness theorems and quantum mechanics. He also presents and clarifies his own conception of the history of Western thought as a creative process of concept-forming. By creating rather than discovering concepts, great mathematicians break new ground for thinking. The book as a whole argues in length for these two main theses.In chapter 3, the author addresses the thought of Bernhard Riemann to make the case that Riemann’s mathematical thought puts an emphasis on creative conceptual thinking and is not defined merely by calculations or logic. Such thinking is problem-posing rather than axiomatic. Riemann’s central innovation in this respect is the concept of manifold (Mannigfaltigkeit), which enabled him to think anew space and geometry in a radical non-Euclidean fashion, implying an infinite number of possible geometries. Plotnitsky argues—borrowing Deleuze and Guattari’s terminology—that this opened up a “plane of immanence,” which is to say that Riemann created a concept-generating movement of thought. However, we cannot grasp the generation of concepts in modern mathematics without giving proper attention to the interplay of algebra and geometry that defines it. Chapter 4 demonstrates this interweaving of the two fields by recounting the history of the idea of the curve in modernity, from the analytic geometry and calculus of Fermat and Descartes to its modification in the concept of a Riemann surface, up to its more recent developments in the algebraic geometry of Weil and Grothendieck. It gives a historical account of the algebraization of the curve that at the same time brings into relief the irreducible geometrical aspect, continuity, of all modern mathematical conceptualizations of the curve.The next two chapters provide further case studies. Chapter 5 expands and solidifies the argument of chapter 2. It discusses three examples of mathematical practice, each displayed by a pair of important [End Page 360] mathematicians: Abel and Galois, Lobachevsky and Riemann, Weil and Grothendieck. Each case... (shrink)

Patrick L. Miller - Archytas of Tarentum: Pythagorean, Philosopher, and Mathematician-King - Journal of the History of Philosophy 46:1 Journal of the History of Philosophy 46.1 165-166 Muse Search Journals This Journal Contents Reviewed by Patrick Lee Miller Duquesne University Carl Huffman, Archytas of Tarentum: Pythagorean, Philosopher, and Mathematician-King. Cambridge-New York: Cambridge University Press, 2005. Pp. xv + 665. Cloth, $180.00. Archytas of Tarentum has in some ages been considered a major philosopher. He was one of the three most important (...) early Pythagoreans—along with Philolaus and Pythagoras himself—and so his reputation has risen and fallen with the popularity of this movement. When Pythagoreanism captivated some Romans at the end of the Republic, they championed him as a native talent; much later, some medieval Europeans inflated him into one of the great wise men of the ancient world. But Pythagoreanism has ebbed in recent centuries, and Archytas is now known only to specialists in ancient philosophy, and even then only by acquaintance from a few passages of Plato and.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:To Think Like God: Pythagoras and Parmenides: The Origins of PhilosophyScott AustinArnold Hermann. To Think Like God: Pythagoras and Parmenides: The Origins of Philosophy. Las Vegas: Parmenides Publishing, 2004. Pp. xxx + 374. Cloth, $32.00.Mr. Arnold Hermann could presumably have used his connection with Parmenides Press to publish anything he wanted. Instead, he has put out a sober, bibliographically well aware, thesis about the origin, nature, and motivations (...) of Parmenides' thought, one evolved in dialogue with scholars such as Cordero, McKirahan, and Curd. The resulting monograph, to be the first of three on subjects connected with Parmenides and Plato's Parmenides, is a legitimate contribution to the field, to be taken very seriously as part of any scholarly bibliography. Best of all, in this reviewer's opinion, is the exploration of Parmenides' legal language and metaphors as they might affect the type of abstract object which can count as subject for the esti, the differences between or among routes and rules of inquiry, the necessity for a Doxa-section, the relationships with Xenophanes and Plato. Less successful, I think, is the attempt to put together a coherent picture out of the hazy evidence on Pythagoras and to distinguish levels of probability within that evidence. Nor, in spite of his praise for Mourelatos's book ("... truly one of the most indispensable works on the Eleatic," 214), does Hermann seem aware (see 189) that the possibility of a pluralistic Eleatic ontology was also present in that book, before it was followed by Curd's book.Hermann's thesis is that Parmenidean philosophy was an answer both to Xenophanes' challenge, to his indication of the limits of our human inquiry—our difference from the gods—and the problem of irrationals in Pythagoreanism. By setting out the rational limits which any purely theoretical inquiry must have, Parmenides intended to show that, in pursuing theory, we can, indeed, think like the gods. (Here excellent use is made of work by Mourelatos and Long.) But by distinguishing theoretical inquiry from empirical inquiry, Parmenides showed the limits of Pythagoreanism, which had been unable to make that distinction. (Units are theoretical entities, but points are empirical, so there are in reality no Pythagorean unit-points; rational numbers are theoretical while irrationals are empirical, and so on.) (Here there are affinities with the recent Popper book.) That is, the Doxa-section, according to Hermann, shows tragically that we do not always think like the gods, especially when we think empirically. Our contemporary science is continuing the inquiry that Parmenides began, but the original Parmenidean philosophy was a methodology about how to inquire into abstract objects, not necessarily a metaphysics, a theory about what those abstract objects are. Here Hermann differs from the standard portrait of Parmenides as theorist of an intelligible world; he thinks that it is Plato who is more responsible than Parmenides for theorizing about which objects correspond to the methodology. (Here he follows Mourelatos and Curd in viewing fragment 2 as setting up types of inquiry rather than types of entity.)One has no quarrel with the claim that Parmenides' originality is at least partly methodological. Yet a methodology whose strictures are this strong was thought, by Plato at least, to require absolute monism; is it an accident that I cannot have a pluralistic ontology without having non-identities between members of pairs of objects, non-identity being an example of saying what is not (fragment 8, lines 36–38)? Does not this particular methodology necessitate a very particular metaphysics? [End Page 481]And, if we do think like the gods, at least at times, are we not then thinking about what is ultimately real, i.e., about a certain type of object? If rules can be given which secure a kind of thought which is exempt from Xenophanean scepticism, do not these rules determine sorts of entities? And, if Pythagoreanism errs in confusing theoretical with empirical entities, must not we ourselves possess the perspective of a god or goddess in order to avoid the confusion?These are some of the questions provoked by this fresh book, which, untrammelled by much scholarly narrowness, is yet capable of contributing much to scholarly... (shrink)

This chapter contains sections titled: Criteria Used for the Concept of the Universal Some Conceptual Barriers to Early Grasp of the Universal Empedocles: Formulae for Compounds; Biological Forms; Type‐Identities across Cycles Philolaus: Genus, Species, and the Relation to Particulars Democritus: An Infinity of Atomic Types, Atomic Tokens Comments by Democritus on the Universal Democritus and Aristotle: Origins of the Type–Token Distinction Democritus and Plato Bibliography.

This chapter elucidates the ways Plato camouflages his critical responses to Pythagoreanism by using mythological figures to refer to Pythagorean philosophical invention in the middle and later dialogues, especially Republic, Timaeus, and Philebus. The theme of “discovery” takes on Pythagorean overtones, and the exploration of the various culture heroes in these dialogues shows that Plato ensconced his responses to the mathematical Pythagoreans by narrating the stories of mythological philanthropists who suffered punishment for their transgressions, such as Prometheus and Palamedes.

The paper is the final, fifth part of a series of studies introducing the new conceptions of “Hilbert mathematics” and “ontomathematics”. The specific subject of the present investigation is the proper philosophical sense of both, including philosophy of mathematics and philosophy of physics not less than the traditional “first philosophy” (as far as ontomathematics is a conservative generalization of ontology as well as of Heidegger’s “fundamental ontology” though in a sense) and history of philosophy (deepening Heidegger’s destruction of it from (...) the pre-Socratics to the Pythagoreans). Husserl’s phenomenology and Heidegger’s derivative “fundamental ontology” as well as his later doctrine after the “turn” are the starting point of the research as established and well known approaches relative to the newly introduced conception of ontomathematics, even more so that Husserl himself started criticizing his “Philosophy of arithmetic” as too naturalistic and psychological turning to “Logical investigations” and the foundations of phenomenology. Heidegger’s “Aletheia” is also interpreted ontomathematically: as a relation of locality and nonlocality, respectively as a motion from nonlocality to locality if both are physically considered. Aristotle’s ontological revision of Plato’s doctrine is “destructed” further from the pre-Socratics' “Logos” or Heideger’s “Language” (after the “turn”) to the Pythagoreans “Numbers” or “Arithmetics” as an inherent and fundamental philosophical doctrine. Then, a leap to contemporary physics elucidates the essence of ontomathematics overcoming the Cartesian abyss inherited from Plato’s opposition of “ideas” versus “things”, and now unifying physics and mathematics, particularly allowing for the “creation from nothing” instead of the quasi-scientific myth of the “Big Bang”. Furthermore, ontomathematics needs another interpretation of arithmetic, propositional logic and set theory in the foundations of mathematics, where the latter two ones are both identified with Boolean algebra, and the former is considered to be a “half of Boolean algebra” in the exact meaning to be equated to it after doubling by a dual anti-isometric counterpart of Peano arithmetic. That unified algebraic realization of the foundations of mathematics is related to Hilbert mathematics in both “narrow and wide senses” where the latter is isomorphic to the qubit Hilbert space, thus underlying all the physical world by the newly introduced substance of quantum information being physically dimensionless and generalizing classical information measured in bits. The substance of information, whether classical or quantum, visualizes the way of the unification of physics and mathematics by merging their foundations in Hilbert arithmetic and Hilbert mathematics: thus how ontomathetics is a “first philosophy”. The relation of ontomathematics to the Socratic “human problematics”, furthermore being fundamental for Western philosophy in Modernity, is discussed. Ontomathematics implies its “substitution” by abstract information (or by “subjectless choice” relevant to it), thus “obliterating the human outline on the ocean beach sand” (by Michel Foucault’s metaphor). A reflection back from the viewpoint of mathematic to Western philosophy as the philosophy of locality ends the study. (shrink)

Kahn tries to do justice to the contribution Pythagoras and his followers might have had for Greek science. Thus he downplays the religious figure so prominent with Burkert's groundbreaking study "Lore and Science". He sees the transformation Pythagorean ideas may have undergone in Plato's Academy as pivotal for the developments of Pythagoreanism in later antiquity as well as in Renaissance speculation, e. g. Kepler. The book offers a good overview for the history of Pythagoreanism from its founder to (...) modern times. (shrink)

This paper traces how the dualism of body and soul, cosmic and human, is bridged in philosophical and religious traditions through appeal to the notion of ‘breath’ (πνεῦμα). It pursues this project by way of a genealogy of pneumatic cosmology and anthropology, covering a wide range of sources, including the Pythagoreans of the fifth century BCE (in particular, Philolaus of Croton); the Stoics of the third and second centuries BCE (especially Posidonius); the Jews writing in Hellenistic Alexandria in the first (...) century BCE (Philo); and the Christians of the first century CE (the gospel writers and Paul). Starting from the early Pythagoreans, ‘breath’ and ‘breathing’ function to draw analogies between cosmogony and anthropogony – a notion ultimately rejected by Plato in the Timaeus and Aristotle in his cosmological works, but taken up by the Posidonius (perhaps following the early Stoa) and expanded into a rich and challenging corporeal metaphysics. Similarly, the Post-Hellenistic philosopher and biblical exegete Philo of Alexandria, who was deeply influenced by both Platonist and Stoic physics, approaches the cosmogony and anthropogony described in Genesis (1:1–3 and 1:7) through Platonist-Stoic philosophy, in his attempt to provide a philosophically rigorous explanation for why Moses employed certain terms or phrases when writing his book of creation. Finally, the chapter sees a determined shift in the direction of rejecting pneumatic cosmology for a revised pneumatic anthropogony in the writings of the New Testament: by appeal to the ‘Holy Spirit’ or ‘Holy Breath’ (πνεῦμα ἅγιον), early Christians effectively adapted the Stoic metaphysics of ‘breath’, with its notions of divine intelligence and bonding, to the prophetic and ecclesiastical project of building a Christian community conceived of as the ‘body of Christ’. Hence, the spiritual cosmogony of the Pythagoreans, Stoics, and Philo is effectively subordinated to the spiritual anthropogony that facilitates the construction of the Christian kosmopolis, only fully realised in the form of New Jerusalem, the ‘bride’ which, in tandem with the Holy Spirit, calls to the anointed. At the end of the Christian worldview, the kosmos of Greek philosophy is supplanted by the pneumatic kosmopolis. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

The dóxa is a major aspect of Parmenides’ “scientific” commitment, as evidenced by studies of its discoveries in various fields of knowledge. Among them in astronomy. However, these studies have ended up identifying a scientific plan separate from the mythicalreligious and philosophical one of the first part. This survey explores the possibility of reconsidering the presence in the Parmenides’ dóxa of ontology and theology. It does so by proposing the hypothesis that the proem contains among the multiple semantic-linguistic layers the (...) mythopoeic presentation of the astronomical theorem of the identity of Eos and Hesperus, with the intent of showing, from the very beginning and in the most authoritative language for his audience, the alḗtheia of «being» on the divine stage of the celestial sphere. In our reading, therefore, the deities that enter the scene give life to a firmament that interconnects every cosmic dimension with each other and leads the one who knows (the εἰδότα φῶτα; B1, 3) – he knows that πᾶν πλέον ἐστὶν ὁμοῦ φάεος καὶ νυκτὸς because ἐπεὶ οὐδετέρωι μέτα μηδέν (B9, 3-4) – on the other side of the sky, in the presence of the theá that will reveal to him the meaning of everything. (shrink)