Pythagoreans

Averiguando la esboza de Filolao, tomando cuenta del recurso acesorio que hizo verodImilmente a datos observacionales.

Mathematics appears to play an explanatory role in science. This, in turn, is thought to pave a way toward mathematical Platonism. A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and their associated physical states. I explain why Platonists (...) should accept that physical systems have mathematical properties, and why a property based account is better than existing accounts of mathematical explanation. I close by considering whether nominalists can accept the view I propose here. I argue that they cannot. (shrink)

O problema deste paper constitui-se a partir da seguinte pergunta: seria Platão um plagiador dos pitagóricos? A suspeita nasce de uma tradição já antiga pela qual Platão é acusado de ter plagiado uma suposta obra atualmente perdida de Filolau de Crotona. Tomou-se, então, como objeto de análise, a obra Timeu, a fim de verificar a existência de trechos que poderiam sugerir um plágio por parte de Platão. Para tanto, realizou-se um levantamento das principais teorias “pitagóricas” identificáveis no Timeu, a partir (...) de uma revisão da história da crítica. Concluiu-se que, inegavelmente, há referência implícita ao pitagorismo na obra, a qual, entretanto, não configura propriamente um plágio, mas, sim, pode ser melhor considerada como uma influência pitagórica. Efetivamente, Platão acaba por transpor diversas teorias “pitagóricas” – bem como de outros filósofos pré-socráticos – para seus próprios interesses filosóficos. (shrink)

La explicación tradicional del concepto de "vacío" (τò κενóν) en la filosofía antigua lo sitúa como una invención del atomismo de Demócrito y Leucipo o, incluso, del eleático Meliso de Samos. De esta manera se ocultan las profundas razones que pudieron llevar a la necesidad y surgimiento de un tal concepto, y aparece como si hubiera sido creado ex nihilo. En este artículo se pretende descubrir tanto el principio mitológico como el origen racional del concepto de "vacío" en la filosofía (...) presocrática, que lo sitúa como anterior a las historias contadas hasta ahora. Mediante dos columnas que intentan aprehender lo mismo desde puntos de vista distintos, se llegará a la conclusión de que el concepto de "vacío" fue definido en el inicio como diferencia y repetición. En contra de una determinada visión de la historia de la filosofía que ve a los "clásicos" y a los "modernos" como dos compartimentos separados e incluso enfrentados, en el curso de esta interpretación se establecerá un diálogo entre la doxografía presocrática y la filosofía contemporánea de Schelling, Hegel, Nietzsche, Heidegger, Schmitt, Irigaray, Derrida o Deleuze. (shrink)

In this paper, we introduce the concepts of Pythagorean fuzzy valued neutrosophic set (PFVNS) and Pythagorean fuzzy valued neutrosophic (PFVNV) constructed by considering Pythagorean fuzzy values (PFVs) instead of numbers for the degrees of the truth, the indeterminacy and the falsity, which is a new extension of intuitionistic fuzzy valued neutrosophic set (IFVNS). By means of PFVNSs, the degrees of the truth, the indeterminacy and the falsity can be given in Pythagorean fuzzy environment and more sensitive evaluations are made by (...) a decision maker in decision making problems compared to IFVNSs. In other words, such sets enable a decision maker to evaluate the degrees of the truth, the indeterminacy and the falsity as PFVs to model the uncertainty in the evaluations. (shrink)

The well-known connection between the extant fragments of Philolaus and Plato’s Philebus is examined in its methodological aspect. By drawing on more texts, it is shown that Plato was aware of an explanatory scheme that can be attributed to Pythagoreanism. His attempt to modify it is also outlined, which sets the historical-philosophical perspective.

This chapter pursues an understanding of what Cicero thought 'Italic' philosophy to be, and proceeds to develop a broader account of how Cicero's version compares with the surviving textual evidence and testimonia from the Hellenistic period of the philosophy of the 'Italic' philosophers, including the Lucanians 'Ocellus', 'Eccelus', and 'Aresas/Aesara', and the Rudian Ennius. Special focus is placed on their theories of cosmology, psychology, and law. Collocation of 'Italic' with 'Pythagorean' philosophy of this era aids in building a more comprehensive (...) view of what Hellenistic Pythagoreanism may have looked like. Uploaded here is the pre-publication version; please email me for the printed version. (shrink)

This book presents a new account of Thales based on the idea that Acheloios, a deity equated with water in the ancient Greek world and found in Miletos during Thales’ life, was the most important cultic deity influencing the thinker, profoundly shaping his philosophical worldview. In doing so, it also weighs in on the metaphysical and epistemological dichotomy that seemingly underlies all academia—the antithesis of the methodological postulate of Marxian dialectical materialism vis-à-vis the Platonic idea of fundamentally real transcendental forms. (...) Unbeknownst to many scholars, there are various Neo-Marxian thinkers that position the origin of coinage as the pivotal technological development giving rise to impersonal “metaphysical cosmology,” suggesting that the value of money was more-or-less projected back onto the cosmos in the form of “ideal substances.” While the arguments are incredibly sophisticated and persuasive, their conclusions (either stated or implied) are rather difficult to swallow: the self is merely an illusion, abstract ideas of an ultimate source of value, like God or the Good, are totally delusional (as is the soul, and presumably any notion of inherent human dignity), and essentially everything is reducible to mankind’s enslavement to commodities and the notion of our own objectified labor, which is the true source of all value according to Marx. Not only is this an alarming belief that many scholars (consciously or unconsciously) have adopted, since essentially any action could be “justified,” it is also demonstrably false, since it rests on a misunderstanding of Thales and misconception of philosophy as such. -/- My work rectifies that misunderstanding. In an important sense, it is an attempt at redefining philosophy as a “love of wisdom,” which I argue was accurate even in the Presocratic setting, and it uses the influence of Acheloios on Thales to do so. Throughout its pages I explore the etymology and historical uses of the word ὔδωρ, examine the archaeological context of 7th to 6th century Miletos, consider various aquatic myths Thales encountered, and highlight a hitherto overlooked tradition stemming from Thales and influencing such thinkers as Pythagoras, Empedokles, and Hippo, which culminates in a completely new reading of Plato’s Phaedrus, a dialogue in which Plato responds to the exact type of thinking employed by the Neo-Marxians. It is there that we find Socrates and Phaedrus surrounded by the iconography of Acheloios and the nymphs, all while they lie reclined like river gods (the sinews of Acheloios) on the banks of the Ilisos. And it is in that dialogue that Plato defines philosophy as a love of wisdom—the beholding of a multiplicity of hermeneutical frameworks—and alludes to the fact that it began with the sacrifice of Acheloios, the initial philosophical maneuver which he attributes to Thales. The book ends with a threefold rejoinder to the Neo-Marxian school, corresponding to the λόγος, μῦθος, and ἔργον of Acheloios. It turns out that, (1) the λόγος of Acheloios contained the ideal preconditions conducive to an abstraction to a more refined philosophical worldview in which divine water operated as the One among the Many; (2) the μῦθος of Acheloios actually encouraged the application of the notion of sacrifice to Acheloios himself (thus revealing his essence as divine water); and, (3), the ἔργον of Acheloios, in which he kneels in assent to sacrifice, is found on a coin that was probably designed by Thales. In the final analysis, I suggest that the tradition of Acheloios is reflective of a greater philosophical truth, and that by following Thales’ lead, we transcend the Marxian hermeneutic of doubt and reorient ourselves toward the οὐσια ὄντως οὖσα. (shrink)

This paper attempts to demonstrate that the conviction about the harmony and order of the world was a fundamental metaphysical principle of the Pythagoreans. This harmony and order were primarily sought in the structures of arithmetics, yet following the discovery of incommensurable magnitudes (irrational numbers, as we now call them), the Pythagoreans began to see geometrical structure as a fundamental part of the world. On the example of the Pythagoreans’ metaphysics and science, the paper shows the mutual relations between metaphysics (...) and science. It demonstrates— on the one hand—the necessity of the first as a guide for the latter, and—on the other—how our scientific research can change its basic metaphysical principles when these are found to be inappropriate. The paper also tries to show the need for a realistic approach in our scientific research by means of the same example of the Pythagoreans, that is, the need to discern something which is below the surface appearance. (shrink)

This paper critically examines the use of the name 'Pseudo-Archytas' to refer to two aspects of the reception of Archytas of Tarentum in antiquity: the 'author-inflection' and the 'authority-inflection'. In order to make progress on our understanding of authority and authorship within the Pythagorean tradition, it attempts to reconstruct Porphyry's views on the importance of Archytas as guarantor of Pythagorean authenticity in the former's lost work On the History of the Philosophers by considering a fragment preserved in Arabic by Ibn (...) Abī Uṣaybi‘a. The article finally argues that a range of problems attend our use of the term 'pseudo-Archytas', which is not fit for purpose when considering the evidence regarding authorship and authority in the Pythagorean tradition. It recommends a more critical approach to the notion of authenticity within the Pythagorean tradition and suggests a new term, 'Archytism', as a more useful point of reference. (shrink)

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite sets and series and quantum (...) information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

This paper examines the relation between Pythagorean and Heraclitean political views. I argue that for Pythagoras, Heraclitus, and Archytas the cosmological and musical notions of harmony (ἁρμονία) and the related notion of concord (ὁμόνοια) have an intrinsic political significance. These thinkers variously reflect upon political harmony and concord, and agree that a crucial condition for it is law (νόμος), which according to Pythagoras and Heraclitus has a divine origin. I begin with the Heraclitean fragments 22 B51, 54, 72, and 114 (...) DK, in which social and political reflection is connected with the theory of the harmony of opposites. In the case of Pythagoras and early Pythagoreanism, the intense, albeit indirect political influence in Magna Graecia—as transmitted by Iamblichus and Porphyry—offers evidence for social and political ideas determined by a reflection on the cosmological role of harmony and number. Finally, Archytas’ political application of rational calculation in fragment 47 B3 DK, which aims at producing concord by establishing the just measures of wealth in the city, together with the testimony on Archytas’ intense and fruitful political activity, can be taken as confirmation that the thinker’s wider cosmological views were indeed intertwined with his political reflection and action. (shrink)

Archytas of Tarentum, a contemporary and associate of Plato, was a famous Pythagorean, mathematician, and statesman of Tarentum. Although his works are lost and most of the fragments attributed to him were composed in later eras, they nevertheless contain valuable information about his thought. In particular, the fragments of On Law and Justice are likely based on a work by the early Peripatetic biographer Aristoxenus of Tarentum. The fragments touch on key themes of early Greek ethics, including: written and unwritten (...) laws; freedom and self-sufficiency; moderation of the emotions and cultivation of virtues; equality and the competence of the majority to participate in government; criticism of “rule by an individual”; a theory of the ideal “mixed constitution”; distributive and corrective justice and punishment, and of the rule of law. The fragments also contain one of the only positive accounts of democracy in ancient Greek philosophy. (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)

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)

The article deals with the doctrines of Orpheus and Pythagoras about the immortality of the soul in the context of the birth of philosophy in ancient Greece. Orpheus demonstrated the closeness of heavenly (divine) and earthly (human) worlds, and Pythagoras mathematically proved their fundamental identity. Greek philosophy was “an investment in the afterlife future”, being the product of the mystical (Orpheus) and rationalist (Pythagoras) theology.

Pythagoras’s number doctrine had a great effect on the development of science. Number – the key to the highest reality, and such approach allowed Pythagoras to transform mathematics from craft into science, which continues implementation of its project of “digitization of being”. Pythagoras's project underwent considerable transformation, but it only means that the plan in knowledge is often far from result.

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)

Traditionally, kairos has been seen as a “timely” concept, and so invention is said to emerge fromthe timeliness of a cultural and historical situation. But what if invention was thought of as thepotential to shift historical courses through the injection of something new or alien into a situation?This essay argues that kairos has not been able to free itself from its historical constraints becauseit has been bound to a human sense of temporality. By evolving along patterns different from print,the apparatus (...) of the cinema developed in a way where it was not bound to illustrating movement ortimeas it occurs in human-centered experience. Following the work of Gilles Deleuze on cinema,this article argues that the outside of a human sense of time is an untapped source of invention,already present yet dormant within kairos. (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.

The advent of logical positivism contributed to the sharp definitional demarcation between what we consider mythical (mythos) and what we take to be a true account (logos). This essay attempts to go back to one of the sources of such a supposed distinction. By analyzing the Gorgias, I will show that even Plato did not make such a distinction. In fact, Plato even constructed a theory of justice that made use of myth as its medium. The Platonic Myth in the (...) Gorgias was used as true logos in order to justify Socrates’ use of the myth as the paradigm of a life that is philosophical in contrast to that of his accusers who espoused mere sophistry. By using the concept of historia in relation with the concept of mythos, Plato regarded the futuristic afterlife in the myth as a sufficient condition to live a just life. This sufficient condition exemplified by the Myth in the Gorgias is a measuring rod by which we can compare our present system of justice. (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. (shrink)

Studying Plato's "unwritten doctrines" in the light of his discussion of limit and unlimited in his dialogue Philebus. The essay raises also the question whether there is too much "atomism" in the usual presentation of Plato's Forms as individual absolute entities, rather than as themselves derived from a more fundamental limit/unlimited ontology.