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

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

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)

Eugene Afonasin highlights the wealth of information on Pythagoras and his tradition preserved in Clement of Alexandria’s Stromateis and presents them against the background of Later Platonic philosophy. He  rst outlines what Clement knew about the Pythagoreans, and then what he made of the Pythagorean ideal and how he reinterpreted it for his own purposes. Clement clearly occupies an intermediate position between the Neopythagorean biographical tradition, rmly based on Nicomachus, and that more or less vague and difuse literary (...) situation which preceded the later developments, and in this respect is a very good source, worth studying for its own sake and as supplementary material which can help to understand the great Pythagorean synthesis attempted by Iamblichus. Developing their variants of the “exhortation to philosophy” (protreptikoi logoi), these men were much concerned with the educational value of the Pythagorean way of life rather than biographical circumstances, designed to place the ancient sage in the proper cultural context. (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 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)

India dropped its target of 500 GW of renewable energy capacity fossil fuel sources by 2030. Its responsibilities the United Nations Framework Convention Climate Change [UNFCCC],and reducing radiations by one billion tonnes by the end of the decade at the COP26 conference, held in Glasgow in November 2022. Researchers are continually searching for inexhaustible and reasonable energy sources. Solar energy is one of the greenest sources of energy and is also one of the cleanest. The most important factor in using (...) solar energy is the location of the solar power plant. The main objective of this study is to find the best location for a new solar power plant in a specific region called Bundelkhand region of Uttar Pradesh in India. Here we offer an extension of ELECTRE III method as two-phase Pythagorean neutrosophic elimination and choice translating reality PN-ELECTRE-III) method to adapt with fuzzy, ambiguous, unsure, and indeterminate criteria. The Pythagorean neutrosophic numbers [PNNs] used by the group decision support system of PN-ELECTRE III to measure performance of the alternatives. The options are entirely outclassed in the subsequent stage in view of the past stage's evaluations of them. By defining PNN we describe the technique of indifference threshold functions, preference treshold and veto threshold functions, which provide a more stable basis to drop outranking relations. By calculating the concordance credibility, discordance credibility and net credibility degrees of each alternative, the ranking module of the PN-ELECTRE III approach is made simpler. In order to confirm the applicability of the strategy suggested in this paper, the location selection problem for solar plants is finaly solved. (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 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.

The polis, as a gathering of various citizens, may be threatened by discord and finally may collapse because of the stasis, the internal conflict between different groups of people with diverging interests. This scheme is tackled by Plato in Gorgias, and more thoroughly in the Republic. Both dialogues were a source of inspiration for the pseudo-Pythagorean writings which flourished between the second half of the 4th century B.C. and the Hellenistic period. Among them, the treaties attributed to Kleinias, Metopus, Theages, (...) Lysis and Hippodamus frequently use the concept of stasis and pleonexia to describe how a city may be governed and what kind of danger may appear if the citizens’ behavior is not controlled. In general, these treaties adapt the vision of Plato concerning conflict to some Pythagorean images and teachings. By mingling both influences, they blur the frontier between Platonism and Pythagoreanism and create a genre of intertwined literature which may be qualified as bricolage, according to Lévi-Strauss’s concept. These philosophical texts use a range of material mostly traced back to the Hellenistic period, but also some fragments related to the conception of conflicts and violence in early Pythagoreanism. (shrink)

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)

This contribution aims to explore the historical predecessors of the Five Percenter model of self-realization, as popularized by Hip Hop artists such as Supreme Team, Rakim Allah, Brand Nubian, Wu-Tang Clan, or Sunz of Man. As compared to frequent considerations of the phenomenon as a creative mythological background for a socio-political struggle, Five Percenter teachings shall be discussed as contemporary interpretations of historical models of self-realization in various philosophical, religious, and esoteric systems. By putting the coded system of the tenfold (...) Supreme Mathematics as one of its core teachings in context with the Pythagorean Tetractys, an arrangement of ten points in four lines, the commonalities between the sequence and concepts attributed to the respective numbers will be demonstrated. (shrink)

Here we propose a discussion about the "mandorla" or "vesica piscis". It is a type of 2-dimensional lens, that is, a geometric shape formed by the intersection of two circles with the same radius, intersecting in such a way that the centre of each circle lies on the perimeter of the other. The aim of the discussion is that of understanding when such a geometric shape became a symbol and when this symbol received a specific name. We will find that (...) the name "mandorla" was used long before the term "vesica piscis", which is the Latin translation of the German "fischblosen" used by Albrecht Dürer in his book on geometry. Therefore, the name invented by Dürer was not used by the painter for a sacred form. However, after the middle of the nineteenth century the term "vesica piscis" exploded in literature. Its use was criticized and, at the same time, it was stressed that the proper term for the symbol is "mandorla". Nonetheless, the "vesica piscis" continues to be largely used in the sacred geometry, which ascribes symbolic and sacred meanings to certain geometric shapes and proportions. In the proposed discussion, we will also show that, recently, the 2-dimensional lenticular symbol has been related to the Pythagorean philosophy. It is told that the followers of this philosophy had the habit of using an apple for symbolic communications. Sliced across, the core of the apple is displaying a pentagram, but sliced lengthwise it forms two intersecting circles, that is a "mandorla". Then, the investigation about terms and related uses of this specific geometric shape is the core of the matter of our discussion or, let us tell, the "core of the apple", in a Pythagorean-like approach to find how a German "fischblosen" evolved into a sacred symbol, thanks to a translation into Latin. (shrink)

In the literature, new types of neutrosophic sets have been introduced in the meantime by the growing neutrosophic community. We present a few: Pythagorean Neutrosophic Set, Fermatean Neutrosophic Set, Generalized Fermatean Neutrosophic Set, n-power Neutrosophic Set, Cubic Spherical Neutrosophic Set, Spherical Neutrosophic Set, n-HyperSpherical Neutrosophic Set, Refined n-HyperSpherical Neutrosophic Set.

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)

I contend that “philosophos” is meant to carry the connotation of a Pythagorean: Euenus is a native from Paros which had a strong Pythagorean community down to the end of the fifth century. Moreover, “philosophos” was used to refer to the Pythagoreans, as can be seen from the story related by Cicero from Heraclides Ponticus (Tusc. Disp. V, iii, 7-8; cp. DL, 1.12; 8.8). I argue (against Burkert) that even if this story is part of the lore surrounding Pythagoras (...) and, hence, without historical value as for Pythagoras, it may still be used as evidence for the use of “philosophos” among latter-day Pythagoreans. (shrink)

This paper aims to construct a new formula that generates a generalized version of congruent numbers based on a generalized version of Pythagorean triples. Here, an elliptic curve equation is constructed from the derived generalized version of Pythagorean triples and congruent numbers and gives some new results.

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)

At Phaedo 86b7–c2 Simmias puts forward the theory that the soul is the attunement of bodily elements. Many scholars have claimed that this theory originates in the Pythagoreans, especially Philolaus. The claim is largely based on their reading of the Phaedo, since we have scarce doxographical evidence. In this paper I show that the dialogue in question does not constitute any evidence for the Pythagorean origin of Simmias’ attunement theory, and that it rather represents the theory as stemming from (...) a materialistic world view that had become widespread among people at that time. I also pursue the question how and why, then, the attunement theory was able to attract many people, by examining its relation to Socrates’ affinity argument. (shrink)

The Pythagorean tradition dominates the understanding of beauty up until the end of the 18th Century. According to this tradition, the experience of beauty is stimulated by certain relations perceived to be between an object/construct's elements. As such, the object of the experience of beauty is indeterminate: it has neither a determinate perceptual analogue (one cannot simply identify beauty as you can a straight line or a particular shape) nor a determinate concept (there are no necessary and sufficient conditions for (...) beauty at the semantic level). By the 13th Century in the West, the pleasure experienced in beauty is characterized as disinterested. Yet, on the basis that all cultural manifestations of the pythagorean theory of beauty recognize that judgments of beauty are genuine judgments, we would want to say that judgments of beauty are lawful. In addition, from ancient times, up until after Kant, philosophers of beauty within this tradition recognize two kinds of beauty: a universal, unchanging beauty coexisting with a relative, dynamic beauty. These two kinds of beauty and the tensions discussed above, are reconciled and dissolved respectively, according to the metaphysical/religious commitments of the particular author. As yet, however, these features of beauty have not been reconciled within a physicalist worldview. This is what I set out to do. (shrink)

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)

I argue that according to Socrates in the Phaedo we should not merely evaluate bodily pleasures and desires as worthless or bad, but actively avoid them. We need to avoid them because they change our values and make us believe falsehoods. This change in values and acceptance of falsehoods undermines the soul’s proper activity, making virtue and happiness impossible for us. I situate this account of why we should avoid bodily pleasures within Plato’s project in the Phaedo of providing Pythagorean (...) and Orphic ideas with clearer meanings and better justifications. (shrink)

A new translation and edition of Aristotle's Protrepticus (with critical comments on the fragments) -/- Welcome -/- The Protrepticus was an early work of Aristotle, written while he was still a member of Plato's Academy, but it soon became one of the most famous works in the whole history of philosophy. Unfortunately it was not directly copied in the middle ages and so did not survive in its own manuscript tradition. But substantial fragments of it have been preserved in several (...) works by Iamblichus of Chalcis, a third century A.D. neo-Pythagorean philosopher and educator. On the basis of a close study of Iamblichus' extensive use and excerption of Aristotle's Protrepticus, it is possible to reconstruct the backbone of the lost work, and then to flesh it out with the other surviving reports about the work from antiquity (for example in Alexander of Aphrodisias and other ancient commentators on Aristotle). It is also possible to identify several papyrus fragments of the work, and many references and literary allusions in later authors, especially Cicero, whose own lost dialogue Hortensius was a defense of philosophy modeleld on Aristotle's. (shrink)

_Theurgy and the Soul_ is a study of Iamblichus of Syria, whose teachings set the final form of pagan spirituality prior to the Christianization of the Roman Empire. Gregory Shaw focuses on the theory and practice of theurgy, the most controversial and significant aspect of Iamblichus's Platonism. Theurgy literally means "divine action." Unlike previous Platonists who stressed the elevated status of the human soul, Iamblichus taught that the soul descended completely into the body and thereby required the performance of theurgic (...) rites—revealed by the gods—to unite the soul with the One. Iamblichus was once considered one of the great philosophers whose views on the soul and the importance of ritual profoundly influenced subsequent Platonists such as Proclus and Damascius. The Emperor Julian followed Iamblichus's teachings to guide the restoration of traditional pagan cults in his campaign against Christianity. Although Julian was unsuccessful, Iamblichus's ideas persisted well into the Middle Ages and beyond. His vision of a hierarchical cosmos united by divine ritual became the dominant world view for the entire medieval world and played an important role in the Renaissance Platonism of Marsilio Ficino. Even Ralph Waldo Emerson wrote that he expected a reading of Iamblichus to cause a "revival in the churches." But modern scholars have dismissed him, seeing theurgy as ritual magic or "manipulation of the gods." Shaw, however, shows that theurgy was a subtle and intellectually sophisticated attempt to apply Platonic and Pythagorean teachings to the full expression of human existence in the material world. (shrink)

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)

The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations of (...) energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality. (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 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)

This academic paper provides a historical reflection on the problem of the origin of science in order to determine the reasons for differences in determining the date and content of the first scientific achievements. The application of historical-genetic research methods in the disciplinary aspect contributes to the distinction of particular scientific programs in the science body frame with a different relationship between the object and subject of cognition, the internal logic of ideas and worldviews. As a result, the existing research (...) concerned with the conditions of knowledge function in different types of society or with the relations between the structure of knowledge and its purpose, was supplemented by an explication of forms of the continuity of program components – ontological, epistemological and axiological basis of specific scientific cognition/knowledge. In particular, it is established that the ancient world atomistic and peripatetic programs follow the structural pattern of the scientific explanation of the Pythagorean one (scientific and philosophical at the same time), contributing a new subject content in it. (shrink)

Plato refers frequently to the views held by the early Greek thinkers we today call ‘the Presocratics’, typically while lining up witnesses for or against a philosophical thesis. His characters speak approvingly of the doctrines of Parmenides and the Pythagoreans but repudiate in the strongest terms the teachings of ‘atheistic materialists’ such as the Milesian inquirers into nature we today regard as the founders of Western philosophy and science. The chief failings of the materialists lay in not acknowledging the (...) priority of soul over matter and not believing that a cosmic intelligence has arranged all things for the best. On occasion, Plato states a view held by a thinker he has elsewhere criticized and he is not above borrowing the ideas of others without identifying his source. Thus, while Plato’s dialogues are an invaluable source of information for the views of earlier thinkers, his representations must be read with caution. (shrink)

The text is a continuation of the article of the same name published in the previous issue of Philosophical Alternatives. The philosophical interpretations of the Kochen- Specker theorem (1967) are considered. Einstein's principle regarding the,consubstantiality of inertia and gravity" (1918) allows of a parallel between descriptions of a physical micro-entity in relation to the macro-apparatus on the one hand, and of physical macro-entities in relation to the astronomical mega-entities on the other. The Bohmian interpretation ( 1952) of quantum mechanics proposes (...) that all quantum systems be interpreted as dissipative ones and that the theorem be thus derstood. The conclusion is that the continual representation, by force or (gravitational) field between parts interacting by means of it, of a system is equivalent to their mutual entanglement if representation is discrete. Gravity (force field) and entanglement are two different, correspondingly continual and discrete, images of a single common essence. General relativity can be interpreted as a superluminal generalization of special relativity. The postulate exists of an alleged obligatory difference between a model and reality in science and philosophy. It can also be deduced by interpreting a corollary of the heorem. On the other hand, quantum mechanics, on the basis of this theorem and of V on Neumann's (1932), introduces the option that a model be entirely identified as the modeled reality and, therefore, that absolutely reality be recognized: this is a non-standard hypothesis in the epistemology of science. Thus, the true reality begins to be understood mathematically, i.e. in a Pythagorean manner, for its identification with its mathematical model. A few linked problems are highlighted: the role of the axiom of choice forcorrectly interpreting the theorem; whether the theorem can be considered an axiom; whether the theorem can be considered equivalent to the negation of the axiom. (shrink)

In this book, we approach different topics related to neutrosophics, such as: Neutrosophic Set, Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set, Picture Fuzzy Set, Ternary Fuzzy Set, Pythagorean Fuzzy Set, Atanassov’s Intuitionistic Fuzzy Set of second type, Spherical Fuzzy Set, n-HyperSpherical Neutrosophic Set, q-Rung Orthopair Fuzzy Set, truth-membership, indeterminacy-membership, falsehood-nonmembership, Regret Theory, Grey System Theory, Three-Ways Decision, n-Ways Decision, Neutrosophy, Neutrosophication, Neutrosophic Probability, Refined Neutrosophy, Refined Neutrosophication, Nonstandard Analysis; (Theory, NeutroTheory, AntiTheory), S-denying an Axiom, Multispace with Multistructure, and so on.

The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after (...) measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question. Number and being (by the meditation of time), the natural and artificial turn out to be not more than different hypostases of a single common essence. This implies some kind of neo-Pythagorean ontology making related mathematics, physics, and technics immediately, by an explicit mathematical structure. (shrink)

This paper seeks to explore the way Giovanni Pico della Mirandola treated the Orphics and the Pythagoreans in his Conclusiones nongentae, his early and most ambitious work, so that he formulates his own philosophy. I do not intend to present and analyze the sum of Pico’s references to Orphics and Pythagoreans, since such an attempt is beyond the scope of this paper. Rather, I aim to highlight certain Pico’s aphorisms that allow readers to understand and evaluate his syncretic (...) method and his goals. In addition, I attempt to trace Pico’s sources and evaluate his proper knowledge, understanding and treatment of the Orphics and the Pythagoreans. Pico resorts to the Orphics and the Pythagoreans because he wants to give a practical and applied dimension to his philosophy. He attempted the revival of the original wisdom that underlies the traditions he combined. He was not a fan of the Aristotelian θεωρίης ἢνεκεν. Philosophy does not aim at proper knowledge, but is the key for the manipulation of the cosmos, physically and metaphysically speaking. Pico probably thinks of himself as the modern Orpheus: he does not intend to reveal the paths which cross the sensible and the intelligible, but he aspires to tread them. (shrink)

In this paper, I present two crucial problems for Wolff’s metaphysics of quantities: 1) The structural identification problem and 2) the Pythagorean problem. The former is the problem of uniquely defining a general algebraic structure for all quantities; the latter is the problem of distinguishing physical quantitative structure from mathematical quantities. While Wolff seems to have a consistent and elegant solution to the first problem, the second problem may put in jeopardy his metaphysical view on quantities as spaces. After drawing (...) a parallelism between Wolff’s treatment of quantitative structures and Frege’s conception of quantitative domain, I propose a solution to the Pythagorean problem based on the idea that mathematical structures are the result of applying an abstraction principle on physical quantitative structures. In particular, I propose the view that abstraction may be seen as the operation of structure determination which transforms concrete physical quantities (i.e. undetermined structures) into abstract mathematical quantities (fully determined structures of thin and shallow objects). (shrink)

In his treatise "Elementa harmonica," Aristoxenus of Tarentum formulated a distinctive philosophical framework within the realm of sound exploration, deviating from the harmonics theory advanced by the Pythagoreans. A pivotal disparity lies in the methodological approach to research. Aristoxenus posited meticulous observation of acoustic phenomena as the foundational step in all inquiries, constituting a fundamental criterion for truth. He asserted that perception alone could authenticate the fidelity of the observed state of affairs. Aristoxenus vehemently opposed the Pythagorean disregard for (...) auditory testimony, deeming it a notable methodological lapse. According to him, the harmonikos must serve as a vigilant observer of acoustic phenomena, equipped with a discerning and cultivated musical ear. (shrink)

The concept of kosmos did not play the leading role in Aristotle’s physics that it did in Pythagorean, Atomistic, Platonic, or Stoic physics. Although Aristotle greatly influenced the history of cosmology, he does not himself recognize a science of cosmology, a science taking the kosmos itself as the object of study with its own phenomena to be explained and its own principles that explain them. The term kosmos played an important role in two aspects of his predecessor’s accounts that Aristotle (...) rejected: first, cosmogeny and kosmopoiia, generation or creation of the kosmos; second, diakosmêsis, arranging of a plurality of kosmoi. Aristotle was extremely critical of accounts involving kosmopoiia and diakosmêsis and he developed general dialectical strategies against them. In emphatically distinguishing his view from all his predecessors (including Plato), he uses the terms ho ouranos (the heaven), to holon (the whole), and to pan (the totality) in preference to ho kosmos (the kosmos or world). There is usually no harm in speaking loosely of ‘Aristotle’s cosmology’ when referring to his concept of the order of nature and the ouranos. Nevertheless, Aristotle’s theoretical philosophy offers something very different from those of his predecessors for whom kosmos was a keyword. (shrink)

In the context of feminist and postmodern thought, traditional conceptions of masculinity and what it means to be a “Real Man” have been critiqued. In Genevieve Lloyd's The Man of Reason, this critique takes the form of exposing the effect that the distinctive masculinity of the “man of reason” has had on the history of philosophy. One major feature of the masculine-feminine dichotomy will emerge as a key notion for understanding the rest of the paper: the dichotomy of reason-feeling, a (...) parallel and a foundational aspect to the dichotomy of masculinity and femininity. In exploring the history of symbolic conceptions of masculinity in ancient Greece, the Renaissance, and the present, one finds that the oppression of women is integrally linked to the traditional tie between masculinity and reason. There have been many efforts in recent feminist philosophy to rewrite or redefine “Woman” in such a way as to alleviate the oppression of women. I argue that the effectiveness of rewriting Woman for this purpose is problematic, primarily because any rewriting of this type must occur in the current historical context of hierarchical dualisms, like Man-Woman, masculine-feminine, male-female. These binary oppositions arguably find their roots in Pythagorean philosophy and can be traced through the Renaissance to our current historical context. It is these dualisms that have traditionally valued the masculine side of the Man-Woman dichotomy more than the feminine. Further, it will be argued that the hierarchical dualism of Man and Woman is so pervasive that if we rewrite or redefine the inferior, deprivileged side of that dualism, we cannot correct its devalued status. Instead, we redefine that which is undervalued but retain its devalued status. This particular aspect of attempts to critique hierarchical dualisms like reason and feeling has been reflected in the writings of many feminists, male and female. This paper will show that in cases where women attempt to redefine the dichotomy by revaluing the traditionally feminine (like feelings and emotions) over the traditionally masculine (like reason) their work is often mistakenly criticized for being purely political; conversely, when men attempt to redefine the same dichotomy in an attempt to allow men to “get back in touch with their feelings,” to be nurturers, their work is described in terms of providing a better epistemology. The current literature on masculinity explores alternatives to rewriting or redefining Woman that try to avoid the problem of status remaining with redefinition. This alternative is rewriting or redefining Man. Through redefining Man, one may be able to reconceptualize the privileged side of the hierarchical dualism in such a way that it is no longer privileged. Deprivileging, as well as redefining Man, is argued by theorists of masculinity to be possible because while the devalued status of the inferior side of a hierarchical dualism tends to keep the same status when redefined, it may be possible to redefine the privileged side of the dualism in such a way that it loses its privileged status. Unfortunately, many of these attempts to rewrite or redefine masculinity have detrimental faults of their own. Finally, this paper will discuss more promising possibilities for new definitions of Man, as well as a vision for better interaction between the work of women and men in general. (shrink)

As Leibniz points out in the Méditation sur la notion commune de la jus tice, justice—defined as charity of the wise and universal benevolence—belongs “to the necessary and eternal truths about the nature of things, as numbers and proportions.” According to the interpretation of Patrick Riley, from this perspective the two manuscripts usually regarded as belonging to the Méditation should be seen as complementary parts of a unitary Platonizing work. According to Riley, the manuscript that now constitutes the first part (...) of the Méditation is concerned with definitions of ethical concepts viewed as “quasi-mathematical, demonstrable ‘eternal verities’”, whereas the manuscript that now constitutes the second part of the Méditation is concerned “with Platonic ‘ascent’, in the manner of Phaedrus and Symposium,” which recommends the transition from mere negative forbearance from harm to doing positive good. In formulating these claims, Riley uses scare quotes to indicate that he uses the terms “eternal verities” and “ascent” in an unusual way that diverges from the views of the historical Plato. According to his interpretation, Leibniz’s modifications of Platonism are restricted to peripheral parts such as the doctrine of the pre-existence of the soul or Pythagorean components such as the doctrine of metempsychosis. Therefore, Riley claims that when Leibniz is talking about knowing eternal truths, what he has in mind is literally the view that we know the same truths as the truths that the gods know and love but do not cause or change. (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 follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” (...) to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices. (shrink)

This essay is a contribution to the historical phenomenology of science, taking as its point of departure Husserl’s later philosophy of science and Jacob Klein’s seminal work on the emergence of the symbolic conception of number in European mathematics during the late sixteenth and seventeenth centuries. Sinceneither Husserl nor Klein applied their ideas to actual theories of modern mathematical physics, this essay attempts to do so through a case study of the conceptof “spacetime.” In §1, I sketch Klein’s account of (...) the emergence of the symbolic conception of number, beginning with Vieta in the late sixteenth century. In §2,through a series of historical illustrations, I show how the principal impediment to assimilating the new symbolic algebra to mathematical physics, namely, thedimensionless character of symbolic number, is overcome via the translation of the traditional language of ratio and proportion into the symbolic language of equations. In §§3–4, I critically examine the concept of “Minkowski spacetime,” specifically, the purported analogy between the Pythagorean distance formula and the Minkowski “spacetime interval.” Finally, in §5, I address the question of whether the concept of Minkowski spacetime is, as generally assumed, indispensable to Einstein’s general theory of relativity. (shrink)

The relationship between logic and ethics is one of the basic and most essential questions of classical philosophical analysis. Since the time of the Pythagoreans, the fundamental unity of the two – whether by means of vague intuition, an elaborate conceptual scheme, or even a carefully crafted lifestyle – has led philosophers to identify truth and virtue. In his critical philosophy Kant put this unity of truth and virtue to extensive and rigorous trial to determine what conditions, if any, (...) a... (shrink)

Recent feminist and postmodern thought has critiqued traditional conceptions of masculinity, describing the effect that the distinctive masculinity of the "man of reason" has had on the history of philosophy, on consciousness, and on the academy. A common characteristic of the recent literature on masculinity is that it reflects the historical and cultural context in which it is written -- a context of binary, hierarchical dualisms which involve certain symbolic associations. These dualisms, such as Man-Woman, masculine-feminine, and reason-emotion, arguably find (...) their roots in Pythagorean philosophy and can be traced through the Renaissance to our current historical context. One example is our conception of reason: the association of maleness with a clear, determinate mode of thought, i.e. reason, was incorporated into the form-matter distinction that was central to Greek thought. This paper will show that in cases where women attempt to redefine the dichotomy by revaluing the traditionally feminine (like feelings and emotions) over the traditionally masculine (like reason) their work is often mistakenly criticized for being purely political; conversely, when men attempt to redefine the same dichotomy in an attempt to allow men to "get back in touch with their feelings," to be nurturers, their work is described in terms of providing a better epistemology. The current literature on masculinity explores alternatives to rewriting or redefining Woman that try to avoid the problem of status remaining with redefinition. This alternative is rewriting or redefining Man. Through redefining Man, one may be able to reconceptualize the privileged side of the hierarchical dualism in such a way that it is no longer privileged. Deprivileging, as well as redefining Man, is argued by theorists of masculinity to be possible because while the devalued status of the inferior side of a hierarchical dualism tends to keep the same status when redefined, it may be possible to redefine the privileged side of the dualism in such a way that it loses its privileged status. Unfortunately, many of these attempts to rewrite or redefine masculinity have detrimental faults of their own. Finally, this paper will discuss more promising possibilities for new definitions of Man, as well as a vision for better interaction between the work of women and men in general. (shrink)

Art as Festival Art has been a privileged sphere in the sense that it appeales to art-receivers because of its power of taking them outside the sphere of everyday life (commonplaceness) and practicality. For the first time the philosophers of ancient Greece propagated this view. In this paper the Pythagorean and Aristotle’s views are considered; however they have different metaphysical bases, there is some aspect common to theirs aesthetics. In 20th century it is Helmut Kuhn, Hans-Georg Gadamer and Władysław Stróżewski (...) who express similar ideas and convictions about an exceptionality of the meeting between art and man. In the first part of the article (the ancient) notion of “leisure” (scholé) is presented, while in the second one the case of “festival”, Festlichkeit and Eigenzeit is taken into account. (shrink)

Regarding the influence of Francois Rabelais on the Giordano Bruno’s works, up to now the criticism have only taken into consideration the lexical and thematic analogies. This article individualizes, in a passage of the "Oratio Valedictoria", a literal quotation from the Gargantua et Pantagruel , showing that Rabelais was adirect source of inspiration for Bruno. The protagonist of the passage is the pythagorean Apollonius of Tyana, a character well known from the Nolan, who mentioned him in many occasions. He represents (...) theideal medium with the oriental wisdom, that had so much influence on the thought of Bruno. Besides, heembodies the model of the "natural magician" in which the philosopher repeatedly identified himself, both inhis works and life. (shrink)

Regarding the influence of François Rabelais on the Giordano Bruno‟s works, up to now the criticism have only taken into consideration the lexical and thematic analogies. This article individualizes, in a passage of the Oratio Valedictoria, a literal quotation from the Gargantua et Pantagruel, showing that Rabelais was a direct source of inspiration for Bruno. The protagonist of the passage is the pythagorean Apollonius of Tyana, a character well known by the Nolan, who mentioned him in many occasions. He represents (...) the ideal medium with the oriental wisdom, that had so much influence on the thought of Bruno. Besides, he embodies the model of the "natural magician" in which the philosopher repeatedly identified himself, both in his works and life. (shrink)