Discussions of the Gallo-Roman dodecahedron often note the significance of the dodecahedron in Platonic and Pythagorean philosophy, but they tend not to relate the archaeology to the textual evidence in any detail. We attempt to do that in this paper. We argue that, whilst it remains the case that there is no contemporaneous description of the Gallo-Roman dodecahedron, there are several texts – including an overlooked passage in Iamblichus’ On the Pythagorean Life – that point to its possible inspiration. We relate these texts to the location of dodecahedra in Gaul and Britain and to the interest of the Druids in Pythagoras, which was frequently remarked upon by Roman commentators.

As an academic discipline, mathematics is more than a set of computing mechanisms; it is argued by scholars that the quality of mathematics education is in decline today in human society. Historically, mathematics and philosophy had a close interrelationship enabling the understanding of various natural phenomena. According to Platonism of pure mathematics, the dynamics of natural phenomena and human societies follow invariant and absolute mathematical principles. This article presents a socio-philosophical argument that the concept of society within the natural space can be classified into natural society and synthetic society based on the concept of mathematical purifications, and mathematics education has a role in it. The existence of logical inversions of various forms in the synthetic societies are analysed and corrective roles of mathematics education are explained. The Pythagorean philosophy of mathematics in building an improved as well as just society is an appropriate solution that calls for a relook into mathematics education in order to reduce utilitarian distortion in mathematics education today and to promote intellectualism as well as harmony while reforming mathematics education. Contribution: This article presents detailed analysis of the existing critical issues related to mathematics education today in human society and it compares various social forms in nature, including human, in light of the concepts of socio-philosophy. It is illustrated that these two aspects are interrelated in view of developments in education and evolution of human society. The historical perspectives of mathematics education forming a free and harmonious society following the Pythagorean philosophy of mathematics and society are presented, which can be a solution to the multidimensional problems in human society and in mathematics education today.

In the Pythagorean tradition, friendship is elevated beyond a mere human relationship, serving as a means to transcend human frailty and attain immortality. This philosophy posits that humans are imprisoned and require liberation through the benevolence of the gods. The Pythagorean way of life is seen as a path to achieving immortality and freedom, where friendship with the gods is the highest form of association. The spiritual practice of theurgy is essential in this process, enabling humans to purify themselves and receive the gift of friendship from the gods. The Pythagoreans distinguished between various forms of friendship, including the highest and most noble understanding between gods and humans, which requires faith, knowledge, philosophy, and theurgy. True friendship is characterized by trust, piety, and scientific worship, and its pursuit necessitates the avoidance of jealousy and conflict, as well as careful judgment and reverence. Additionally, purification, self-control, and a healthy diet are crucial in the pursuit of wisdom and friendship. Ultimately, the Pythagorean philosophy on friendship offers a profound understanding of human relationships, emphasizing the importance of spiritual growth, self-transcendence, and the pursuit of wisdom, leading to the cultivation of true and lasting friendships that bring about wholeness, reconciliation, and harmony.

The article discusses how Pythagorean philosophy evolved from seeing numbers as the basis of everything to understanding them as essential principles shaping the universe. It emphasizes the practical and philosophical significance of numbers, exploring their role in various fields like math, science, and ethics. Pythagoreans viewed numbers not just as tools for counting but as fundamental aspects of existence itself. They focused on two key principles: unity (monad) and duality (duad), seeing them as foundational concepts in understanding both numbers and reality[23]. The text also compares Pythagorean ideas with those of later philosophers like Plato, Iamblichus and others. Overall, it explores the profound implications of seeing the world through the lens of numbers.

Pythagoras ve Pythagorasçılık, antik felsefe geleneğinde farklı bir yere sahiptir. Bunun sebebi Pythagorasçılığın Orfizm, Hint ve Mısır dinlerinin felsefe ile harmanlanmasından oluşan eklektik yapısıdır. Bu dinlerden Orfizm, Pythagorasçılığı en çok etkilediği ve antik Yunan dinine mistik bir muhteva kazandırdığı için bilhassa önemlidir. Bir sır dini olan Orfizm, felsefe ve dinler tarihinde kimi zaman Pythagoras ile özdeşleştirilen Ozan Orfeus’a dayanır. Sesinin güzelliği ve ilahileri ile antik dönemde kendisine ilahi bir nitelik atfedilen Orfeus, reenkarnasyon ve Tanrı’yla bir olma gibi pek çok ögeyi Yunan dinine kazandırmıştır. Orfizm’in katı çilecilik ve hikmet anlayışına dayalı yaşam tarzı ve öğretileri Yunan dininden daha güçlü bir teolojik altyapıya sahip olduğu için Pythagoras’tan itibaren önce İtalyan felsefe geleneğini etkilemiş, oradan da Platon aracılığıyla gerek felsefe gerekse Yahudilik, Hıristiyanlık gibi diğer dinleri de etkilemiştir. Böylece Milet filozoflarının homerik dinden arındırmaya çalıştığı felsefe, Orfizm ve Pythagoras sayesinde yeniden bir nebze mistik ruh kazanmıştır. Bir filozof aynı zamanda da dinî bir cemaatin de lideri olan Pythagoras, Kroton’da kurduğu “Düzen” isimli felsefe okuluna inisiye usulü ile öğrenci kabul etmiştir. Böylece Pythagorasçılık hem bir felsefi ekol hem de inisiyetik bir sır cemaati olma niteliği kazanmıştır. Pythagoras, bu okulda, sayılardan oluşan arkhe düşüncesinin yanı sıra matematik ve müzik sistemine dayanan güçlü bir doğa felsefesi kurmuştur. Aynı zamanda o dinî bir lider olduğu için okulunu hem felsefe öğretmek hem de orfik bir ritüel olan teletaik sır ayinlerini yönetmek için de kullanmıştır. Pythagorasçılara göre o, bir üstad olarak bu ayinlerde Dionysos’un ruhu ile birleşerek Tanrı’nın Külli Ruh’una yükselme ayrıcalığı kazanmıştı. Bu birleşme sayesinde Pyhtagoras’ın Dionysos’tan vahiy aldığına ve Külli Ruh’un sırlarına vakıf olduğuna inanılıyordu. Benzer şekilde Pythagoras’ın felsefe öğrencileri de bu ayinlere katılıyordu. Onlar bir yandan Üstad Pythagoras’ın tanrısal birleşmesine şahit oluyor diğer yandan da Dionysos’a bağlı ikincil tanrılar yani demonlarla iletişime geçmeyi deneyimleyebiliyorlardı. Pythagorasçılık’ta icra edilen bu mistik ayinler sayesinde elde edilen felsefi bilginin sadece bir akıl yürütme olmadığı aynı zamanda ilahi kelamdan edinilen tanrısal bir ilim olduğuna inanılıyordu. Bu sebeple Pythagorasçılık felsefeyi, kaynağı Tanrı’da olan ve insanı tanrısallık makamına ulaştırma işlevine sahip bir yaşam tarzı olarak kabul ediyordu. Pythagoras felsefesine göre Tanrı ile ilgili düşünmekle doğa felsefesiyle uğraşmak aynı şeydi. Çünkü panteist bir Tanrı inancına sahip olan Pythagorasçılık, Tanrı ile doğayı aynı varlık olarak kabul ediyordu. Buna bağlı olarak ekol, doğa felsefesini Külli Ruh kabul ettikleri Monad anlayışı üzerine kurmuştu. Varlığın temel ilkesini sayı arkhesine dayandıran Pyhtagorasçılık için Monad, aslen bir sayı değil, bütün sayıların kaynağı ve teolojik bir ilke olduğuna inanılan BİR’di. Külli Ruh olan Tanrı’ya tekabül eden ve kozmik bir vahdeti simgeleyen BİR, karşıtı olan İKİ’nin yol açtığı kaosu sınırlandırmasıyla hayat var ediyordu. Benzer şekilde bir sayıdan değil ilke olan İKİ, Anaksimandros’ta olumlu anlamda kullanılmış ve “sınırsızlık” anlamındaki apeiron şeklinde ifade edilmişti. Fakat Pythagoras için sınırsızlık negatif bir hâl olduğu için İKİ/Apeiron, varlık öncesi sınırsız yokluk ve kötülük haline denk düşen dişil bir varlık kabul edilmişti. Monad’ın, Apeiron’u sınırlamasıyla kozmosa geçilmiş, kozmik denge (harmoni) kurulmuş ve küresel bir varlık hali ortaya çıkmıştır. İşte bu, Pythagorasçı doğa felsefesinde hem Külli Ruh Monad’a hem de doğanın kendisine tekabül eden “vahdet” halidir. Pythagorasçı felsefede Külli Ruh ve Tanrı’yı ifade eden Monad kavramının ele alınacağı bu çalışma, felsefe tarihinin konusu olan bir meseleyi dinler tarihi perspektifiyle ele alacaktır. Bu bağlamda felsefe tarihinin kaynak ve bilgi birikimi, karşılaştırmalı dinler tarihi yöntemiyle kullanılacak ve Pythagorasçı Monad Orfizm’den başlayarak mitlerdeki benzer ögelerle ele alınacaktır. Böylece Pythagorasçı felsefenin antik dinlerdeki kökeni tespit edilerek teolojik altyapısı ortaya konacaktır.

Sir Francis Galton singlehandedly instigated the navigational settings for the discipline of psychometrics by presupposing that mental attributes are measurable. In turn, this presupposition became the defining pillar of the psychometric paradigm. There were no scientifically sound reasons for adopting this presupposition and those Galton gave beg the question every time. So, what drove him to endorse this presupposition? Two considerations steered him in this direction: first, his Pythagorean philosophy of science according to which measurement is a necessary feature; and second, his desire to present eugenics as a science, which, given his Pythagorean vision, entailed that eugenics must involve measurement of relevant mental attributes. The quantitative presupposition guiding psychometrics throughout its history was, therefore, a spin-off from Galton’s marketing strategy for the pseudoscience of eugenics.

The article explores the informal side of the idea of the proof. The word "idea" is used in a sense dating back to Plato. Proof is understood as a precisely established connection of precisely formulated and objectively existing ideas. This connection of ideas belongs to the realm of the possible and can be present in some possible worlds and absent in others. Attempts to interpret the proof as a procedure of convincing argumentation are criticized. It is shown that the proof is not necessarily convincing, and that persuasiveness may be inherent outside the proof-based forms of argument. The genesis of the idea of the proof is briefly traced from its origins to the present day. Arguments are made in favor of the thesis of the idea of the proof in Pythagorean philosophy. The question of why the idea of the proof has not been rediscovered anywhere and never is discussed. The problem of the time gap between the appearance of the proof and the exact definition of the concept of the proof in modern logic is considered. An accessible example substantiates the inseparability of the idea of the proof from its formal presentation in one or another logic. A list of some basic informal predicates of the proof is given and a brief description is given.

Abstract We critically reconsider an old hypothesis of the role of the dekad in Pythagorean philosophy. Unlike Zhmud, we claim that: 1) the dekad did play a role in Philolaus’ astronomical system, and 2) Aristotle did not project Plato’s theory of the ten eidetic numbers onto the Pythagoreans. We claim that the dekad, as the τέλειος ἀριθμός, should be understood in Philolaus’ philosophy as completeness and the basis of counting in Greek – as in most other languages – in a decimal system. Additionally, we argue that the number ten is not even a candidate for the τέλειος ἀριθμός in Plato’s philosophy. As a final result of our discussion, we compare and contrast Philolaus’, Plato’s, and Speusippus’ accounts of completeness in relation to numbers.

In the sixth and fifth centuries BC, Pythagorean philosophers managed to hold power in several cities of southern Italy. Subsequently, they were victims of attacks from political opponents, which killed most of them. The surviving Pythagoreans had to seek asylum elsewhere, becoming political refugees. There are several different ancient traditions and modern theories about the reasons of the violent opposition who turned the powerful Pythagoreans in exiles. However, in this paper it is argued that in, any case, the root cause of this opposition is the elitist attitude that marked Pythagorean philosophy, way of life and politics.

<p>This work is an analysis of the number concept based on the thoughts of Pythagoras, well as to seek an approximation of the historical reflections of the Pythagorean School in the Greek context of Classical Antiquity. The guiding objective of the study is to analyze the theory of the integers in the Pythagorean philosophy as ultimate cause of man and matter. From Methodological procedures point of view, the research was the bibliographical type, developed from already published material constituted, primarily, of books and journal articles through which we sought a dialogue with the developed studies concerning the subject in question. Through research it was reached the following findings: First, Pythagoras considered the number as the forming principle of the universe. However, he believed that the number constitutes the agreement between an indeterminate or unlimited element and another determinant or limiting and from there had been origin all things; Second, to the Pythagoreans the structure of numbers started from the assumption that in even numbers in even numbers predominates the indeterminate element and in the odd numbers the limiting element, for this reason they had the odd numbers as perfects, because in their distribution the odd numbers always had a limiting element. In sum, both Pythagoras as the Pythagoreans had in the cult of the number the basis of both his philosophy and his way of life.</p>

The preserved fragments of the Parmenides’ work is one of the oldest testimonies of the mystical experience in philosophy. Mystique meets here with metaphysics. The poem’s prologue is a transcript of the mystic passage of a young man to the goddess who symbolizes the truth of being. By knowing the goddess and her message, young man learns the nature of being. The ecstatic experience opens him on the metaphysical dimension of reality. The sources of the Parmenides’ mysticism are following: the pythagorean philosophy, orphism, shamanism and the cult of Apollo at Delphi.

The question-statement, “All is number?”, the title of Zhmud’s famous 1989 article in Phronesis , opens a challenge to the extremely important Aristotelian testimony that “all is number” was the fundamental definition of Pythagorean philosophy. Such a challenge is anything but easy, especially when one considers that, so far, the histories of both ancient philosophy and ancient mathematics seem to have no doubts that this definition is correct. This paper aims to submit Aristotle’s claim that the Pythagoreans believed that “all is number” to critical review. Our analysis of the many ways in which Aristotle states the thesis that “all is number” will reveal, beyond merely semantic variations, a fundamental theoretical contradiction that Aristotle himself seems incapable of solving. Three different versions of the doctrine are in fact present in the Aristotelian doxography: (a) an identification of numbers with the sensible objects; (b) an identification of the principles of numbers with the principles of things that are; (c) an imitation of objects by numbers. While versions (a) and (c) seem to identify numbers with the material cause of reality, in terms (“imitation”) reminiscent of Plato, version (b), numbers as formal causes of reality, is an Aristotelian reconstruction of the Pythagorean theory. Aristotle would have been pushed to such a reconstruction by the difficulty he found in accepting the Pythagorean material notion of number, and by considering it closer to its sensitivity, strongly marked by the reception of that same theory in the Academic realm. Keywords: Ancient Philosophy, Pythagoreanism, Aristotle, numbers, ontology.

Acknowledgements Abbreviations Introduction: Toward a Genealogy of Christian Neopythagoreanism PART ONE The Genesis of Neopythagoreanism: A Synopsis 1. Platonic Transformations of Early Pythagorean Philosophy 2. The Neopythagorean Revival: Henology and Mediation 3. The Late Antique Preservation of Neopythagoreanism PART TWO The Pearl Diver: Thierry of Chartres's Theology of the Quadrivium 5. The Discovery of Folding 6. Thierry's Diminished Legacy PART THREE Bright Nearness: Nicholas of Cusa's Mathematical Theology 7. The Accidental Triumph of De docta ignorantia 8. Chartrian Theology on Probation in the 1440s 9. The Advent of theologia geometrica in the 1450s 10. Completing the Circle in the 1460s Epilogue Bibliography

Was Plato a Pythagorean? Plato's students and earliest critics thought so, but scholars since the 19th century have been more skeptical. In Plato and Pythagoreanism, Phillip Sidney Horky argues that a specific type of Pythagorean philosophy, called 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 new approaches to science that sought to reconcile empirical knowledge with abstract mathematical theories. Plato and Pythagoreanism shows 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 over the development of philosophical and scientific methods have 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 concerning the essential relationships between the cosmos and the human being.

This article shows how in two texts of Ennius, the opuscule entitled Epicharmus and the prologue of the Annals, the experience of a revelatory dream is the setting of a disquisition of some aspects of Pythagorean philosophy (the elements doctrine, metempsychosis). In doing so, this article aims at making known fragmentary texts hardly studied in France, that nonetheless happen to be significant in the history of ideas, in particular on account of the posterity in Latin literature of this alliance between the apocalyptic paradigm and Pythagorean philosophy.

Prispevek je posvečen filozofskim vidikom Beroaldovega komentarja k Apulejevemu Zlatemu oslu, zlasti k 11. knjigi. Beroaldo večkrat namigne na mistične obrede boginje Izide, ki jih razume kot odsev pitagorejskih in platonističnih skrivnih naukov. Tako se kot »Lukijev« komentator in hkrati aemulator sooči s težko nalogo interpretacije, ki ne sme razkriti preveč.

One tends to think of the study of pure number as an esoteric pursuit. But for the ancient Maya, particularly when it came to temporal matters, numbers were more than mere devices to tally units of time. In stark contrast to the Western calendar, in the Maya realm of timekeeping the duration between ritual events seems to have mattered as much as the times when the events themselves occurred. Moreover, the manner in which the daykeepers of the Maya codices sequenced the intervals followed well-defined patterns, which reveal an array of motives for the Maya way of structuring time. Among these motives were the need to: a) arrive at or avoid particular lucky or unlucky days; b) accommodate changing seasonal or other astronomical events; and c) set up numerological mirror symmetries, a characteristic that resonates with the Pythagorean philosophy of number. Thus the long-neglected study of Maya number offers insight into the chronological structure underlying ritual process, which finds parallels in contemporary Maya culture.

Plato was not just a ´Socratic´. According to Diogenes Laertius, he started with Heraclitean philosophy before he met Socrates. His dialogues show that he was eager to take up other approaches as well. Next to the Eleatic influence, visible in the Parmenides and other late dialogues, Pythagorean philosophy seems to have had the strongest impact on his thinking. The biographical tradition asserts that he had repeatedly personal contact with the Pythagoreans from Tarentum, and Aristotle´s chapters 5 and 6 of Metphysics, Book I show that Plato´s philosophy of number and of the principles had great similarity to Pythagorean thinking. In this article, Aristotle´s statements about this part of Plato´s philosophy are examined one by one. It can be shown that in their entirety they form a coherent, meaningful whole.

Mathematical Constants of Natural Philosophy

The present essay will cross, inside the matter of the sources of the platonic thought, the suggestion of Damascius of Damascus, with the intention to draw clear understanding, unless in this particular point, of the relationship between the ancient pythagoreanism and the platonic philosophy. In this, the study of the matter of the dialectics of the limiters/unlimited one is central. The page 16c of the Philebus is the crucial point of this discussion: here Socrates introduces the theme of the unity/multiplicity as a very beautiful hodos, a run that comes from a very distant point of departure: it’s a “gift of the gods” and a discovery of the ancient ones. The comparative study of these footsteps of the Philebus and the fragments of Philolaus, especially В 2, 3, 6 and 7 bring us to conclude that the presence in the pages of the Philebus of the theory of the limiter/unlimited be taken as a pre-platonic “discovery” of the more ancient pythagorean philosophy.