arXiv subjects

Stelios Negrepontis

Publications and source records attributed to Stelios Negrepontis.

8 recordsLinked to original sources

arXiv2026

The Mysteries of Plato's Parmenides Dispersed: From musical intervals to contacts/"hapseis" to "logoi"

Plato's Parmenides has long been considered the most difficult and enigmatic of Platonic dialogues, marked by seemingly contradictory arguments. We present an interpretation showing that the One of the Second Hypothesis (142b-155e) is an intelligible Being whose structure constitutes the philosophical analogue of a dyad in periodic anthyphairesis. First, by linguistic analysis and Proclus' Platonic Theology, we show that the One and the Being form an indefinite dyad satisfying the philosophical analogue of infinite anthyphairesis. We analyze Plato's initial introduction and eventual rejection of eristic numbers formed with unequal units from the infinite sequence of remainders (143c-144c). Second, we reveal that the presence of the One in the Being (144c-d, 138a) is achieved via contacts ("hapseis", 148d-149d). Based on the Pythagorean musical rule relating terms and intervals, a contact is identified with the ratio between consecutive anthyphairetic remainders. The circularity of contacts (138a) expresses Theaetetus' Logos Criterion for anthyphairetic periodicity, defining dialectical numbers with equalized units and establishing the finiteness of the dialectical number of parts. This framework resolves key Platonic enigmas: Plato's Indivisible Line is identified with the intelligible Being; the Sophist's assertion that "not-Being is a Being" corresponds to the equalization of the dyad; and the compresence of opposite properties separates intelligibles from sensibles. Finally, the self-similar Oneness of an intelligible Being refutes Vlastos' Non-Identity principle, resolving the Third Man Argument.

math.HO↗

arXiv2026

The Birth of Number Theory (Book VII of Euclid's Elements) from the Arithmetization of Pythagorean Music

Book VII of Euclid's Elements represents a remarkable mathematical achievement, marking the birth of number theory. Although it falls short of explicitly stating the Fundamental Theorem of Arithmetic (that every natural number is uniquely a product of primes), it contains all the tools necessary for its proof: the Principle of the Least (equivalent to Mathematical Induction) and arithmetical anthyphairesis for finding the greatest common divisor. Our work presents novel arguments that Book VII evolved directly from early Pythagorean arithmetized music. The accounts attributing this arithmetization to Pythagoras' acoustical experiments were shown to be fictitious by Vincenzo Galilei. In contrast, the alternative experiments and Hippasus' 4-chord (bronze cylinders of heights 6, 8, 9, 12) are validated as physically correct by Euler's Law for pipes. This experimental foundation led to the arithmetization of musical intervals and to the discovery of musical (multiplicative) anthyphairesis. Applying Aristotle's Topics 158b24-29 Principle, we reconstruct how musical anthyphairesis was transferred - through an inductive ladder of ratios starting with multiple and epimoric ratios recounted by Theon of Smyrna - to its arithmetical (additive) counterpart. Furthermore, we show that the mathematical peculiarities in the definitions and proofs of Book VII find a convincing explanation only through their musical origin. Ultimately, Hippasus emerges as the pivotal figure who discovered musical anthyphairesis (which evolved into Philolaus' Fragment 6) and transferred arithmetical anthyphairesis to geometry, crucial for Pythagorean incommensurability and the principles of the Infinite and Finite.

math.HO↗

arXiv2026

Notes on angles and solid angles, in relation with Euler's memoir De mensura angulorum solidorum

We provide some historical context to the study of solid angles carried out by Euler in his memoir \emph{De mensura angulorum solidorum} (On the measure of solid angles). We extend our study to the general notion of angle (not only solid). While doing so, we explore some works by Ancient Greek mathematicians and others by Arabs mathematicians of the Middle-Ages as well as some later Western authors from the Renaissance. In particular, we review the Pythagorean anthyphairetical perspective on angles which establishes the basis of the important relation between the mathematical notion of angle and the philosophical concept of finitization of the Infinite. In doing so, we shall show that questions addressed by Euler lead us to questions raised about 2500 years ago. At the same time, we highlight the fact that mathematics in those times is also today's mathematics. The reader can also see in this study the intermingling between mathematics and philosophy. This paper will appear in the book \emph{Spherical geometry in the Eighteenth Century, I: Euler, Lagrange and Lambert}, ed. R. Caddeo and A. Papadopoulos, Springer, 2026.

math.GT↗

arXiv2025

The mathematics of periodic anthyphairesis as a basis for the full understanding of Plato's philosophy

Even though Plato's philosophy in ancient times was always closely associated with mathematics, modern Platonic scholarship, during the last five centuries, has moved steadily toward de-mathematization. The present work aims to outline a radical re-interpretation of Plato's philosophy, according to which the Platonic Idea, that is, the intelligible Being, has the structure of the philosophical analogue of a geometric dyad in a philosophic anthyphaeresis -- the precursor of modern continued fractions -- which was studied by the Pythagoreans, Theodorus and Theaetetus in relation with the discoveries of quadratic incommensurabilities. This mathematical structure is clearly visible in the Platonic method of Division and Collection, equivalently Name and Logos, equivalently True Opinion plus Logos, in the dialogues Theaetetus, Sophist, Statesman, Meno, and Parmenides. Equipped with this structure of an intelligible Being, we provide definitive answers to fundamental questions, that were not be resolved by Platonists, concerning the following topics: the dialectic numbers, which are based on the anthyphairetic periodicity and the plus one rule, stating that the dialectic number of terms of a sequence is the (number of) ratios of successive terms plus one (stated in the Parmenides 148d-149d); the description of the intelligible being as an Indivisible Line, a statement bordering on the contradictory; the also seemingly contradictory Sophist 's statement that ``the not-Being is a Being'', based on the equalization of the two elements of the dyad defining an intelligible Being; the more general self-similar Oneness of an intelligible Being, based on the equalization of all parts generated by the anthyphairetic division of an intelligible Being; and finally the Third Man Argument in the Introduction to the Parmenides, appearing as a threat for Plato's theory, but essentially innocuous because of the self-similar Oneness. The third part of our study aims to prove that, contrary to the presently dominant interpretation of Zeno's arguments and paradoxes as being devoid of mathematical content, the analysis of Zeno's presence in the Parmenides, Sophist (via the Eleatic Stranger), and Zeno's verbatim Fragments preserved by Simplicius, show that Plato's intelligible Beings essentially coincide with Zeno's true Beings, and hence that Zeno's philosophical thought was already anthyphairetic, and hence heavily influenced by the Pythagorean's Mathematics. These findings run against Burkert's claim that ``ontology is prior to mathematics''. Modern Platonists have never obtained a clear description of the structure of an intelligible Idea in terms of the mathematics of periodic anthyphairesis, and thus were not able to answer fundamental questions, nor to realize the close connection of Zeno's intelligible beings with Zeno's true Beings.

math.HO↗

arXiv2025

The restoration of Book X of the Elements to its original Theaetetean form

In the present work, we aim to restore Book X of the $\it{Elements}$ to its original Theaetetean, pre-Eudoxean form in two separate ways. First, we restore the considerable mathematical content of Book X, by correlating Book X with Plato's account of Theaetetus' mathematical discoveries and Plato's imitations of these discoveries for his philosophy. Thus, Theaetetus proved (i) the eventual periodicity of the anthyphairesis of lines a to b, satisfying $Ma^2 = Nb^2$, for MN not square number; (ii) the eventual periodic anthyphairesis of lines a to b, satisfying more general quadratic expressions, including the Application of Areas in defect, and employing this to show that the 12 classes of alogoi lines, including the minor, despite being alogoi, are determined by an eventually periodic Application of Areas in defect; (iii) the anthyphairetic palindromic periodicity of the anthyphairesis of the surds $\sqrt{N}$ for any non-square number N, of relevance to the general Pell's Diophantine problem. Secondly, we restore the proofs of all propositions of Book X, in such way that these are proofs based on Theaetetus', and not on Eudoxus' theory of proportion of magnitudes, in particular not making any use of Eudoxus' condition (namely of definition 4 of Book V). The restoration is based on our reconstruction of Theaetetus' theory of proportion for magnitudes, for the limited class of ratios a/b such that either a, b are commensurable or the anthyphairesis of a to b is eventually periodic, without employing Eudoxus' condition, and its success provides a confirmation of our reconstruction. The final version of this paper will appear as a chapter in the journal Ganita Bh\=arat\=i, Bulletin of the Indian Society for History of Mathematics, (2) 45 (2023).

math.HO↗

arXiv2025

The Reconstruction of Theaetetus' Theory of Ratios of Magnitudes

In the present chapter, we obtain the reconstruction of Theaetetus' theory of ratios of magnitudes based, according to Aristotle's Topics 158b, on the definition of proportion in terms of equal anthyphairesis. Our reconstruction is built on the anthyphairetic interpretation of the notoriously difficult Theaetetus 147d6-e1 passage on Theaetetus' mathematical discovery of quadratic incommensurabilities, itself based on the traces it has left on Plato's philosophical definition of Knowledge in his dialogues Theaetetus, Sophist and Meno. Contrary to earlier reconstructions by Becker, van der Waerden and Knorr, our reconstruction reveals a theory that (a) applies only to the restricted class of pairs of magnitudes whose anthyphairesis is finite or eventually periodic, and (b) avoids the problematic use of Eudoxus' definition 4 of Book V of Euclid's Elements. The final version of this paper will appear as a chapter in the book Essays on Topology: Dedicated to Valentin Po\'enaru, ed. L. Funar and A. Papadopoulos, Springer, 2025.

math.HO↗

arXiv2025

The anthyphairetic reconstruction of the original Pythagorean proof of incommensurability, by means of the restoration of Book II of the Elements to its original Pythagorean form

Unquestionably the greatest discovery of the Pythagoreans is the existence of incommensurable magnitudes, most probably the incommensurability of the diameter to the side of a square, but there is no agreement among historians of Greek mathematics on their method of proof. In this chapter we present novel arguments not only for an anthyphairetic reconstruction of the original Pythagorean proof of incommensurability, but also in favor of one that employs the Pythagorean Application of Areas in Excess and in fact Geometric Algebra. The main tool for this reconstruction is the restoration of Book II of the Elements to its original Pythagorean form. The final version of this paper will appear as a chapter in the book Essays on Geometry: Celebrating the 65th Birthday of Athanase Papadopoulos, ed. A. Muhammed Uluda\u{g} and A. Zeytin, Springer International Publishing, 2025.

math.HO↗

arXiv2014

The Anthyphairetic Revolutions of the Platonic Ideas

In the present work it is shown, by an examination of the Platonic dialogues Theaetetus, Sophistes, Politicus, and Philebus, that (a) a Platonic Idea is the philosophic analogue of a pair of lines incommensurable in length only, (b) the Division and Collection, the method by which humans obtain knowledge of a Platonic Idea, is the philosophic analogue of the palindromically periodic anthyphairesis of this pair, and (c) a Platonic Idea is One in the sense of the self-similarity induced by periodic anthyphairesis. A byproduct of the above analysis is that (d) Theaetetus had obtained a proof of the Proposition: The anthyphairesis of a dyad of lines incommensurable in length only is palindromically periodic. It is further verified that the concepts and tools contained in the Theaetetean Book X of the Elements suffice for the proof of the Proposition.

math.HO↗