Homeomeric Lines in Greek Mathematics

AN ANTHOLOGY OF PHILOSOPHICAL STUDIES, 2011

In the 4th century BC Euclid composed his well-known text the Elements, introducing to the world the first known example of an axiomatic treatise and system that was also considered to be the first ‗textbook' designed for students. These thirteen books on mathematics, and primarily geometry, however, do not at any point define or even utilize the term ‗space'. The most obvious reason for this omission is that the ancient Greeks as a whole did not have a term exactly equivalent to that of space. Instead, the Greeks used either topos, meaning ‗place', or diastēma, meaning, for the most part, ‗distance', chōra, meaning ‗locus', schēma, meaning ‗figure', and atopos meaing ‗void' or more literally ‗noplace'. Of course, it is not necessarily the case that one must utilize an actual definition of space in order to present a viable account of geometric objects, yet it seems we would still be inclined, at least in modern thinking, to consider or else just assume geometrical objects to be spatial objects of some sort. Euclid's response is in no way mysterious; all we have to do is revisit his definitions of the different kinds of objects that he identifies in order to see the manner in which he characterizes them without utilizing the concept of space, at least not explicitly. Instead, perhaps the broadest or most general summation of Euclid's definitions of geometrical objects is that he relies on a principle of relation, whereby each object involves a particular kind of relation between its constituent parts, particularly, in this case, those of magnitudes and angles. Euclid, however, also does not define or really utilize a term equivalent to ‗relation' either, but rather more specifically utilizes and defines the concept of logos, i.e., ‗ratio', yet this is not found in his definitions of such objects themselves. The terms Euclid does mention are directional, e.g., lines that approach one another or that tend away from one another according to angle, as well as, of course, those that are parallel. One point worth noting here is that in Modern Greek chōra can indeed be translated as ‗space', but again in Ancient Greek the term is usually translated as a ‗locus' or spot in which an object is located. Euclid first employs another term, namely schēma or ‗figure' in Book I Def. 14:-A figure is that which is contained by any boundary or boundaries.‖ We can see how none of these terms can be the

The aim of this paper is to employ the newly contextualised historiographical category of ''premodern algebra'' in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on ''geometrical algebra''. Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid’s geometrical proofs, the so-called ''semi-algebraic'' alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron’s practice, highlights the significance of contextualizing ''premodern algebra'', and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

Symmetry, 11(1) 98, 2019

The objective of this paper is to propose a mathematical interpretation of the continuous geometric proportion (Timaeus, 32a) with which Plato accomplishes the goal to unify, harmonically and symmetrically, the Two Opposite Elements of Timaeus Cosmos—Fire and Earth—through the Mean Ratio. As we know, from the algebraic point of view, it is possible to compose a continuous geometric proportion just starting from two different quantities a (Fire) and b (Earth); their sum would be the third term, so that we would obtain the continuous geometric proportion par excellence, which carries out the agreement of opposites most perfectly: (a + b)/a = a/b. This equal proportion, applied to linear geometry, corresponds to what Euclid called the Division into Extreme and Mean Ratio (DEMR) or The Golden Proportion. In fact, according to my mathematical interpretation, in the Timaeus 32b and in the Epinomis 991 a–b, Plato uses Pingala’s Mātrāmeru or The First Analogy of the Double to mould the body of the Cosmos as a whole, to the point of identifying the two supreme principles of the Cosmos—the One (1) and the Indefinite Dyad (Φ and1/Φ)—with the DEMR. In effect, Fire and Earth are joined not by a single Mean Ratio but by two (namely, Air and Water). Moreover, using the Platonic approach to analyse the geometric properties of the shape of the Cosmos as a whole, I think that Timaeus constructed the 12 pentagonal faces of Dodecahedron by means of elementary Golden Triangles (a/b = Φ) and the Mātrāmeru sequence. And, this would prove that my mathematical interpretation of the platonic texts is at least plausible.

... of distant mathematics JENS HØYRUP FILOSOFI OG VIDENSKABSTEORI PÅ ... being represented by lines. Another, more complex case is found in the problem TMS XIX #2 [Høyrup 2002a: 195–200]: It deals with a rectangle for which is given, ...

NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin, 2013

Reading this book should start with the Appendix (Anhang, pp. 395-414), containing the Aristotelian and Platonic Greek sources, and their Latin and German translations, the textual foundations upon which Toth erects his entire interpretive échafaudage. Of these nineteen odd pages, the Greek originals occupy a mere seven (pp. 395-401). The attentive, open-minded reader will not discover any traces whatsoever of non-Euclidean geometry in these sources. Nothing at all. And this should not be surprising: Like Marx, who was no Marxist, Euclid, too, was without party affiliation, he was, of course, no follower of Euclid, he was no Euclidean. Greek geometry was just that, geometry, with no additional qualifiers. Hence, even on the basis of this simple, obvious, semantic consideration, there could not have been any Greek non-Euclidean geometry; but, there is more to it. Tóth's creation is, largely, godlike, de nihilo. He brings to his sources, of course, like all of us, his biases, but, unlike most historians worth their salt, he does not leave his prejudices at the door before entering the textual world; on the contrary, he invests his sources with all of his preconceived ideas, reading

Revue de Synthèse, 2003

Enriques, et autres-ont joué un rôle important dans la discussion sur les fondements des mathématiques. Mais, contrairement aux idées d'Euclide, ils n'ont pas identifié « l'espace physique » avec « l'espace de nos sens ». Partant de notre expérience dans l'espace, ils ont cherché à identifier les propriétés les plus importantes de l'espace et les ont posées à la base de la géométrie. C'est sur la connaissance active de l'espace que les axiomes de la géométrie ont été élaborés ; ils ne pouvaient donc pas être a priori comme ils le sont dans la philosophie kantienne. En outre, pendant la dernière décade du siècle, certains mathématiciens italiens-De Paolis, Gino Fano, Pieri, et autres-ont fondé le concept de nombre sur la géométrie, en employant des résultats de la géométrie projective. Ainsi, on fondait l'arithmétique sur la géométrie et non l'inverse, comme David Hilbert a cherché à faire quelques années après, sans succès.

Science in Context, 2001

This paper reviews contemporary interpretations of classical Greek geometry, and offers a Deleuzian, post structural alternative. We emphasize the embodied aspect of Greek mathematical practice, and indicate how it conflicts with ideal linear-modular accounts. Our main theoretic tools are the notions of 'catastrophe' and 'haptic eye' as reconstructed by Deleuze in his analysis of the artistic practices of painter Francis Bacon.