History of Ancient Greek Mathematics

The ‘mistakes’ in the arithmetical problem of SB XXIV 16038 A turn out to be most likely explained not as calculation mistakes but as errors occurred in a copying process, both considered in themselves and in the light of the transmission tendencies that can be detected in the papyrological tradition of this kind of mathematical texts. These errors appear to be strictly analogous or maybe in some way connected with the similar inconsistencies that affect the text of the closest parallel, namely, two loan problems of P.Mich. III 145 fr. 3 col. vii.1–4 (probl. 1) and 5–7 (probl. 2).

Contrary to popular belief, Pythagoras' investment in mathematics remained very small or non-existent. In the same way, the Pythagoreans are far from having a uniform discourse in this area, and even paid no attention to it for the vast majority. While there are indeed Pythagorean mathematical works in the literal sense, other speculations on numbers have led to the construction of philosophical-religious discourses on numbers. This arithmological tendency has long been considered late, but there are a large number of fragments that testify to what these speculations may have been between the fifth and fourth centuries BCE. A reprint of the record shows the Pythagorean attempt to connect abstract concepts with numbers and deities, in order to formulate ethical, moral, or political propositions. Number or mathematical reasoning thus becomes a support for philosophical demonstration, without being a principle.

Johannes Clauberg (1622-1665) was a prominent figure in the field of philosophy both during his lifetime and after his death, on a continuum that extends at least as far back as Christian Wolff. Even before his death, a not inconsiderable number of works on physics, logic and, primarily, metaphysics openly inspired by him began to spread. The man whom historiography today describes as a minor author was a leading figure in his time, who initiated a veritable philosophical current that bears his name. The aim of this work is to show that the most radical Claubergianism is above all that which arose and developed within the academic milieu in Zurich from the 1660s onwards and which belongs thematically to the history of ontology.

We may take tables for granted. However, due to a variety of factors, tables were a rarity in the history of ancient Greek culture, used only limitedly in very special contexts and generally in a non-systematic way, except in astronomy. In this paper I present the main types of tables that can be found in ancient Greek texts: non-ruled columnar lists (accounts and other types of informal tables), ruled columnar lists (mostly astronomical tables), and symmetric tables (mainly Pythagorean displays of numbers).

The little-known Venetian mathematician Matteo Macigni (c. 1510-1582), who graduated in Padua, pursued the academic career teaching in Salerno, Paris and possibly Rome, before coming back to the Venetian Republic. In his lifetime, Macigni never published a single line and his legacy rests on his sizable library, full of Greek and Byzantine mathematical manuscripts, now for the most part preserved in Wolfenbüttel, Germany. The present study starts by offering a biographical sketch of Macigni, based on the reassessment of what has been published so far and on new archival evidence. Macigni’s involvement in the life of the Accademia degli Infiammati and within the circle of Pietro Bembo, Sperone Speroni and Gianvincenzo Pinelli is fully explored. The core of this paper is the edition of three inventories of Macigni’s Greek manuscripts. In several cases, we were able to identify the still existing manuscripts, now scattered all over Europe, virtually reconstructing the now dispersed library. For each item we supply all the essential information, clarifying the textual value of Macigni’s manuscripts and exploring their connections to other extant witnesses of the same work. Such an enterprise allows us to investigate the humanistic reception of several collections of Greek mathematical texts, f rom antiquity to the Palaeologan era, often reshaping their stemmata codicum.

Except for the fraction 2/3, Egyptian scribes used only unit fractions, i.e. fractions with numerators equal to unity. They expressed all other fractions as a sum of these unit fractions. They used tables such as the Rhind table, providing decompositions of numbers written as 2/n with n an odd integer (from three to 101) into a sum of unit fractions, for which the decomposition procedure remains today mysterious. Here, we propose a simple and unique procedure based on elementary integer summations with which to reproduce the entire Rhind table. We add two prioritized selection criteria and discuss some of the Egyptian scribes' possible preferences, finally resulting in the Rhind table decompositions in 87.8% of cases at the first attempt. All the remaining cases can also be calculated using our procedure, with the same criteria and preferences, but differently prioritised. In addition to understanding how ancient Egyptians computed, this simple procedure improves our understanding of perceptions of numbers in ancient Egypt.

The article introduces a new concept of structure, defined, echoing Wheeler's concept of "law without law," as a "structure without law", and a new philosophical viewpoint, that of structural nonrealism, both of which, the article argues, emerged with Heisenberg's discovery of quantum mechanics and Bohr's interpretation of it in terms of complementarity. The article takes advantage of the circumstance that any instance of quantum data or, in presentday terms, quantum information is a "structure"-an organization of elements, ultimately bits, of classical information, manifested in measuring instruments. While, however, this organization can, along with the observed behavior of measuring instruments, be described by means of classical physics, it cannot be predicted by means of classical physics, but only probabilistically or statistically by means of quantum mechanics or quantum field theory (or possibly some alternative theories within their scope). By contrast, the emergence of this information and of this structure cannot, in the present view, be described by either classical or quantum theory, or possibly by any other means, which leads to the concept of "structure without law" and the viewpoint of structural nonrealism.

Books M and N of the Metaphysics, which form a unity, criticize Plato and the Academy and present Aristotle's views regarding the mathematical objects. In the major part of M 1 the structure of books M and N is presented. Then, starting from M 1, 1076a32, up to M 3, Aristotle deals with the existence of the mathematical objects. At the end of M 1, Aristotle mentions four possible cases regarding the existence of mathematical objects. As he mentions, mathematical objects either a) exist in sensible objects, or b) exist as separate in substance from the sensible objects, or c) they do not exist at all, or d) they exist in some other way. In M 2, he presents arguments against the first two of the aforementioned cases. At first, in lines 1076a38-1076b11, he presents two arguments against the first case. Then, starting from line 1076b11, he presents a series of arguments against the second case. After refuting those two cases regarding the existence of mathematical objects, he proceeds, in chapter M 3, to present his own theory which falls under the fourth case, namely that the objects of mathematics exist in some other way 1. In this paper I am going to discuss and offer an interpretation to the Aristotelian argument in lines 1076b36-39 from M 2, which targets the view that the objects of arithmetic exist as separate in substance from the sensible objects. Since this argument functions in the same way as the preceding one, in lines 1076b11-36, I am first going to, briefly, present this argument and then focus on the next argument in lines 1076a36-39. The argument in lines 1076b11-36 is from the case of geometry. There Aristotle attempts to show that the view, according to which geometrical objects exist as separate in substance from sensible objects, leads to some absurd consequences. What he argues is that, if we posit geometrical objects as separate

Aristotle counts as the founder of formal logic. The logic he develops dominated until Frege and others introduced a new logic. This new logic is taken to be more powerful and better capable of capturing inference patterns. The new logic differs from Aristotelian logic in significant respects. It has been argued by Fred Sommers and Hanoch Ben-Yami that the new logic is not well equipped as a logic of natural language, and that a logic closer to Aristotle's is better suited for this task. Each of them developed their own formalism - Sommers in form of term logic, Ben-Yami in form of his Quantified Argument Calculus (QUARC). I discuss Aristotle's logic - a term logic - and attempt a comparison between Aristotelian logic and (i) the new logic, (ii) Sommers' term logic, and (iii) Ben-Yami's QUARC. I consider the differences between the systems, and show how they are related to and diverge from the new logic.

This article reconsiders E. Schrödinger’s cat paradox experiment from a new perspective, grounded in the interpretation of quantum mechanics that belongs to the class of interpretations designated as “reality without realism” (RWR) interpretations. These interpretations assume that the reality ultimately responsible for quantum phenomena is beyond conception, an assumption designated as the Heisenberg postulate. Accordingly, in these interpretations, quantum physics is understood in terms of the relationships between what is thinkable and what is unthinkable, with, physical, classical, and quantum, corresponding to thinkable and unthinkable, respectively. The
role of classical physics becomes unavoidable in quantum physics, the circumstance designated as the Bohr postulate, which restores to classical physics its position as part of fundamental physics, a position commonly reserved for quantum physics and relativity. This view of quantum physics
and relativity is maintained by this article as well but is argued to be sufficient for understanding fundamental physics. Establishing this role of classical physics is a distinctive contribution of the article, which allows it to reconsider Schrödinger’s cat experiment, but has a broader significance
for understanding fundamental physics. RWR interpretations have not been previously applied to the cat experiment, including by N. Bohr, whose interpretation, in its ultimate form (he changed it a few times), was an RWR interpretation. The interpretation adopted in this article follows Bohr’s
interpretation, based on the Heisenberg and Bohr postulates, but it adds the Dirac postulate, stating that the concept of a quantum object only applies at the time of observation and not independently.

Κείμενο διάλεξης που έγινε στην Αίγινα, στις 30/9/2023, στο πλαίσιο μιας εκδήλωσης που διοργάνωσε το πολιτιστικό περιοδικό "Αιγιναία" για  τον βυζαντινό ιατρό Παύλο τον Αιγινήτη. Το θέμα της εκδήλωσης ήταν "Από το Βυζάντιο και τον Ισλαμικό κόσμο στη Δύση: το έργο του γιατρού Παύλου Αιγινήτη".

It is often said that Aristotle takes geometrical objects to be absolutely unmovable and unchangeable. However, Greek geometrical practice does appeal to motion and change, and geometers seem to consider their objects apt to be manipulated. In this paper, I examine if and how Aristotle’s philosophy of geometry can account for the geometers’ practices and way of talking. First, I illustrate three different ways in which Greek geometry appeals to change. Second, I examine Aristotle’s ontology of geometrical objects and argue that although the truth-makers of geometrical statements are in fact unmovable because they are properties of sensible objects, geometers ‘separate them in thought’ and treat them as substances apt to be modified. Finally, I examine whether allowing for the possibility of manipulating these semi-fictional geometrical individuals creates problems for the applicability of geometry. I find that it does not, insofar as one accepts that geometry is not meant to track physical change but merely to study the instantaneous geometrical configuration of sensible bodies, and is thus only applicable at the instant.

On ne peut qu'admirer, ici comme ailleurs, l'impeccable rigueur philosophique d'André Stanguennec. Si le style du professeur de l'université de Nantes, traitant dans ses cours de la question du rationnel et de l'irrationnel, diffère apparemment de celui de l'auteur de La Dialectique réflexive (trois tomes publiés en 2006, 2008 et 2013 aux Presses Universitaires du Septentrion), dans l'un et l'autre cas, le propos est foncièrement du même ordre, c'est-à-dire de part en part métaphysique. Tant mieux ! Comment, en effet, ne pas se réjouir de voir coïncider l'enseignant et le philosophe, les cours et l'oeuvre ? Sous cet angle, la publication de cours faits par des professeurs de philosophie qui sont aussi des philosophes nous rend même, si l'on ose dire, triplement service. D'une part, elle contribue à éclairer les étudiants de la plus belle manière, celle qui, sans concession, consiste à aller aux textes mêmes (ici, ceux de Leibniz, de Kant, de Popper, d'É. Weil, de Cassirer, etc.). D'autre part, elle nous montre que le partage institutionnel des savoirs et des pratiques, tendant aujourd'hui à faire la part belle aux sciences sociales, n'a pas tari les sources de la pensée pensante. Enfin, elle témoigne, toutes proportions gardées, de la force d'une tradition qui, semble-t-il, ne s'éteint pas : celle de ces grands philosophes qui, en France, furent aussi, dans les lycées, à l'université ou au Collège de France, de grands professeurs (Bergson, Brunschvicg, Bachelard, J. Wahl, etc.). Dans ces conditions, ce que l'on pourrait paradoxalement reprocher ici à A. Stanguennec, c'est l'irrépressible élan spéculatif qui le caractérise. Certes, l'A., en pédagogue, prend soin de rappeler, dans son chapitre inaugural, le contexte grec de l'émergence de la notion de nombre « irrationnel » (alogos). Mais très vite-et c'est en cela que consiste l'élan spéculatif que nous évoquons-A. Stanguennec montre que le propre d'une expérience de l'irrationnel est non seulement de surprendre ou d'étonner, mais, plus profondément, et à l'instar de l'épreuve du sublime, de provoquer un saisissement du Sujet qui est aussi un dessaisissement de lui-même. Confronté alors à une extériorité qui le défie ou à une étrangeté qui le constitue, en tout cas à une altérité qu'il ne peut arraisonner, notre pouvoir de connaître (nos sens, notre entendement mais aussi notre imagination) est ainsi sommé, sinon de se faire esprit et d'outrepasser toute borne, du moins de se transmuter en une capacité de penser ses propres limites. Source de différences ou d'altération, l'irrationnel apparaît alors, dans son essence la plus pure, et donc de façon quasi irréfutable, comme l'aiguillon, sinon le moteur, de toute dialectique spirituelle devenant pleinement consciente de ses présuppositions. Ainsi prévenu, le lecteur ne s'étonnera pas d'entrer directement, dès la « leçon 2 », dans les arcanes de l'onto-théologie mais aussi de la théo-ontologie (qui, elle, pense la relation métaphysique de Dieu avec le monde, en partant de Dieu). Il ne s'étonnera pas non plus de constater que le concept de « rationalité métaphysique » (voir la seconde partie de l'ouvrage, intitulée « Les critiques de la rationalité métaphysique ») soit au centre des débats. Car il ne s'agit jamais ici de découvrir, comme dans un dictionnaire, les définitions usuelles des termes « rationnel » et « irrationnel », ni d'ailleurs de passer en revue, comme dans certains manuels, des oppositions stéréotypées (sagesse/savoir ; raison/passion ; postulat/foi ; conscience/inconscient ; nécessité/contingence ; calculabilité/imprévisibilité ; diurne/nocturne ; argumentation/persuasion ; abandon/action ; mérite/grâce ; présymbolique/symbolique ; violence/sens ; normativité/folie ; etc.), mais plutôt de penser l'articulation originaire, toujours et déjà dialectiquement réfléchie, du rationnel et de l'irrationnel. Et sur ce point, force est de reconnaître que le livre tient ses promesses, puisqu'il nous donne à entrevoir, sous un jour nouveau, certaines oppositions conceptuelles que l'on croyait radicales (sans doute sous l'influence de certaines manières de penser à la mode) et qui, en vérité, se révèlent être secondaires, superficielles ou artificielles (voir, par exemple, p. 179 et suivantes), et, qui plus est, sans véritables enjeux théoriques ou pratiques.

Ba~rTz and Thábit b. Qurra translated ioto Arabic {he Arirhmetical lntroduction of Nicomachus, the chapters 00 figurate numbers, based 00 the propenies ilIustrated in rhe table below, were further developed and were introduced in the teaching of marhematics along with other problems of Euclidian number theory. I For the 10 th century, ¡his interest in lhe figurare numbers is confirmed by Ihe mathematical wrilings of al-An!akT 2 and Ibn STna (lbn SIna 1975, 53-61; 1976, 3) and by the encyclopedia of Abii tAbd Alliih al-Khwarizmf (al-KhwiirizmI 1968, 188-91). But lhe cantents of these treatises do nal give us much information 00 early deve[opments in this subject. Also we know that in the central part of the Muslim Empire this tradition was continued for several centuries, as evidenced by the writings of CAbd al-Qahir b. The translation of Habib b. Bahrlz was made (at the end of the 8"' or the begining of the 9'• century) from a Syriae Yersi~n (F. Sezgin 1974, 164-66). For the translation of Thabit b. Qurra, ef. Kutsch 1958. 67-82. 1 Abü'I-Qaslm al-An!akl (ef. F. Sezgin 1974, 30) wrote the KiUib la/srr al-arilhmá!rqr whieh is really a cornmemary on the Arilh!nelical IntroduClion of Nicomachus. Only the third pan has come down to us (Ms. Ankara Saip 5311, ff. 1b-36a).

The study examines the origins and characteristics of a unique Byzantine “epic” in the Chronicle of Morea: a description of the Battle of Pelagonia and the exploits of Sir Geoffroy de Briel. Although the professional literature concurs that the episode is likely to be based on folk poetry and oral tradition, many questions (concerning literary crosstalk, intertextual connections, the author’s intentions, etc.) remain unanswered. The study examines the person of Sir Geoffroy de Briel and the possible intertextual relations of this epic, and analyses the Hungarian role in the Battle of Pelagonia.

Una presentazione dell'opera di Archimede e dell'impatto che ebbe la sua riscoperta sull'origine della matematica moderna. ...

A presentation of Archimedes' work and the impact that its rediscovery had on the origins of modern mathematics.

The common association between the teaching of numbers and basic arithmetical operations, on the one hand, and the earliest levels of linguistic-literary education, on the other, is a datum on which interpreters of Greek papyrological sources are now essentially agreed. A more controversial and less often considered question is whether, in these same ‘scholastic courses’, this basic knowledge also encompassed geometrical and metrological calculations in the form of ‘problems’ (which are typical of the papyrological mathematical tradition). Only a few witnesses to the latter type of texts in Greek have characteristics that would most likely associate them with instructional contexts. The four oldest of these date between the 1st and 3rd centuries AD and are of proven or hypothetical Fayumic provenance. Although they are few in number, these witnesses attest to certain practices in the teaching of mathematics, at least in this historical-geographical context, which are worthy of note.

In his Metrica, Hero provides four procedures for finding the area of a circular segment (with b the base of the segment and h its height): an Ancient method for when the segment is smaller than a semicircle, $$(b + h)/2 \, \cdot \, h$$ ( b + h ) / 2 · h ; a Revision, $$(b + h)/2 \, \cdot \, h + (b/2)^{2} /14$$ ( b + h ) / 2 · h + ( b / 2 ) 2 / 14 ; a quasi-Archimedean method (said to be inspired by the quadrature of the parabola) for cases where b is more than triple h, $${\raise0.5ex\hbox{$\scriptstyle 4$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}}(h \, \cdot \, b/2)$$ 4 / 3 ( h · b / 2 ) ; and a method of Subtraction using the Revised method, for when it is larger than a semicircle. He gives superficial arguments that the Ancient method presumes $$\pi = 3$$ π = 3 and the Revision, $$\pi = {\raise0.5ex\hbox{$\scriptstyle {22}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 7$}}$$ π = 22 / 7 . We are left with many questions. How ancient is the Ancient...

Geschichte der Arithmetik 1 1.1 Über die Anfänge des Rechnens Für die meisten Menschen sind Zahlen die erste Assoziation zum Thema Mathematik. Tatsächlich kann man sagen, dass die ersten mathematischen "Tätigkeiten" in der Geschichte der Menschheit mit dem Zählen zu tun haben. Die ältesten "mathematischen" Artefakte, Beweise menschlicher Zähltätigkeit, sind Knochen und Steine mit Einritzungen und Kerben, die 30.000 oder mehr Jahre alt sind, also aus der Steinzeit stammen. Die zwei berühmtesten dieser Artefakte sind zwei Pavianknochen mit Einkerbungen: Der Lebombo-Knochen, über 43.000 Jahre alt und in den Lebombobergen im südlichen Afrika gefunden, sowie der Ishango-Knochen, ein etwa 20.000 Jahre alter Knochen der in Zentralafrika gefunden wurde. Der Knochen von Ishango ist besonders interessant, da seine Einkerbungen deutlich in drei Spalten gegliedert sind und innerhalb der Spalten gruppiert sind. Deswegen hat der Knochen viele Vermutungen und Theorien über sein Zweck ausgelöst [16, 29]. Knochen, Steine und Hölzer, die durch Einritzungen zum Zweck des Zählens verwendet wurden, sind allgemein als Kerbhölzer bekannt. Die Verwendung von Kerbhölzern erstreckt sich bis in die Neuzeit, in ländlichen Gegenden sogar bis heute. Interessant ist auch die Tatsache, dass die englische Institution The English Exchequer vom 12. Jh. bis zum Jahre 1826 zweiteilige Kerbhölzer bei Steuererhebungen und Darlehen verwendete [17]. Genauso alte, oder vielleicht sogar von Kerbhölzern ältere, Zählhilfsmittel sind Jetons (kleine Steine, Muscheln, ...). Jetons sind nicht für große Zahlen geeignet, aber es wird angenommen, dass gerade die Verwendung von Jetons zur Entwicklung des Rechnens führte, z. B. zur Subtraktion durch Wegnehmen von Jetons. Nach Entdeckung der Tonverarbeitung konnte man durch Jetons verschiedener Größen und Formen auch große Zahlen darstellen; viele solcher Tonjetons sind im Nahen Osten, aber auch andernorts gefunden [17].

Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This sourcebook presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time.
Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos’s elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon’s use of mathematical induction in combinatorial proofs; Al-Mu’taman Ibn Hūd’s extensive survey of mathematics, which included proofs of Heron’s Theorem and Ceva’s Theorem; and Muhyī al-Dīn al-Maghribī’s interesting proof of Euclid’s parallel postulate. The book includes a general introduction, section introductions, footnotes, and references.
The Sourcebook in the Mathematics of Medieval Europe and North Africa will be indispensable to anyone seeking out the important historical sources of premodern mathematics.

On January 3, 2001, Isabella Grigoryevna Bashmakova, a renowned authority on history of mathematics, celebrated her eightieth birthday. Bashmakova was born in Rostov-on-Don, to Grigory Georgevich Bashmakov and Anna Ivanovna (maiden name Aladzhalova). Her father was a lawyer, known for his oratory, a graduate of the Moscow M. V. Lomonosov University and a pupil of the distinguished jurist and philosopher Pavel I. Novgorodtsev (1868-1924). He was a remarkable personality endowed with many talents. He loved philosophy and history. In 1932, the Bashmakov family moved to Moscow, where Isabella finished secondary school. The family established lively links with Moscow cultural life and the atmosphere in their home was very intellectual. The circle of family friends included eminent scientists, such as the physiologist and academician (since 1968) Mikhail Kh. Chailakhyan, as well as men of arts, such as the famous poets Boris L. Pasternak (1890-1960) and Samuel Ya. Marshak (1887-1964). Very soon, she showed a turn for mathematics. Thus, she enrolled in the Faculty of Mathematics and Mechanics of the Moscow M. V. Lomonosov State University in 1938. All her subsequent life has been closely associated with the activities of this faculty. During World War II, she evacuated Moscow, together with the personnel from the University, to Samarqand, where she served as a nurse. Bashmakova's intellectual interests in the history of mathematics were formed within the wider environment of the Moscow Mathematical School of the first half of the 20th century and particularly the Research Seminar on History of Mathematics, initiated 1 Editorial note. A retrospective of the life and work of Isabella Bashmakova on the occasion of her sixtieth birthday was published in 1981 in Historia Mathematica 8, 389-392. The present article provides an updated and more detailed account, supplemented with a complete bibliography of her extensive writings in the history of mathematics.

In this book Diana Quarantotto carries out an analysis of Aristotle's conception of place. The title of the text, L'universo senza spazio. Aristotele e la teoria del luogo (The universe without space: Aristotle and the theory of place) briefly encapsulates how Aristotle conceives of things' location: space, as a three-dimensional extension existing independently of the things that occupy it, is not Aristotle's considered understanding of location; rather for Aristotle things are in a place, for it is true of them to say that they are somewhere. Conversely, every portion of space turns out to be occupied by a body-if only by air-so that Aristotle's universe is a plenum, that is, a universe where no space is empty. In such a universe, explains Quarantotto, to hypothesise the existence of space would be to multiply entities unnecessarily (p. 38). Since antiquity Aristotle's theory of place has never failed to provoke controversy, and in fact has been rarely accepted as it stands. The theory has been deemed irremediably unsatisfying, 'open to serious objections' (Ross),1 'inadequate' (Hussey),2 vel sim. In the recent debate, however, interpreters have started to reassess Aristotle's theory and to pay due respect to its own philosophical merits. Quarantotto belongs to this camp in so far as she attempts to bring out the virtues of Aristotle's theory; but she also highlights the reasons why such a theory may well fail to command agreement to contemporary readers. Quarantotto's book consists of a close analysis of Aristotle's Physics IV.1-5.3 Accordingly, the volume breaks down into 5 chapters, each chapter being devoted to its correspondent chapter in Aristotle's Physics. The reader will find a glossary at the end, with key terms, as well as an index locorum. An introduction is devoted to frame Aristotle's investigation within the larger project of his Physics. This makes for a succinct but highly illuminating lead-in to Aristotle's treatise as

In his Commentary on Book I of Euclid's Elements, Proclus Lycius, a famous Neoplatonic philosopher of the fifth century CE, proposed a scheme for analyzing complete mathematical propositions into six parts: proposition, enunciation, setting out, determination, construction, proof and conclusion. Leaving content aside, most of these parts can be identified based on specific formulaic expressions. Still, neither all parts are always identifiable by such expressions, nor do they always appear in a strictly sequential order with no intermingling with each other. This paper focuses on developing a tool which will allow scholars to semantically annotate ancient Greek mathematical texts according to Proclus' scheme. To this end, we apply machine learning algorithms to the first seven books of Euclid's Elements.

Pour fonder et introduire la theorie exposee dans son traite d'algebre, al-Khayyam evoque longuement au debut de son texte les notions de dimension et de grandeur mesurable. II ecrit notamment: Ce qui fait partie des grandeurs, c'est d'abord une seule dimension, c'est-adire la racine, ou, rapporte a son carre, le cote. Puis les deux dimensions, c'est-a-dire la surface; le carre dans les grandeurs est done la surface carree. Enfin, les trois dimensions, c'est-a-dire le corps; le cube dans les grandeurs est le solide limite par six carres. Or, comme il n'existe aucune autre dimension, ne font partie des grandeurs ni le carre-carre, ni, a plus forte raison, ce qui lui est superieur. Et si on parle de carre-carre dans les grandeurs, on le dit pour le nombre de leurs parties, quand on les mesure, mais pas pour elles-memes en tant que grandeurs mesurables, ce qui est different. 1

L'utilisation de plus en plus fréquente des transformations-homothéties, affinités orthogonales ou similitudes-constitue sans aucun doute l'un des faits les plus marquants de l'histoire de la géométrie entre les IX e et XI e siècles. Parce qu'une telle utilisation s'est étendue à des chapitres de plus en plus nombreux, parce qu'elle incitait à s'intéresser aux relations entre les figures plus qu'à les considérer isolément, parce qu'elle interrogeait dès lors la légitimité du recours au mouvement et, par là, faisait surgir la question des fondements, elle a pu conduire en effet à infléchir la nature même de l'objet de la discipline. Roshdi Rashed a brossé ailleurs les grandes lignes de cette évolution qui, partant de l'oeuvre des Banū Mūsā en passant par celles de Thābit ibn Qurra, de son petit-fils Ibrāhīm ibn Sinān et de bien d'autres, aboutit notamment à la remise en ordre d'Ibn al-Haytham dans ses traités sur les Connus et sur le Lieu 1. De cette histoire, le moment qui correspond à la contribution d'al-Siğzī, à la fin du X e siècle, est l'un des plus importants. On ne trouve certes rien dans cette oeuvre qui invite à repenser les fondements de la géométrie : contrairement à son successeur Ibn al-Haytham, al-Siğzī ne cherche pas, du moins explicitement, à dépasser le cadre légué par Euclide. Pourtant, deux points particulièrement remarquables suffisent à distinguer les écrits d'al-Siğzī de ceux de ses prédécesseurs : en premier lieu, le fait que, pour la première fois, il introduise un terme générique pour désigner les transformations et la méthode qui leur est attachée ; et, en second lieu, la recherche auquel il se livre explicitement pour dégager des invariants géométriques en faisant varier certains éléments d'une figure. C'est à expliciter le premier point et à mettre en évidence le second que nous voudrions nous consacrer ici. I-LES TRANSFORMATIONS COMME MÉTHODE GÉOMÉTRIQUE C'est d'une façon pour le moins succincte qu'al-Siğzī introduit le mot transformation (naql). Dans son traité intitulé Pour aplanir les voies en vue de déterminer les propositions géométriques 2 , où il entreprend d'une façon générale de présenter l'ensemble des méthodes dont dispose le géomètre, il écrit en effet simplement :

La forme et l'objet du De Possest placent cet opuscule dans la ligne d'un platonisme christianisé ou christianisant-illustrant, à sa façon, le mot de Pascal, « Platon pour disposer au christianisme » (Pascal, liasse XXXIV, Sellier 505). On peut même préciser, pour situer plus exactement le dialogue dans ce contexte de renaissance platonicienne, que sa rédaction (vers la fin de l'année 1459), coïncide très exactement avec le moment où George de Trébizonde achève sa traduction du Parménide de Platon, traduction dont Nicolas de Cues lui avait passé commande. Il s'agit de la première traduction latine de l'intégralité du dialogue-car jusqu'alors, le Parménide était plus et mieux connu à travers le commentaire de Proclus, lui-même traduit par Guillaume de Moerbeke, et sur lequel Nicolas de Cues avait activement travaillé une quinzaine d'années auparavant.

I argue that intelligible matter, for Aristotle, is what makes mathematical objects quantities and divisible in their characteristic way. On this view, the intelligible matter of a magnitude is a sensible object just insofar as it has dimensional continuity, while that of a number is a plurality just insofar as it consists of indivisibles that measure it. This interpretation takes seriously Aristotle’s claim that intelligible matter is the matter of mathematicals generally—not just of geometricals. I also show that intelligible matter has the same meaning in all three passages where it is explicitly invoked: Z.10, Z.11, and H.6. Since the H.6 passage involves a mathematical definition, this requires determining what the mathematician defines and how she defines it. I show that, as with natural scientific definitions, there must be a matterlike element in mathematical definitions. This element it is not identical with, but rather refers to, intelligible matter.

Le théorème de Thalès pour les cercles et sa réciproque (plus connue sous le nom de théorème du cercle circonscrit à un triangle rectangle) sont des propriétés de géométrie élémentaire enseignées dans les collèges français. Nous nous intéressons non seulement à différentes descriptions possibles de ces propriétés en Coq mais aussi aux preuves correspondantes. Nous étudions notamment comment le choix d’une représentation des objets géométriques contenus dans l’énoncé influencent les arguments de preuve. Nous présentons plusieurs approches, une basée sur des calculs d’angles, une autre basée sur la géométrie analytique et finalement deux démonstrations différentes dans l’axiomatique de Tarski en utilisant dans un cas le théorème de la droite des milieux et dans l’autre la notion de symétrie centrale et ses propriétés.

According to Aristotle, those who seek mathematical principles of sensible things are looking in entirely the wrong place. But despite his strong opposition to mathematized metaphysics, Aristotle does not outright reject mathematical explanation of the natural world. In fact, he argues that mathematics does explain certain sensible phenomena, that the natural world has many mathematical patterns and features, and that this is often not mere coincidence. That he devotes two books of his Metaphysics to shoring up the boundary between mathematics and metaphysics indicates that this issue is crucially important to him. Yet it receives little systematic scholarly discussion. I take up the issue by examining Aristotle's treatment of the sensible phenomena that have mathematical explanations. This examination shows that-and why-on his view, mathematics explains many things about the natural world, but not the most fundamental things.

Die Ionier. Die Geburt der Philosophie - Lecture held at the Volkshochschule Wien West, Vienna within the Programme UMP-University Meets Public, 3rd December 2009.Thales, Anaximander und Anaximenes sind der Anfang der altgriechischen Philosophie. Der Vortrag will sich mit deren Positionen auseinander setzten und erklären, dass die Positionen dieser drei Denker wahrscheinlich als gemeinsamen Punkt haben, dass sie eine universelle Kraft als Fundament der Wirklichkeit haben individuieren wollen.

The argument of this chapter is governed by the double meaning of its title:
(a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum
mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not
disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over C and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity,
and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck,
following B. Riemann’s comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships
between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.

The argument of this article is grounded in the irreducible interference of observational instruments in our interactions with nature in quantum physics
and, thus, in the constitution of quantum phenomena versus classical physics, where this interference can, in principle, be disregarded. The irreducible character of this interference was seen by N. Bohr as the principal distinction between classical and quantum physics and grounded his interpretation of quantum phenomena and quantum theory. Bohr saw
complementarity as a generalization of the classical ideal of causality, which defined classical physics and relativity. While intimated by Bohr, the relationships among observational technology, complementarity, causality and the arrow of events (a new concept that replaces the arrow of time commonly used in this context) were not addressed by him either. The article introduces another new concept, that of quantum causality, as a form of probabilistic causality. The argument of the article is based on a particular
interpretation of quantum phenomena and quantum theory, defined by the concept of ‘reality without realism (RWR)’. This interpretation follows Bohr’s
interpretation but contains certain additional features, in particular the Dirac postulate. The article also considers quantum-like (Q-L) theories (based
in the mathematics of QM) from the perspective it develops.

Cet article décrit des stratégies de poursuite pure sur le plan, y compris les théorèmes récemment prouvés sur l'optimalité de la stratégie récursive de la poursuite pure (1979) et la E-stratégie de la poursuite pure (1987). Les propositions nécessaires pour prouver les théorèmes ci-dessus sont présentées à partir de la source la plus prestigieuse-les «Eléments » d'Euclide, écrits vers 300 av. JC.

A Thesis Submitted to The Faculty of Thomas Aquinas College In Partial Fulfillment of the Requirement For the Degree of Bachelor of Arts

This thesis investigates and offers a critique of Descartes' view of "mechanical curves," as enunciated in his The Geometry. It was written and defended in 2023, under the advisory of Dr. Paul Shields, at Thomas Aquinas College, in Northfield MA.
Any questions, critiques, or commentary are very welcome!

Guardare il mare. Sul Talete di Florenskij antinomie.it/index.php/2022/03/22/guardare-il-mare-sul-talete-di-florenskij/ Su certe antiche anfore minoiche abita una piovra immensa quanto l'Egeo, molle come l'argilla da cui è sorta. Lungo i flutti dei suoi tentacoli ondulati pullula la vita animale, quella vorticante massa polimorfa in cui il pesce diventa riccio, il riccio cavallo, il cavallo uccello: nel suo grembo salmastro deflagra un Cambriano delle immagini.

This article considers a rarely discussed aspect, the no-cloning principle or postulate, recast as the uniqueness postulate, of the mathematical modeling known as quantum-like, Q-L, modeling (vs. classical-like, C-L, modeling, based in the mathematics adopted from classical physics) and the
corresponding Q-L theories beyond physics. The principle is a transfer of the no-cloning principle (arising from the no-cloning theorem) in quantum mechanics (QM) to Q-L theories. My interest in this principle, to be related to several other key features of QM and Q-L theories, such as the irreducible
role of observation, complementarity, and probabilistic causality, is connected to a more general question: What are the ontological and epistemological reasons for using Q-L models vs. C-L ones? I shall argue that adopting the uniqueness postulate is justified in Q-L theories and adds an important new motivation for doing so and a new venue for considering this question. In order to properly ground this argument, the article also offers a discussion along similar lines of QM, providing a new angle on Bohr’s concept of complementarity via the uniqueness postulate.

Aristotle says that ὑπαρχειν has as many senses as ‘to be true’ (PrA. , A, 36, 48b2-9) and as many ways as there are different categories. (PrA., A, 37, 49a6-9) This may mean that for every ‘is’ there is a ὑπαρχειν. Τhe reason is that Aristotle uses ὑπαρχειν in converse direction of ‘is’. The equal statement of ‘A is B’ with ὑπαρχειν is ‘B ὑπαρχει to A.’ Allen Bāck  points to the difference between the use of the verb with dative case and its use with a subject alone in Greek language. When it is used with the dative, it retains its basic meaning, that is, ‘be already present’ or ‘exist really.’ ‘So to say that P belongs to S is to say that P exists in S, or, if you like, that P has its being in S.’ He believes that Aristotle uses this construction to insist that primary substances alone are the fundamental being and all other things only are ‘in’ substances. Thus, when the verb is used with the dative, it expresses dependent substance relation. But when it is used with a subject alone, e.g. in ‘S ὑπάρχει,’ it expresses that the subject really exists. (OI., 17a24 and 17b2) Thus, while it is the subject which is said to be the predicate, it is the predicate which ὑπάρχει to the subject. The formula of ‘τὸ Α ὑπάρχει τῷ Β’ is equivalent to ‘τὸ Β ἐστι Α.’  He uses the word for the predication of a category on substance (e.g. Met., α, 993b24-25; Met., Z, 1029a15-16; Cat., 5, 3b24-25), the predication of secondary substance on primary substance (e.g. Met., Z, 1038b21-23; Cat., 5, 2a14-19), the predication of primary substance on nothing else but itself (e.g. Met., Z, 1040b23-24), the belonging of a verb to the subject, the belonging of axioms to a single science (Met., 1005a22-23) , the belonging of PNC to all things that are (Met., IV, 3)  , in the sense of really existing (Cael. 297b22, Met., 1041b4. Cf. B503, 125-126) or merely in the sense of the predication of a predicate on a subject (e.g. OI., I, 5, 17a22-24) and especially and much more repeatedly than anywhere else in his discussion of syllogism. (e.g. PrA., A, 36, 48a40-b2; PrA., B, 22, 67b28-30; PrA., A, 24a26-28) It is also used in some related senses like mere belonging (e.g. Met., A, 989a10-14; K, 1060a9-10) or to be there. (So., B, 5, 417b23-26)