If one wishes to understand medieval philosophical descriptions of the nature of the various sciences, a familiarity with the scientific knowledge possessed at that time is necessary. An excellent example of the basic characteristics of medieval mathematics can be found in this critical edition of De numeris datis. The work itself is the first advanced algebra text produced in Western Europe. It is a study of the general methods of solving problems for unknown numbers if one knows, is given, other (...) numbers and the relationships between them. Although the mathematical ideas Jordanus presents will be familiar to readers of this book, their development differs from that of contemporary algebra. The differences illuminate the medieval approach to mathematics. Medieval mathematicians approached this sort of problem by starting with the known numbers and working through to the unknown rather than concentrating on manipulation of the unknown variables as is done in modern approaches to mathematics. Medieval mathematics also viewed numbers as semi-geometric things. Solutions to algebraic problems were often seen in terms of geometric approaches to the areas and sides of rectangles. In this work Jordanus follows a more "algebraic" approach, but this approach did not replace the dominant medieval geometric approach to mathematics. In fact, as Hughes notes, the marginal notations in later copies of the work are usually geometric diagrams of the concepts. One can read such statements about medieval mathematics easily enough, but if one reads medieval mathematical thought in its original form the sort of thinking involved in medieval mathematics is much clearer. Such a background is useful in interpreting the descriptions writers like Aquinas offer of the nature of mathematics. (shrink)

This article proposes a reconstruction of the Sextan critique against the Pythagorean conception of number as principle of all existing things. Taking into consideration all the parallel passages concerning the presentation and the polemic about the Pythagorean doctrine, the present article proposes a division of the sceptical critique into two sections, the first one presenting the Pythagorean reasons to consider the numbers as principle existing apart from numbered things, and the second one presenting the arguments that Sextus himself built against (...) them. This paper will show that this structure is consistent with the sceptical method, which consists in pointing out the ἰσοσθένεια among different positions which leads to suspension of judgment. Regarding the Pythagorean doctrine that emerges from these passages, the present article discusses the way in which Sextus testified a juxtaposition between Pythagorean and Platonic doctrines, probably due to a Middle-Platonic or Neopythagorean elaboration, against which Sextus used Platonic concepts, adopting in original way arguments taken from the Parmenides and the Phaedo. Despite the ἐποχή about the nature of the number, this article will argue that Sextus did not avoid to use the number in practical daily life, in a way that is consistent with the sceptical practical criterion. (shrink)

This short introduction to the philosophy of mathematics focuses on the main ontological and epistemological concerns raised by any realist position on the existence of abstract objects. The first half of the book outlines Plato's and Kant's views on mathematics. The second half discusses the ontological and epistemological problems discussed in Paul Benacerraf's two seminal articles "What Numbers Could not Be" (1965) and "Mathematical Truth" (1973).

The hermeneutical process underlying Paul’s exegesis of Exodus 17:6 and Numbers 20:7–11 in 1 Corinthians 10:1–4. In this article, Paul’s use of the Old Testament in 1 Corinthians 10:1–4 comes under scrutiny. In contrast with the theory of some modern scholars that Paul uses, ‘fanciful analogies’, ‘startling figurative claims’ and metaphors that ‘should not he pressed’, in reaching his conclusion that ‘the rock was Christ’, in 1 Corinthians 10:4c, it is indicated that Paul is indeed taking the original text, the (...) Old Testament’s interpretation of the text, and the Jewish tradition of the interpretation of the text, seriously, in the light of the Christ-event. To prove this claim, research of the text (Exodus 17:6 & Numbers 20:7–11), that Paul uses in 1 Corinthians 10:1–4, is followed by research of the ‘world in front of that text’ (Deuteronomy 32, the Psalms and Second Temple Judaism). Contribution : The conclusion that is reached indicates that Paul established within the context of contemporary Jewish practices, a true dialogical relationship between an intertextual handling of the text, and his interpretation thereof in the light of the relevance of the Christ-event for the conflict in the Church of Corinth. (shrink)

The Treatise On Numbers develops a typical Plotinian inquiry about oneness and intelligible plurality, in which the analysis is based on the problem of the infinite inside the intelligible realm, as it seems to be presented in the socalled generation of numbers in Plato's Parmenides, where numbers are linked to Being, identified by Plotinus with the hypostasis of Intellect, and produce a process of unlimited division. In order to avoid this interpretation, Plotinus finds in other Plato's dialogues the proof that (...) number is an eidetic nature, encompassing all the intelligible entities, so that the infinite has to be interpreted as what is produced by calculus. Other arguments are rather depending on the model of the Old Academy, concerning the relationship between forms and numbers: Plotinus wonders if numbers come ontologically before forms and indeed he states right this position. Other aspects of Plotinus' investigation, which can be traced back to the Old Academy, are the distinction between two kinds of numbers and the doctrine of the unified number. This model of number, quite ambiguous, may be identified with the indefinite dyad, even though the former has to be conceived, differently from that of Old Academy, within a mono-polar framework, according to which the sole principle is the One. (shrink)

Presentamos la traducción con comentario de un fragmento en lengua armenia, publicado por primera y única vez, con traducción al inglés, por Abraham Terian, en 1984, sin paralelo en el corpus griego de Filón. Describimos el largo camino que recorrió el texto desde su traducción desde la lengua original al armenio en el siglo VI hasta llegar a nuestro conocimiento y hacemos una breve referencia a la importancia de la concepción filónica sobre la década propia de la aritmología pitagórica. Concluimos (...) con las razones de la adscripción de este fragmento al tratado perdido _De numeris_. (shrink)

Presentamos la traducción con comentario de un fragmento en lengua armenia, publicado por primera y única vez, con traducción al inglés, por Abraham Terian, en 1984, sin paralelo en el corpus griego de Filón. Describimos el largo camino que recorrió el texto desde su traducción desde la lengua original al armenio en el siglo VI hasta llegar a nuestro conocimiento y hacemos una breve referencia a la importancia de la concepción filónica sobre la década propia de la aritmología pitagórica. Concluimos (...) con las razones de la adscripción de este fragmento al tratado perdido _De numeris_. (shrink)

Franco Eugeni remembers Bruno Rizzi: in this brief introduction, I would like to remember an afternoon spent in “ Roma Tre ” with Bruno, since we were both Ordinary Professors at that University. We passed it doing a dense program of work for the next three years. At 6.00 pm, I left for “Roseto degli Abruzzi”. At six o'clock a.m. of the next morning, I still have the voice in my ears. A phone call from the Headmaster Ciro d'Aniello, who (...) told me " The Professor is dead" In that afternoon, Emilio Ambrisi and I went to Rome; we were stunned and desperate. Not many days later, a conference was held at the University of Teramo; it was the conference that we had prepared for Bruno, to celebrate his 60th birthday, but it was his memory. In the opening conference, Prof. Antonino Giambo, also fraternal friend of Bruno, burst into a tearful cry, which expressed in a moment, the senses of friendship and love that we all had for the missing friend! Starting from this work with Fabrizio Maturo, the study of some problems left open by the works of Eugeni and Rizzi will be investigated. Sunto Ricordo di Franco Eugeni su Bruno Rizzi: in questa brevissima introduzione vorrei ricordare un pomeriggio passato a Roma Tre con Bruno, visto che allora eravamo entrambi Professori Ordinari in quella Università. Lo passammo a fare un denso programma di lavoro per i successivi tre anni. Alle 18.00 ripartii per Roseto degli Abruzzi. Alle sei del mattino successivo, ho ancora la voce nelle orecchie, una telefonata del Preside Ciro d’Aniello, che mi diceva “ Il Professore è morto” Nel pomeriggio Emilio Ambrisi ed io ci recammo a Roma, eravamo attoniti e disperati. Non molti giorni dopo si tenne presso l’Università di Teramo un Convegno, era il Convegno che avevamo preparato per Bruno, per festeggiare i suoi 60 anni, fu invece il ricordo. Nella conferenza di apertura il prof. Antonino Giambò, anche lui, come noi, fraterno amico di Bruno, scoppiò in un pianto dirotto, che espresse in un attimo, i sensi dell’amicizia e dell’amore, che tutti noi avevamo per l’amico scomparso! A partire da questo lavoro con Fabrizio Maturo verrà approfondito lo studio di alcune problematiche rimaste aperte dai lavori di Eugeni e Rizzi. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A Humanist History of Mathematics?Regiomontanus's Padua Oration in ContextJames Steven ByrneIn the spring of 1464, the German astronomer, astrologer, and mathematician Johannes Müller (1436–76), known as Regiomontanus (a Latinization of the name of his hometown, Königsberg in Franconia), offered a course of lectures on the Arabic astronomer al-Farghani at the University of Padua. The only one of these to survive is his inaugural oration on the history and utility (...) of the mathematical arts.1 Regiomontanus tells his audience that the purpose of the oration is torelate first the origin of our arts, and among which nations they first began to be cultivated, in what way they were at last translated from various foreign tongues into Latin, which of our ancestors were famed in these disciplines, and to whom in our lifetimes recognition should be granted.2To this end, he offers a history of the quadrivial arts (arithmetic, geometry, music, and astronomy) and other important mathematical disciplines from [End Page 41] antiquity to his own time, praises their utility, and exhorts his audience to revive the languishing study of mathematics at Padua. Astrology, a discipline with which Regiomontanus himself was closely associated, is singled out for particular praise.Traditionally, Regiomontanus's Padua oration has been seen through the lens of its rather obvious humanism. In particular, the Padua oration has come to be understood as the rhetorical embodiment of the fifteenth-and-sixteenth-century revival of ancient (Greek) mathematics. Just as this revival is inextricable from the rise of humanism, so Regiomontanus has come to be seen as an exemplar of humanist mathematics.3 It is the aim of this paper to examine the Padua oration in the context both of contemporary humanist rhetoric and of Regiomontanus's own intellectual background in order to argue that, while the oration is stylistically consistent with humanist norms, the vision of mathematics presented in it is also deeply grounded in the university mathematical curriculum and in Regiomontanus's own reading of mathematical texts.Regiomontanus was educated primarily at the University of Vienna (he was also briefly at the University of Leipzig), where he enrolled in 1450, completed his baccalaureate in 1452 and became a master in 1457.4 He remained at Vienna until 1461, when the death of his friend and teacher Georg Peurbach prompted him to travel to Italy with his patron, Cardinal Bessarion.5 In Regiomontanus's day Vienna was probably the most important of the German universities, rivaled only by Prague, whose prestige had declined after it was stripped of its theology faculty in the aftermath of the Hussite Wars.6 The curriculum at the University of Vienna was modeled [End Page 42] after that of Paris, and Parisian scholars played a major role both in its founding in 1365 and in its re-establishment (this time with a theology faculty) in 1384.7Most importantly for the purposes of this paper, the Viennese mathematical curriculum of Regiomontanus's day included all of the traditional authorities taught at Paris in the fourteenth century. For arithmetic and algebra, various "algorisms" (prose or poetry instructions for carrying out arithmetical operations that often also included a small amount of number theory) were the basic texts, supplemented in the fourteenth century by the Quadripartitum numerorum of Jean de Murs. Jordanus de Nemorare's De numeris datis and al-Khwarizmi's Algebra were common sources for those engaged in more advanced studies (i.e., they were not normally the subject of ordinary lectures, but were readily available to interested students, and would perhaps have been the subject of occasional extraordinary lectures). Euclid's Elements, supplemented by the commentaries of Pappus and Campanus, was the central text for geometry, and a number of medieval texts on practical and speculative geometry were in circulation as well. For astronomy, the Sphere of Sacrobosco and the Theorica planetarum were the most commonly used teaching texts, sometimes supplemented by al-Farghani's Elements of Astronomy (the subject of Regiomontanus's Padua lectures). Advanced students could read numerous more specific treatises by Arabic and Latin authors. For optics, the Perspectiva communis of John Peckham was the most common basic text, with texts... (shrink)

I criteri neurologici per accertare il decesso, da impiegare in alternativa a quelli cardiorespiratori se il paziente ha subìto lesioni cerebrali e si trova collegato alle apparecchiature per la ventilazione artificiale, sono entrati nell’uso comune della pratica medica occidentale da circa quarant’anni ed il consenso di cui essi godono nella comunità scientifica sembra, a prima vista, essere ancora oggi molto solido. Si diceva a prima vista, perché se si esamina con attenzione la letteratura dal 1992 ad oggi, si possono scoprire (...) i crescenti numeri del dissenso: ripensamenti e critiche provengono soprattutto da esperti di area medico-biologica, che associano le loro voci a quelle di filosofi, teologi, studiosi di etica. I motivi del dissenso sono molteplici, e l’ampia letteratura che li documenta riguarda la teoria scientifica posta a fondamento dei criteri neurologici, ed evidenzia le contraddizioni, teoriche ed applicative, derivanti dall’uso di quei criteri. (shrink)

Gell. NA 3.10.13 citing Varro's Hebdomades uel de imaginibus, reports: Venas etiam in hominibus, uel potius arterias, medicos musicos dicere ait numero moueri septenario, quod ipsi appellant τν δι τεσσρων συμφωναν, quae fit in collatione quaternarii et ternarii numeri. He also states that doctors who make use of music theory declare that the veins, or rather arteries, in human beings move in accordance with the number seven; they call this motion ‘the consonance of the fourth’, which is produced by (...) the proportional relation of the numbers four and three. (shrink)

Fabio Acerbi Diofanto, “De polygonis numeris”. Introduzione, testo critico, traduzione italiana e commento di F. Acerbi», Pisa, Roma, Fabrizio Serra Editore, 2011. 252 p. Dany Amiot - [avec Dejan Stosic], « Sautiller, voleter, dansoter : évaluation, pluriactionnalité, aspect », Temps, aspect et classes de mots : études théoriques et didactiques. Actes du septième colloque international de linguistique française et roumaine, E. Arjoca-Ieremia, C. Avezard-Roger, J. Goes, E. Moline & A. Tih..

Nel sesto libro del De musica Agostino pone il problema della natura del conoscere a partire da una descrizione, che si potrebbe definire fenomenologica, dei movimenti ritmici di cui il soggetto fa esperienza. Inizia così un percorso conoscitivo, che prendendo le mosse dalle tracce dei numeri arriva a indagare le condizioni di possibilità della conoscenza, rilanciando continuamente l’indagine su molteplici livelli. Con questo articolo si intende comprendere le più rilevanti implicazioni del paradigma conoscitivo agostiniano, il quale non appare riducibile (...) a una mera logica numerica ma permette di gettare uno sguardo filosofico fino alla realtà metafisica della creatura. In the De musica’s sixth Book Augustin puts the question of the knowledge’s nature from the description of the rhytmical movements experienced by the subject. This is the beginning of a cognitive process that starts from the numers’ traces and ends with the inquiry about the conditions of knowledge’s possibility, giving impulse to a research on several levels. The article’s purpose is to understand the most important implications of the Augustinian cognitive paradigm, that doesn’t seem to be a mere numerical logic but allow us to take a philosophical glance to the creature’s metaphysical truth. (shrink)

Excerpt from Caroli Lachmanni in T. Lucretii Cari De Rerum Natura Libros Commentarius Iterum Editus Itaque recensendi mnnere ita functus sum nt quod Gumque bonum et verum esset'aut in utroque aut in alter utro codice id sine ullo dubitationis indicio exhiberem ver suum autem erdinem eum quo essent a veteribus librariis scripti, numeris appositis indicarem: emendafionem his re bus contineri arbitratus sum primum ut versus a. Librariis traiecti in suum locum reducerentur; quod ubi evenit nu meri sese naturali ordine excipere (...) non petuerunt: deinde nt ipsa versuum vocabula ubi vel perissent vel depravata es sent, quantum fieri posset, reciperarentur, ita tamen nt sin gula ad codicum fidemscripta cum singnlis commentariernm particulis praefigerentur, tum in imis paginis carminum enn merarentur. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Growth in popularity of computer games is a noticeable change in recent years. Electronic entertainment increasingly engages the wider society and reaches to new audiences by offering them satisfy of wide variety of needs and aspirations. As a mass media games not only provide entertainment, but they are also an important source of income, knowledge and social problems. Article aims to bring closer look on the common areas of games and comics. On the one hand designers and artists working on (...) games are often inspired by comic books, as well as they create their licensed adaptations and separate „interactive issues”. On the other hand more and more often we can see comics based on popular games. Study present the areas of agreement, cooperation or dependence like: technologies used to create games and comic books, use of comic books to comment events in the gaming industry and organization of exhibitions or events popularizing the works from both fields. ** W ostatnich latach dostrzegalny jest wzrost popularności gier komputerowych. Elektroniczna rozrywka angażuje uwagȩ coraz to szerszych krȩgów społecznych i dociera do nowych grup odbiorców oferuj¸a}c im zaspokojenie najrozmaitszych potrzeb i aspiracji. Jako środek masowego przekazu gry dostarczaj¸a} nie tylko rozrywki, ale stanowi¸a} też istotne źródło dochodów, wiedzy i problemów społecznych. Celem artykułu jest przybliżenie wspólnych obszarów gier i komiksu. Z jednej strony projektanci i artyści pracuj¸a}cy nad grami czȩsto czerpi¸a} inspiracjȩ z komiksów, jak również tworz¸a} ich licencjonowane adaptacje i odrȩbne, „interaktywne numery”. Z drugiej zaś coraz czȩściej publikowane s¸a} komiksy bazuj¸a}ce na popularnych grach. Opracowanie przedstawia również takie obszary porozumienia, współpracy, b¸adź zależności jak: technologie wykorzystywane przy tworzeniu gier i komiksów, zastosowanie komiksów w komentowaniu wydarzeń branży gier oraz organizowanie wystaw i imprez popularyzuj¸acych twórczość z obu dziedzin. (shrink)

Before venturing to emend 39 it is as well to be certain of the meaning of 40. Here are some suggestions: ‘fill out their tale of years’ ; ‘accomplissent toute [om. Ripert] leur destinee’ ; ‘compiono interamente il loro destino’ ; ‘¨berdauert die Frist’ . This is no doubt what the words, in the context of 41–42, might be expected to mean, but is this sense in the Latin? Riley was bolder with his ‘fills its destined numbers’, but it was (...) left to Némethy to state explicitly that the passive and the active voices of the Latin verb are interchangeable and that inplentur numeris suis means ‘implent numeros annorum suorum’. (shrink)

Può la matematica spiegare il mondo che ci circonda, o addirittura sé stessa? Possono i numeri, e più in generale le teorie matematiche, dirci perché alcuni fenomeni naturali e sociali avvengono o perché alcuni risultati matematici siano da considerarsi veri? Che cosa si intende esattamente per spiegazione matematica? Attraverso numerosi esempi, l’autore offre una risposta a queste domande e illustra le principali posizioni filosofiche elaborate per la nozione di spiegazione matematica, nozione che è alla base di dibattiti riguardanti aree (...) diverse della filosofia della scienza e della filosofia della matematica. (shrink)