The Partial Eclipse of Plotinus in the Middle Ages and His Recovery in the Renaissance
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The Modal Logic of Potential Infinity, with an Application to Free Choice
The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. -
Cantor on Infinity in Nature, Number, and the Divine Mind
Cantor on Infinity in Nature, Number, and the Divine Mind Anne Newstead Abstract. The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristote- lian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian volunta- rist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in Can- tor’s thought with reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspec- tives on the Actual Infinite”) (1885). I. he Philosophical Reception of Cantor’s Ideas. Georg Cantor’s dis- covery of transfinite numbers was revolutionary. Bertrand Russell Tdescribed it thus: The mathematical theory of infinity may almost be said to begin with Cantor. The infinitesimal Calculus, though it cannot wholly dispense with infinity, has as few dealings with it as possible, and contrives to hide it away before facing the world Cantor has abandoned this cowardly policy, and has brought the skeleton out of its cupboard. He has been emboldened on this course by denying that it is a skeleton. Indeed, like many other skeletons, it was wholly dependent on its cupboard, and vanished in the light of day.1 1Bertrand Russell, The Principles of Mathematics (London: Routledge, 1992 [1903]), 304. -
Leibniz and the Infinite
Quaderns d’Història de l’Enginyeria volum xvi 2018 LEIBNIZ AND THE INFINITE Eberhard Knobloch [email protected] 1.-Introduction. On the 5th (15th) of September, 1695 Leibniz wrote to Vincentius Placcius: “But I have so many new insights in mathematics, so many thoughts in phi- losophy, so many other literary observations that I am often irresolutely at a loss which as I wish should not perish1”. Leibniz’s extraordinary creativity especially concerned his handling of the infinite in mathematics. He was not always consistent in this respect. This paper will try to shed new light on some difficulties of this subject mainly analysing his treatise On the arithmetical quadrature of the circle, the ellipse, and the hyperbola elaborated at the end of his Parisian sojourn. 2.- Infinitely small and infinite quantities. In the Parisian treatise Leibniz introduces the notion of infinitely small rather late. First of all he uses descriptions like: ad differentiam assignata quavis minorem sibi appropinquare (to approach each other up to a difference that is smaller than any assigned difference)2, differat quantitate minore quavis data (it differs by a quantity that is smaller than any given quantity)3, differentia data quantitate minor reddi potest (the difference can be made smaller than a 1 “Habeo vero tam multa nova in Mathematicis, tot cogitationes in Philosophicis, tot alias litterarias observationes, quas vellem non perire, ut saepe inter agenda anceps haeream.” (LEIBNIZ, since 1923: II, 3, 80). 2 LEIBNIZ (2016), 18. 3 Ibid., 20. 11 Eberhard Knobloch volum xvi 2018 given quantity)4. Such a difference or such a quantity necessarily is a variable quantity. -
Measuring Fractals by Infinite and Infinitesimal Numbers
MEASURING FRACTALS BY INFINITE AND INFINITESIMAL NUMBERS Yaroslav D. Sergeyev DEIS, University of Calabria, Via P. Bucci, Cubo 42-C, 87036 Rende (CS), Italy, N.I. Lobachevsky State University, Nizhni Novgorod, Russia, and Institute of High Performance Computing and Networking of the National Research Council of Italy http://wwwinfo.deis.unical.it/∼yaro e-mail: [email protected] Abstract. Traditional mathematical tools used for analysis of fractals allow one to distinguish results of self-similarity processes after a finite number of iterations. For example, the result of procedure of construction of Cantor’s set after two steps is different from that obtained after three steps. However, we are not able to make such a distinction at infinity. It is shown in this paper that infinite and infinitesimal numbers proposed recently allow one to measure results of fractal processes at different iterations at infinity too. First, the new technique is used to measure at infinity sets being results of Cantor’s proce- dure. Second, it is applied to calculate the lengths of polygonal geometric spirals at different points of infinity. 1. INTRODUCTION During last decades fractals have been intensively studied and applied in various fields (see, for instance, [4, 11, 5, 7, 12, 20]). However, their mathematical analysis (except, of course, a very well developed theory of fractal dimensions) very often continues to have mainly a qualitative character and there are no many tools for a quantitative analysis of their behavior after execution of infinitely many steps of a self-similarity process of construction. Usually, we can measure fractals in a way and can give certain numerical answers to questions regarding fractals only if a finite number of steps in the procedure of their construction has been executed. -
Galileo and Leibniz: Different Approaches to Infinity
Arch. Hist. Exact Sci. 54 (1999) 87–99. c Springer-Verlag 1999 Galileo and Leibniz: Different Approaches to Infinity Eberhard Knobloch Communicated by J. J. Gray Introduction “God exists since mathematics is consistent, and the devil exists since we cannot prove it.” In this way Andre´ Weil, one of the leading mathematicians of this century, characterized the mathematics of our time (Meschkowski 1969, p. 112; Rosenbloom 1950, p. 72). This connection between mathematics, God and the devil is not new: I would like to recall here only the study scene in Goethe’s Faust: Faust has the devil temporarily in his hand, because the pentagram is badly drawn. What is new in Weil’s saying is the insight that even mathematics cannot provide complete foundations for itself. Metamathematics and Hilbert’s proof theory are disciplines of our century. When Hilbert set about showing how the antinomies of the set theory created by Cantor could be avoided without betraying mathematics, as described in his lecture “On the Infinite” in 1925, he emphasized the necessity of clarifying the nature of the infinite (Hilbert 1926, p. 188). The infinite plays a crucial role in analysis, this most ingenious product of mathematical science, being ramified in the most refined way. As Hilbert fittingly formulated, mathematical analysis is, so to speak, a pure symphony on the infinite (Hilbert 1926, p. 184). No wonder that three years after Hilbert’s lecture his pupil Hermann Weyl character- ized mathematics as the science of the infinite (Weyl 1925–1927, p. 1; Laugwitz 1986, p. 233). He thus defined mathematics in a completely different way than his predecessors of earlier centuries. -
Plotinus on the Soul's Omnipresence in Body
Plotinus on the Limitation of Act by Potency Gary M. Gurtler, S.J. Boston College The limitation of act by potency, central in the metaphysics of Thomas Aquinas, has its origins in Plotinus. He transforms Aristotle’s horizontal causality of change into a vertical causality of participation. Potency and infinity are not just unintelligible lack of limit, but productive power. Form determines matter but is limited by reception into matter. The experience of unity begins with sensible things, which always have parts, so what is really one is incorporeal, without division and separation. Unity is like the esse of Thomas, since it is the act that makes a thing what it is and has its fullness in God. 1. Introduction Over fifty years ago Norris Clarke, S.J., published “The Limitation of Act by Potency: Aristotelianism or Neoplatonism,” The New Scholasticism, 26 (1952) 167–194, which highlighted the role of Plotinus in formulating some of the fundamental distinctions operative in medieval philosophy. He singled out the doctrine of participation and the idea of infinity and showed how they went beyond the limits of classical Greek thought. At the same time, his work challenged some of the basic historical assumptions of Thomists about the relation of scholastic philosophy, especially St. Thomas’s, to Plato and Aristotle. I have appreciated more and more the unusual openness and self critical character of Fr. Clarke’s historical method, and am pleased to have the chance in this response to Sarah Borden’s paper to explore some of the issues raised by Fr. Clarke’s seminal article in light of my own work on Plotinus. -
Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle. -
Trans. Greek Thot Handout
11/14/19 TRANSMISSION OF GREEK THOUGHT TO THE WEST PLATO & NEOPLATONISM Chalcidius (late 3rd-early 4th cent. Christian exegete): incomplete translation & commentary of Timeaus Henricus Aristippus in Sicily (12th c.): translated the Meno and Phaedo Leonardo Bruni (c. 1370-1444/Florence) translated a selection of Plato’s dialogues (from Greek to Latin). Marsilio Ficino (1433-1499/Florence): 1st complete translation into Latin of Plato’s works (publ. 1496), and translation of Plotinus’s Enneads into Latin (1492). Neoplatonic thought was transmitted in the following: (a) Boethius’ Consolation of Philosophy (written 524, in prison) (b) Macrobius’ Commentary on Cicero’s Dream of Scipio (written c. 400 CE). (c) Pseudo-Dionysius. A collection of writings attributed to Dionysius the Aeropagite (see Acts 17:34), but 19th century scholarship determined to be written c. 500 by a disciple of Proclus, held considerable authority throughout the middle ages and was a Christian Neoplatonism. (d) Theologica Aristotelis: this summary of Books 4-6 of Plotinus’s Enneads had been wrongly attributed to Aristotle (until 13th century) (e) Liber de Causis: this work based on Proclus’s Elements of Theology was wrongly attributed to Aristotle (until 13th century). ARISTOTLE Victorinus (4th century): Latin translations of Aristotle’s Categories and De interpretatione, as well as of Porphyry’s Isagoge. Boethius (470-524/Padua?): translated the entire Organon and wrote commentaries on all but the Posterior Analytics), as well as a translation of Porphyry’s introduction (Isagoge) to the Categories, but only De Interp. and Categories were readily available until 12th century. James of Venice (c.1128): translated Posterior Analytics; with the rediscovery of other translations by Boethius, this completed the Organon. -
The Crisis of Infinity: Mathematics, Philosophy, and Nineteenth-Century Poetry
The Crisis of Infinity: Mathematics, Philosophy, and Nineteenth-Century Poetry by Rachel Elizabeth Feder A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (English Language and Literature) in The University of Michigan 2012 Doctoral Committee: Professor Marjorie Levinson, Chair Professor Adela N. Pinch Professor Yopie Prins Assistant Professor Gillian C. White ! ! ! Acknowledgements This work is dedicated to Marjorie Levinson with infinite thanks. Ever since this project began to take shape in Marjorie’s 2007 seminar on Romantic poetry and philosophy, she has worked to bring out the best in both the scholar and the project. I am thankful to Marjorie for her tireless mentorship, keen insight, honest critique, and endless encouragement. I am lucky to have an advisor who is such a constant source of inspiration and support. Neither this dissertation nor I would be the same without her. Yopie Prins has shared her critical interventions and intellectual communities with me and encouraged the development of my own, expanding my sense of what is possible. Time and again, she has illuminated my scholarly endeavors, showing me how seemingly superfluous points of interest and obsession were actually crucial to my argument. Yopie encouraged experiments, organized ideas, and always saw the good. If my ideas have achieved elegance, they have done so under her watch. I am grateful to Adela Pinch for the constancy and clarity of her counsel. Adela provided an encouraging word when needed and an honest word always. She fostered critical openness, and her generous feedback and painstaking attention to detail fortified my writing process. -
AUGUSTINE of HIPPO and HANNAH ARENDT Antonio Calcagno
PARRHESIA NUMBER 13 • 2011 • 14-27 THE ROLE OF FORGETTING IN OUR EXPERIENCE OF TIME: AUGUSTINE OF HIPPO AND HANNAH ARENDT Antonio Calcagno Hannah Arendt’s work is deeply marked by a rich analysis of temporality. One finds references to it from her earliest works on Augustine of Hippo1 to her last major work, The Life of the Mind. Scholars have rightly explored the significance of temporality and its constitutive centrality for Arendt’s thought.2 Her views of temporality can be divided grosso modo along three axes. First, there is time-lived or the times-experienced, which comprise or touch upon such aspects as history, biography, narrative, and a timely engagement with the issues of the day, which include, for example, various moments in American Cold-War culture and civil-rights movements.3 Second, there is a sense of time understood as spontaneity, as natality, in which a new beginning is made possible today, hodiernus. Arising spontaneously, this time marks a new promise, as in the case of forgiveness.4 Perhaps we can call this the time of the kairos, the opportune time, the now-time that emerges from a set of circumstances but is not merely a result of a series of causal events. Finally, one finds in Arendt’s work a philosophical position on the very nature of time itself, understood as the past and the future that make possible the present, that condition the very existence of the human being. Particularly striking is the account of time that Arendt gives in the concluding parts of her work on “Thinking” in The Life of the Mind. -
Forgetful Platonism: Misrepresenting Proclus and Twentieth-Century
Misrepresenting Neoplatonism in Contemporary Christian Dionysian Polemic: Eriugena and Nicholas of Cusa versus Vladimir Lossky and Jean-Luc Marion American Catholic Philosophical Quarterly 82:4 (2008): 683–703. Wayne Hankey Part I: From Eriugena to Nicholas of Cusa The Corpus Areopagiticum in Greek—or at least a part of it—circulated in Rome, probably from the time of Gregory the Great‘s Homily XXIV (592 or 593), but certainly from the middle of the seventh up to the second half of the ninth century.1 Nonetheless, there was no profound encounter by the Latins with the philosophical and mystical content of the corpus, until the translations by Hilduin and Eriugena consequent on its arrival at the Frankish court in 827.2 John the Scot‘s grasp and development of what the Areopagite passed on from his sources has only rarely been equalled. After Boethius, he was the first to draw together the Greek and Latin Platonists. Among the Latins, Augustine and Boethius, together with Ambrose, because, as Eriugena wrote, he so often followed the Greeks, were pre-eminent .3 He learned most from the Greek texts he translated—the Dionysian corpus, the Ambigua and Scoliae of Maximus the Confessor, and the De hominis opificio of Gregory of Nyssa—; what he took from the Latins was generally assimilated to their logic.4 His sources were so nearly exclusively Christian, and the texts so predominantly theological and mystical—despite such ―secular‖ and pagan authorities as the De nuptiis Philologiae et Mercurii of Martianus Capella from his Irish education5—that no real confrontation between Hellenic and Christian Neoplatonisms occurs for him. -
Reading Proclus and the Bookof Causes
introduction Reading Proclus and the Book of Causes: Notes on the Western Scholarly Networks and Debates Dragos Calma University College Dublin / Newman Centre for the Study of Religions, Dublin The majority of contributions reunited in this volume were first presented during the first of the three sessions of the conference “Les Éléments de thé- ologie et le Livre des causes du Ve au XVIIe siècle”. It took place at the École pratique des hautes études, Paris, on 13–14 November 2015, during the ter- rorist attacks.1 The second took place on 12–13 February 2016, and the third on 14–15–16 April 2016; all within the framework of the project LIBER (ANR- 13-PDOC-0018-01) which I directed between 2013 and 2017 at the École pra- tique des hautes études, Paris, and which generously financed these meet- ings and the publication of the proceedings.2 The conferences were organ- ised by Marc Geoffroy and myself, under the auspices of Olivier Boulnois, Philippe Hoffmann, Ruedi Imbach, Zénon Kaluza and Dominique Poirel. I wish to acknowledge the important support of the following bodies in the organisa- tion of the conference: École pratique des hautes études, Équipe “philosophie arabe” of the Centre “Jean Pépin”—CNRS (UMR 8230), Laboratoire d’études sur les monothéismes—CNRS (UMR 8584), Labex haStec (Laboratoire européen d’histoire et anthropologie des savoirs, des techniques et des croyances), Insti- tut de recherche et d’histoire des textes—CNRS, Centre “Pierre Abélard”— Université Paris Sorbonne. Compared to the original program of the three conferences, minor thematic rearrangements have been made for publication.