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Structural Reflection and the Ultimate L As the True, Noumenal Universe Of
Scuola Normale Superiore di Pisa CLASSE DI LETTERE Corso di Perfezionamento in Filosofia PHD degree in Philosophy Structural Reflection and the Ultimate L as the true, noumenal universe of mathematics Candidato: Relatore: Emanuele Gambetta Prof.- Massimo Mugnai anno accademico 2015-2016 Structural Reflection and the Ultimate L as the true noumenal universe of mathematics Emanuele Gambetta Contents 1. Introduction 7 Chapter 1. The Dream of Completeness 19 0.1. Preliminaries to this chapter 19 1. G¨odel'stheorems 20 1.1. Prerequisites to this section 20 1.2. Preliminaries to this section 21 1.3. Brief introduction to unprovable truths that are mathematically interesting 22 1.4. Unprovable mathematical statements that are mathematically interesting 27 1.5. Notions of computability, Turing's universe and Intuitionism 32 1.6. G¨odel'ssentences undecidable within PA 45 2. Transfinite Progressions 54 2.1. Preliminaries to this section 54 2.2. Gottlob Frege's definite descriptions and completeness 55 2.3. Transfinite progressions 59 3. Set theory 65 3.1. Preliminaries to this section 65 3.2. Prerequisites: ZFC axioms, ordinal and cardinal numbers 67 3.3. Reduction of all systems of numbers to the notion of set 71 3.4. The first large cardinal numbers and the Constructible universe L 76 3.5. Descriptive set theory, the axioms of determinacy and Luzin's problem formulated in second-order arithmetic 84 3 4 CONTENTS 3.6. The method of forcing and Paul Cohen's independence proof 95 3.7. Forcing Axioms, BPFA assumed as a phenomenal solution to the continuum hypothesis and a Kantian metaphysical distinction 103 3.8. -
(De)Constructing Boundaries
21st Harvard East Asia Society Graduate Conference (DE)CONSTRUCTING BOUNDARIES 9-10 February, 2018 21st Annual Harvard East Asia Society Graduate Conference (DE)CONSTRUCTING BOUNDARIES CGIS South, Harvard University 9-10 February, 2018 Abstract Booklet 1 Table of Contents Welcome Note 3 Sponsors 4 Keynote Speakers 5 Campus Map 6 Harvard Guest Wi-fi access 6 Panel Information 7 Panel A: (De)constructing Nation: Gendered Bodies in the Making of Modern Korea 7 Panel B: Urban Fabrics Unraveled 9 Panel C: Reimagining the boundary of novelistic styles in Pre-modern East Asia 10 Panel D: Transmission and Displacement in Literature 12 Panel E: Reframing Regionalism in East Asia 14 Panel F: Art and Visual Culture in Context 16 Panel G: Traversing Boundaries in Education 18 Panel H: Transnationalism in the Age of Empire 20 Panel I: Re-examining Boundaries in Chinese Politics in Xi Jinping's "New Era" 23 Panel K: Media Across Boundaries 28 Panel L: De(constructing) Myths of Migration 29 2 Welcome Note Welcome to the 21st annual Harvard East Asia Society Conference! It is our privilege to host graduate students working across all disciplines to exchange ideas and discuss their research related to Asia. In addition to receiving feedback from their peers and leading academics, participants have the opportunity to meet others doing similar research and forge new professional relationships. This year’s theme, “(De)constructing Boundaries”, critically assesses boundaries - physical, national, cultural, spatial, temporal, and disciplinary - between different spatial- temporal areas of study. As the concept of “Asia” continues to evolve, the construction and deconstruction of boundaries will enable redefinitions of collective knowledge, culture, and identity. -
A Cardinal Sin: the Infinite in Spinoza's Philosophy
Macalester College DigitalCommons@Macalester College Philosophy Honors Projects Philosophy Department Spring 2014 A Cardinal Sin: The nfinitI e in Spinoza's Philosophy Samuel H. Eklund Macalester College, [email protected] Follow this and additional works at: http://digitalcommons.macalester.edu/phil_honors Part of the Philosophy Commons Recommended Citation Eklund, Samuel H., "A Cardinal Sin: The nfinitI e in Spinoza's Philosophy" (2014). Philosophy Honors Projects. Paper 7. http://digitalcommons.macalester.edu/phil_honors/7 This Honors Project is brought to you for free and open access by the Philosophy Department at DigitalCommons@Macalester College. It has been accepted for inclusion in Philosophy Honors Projects by an authorized administrator of DigitalCommons@Macalester College. For more information, please contact [email protected]. A Cardinal Sin: The Infinite in Spinoza’s Philosophy By: Samuel Eklund Macalester College Philosophy Department Honors Advisor: Geoffrey Gorham Acknowledgements This thesis would not have been possible without my advisor, Professor Geoffrey Gorham. Through a collaborative summer research grant, I was able to work with him in improving a vague idea about writing on Spinoza’s views on existence and time into a concrete analysis of Spinoza and infinity. Without his help during the summer and feedback during the past academic year, my views on Spinoza wouldn’t have been as developed as they currently are. Additionally, I would like to acknowledge the hard work done by the other two members of my honors committee: Professor Janet Folina and Professor Andrew Beveridge. Their questions during the oral defense and written feedback were incredibly helpful in producing the final draft of this project. -
Qualitative Freedom
Claus Dierksmeier Qualitative Freedom - Autonomy in Cosmopolitan Responsibility Translated by Richard Fincham Qualitative Freedom - Autonomy in Cosmopolitan Responsibility Claus Dierksmeier Qualitative Freedom - Autonomy in Cosmopolitan Responsibility Claus Dierksmeier Institute of Political Science University of Tübingen Tübingen, Baden-Württemberg, Germany Translated by Richard Fincham American University in Cairo New Cairo, Egypt Published in German by Published by Transcript Qualitative Freiheit – Selbstbestimmung in weltbürgerlicher Verantwortung, 2016. ISBN 978-3-030-04722-1 ISBN 978-3-030-04723-8 (eBook) https://doi.org/10.1007/978-3-030-04723-8 Library of Congress Control Number: 2018964905 © The Editor(s) (if applicable) and The Author(s) 2019. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. -
On the Infinite in Leibniz's Philosophy
On the Infinite in Leibniz's Philosophy Elad Lison Interdisciplinary Studies Unit Science, Technology and Society Ph.D. Thesis Submitted to the Senate of Bar-Ilan University Ramat-Gan, Israel August 2010 This work was carried out under the supervision of Dr. Ohad Nachtomy (Department of Philosophy), Bar-Ilan University. Contents א.……………………………….…………………………………………Hebrew Abstract Prologue…………………………………………………………...………………………1 Part A: Historic Survey Methodological Introduction…………………………………………………………..15 1. Aristotle: Potential Infinite………………………………………………………….16 2. Thomas Aquinas: God and the Infinite………………………………………..…….27 3. William of Ockham: Syncategorematic and Actual Infinite……………………..….32 4. Rabbi Abraham Cohen Herrera: Between Absolute Unity and Unbounded Multitude………………………………………………………………………..….42 5. Galileo Galilei: Continuum Constructed from Infinite Zero's………………………49 6. René Descartes: Infinite as Indefinite…………………………………………….…58 7. Pierre Gassendi: Rejection of the Infinite…………………………………………...69 8. Baruch Spinoza: Infinite Unity…………………………………………………...…73 9. General Background: Leibniz and the History of the Infinite……………………....81 Summary…………………………………………………………………………….…94 Part B: Mathematics Introduction…………………………………………………………………………….99 1. 'De Arte Combinatoria' as a Formal Basis for Thought: Retrospective on Leibniz's 1666 Dissertation………………………………………………………………....102 2. Leibniz and the Infinitesimal Calculus……………………………………….……111 2.1. Mathematical Background: Mathematical Works in 16th-17th Centuries…..111 2.2. Leibniz's Mathematical Development…………………………………….…127 -
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. -
Jürgen Habermas and the Third Reich Max Schiller Claremont Mckenna College
Claremont Colleges Scholarship @ Claremont CMC Senior Theses CMC Student Scholarship 2012 Jürgen Habermas and the Third Reich Max Schiller Claremont McKenna College Recommended Citation Schiller, Max, "Jürgen Habermas and the Third Reich" (2012). CMC Senior Theses. Paper 358. http://scholarship.claremont.edu/cmc_theses/358 This Open Access Senior Thesis is brought to you by Scholarship@Claremont. It has been accepted for inclusion in this collection by an authorized administrator. For more information, please contact [email protected]. Introduction The formation and subsequent actions of the Nazi government left a devastating and indelible impact on Europe and the world. In the midst of general technological and social progress that has occurred in Europe since the Enlightenment, the Nazis represent one of the greatest social regressions that has occurred in the modern world. Despite the development of a generally more humanitarian and socially progressive conditions in the western world over the past several hundred years, the Nazis instigated one of the most diabolic and genocidal programs known to man. And they did so using modern technologies in an expression of what historian Jeffrey Herf calls “reactionary modernism.” The idea, according to Herf is that, “Before and after the Nazi seizure of power, an important current within conservative and subsequently Nazi ideology was a reconciliation between the antimodernist, romantic, and irrantionalist ideas present in German nationalism and the most obvious manifestation of means ...modern technology.” 1 Nazi crimes were so extreme and barbaric precisely because they incorporated modern technologies into a process that violated modern ethical standards. Nazi crimes in the context of contemporary notions of ethics are almost inconceivable. -
Cantor-Von Neumann Set-Theory Fa Muller
Logique & Analyse 213 (2011), x–x CANTOR-VON NEUMANN SET-THEORY F.A. MULLER Abstract In this elementary paper we establish a few novel results in set the- ory; their interest is wholly foundational-philosophical in motiva- tion. We show that in Cantor-Von Neumann Set-Theory, which is a reformulation of Von Neumann's original theory of functions and things that does not introduce `classes' (let alone `proper classes'), developed in the 1920ies, both the Pairing Axiom and `half' the Axiom of Limitation are redundant — the last result is novel. Fur- ther we show, in contrast to how things are usually done, that some theorems, notably the Pairing Axiom, can be proved without invok- ing the Replacement Schema (F) and the Power-Set Axiom. Also the Axiom of Choice is redundant in CVN, because it a theorem of CVN. The philosophical interest of Cantor-Von Neumann Set- Theory, which is very succinctly indicated, lies in the fact that it is far better suited than Zermelo-Fraenkel Set-Theory as an axioma- tisation of what Hilbert famously called Cantor's Paradise. From Cantor one needs to jump to Von Neumann, over the heads of Zer- melo and Fraenkel, and then reformulate. 0. Introduction In 1928, Von Neumann published his grand axiomatisation of Cantorian Set- Theory [1925; 1928]. Although Von Neumann's motivation was thoroughly Cantorian, he did not take the concept of a set and the membership-relation as primitive notions, but the concepts of a thing and a function — for rea- sons we do not go into here. This, and Von Neumann's cumbersome nota- tion and terminology (II-things, II.I-things) are the main reasons why ini- tially his theory remained comparatively obscure. -
A Study of Alterity and Influence in the Literary and Philosophical Neighbourhood of Jean Genet and Emmanuel Levinas
A study of alterity and influence in the literary and philosophical neighbourhood of Jean Genet and Emmanuel Levinas Thomas F. Newman University College London A thesis submitted for the degree of Doctor of Philosophy January 2008 UMI Number: U593B63 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U593363 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 Declaration I declare that the work presented in this PhD is my own. This thesis is the one on which I wish to be examined. Thomas F. Newman 29 January 2008 2 Thesis Abstract The dissertation is a chiasmic reading of the works of Jean Genet and Emmanuel Levinas, examining the way they each address the relation to the Other in terms of ethics and subjectivity. Whereas a straightforward association between the two writers might seem paradoxical because of the differences in their approaches and rhetoric, a chiasmic reading allows intricate approaches, moments of proximity and departures to be read both conceptually and aesthetically. We show that these two writers share a tightly-woven discursive neighbourhood, and examine that neighbourhood through detailed analysis of various textual encounters. -
Lost in the Labyrinth: Spinoza, Leibniz and the Continuum Lost in the Labyrinth: Spinoza, Leibniz and the Continuum
LOST IN THE LABYRINTH: SPINOZA, LEIBNIZ AND THE CONTINUUM LOST IN THE LABYRINTH: SPINOZA, LEIBNIZ AND THE CONTINUUM By PATRICK RIESTERER, B.A. A Thesis Submitted to the School of Graduate Studies In Partial Fulfillment ofthe Requirements For the Degree Master ofArts McMaster University © Copyright by Patrick Riesterer, August 2006 MASTER OF ARTS (2006) McMaster University (Philosophy) Hamilton, Ontario TITLE: Lost in the Labyrinth: Spinoza, Leibniz and the Continuum AUTHOR: Patrick Riesterer, B.A. (Trinity Western University) SUPERVISOR: Professor Richard Arthur NUMBER OF PAGES: vi, 110 ii Abstract In this thesis, I address the extent ofSpinoza's influence on the development of Leibniz's response to the continuum problem, with particular emphasis on his relational philosophy oftime and space. I expend the first chapter carefully reconstructing Spinoza's position on infinity and time. We see that Spinoza developed a threefold definition ofinfinity to explain the difference between active substance and its passive modes. Spinoza advances a syncategorematic interpretation ofinfinity, and founds a causal theory oftime directly on this conception ofinfinity. In the second chapter, I examine the changes Leibniz's understanding ofthe continuum problem underwent during 1676 and immediately thereafter. During this period, Leibniz's interacted extensively with Spinoza's ideas. We see that several fundamental features ofLeibniz's philosophy oftime take shape at this time. Leibniz adopts a Spinozistic definition ofdivine eternity and immensity, he reevaluates several analogies in an attempt to understand how the attributes ofa substance interrelate, and he develops the notion ofthe law of the series that will become an essential feature ofmonadic appetition. Leibniz synthesizes several ofthese discoveries into a first philosophy ofmotion. -
A Mathematical Model of Divine Infinity
A Mathematical Model of Divine Infinity Prof. Eric Steinhart, Department of Philosophy, William Paterson University, Wayne NJ 07470. Email: <[email protected]>, <[email protected]>. Published in 2009 in Theology and Science 7 (3), 261 – 274. ABSTRACT: Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. KEYWORDS: God; mathematics; perfection; infinity; Cantor; transfinite recursion. 1. Introduction All will agree that science makes extensive use of mathematics. At least in physics, scientific progress seems to go hand in hand with increasing formalization. More sophisticated physical theories are also more highly mathematical. And the correlation of progress with increasing formalization also appears sciences like chemistry and biology as well. Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. Nevertheless, mathematics has seen little application in theology. -
Self and Other in Critical International Theory: Assimilation, Incommensurability and the Paradox of Critique*
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository 1 SELF AND OTHER IN CRITICAL INTERNATIONAL THEORY: ASSIMILATION, INCOMMENSURABILITY AND THE PARADOX OF CRITIQUE* Vassilios Paipais Paper accepted for publication in the REVIEW OF INTERNATIONAL STUDIES Abstract: This paper is principally concerned with the way some sophisticated critical approaches in International Relations (IR) tend to compromise their critical edge in their engagement with the self/other problematic. Critical approaches that understand critique as total non-violence towards, or unreflective affirmation of, alterity risk falling back into precritical paths, i.e. either a particularistic, assimilative universalism with pretensions of true universality or radical incommensurability and the impossibility of communication with the other. This is what this paper understands as the paradox of the politics of critique. Instead, what is more important than seeking a final overcoming or dismissal of the self/other opposition is to gain the insight that it is the perpetual striving to preserve the tension and ambivalence between self and other that rescues both critique’s authority and function. Vassilios Paipais is a research student at the International Relations Department of the London School of Economics and Political Science and an LSE100 Tutorial Fellow. He is currently co-Convenor of CRIPT, a BISA working group, and his interests include International Relations theory, political theory, international political theory, international ethics, strategic studies and military history. *I would like to thank George Papanicolaou, Felix Berenskoetter and two anonymous referees of this journal for their helpful and constructive criticism.