INFINITE SETS and NUMBERS by Robert Bunn B.A., University Of
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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. -
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. -
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. -
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. -
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. -
The History of Logic
c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). -
How Peircean Was the “'Fregean' Revolution” in Logic?
HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0. -
Two Pictures of the Iterative Hierarchy
TWO PICTURES OF THE ITERATIVE HIERARCHY by Ida Marie Myrstad Dahl Thesis for the degree of Master in Philosophy Supervised by Professor Øystein Linnebo Fall 2014 Department of Philosophy, Classics, History of Arts and Ideas University of Oslo Two pictures of the iterative hierarchy ©2014 Two pictures of the iterative hierarchy Ida Marie Myrstad Dahl http://www.duo.uio.no Print: OKPrintShop iii Contents Acknowledgements . v Abstract . vi Introduction 1 1 The iterative conception of set 4 1.1 What it is . 4 1.2 Why the iterative conception? . 8 2 Actualism and potentialism on the iterative conception 15 2.1 The actualist picture . 16 2.2 The potentialist picture . 19 3 From the ancient to the contemporary concept of infinity 24 3.1 Aristotle on infinity . 24 3.2 Three important developments . 30 3.3 Cantor’s theory of the infinite . 31 3.4 A tension between actualism and potentialism . 34 4 Two tenable pictures? 40 4.1 (1) An actual conception . 41 4.2 (2) An intuitive conception . 47 4.3 (3) Explaining paradox . 50 4.4 Two tenable interpretations . 53 Conclusion 56 iv Acknowledgements I owe special thanks to Øystein Linnebo, for helpful supervision of my project, but also for introducing me to the philosophical ideas about the infinite in a phi- losophy of language seminar, at the University of Oslo, autumn, 2012. His ideas and teaching has been of great inspiration. Also, attending the PPP-seminars (Plurals, Predicates and Paradox), led by Linnebo, in 2013, was of great interest, and helped me single out the topic I wanted to write about. -
Paul M. Livingston
C ABSTRACT: C R R I Partially following Gilles Deleuze, I articulate six criteria for a strong I S variety of critique: one which affirms the power of thought in going all S I I How do we S the way to the limit of existing societies, situations, institutions and S practices. The form of this strong critique is a complex unity of thought & & and life that can be indicated, as I argue, on the basis of a twofold C C R condition: a contemporary repetition of the classical structuralism R I that Deleuze develops in the 1967 article “How do we recognize I recognize strong T T I structuralism?” and a formally based reflection on the properly infinite I Q dimension of structure and sense. I develop the implications of this Q U U E strong critique under contemporary conditions, distinguishing it E from various alternative current forms of sociopolitical critique and critique? # # 3 non-critique. In particular I argue that through its articulation of the 3 consequences of constitutive paradox, the structure of the situationally undecidable, and the ineffectivity characteristic of the constitution of sense, strong critique offers appropriate forms of response in thought and action to the structural problems and antagonisms characteristic of contemporary global capitalism. Keywords: Paul M. Critique, Deleuze, infinite, paradoxico-critical, undecidable, ineffective Livingston In his 1965 short monograph Nietzsche, Gilles Deleuze indicates the complex condition of a strong variety of critique: The philosopher of the future is the explorer of ancient worlds, of peaks and caves, who creates only inasmuch as he recalls something that has been essentially forgotten. -
João Vitor Schmidt on Frege's Definition of the Ancestral Relation
Universidade Estadual de Campinas Instituto de Filosofia e Ciências Humanas João Vitor Schmidt On Frege’s definition of the Ancestral Relation: logical and philosophical considerations Sobre a definição fregeana da Relação Ancestral: considerações lógicas e filosóficas CAMPINAS 2017 João Vitor Schmidt On Frege’s definition of the Ancestral Relation: logical and philosophical considerations Sobre a definição fregeana da Relação Ancestral: considerações lógicas e filosóficas Dissertação apresentada ao Instituto de Filosofia e Ciências Humanas da Universidade Estadual de Campinas como parte dos req- uisitos exigidos para a obtenção do título de Mestre em Filosofia. Dissertation presented to the Institute of Phi- losophy and Human Sciences of the University of Campinas in partial fulfillment of the re- quirements for the degree of Master in the area of Philosophy. Orientador: Marco Antonio Caron Ruffino Este exemplar corresponde à versão final da dissertação defendida pelo aluno João Vitor Schmidt, e orientada pelo Prof. Dr. Marco Antonio Caron Ruffino. Campinas 2017 Agência(s) de fomento e nº(s) de processo(s): CNPq, 131481/2015-0 Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Filosofia e Ciências Humanas Paulo Roberto de Oliveira - CRB 8/6272 Schmidt, João Vitor, 1987- Sch52o SchOn Frege's definition of the ancestral relation : logical and philosophical considerations / João Vitor Schmidt. – Campinas, SP : [s.n.], 2017. SchOrientador: Marco Antonio Caron Ruffino. SchDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas. Sch1. Frege, Gottlob, 1848-1925. 2. Kant, Immanuel, 1724-1804. 3. Filosofia contemporânea. 4. Lógica simbólica e Matemática. 5. Matemática - Filosofia. I. Ruffino, Marco Antonio Caron,1963-.