Mathematics of the transcendental : onto-logy and being-there

In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of Category Theory, demonstrating their internal logic and veracity, their derivation and distinction from Set Theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds. This important book combines both his elaboration of the disjunctive synthesis between ontology and onto-logy (the discourses of being as such and being-appearing) from the perspective of Category Theory and the categorial basis of his philosophical conception of 'being there'. Hitherto unpublished in either French or English, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of Category Theory. The book is an essential aid to understanding the mathematical and logical basis of his theory of appearing as elaborated in Logics of Worlds and other works and is essential reading for his many followers. In Mathematics Of The Transcendental, Alain Badiou Painstakingly Works Through The Pertinent Aspects Of Category Theory, Demonstrating Their Internal Logic And Veracity, Their Derivation And Distinction From Set Theory, And The 'thinking Of Being'. In Doing So He Sets Out The Basic Onto-logical Requirements Of His Greater And Transcendental Logics As Articulated In His Magnum Opus, Logics Of Worlds. Previously Unpublished In Either French Or English, Mathematics Of The Transcendental Provides Badiou's Readers With A Much-needed Complete Elaboration Of His Understanding And Use Of Category Theory. The Book Is Vital To Understanding The Mathematical And Logical Basis Of His Theory Of Appearing As Elaborated In Logics Of Worlds And Other Works And Is Essential Reading For His Many Followers. Translators' Introduction : The Categorical Imperative -- Part One. Topos, Or Logics Of Onto-logy : An Introduction For Philosphers. General Aim ; Preliminary Definitions ; The Size Of A Category ; Limit And Universality ; Some Fundamental Concepts ; Duality ; Isomorphism ; Exponentiation ; Universe, 1 : Closed Carteisian Categories ; Structures Of Immanence, 1 : Philosophical Considerations ; Structures Of Immanence, 2 : Sub-object ; Structures Of Immanence, 3 : Elements Of An Object ; 'elementary' Clarification Of Exponentiation ; Central Object (or Sub-object Classifier) ; The True, The False, Negation And More ; The Central Object As Linguistic Power ; Universe, 2 : The Concept Of Topos ; Ontology Of The Void And Difference ; Mono., Epi., Equ., And Other Arrows ; Topoi As Logical Places ; Internal Algebra 1 ; Ontology Of The Void And Excluded Middle ; A Minimal Classical Model ; A Minimal Non-classical Model -- Part Two. Being There : Mathematics Of The Transcendental. Introduction ; A. Transcendental Structures ; B. Transcendental Connections. B.1. Connections Between The Transcendental And Set-theoretic Ontology : Boolean Algebras -- B.2. Connections Between The Transcendental And Logic In Its Ordinary Sense (propositional Logic And First Order Predicate Logic) -- B.3. Connection Between The Transcendental And The General Theory Of Localizations : Topology ; C. Theory Of Appearing And Objectivity ; E. Theory Of Relations : Situation As Universe -- Appendix : On Three Different Concepts Of Identity Between Two Multiples Of Two Beings. Alain Badiou ; Edited, Translated And With An Introduction By A. J. Bartlett And Alex Ling. Includes Bibliographical References And Index. Cover......Page 1 Half Title......Page 2 ALSO AVAILABLE FROM BLOOMSBURY......Page 3 Title......Page 4 Copyright......Page 5 TABLE OF CONTENTS......Page 6 TRANSLATORS’ INTRODUCTION: THE CATEGORIAL IMPERATIVE......Page 10 PART ONE TOPOS, OR LOGICS OF ONTO-LOGY: AN INTRODUCTION FOR PHILOSOPHERS......Page 20 1 GENERAL AIM......Page 22 2 PRELIMINARY DEFINITIONS......Page 26 3 THE SIZE OF A CATEGORY......Page 30 4 LIMIT AND UNIVERSALITY......Page 36 5 SOME FUNDAMENTAL CONCEPTS......Page 38 6 DUALITY......Page 46 7 ISOMORPHISM......Page 50 8 EXPONENTIATION......Page 54 9 UNIVERSE, 1: CLOSED CARTESIAN CATEGORIES......Page 60 10 STRUCTURES OF IMMANENCE, 1: PHILOSOPHICAL CONSIDERATIONS......Page 64 11 STRUCTURES OF IMMANENCE, 2 : SUB-OBJECT......Page 68 12 STRUCTURES OF IMMANENCE, 3: ELEMENTS OF AN OBJECT......Page 72 13 ‘ELEMENTARY’ CLARIFICATION OF EXPONENTIATION......Page 76 14 CENTRAL OBJECT (OR SUB-OBJECT CLASSIFIER)......Page 80 15 THE TRUE, THE FALSE, NEGATION AND MORE......Page 86 16 THE CENTRAL OBJECT AS LINGUISTIC POWER......Page 94 17 UNIVERSE, 2: THE CONCEPT OF TOPOS......Page 98 18 ONTOLOGY OF THE VOID AND DIFFERENCE......Page 104 19 MONO., EPI., EQU., AND OTHER ARROWS......Page 108 20 TOPOI AS LOGICAL PLACES......Page 122 21 INTERNAL ALGEBRA OF 1......Page 132 22 ONTOLOGY OF THE VOID AND EXCLUDED MIDDLE......Page 150 23 A MINIMAL CLASSICAL MODEL......Page 156 24 A MINIMAL NON-CLASSICAL MODEL......Page 160 PART TWO BEING THERE: MATHEMATICS OF THE TRANSCENDENTAL......Page 172 INTRODUCTION......Page 174 A. TRANSCENDENTAL STRUCTURES......Page 180 B.1. Connections between the transcendental and set-theoretic ontology: Boolean algebras......Page 192 B.2. Connections between the transcendental and logic in its ordinary sense (propositional logic and first order predicate logic)......Page 204 B.3. Connection between the transcendental and the general theory of localizations: Topology......Page 211 C. THEORY OF APPEARING AND OBJECTIVITY......Page 226 D. TRANSCENDENTAL PROJECTIONS: THEORY OF LOCALIZATION......Page 244 E. THEORY OF RELATIONS: SITUATION AS UNIVERSE......Page 258 APPENDIX : ON THREE DIFFERENT CONCEPTS OF IDENTITY BETWEEN TWO MULTIPLES OR TWO BEINGS......Page 274 TRANSLATOR’S ENDNOTES......Page 278 INDEX......Page 286 Cover -- Half Title -- ALSO AVAILABLE FROM BLOOMSBURY -- Title -- Copyright -- TABLE OF CONTENTS -- TRANSLATORS' INTRODUCTION: THE CATEGORIAL IMPERATIVE -- PART ONE TOPOS, OR LOGICS OF ONTO-LOGY: AN INTRODUCTION FOR PHILOSOPHERS -- PART TWO BEING THERE: MATHEMATICS OF THE TRANSCENDENTAL -- APPENDIX : ON THREE DIFFERENT CONCEPTS OF IDENTITY BETWEEN TWO MULTIPLES OR TWO BEINGS -- TRANSLATOR'S ENDNOTES -- INDEX -- 1 GENERAL AIM -- 2 PRELIMINARY DEFINITIONS -- 3 THE SIZE OF A CATEGORY -- 4 LIMIT AND UNIVERSALITY -- 5 SOME FUNDAMENTAL CONCEPTS -- 6 DUALITY -- 7 ISOMORPHISM -- 8 EXPONENTIATION 22 ONTOLOGY OF THE VOID AND EXCLUDED MIDDLE -- 23 A MINIMAL CLASSICAL MODEL -- 24 A MINIMAL NON-CLASSICAL MODEL -- INTRODUCTION -- A. TRANSCENDENTAL STRUCTURES -- B. TRANSCENDENTAL CONNECTIONS -- C. THEORY OF APPEARING AND OBJECTIVITY -- D. TRANSCENDENTAL PROJECTIONS: THEORY OF LOCALIZATION -- E. THEORY OF RELATIONS: SITUATION AS UNIVERSE -- B.1. Connections between the transcendental and set-theoretic ontology: Boolean algebras -- B.2. Connections between the transcendental and logic in its ordinary sense (propositional logic and first order predicate logic)