H/T to NewAPPS blog for the reminder.
Showing posts with label Happenings. Show all posts
Showing posts with label Happenings. Show all posts
Tuesday, August 2, 2011
Conference Announcement: Classical Model of Science II
Starting this evening at 20:15 with a lecture by the great philosopher of mathematics Stewart Shapiro [[UPDATE: SHAPIRO'S TALK HAS BEEN MOVED TO TOMORROW (WED.) MORNING AT 9:45. SCHLIESSER WILL BE SPEAKING TONIGHT ON SPINOZA, INSTEAD]], and running through to Friday (when Paolo Mancosu, historian of mathematics and editor of the indispensable anthology, From Brouwer to Hilbert, and Hourya Sinaceur, a brilliant philosopher of mathematics in the tradition of Cavaillès and author of Fields and Models, will both be speaking, among many others), the AXIOM group is hosting its second conference on "the Classical Model of Science", subtitled, "The Axiomatic Method, the Order of Concepts and the Hierarchy of Sciences from Leibniz to Tarski". The conference is taking place at the Vrije Universiteit Amstersdam, in the Philosophy Department. The conference programme, with schedule and abstracts, can be found here in pdf form.
Wednesday, July 13, 2011
Fernando Zalamea Seminar in Maastricht, Netherlands
It's official: Fernando Zalamea will be delivering a long-awaited English-language seminar on his recent work at the Jan van Eyck Academie in Maastricht (The Netherlands) this September, through the Versus Laboratory. As some of this blog's readers may know, I'm currently working on a translation of his 2009 book, Filosofía sintética de las matemáticas contemporaneas, for Urbanomic press, a few chapters of which we'll be reading for the seminar. Zalamea is an extraordinarily original philosopher with a vision of mathematical activity as deep and sweeping as any philosopher since Albert Lautman. If the resurgence of the mathematical condition into contemporary 'continental' philosophy is something that interests you, then this seminar is well worth the train to Maastricht. Like all JVE events, admission is free and open to the public. Here is the announcement from the Versus Laboratory page:
Sheaf Logic & Philosophical Synthesis
Thursday 29 September 2011
14h to 17h Auditorium
Fernando Zalamea (Universidad Nacional de Columbia), philosopher and mathematician, will illustrate how today’s sheaf logic carries a potential to revolutionize contemporary philosophical practice in a manner no less radical than the force with which the early 20th Century formalization of classical logic conditioned the practice of Analytic Philosophy. In particular, we will study a series of Grothendieck transformations of mathematical concepts, and come to understand how analogous procedures may be employed to the end of overturning a number of traditional philosophical polarities (realism/idealism, statics/dynamics, form/content, etc.).
Texts: Fernando ZALAMEA, Synthetic Philosophy of Contemporary Mathematics, trans. Z.L. Fraser, London: Urbanomic, 2012. Chapters 4, 8 and 9. Copies of the readings will be made available here at the end of August.
Information on the book (forthcoming from Urbanomic) can be found here.
* In other Zalamea-related news, Reza Negarestani has recently mentioned plans for a series of introductory posts on Zalamea's work on his own blog, Eliminative Culinarism, which I highly recommend readers of this blog take a look at when they're up. Zalamea also has an article in the most recent volume of COLLAPSE, which I'm still waiting to get my hands on. [[UPDATE: NEGARESTANI'S INTRODUCTORY POST ON ZALAMEA'S PROJECT IS NOW UP, AND CAN BE FOUND HERE.]]
Saturday, May 21, 2011
Mexico City Seminars on Mathematical Logic, Badiou & Girard
The slides for the seminars (which are predominantly in Spanish, or something resembling Spanish) are available HERE.
The dominant theme in all three seminars was the gap between syntax and semantics, conceived as site for conceptual and mathematical invention rather than as a call to mimicry. This took us from a study of Badiou's Concept of Model (read in the light of Althusser's critique of the "mirror myth of knowledge" in the introduction to Reading Capital, and as a more or less successful attempt to elaborate and refine this critique in order to intervene against the empiricist use of the logico-mathematical theory of models), a case study of the Löwenheim-Skolem theorem (the first really significant theorem dealing with the concept of model, and also the one which opened the abyss between syntax and semantics, and produced the concept of "non-standard" models), in Seminar 1, to a examination, in Seminar 2, of Badiou's category of "forcing", focussing on how it draws on the two mathematical "conditions" of Robinson's method for producing non-standard models for analysis (as analysed in Badiou's early text, "La Subversion infinitesimale"), and, of course, Paul Cohen's "forcing" technique in the controlled production of models for set theory, which was the key condition for the theory of truth and subjectivity in Being and Event. Seminar 3 dealt with Girard's work, and its conceptual and historical context. We looked at his critique of Tarski-style semantics, and his more or less tacit philosophical concepts of blind spot and of logic as essentially 'productive' -- themes which helped to link this material back to Althusser, etc. The shift from a set-theoretic paradigm to a procedural one -- the neglect of which would make Girard's project almost impossible to understand -- was examined in terms of the move from a set-theoretic conception of functions to a conception informed by the lambda calculus. A bit of time, not much, was spent on the parallel difference between Tarskian and denotational semantics (where what is modelled is the dynamics of proofs, not mere 'provability' -- the notion which semantic 'truth' roughly and imperfectly captures, in a garment which leaves nothing to the imagination and yet is far too bulky). We moved from there through the the sequent calculi and the Curry-Howard isomorphism (the isomorphism between proofs (in the sequent calculus, or natural deduction systems) and programmes (in the lambda calculus, the Turing machine formalism, or any actual computer), and so on, to linear logic, looking at the subtle tensions which emerge in our understanding of logic as we direct our attention to its hidden symmetries, its procedural aspects, etc. Another important conceptual distinction that we dealt with was that between "typed" and "untyped" systems -- focussing again on the lambda calculus, but with the intention of asking (in light of the Curry-Howard isomorphism) what an untyped logic might look like.
(Explanation, by way of example: in the untyped lambda calculus, every lambda term -- every programme or function -- can interact with every other lambda term, even if the result is a non-terminating procedure (a 'crash') -- "plus 1", for example, can act not only on the numerals for which it was designed, but even on functions which have nothing to do with numbers. The result's not always pretty, but something always happens. The untyped lambda universe is a wild world, and this leads to some very strange facts -- such as every function possessing a fixed point (for all F there exists an X such that F(X) = X) even if this fixed point is monstrous. In the typed lambda calculus, by contrast, everything is domesticated: the functions are saddled with a "superegoic" apparatus of types (Girard's metaphor, I think, if not Joinet's) which limits interaction, and allows terms to act only upon terms of the appropriate "type". The upshot is that every function eventually "terminates" or reaches "normal form" -- nothing crashes -- in the typed calculus, but the control by which this peace is won seems a bit artificial, or at least superficial, and doesn't really seem to proceed from the deeper structure of the calculus.)
[ADDENDUM: What is a 'type' in logic, you ask? A type is the name of a proposition. "A & B", for example, is a proposition of type A&B. The Curry-Howard isomorphism maps proofs to programmes, and propositions to types of programmes. So the question, "What would an untyped logic look like?" becomes something like "Can we do logic without casting our propositions in types prior to the demonstrative work that explicates them and tests them for consequences?" Can we have a logic where we don't begin with a battery of atomic sentences and pre-fabricated connectives? That's the gist of it.]
Finally, we looked at ludics, which is just such a logic (an untyped logic, that is), and which in Girard's eyes succeeds in sublating the gap between syntax and semantics. This section was pretty much improvised. I'll try and write something more precise about and thorough it soon, and post it here. [ADDENDUM: For now, I'll just say: cut-elimination, the algorithmic procedure by which appeals to lemmas are eliminated from a proof, rendering the proof wholly explicit, without 'subroutines', is the key. Cut-elimination is always possible for classical logic, always yields a unique result for intuitionistic and linear logic, but only in ludics does the dynamic of cut-elimination find its full scope, becoming the real engine of the entire system. In 'pre-ludic' logics, many characteristically 'semantic' properties can be expressed in terms of syntactic properties of cut-free proofs. Ludic 'interaction' -- a generalized form of cut-elimination -- reaches into crannies that ordinary cut-elimination can't.] In the meantime, curious readers can find some of my rough sketches of this subject matter (in English this time) here and here.
I'm happy to say that the seminars went extremely well, better than I could have hoped. I'm incredibly grateful for the boundless hospitality and generosity of Carlos Gomez, the Lacanian psychoanalyst who not only, through some incomprehensible faculty of persuasion, convinced the Department of Mathematics and Physics to invite me to come give the seminars, but ensured that my wife and I received full royal treatment while in the city. (And what a city!)
The participants in the seminars were few, but brilliant, and I left with several loose threads which I hope to follow up soon in my research. Among the most interesting of these concerned the sense that should be read into Girard's project for a "transcendental syntax" -- of which ludics is just one adumbration -- with one participant, named Cristina, pointing out that this sounds like Deleuze's conception of the transcendental more than anything (productive of what it conditions, untyped or 'wild', not already sorted into kinds, not resembling the conditioned -- unlike the Tarskian "meta"). This is something I'll have to look at more closely, so, readers, where should I start for a clear treatment of Deleuze's concept of the transcendental? Deleuze has always been someone I've liked quite a bit, but who I've read more or less casually. I'm thinking that Difference and Repetition would be the key text on this topic, but I welcome other suggestions.
The dominant theme in all three seminars was the gap between syntax and semantics, conceived as site for conceptual and mathematical invention rather than as a call to mimicry. This took us from a study of Badiou's Concept of Model (read in the light of Althusser's critique of the "mirror myth of knowledge" in the introduction to Reading Capital, and as a more or less successful attempt to elaborate and refine this critique in order to intervene against the empiricist use of the logico-mathematical theory of models), a case study of the Löwenheim-Skolem theorem (the first really significant theorem dealing with the concept of model, and also the one which opened the abyss between syntax and semantics, and produced the concept of "non-standard" models), in Seminar 1, to a examination, in Seminar 2, of Badiou's category of "forcing", focussing on how it draws on the two mathematical "conditions" of Robinson's method for producing non-standard models for analysis (as analysed in Badiou's early text, "La Subversion infinitesimale"), and, of course, Paul Cohen's "forcing" technique in the controlled production of models for set theory, which was the key condition for the theory of truth and subjectivity in Being and Event. Seminar 3 dealt with Girard's work, and its conceptual and historical context. We looked at his critique of Tarski-style semantics, and his more or less tacit philosophical concepts of blind spot and of logic as essentially 'productive' -- themes which helped to link this material back to Althusser, etc. The shift from a set-theoretic paradigm to a procedural one -- the neglect of which would make Girard's project almost impossible to understand -- was examined in terms of the move from a set-theoretic conception of functions to a conception informed by the lambda calculus. A bit of time, not much, was spent on the parallel difference between Tarskian and denotational semantics (where what is modelled is the dynamics of proofs, not mere 'provability' -- the notion which semantic 'truth' roughly and imperfectly captures, in a garment which leaves nothing to the imagination and yet is far too bulky). We moved from there through the the sequent calculi and the Curry-Howard isomorphism (the isomorphism between proofs (in the sequent calculus, or natural deduction systems) and programmes (in the lambda calculus, the Turing machine formalism, or any actual computer), and so on, to linear logic, looking at the subtle tensions which emerge in our understanding of logic as we direct our attention to its hidden symmetries, its procedural aspects, etc. Another important conceptual distinction that we dealt with was that between "typed" and "untyped" systems -- focussing again on the lambda calculus, but with the intention of asking (in light of the Curry-Howard isomorphism) what an untyped logic might look like.[ADDENDUM: What is a 'type' in logic, you ask? A type is the name of a proposition. "A & B", for example, is a proposition of type A&B. The Curry-Howard isomorphism maps proofs to programmes, and propositions to types of programmes. So the question, "What would an untyped logic look like?" becomes something like "Can we do logic without casting our propositions in types prior to the demonstrative work that explicates them and tests them for consequences?" Can we have a logic where we don't begin with a battery of atomic sentences and pre-fabricated connectives? That's the gist of it.]
Finally, we looked at ludics, which is just such a logic (an untyped logic, that is), and which in Girard's eyes succeeds in sublating the gap between syntax and semantics. This section was pretty much improvised. I'll try and write something more precise about and thorough it soon, and post it here. [ADDENDUM: For now, I'll just say: cut-elimination, the algorithmic procedure by which appeals to lemmas are eliminated from a proof, rendering the proof wholly explicit, without 'subroutines', is the key. Cut-elimination is always possible for classical logic, always yields a unique result for intuitionistic and linear logic, but only in ludics does the dynamic of cut-elimination find its full scope, becoming the real engine of the entire system. In 'pre-ludic' logics, many characteristically 'semantic' properties can be expressed in terms of syntactic properties of cut-free proofs. Ludic 'interaction' -- a generalized form of cut-elimination -- reaches into crannies that ordinary cut-elimination can't.] In the meantime, curious readers can find some of my rough sketches of this subject matter (in English this time) here and here.I'm happy to say that the seminars went extremely well, better than I could have hoped. I'm incredibly grateful for the boundless hospitality and generosity of Carlos Gomez, the Lacanian psychoanalyst who not only, through some incomprehensible faculty of persuasion, convinced the Department of Mathematics and Physics to invite me to come give the seminars, but ensured that my wife and I received full royal treatment while in the city. (And what a city!)
The participants in the seminars were few, but brilliant, and I left with several loose threads which I hope to follow up soon in my research. Among the most interesting of these concerned the sense that should be read into Girard's project for a "transcendental syntax" -- of which ludics is just one adumbration -- with one participant, named Cristina, pointing out that this sounds like Deleuze's conception of the transcendental more than anything (productive of what it conditions, untyped or 'wild', not already sorted into kinds, not resembling the conditioned -- unlike the Tarskian "meta"). This is something I'll have to look at more closely, so, readers, where should I start for a clear treatment of Deleuze's concept of the transcendental? Deleuze has always been someone I've liked quite a bit, but who I've read more or less casually. I'm thinking that Difference and Repetition would be the key text on this topic, but I welcome other suggestions.
Friday, May 6, 2011
Taller Sobre Badiou y la Lógica Matemática en la Ciudad de México
As I mentioned in an earlier post, I will be holding a three-seminar workshop this Tuesday and Wednesday at La Universidad Iberoamericana, in Mexico City. The seminars will be held in a combination of English and Spanish (I took the trouble to putting together my slides in Spanish, but will probably swing back and forth between the two languages as necessary. It's been a while since I my Spanish was at a fluent conversational level). I'll post my powerpoint slides to the blog after the seminars, if anyone's interested.
1. Sobre El Concepto de Modelo de Alain Badiou, (martes 10 de mayo, de 9am a 10:30 am y de 11 am a 12 pm)
2. El Concepto y La Categoría de Forcing (Forzamiento), de La Subversión Infinitesimal al Ser y el Acontecimiento, (Miércoles 11 mayo, de 9 am a 10:45 am)
3. El Proyecto Lógico de Jean-Yves Girard, como Radicalización Lógico-Matemático de la Critica de el 'Espejo-Mito' de Saber Criticado por Althusser y Badiou, y como una Condición Contemporánea para la Filosofía, (Miércoles 11 mayo, de 11:15am a 1:00pm)
1. Sobre El Concepto de Modelo de Alain Badiou, (martes 10 de mayo, de 9am a 10:30 am y de 11 am a 12 pm)
2. El Concepto y La Categoría de Forcing (Forzamiento), de La Subversión Infinitesimal al Ser y el Acontecimiento, (Miércoles 11 mayo, de 9 am a 10:45 am)
3. El Proyecto Lógico de Jean-Yves Girard, como Radicalización Lógico-Matemático de la Critica de el 'Espejo-Mito' de Saber Criticado por Althusser y Badiou, y como una Condición Contemporánea para la Filosofía, (Miércoles 11 mayo, de 11:15am a 1:00pm)
LECTURAS PARA LOS SEMINARIOS
El seminario donde el text se va usar se indica en abrazaderas, en la forma [S#].
Textos citados por negritos son fuertemente recomendados. Los otros son algo opcionales.
Unos de estes textos se pueden encontrar al sitio de Jean-Yves Girard: http://iml.univ-mrs.fr/~girard/Articles.html
Althusser, Louis. “Prefacio: De El Capital a la filosofía de Marx,” in Para leer el Capital. Buenos Aires: Siglo XXI. [S1]
Althusser, Louis. Curso de filosofía para científicos (introducción: Filosofía y filosofía espontánea de los científicos, 1967). [S1]
Badiou, Alain. 2009 (1969). El Concepto de modelo: Introducción a una epistemología materialista de las matemáticas. Trad. Vera Waksman. Buenos Aires: La Bestia Equilátera. [S1, S2, S3]
———. 1967. La Subversion infinitesimale. En Cahiers pour l’analyse, Vol. 9. [S1]
———. 1968. Marque et manque: à propos de zéro. En Cahiers pour l’analyse, Vol. 10. [S2, S3]
———. 1999 (1988). El Ser y el acontecimiento. Trad. R. Cerdeiras et al. Buenos Aires: Manantial. Meditaciones 31, 33, 34, 35, 36. [S2]
Cohen, Paul. 2008 (1966). Set Theory and the Continuum Hypothesis. Mineola, NY: Dover. [S2]
Miller, Jacques-Alain. 1987 (1967). Acción de la estructura. En Matemas I. Buenos Aires: Manantial. (En francés: Action de la structure. En Cahiers pour l’analyse, Vol. 9.) [S2]
Girard, Jean-Yves. Proofs & Types. Trans. P. Taylor & Y. Lafont. Cambridge: Cambridge University Press, 1989. Vean especialmente Chapters 1-5. [S3]
———. Linear Logic, Theoretical Computer Science, London Mathematical 50:1, pp. 1-102, 1987. Restored by Pierre Boudes. [S3]
———. On the meaning of logical rules I: syntax vs. semantics, Computational Logic, eds Berger and Schwichtenberg, pp. 215-272, SV, Heidelberg, 1999. [S3]
———. Locus Solum, Mathematical Structures in Computer Science 11, pp. 301-506, 2001. (Vean la “Dictionary”, en particular.) [S3]
———. Le fantôme de la transparence, pour les 60 ans de Giuseppe Longo.
Identité, égalité, isomorphie ; ou ego, individu, espèce. D'après une exposé à la réunion LIGC opus 10, Firenze, villa Finaly, 18 Septembre 2009. [S3]
Identité, égalité, isomorphie ; ou ego, individu, espèce. D'après une exposé à la réunion LIGC opus 10, Firenze, villa Finaly, 18 Septembre 2009. [S3]
———. La syntaxe transcendantale, manifeste, Février 2011. [S3]
(Todos éstos textos de Girard son para examinar ligeramente. No se preocupen por los detalles muy difícils o técnicos. Están disponible a http://iml.univ-mrs.fr/~girard/Articles.html)
Joinet, Jean-Baptiste. 2009. ‘Introduction’ a J-B. Joinet y S. Tronçon (eds.), Ouvrir la logique au monde: Philosophie et mathématique de l’interaction. Paris: Hermann. [S3]
Thursday, April 21, 2011
Everything you needed to know about forcing, but were afraid to ask Alain Badiou
Hat tip to Fabio at Hypertiling and Tzuchien Tho for bringing this to my attention. Wish I could be there. If the reason you can't attend this magnificent, London-based workshop is that you'll be in Mexico City this May, stay tuned...
===================
A Set Theory Postgraduate Workshop for Readers of Being and Event or: Everything You Needed to Know about Forcing but were Afraid to Ask Alain Badiou
24 May, 31 May and 7 June 2011 in Room G03, 28 Russell Square
As part of its ongoing Seminar Series on Mathematics for the Humanities and Cultural Studies, the London Consortium will hold a series of three postgraduate students workshops that aims to discuss and provide a quick overview of some of the mathematics that needs to be known in order to follow the use of set theory
in Alain Badiou’s Being and Event [L'Être et l'Événement]. The main focus of these ‘bootcamps’ will be on introducing and discussing, in a friendly manner, the technicalities concerning the “mathematical bulwarks” mentioned in Badiou’s philosophical masterpiece:
(1) the Zermelo-Fraenkel Axioms of Set Theory
(2) the Theory of Ordinal and Cardinal Numbers
(3) Kurt Gödel and Paul Cohen’s work on Consistency and Independence.
Particular attention will be paid towards providing a sufficiently detailed, rigorous and clear explication of Paul Cohen’s technique of forcing and generic models, a mathematical result that Peter Hallward has called “the single most important postulate” in the whole of Being and Event. If time permits, we might also be touching on some recent mathematical developments in the field as well as trying to understand how Badiou contributes towards the contemporary re-intervention of mathematical thinking into philosophy.
The workshops will be held in room G03 at 28 Russell Square, Bloomsbury, London. The intended audience are postgraduate students or researchers who are interested in understanding Badiou’s philosophy but who lack the mathematical background. There are no assigned compulsory readings for the bootcamps,
and there are no pre-requisites save for some minimal familiarity with mathematics at the pre-university level. The sessions are free and open to the public but please register by sending your name, email and affiliation to forcing.badiou@gmx.com so as to give us an idea of the numbers since the classroom size, unfortunately, will be limited.
Workshop Convener: Burhanuddin Baki
Schedule and List of Topics
Session I (Tuesday, 24 May)
2-4pm – Basic Mathematics and Short Introduction to Forcing
Short Introduction to Badiou’s Being and Event; Basic Arithmetic, Abstract Algebra and First-Order Logic;
Idea of Mathematical Proof; Induction; Naïve Set Theory, Cantor’s Theorem and Russell’s Paradox
6-8pm – Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel Axioms plus the Axiom of Choice (ZFC); Peano Axioms of Arithmetic; Gödel’s
Incompleteness Theorems; Being and Event Parts I, II, IV and V; Some Alternative Axiomatizations of
Mathematics and Set Theory
Session II (Tuesday, 31 May)
2-4pm – Cardinal and Ordinal Numbers
Cardinal Numbers; Well-Orderings; Ordinal Numbers; Continuum Hypothesis; Being and Event Part III
and Meditation 26; Introduction to Surreal Numbers
6-8pm – Kurt Gödel and the Constructible Hierarchy
Some Relevant Model Theory; Formal Semantics of Situations; Formal Semantics of Events; Gödel’s
Completeness Theorem; Compactness Theorem; Löwenheim-Skolem Theorem and Paradox; Transfinite
Induction; Cumulative and Constructible Hierarchies; Axiom of Constructibility; Gödel’s Proof of
Consistency; Being and Event Meditation 29; Introduction to Large and Inaccessible Cardinals
Session III (Tuesday, 7 June)
2-4pm – Paul Cohen, Forcing and Generic Models
General Machinery of Forcing; Analogy between Forcing and Field Extensions; Cohen’s Proof of
Independence
6-8pm – Beyond Cohen’s Forcing
Cohen’s Proof of Independence (con’t); Being and Event Meditations 33, 34 & 36; Forcing in terms of
Kripkean Semantics; Forcing in terms of Boolean-valued models; Forcing Axioms and Generic
Absoluteness; Introduction to Categorial Logic and Topos Theory; Forcing in terms of Topos Theory;
Short Introduction to Forcing in Badiou’s The Logics of Worlds
Suggested Introductory Reading List
Avigad, J. (2004). “Forcing in Proof Theory”. http://www.andrew.cmu.edu/user/avigad/Papers/forcing.pdf.
Badiou, A. (2007). Being and Event. Translated by Oliver Feltham. London: Continuum.
Badiou, A. (2007). The Concept of Model. Translated by Zachary Fraser & Tzuchien Tho. Victoria: re:press.
Badiou, A. (2008). Number and Numbers. Translated by Robin Mackay: Cambridge: Polity.
Badiou, A. (2009). The Logics of Worlds. Translated by Alberto Toscano. London: Continuum.
Bowden, S. (2005). “Alain Badiou: From Ontology to Politics and Back”.
http://jffp.pitt.edu/ojs/index.php/jffp/article/view/244/238.
Chow, T. (2004). “Forcing for Dummies”. http://math.mit.edu/~tchow/mathstuff/forcingdum.
Chow, T. (2008). “A Beginner’s Guide to Forcing”. http://arxiv.org/abs/0712.1320.
Cohen, P. (2002). “The Discovery of Forcing”.
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.rmjm/1181070
010.
Cohen, P. (2008). Set Theory and the Continuum Hypothesis. New York: Dover.
Crossley, J., et. al. (1991). What is Mathematical Logic?. Mineola: Dover.
Devlin, K. (1991). The Joy of Sets: Fundamentals of Contemporary Set Theory. New York: Springer.
Doxiadis, A., et. al. (2009). Logicomix: An Epic Search for Truth. Bloomsbury Publishing PLC.
Easwaran, K. (2007). ” A Cheerful Introduction to Forcing and the Continuum Hypothesis”.
http://arxiv.org/abs/0712.2279.
Fraser, Z. (2006). “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of
Forcing and the Heyting Calculus”. http://www.cosmosandhistory.org/index.php/journal/article/view/30.
Goldblatt, R. (2006). Topoi: The Categorial Analysis of Logic. Mineola: Dover.
Hallward, P. (2003). Badiou: A Subject to Truth. Minneapolis: U. of Minnesota Press.
Halmos, P. (1974). Naive Set Theory. New York: Springer.
Jech, T. (2004). Set Theory: The Third Millennium Edition, Revised and Expanded. New York: Springer-Verlag.
Jech, T. (2008). “What is Forcing?”. http://www.ams.org/notices/200806/tx080600692p.pdf.
Kanamori, A. “Cohen and Set Theory”. Bull. Symbolic Logic 13(3) (2008), 351-78.
Kunen, K. (1992). Set Theory: An Introduction to Independence Proofs. New York: North Holland.
Norris, C. (2009). Badiou’s Being and Event. London: Continuum.
Pluth, E. (2010). Badiou: A Philosophy of the New. Cambridge: Polity Press.
Potter, M. (2004). Set Theory and its Philosophy. Oxford: Oxford U. Press.
Smullyan, R. & M. Fitting. (2010). Set Theory and the Continuum Problem. Mineola: Dover.
Tiles, M. (1989). The Philosophy of Set Theory. Mineola: Dover.
The London Consortium is a multi-disciplinary graduate programme in Humanities and Cultural Studies. We are a collaboration between five of London’s most dynamic cultural and educational institutions: the Architectural Association, Birkbeck College (University of London), the Institute of Contemporary Arts, the Science Museum, and Tate.
===================
A Set Theory Postgraduate Workshop for Readers of Being and Event or: Everything You Needed to Know about Forcing but were Afraid to Ask Alain Badiou
24 May, 31 May and 7 June 2011 in Room G03, 28 Russell Square
As part of its ongoing Seminar Series on Mathematics for the Humanities and Cultural Studies, the London Consortium will hold a series of three postgraduate students workshops that aims to discuss and provide a quick overview of some of the mathematics that needs to be known in order to follow the use of set theory
in Alain Badiou’s Being and Event [L'Être et l'Événement]. The main focus of these ‘bootcamps’ will be on introducing and discussing, in a friendly manner, the technicalities concerning the “mathematical bulwarks” mentioned in Badiou’s philosophical masterpiece:
(1) the Zermelo-Fraenkel Axioms of Set Theory
(2) the Theory of Ordinal and Cardinal Numbers
(3) Kurt Gödel and Paul Cohen’s work on Consistency and Independence.
Particular attention will be paid towards providing a sufficiently detailed, rigorous and clear explication of Paul Cohen’s technique of forcing and generic models, a mathematical result that Peter Hallward has called “the single most important postulate” in the whole of Being and Event. If time permits, we might also be touching on some recent mathematical developments in the field as well as trying to understand how Badiou contributes towards the contemporary re-intervention of mathematical thinking into philosophy.
The workshops will be held in room G03 at 28 Russell Square, Bloomsbury, London. The intended audience are postgraduate students or researchers who are interested in understanding Badiou’s philosophy but who lack the mathematical background. There are no assigned compulsory readings for the bootcamps,
and there are no pre-requisites save for some minimal familiarity with mathematics at the pre-university level. The sessions are free and open to the public but please register by sending your name, email and affiliation to forcing.badiou@gmx.com so as to give us an idea of the numbers since the classroom size, unfortunately, will be limited.
Workshop Convener: Burhanuddin Baki
Schedule and List of Topics
Session I (Tuesday, 24 May)
2-4pm – Basic Mathematics and Short Introduction to Forcing
Short Introduction to Badiou’s Being and Event; Basic Arithmetic, Abstract Algebra and First-Order Logic;
Idea of Mathematical Proof; Induction; Naïve Set Theory, Cantor’s Theorem and Russell’s Paradox
6-8pm – Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel Axioms plus the Axiom of Choice (ZFC); Peano Axioms of Arithmetic; Gödel’s
Incompleteness Theorems; Being and Event Parts I, II, IV and V; Some Alternative Axiomatizations of
Mathematics and Set Theory
Session II (Tuesday, 31 May)
2-4pm – Cardinal and Ordinal Numbers
Cardinal Numbers; Well-Orderings; Ordinal Numbers; Continuum Hypothesis; Being and Event Part III
and Meditation 26; Introduction to Surreal Numbers
6-8pm – Kurt Gödel and the Constructible Hierarchy
Some Relevant Model Theory; Formal Semantics of Situations; Formal Semantics of Events; Gödel’s
Completeness Theorem; Compactness Theorem; Löwenheim-Skolem Theorem and Paradox; Transfinite
Induction; Cumulative and Constructible Hierarchies; Axiom of Constructibility; Gödel’s Proof of
Consistency; Being and Event Meditation 29; Introduction to Large and Inaccessible Cardinals
Session III (Tuesday, 7 June)
2-4pm – Paul Cohen, Forcing and Generic Models
General Machinery of Forcing; Analogy between Forcing and Field Extensions; Cohen’s Proof of
Independence
6-8pm – Beyond Cohen’s Forcing
Cohen’s Proof of Independence (con’t); Being and Event Meditations 33, 34 & 36; Forcing in terms of
Kripkean Semantics; Forcing in terms of Boolean-valued models; Forcing Axioms and Generic
Absoluteness; Introduction to Categorial Logic and Topos Theory; Forcing in terms of Topos Theory;
Short Introduction to Forcing in Badiou’s The Logics of Worlds
Suggested Introductory Reading List
Avigad, J. (2004). “Forcing in Proof Theory”. http://www.andrew.cmu.edu/user/avigad/Papers/forcing.pdf.
Badiou, A. (2007). Being and Event. Translated by Oliver Feltham. London: Continuum.
Badiou, A. (2007). The Concept of Model. Translated by Zachary Fraser & Tzuchien Tho. Victoria: re:press.
Badiou, A. (2008). Number and Numbers. Translated by Robin Mackay: Cambridge: Polity.
Badiou, A. (2009). The Logics of Worlds. Translated by Alberto Toscano. London: Continuum.
Bowden, S. (2005). “Alain Badiou: From Ontology to Politics and Back”.
http://jffp.pitt.edu/ojs/index.php/jffp/article/view/244/238.
Chow, T. (2004). “Forcing for Dummies”. http://math.mit.edu/~tchow/mathstuff/forcingdum.
Chow, T. (2008). “A Beginner’s Guide to Forcing”. http://arxiv.org/abs/0712.1320.
Cohen, P. (2002). “The Discovery of Forcing”.
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.rmjm/1181070
010.
Cohen, P. (2008). Set Theory and the Continuum Hypothesis. New York: Dover.
Crossley, J., et. al. (1991). What is Mathematical Logic?. Mineola: Dover.
Devlin, K. (1991). The Joy of Sets: Fundamentals of Contemporary Set Theory. New York: Springer.
Doxiadis, A., et. al. (2009). Logicomix: An Epic Search for Truth. Bloomsbury Publishing PLC.
Easwaran, K. (2007). ” A Cheerful Introduction to Forcing and the Continuum Hypothesis”.
http://arxiv.org/abs/0712.2279.
Fraser, Z. (2006). “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of
Forcing and the Heyting Calculus”. http://www.cosmosandhistory.org/index.php/journal/article/view/30.
Goldblatt, R. (2006). Topoi: The Categorial Analysis of Logic. Mineola: Dover.
Hallward, P. (2003). Badiou: A Subject to Truth. Minneapolis: U. of Minnesota Press.
Halmos, P. (1974). Naive Set Theory. New York: Springer.
Jech, T. (2004). Set Theory: The Third Millennium Edition, Revised and Expanded. New York: Springer-Verlag.
Jech, T. (2008). “What is Forcing?”. http://www.ams.org/notices/200806/tx080600692p.pdf.
Kanamori, A. “Cohen and Set Theory”. Bull. Symbolic Logic 13(3) (2008), 351-78.
Kunen, K. (1992). Set Theory: An Introduction to Independence Proofs. New York: North Holland.
Norris, C. (2009). Badiou’s Being and Event. London: Continuum.
Pluth, E. (2010). Badiou: A Philosophy of the New. Cambridge: Polity Press.
Potter, M. (2004). Set Theory and its Philosophy. Oxford: Oxford U. Press.
Smullyan, R. & M. Fitting. (2010). Set Theory and the Continuum Problem. Mineola: Dover.
Tiles, M. (1989). The Philosophy of Set Theory. Mineola: Dover.
The London Consortium is a multi-disciplinary graduate programme in Humanities and Cultural Studies. We are a collaboration between five of London’s most dynamic cultural and educational institutions: the Architectural Association, Birkbeck College (University of London), the Institute of Contemporary Arts, the Science Museum, and Tate.
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