{"id":1058,"date":"2012-02-22T22:39:51","date_gmt":"2012-02-22T22:39:51","guid":{"rendered":"http:\/\/phaidon.philo.at\/qu\/?p=1058"},"modified":"2018-11-18T10:50:25","modified_gmt":"2018-11-18T10:50:25","slug":"the-matheme-of-the-event","status":"publish","type":"post","link":"https:\/\/quatsch.philo.at\/?p=1058","title":{"rendered":"The Matheme of the Event"},"content":{"rendered":"<p><em>I am a fan of the (probably hermeneutic) idea that if one aims to write an adequate critics\/response of a text, one needs to initially assume that the text, this crude and crazy production of thought \u2013 is actually adequate. Let&#8217;s undertake this apparently naive experiment, that everything written in the text in front of us is correct and if we work hard enough we can make sense out of it. The question then is to observ, pronounce and \u2013 if possible \u2013 overcome the resistances that our understanding produces when we read the text. With such an assumption we get the chance to describe arguments in the text in a way that makes it more accessible to others (who may have made similar reading-experiences). But secondly &#8211; I am convinced &#8211; it is one way to better locate flaws, when we at the same time keep in mind that this is an experiment \u2013 and that our initial assumption can arguable turn out to be wrong.<\/em><\/p>\n<p><a href=\"http:\/\/phaidon.philo.at\/qu\/wp-content\/uploads\/2012\/02\/2007-08-24-contain-yourself.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-1060\" src=\"http:\/\/phaidon.philo.at\/qu\/wp-content\/uploads\/2012\/02\/2007-08-24-contain-yourself-300x221.jpg\" alt=\"\" width=\"300\" height=\"221\" \/><\/a><\/p>\n<p>The following &#8211; very limited &#8211; examination deals with one aspect of &#8220;Being and Event&#8221; (BE), a major work of Alain Badiou, namely the &#8220;Matheme of the Event&#8221;. Badiou uses a set-theoretical framework in order to analyze how it is possible that a situation &#8211; shaped by structures, rules, habits, stabilized knowledge &#8211; gets disturbed such that novelties &#8211; new perspectives &#8211; emerge that were not thinkable within the situation before. The flash that disturbs the situation is called the event.<\/p>\n<p>The context and motivation of this post are (1) recent, unsatisfying\u00a0critiques and repliques about the status of mathematics in BE &#8211; published in Critical Inquiry and (2) the collection of <a href=\"http:\/\/phaidon.philo.at\/qu\/?s=Badiou\">Badiou-related postings<\/a>\u00a0in this blog:<\/p>\n<ul>\n<li>Ricardo and David Nirenberg: &#8220;<a href=\"http:\/\/criticalinquiry.uchicago.edu\/uploads\/pdf\/nirenbergs_badiousnumber_complete.pdf\">Badiou&#8217;s Number: A Critique of Mathematics as Ontology<\/a>&#8220;.\u00a0<em>Critical Inquiry<\/em>, Vol. 37\u00a0(Summer 2011), pp. 583-614<\/li>\n<li>Alain Badiou: &#8220;<a href=\"http:\/\/pastehtml.com\/view\/bo1zmm4o3.html\">To Preface the Response to the \u2018Criticisms\u2019 of Ricardo Nirenberg and David Nirenberg<\/a>&#8220;.\u00a0<em>Critical Inquiry<\/em>, Vol. 38, No. 2. (January 2012), pp. 362-364<\/li>\n<li>A. J. Bartlett and Justin Clemens: &#8220;<a href=\"http:\/\/pastehtml.com\/view\/bo20snbz7.html\">Neither Nor<\/a>&#8220;.\u00a0<em>Critical Inquiry<\/em>, Vol. 38, No. 2. (January 2012), pp. 365-380<\/li>\n<li>Richardo and David Nirenberg: &#8220;<a href=\"http:\/\/pastehtml.com\/view\/bo219i443.html\">Reply to Badiou, Bartlett, and Clemens<\/a>&#8220;<\/li>\n<\/ul>\n<p><!--more--><\/p>\n<p>Badiou introduces the \u201eMatheme of the Event\u201c in Meditation 17:<\/p>\n<blockquote><p><strong>e<sub>x\u00a0<\/sub>= {x \u2208\u00a0X,\u00a0e<sub>x<\/sub>}<\/strong><\/p><\/blockquote>\n<p>The event of the site X ( X \u2208\u00a0S, where S is a situation) is composed of the elements of X and &#8230;. the event itself. Two questions arise: First, a.) what is a site? And more crucial, b.) is the formula above indeed a matheme, i.e., something that can be accepted amongst mathematicians?<\/p>\n<p>ad a.) An evental site X is defined in Meditation 16 as \u201eentirely abnormal multiple; that is, a multiple such that none of its elements are presented in the situation. The site, itself, is presented, but &#8216;beneath&#8217; it nothing from which it is composed is presented.\u201c (BE, p.175) Whenever a situation contains at least one site, it is called historical situation. Otherwise, neutral or natural situation. The event however, does not affect the whole historical situation, only the site(s).<\/p>\n<p>ad b.) As we know from Russels paradox, set theory has to avoid sets that contain itself, otherwise there is the possibility to create paradoxes, e.g. the set A that contains all sets that do not contain itself. Does A contain itself?<\/p>\n<ul>\n<li>If yes, then according to the property that A expresses, A does not contain itself. But in fact, A contains itself.<\/li>\n<li>If no, A should contain itself. But in fact, A does not contain itself.<\/li>\n<\/ul>\n<p>To come back to the formula: e<sub>x<\/sub>\u00a0is not a well-founded set. In one of the most common axiomatizations of set theory &#8211; ZF &#8211; the <a href=\"http:\/\/mathworld.wolfram.com\/AxiomofFoundation.html\">axiom of foundation<\/a>\u00a0(aka: axiom of regularity)\u00a0is installed to avoid such sets. Badiou based his ontology on ZF. And at a crucial point of the argumentation he violates one axiom of ZF. How to respond to this cheek? Ricardo and David Nirenberg phrase it that way:<\/p>\n<blockquote><p>\u201eRather than being defined in terms of objects previously defined, e<sub>x<\/sub>\u00a0is here defined in terms of itself; you must already have it in order to define it. Set theorists call this a not-well-founded set. This kind of set never appears in mathematics\u2014not least because it produces an unmathematical mise-en-ab\u00eeme: if we replace e<sub>x<\/sub>\u00a0inside the bracket by its expression as a bracket, we can go on doing this forever\u2014and so can hardly be called \u201ca matheme.\u201d (Nirenberg, p. 598f)<\/p><\/blockquote>\n<p>That&#8217;s how the Nirenbergs end the examination about the matheme of the event. So, let&#8217;s throw BE in the bin? Wait a moment. We are still undertaking an experiment with the assumption that Badiou&#8217;s text is adequate. Should we already give up here? No, that&#8217;s too easy. In 2008, after the translation of BE into English, Paul Livingston wrote a <a href=\"http:\/\/philpapers.org\/rec\/LIVROB\">review of \u201eBeing and Event\u201c<\/a>, where he discusses the same problem. Actually, he has\u00a0another idea:<\/p>\n<blockquote><p>\u201eAs further set-theoretical reflection has shown, however, the axiom of foundation, though the most direct way to avoid Russell\u2019s paradox, is not strictly necessary for the logical coherence of an axiomatization of the nature of sets; various versions of \u2018\u2018non-well founded\u2019\u2019 set theory take up the consequences of its suspension. Most directly, suspending the axiom of foundation means that sets can be, as Badiou suggests they inherently are, infinite multiplicities that never \u2018\u2018bottom out\u2019\u2019 in a simplest or most basic element. And this infinite multiplicity is indeed essential, on Badiou\u2019s accounting, to the potentiality of the event to produce novelty. The schema that portrays this infinite potentiality breaks with the axiom of foundation by explicitly asserting the self-membership of the event. For Badiou, however, this is not the basis of a rejection of the axiom itself as a fundamental claim of ontology, but rather an index of the event\u2019s capability to go beyond ontology in introducing happening into the intrinsically non- evental order of being. \u201c (Livingston, p.225)<\/p><\/blockquote>\n<p>If Livingston is right, then e<sub>x<\/sub>\u00a0is not\u00a0<em>necessarily<\/em>\u00a0un-mathematical, just un-conventional. (Do the Nirenbergs maybe are\u00a0plagued\u00a0by\u00a0the same issue that they assign to Badiou: to base their arguments on a conventional mathematical basis and ignore the alternatives? But that&#8217;s not our business here). It seems that non-well-founded sets can appear in mathematics &#8211; not in most common axiomatization, but in <a href=\"http:\/\/en.wikipedia.org\/wiki\/Non-well-founded_set_theory\">non-well-founded-set theory<\/a>\u00a0(which probably do not play a central role in every-day mathematical working).<\/p>\n<p>Anyway, this finding gives us enough patience to read further.\u00a0It fortunately turns out that Badiou is aware that this matheme of the event does not satisfy the axiom of foundation:<\/p>\n<blockquote><p>\u201eWith the event we have the first concept external to the field of mathematical ontology. Here, as always, ontology decides by means of a special axiom, the axiom of foundation. [&#8230;] If there existed an ontological formalization of the event it would therefore be necessary, within the framework of set theory, to allow the existence, which is to say the count-as-one, of a set such that it belonged to itself: a \u2208\u00a0a [&#8230;] Sets which belong to themselves were baptized <em>extraordinary<\/em> sets by the logician Mirimanoff. We could thus say the following: an event is ontologically formalized by an extraordinary set.\u00a0We could. But the axiom of foundation forecloses extraordinary sets from any existence, and ruins any possibility of naming a multiple-being of the event. Here we have an essential gesture: that by means of which ontology declares that the event is not.[&#8230;] Ontology has nothing to say about the event.\u201c (BE, Med 18, p. 184, 189f.)<\/p><\/blockquote>\n<p>An ontology that does not cover all important concepts of reality? That is irritating and interesting (maybe itself an event), but a problem for another posting. Experiment paused. Interim report: The &#8220;matheme of the event&#8221; can possibly be accepted as an exotic formula that can appear in mathematics, but is not allowed in ZF, which is the basis for Badious ontology. What remains open after this examination is: Why is Badiou not choosing a non-well-founded set theory to embed the event into ontology? Or is it not a matter of choosing?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I am a fan of the (probably hermeneutic) idea that if one aims to write an adequate critics\/response of a text, one needs to initially assume that the text, this crude and crazy production of thought \u2013 is actually adequate. Let&#8217;s undertake this apparently naive experiment, that everything written in the text in front of<a class=\"more-link\" href=\"https:\/\/quatsch.philo.at\/?p=1058\">Read more<\/a><\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[34,74,94,159],"class_list":["post-1058","post","type-post","status-publish","format-standard","hentry","category-philosophie-klassisch","tag-badiou","tag-event","tag-hermeneutics","tag-set-theory"],"_links":{"self":[{"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=\/wp\/v2\/posts\/1058","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1058"}],"version-history":[{"count":1,"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=\/wp\/v2\/posts\/1058\/revisions"}],"predecessor-version":[{"id":2272,"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=\/wp\/v2\/posts\/1058\/revisions\/2272"}],"wp:attachment":[{"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1058"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1058"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/quatsch.philo.at\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}