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 – is actually adequate. Let’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 – if possible – 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 – I am convinced – it is one way to better locate flaws, when we at the same time keep in mind that this is an experiment – and that our initial assumption can arguable turn out to be wrong.
The following – very limited – examination deals with one aspect of “Being and Event” (BE), a major work of Alain Badiou, namely the “Matheme of the Event”. Badiou uses a set-theoretical framework in order to analyze how it is possible that a situation – shaped by structures, rules, habits, stabilized knowledge – gets disturbed such that novelties – new perspectives – emerge that were not thinkable within the situation before. The flash that disturbs the situation is called the event.
The context and motivation of this post are (1) recent, unsatisfying critiques and repliques about the status of mathematics in BE – published in Critical Inquiry and (2) the collection of Badiou-related postings in this blog:
- Ricardo and David Nirenberg: “Badiou’s Number: A Critique of Mathematics as Ontology“. Critical Inquiry, Vol. 37 (Summer 2011), pp. 583-614
- Alain Badiou: “To Preface the Response to the ‘Criticisms’ of Ricardo Nirenberg and David Nirenberg“. Critical Inquiry, Vol. 38, No. 2. (January 2012), pp. 362-364
- A. J. Bartlett and Justin Clemens: “Neither Nor“. Critical Inquiry, Vol. 38, No. 2. (January 2012), pp. 365-380
- Richardo and David Nirenberg: “Reply to Badiou, Bartlett, and Clemens“
Badiou introduces the „Matheme of the Event“ in Meditation 17:
ex = {x ∈ X, ex}
The event of the site X ( X ∈ S, where S is a situation) is composed of the elements of X and …. 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?
ad a.) An evental site X is defined in Meditation 16 as „entirely abnormal multiple; that is, a multiple such that none of its elements are presented in the situation. The site, itself, is presented, but ‘beneath’ it nothing from which it is composed is presented.“ (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).
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?
- If yes, then according to the property that A expresses, A does not contain itself. But in fact, A contains itself.
- If no, A should contain itself. But in fact, A does not contain itself.
To come back to the formula: ex is not a well-founded set. In one of the most common axiomatizations of set theory – ZF – the axiom of foundation (aka: axiom of regularity) is 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:
„Rather than being defined in terms of objects previously defined, ex is 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—not least because it produces an unmathematical mise-en-abîme: if we replace ex inside the bracket by its expression as a bracket, we can go on doing this forever—and so can hardly be called “a matheme.” (Nirenberg, p. 598f)
That’s how the Nirenbergs end the examination about the matheme of the event. So, let’s throw BE in the bin? Wait a moment. We are still undertaking an experiment with the assumption that Badiou’s text is adequate. Should we already give up here? No, that’s too easy. In 2008, after the translation of BE into English, Paul Livingston wrote a review of „Being and Event“, where he discusses the same problem. Actually, he has another idea:
„As further set-theoretical reflection has shown, however, the axiom of foundation, though the most direct way to avoid Russell’s paradox, is not strictly necessary for the logical coherence of an axiomatization of the nature of sets; various versions of ‘‘non-well founded’’ 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 ‘‘bottom out’’ in a simplest or most basic element. And this infinite multiplicity is indeed essential, on Badiou’s 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’s capability to go beyond ontology in introducing happening into the intrinsically non- evental order of being. “ (Livingston, p.225)
If Livingston is right, then ex is not necessarily un-mathematical, just un-conventional. (Do the Nirenbergs maybe are plagued by the same issue that they assign to Badiou: to base their arguments on a conventional mathematical basis and ignore the alternatives? But that’s not our business here). It seems that non-well-founded sets can appear in mathematics – not in most common axiomatization, but in non-well-founded-set theory (which probably do not play a central role in every-day mathematical working).
Anyway, this finding gives us enough patience to read further. It fortunately turns out that Badiou is aware that this matheme of the event does not satisfy the axiom of foundation:
„With 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. […] 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 ∈ a […] Sets which belong to themselves were baptized extraordinary sets by the logician Mirimanoff. We could thus say the following: an event is ontologically formalized by an extraordinary set. We 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.[…] Ontology has nothing to say about the event.“ (BE, Med 18, p. 184, 189f.)
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 “matheme of the event” 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?
Here are some remarks triggered by Andy’s careful exposition.
(1) He invokes the “principle of charity”. In reading strange-sounding text you should aim at maximum plausibility on part of the author, otherwise you will “ruin the game”, even if you want to critizes the text in the end. This is fine, but there is a caveat. Assuming that the text “is actually adequate” (ak) does not tell you adequate to what. In particular: adequate to set theory? Which set theory? There is a serious danger of a reader faking to understand a text just by nodding and disregarding her own horizon of understanding.
(2) Mathemes. There is no way for mathematicians to accept mathemes a la Lacan or Badiou into mathematics. A “historical situation” is simply no topic for a formalized deductive system. (Which, of course, is Badiou’s point.) A certain syntax is used to present content that is not usually expressed this way. Think of a traffic light in comparison to passer-by who directs traffic at an intersection in case of an emergency. The laywoman’s gestures are a direct expression of her intentions, whereas the lights are an artificial syntax imposed upon the situation. They can express “stop” by a certain convention, but this is not the meaning of a red light per se. The lights can be made to express concepts of traffic regulations, but they have no immediate relationship to ongoing traffic.
(3) “with the assumption that Badiou’s text is adequate”. Again: to standard or non-standard mathematics? One can, of course, be quite pragmatic about this, relativising the criterion. But there is a deeper issue. One can discuss extraordinary sets within mathematics and accept or reject them for professional use. But if mathemes are extra-mathematical exercises to capture human experiences by means of mathematical syntax, another problem arises. We should probably not let competing mathematical theories decide on our notions of entirely non-mathematical phenomena (The Nirenberg’s point.) Some essential functions of a bridge can be expressed in elegant, “adequate” mathematical formulae. Should one therefore try to put their symbolic appeal, i.e. setting up a connection between different countries, into a formulaic expression?
One of Herberts points is: When you say “text X is adequate”, it is incomplete to say “X is adequate (at all)”, you have to specify the matter, for which the text is adequate. Otherwise, no reasonable comparison/judgement is possible. In my post, I have indicated that e_x is adequate in non-well-founded set theory and that the proposition that such a “kind of set never appears in mathematics” is not an adequate judgement of e_x, if non-well-founded set theory belongs to mathematics. I admit, that this is not the most important point in the argumentation of the Nirenbergs and does not cover the whole problem of Badious mathemes entirely, e.g. are axiomatic systems like ZF adequate to summarize situations in our life?
Moreover, the answer to the question “What Badiou is talking about (in general)?” is not obvious. What is the corresponding matter of subject? Depending on the meditation, it’s a mixture and sometimes an overlay of mathematics, traditional ontological discourse and situations of our Lebenswelt or interesting dynamics in history.
I also have to admit, that my understanding of mathemes in the posting (“something that can be accepted amongst mathematicians”) is too flat for Badiou’s mathems. But moreover, Jaques Lacan’s mathems are different from Badious. Therefore it might be useful to be more precise on what matheme means in these contexts. A few thoughts on that:
For Lacan, I looked it up – a Matheme is basically a didactic tool, a “formulae, designed as symbolic representations of [Lacans] ideas and analyses. They were intended to introduce some degree of technical rigour in philosophical and psychological writing, replacing the often hard-to-understand verbal descriptions with formulae resembling those used in the hard sciences, and as an easy way to hold, remember, and rehearse some of the core ideas of both Freud and Lacan.” (Wikipedia: http://en.wikipedia.org/wiki/Matheme)
I think of what software engineers do when creating UML diagrams in dialog with a customer. UML diagrams are often used as an intermediate step between a.) natural language descriptions of customers and b.) final programs. It is worth to mention that such UML diagrams suit two purposes:
(1) UML diagrams make implementation easier, because they are defined with a formalized syntax that is less ambigious than some natural language descriptions. They simplify communication amongst experts and allow for task separating in bigger projects. (2) Moreover, they allow laywoman to sharpen their thought and to refine their wishes/specifications, although they do not know the formalized syntax behind these diagrams. When a customer sees boxes connected via arrows plus (!) when a software engineer guides the customer by asking questions, she intuitively refines her specifications based on the appearance of the diagram and the guidance of the software engineer.
In this kind of data modelling we have already a janus-like role of mathemes in form of UML diagrams. Two kinds of usage are superposed. On the one hand it enables rigorous and exact implementation. On the other hand the customer in the worst case knows nothing about the formalism behind the diagram, she just thinks along the surface, the appearance – and the guidance of the engineer helps her not to get it too wrong. Given the guidance of the software engineer this is fruitful because the programming langue requires precise specifications that are more close to mathematical formalism than to natural language. In this case at least, the janus-like role makes sense, because in the end software engineers do not need to explain their software (to everybody), it’s main purpose is automation and assistance (this two-level-understanding on software causes problems on it’s own that would lead me too far).
And now a remark about the matheme of the event (and probably similar for other mathemes in BE 1), which is also janus-like but in another sense:
* On the one hand, the mathem of the event can be seen as a didactic tool like Lacans mathemes. It is practical to have a compact, catchy, summary of what is meant by events like “The French Revolution”. One could endlessly number and collect moments, opinions, movements that belong to The French Revolution, but at some point, there is the need to have a “immanent résumé” available that indicates the need to escape the enumeration of facts to understand what has happened. And so, the description of ‘The French Revolution’ includes also ‘The French Revolution’, i.e., that what should be described.
* But when we take Badiou serious, then what one can say about an event is primarily expressed in e_x and not in an ‘image’ of “The French Revolution”, that Badiou gives only as “empirical evidence” (BE, p.181). So when we ignore this image, we are left with a formulae that describes a non-well-founded set.
* These two sides are related, but the relation is not mathematical, it is not an “immediate relation” of mathematics. But every comparison has its borders. Not everything (and even not all important things) that can be said of e_x within mathematics can be applied to The French Revolution. On the other hand, not everything that is important to grasp The French Revolution can be expressed in e_x.
To the point: “We should probably not let competing mathematical theories decide on our notions of entirely non-mathematical phenomena”. Every phenomenon is non-mathematical. And mathematical objects do not appear. At the same time – for certain traditions – being (as being) does not appear. When there is something that can be said about being at all, it is probably mathematics. So there is a undeniable tension between the ontological discourse and how it relates to our phenomena. If one accepts this framing, I do not see that the tension can ever disappear.
adequate
Andreas is right: it was quite clear that he used “adequate” relativized to two kinds of set theory. But this does not remove the underlying problem. It is only in the process of trying to interpret a given text that the options of horizons of understanding open up. It is no help to just be told “relative to so and so”. There is a decision involved, namely that the text makes most sense viewed in a particular light. And this is generally not a naive move. (But I, of course, accept the principle of charity.)
UML
Reference to UML is really helpful. I think that this example explains the motives behind mathemes nicely and that Andreas’ description about the semi-formal character of the UML notation is to the point. Still, on reflexion I have a difficulty with this comparison. One may derive actual source code from an UML diagram, precisely because it “masks” a conventionally determined use of all of its components. UML components are standardized and accepted by the community to behave in certain ways. The “scandal” of Lacan’s and Badiou’s mathemes might be illustrated by imagining that someone uses UML diagrams in an idiosyncratic way.
French Revolution
It is indeed the case that there is a circularity in concept use. In order to call something a supermarket I have to have knowledge of certain buildings and of what they are implementations of, namely the concept SUPERMARKET. This is widely discussed in Analytic Philosophy, which Badiou rejects. I very much doubt that putting this circularity into a set theoretical reconstruction of “situations” and “events” is the most helpful procedure here. (I can back this up with a paper on Davidson and Wittgenstein.)
phenomena
In the most general sense mathematics is a phenomenon like everything else. Like the supermarket above you have instances and a concept of, e.g. an addition. Now, if one holds that Being does not appear and Mathematics is ontology I confess to not quite knowing which way to approach such claims as adequate 🙂
Let me start my reaction to this discussion by confirming that an appeal to the principle of charity in face of such a challenging, provocative and controversial text like Badiou’s B&E is an adequate and useful hermeneutic strategy. Moreover, Badiou’s “matheme of the event” e_x = {x e in X, e_x} in Meditation 17 is indeed a neuralgic and paradigmatic point in his endeavor to demonstrate that (the language as well as substance matter of) mathematics provides the key to understanding “being” and “event”. It is neuralgic since it relies on a strategy that Badiou invokes on several places: he appeals to the equation ontology = mathematics. (If I understand Badiou correctly, as I honestly strive to do, “Ontology” should be read here as something like “what there is to say about all that exists, if we completely abstract away from what exists in particular”.) But actually (without discussing this explicitly) Badiou replaces mathematics by ZFC in this equation. This move allows him to claim (among other central thesis) that “ontology has nothing to say about the event”. I want to emphasize that statements like the just cited one indeed crucially depend on equating ontology with ZFC and not with mathematics in general – or even with set theory in general (more on that below). I take this to be an important observation in the context of the Nirenberg’s main point of criticism, which they call “Pythagoric snare”: drawing conclusions from a particular interpretation of axioms and selling them as consequences of these axioms. However, note that I don’t want to simply confirm the Nirenbergs’ objection here. (Although I plead to read them with MUCH more charity as Badiou and his Australian lieutenants are willing or able to do. I think we all agree on that point.) In the Nirenbergs’ understanding of the “Pythagoric snare” the “axioms” are simply the axioms of ZFC and the problem is that Badiou insists on a certain – contingent – interpretation of those axioms. In contrast, on my reading the crucial “axiom” here is “ontology = mathematics” and the snare consists in confusing mathematics with ZFC, i.e., with a particular version of set theory.
Let me try to be a bit more explicit about the issue of the “e_x-formula” (bombastically called “matheme of the event” by Badiou – but let’s not be polemic, even if admittedly I often have to work hard to retain charity in face of Badiou’s language and attitude regarding other philosophies and philosophers). Of course, Badiou knows very well that the formula is simply meaningless within ZFC. He is fully aware of the fact that the expression e_x = { …, e_x} cannot denote a set since it violates the axiom of foundation. (I don’t want to be nitpicking about Badiou’s notation at this point, which is not respecting mathematical conventions. However that’s rather inessential.) Even more: in order to draw the indicated conclusions about the status of an “event” he has to INSIST that it is meaningless within ZFC itself. However he also has to insist that the expression carries meaning nevertheless – as indeed it does (disregarding minor notational problems) in a more general set theoretic setting. In particular, sets containing themselves as elements are perfectly clear concepts within versions of set theory that drop the axiom of foundation from the ZF(C) list of axioms and replace it by some “anti-foundation axiom” (as indicated by Livingston). In fact, highly respected mathematicians, like Peter Aczel, Dana Scott, and Maurice Boffa, among many others, have developed such set theories. Moreover at least Aczel’s and Scott’s theories turned out to be very useful mathematical tools in different application scenarios. One may (with some historic justification call the theory of non-well founded sets “non-standard” or “non-convential” set theory. However, it is untenable to call it “non-standard mathematics” (or “non-convential mathematics”). This were like calling, say, non-abelian geometry “non-standard mathematics” (completely disregarding its clear connections to mainstream mathematical research and its applications to contemporary physics); or – to draw an analogy beyond mathematics – to call Robert Brandom’s philosophy non-standard, since it re-interprets and augments important tenets of analytic as well as continental philosophy.
Why do I think that this reminder on developments in (ordinary!) contemporary mathematics to be relevant in connection with Badiou? Because (currently) my only successful (and I claim: charitable) attempt to understand and appreciate the crucial assertions in Meditations 17 and 18 is to read Badiou’s use of “ontology” as “standard model of ZFC” and thus to reject his much more exiting and claim that the essence of mathematics itself or at least of different kinds of mathematical activities that mathematicians would recognize as such are the key to what we can or should call “ontology”. The implications of identifying (merely) ZFC (or rather, even more narrowly, a certain model of ZFC – as correctly pointed out by the Nirenberg) with ontology remain open to discussion. Certainly Badiou has a lot offer on this (“corrected”) premise. However the premise itself is neither explicitly discussed by Badiou nor is it easy to swallow from a mathematical point of view. (Nor do I see what it does to French revolution…)