Mathematics as the Non-Object?

The existence axiom at the basis of ZF set theory postulates the existence of the empty set, the set without elements. But being without elements, thus also without parts, the empty set qua mathematical cannot be an object according to OOO (Object-Oriented Ontology), since an object is defined on the basis of specific relations (emergence, submergence) between the whole and its parts. It cannot even be a pseudo-object, since a pseudo-object is also defined on the basis of specific relations (lack of emergence, etc) between the whole and its parts. (See this post for a summary of Graham Harman’s views on objects and pseudo-objects).

Insofar as one can construct basically the whole of mathematics out of ZF set theory, it might seem as if mathematical structures in general are neither objects nor pseudo-objects. So what are they? What is mathematics, according to OOO?

In itself, classifying mathematics as non-objective need not be a problem. For instance, the point that mathematics is not “objective” and has no objects is central to Badiou’s account of mathematics. In fact, it is a precondition for his central claim that mathematics is ontology. Mathematical structures are not objects, they are the being of objects. For object-oriented ontology, on the other hand, it must mean that mathematics has no place within ontology. Of course, there are many ontologies which do not include mathematics and mathematical objects. But object-oriented ontology includes societies, ideas, phonemes, organizations, and so much else (not by coincidence can it be referred to as promiscuous ontology), so it is fairly surprising that it does not seem to include mathematics!

I do not mean, of course, that it does not include mathematics at all. It obviously does. Mathematics qua signs, for instance the name of the empty set, ø, is presumably an object. So is mathematics qua scientific practice, mathematics qua mental and physical ideas and constructions, and so on. But mathematics qua mathematics is, I suspect, not. If this is correct, then OOO is reductionistic with respect to mathematics, insofar as the existence of the subject matter of mathematics cannot be affirmed as such, but only as something else, whether signs, ideas, physical patterns, practices, or something else.

I’m fully aware that I’ve drastically simplified things here, basically reducing the mathematics-OOO interface to the encounter between the statements “the empty set exists” and “an object has parts”. My primary wish is to see the partisans of object-oriented ontology engage with mathematics, and I would be perfectly happy to stand corrected. As per now, whereas I am not convinced by either Badiou’s or Meillassoux’ account of mathematics (though seeing as Meillassoux has not yet published his thoughts on mathematics I can hardly expect to, in his case), when it comes to OOO I have only the vaguest idea of what their account would even be, as this post no doubt shows.

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Badiou, Hegel, and the Infinite – part I: Introduction

Meditation 15 of Badiou’s Being and Event is one of the most forbidding encounters of that book, pitching the Hegelian doctrine – or, more precisely, doctrines – of the infinite against Badiou’s own Cantorian conception. Keeping both these competing views in focus while attempting to grasp their precise differences and perhaps even adjudicate between them….. I remember someone stating that reading Difference and Repetition is intellectually akin to running into the ocean – splashing out those exhilarating first few steps before you abruptly come to a halt and tumble into the deep water of “sink or swim” – but I’ve read this meditation a few times before, and each time it feels more akin to running into a brick wall. And so, previously, I’ve settled for the bland “Badiou thinks Hegel cannot do justice to the mathematical infinite” conclusion. Which is clearly unsatisfactory in the long run, so this post is a first effort at dismantling the “wall” aspect of those nine pages.

Some of the obvious  – too obvious – disagreements between Badiou and Hegel can be stated at the outset, simply to get a preliminary grasp of the battlefield:

Hegel thinks that infinity is 1) generated, i.e., immanently derived from the dialectical process, and 2) ‘good’, i.e., qualitative, so it is the qualitative essence of quantity that constitutes its true infinity.

Badiou, on the other hand, thinks that infinity is 1) postulated, i.e., that the existence of an infinite set must be axiomatically decided upon, and 2) ‘bad’, i.e., quantitative, since Cantorian infinity allows for “the very proliferation that Hegel imagined one could reduce” (B&E, 170).

Both these disagreements can be found more generally in Badiou’s engagement with the philosophical and mathematical tradition. For instance, Badiou likewise criticizes Dedkind’s attempted deduction of the existence of the infinite, in Number and Numbers, ch. 4. (“Now, just like the empty set, or zero, the infinite will not be deduced: we have to decide its existence axiomatically” (N&N, 44). And the qualitative conception of infinity is something Badiou takes the philosophical tradition in general to task for, seeing in it a latent religiosity.

We must immediately complicate the above assertions, however, for they do not adequately represent Badiou’s particular line of argumentation in meditation 15. It might, from the above, seem as if Badiou gainsays any deduction of the infinite, and that he opposes the proper, quantitative infinite to the spurious qualitative infinite. But in fact, Badiou does not object to Hegel’s deduction of the good qualitative infinite. And in fact, Badiou recognizes that there is not, in Hegel, a simple dichotomy between qualitative and quantitative infinity, but rather a fourfold of infinities: the bad qualitative infinity, the good qualitative infinity, the bad quantitative infinity, and the good quantitative infinity.

What Badiou seems to argue is 1) that Hegel cannot properly ground the third infinity, the bad quantitative infinity: “One must recognize that the repetition of the One in number cannot arise from the interiority of the negative” (B&E, 169). 2) That in any case, what Hegel puts forward as the fourth infinity, the good quantitative infinity, cannot properly be called “infinity” at all: “I have no quarrel with there being a qualitative essence of quantity, but why name it ‘infinity’?” (B&E, 169).

I must therefore demur from Hallward when he writes, in Badiou: A Subject to Truth, that “The crux of Badiou’s argument is that without tacit recognition of the disjunctive decision to affirm the infinite, the good, ‘qualitative’ infinity cannot join up with a properly quantitative notion of infinity at all.” (173). The problem is not that there are two different infinities, and they cannot be joined. Rather, the problem is as follows: there are two dialectics, quality and quantity, which Hegel connects by means of a “fragile verbal footbridge thrown from one side to the other: ‘infinity’.” (B&E, 170) But this footbridge is fragile indeed: It is in fact merely verbal, since on one side it is anchored to an ungrounded hallucination – the Hegelian ‘good quantitative infinity’.

I’ll try, next, to investigate more in detail Badiou’s analysis of Hegel, and perhaps even delve properly into the Science of Logic to see whether his objections can be met from a Hegelian standpoint.

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The Limits of Entropy

In this post, Levi Bryant argues that object-oriented ontology takes into account the problem of entropy in a way that classical structuralism is unable to.

This seems to me a plausible claim, one that I will not dispute. Rather, I am interested in the ramifications of bringing entropy into an object-oriented ontology. I realize of course that these ramifications are far from obvious or settled (which is why I’m interested). But entropy seems to be one of those problematic concepts where the boundaries between metaphysics and science is less than clear, and even more interestingly, I have a feeling that the concept of entropy might very well attract philosophy towards ideas it does not want.

For instance, I have one latent worry concerning the compatibility of scientific conceptions of entropy and object-oriented ontology: Recent physics consider entropy to have a limit, a maximum, and claims that this maximum is in fact evinced by black holes. What does this imply? According to wikipedia, at least, it suggests that “matter itself cannot be subdivided infinitely many times and there must be an ultimate level of fundamental particles” (see this article). And it is indeed difficult to see how an entropy limit would not signify a limit in the degrees of freedom of the components of the entropic system, and each such basic, uncomposed degree of freedom signify something like a fundamental particle or a discrete state of such a particle (or could it be asymptotical?)

From what I know about the object-oriented ontologists like Bryant and Harman, they do not look favorably upon materialism and atomism, so I suppose that they are not eager to embrace entropy limits either. But what are the options? Deleuze’s account and critique of “thermodynamical illusion” (in chapter 5 of Difference and Repetition)?

In any case, I would like to propose black holes as one of the paradigm cases of “object”; not because they are typical but precisely because they are such extreme tests of any theory of objects. You have a theory of objects, you should at some point plug “black hole” into the litanizer and see what results come out. Any bias towards anthropocentrism or common sense should have a good chance of being exposed.

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Hägglund and Meillassoux on Time

Martin Hägglund’s critique of Meillassoux has come up now and again (for instance here), and I just now listened to his presentation at the 21st Century Materialism conference. (the talk can be downloaded here).

Their main point of contention is their views on time, and I think Hägglund has drawn the shortest straw. He argues that temporal succession is a presupposition of contingency, and I think that he is wrong – Meillassoux cannot, should not, and does not need to invoke temporal succession as an absolute, as a condition of absolute contingency.

Hägglund argues that his temporal logic gives a more plausible account of the emergence of life than Meillassoux’ eruption of the new for no reason. I in fact agree. There is no good reason, as far as I can see, for Meillassoux to invoke absolute contingency as the source of the emergence of life. That seems (I have not read Meillassoux’ actual statements about it) to be a terrible argument. But it is confusing for Hägglund to bring this into the discussion about time and the necessity of succession. For however unfortunate this particular argument from Meillassoux is, one easily sees the advantage of Meillassoux’ conception of time by turning to another example: from the emergence of life to the emergence of the universe.

What Meillassoux gives us is a framework where physical time is a function of physical laws. The physical laws of our universe may be laws that provide us with a time of temporal succession, with an arrow of time, or so on, or they may not. The jury is still out, as far as contemporary physics is concerned. In any case, the absolute time, the “mad time” of Meillassoux, is the time assuring that the physical laws might change for no reason, and physical time with them. And the following should be correct, if I have understood any contemporary physics at all: If the laws of nature as such were to change, not only the present and the future but also the past would change. And it seems possible that the laws of nature might change (of course, Meillassoux claims to have demonstrated that it is absolutely possible). So Meillassoux’ absolute, chaotic time is “not governed by physical laws because it is the laws themselves which are governed by a mad time” (Time without Becoming, p. 10). Hägglund calls this to “posit an instance that has power over time”, and if by time he means physical time then he is of course correct. But that seems quite reasonable to me.

Hägglund’s framework where temporal succession is absolutely necessary, on the other hand, I am not able to make sense of at all. Does Hägglund want us to rule out a priori the possibility that the beginning of the universe was a contingent event that took place outside of temporal succession? Moreover, it is becoming increasingly common for physicists to suggest that our future understanding of the physical universe will not even include space and time in the theory (e.g., Julian Barbour). Must Hägglund say that the physicists are wrong a priori if they claim that irreversible succession has no place in our best theories of the physical universe? Will he not, then, force us into an equally untenable picture as the one upheld by correlationism, where the unaccountable phenomena are not archi-fossils but atemporal or weirdly temporal accounts of the physical universe?

I have not read Radical Atheism, so I do not really know the details of Hägglund’s theory of temporality. But from the talk, it sounds like the temporal succession involved is something like a total ordering of moments. Which I simply cannot bring myself to believe as an a priori principle.

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Meillassoux and the Cahiers

Apropos my previous post on the Cahiers pour L’Analyse and the website dedicated to it:

Fabio Cunctator at Hypertiling has posted a partial translation of Meillassoux’ article “Contigence et Absolutisation de l’Un”. What is fascinating about the translated part is that in it, Meillassoux sketches the outline of his argument for the absolute possibility of mathematical statements, the argument that After Finitude only hints at. I don’t yet have anything to say about the strength of Meillassoux’ argument (Pete at deontologistics suspects that it might involve a conflation of aletheic modality and deontic modality), but merely want to note that it seems to indicate the contemporary relevance of the Cahiers:

Meillassoux claims that there is an essential difference between the “ontic one” and the “semiotic one”, between the object or mark and the sign. The sign devoid of meaning, as utilized in mathematical formalization, is radically self-identical, as type. So for instance, in the series 11111111, if the 1s are considered as marks they are all different from one another. However, considered as signs they are completely identical instances of the type 1. This identity is, according to Meillassoux, due to the contingency of the signs as signs – the particular shapes of the signs used as variables, constants, etc., in mathematics has no bearing on the argument and can be chosen at random. And presumably, this contingency of the signs of mathematical formalization is the same absolute contingency that Meillassoux’ principle of factiality establishes.

Now, is this argument an attempt to ground the view of mathematics put forward by the early Badiou in articles like “Marque et Manque: à propos de Zéro”? This quote, from the synopsis of that article, might suggest as much:

The whole of ‘Mark and Lack’ presumes, then, an ‘inaugural confidence in the permanence’ and self-identity or self-substitutability of logico-mathematical marks or graphemes (156). Given any mark x, logic must always treat x as strictly and unequivocally identical with itself. Badiou thus takes for granted a position that Miller associates, in ‘Suture’, with Leibniz and Frege (CpA 1.3:43): scientific knowledge depends on the exclusion of the non-identical, with the proviso that ‘the concept of identity holds only for marks’, i.e. mathematical inscriptions. ‘Science as a whole takes self-identity to be a predicate of marks rather than of the object’, a rule which applies to the ‘facts of writing proper to Mathematics as it does for the ‘inscriptions of energy proper to Physics’, along with the instruments used to measure them. (The entire synposis can be found here)

The “fact of science” which Badiou claims to be a groundless, contingent breach in the fabric of ideology, would then turn out to be “grounded”, if that is the right word, in the principle of factiality.

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Cahiers pour L’Analyse

I just found out that the complete texts of the Cahiers pour L’Analyse, along with detailed synopses in English of each article, has been published online. This French journal published ten issues from 1966 to 1969, with articles from Alain Badiou, Jacques Derrida, Jacques Lacan, Jacques-Alain Miller, among others.

The website is really a treasure trove. From what I’ve seen so far, the synopses are wonderfully detailed and helpful, and in addition there is also comprehensive information on all the names involved in the journal, and many of the most important concepts.

Recent interviews with many of the members of the editorial board of the Cahiers, and translations of many of the articles into English, will appear. Or at least, so I hope. The website is a result of the work of CRMEP, previously at Middlesex University, now at Kingston. The recent events concerning the closure of philosophy at Middlesex and the relocation of the CRMEP to Kingston leave me ambivalent, as do the invitation at the Cahiers website to send suggestions or corrections to Christian Kerslake, who will not be joining the four ‘senior’ members of the CRMEP staff… Judging from what I’ve seen of his articles and books on Deleuze, as well as this website, Kerslake’s research is stunningly good and original, and I hope that his research and teaching situation will look up in the future.

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Objects and Joints

I realized today, while browsing Philip Kitcher’s book “Science, Truth and Democracy”, one of the things I find fascinating about object-oriented philosophy: That it inverses the claims of the kind of “weak realism” so common in the kind of analytic philosophy that is sensitive to pragmatism without being Richard Rorty. Philip Kitcher is a classic example.

Kitcher defends a version of a correspondence theory of truth, it seems, and presents an extended (and I really mean extended, it constitutes an entire chapter) analogy to cartography in order to make his point. Quite interesting, in fact. Now, as far as I understand no version of object-oriented philosophy accepts representation and truth as correspondence, favoring instead concepts like translation and so on.

However, Kitcher rejects the possibility of “carving nature at the joints”. Kitcher describes the view he opposes as follows: “The world comes to us prepackaged into units, and a proper account of the truth and objectivity of the sciences must incorporate the idea that we aim for, and sometimes acheive, descriptions that correspond to the natural divisions.” (Kitcher, p. 43). Such a view cannot be maintained, he claims, since we divide the world into units based on our own interests, and any claim that one can pick out “natural” objects or divisions is inevitably colored by our own interests and capacities. Dividing the space around me into tables, chairs, windows and walls is natural for me, but I cannot claim that it is natural tout court.

Against this, object-oriented philosophy claims that one can distinguish objects from pseudo-objects. The point has been made on several occasions, but perhaps most clearly by Graham Harman in this post on his blog. Here, he makes two points: 1) That one can never know for sure whether an object is a real object or a pseudo-object. 2) That this does not rule out the possibility of distinguishing between them, on an a posteriori and revisable basis. In other words, it is possible to carve objects at their joints, though one will always remain fallible in doing so.

In other words, Kitcher’s weak realism keeps correspondence truth, and rejects joints of nature. Object-oriented philosophy, on the other hand, rejects correspondence truth, but keeps the joints of nature, i.e., of objects.

But Philip Kitcher’s way of arguing against the “carving nature/objects at their joints” thesis highlights something interesting. For his main point is that the carving can only ever be “natural” for us, with our specific interests in mind. That conclusion obviously cannot be admitted by object-oriented philosophy, since the objects it aims to understand are supposed to be (except obviously human-related objects like nations and so on) independent of humans and human existence. Nonetheless, I think there is a pretty strong pragmatist argument against the “joints of nature/objects” thesis. And I am not sure those arguments can be countered simply by the standard anti-correlationist rejoinders, and will try to outline why in another post. In any case I’m curious to see what arguments there are that objects have natural joints, as well as what criteria can be mounted for distinguishing the joints, i.e., separating the real objects from the pseudo-objects. In other words: 1) Can an object-oriented philosophy afford a distinction between object and pseudo-object? 2) If so, can the distinction be used in practice, and by what means?

In Harman’s reading of Delanda, he suggests that the main criterion of a real object as opposed to a pseudo-object is redundant causation. He says: “The basic idea is that the same emergent assemblage might have arisen from any number of different processes, rendering the exact details of its history irrelevant”. I have to admit that I fail to grasp how redundant causation is to function as a criterion of a real object. But I’ve ordered Delanda’s book from the library, and look forward to reading it. I’ve just noticed a reading group dedicated to the book has begun, which I will now want to follow closely, I might even try to participate if I can get the book in time.

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