My specific argument against Badiou’s foundational treatment of mathematics goes one step further than general arguments against the validity of scientism and foundationalism that Badiou himself lays out convincingly.

Badiou’s philosophy is profoundly split between a specifically mathematical exploration that does not, and cannot, have the philosophical consequences that he wants and a more general metaphorical extrapolation of concepts he has derived from his reflection on the mathematics. Badiou declares that mathematics is ontology but the burden of proof is on him.

In the specifics of the working out of his system he will derive certain concepts from the mathematics. For example in his reflexion on omega, the first infinite ordinal, he extracts a typology of infinities: the infinite as point, as place, as horizon, and as repetition. He then turns to poetry to “explore the labyrinth of the different forms taken by the couple finite/infinite”, admitting explicitly that this turn is metaphorical.

The mathematics is a very interesting theoretical tool that is well worth working out, but that has only suggestive value, heuristic rather than apodictic force.

Badiou has aptly remarked that the contemporary transcendental is a mixture of “crass positivism” (his expression) and vacuous moralism. I think that the nadir of this mix in theoretical circles is the rise of mathematical reductionism, an ideology which combines the speculative privileging of mathematics and a sort of hard-nosed pathos reminiscent of what in science fiction used to be called the “cold equations”.

People whose philosophical penetration does not go beyond a glib juggling with jargon but who have some mastery of theoretically fashionable sectors of mathematics pop up and descry, or so they think, their own image writ large in Badiou’s philosophy. Formerly this combination of bland moralism and crass positivism was embodied in an obsession with physics as foundational. We can see relics of this in Laruelle’s appeal to a vague “quantum thought” of his own imagination, that would be better described as a quantum Stimmung.

The advantage of mathematics as de-realised foundational discipline is that any tie with actual scientific theories and empirical evidence. Mathematical reductionism irepresents an attempt at an ontology compatible with any scientific discovery while yet maintaining an aura of scientificity. I do not think that Badiou himself fully falls into this trap, but he is not fully free from it either.

Unfortunately, I’m still failing to see the exact nature of the argument. On the one hand, your idea of “the cold equations” seems to be one of the very standard critique of scientism – the scientistic thinker believes he is reaching a kind of “objectivity” with his “arid” “rigorous” thinking, but his very attitude is actually *itself* just a pathos or pathology. But we need to pay attention to the content, the “aridness” of mathematics for Badiou, comes from the fact that it is subtractive: i.e it literally denies any invocation of presence or life.

Again, with regard to content. To say that Badiou tries to produce apodictic conclusions from mathematics while really only employing it as metaphor, is missing the subtlety of the argument. Badiou’s conception of philosophy is that it creates concepts (like Deleuze), though specifically with an idea to outlining the concept of Truth in general. This involves yes an *interpretation* of each truth procedure, with math being a primary condition, in so for as, through a symptomatic reading of ontology, the ontological structure of Truth can actually be discovered, giving philosophy a foothold in the Real. This assures that its concept of Truth is not a mere fiction.

However, the interpretation of math specifically is meta-ontology, not ontology. Thus, philosophy cannot have the same apodictic character as formalized deduction itself. Philosophy nonetheless poses hypotheses and rational argumentation. At the same time, it must have a foothold not only in the Real of Being, but in the existentiality of the subject, who is the agent of truth (otherwise how would it interpret truths?). This brings it from matheme to poeme. It must touch the real on both sides. But I struggle to see how this is merely “metaphorical” or a kind of heuristic tool. I see these as conceptual creations which attempt to think real structures.

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I think that this is a superficial reading of Badiou, and that it does not even come to grips with all that Badiou himself says. Badiou explicitly says that the transfer of his four conceptualisations of infinity from mathematics to poetry is a “metaphorisation”, but some people have trouble understanding how an idea can be both conceptual and metaphorical at the same time. See here, comments on Mallarmé: http://www.entretemps.asso.fr/Badiou/12-13.htm

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will look at this. My trouble is not your claim that it is metaphorical in itself, nor how something can be metaphorical and conceptual at the same time. I DO take Badiou to be doing just that. But the point is this metaphoricity can only gaurantee a genuine conceptual *content* IF it has a foothold in the real, and thus only if math is literally taken as ontology.

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the question is the difference between ontology and (meta)ontology and philosophy. Metaphorization is the movement between the different truth procedures.

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from the link you sent me. This sums up everything i’ve been saying: empiricity of typology of truths, generality of concept of Truth, literal ontological structure of Truth. “mon souci, à l’époque, était de garantir une pensée possible de l’être des vérités *à partir de la particularité des situations, sans avoir à accorder aux vérités, donc à l’universalité possible de la pensée, un type d’être irréductible.”

but i will read the rest and get back to you.

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Hi Adam,

“This involves yes an *interpretation* of each truth procedure, with math being a primary condition, in so for as, through a symptomatic reading of ontology, the ontological structure of Truth can actually be discovered, giving philosophy a foothold in the Real. This assures that its concept of Truth is not a mere fiction”

Can there an “ontological structure of truth”. I would prefer: mathematical knowledge of the ontological structure of being, which, after a degree of consensus has been arrived at by mathematicians, achieves an apodictic force; until, that is, a new theoretical formulation appears to overturn or supplement the existing paradigm.

Truth, on the other hand, is the effect of an event arising within a “state of the situation” to which a subject is faithful. It can never have apodictic force conditioned on mathematical evidence since there is no connection between truth as event and mathematical knowledge. The former simply arises as truth for a subject. The latter is a matter of empirical investigation of the materials of mathematics – numbers and their relation, grounded on specialization by way of an empirically accessible division of labour. ( the scientific academy’s technological and procedural networks )

I don’t think Badiou attempts to ground his truth in mathematics but thinks from mathematics by way of a metaphorical or poetic procedure that tries to understand how something (a truth) can arise from nothing. In other words he tries to provide an apodictic force for a truth other than an appeal to mathematical knowledge but arising by way of a poetic procedure that uses mathematical materials for other ends. This apodictic force or truth functions, however, only between a community of faithful subjects. To those on the “outside” it must always appear unsubstantiated and any action premised on such a truth must always appear ill-advised or even scandalous. This is particularly so for the academy of mathematicians, and rightly so.

“This assures that its concept of Truth is not a mere fiction”

Truth, from the perspective of another “mode” will always appear “fictitious”. However “fiction” is exactly a truth validated according to a procedure different but analogous to the empirically grounded knowledge procedures of the mathematician. Latour recognises this by specifying fiction as one mode of being among others.

I say all of this, of course, with the proviso that I am way off the mark. That in itself is a sort of conundrum for me since it seems to undermine the whole idea of trying for a truth as such, rather than for knowledge pure and simple. My problem is that one can only assuage one’s doubts about the truth by way of exchange between the already and always faithful community. Which is why I suspect that Badiou, for all his materialism, would like to have his religion and eat it too.

Unless Adam you are right and we can assure ourselves via math that our truth is not a “mere” fiction. How, though, is that not a return to “mere” positivism?

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I would say that Badiou very much explicitly says there is an ontological structure of Truth, specifically in EE. He aims to argue that a generic set formalizes the Being of Truth. The key is to understand Truth (at a general level) and truths, those specific kinds of truths which trace situations (and can be typologized). Maybe we could venture truths —> typology of truths —-> concept of Truth, from general to specific. Math provides this concept, not in its actual working out by mathematicians (Badiou would say this is its *specific* truth, fully historical and empirical as you said) but that this set, this “object” of mathematics gives the most formal, bare structure of what any truth must be.

This can help us understand the idea of “truth as fiction.” Badiou does think “truth has the structure of a fiction” (which he takes from Lacan). But this means that the subject of truth fictionalizes the infinite completed truth as she traces the finite situation. She uses the fiction as a kind of future anterior in order to have a standard of evaluation for the new kind of multiples the procedure presents. But *that doesn’t mean Truth itself is a fiction* – there is an ontological structure that any truth must meet if it is to be really a truth (that of being a generic multiple). Thus, Badiou argues for the *real possibility* of truths, even if every specific truth can only be validated empirically. And this criteria of the generic multiple, I argue, also allows him to enumerate a typology of truths in way that isn’t arbitrary.

This is why I don’t see Badiou’s understanding of mathematics as merely metaphorical or poetic. I see the metaphorical element in the movement between truth procedures that philosophy must engage in order to think their compatibility, as well its exigency of mediating between matheme (the Real on the side of Being) and poeme (the Real on the side of the engaged Subject).

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excuse me, in the first paragraph, when I said general to specific I meant specific to general.

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With regard to the question of “mere” positivism I would say its more complicated. Badiou (at least in some places) seems to think math will always be a battleground. Even though a generic multiple is the form of any truth, and Badiou thinks math itself is a truth, math is always traced by its own desire to close down the generic in “the constructivist orientation of thought,” formalized by the theory of constructible sets. The axiom of choice (which is a necessary component of formalizing a truth procedure) is generally considered independent from ZF. You can have ZF or ZFC, and most mathematicians (as far as I know) seem to consider this a question of “taste.” For Badiou in reality this is a political battle, in the heart of ontology, over the existence of Truth.

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Hi Terence,

“We can see relics of this in Laruelle’s appeal to a vague “quantum thought” of his own imagination, that would be better described as a quantum Stimmung.”

Would you describe Badiou’s metaphorical use of mathematics also as a stimmung; as a “mood” or a “poetic mode” or even a “fiction” in Laruelle’s sense?

Thanks once again for a long series of stimulating posts.

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Hi Adam,

Thanks a lot for your reply. Firstly I have to again offer you this proviso. I may need to be taken by the ear and forced to read Badiou and not “read into” Badiou. Secondly, I don’t want to clog up Terence’s comment tread with my attempts at coherence. Badiou’s thought, I have to admit, wears me out and often leaves me at a loss. So I won’t continue here. I have just discovered your blog so I will read there for a while. That said:

“He aims to argue that a generic set formalizes the Being of Truth.”

I think this statements puts Badiou among the idealist philosophers; it makes him commit the mistake which Marx, especially in his thesis on Feuerbach, tries to warn us against— he begins not with the concrete biological /socialised individual but with a synchronic abstract structure — theoretical/empirical, Being/beings, Truth/truths. The antidote , as Marx was at pains to point out, was not a form of positivistic materialism or anti-theoretical empiricism (a disguised idealism) but a necessarily axiomatic insistence on the individual as a sort of transcendental ploy (fiction?), so as to do something with one’s thinking other than make of it a scholastic deployment of philosophical terms.

If you are right with the above I can see little use for Badiou’s thought other than as what Laruelle thinks his thought is– philosophical materials on which non-philosophy can set to work.

“But this means that the subject of truth fictionalizes the infinite completed truth as she traces the finite situation. She uses the fiction as a kind of future anterior in order to have a standard of evaluation for the new kind of multiples the procedure presents.”

Again, this presents Being as undergoing, by way of the Subject, a sort of unfolding validation in real time. One begins with the “infinite completed truth” traced into the finite situation. The “future anterior”, in other words, offers a sort of generic of what a truth should look like to the concrete living individual who thinks and acts in the real-time historical situation. This locks Badiou’s thought into a synchronic structure and uses a pseudo-time to extricate it by way of an unfolding of a potentiality already and always present (infinite completed truth) as an absolute deployed as history. This goes no further than Hegel’s antidote to Kant.

You might be right. I hope not.

“Maybe we could venture truths —> typology of truths —-> concept of Truth, from general to specific.”

“excuse me, in the first paragraph, when I said general to specific I meant specific to general”

What difference could the reversal of terms make? Exactly a“world”of difference” – to the Philosopher. That “thought/world” conditioned on a choice between abstract terms and their (reversible) deployment is exactly what, in my reading, Badiou tries to extricate himself from so as to escape scholastic Idealism. His problem is exactly expressed in the term “an ontological structure of Truth”(sic!).

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While the particularization of reality by the universalizing of math can be sanitized by considering Badious’s use of math a metaphor, it does not eliminate his universalizing dogmatic logic that permeates his reasoning. Nevertheless, we salute terence blake’s efforts at sanitization and await to hard won fruits.

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My problem is that I think Badiou is very worthwhile reading but that I don’t accept what the others who are favourable to Badiou appreciate: his mathematical foundationalism. So I would say that their reading is the sanitised one whereas I practice a less mathematico-hygienic reading. But this is merely a difference in terminology. I read Badiou’s “universal” as akin to Deleuze’s “untimely”. I give greater weight than other readers to Badiou’s occasional qualifications, that universal is not meant in the sense of the universal quantifier. But I think Badiou himself is only half-aware of this attenuation of the sense of universality.

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sorry that’s odd tried to post

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Yes, that’s strange. After approving I checked it, and it was the one on Nietzsche, but somehow it has changed since!

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https://progressivegeographies.com/2017/02/02/alain-badiou-no-limit-a-report-on-his-final-seminar/

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