What has mathematics to do with politics; and, by default, with our survival as a species, given the apparently unstoppable acceleration of capitalism, the collapse of a realistic left alternative, and the coming climate-apocalypse?

A lot, according to Alain Badiou.

Only the appearance of a new political subject, a subject subjected to a truth and to a practice of fidelity to that truth, can make the changes necessary to the survival of the species. This survival, for Badiou, is necessarily a process of collective revolutionary renewal enabled on the global reach and accelerated potential of capitalist structures and processes.

Badiou’s thinking is conditioned on the way he parses his universe: here a synchronic structure of thought – Cantorian set theory and the ontology extrapolated from it; there a diachronic process — biological evolution, human history, social revolutions. This ontological divide is paralleled by an epistemological one: the distinction between truth – the logic of number which always and already coincides with its “object” and knowledge – psychics, evolutionary biology, anthropology, sociology, historical research etc.

Having established this duality Badiou sets himself the task of undoing it in a strangely platonic way; platonic because he must find a way of ascending and descending between these two realms– an absolute and invariant realm of number and a relative and variant realm of empirically accessible material entities; strange because he makes this platonic move not in the name of the One but of the Multiple.

Badiou uses the idea of “truth procedures” to bridge the empirically accessible world in which we live, what he calls the “state of the situation”, and the foreclosed or subtracted void of “pure multiplicity”. It is Cantorian set theory which provides the ideal template for how we should think the ontological by showing how a count as one can be premised on the subtracted void of pure multiplicity. He names this insight a truth event and names the subject of this truth as an effect of the event. The biological, social, existential individual (bodies and language) “receives” this truth by way of the subject who comes into existence conditioned on an event risen from the void, much as the faith of the believing Christian is conditioned on the risen Christ. As concrete individuals, we can only try to remain faithful to a truth, much as a believer, in his fleshy weakness, must try to remain faithful to a grace miraculously bestowed on her by God via the risen Christ.

Received truth, in Badiou’s case, is not the result of God’s ineffable action nor is it mysteriously bestowed. It is won by a rigorous mathematical practice which can access a truth already and always co-existent with mathematical thought. One must enter into the realm of numbers; or rather one must become an individual subjected to the truth of mathematics via an event risen from the void of pure multiplicity– a “count as one”, the re-presentation of the pure multiple in thinking not as passive reflection, but as an “effect of the One which is not” on a mind ready to seize it.

Badiou points to decision as the only option in the face of a new truth; a decision to remain faithful to a truth that has seized one as forcefully as one has made the effort to seize it. In that crucible something new appears in a world.  In the case of ontological truth the philosopher must, according to Badiou,  defer to the truth of the matheme. Further, this truth must be concretely situated  in a world in the form of a political practice of fidelity to that truth requiring  a revolution in the existing state of the situation.

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Badiou invokes what he calls a place of absolute reference, a realm of number exhibiting four qualities:

Immobility– a purely synchronic positing of identity as absolute stasis. The identity of a set is invariant since any inclusion or exclusion of a member constitutes the creation of a new set. That just is the meaning of the act of bracketing — what Badiou calls “count as one”. You cannot change a set, nor does it evolve out of its own  dynamic. You can add to it or subtract from it, yes, but each time you do so you create a new set.

Radical non-positivity—thinkable on the basis of nothing or voidness; the positing of the empty set (o) such that any existing set x is by definition not a member of the empty set; any set x which is also a member of the empty set deprives the empty set of its self established identity. One immediately appears out of the void as an operation on number. In other words the existence of multiples is a matter of a confluence of the void and a count as one: mathematical thinking, or the count-as-one, is co-equivalent with the be-ing of number.

Axiomaticity– based on radically non-empirical prescription; we can start with prescribed axioms and verify the existence of entities by proving whether they violate the logicality of the axiom. For example, the sequence of natural numbers 0, 1, 2, 3, . . . . are proved to exist by way of the axiom of extension: set x equals set y if and only if z belongs to x and z belongs to y; or 1+1=2 if and only if logical extension is provable: (0), [(0)], [[(0),(0)]], [[[(0), (0),(0)]]] ….. By counting the brackets in this extension we see that we have extracted the singularities 1, 2, 3 from the void.

Maximality—capable of extension to infinity on the basis of an axiom of infinity: From the empty set, following the axiom of extension, we can proceed to the “count as one” from 0, 1, 2, 3 . . . . From the set of natural numbers ( points along an unbroken line) we can infer another infinity by division of the space between the natural numbers (rational numbers) and so on all the way to an infinity of infinites.

Absolute ontological referentially is, according to Badiou, possible only on the basis of Cantorian set theory. Cantorian set theory thinks the referential horizon by positing the empty set and inferring an infinity of sets by extension following the law of non-contradiction– if a set conforms to the logicality governing the production of sets it will be deemed to exist, either exnihlo, or as a rigorous extension from the axioms. In this way an infinity of infinite sets or a pure multiplicity of sets can be inferred without contradiction.

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 Badiou decides for what he calls the absolutisation of set theory.

“By saying that set theory constitutes an absolute reference, I am assuming that there exists a system of axioms, incompletely discovered as yet, which defines the universe V, the rational fiction in which all sets are thinkable, and defines it alone. In other words, no important, significant, useful property of sets will remain undecidable once we have been able to fully identify the axioms.”

Having established the veracity of Cantor’s discovery Badiou posits the truly existing infinity of infinite sets as the pure being of the infinity of material entities by extrapolating from abstract number to physical entity. For Badiou the most rigorously thinkable aspect of any physical entity is its countability. All else, as mere empiricity, drains away, as it were; both inanimate objects and living beings succumb to decomposition, disappearance, death, negation. Number, as the synchronically immobile, unchanging but infinitely extendible referential horizon is the absolutely true and unchanging mode of being of actually existing entities. A form of ontological re-thinking of mathematical thinking is, Badiou wagers, co-equivalent with being.

This is precisely what my own philosophical wager has been since the 1980s: to construct a theory of worlds such that modifications in it are only intelligible to the extent that the invariance of the real concept of multiplicity is assumed. To assume, to that end, that the immanent mobility of worlds and the instability of appearing are what happens locally to multiplicities that are in other respects mathematically thinkable in terms of their non-localized being, their pure being, in the determinate framework of set theory, hence in a place where being and being-thought are identical. On this basis, it will be said that to appear is only to come, as a multiple, to a place where absoluteness is topologically particularized.

When incorporated into philosophical thinking, says Badiou, such a truth enables a thinking of being qua being as, essentially, infinitely dividable and extendible. The count as one becomes a truth-event arising in the situation of a World to counteract the nihilistic predominance of the finite as constituting the horizon of that world. Cantorian set theory, extrapolated at the philosophical level as the referential horizon against which we can conceptualise our collective experience,  offers a structure of invariance as an antidote to the relativity of all theories as necessarily falsifiable over time conditioned on the mere physicality of the perceiver, her symbolic systems, social relations and material objects. As such, Cantorian set theory offers an invariant ontological truth as distinct from mere knowledge.

Despite the avowed constraints under which philosophy must produce its concepts – its undeniable historicity- one can rescue philosophy by a supplement – Cantorian set theory, itself a pure form of the set-theoretic escaping history by way of an essential quality peculiar to number — the absolute stasis of the structure of its relations. The matheme is key to the essential nature of every entity counted as one. And the origin of entities as count as one is a subtractive operation on the one – nil, or the empty set.

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Badiou’s absolutisation of the referential horizon premised on Cantorian set theory is philosophical; or, as Laruelle would put it, decisional. In that sense it is unavoidably circular. If we ask the naive question – who thinks number? – the answer according to Badiou is: number thinks number by way of the Subject of number. The philosopher and indeed the mathematician– those mere historical beings, those mere bodies, can only labour within time and produce knowledge of number—only the subject of number as an operation on number risen from the void of pure multiplicity makes accessible the truth of number as absolute referential horizon. In that sense it was not Cantor who conceptualised a Cantorian set theoretic, rather, Cantor the man receives by way of the mathematical subject a truth already and always co existent with number.

It is of no use to insist that Cantors set theoretic is premised on mathematical insights developed by Copernicus and Galileo or that Mathematics and the set theoretic in particular have a long and controversial history ending in a crisis that still needs to be resolved. Badiou has already decided on an essential difference between mere number and the matheme, between knowledge of number and and truth of mathematics. This difference is arrived at not via mathematics but by a philosophical decision exterior to mathematics. While Badiou insists that philosophy needs a mathematical supplement, he has, in fact, supplemented mathematics by way of philosophy; he had decided on the truth of the matheme and subsumed the biologically evolved and historically conditioned operator on number, the mathematician, under an abstract philosophical posit.

And yet, of course, all one can say with certitude is that the presence of the mathematician is a necessary occasion for the operation on number. That strange quality of number, what Badiou calls it’s invariance, cannot be reduced by fiat to its material correlates without making a leap from the raw scientific data to a philosophical decision for materialism, pre-empting the scientific evidence which, at the moment at any rate, provides only circumstantial evidence about the relation between mathematical thought, or any other thought, and brain activity. That same stipulation, on the other hand, also pre-empts an outright idealisation of the matheme, an Idealism that would allow one to insist that “to appear is only to come, as a multiple, to a place where absoluteness is topologically particularized”. Perhaps it is feasible to conclude that the “oneing” produced by an operation on number is such that the multiple survives as a singularity impervious to absolute capture by philosophical decision, if not the scientific-mathematical “count as one”. Such a “weak” form of Non-Idealism seems not to be excluded by the scientific evidence.

There are countless mathematicians who regard the relation between the matheme and physical particles to be a form of reciprocal causation, and many ongoing attempts to bind mathematics and theoretical psychics in a seemless way. Considering that experimental psychics has, for the most part, confirmed the equations of theoretical psychics, Badiou’s extrapolation doesn’t seem as far-fetched as one might imagine.

As an example of one of innumerable enquiries into the relation between number and particle psychics, consider the work of mathematician Cohl Furey. One does not have to be a mathematical specialist to appreciate the significance of such work and it’s philosophical implications. Or consider how the work of the mathematician Alan Turing sheds light on the generation of the myriad amorphous patterns found in the biological world.

At any rate, Badiou’s  philosophical system allows for an epistemological pluralism which makes it possible to arrive at a truth by way of procedures other than the mathematical. Art, politics and love are, according to Badiou, equally valid ways of arriving at “an horizon of the absolute” able to free one from the grossly materialistic and consumerist life style of a triumphant and ubiquitous “democratic materialism”. That fact alone, coupled with Badiou’s uncompromising anti-capitalism, makes of Badiou’s Platonism a practice of liberation, at least in his own case, and, perhaps, in the case of anyone who subjects himself to the appearance of Badiou’s thought in their World.

The great tragedy, none-the-less, as Badiou himself concedes, is that fidelity to a truth would ask of us an abasement before dogmatic authorities  and an abandonment of our allegiance to an immanence of truths in a plurality of worlds.

You can read further on Badiou’s concept of the absolute horizon here

Or read Terence Blake’s incisive and accessible posts on Badiou here