Note: this is an English version of the first draft of my intervention at the IMMANENCE OF TRUTHS event (pdf here). Each version contains developments missing in the other but the overall argument is the same.


Philosophical intensities and language of the infinite

by Terence Blake

ABSTRACT: The aim of Badiou’s book is to convince us that we live in L, the constructible universe, without being aware that this is the case ; guide us progressively from L to V, the class of all sets, the universe containing the large non-constructible infinities, and elevate us to the “Absolute” (the philosophical name for V); take us down again into the finitude of L equipped with perceptions drawn from V, thus permitting us to re-index the world according to absoluteness and no longer in terms of relativity.

We could summarise the global movement with the slogan: L is Hell, but V is free.

My path through the book is multiple, and comprises the following aspects:

1) a conceptual path centred on philosophical pluralism and on the book’s concepts concerning the multiple types of infinites;

2) a poetic path centred on the movements of descent into the hell of the finitude of the waste product cut off from infinity, of purgation, and of ascension to Paradise (the paradise of the Absolute, or “Cantor’s paradise”), and finally the re-descent into the finitude of the work “touched” by the infinite;

3) a linguistic path centred on the question of the type of language that is the most appropriate for speaking of our world and our life – Alain Badiou demonstrates the depth and fecundity of a philosophical language drawn from the mathematics of the higher infinite;

4) a contextual path centred on a comparison of Badiou’s rigorous mathematically-based analysis and division of the concept of the infinite with Deleuze’s more intuitive and poetic conceptions of the infinite in WHAT IS PHILOSOPHY?;

5) a subjective path centred on what the book can change both in our relation to the infinite and in the reader’s philosophical unconscious.


In this first section I will talk about the search for a pluralism that does not fall into the trap of relativism.

Alain Badiou has pursued his ontological research programme for over fifty years. During this period he has published hundreds of texts. One of the guiding threads of this abundant production is pluralism: a thought of the pure multiple, of the plurality of truth-procedures, and of the plurality of infinites. Badiou is a pluralist, and it is as such that he has attracted and held my attention. I have been following his work for forty years.

On my blog AGENT SWARM I discuss the ideas of a number of contemporary thinkers who work in the domain of epstemological and ontological pluralism. This analysis is based on the idea that one can better understand a philosophy by considering it not in isolation as a static system, but as a research programme evolving in the context of other programmes of the same type.

I employ this expression of “research programme” in the wide sense that Karl Popper gives it: a metaphysical research programme is a general conceptual framework comprising both non-testable speculative elements and testable empirical elements..

The theoretical projects which interest me are all « pluralist », in the sense that that escape the hegemony of the One, and give speculative priority to the multiple. I seek to analyse these projects, to compare them, and to put them in dialogue in terme of a set of shared criteria. In particular, one can examine a pluralist research programme in terms of its type and degree of immanence and of its testability.

From this point of view, Badiou’s BEING AND EVENT trilogy is exemplary. It constitutes a « Summa Ontologica » in three volumes:

I) L’être et l’événement – 570 pages, II) Logiques des mondes – 630 pages,

III) L’immanence des vérités – 710 pages.

Over almost two thousand pages, Badiou leads us on the path of an extended meditation on the dialectic of the finite and the infinite, guiding us towards an immanent understanding of Being, of Truths, Worlds, the Subject, and the Absolute. This project is highly speculative, but it is elaborated under the conditions of the four truth-procedures, which assure a degree of indirect testability for his thought.

L’immanence des vérités is an invitation to embark on a passionately inspiring adventure combining movements of vertiginous ascension, thanks to the abstaction of mathematical language, and of precipitous descent towards concrete life, empowered by poetic language.

These movements are not unique to the mathematical procedure. According to Badiou’s system we can find these movements also in the the other truth-procedures (i.e. the poem, love, and politics).


In this second section I will compare L’Immanence des vérités with a poem, Dante’s DIVINE COMEDY to help us see the global and specific movements of Badiou’s project.

When mathematicians and philosophers talk about the beauty of « Cantor’s paradise » of higher and higher infinities or transfinite numbers, we should understand this in terms of Dante’s PARADISO as abstract poem. Dante’s ascent finishes on the poet’s vision of “the love that moves the Sun and the other stars”.

This allows us to establish the link between Dante’s ascent to paradise and Plato’s ascension to the Ideas. It also highlights that Dante’s poem is incomplete, because after this ascent to infinite abstraction we need to descend to finite life again.

Otherwise we do not include the finite incarnated Dante, the poet who has visited these realms and who has returned to tell us of them. The poem needs to be completed by a fourth part: the RITORNO.

Badiou starts his book with the analysis of the operators of finitude as the basis of the system of oppression. This is his INFERNO: the hell of living under the forced regime of finitude. The modern form of finitude is what he calls « covering ». It is based on the idea that the universe of sets is L, the constructible universe.

The book then ascends, taking us on a voyage through the different types of infinity: inaccessible, resistant to partition, complete, and approaching the Absolute. In this ascent we are in Purgatory. The turning point comes when Badiou expounds a theorem allowing the « defeat » of covering. He then moves up the hierarchy of higher and higher infinities, mounting to the Absolute, « V », the class of all sets, which is not itself a set.

The analogy between Badiou’s book and Dante’s DIVINE COMEDY is striking, except that THE IMMANENCE OF TRUTHS provides the missing fourth part, as it accomplishes the re-descent, not only mathematically but also philosophically, into finitude « touched » by the infinite. Each of the four truth procedures (art, science, love, invented politics) are examined to determine their « index of absoluteness ». This is the point in which, while still remaining within the regime of finitude, the « work » produced by each truth procedure can be said to touch the Absolute.

I read the DIVINE COMEDY as an onto-theological poem precisely because it ascends to the Absolute of love by means of poetic abstraction and remains there, unable to return from transcendence to concrete human life.

One can compare this with Deleuze’s emphasis in his cinema seminars on the danger of a possible dualist dead-end in the move towards complete abstraction. He explicitly favoured the works of transformative re-descent, that comport what he called « re-injection », or the inclusion of elements of pure abstraction into re-worked mundane forms.


In this third section I will speak about the problem of the choice of an appropriate language to speak about the contemporary world and our life.

Badiou’s starting point is the “secularisation” of the infinite brought about by Cantor’s revolutionary approach, which also brought about a pluralisation of the infinite. Badiou’s research programme adopts the hypothesis that it is possible to speak of our world and our life in terms of a language of infinities derived from Cantor’s breakthrough.

The popular stereotype about Cantor is that he invented transfinite arithmetic, but that he was also mentally deranged and theologically obsessed, in a naive autodidactic sort of way. However, Cantor was quite philosophically literate, and I find it interesting that he had a well-worked out philosophy of the infinite, and that he read Spinoza and Leibniz very closely. His discoveries led him to distinguish different « sizes » of infinity, allowing him to re-read the classical philosophers from a new point of view.

« Cantor’s inquisitive « how infinite » was an impossible question. To minds like Spinoza and Leibniz, the infinite in this absolute sense was incomprehensible, as was God, and therefore any attempt to assign a basis for determining magnitudes other than merely potential ones was predestined to fail »

(from GEORG CANTOR His Mathematics and Philosophy of the Infinite, Joseph Warren Dauben, 123).

Unlike Badiou, Cantor does not identify the Absolute with “V”, the class of all sets. On the contrary, he distinguishes them for theological reasons, and so subsumes the transfinite to the regime of the One.

However, I think that there is still good reason to distinguish the Absolute from V, and that Badiou does so implicitly in that he carefully distinguishes philosophical concepts as metaphorisations or poetisations of mathematics from mathematics itself.

(This is in reference to Badiou’s thesis that philosophy is a “poetisation” of mathematics).

In contrast, from a Deleuzian perspective, Badiou’s concepts, which are based on the mathematical hierarchy of infinite cardinals, are insufficiently philosophical. While still “poetic” they are much closer to the mathematical end of the spectrum, and so represent a slowing down of the plane of consistency. However, even this Deleuzian critical term of “slowing down” is itself intuitive and poetical.


In this fourth section I will compare the different types of infinity described and analysed by Badiou with Deleuze’s treatment of this theme.

The infinite plays an important role in Deleuze’s thought, but it has not been the object of much discussion. Badiou’s set theoretic conceptual creations can provide us with a useful point of comparison. Contrary to Badiou, Deleuze does not make use of set theory, but prefers an intuitive and qualitative approach to the infinite. Despite this difference, there are important points of convergence between the two theoretical projects.

In his “Immanence of Truths” project Badiou distinguishes

(1) inaccessible infinites

(2) infinites by resistance to division or partition

(3) infinites by immanent power

(4) infinites by increasing proximity to the absolute.

In Deleuze’s work we can find

(1) the outside further than any exteriority – the inaccessible outside

(2) resistance to stratification and segmentation – the body without organs, resistance is prior to power

(3) immanent affirmative powers – distributions of becomings, affects, singularities, and intensities on a plane of immanence

(4) approaching the “absolute horizon” of the non-totalisable plane of immanent consistency, by way of absolute deterritorialisation, the horizon of convergence of the bodies without organs and their planes of immanence.

Deleuze and Guattari’s WHAT IS PHILOSOPHY? is a work which is to a large extent occupied with the ontology of infinity. The words ”infinite”, “infinity”, and their synonyms (the “absolute”, the “outside”, the “plane of consistency”, the “absolute horizon”) are to be found on nearly every page, often in multiple occurrences.

On the one hand one could argue from a Badiousian perspective that Deleuze’s concepts of infinity are too qualitative, too vague and imprecise, and that they remain too intuitive and insufficiently theorised, too close to the “poetic” end of the spectrum.

On the other hand one could invert the arrow of comparison and argue from a Deleuzian point of view that Badiou’s concepts, which are based on the mathematical hierarchy of infinite cardinals, are insufficiently philosophical. While still “poetic” they are much closer to the mathematical end of the spectrum, and so represent a slowing down of the plane of consistency. However, this Deleuzian critical term of “slowing down” is itself intuitive, qualitative and poetic.


In this fifth section I will consider the question of what underlies the common ground between Deleuze and Badiou, and of the subjectivation of infinities.

Ian Hunter’s work on Badiou is very interesting and useful, but I see no real problem or uniqueness in the need for subjective conversion that Hunter finds present in Badiou’s discourse, as it is a common requirement for any general philosophical research programme. Thomas Kuhn has argued that such processes of subjectivation are at work even in science, as we know they are in politics and in the arts.

If there were a performative contradiction in Badiou’s theoretical practice, it would not be between the conflicting requirements of conversion and of philosophical thought, but that between the putative creation of a philosophical élite of Platonic initiates or guardians and his own insistence that anyone can become subject by entering into at least one truth process.

The correspondence with Christian theology is another problem. Badiou systematically downplays the force and influence of religion in his concrete analyses and this attitude is reflected in his refusal to grant religion the status of a fifth condition. The result is that there is a pervasive atmosphere of religiosity in Badiou’s works. Badiou himself theorises this dimension as the need to “subjectivate” a philosophy.

The question of empirical testability is another crucial problem, and despite his talking in terms of “hypotheses” (e.g. the communist hypothesis) Badiou often uses the empirical world as a source of examples and illustrations, not of tests. However, his placing his philosophy under the condition of the truth procedures is a way of guaranteeing its indirect testability.

This redefines the problem of the role of examples in Badiou’s text. Direct empirical tests are not the only possible form of validation. Badiou can consider that his configuration of a space of compossibility for the four truth procedures that are themselves testable is “test enough”. Badiou’s method of verifying a hypothesis proposed with reference to one truth procedure by searching for confirming examples in each of the other truth procedures

It may be that Badiou considers that the network of correspondences he finds between the productions of different truth procedures is test enough. Privileging “Truths” over “facts” can be seen as the application of a hypothetico-deductive method (as against an inductive method).

The problem is not that of Badiou’s proceeding hypothetico-deductively (how could that be a problem?) but of whether Badiou makes use of this method to stimulate critical discussion or to close it off. Badiou’s books on theoretical rivals, such as Deleuze and Heidegger, have shown that he is capable of opening a discussion where previously it seemed impossible.

Deleuze explicitly states that “science renounces the infinite”, which is left to philosophy. This renunciation (which can be extended to art, in the terms of WHAT IS PHILOSOPHY?) leads by way of the absolutisation of philosophy to a relativisation or finitisation of science and art. Thus in Deleuze’s system art and science cannot provide philosophy with indirect testability.

Badiou’s final evaluation of Deleuze’s solution is that it is torn between the absolutism of philosophy and of its great inorganic life close to chaos on the one hand, and on the other the relativism of bodies and languages. Deleuze’s plane of immanence is too fragile, as Deleuze’s idea of the infinite is too vague, through lack of formalisation. Finally, Deleuze’s subjectivation is too empirical, one must not begin by subjectivating. Subjectivation comes after. Badiou’s solution is to begin with a formal elaboration of the dialectic between finite and infinite, and only then to subjectivate.

The conditions for this subjective appropriation exist, they are the object and the stake of all living philosophy. According to Badiou, infinite subjective intensities reside in reserve in the unconscious, both as an immanent infinite resource of energy and ideas and also as an inchoate perception of new possibilities of which one is as yet unaware that one is capable.

One can always, within the wanderings of a life lived without an Idea, fall by chance on an infinite intensity incommensurable with our conscious expectations. However, in this case it is improbable that we recognise the new intensity if we don’t already have within us the idea of such an infinity. This is the paradox of Plato’s MENON. We must already have the idea of the infinites inside us, as immanent intensities, in order to perceive the infinities we encounter in the world.

At the end of my own path through the book I can ask myself the question of conversion in its Badiousian form: what has this book made me capable of, that I did not yet know I was capable of?

Yes, the book has made me capable of revisioning pluralism from the perspective of the infinite, of following a quite technical exposition of the mathematics of the higher infinite cardinals and of ascending up to V. It has inspired me with a poetic vision, it has taught me to speak the language of the infinites and it has allowed me to come to a new understanding of philosophies that I thought I knew. That’s already quite a lot.

But the most important for me is that the book moves and transforms something deep in the reader’s subjectivity. In my case, it was able to speak to my philosophical unconscious, and make me more clearly conscious of infinite intensities in me of which I had only an intuitive perception (and conception) before reading the book.

I have come out of my path of reading the book totally re-indexed.

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14 commentaires pour MY PATH THROUGH BADIOU’S “THE IMMANENCE OF TRUTHS”: full English text

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