Translation of: Alain Badiou, The Immanence of Truths, Section III, Chapter c11, parts 1-4 (out of 12)
Four accesses to the infinite
1. Objections concerning the set-theoretical concept of the infinite
I demonstrated previously that the ontology of every oppressive figure organizes itself based on an imperative of finitude. Now I launch into the counterpart of this negative observation: the aim is to establish that wherever human action liberates itself from the order that constraints it, it is a matter of an encounter with the infinite, in the figure of a work.
It is only natural to begin with what in any case we know about the infinite, a knowledge constitutive of mathematical thinking. This initial course will still be very approximative for two reasons. First, the dialectic finite/infinite is at the heart of the system [dispositif] of this entire book, and we will only see this clearly little by little. Second, the mathematical theory of the infinite is not only complex, but it is still today in the midst of evoltion.
For now it is only a matter of considering “in broad strokes” the challenge we are faced with, in its massiveness.
Objections concerning the infinite such as this book presents it – in the framework of set theory, the concept and the adequate formalisms – appear in two principle forms, instances of which we detailed in the first two sections but which deserve to be recalled here.
The first objection is the “naïve” thesis of finitude which, like the thesis on the sovereignty of opinions, is quite appropriate for the prevailing democratic world, for its prescription to consume as much as the meagerness of its speculative ambitions, and so for the constant practice of covering. This theory is content to declare that our animal-human-moral experience is finite and that, to the extent that science can prove it, the universe itself is finite, because it has a beginning, the big bang; a geometrical closure, because it is curved; and a mass, today barely understood, given the importance of an ungraspable “dark matter.” Once the infinite is expelled from the material universe, two hypotheses concerning it remain. Either the infinite realizes itself as divine transcendence, which is normal notably in the Anglo-Saxon world, where “scientific” empiricism and minimalist religion have always made a good couple. Or philosophy must shut itself away in an analytic of finitude which is able to complete, as we’ve seen, a phenomenology of being-for-death; this is also normal today, notably in the world called “continental,” where analytic philosophy and phenomenology divide up the academic power today.
For what concerns me, faced with these assertions I remain calmly Cartesian. Because a rational and ramified thought of the infinite exists, it is senseless to support the thesis of finitude. To justify this classical stubbornness, I will rely on what I find new in the contemporary conditions of philosophy. Coming to my aid indeed will be the sophisticated mathematics of entangled infinities, the artistic appreciation for the finished work without finitude, the existential experience of amorous infinity, and the finished political sequences of the infinite communist strategy. The contemporary mathematical theory of the infinite is called the “theory of grand cardinals.” Well, let’s just say that the theory of grand cardinals is certainly just as real, if not more, than the Big Bang or global warming.
The second objection concerning the infinite, considered to be a dimension of the multiple or a singular type of multiplicity and thus detached from the One, is much more interesting. It comes down to saying that this way of thinking the multiple does not manage to really grasp the infinite because it bends it to a limited thinking, that of numbers. Cantor certainly secularized the thought of the infinite by freeing it from the power of the One, by showing that an infinite plurality of different infinities exists. However, so goes the objection, by submitting this infinity to calculation Cantor missed the qualitative essence of the infinite. Proof of this is that in set theory there is no terminal infinity, no stop, and so one is in numerical indifference. Just as one number comes after another, one type of infinity comes after another. It’s a sort of open zoology infinities, which makes one see in succession all sorts of monsters of greater and greater infinities, which shirks an authentic thinking of the multiple.
Earlier, Duns Scotus or Saint Thomas argued over the increasing dematerialization of the angels. The latter ranged from simple angels, messengers of God before humans, to the ultimate enjoyments of love that dissolve the seraphim in the light, passing through the rationally defined gradations of archangels, principalities, powers, virtues, dominations, thrones and cherubim. The modern theory of types of infinity, or the theory of grand cardinals, does better still, and its rationality is remarkably more subtle and rigid at the same time. The details are extremely exciting, but they involve the greatest demonstrative difficulty, which will be braved here, notably in the suites that belong to the long course. Let us enjoy, for a moment, the terms: an infinity can be, if I may say so from bottom to top, weakly inaccessible, inaccessible, weakly compact, indescribable, ineffable, compact, Jonsson, Rowbottom, Ramsey, measurable, strong, Woodin, superstrong, strongly compact, supercompact, extensible, Vopenka, nearly enormous, enormous, super-enormous… There is something baroque in these descriptions that seems to validate the objection against the ultimately indifferent character of this succession, right where a very determined, differential and enveloping concept ought to stand. In fact, the objection comes down to proposing a qualitative concept of the infinite, one that precisely evades the extensive and spread out rationality of sets. It is obviously this qualitative infinity that Deleuze has in mind, whether it be when he defines the concept as the trajectory of its components “at infinite speed,” or when he evokes the primordial infinity of Chaos. And on this basis, he believes it possible to say that Cantor lacked the true concept of infinite multiplicity and reduced it to the monotony of numbers.
A substantial part of the examination of these “modern” infinities is designed to refute this belief.
2. The concepts of the infinite: historical and conceptual approach
Theologians identified very early on that there were at least four modes of access of thought to infinity, four conceptual procedures permitting one to produce, if not a definition, at least a sort of context where infinity – which they thought to be in the form of the One, obliging the transcendent orientation – lets itself be integrated in a rational or argumentative system, namely in the form of the proofs of the existence of God.
In the properly transcendent, or negative, mode of access, the infinite is in a position of supereminence in relation to every predicate one would try to use to define it, as well as every operation whereby one would try to construct it. Paradoxically, the only access open for a thought of infinity is to experience that one cannot access it by any of the ways that seem precisely to introduce [it] to one’s thought. This can also be said: the infinite is incommensurable with the finite, and as our world and our thought belong to the finite, the infinite is only accessible to us negatively. This negation is an approximation lacking the thought of the infinite: the infinite is defined as inaccessible by any other definition except that of its inaccessibility. This is the domain of negative theology and was widely explored by the theologians of the Middle Ages, earlier by – but under the name of the One – Plato in the Parmenides dialogue, where it is shown that the One is that about which it cannot be said that it is, nor that it is not, or by the neo-Platonic exegesis of Christianity.
b) Resistance to every division
The second mode of access turns directly around the link between the unity of infinity and its different forms of power: supposing diverse attributes of infinity – or of God – can be thought, how does its transcendent unity withstand [résiste] this multiplicity? The infinite-One is at the same time all wise and all-powerful, it is infinite good but also infinite justice, it is integral knowledge but also action without limits, etc. What do these determinations signify, even when pushed beyond their ordinary earthly meaning, if the infinite is first of all a unity of indivisible power and, for this very reason, heterogenous to the finite, which ends up being annihilated under the effect of real divisions and of corruption? The question is even more pressing if one admits, with Christianity, the division of the divine infinity into multiple persons, one of which, the Son, Christ, has to experience finite existence down to its extreme forms, torture and death. And yet, rightly, the infinite will be determined by the in some way ontological resistance of its own being to all the divisions that our imperfect thought might suppose, and which the unity of a finite being cannot stand up against. The infinite is that which maintains the unity of its power in the apparently most extreme forms of division, of partitioning of its being-One. It will be for example at once pure Thought and unlimited Action, its infinity proving precisely that the opposition of thought and action, however effective it may be, does not undermine its ontological unity. Only by thinking the divine infinity in this way – in a manner undoubtedly still implicit – could the council of Nicaea conclude, in 325, that the Father and the Son were “of the same substance.” This dogma in fact signifies with precision the affirmative resistance of infinity against every partition, not only apparent but also real partitions, for it recognizes the effective existence, in God, of two persons. From this time on, the “reasonable” tendency, headed by the bishop Arius and consequently referred to as Arianism, will show itself to be surreptitiously controlled by an imperative of finitude. It proposes in effect to say that the Father and the Son were “of similar substance,” which was apparently obvious, but which sounded like an abdication of divine power and its unity, put to the test of the divisions rendered necessary by the economy of salvation. If indeed the resistance to division of the infinite is not of an ontological type (the Father and the Son are ontologically identical, thus respecting the law of the One) but belongs only to the order of appearances (the Father and the Son are ontologically similar, which imposes the law of the Two), what results is an obvious weakness in the divine: the infinite, at the point of its identity, obeys laws quote similar to those of finitude, for which what is different cannot be identical but bears only a resemblance. Emperor Constantin was from this point of view right to excommunicate Arius and his group.
It may be observed that Spinoza radicalized this form of access to infinity to the extent that he infinitizes the resistance into substantial divisions. He claims in fact that the infinite properly speaking, which is Substance, or God, or Nature, however radically one it is, possesses an infinity of different attributes, the two examples we know about (though what we know on this matter is not much) being extension and thought. These attributes nonetheless all “express” substance, in such a way that the laws of their becoming are the same. This is the famous statement “Ordo et connexio rerum idem est ac ordo et connexio idearum”: things (being extended) and ideas (being thought) compose an order, an order of identical relations. It is this order and these relations that attest in every attribute to the substantial presence of the divine infinity. For Spinoza, the infinity of God’s attributes is not only not a limitation of the power of his unique infinity but resolves itself in the power [pouvoir] to express through limitless differences the unbreakable and fecund unity of this infinite power itself.
c) Power of internal determinations
The third mode of access to infinity is intrinsic, or immanent: the infinite is such that it possesses singular internal determinations, clearly distinct from all those possessed by finite realities. These determinations prescribe a type of being; they are thus ontological and authorize a differential identification of the infinite, which is then thinkable on the basis of itself as being affirmatively other than the finite. One could speak this time of a conceptual path. Theologians looked for these immanent determinations on the side of totalization, for example God as “all-powerful” or “all good,” or again from the side of the fusion of all predicates that remain separate in the finite, for example God as unity of Knowing and Acting. Descartes works in this way with the concepts of being and existence, because, in God, contrary to what happens in the finite, the concept of what he is brings with it, one might say, the obligation to exist: this is the famous ontological argument. Inversely, the infinite can be sought on the side of a separated predicated, whereas in the finite it is always conjoined to another. Already Aristotle thinks God as pure act without potentiality, or pure form without material, whereas a real finite being is always one composed of action and potential, as of form and material. Theologians also looked on the side of metaphors suggesting an identity “outside the empirical,” hence God as Pure Light.
More abstractly, one might imagine that the infinite does not turn out like the first two modes of thought’s relation to it suppose, i.e., through a relation to its exteriority. A relation that, in the first mode, is negative, to wit, an absolute transcendence in comparison to finite constructive operations. And that, in the second, is rather affirmative, like the resistance of the One-infinite to all the partitions one tries to inflict upon it and that it transforms in expressing its own power. The third path of access to the infinite no longer presumes to characterize it by its relation to that which it is not but by interior sui generis determinations, which are like an immanent ontological push demanding the infinite extension of the being that supports these determinations.
d) Intimate relation to the Absolute as such
The fourth mode of access is tendential, or by successive approximations. The infinite is thought as an approximation [rapprochement] of a figure of the Absolute, ultimately identified with Nothing or with the unthinkable as such. In this case, it remains in a sense inaccessible, but what is accessible is a rational movement toward it. In other words, access to the infinite is created by a sort of infinitization process, where the infinite is approached in such a way that it orientates the process itself. This path admits a hierarchical thinking, for there are degrees of proximity with the One-infinity as inaccessible absolute. But it animates this structure through a sort of mystical ecstasy, to the extent that the polarity of the ascendant trajectory of gradations also approximates an annihilation in ecstasy. One finds precise indications on this point in the angelology of the scholastic theologians and in its paradoxical correlation with the mystical. On one hand, the hierarchy of angels, from quasi-terrestrial guardian angels to the immaterial seraphim, are actualized conceptually by types of proximity more and more tight with the divine One-infinite. On the other, the mystical relates in poems the experienced figures of ecstatic annihilation towards which the hierarchy of angels points. For this reason Rilke was correct to write that “every angel is terrible”: it is in effect a gradation of being that measures our proximity to nothingness. One recognizes in this orientation a conception of infinity as absolute referent (to take up our vocabulary once more), which is in a sense inexistent but in another sense the norm of existence and which lets itself in this respect be approached through experiences or forms of thought that are more and more intense, or more and more synthetic.
3. Rationalisation of the formal achievements of theology
Let us sum up. One can think the infinite from below: it is that which surpasses the finite given and the operations that authorize this given. One can think it through its resistance to the divisions it nonetheless sustains. One can think it through itself: it is what possess a predicative identity that distinguishes it intrinsically from finitude. One can finally think it from above: it is that which tends, by successive degrees, toward a limit-point beyond of which nothing is.
The mathematical secularization – by Cantor – of the infinite immediately implies the separation of the infinite from the One. That there are totally different infinites is without a doubt the most fundamental of discoveries of this secularization. It is all the more remarkable that this novel thought has, in essentials, not only guarded but rationalized and clarified the four modes of access to this notion. One will not be surprised, given these conditions, that Cantor, who was truly very troubled mentally by his own genius and by the radicality of his inventions, asked Rome – the Pope – if his mathematical concept of actual infinity wasn’t in the end blasphemous. I think in fact that it is, because, remaining foreign to the One, the diverse forms of the multiple-infinite cannot be pictured as persons, not even analogically. However, if the being of the infinite, being a singular type of multiplicity, cannot be thought under the term “God,” it is nevertheless truth that it is situated between finitude and the absolute reference, and it is also true that there are four different ways of effectuating this situation. That is what the rest of this chapter will examine from the point of view of ontology properly speaking, such as it takes its instruction from the thought of the possible forms of multiple being, namely, from the formalized theory of infinites, or the theory of the power of V.
4. The four cases of infinity revisited in their formal style
To introduce ourselves into the domain of the modern theory of infinities, we need to recall for the moment that there exists a way of measuring the dimension of any set whatsoever, whether it is infinite or finite. This measurement is guaranteed by the special numbers that Cantor invented and which are called cardinal numbers. We repeated their definition in the Prologue. The underlying intuition rests upon the finite numbers, the sequence 1, 2, 3…, with which we are all familiar. The number 11, for example, has two meanings. It designates a possible order in which objects range, for example the players of a football team, from 1 to 11 (one of these numbers and only one is written on the player’s jersey). Taken in this way – as an ordered count—, 11 is an ordinal number. It allows for a term by term correspondence between the players of two teams, and “11” summarizes the possibilities of this correspondence, which is called one-to-one.
Cantor projected these considerations into the infinite and thus laid the groundwork for a general “measure” of the extension of any multiplicity. The purely quantitative type of a set is the cardinal number with which this set is in one-to-one correspondence. This number is called the cardinality of the set under consideration. One notes |A| the cardinality of the ensemble A.
A parenthesis. In the Prologue, we spoke about the polemic over the eighth axiom of ZFC, the axiom of choice. I proposed diverse arguments in its defense, among which the clear kind of order this axiom introduced in many parts of mathematics, even if otherwise there are some surprising consequences for our – weak – intuition. It is time to recall that, for example, the fact that all infinite sets (in the sense of being equal or superior to ω) can be measured by one cardinal is a consequence of the axiom of choice. Without this axiom, the universe of sets is essentially deprived of order. This is intuitively comprehensible: without the axiom of choice, one cannot “represent” what belongs to a set E by an in some way anonymous element from each of the sets that compose E. I say anonymous, because the axiom of choice says that one can do it, without saying precisely how. But if this resource is disallowed, if the representations of the elements of E has to be guaranteed by an explicit procedure, the shadow that covers the notion of any multiplicity whatsoever is then such that even to compare two sets quantitatively becomes generally impossible. In practice, all the brilliant calculations for any multiplicity, invented by Cantor and of which we are going to make continual use in this book, is annihilated if one refuses the axiom of choice. And thus equally so if one claims to impose an intuitionist logic on the theory, for that would amount to rejecting it.
If, thanks to the axiom of choice, we instead make use of the ordered theory of cardinals, we can perfectly rediscover our four concepts of infinity, which are given in the following four questions:
- Can the infinite under consideration be defined as inaccessible to the operations whereby a set can be constructed “from below,” that is, on the basis of sets whose cardinality is inferior to its own? This is the infinite by transcendence.
- Can the infinite under consideration, if submitted to a division, a partition, of its multiple-substance, resist in the following way: at least one part of the infinite considered, which has the same cardinality as the entire set, possesses all its elements in the very “piece” created by the partition? This part, albeit submitted to the general operation of division, preserves the extension of the entire set. This is the infinite by resistance to division.
- Is the infinite under consideration endowed with internal properties, concerning the elements or parts that compose it, such that they put it in a position of superior power over all those [properties] already obtained? This is the infinite by immanence, or in interiority.
- Is the infinite under consideration “nearer” to the absolute (defined as the set of all sets, or general thought of the possible forms of being-multiple) than other types of infinites are? This is the infinite through hierarchical proximity with the absolute.
We have there only a formal problematic that still maintains itself very close to the medieval theories concerning the divine infinity. One rediscovers in effect the four roads of access to the thought of God proposed by rational theology: infinity by negative transcendence, by maintenance of the substantial unity in the test of real divisions; through immanent determinations of the “faculties” of God; and through ineffable proximity to the incomprehensibility of his designs. To go farther, and to obtain a secular vision of the different types of the infinite, one will have to “harden” the formalism considerably and enter into mental operations that, while including bright spots of clarity at the end, cannot mask the complexity of the steps.
For now, let us be satisfied with these preliminary remarks on each of the four concepts of infinity, of which we propose to construct the unequivocal form.
[Note: the part titles for the rest of the chapter are as follows: 5. Case 1: inaccessibility, 6. Case 2: partitions of an infinite and resistance to these partitions, 7. Case 3: immanent power of large parts of an infinity, 8. Case 4: proximity to absolute V, 9. The notion of class, 10. Relation j (elementary plunge) between V and a transitive class M, 11. Towards the very grand infinities, 12. Provisional conclusion]