Badiou on Forcing

“There are two labyrinths of the human mind: one concerns the composition of the continuum [consistent multiplicities], and the other the nature of freedom [the event], and both spring from the same source—the infinite [inconsistent multiplicities].”

—G.W. Leibniz (“On Freedom,” in G.H.R. Parkinson (ed.), Philosophical Writings, trans. Mary Morris and G.H.R. Parkinson [London: J.M. Dent, 1973] 107)

Fraser on the subject

Zachary Fraser, “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus,” Cosmos and History: The Journal of Natural and Social Philosophy 2.1-2 (2006): 94-133:

Abstract [reformatted]: One of the central tasks of Badiou’s Being and Event is to elaborate a theory of the subject in the wake of an axiomatic [assumed] identification of ontology with mathematics, or, to be precise, with classical Zermelo-Fraenkel set theory. The subject, for Badiou, is essentially a free project that originates in an event [a rupture in the situation as it is], and subtracts itself from both being qua being [i.e., because the event is “not one”/”zero” (i.e., it hasn’t been “counted-as-one” in the situation—by the “state”), whereas elements of the situation are “one”], as well as the linguistic and epistemic apparatuses that govern the situation [i.e., it is “indiscernable,” viz., neither recognizable nor nameable]. The subjective project is, itself, conceived as the temporal unfolding of a “truth” [i.e., a response to an event, a.k.a. a “generic procedure,” be it scientific, political, aesthetic, or “amatory”]. Originating in an event and unfolding in time, the subject cannot, for Badiou, be adequately understood in strictly ontological, i.e. set-theoretical, terms, insofar as neither the event nor time have any place in classical set theory [i.e., events and history are outside ontology].

While the state orders (counts) all officially sanctioned elements of the situation, a truth gathers together (accounts for) those that relate to the event.

N.B.

a. elements, i.e., multiples, are

  1. normal: presented in the situation and represented by the state, i.e., counted-as-one twice;
  2. excrescent: represented by the state—but not presented; or
  3. singular: presented in the situation—but not represented.

b. situations are

  1. natural if they contain only normal and excrescent multiples (e.g. an ecosystem), i.e., there can’t be anything new in the situation;
  2. neutral if they contain normal, excrescent and singular multiples; or
  3. historical if they contain at least one singular multiple, i.e. an event (e.g. what it is to be an Australian—the singular multiple is “Aboriginals”), i.e., there can be something new in the situation (although it must be recognized and named).

The truth must be constructed; the operation has several moments, some of which presumably overlap:

  1. nomination: the event is named, e.g., “I love you” or set theory;
  2. intervention: the multiple is recognized as event, e.g., [if] we did fall in love or [if] we apply set theory (= a leap of faith or wager—an “as if,” though we tend to ignore its hypothetical or axiomatic nature), thus the name is imposed on the situation;
  3. investigation [enquiry]: the elements of the situation are divided one by one into those that fit the evental name (= connexion [+]) and those that don’t (= disconnexion [-]), e.g., . . . and now the strange tension between us makes sense or . . . and set theory can legitimately be applied to the ontology of the situation, thus the situation is reconstructed;
  4. fidelity: the virtue required to sustain an  endless sequence of investigations, e.g., we continue to understand our life together in terms of that love or set theory now founds our other enquiries, thus, the situation is thought according to the event;
  5. forcing: the event that was included (as a part) in the situation is made to belong there (as an element) by us reorganizing the situation to make room for it, e.g., thus our life is that love or thus set theory is proved, thus its truth (vérité) can be confirmed, i.e. verified (i.e., véridique).

“Force,” according to Jason Barker,

is the historical power of producing something new, or of forcing truth to be true (truthful) in the new situation . . . [thereby] transforming—or, better still, deforming—the elements of the original situation in the process. (Alain Badiou: A Critical Introduction 107)

Thus, the subject can “force” truth (to continue Fraser’s abstract) . . .

Badiou nevertheless seeks to articulate the ontological infrastructure of the subject within set theory, and for this he fastens onto Cohen’s concepts of genericity and forcing: the former [genericity] gives us the set-theoretic structure of the truth to which the subject aspires [as the generic extension of the situation by supplementing it (cf. with conditional [if . . . then], rather than declarative statements)], the latter [forcing] gives us the immanent logic of the subjective procedure, the “law of the subject” [as the creation of generic sets]. Through the forcing operation, the subject is capable of deriving veridical statements from the local status of the truth that it pursues [Badiou calls it “fictioning”—see below].

(See also Peter Hallward’s more precise definition of forcing in Alain Badiou, “The Problem of Evil,” Ethics: An Essay on the Understanding of Evil [Verso, 2002] 87-88 [58-89].)

Fraser continues:

Between these set-theoretic structures, and a doctrine of the event and temporality, Badiou envisions the subject as an irreducibly diachronic unfolding of a truth subtracted from language, a subject which expresses a logic quite distinct from that which governs the axiomatic deployment of his classical ontology. This vision of the subject is not unique to Badiou’s work. we find a strikingly similar conception in the thought of L.E.J. Brouwer, the founder of intuitionist mathematics. Brouwer, too, insists on the necessary subtraction of truth from language, and on its irreducibly temporal genesis. This genesis, in turn, is entirely concentrated in the autonomous activity of the subject. Moreover, this activity, through which the field of intuitionistic mathematics is generated, expresses a logical structure that, in 1963, Saul Kripke showed to be isomorphic with the forcing relation.

In the following essay, I take up an enquiry into the structure of these two theories of the subject, and seek to elucidate both their points of divergence and their strange congruencies; the former, we will see, primarily concern the position of the subject, while the latter concern its form. The paper ends with an examination of the consequences that this study implies for Badiou’s resolutely classical approach to ontology, and his identification of ontology as a truth procedure. (94; emphases and italics added)

As Badiou puts it, “[t]he forcing is the powerful fiction of a completed truth. A completed truth is a hypothesis, it’s a fiction, but a strong fiction” (“Alain Badiou—’On the Truth-Process,’ August 2002, European Graduate School, EGS, Saas-Fee, Switzerland,” egs.com, 28 Nov. 2009). Here is his ramble on genericity:

What happens is only that we can anticipate the idea of a completed generic truth. It’s an important point. The being of a truth is a generic subset of knowledge, practice, art and so on, but we can’t have a unique formula for the subset because it’s generic, there is no predicate for it, but you can anticipate the subset’s totalization not as a real totalization but as a fiction. The generic Being of a truth as a generic subset of the situation is never presented. You have no presentation of the completeness of a truth, because truth is uncompletable. However, we can know formally that the truth will always have taken place as a generic infinity. We have a knowledge of the generic act and of the infinity of a truth. Thus the possible fictioning of the effects of its having-took-place is possible. The subject can make the hypothesis of the situation where the truth of which the subject is a local point will have completed its generic totalization. Its always a possibility for the subject to anticipate the totalization of a generic being of that truth. I call the anticipating hypothesis a forcing. The forcing is the powerful fiction of a completed truth. A completed truth is a hypothesis, it’s a fiction, but a strong fiction. Starting with such a fiction, if I am the subject of the truth, I can force some bits of knowledge without verifying this knowledge. Thus, Galileo could make the hypothesis that all nature can be written in mathematical language, which is exactly the hypothesis of a complete physics.

We’re talking a generic “totalization” of truth—though truth can’t be totalized, i.e., completed. Note the terms: “anticipation,” “fiction,” “fictioning,” hypothesis,” “possibility”; they indicate we’re in Badiou’s axiomatic Wonderland, where we act as if on the basis of the coherency of his mathematical thought-experiment.

Genericity is a complex matter, but Badiou’s examples are straightforward—see Peter Hallward, Badiou: A Subject to Truth (U of Minnesota Press, 2003) xxix (and 131):

Badiou’s contemporary examples from the four “conditions of philosophy” (truth procedures) are, somewhat predictably, “democratic” or anti-hierarchical phenomena:

  1. in maths, set theory from Cantor to Cohen;
  2. in politics, popular uprisings from the Cultural Revolution in China to May ’68 in France;
  3. in poetry, the writings of Mallarmé, Pessoa and Celan;
  4. in love, the work of Lacan (xxx).

The discussion of Rousseau’s concept of the “general will” is useful (Meditation 32); see Nina Power, “Towards an Anthropology of Finitude: Badiou and the Political Subject,” The Praxis of Alain Badiou, ed. Ashton et al. (Melbourne: re.press, 2006) 314ff.

Presumably, by situating his philosophy in “a thinking space wherein the four conditions of philosophy can be gathered and located as compossible,” as he does in the Manifesto for Philosophy (105), he would consider his philosophy a “generic conception of reality.” If maths, i.e., set theory, describes the being-qua-being of the situation (as ontology it concerns itself with “the genericity of multiple-being”), philosophy describes what is not being-qua-being, i.e. the event, in particular, the effects of an event that supplements a situation, e.g., set theory (it concerns itself with genericity per se).

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  1. Pingback: Stirner’s Subject | dingpolitik

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