The work of Graham Priest and Gilles Deleuze (and Félix Guattari) converge in significant ways on the concept of the nondenumerable.
Turning to Priest first, and to his Beyond the Limits of Thought especially, one finds in this book an interesting history of philosophy, and one with a particular narrative at work; namely, he uncovers numerous contradictions that are encountered as certain unthinkable limits to thought become the subject of thought itself (e.g., primary substance for Aristotle, God for Cusanus, the noumenon for Kant, among other examples). In the history of thought prior to Hegel, according to Priest, these contradictions were largely denied, primarily through a denial of the very limits that gave rise to them. But with Hegel there is an open recognition and affirmation of the contradictory nature of the limits of thought. It is for this reason that Priest claims that the ‘chapter on Hegel [in Beyond the Limits of Thought] is therefore the lynch-pin of the book.’ (7).
Priest cites Hegel’s own recognition of these contradictory limits in his Science of Logic where he argues that
great stress is laid on the limitations of thought, of reason, and so on, and it is asserted that the limitation cannot be transcended. To make such an assertion is to be unaware that the very fact that something is determined as a limitation implies that the limitation is already transcended. For determinateness, a limit, is determined as a limitation only in opposition to its other in general, that is, in opposition to that which is free from the limitation; the other of a limitation is precisely the being beyond it. (Logic, 134, cited by Priest, 108)
In writing of Hegel, for instance, Priest argues that Hegel’s understanding of the limits of thought, and the contradiction attendant upon this understanding, was disadvantaged because Hegel ‘had only a rudimentary understanding of the boundary-tearing mechanism which transcends limits,’ a mechanism such as that found with ‘diagonalisation’. (109).
Central to understanding the mechanism of diagonalization is the distinction between denumerable and nondenumerable sets. Put simply, if we take the set of natural numbers then any subset of natural numbers, whether finite or infinite, is countable if it can be paired one for one with a natural number. For example, if we take an infinite sequence of sets, (s1, s2, s3, s4,…), where each of these sets consists of an infinite sequence of 1s and 0s, these sets are nonetheless countable since they can be paired one for one with the natural numbers, and hence such sequences are said to be countably infinite. What Cantor was able to show was that one could construct a unique set, s0 let us say, that cannot be paired up with any of the natural numbers and hence is uncountable—or, it is nondenumerable. This set is nonetheless constructed from within the countable sets and hence the elements of the construction of the set are included within the infinite sequences but the set of these elements does not belong, or cannot be counted among one of the infinite sequence of sets. There is thus a paradox, Cantor’s paradox, where there is a set, s0, that both belongs to the set X (of natural numbers) in that it was constructed by drawing from the members of the sets that do belong to X, and at the same time this set does not belong to X since it cannot be countably related to the natural numbers. Hence the paradox: s0 ∈ X and s0 ∉ X. Such paradoxical sets are also referred to as inconsistent sets since it is not possible to construct a functional relationship that would map their elements one for one with the natural numbers, unlike consistent sets where this can be done.
For Priest this method of diagonalization assumes a prominent role in the arguments of the latter two thirds of his book, and he claims that there is a common paradox at the heart of much of ancient and modern philosophy, and in Nagarjuna as well, when it comes to attempts to think the limits of thought, a thought that entails a move beyond the limits of thought. Priest refers to this as the inclosure schema, or the inclosure paradox, which he states as follows:
(1) Ω = {y; φ(y)} exists, and ψ(Ω)
(2) if x is a subset of Ω such that ψ(x): (a) δ(x) ∉ x
(b) δ(x) ∈ Ω
Stated in layperson’s terms, and tying in with the diagonalization argument from above, (1) corresponds to what Priest calls the Existence condition, by which he means there is a set of elements that exists and is definable. (2) brings us the paradox of Transcendence and Closure (which is a prominent and continuous theme in Priest’s book). By virtue of the diagonalization function, δ(x), we end up with a diagonalization of a subset that gives us a set that does not belong to that subset although it may belong to a larger, more inclusive set, denoted by the Ω. Where the inclosure paradox of limits comes in is when we consider the limit set itself, Ω, and apply the diagonalization function to it. What we end up with, Priest points out, is a case whereby we have a set that both belongs to Ω (or we have Closure) and yet does not belong to Ω (Transcendence). This is a case of a true contradiction, according to Priest, and it is such true contradictions that are central to what Priest refers to as dialetheism (see earlier post).
Turning to Deleuze now, and in particular to Deleuze and Guattari’s A Thousand Plateaus, we find that here too the nondenumerable assumes an important position. As Deleuze and Guattari put it in A Thousand Plateaus,
What characterizes the nondenumerable is neither the set nor its elements; rather it is the connection, the “and” produced between elements, between sets, and which belongs to neither, which eludes them and constitutes a line of flight. (470).
First off, there’s a clear similarity between the Deleuzo-Guattarian notion of a line of flight and the function of diagonalization. In both cases the result is something that eludes belonging to countable sets. In the case of DG, however, they place the emphasis neither on sets nor on the elements of sets, but rather on the ‘“and” produced between elements, between sets.’ This emphasis upon the “and” connects with Deleuze’s own concern with empiricism, which he believes is best characterized by the “and” (I’ll turn to Hume below). But as we have seen, the diagonalization method was a method of producing a nondemumerable set by taking an element from one set, “and” another, “and” another, such that the result is a nondenumerable set (though Deleuze will prefer the term multiplicity). DG are thus not offering a heterodox reading of the nondenumerable. Moreover, we gain an important insight concerning ‘line of flight’ by comparing it to the diagonalization function. A line of flight, contrary to Peter Hallward’s reading of Deleuze (in Out of this World) is not an attempt to get out of this world, to escape this world, but it is rather an effort to construct and compose a nondenumerable multiplicity while remaining fully within the world, much as Cantor was able to construct a nondenumerable set by drawing solely from countable sets.
We can also better understand DG’s critique of Badiou in What is Philosophy? For Badiou what is the process whereby an inconsistent set becomes consistent and definable. For Badiou Cohen’s notion of forcing plays a key role in this understanding. What this entails, however, is a subjective intervention that imposes consistency upon the inconsistent. The problem with this perspective, and one that echoes Priest’s own arguments, is that whatever consistency is produced by way of a subject’s faithfulness and forcing can itself, by way of the diagonalization function, become yet another inconsistent set. Rather than a subject that needs to intervene from outside a given, countable situation in order to “force” a consistency that heretofore did not exist, for Deleuze and Guattari the revolutionary project entails creating the connections, the between of elements and sets, the “and,” that gives rise to the nondenumerable that problematizes any given situation and prompts its possible transformation (this is all too brief, but for more on my critique of Badiou see my Deleuze’s Hume). With this in mind we can turn to DG’s analysis of Badiou in What is Philosophy?.
After describing in their own terms much of what we have detailed above about Badiou and nondenumerable sets, DG refer to Badiou’s invocation of the void as an effort to render such sets consistent, an effort which DG claim reintroduces ‘the transcendent’. We can see why they would interpret Badiou in this way. Rather than working immanently from within situations and states of affairs to compose a nondenumerable that leads to the efforts to create a consistency, Badiou introduces the void of a situation, that which cannot be related to any situation, and calls upon a faithful subject to ‘naturalize’ such events so that they can acquire the consistency that enables the events to become placed within the continuity of historical processes. Another important difference, as DG read Badiou, is that whereas Badiou begins and ends with functional relationships between states of affairs, situations, and sets (e.g., functional relationships between countable sets) that are characteristic of science, while DG begin with problematic, nondenumerable multiplicities that are inseparable from countable sets and states of affairs (recall that δ(x) ∈ Ω). In doing the former, DG believe that Badiou ultimately invokes the transcendent, despite his claims to the contrary (which is also probably why Badiou could not see himself in DG’s characterization), whereas the latter approach tracks the path of immanence. To state the difference as this is developed throughout What is Philosophy?, DG are interested in developing philosophical concepts while Badiou is interested in formulating scientific concepts.
I’ll close with one more example – Hume. On my reading of Deleuze’s Hume, or the Deleuzian Hume that accounts for the profound sense in which Deleuze was a Humean throughout his career, the impressions and ideas are not to be understood as a countable set but are rather a nondenumerable multiplicity that becomes, when actualized, the bifurcations of conceptual thought. In Priest’s terms, the multiplicity Hume thinks in his Treatise is a limit to thought itself and gives rise to contradictions when it is thought. For example, there is the well-known contradiction in Hume’s thought, and one Hume himself despaired of in the appendix to the Treatise, regarding the self. On the one hand the self is understood to be nothing but a bundle of impressions and yet, through much of the latter half of the Treatise and in his essays, the self is the assumed and unquestioned condition for many of his analyses of the passions, justice, politics, etc. If the multiplicity of impressions and ideas is understood as a nondenumerable condition inseparable from thought, then as this condition is thought it gives rise to the bifurcations and contradictions that attend conceptual identifications (or intellectual mitosis as this was discussed in an earlier post), as is evidenced in this case with Hume’s attempts to delimit the identity of the self – he too encountered the contradictory limit.
With the notion of the nondenumerable, therefore, there is indeed an interesting and important convergence between Priest and Deleuze