Concept and Form:
The Cahiers pour l’Analyse
and Contemporary French Thought

The concept of ‘stratification’ is a relatively common one in modern logic and mathematics as a result of W.V.O. Quine’s efforts to simplify Russell and Whitehead’s ‘theory of types’ in the Principia Mathematica via an axiomatic set theory in his 1937 article, ‘New Foundations for Mathematical Logic’.

In a later text incorporating the insights of this article, Quine explained the basic ideas as follows: ‘Let us speak of a formula as stratified if it is possible to put numerals for its variables (the same numeral for all occurrences of the same variable) in such a way that "€" comes to be flanked always by ascending numerals ("(n € n+1)").’¹ Quine’s innovation involves thinking of mathematical and logical sequences in terms of a layering with neither remainder nor recursion. Following upon a question he put to Frege, Russell had posed a challenge to the foundation of mathematics in which the problems generated by the recursive properties of a predicate that was both function and term came to the fore. Russell’s paradox turns on consideration of the ‘set of all sets who are not members of themselves’. To satisfy (as function) this formulation is at the same time to be excluded (as term) from the domain it names.² Russell sought to avoid this problem by creating a hierarchy of types, in which each mathematical entity would be assigned a ‘concept’, in Frege’s rubric, or ‘type’, in Russell’s. In order to prevent recursive loops such as the one found in his paradox, objects of a given type must be built exclusively from objects of preceding types, i.e., those lower in the hierarchy. Lacking an upper limit, the hierarchy of types did have a lower limit, namely, the ‘type’ meaning ‘describing nothing’. The mere description of this lowest type is what then generates a second type.³

Quine’s account of stratification is at once a simplification of Russell’s theory of types and an effort to avoid the problems that it generates in turn. Perhaps the most crucial flaw in Russell’s framework - at least according to Ludwig Wittgenstein - can be described as a problem of metalanguage. In each instance, Russell must presuppose the meaning or the sense of the functional terms he uses. How do we get the initial ‘object’ that allows us to build new types of ‘objects’ out of a previous type?⁴ Stratification in the Quinean sense responds to this challenge by accepting, with an eloquent simplicity, that some formulae are ‘stratified’ and others are not. If it can be established that a formula is generative of an ascending sequence, lacking recursion and excluding remainder, then that formula is stratified.

The technical exigencies of logical stratification are complex and remain debated by logicians today. The crucial importance of the concept for the Cahiers pour l’Analyse lies in its descriptive and functional opposition to any sense of closure, or indeed suture, within a formalised logical sequence.

In the Cahiers pour l’Analyse

The concept of stratification is central to Alain Badiou’s argument in ‘Marque et manque: À propos du zero’ (CpA 10.8), which is a critique of Jacques-Alain Miller’s attempt to articulate a ‘logic of the signifier’ that subtends science and accounts for the ideological suture that establishes the subject (CpA 1.3). Badiou proposes to marshal ‘the stratification of the scientific signifier’ (CpA 10.8:150) against Miller’s attempts to preserve the ideological category of the True with recourse to an inaugural marking of lack (i.e., Frege’s generation of numbers via the marking with zero of the category of lack itself, ‘the non-identical to itself’).

In a word, Badiou suggests that Miller does not understand the stratified nature of logical syntax, a syntax that excludes lack – that is, is foreclosed to any possibility of lack - and is without remainder. Badiou does not deny the ideological phenomenon of suturation, and the fact that it is always in play with individual scientists. Rather, his concern is with the concatenation of the ‘absolutely primary raw material of the logical process’, that is, the marks that comprise writing and their formation into a syntax riven by a singular and constitutive scission that divides ‘well-formed’ from ‘ill-formed’ expressions. This division that establishes the category of the ‘well-formed’, by the same measure, excludes lack from this domain. Referring to Gödel’s undecidability theorems, Badiou claims, ‘The aporias of derivation can be ascribed only on condition that there be a perfect syntax […]. [T]he existence of an infallible closed mechanism is the condition for the existence of a mechanism which can be said to be unclosable, and therefore internally limited. The exhibition of a suture presupposes the existence of a foreclosure’ (CpA 10.8:153).

More emphatically, ‘the logico-mathematical signifier is sutured only to itself. It is indefinitely stratified’ (CpA 10.8:156). What this means is that sequences formed by the marks that constitute logical syntax are in no instance acts of a regressive or recursive suturation covering over a previous lack. Badiou affirms that his concern is ‘the strictly functional essence of the logical mechanism’s internal references. Nothing here warrants the determination of object. The thing is null here; no inscription can objectify it’ (CpA 10.8:156). This notion of ‘internal reference’ is key, for the point is precisely that the marks that constitute the syntax of scientific writing only refer to other marks. What is more, it is absolutely crucial that all the marks are self-identical. External reference is not the concern of scientific writing. In a footnote in which he contrasts his position with Quine’s and that of Quine’s student Hao Wang, Badiou writes: ‘For our part, we are convinced that the stratified multiplicity of the scientific signifier, which is inherent to the process of scientific production, is irreducible to any of its orders. The space of marks does not allow itself to be projected onto a plane. And this is a resistance (or limitation) only from the viewpoint of a metaphysical will. The will of a science is the transformation-traversal of a stratified space, not its reduction’ (CpA 10.8:161). The insistence on productive stratification against reductive (or recursive) suture can be read as an instance of Badiou against Miller’s Lacan.

Indeed, ‘there is no subject of science’ for Badiou, a fact which is a consequence of stratification: ‘infinitely stratified, regulating its passages, science is pure space, without inverse or mark or place of what it excludes’ (CpA 10.8:161). Badiou is ultimately direct in his rebuke of Miller: ‘the scientific signifier is neither sutured nor split, but stratified. And stratification repeals the axiom by which Miller, in another text, characterized foreclosure: the lack of a lack is also a lack (CpA 9.6). No; not if that which comes to be lacking was always already marked: then the productive difference of strata suffices to name the interstice’ (CpA 10.8:161). If the subject is predicated on the suturing of a lack, and the subject is the key to ideology, then science accomplishes the epistemological break that distinguishes it from ideology precisely through the foreclosure of lack. Stratification is the mechanism of this foreclosure.

Miller responded to Badiou’s charges in a text titled ‘Matrice’, composed in 1968 but only published a few years later in Ornicar ? (1975/76). There Miller refines his position, incorporating a logic of stratification into his own framework, while reaffirming the univocal quality of lack. Miller grants a measure of primacy to the mark but insists that it is always accompanied by a lack, a lack that takes the form of the mark’s ‘place’. ‘To inscribe a mark is to set down two things: the mark (its materiality, the trace of ink for example) and its place. If one erases the mark, the trace remains, under the form of place. From which derives - does it not? - a minimum of two series: that of marks, that of lacks.’⁵ Further in his analysis, ‘stratification’ comes to be codified as the ‘infinite repetition’ of this sequence.⁶ But this sequence must take place somewhere, and it is the ‘universe of discourse’ itself that is an ‘unstratifiable […] dispersed totality’. The impossibility of this totalisation is itself an element in the sequence for Miller: ‘It is with the fall of the lack, of the impossible element, that the stratification of the universal language becomes possible. Seen from the perspective of stratified discourse, this is nothing but an illusion, an illegitimate operation, a non-being, an appearance, a mirage, a semblance of being, which disappears.’⁷

Select bibliography

• Miller, Jacques-Alain. Un début dans la vie. Paris: Gallimard, 2002.
• Quine, W.V. Mathematical Logic. Cambridge: Harvard University Press, 1940.
• ---. ‘New Foundations for Mathematical Logic’, in From a Logical Point of View, 2nd ed., revised. Cambridge: Harvard University Press, 1980.
• Russell, Bertrand. Essays in Analysis, ed. Douglas Lackey. London: George Allen and Unwin Ltd., 1973.
• Van Heijenoort, Jean. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge: Harvard University Press, 1967.