Concept and Form, Volume 2: Interviews and Essays on Cahiers Pour L'Analyse by Peter Hallward/Knox Peden

Author:Peter Hallward/Knox Peden [Peter Hallward/Knox Peden]
Language: eng
Format: epub
Tags: Humanities, Philosophy, Politics & Social Sciences, Movements, Structuralism
ISBN: 9781844678730
Amazon: 1844678733
Publisher: Verso
Published: 2012-12-12T05:00:00+00:00

It is the technical invariance of traces and instruments that subtracts itself from all ambiguity in the substitution of terms. Thus determined, the rule of self-identity allows of no exceptions and does not tolerate any evocation of that which evades it, not even in the form of rejection. What is not substitutable-for-itself is something radically unthought, of which the logical mechanism bears no trace [. . .]. What is not substitutable-for-itself is foreclosed without appeal or mark (CpA 10.8:157).

On the basis of this scriptural self-identity, Badiou goes on to show how the essential operation of (genuine or scientific) logic – for instance the sort of logic at work in formal propositional calculus – involves the ‘separation’ or distinction of statements as valid or invalid on several distinct levels, or according to several stratified ‘mechanisms’. The primary level or mechanism (M1), the mechanism of ‘concatenation’, produces arrangements of a discrete set of elementary marks or letters, as a sort combinatorial ‘alphabet’ – by way of illustration, consider for instance all the possible combinations of the letters used in the English language. At a second level, the level of ‘formation’ (M2) or ‘syntax’, certain expressions are deemed acceptable or ‘well-formed’, and separated from those that are rejected as ill-formed (152); to pursue our linguistic analogy, at this second level we might distinguish between syntactically valid (though not necessarily coherent or meaningful) sequences of words, as opposed to random series of letters. The third level (M3), the mechanism of ‘derivation’, will further separate sequences that can be derived or ‘proved’ as valid theses from sequences or statements that cannot be thus proved, i.e. from ‘non-theses’. We might call this third level the domain of intelligibility or coherence, the level at which coherent expressions are distinguished from unintelligible albeit ‘well-formed’ strings of words.

Now whereas the distinction of well-formed from ill-formed expressions at level two is absolute and straightforward, the relation between derivable and non-derivable expressions at level three may, in any system complex enough to formulate basic arithmetical expressions, be either decidable or non-decidable. An undecidable statement is formulated in such a way that ‘neither it nor its negation is derivable’ or provable (153) – and although the details need not detain us here, Badiou proceeds to show how Gödel’s demonstration of the incompleteness of any relatively complex logical system applies to this level three (M3) alone. The important thing is that admission of such incompleteness or undecidability, pace Miller and Lacan, in no way threatens the self-sufficient or subject-less status of science, i.e. its exclusion of all lack or non-self-identity. No matter what is expressed of it, any mark x must itself always be identical to itself: as a sign or mark, every x must be and remain this same x. In other words, expression of non-self-equality at level two does not carry any ontological implications of non-self-identity at level one. It is certainly possible to express a lack of equality between x and itself, with the well-formed (though unintelligible) statement ‘x ≠ x’ – both x = x and x ≠ x are legitimate expressions at level two, i.

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