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Wittgenstein and Turing
One of the aims of this chapter, Livingston says, is to see how language is finite and infinite for Wittgenstein. It has important philosophical and critical implications for how we should understand the social, political, and technical consequences of the development and spread of information and computation technologies. The relationship between finitude and infinity in symbolism play a central role in the rule following considerations for Wittgenstein and the proof of the insolubility of Hilbert’s decision problem for Turing. Livingston says the two reveal a problem that is of critical importance for contemporary thought: the relationship of language’s finite symbolic corpus to the (seemingly) infinity of its meaning.
Wittgenstein’s philosophy of mathematics is sometimes called finitist. Livingston thinks this should be revised based on the following statement by Wittgenstein: “Remember that the ‘propositions about infinite numbers’ are all represented by means of finite signs”. He says the point isn’t that signs cannot refer to infinite numbers or that propositions referring to them are meaningless. The point is that even propositions referring to infinite numbers must have their sense through a finite symbolization (such as a proof of finite length). These proofs give us whatever epistemic access we can have to infinite quantities and numbers. Wittgenstein says that all the forms of possible meaning must already show up in the (formal) possibilities of significance in a finite language. So, the problem of the meaning of the infinite is a problem of the logic or grammar of finite signs, how the possibilities of signification in a finite language can give us access to infinite structures. These reasonings are similar to the ones that underly Hilbert’s project.
In a discussion with Turing, Wittgenstein distinguishes himself from the finitist who would argue that we cannot say that my capacity includes any more than actually has occurred or will occur. It limits what we as human beings are capable of. The question of how this infinitary capacity is based on our actual contact with a finite number of signs and rules was central to the Philosophical Investigations and in Turing’s development of the definition of a universal computing machine (which demonstrated the unsolvabillity of Hilbert’s decision problem: the problem of whether there is a finitely specifiable algorithm that can determine the truth or falsity of any arbitrary mathematical statement).
The Turing or Turing-Church thesis suggests that every number or function that is effectively computable at all is computable by some Turing machine. So, the architecture of the Turing machine captures, replaces, or formalizes the intuitive notion of computability. This this is nearly universally accepted today, but Livingston points out, this should not blind us to the philosophical issues that are involved in this particular way of understanding the nature of a technique and the kind of relation between a finite calculus and its infinite application. For example, according to Turing’s thesis, what it means to be calculable at all is to be calculable by finite means. Turing’s justification for this is in the finitude of human cognition and states of mind.
The negative result of Turing’s thesis is an application of Cantor’s diagonalization, which underlies Godel’s incompleteness theorems, which is similar to Turing’s results. Diagonalization is essentially an intervention on symbolic expressions. It depends on a meaningful procedure that is necessarily captured an a symbolic expression which itself combines signs according to definite rules. So, diagonalization is always a procedure that is concerned with not only numbers (or other mathematical objects) but the ways in which procedures and numbers are expressed by finite strings of finite symbols through which language accomplishes its infinitary powers of symbolization. Now, Wittgenstein was skeptical of the results of diagonalization but that does not mean that this same skepticism pervades his view of the relationship of finite symbolism to infinite applications. Since, diagonalization is the “outcome” of an infinite procedure it cannot be said to have finished, but Wittgenstein doesn’t deny that there is such a procedure.
Livingston then offers two preliminary conclusions. First, Wittgenstein was probably never a finitist. He never held that the finite character of language implies the non-existence of infinite procedures. He focused on the problem of the grammar of the infinite procedure and how finite signs handled by finite beings gain the sense of infinity. This is the question of the nature of a technique or practice, which was central to later Wittgenstein’s thought. Second, the infinite of technique is not an extension or intensification of the finite, nor is it a transcendent object beyond all finite procedures. The infinity of technique enters human life at its capture in finite signs, the crossing of syntax and semantics.
Wittgenstein and Godel
The next question Livingston addresses is how we should view Wittgenstein’s critical attitude towards Godel’s incompleteness theorems. Many have taken them as reason to think Wittgenstein a finitist, but Wittgenstein’s purpose wasn’t to address Godel’s theorems but to bypass them or suggest alternative possibilities of interpretation. On this alternative interpretation, there isn’t a unified sense of truth that subsumes the use of this predicate within formalism and in ordinary language in which the informal, metalogical argument is given. If the assumption of a unified sense of truth is relaxed then the results have a different significance. Wittgenstein suggests, on an alternative reading of the proof, we would have to give up the interpretation of the statement P as saying that it itself, is unprovable. We would have to distinguish between what is actually established by the mathematical result itself and the metaphysical claims that are made on its behalf. And, in turn, the proof would bear no implications for the powers or structure of the system as a whole. The Godel sentence, on another account, might be equivalent to the Liar paradox.
Summary of Paradoxico-Criticism
The main elements or PC are summarized by Livingston as follows:
- Critical thought is possible only on the condition of a formal-syntactical consideration of language as a whole in its problematic relation with facts and objects.
- This formal-structural consideration of the totality of language suggests the possibility of a reflexive inscription of the total syntactical structure of language within itself which leads to an essential crossing of syntax and semantics.
- The syntactic possibility of reflexive inscription demonstrates that there is an essential structural inadequation between language and the world that is manifest as a structural excess of signification over the signified, or of sense over reference.
- This structural inadequation yields structurally necessary points of paradox where the total logic of the system is coded into itself (floating signifiers).
- These points of paradox and contradiction are undecidable between an “intra-systematic” (semantic) meaning and an “extra-systematic” (syntactic) one.
- This renders the position of the “subject of language” with respect to the “boundaries of language” itself undecidable between “inside” and “outside.”
So, what are the critical implications of these structures and how does PC grasp them?
- Paradoxico-criticism understands the excess of signification, and hence the paradoxes of reflexivity, not as problematic effects of language but as constitutive conditions for (the possibility of) “meaning” as such. . .
- . . . And thus the ultimate basis of the “ideal dimension” of language, and hence of the Idea as such.
- This identification of structural paradox has the critical effect of locating the structural contradictions inherent to reflexive linguistic reason as such, and hence of diagnosing the “weak points” of ineffectivity in any existing sovereign regime. These are the points at which any such regime can be resisted, or transformed.
So, if structural and formal paradox is at the basis of the “ideal dimension” of reflexive language itself, then this implies that language is also inherently ideological in the sense that it produces the ideality of meaning and the excess of signification as internal structural effects. Through discerning the structural origin and pervasiveness of these contradictions PC offers the possibility of diagnosing and criticizing these imaginary compensations and interrogating and removing the force of the superlative figures of power they engender. The PC orientation discerns in the consequences of formalism, the radical possibility of resistance to the ruling figures of unity, normalcy, progress, and effectiveness.