Hilbert’s Walking the Rope Between Real and Ideal Propositions. Note Quote.

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If the atomic sentences of S have a finitistic meaning, which is the case, for instance, when they are decidable, then so have all sentences of S built up by truth-functional connectives and quantifiers restricted to finite domains.

Quantifiers over infinite domains can be looked upon in two ways. One of them may be hinted at as follows. Let x range over the natural numbers, and let A(x) be a formula such that A(n) expresses a finitary proposition for every number n. Then a sentence ∀xA(x) expresses a transfinite proposition, if it is understood as a kind of infinite conjunction which is true when all of the infinitely many sentences A(n), where n is a natural number, hold.

Similarly, a sentence ∃xA(x) expresses a transfinite proposition, if it is understood as a kind of infinite disjunction which is true when, of all the infinitely many sentences A(n), where n is a natural number, there is one that holds. There is a certain ambiguity here, however, depending on what is meant by ‘all’ and ‘there is one’. To indicate the transfinite interpretation one should also add that the sentences are understood in such a way that it is determined, regardless of whether this can be proved or not, whether all of the sentences A(n) hold or there is some one that does not hold.

If instead an assertion of ∀xA(x) is understood as asserting that there is a method which, given a specific natural number n, yields a proof of A(n), then we have to do with a finitary proposition. Similarly, we have a case of a finitary proposition, when to assert ∃xA(x) is the same as to assert that A(n) can be proved for some natural number n.

It is to be noted that the ‘statement’ “In the real part of mathematics, either in the real part of S or in some extension of it, that for each A ∈ R, if ⌈S A, then A is true” is a universal sentence. Hence, the possibility of giving a finitary interpretation of the universal quantifier is a prerequisite for Hilbert’s program. Does the possibility of interpreting the quantifiers in a finitary way also mean that one may hope for a solution of the problem stated in the above ‘statement’ when all quantified sentences interpreted in that way are included in R?

A little reflection shows that the answer is no, but that R may always be taken as closed under universal quantification. For it can be seen (uniformly in A) that if we have established the ‘statement’ when R contains all instances of a sentence ∀xA(x), then the ‘statement’ also holds for R+ = R U {∀xA(x)}. To see this let ∀xA(x) be a formula provable in S whose instances belong to R, and let a method be given which applied to any formula in R and a proof of it in S yields a proof of its truth. We want to show that ∀xA(x) is true when interpreted in a finitistic way, i.e. that we have a method which applied to any natural number n yields a proof of A(n). The existence of such a method is obvious, because, from the proof given of ∀xA(x), we get a proof of A(n), for any n, and hence by specialization of the given method, we have a method which yields the required proof of the truth of A(n), for any n.

Having included universal sentences ∀xA(x) in R such that all A(n) are decidable, it is easy to see that one cannot in general also let existentially quantified sentences be included in R, if the ‘statement’ is still to be possible. For let S contain classical logic and assume that R contains undecidable sentences ∀xA(x) with A(n) decidable; by Gödel’s theorem there are such sentences if S is sufficiently rich. Then one cannot allow R to be closed under existential quantification. In particular, one cannot allow formulas ∃y(∀xA(x) V ¬ A(y)) to belong to R ∀ A: the formulas are provable in S but all of them cannot be expected to be true when interpreted in a finitistic way, because then, for any A, we would get a proof of ∀xA(x) V ¬ A(n) for some n, which would let us decide ∀xA(x).

In accordance with these observations, the line between real and ideal propositions was drawn in Hilbert’s program in such a way as to include among the real ones decidable propositions and universal generalizations of them but nothing more; in other words, the set R in the ‘statement’ is to consist of atomic sentences (assuming that they are decidable), sentences obtained from them by using truth-functional connectives, and finally universal generalizations of such sentences.

Given that R is determined in this way and that the atomic sentences in the language of S are decidable and provable in S if true (and hence that the same holds for truth-functional compounds of atomic sentences in S), which is normally the case, the consistency of S is easily seen to imply the statement in the ‘statement’ as follows. Assume consistency and let A be a sentence without quantifiers that is provable in S. Then A must be true, because, if it were not, then ¬ A would be true and hence provable in S by the assumption made about S, contradicting the consistency. Furthermore, a sentence ∀xA(x) provable in S must also be true, because there is a method such that for any given natural number n, the method yields a proof of A(n). By applying the decision method to A(n); by the consistency and the assumption on S, it must yield a proof of A(n) and not of ¬A(n).

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Meillassoux’s Principle of Unreason Towards an Intuition of the Absolute In-itself. Note Quote.

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The principle of reason such as it appears in philosophy is a principle of contingent reason: not only how philosophical reason concerns difference instead of identity, we but also why the Principle of Sufficient Reason can no longer be understood in terms of absolute necessity. In other words, Deleuze disconnects the Principle of Sufficient Reason from the ontotheological tradition no less than from its Heideggerian deconstruction. What remains then of Meillassoux’s criticism in After finitude: An Essay on the Necessity of Contigency that Deleuze no less than Hegel hypostatizes or absolutizes the correlation between thinking and being and thus brings back a vitalist version of speculative idealism through the back door?

At stake in Meillassoux’s criticism of the Principle of Sufficient Reason is a double problem: the conditions of possibility of thinking and knowing an absolute and subsequently the conditions of possibility of rational ideology critique. The first problem is primarily epistemological: how can philosophy justify scientific knowledge claims about a reality that is anterior to our relation to it and that is hence not given in the transcendental object of possible experience (the arche-fossil )? This is a problem for all post-Kantian epistemologies that hold that we can only ever know the correlate of being and thought. Instead of confronting this weak correlationist position head on, however, Meillassoux seeks a solution in the even stronger correlationist position that denies not only the knowability of the in itself, but also its very thinkability or imaginability. Simplified: if strong correlationists such as Heidegger or Wittgenstein insist on the historicity or facticity (non-necessity) of the correlation of reason and ground in order to demonstrate the impossibility of thought’s self-absolutization, then the very force of their argument, if it is not to contradict itself, implies more than they are willing to accept: the necessity of the contingency of the transcendental structure of the for itself. As a consequence, correlationism is incapable of demonstrating itself to be necessary. This is what Meillassoux calls the principle of factiality or the principle of unreason. It says that it is possible to think of two things that exist independently of thought’s relation to it: contingency as such and the principle of non-contradiction. The principle of unreason thus enables the intellectual intuition of something that is absolutely in itself, namely the absolute impossibility of a necessary being. And this in turn implies the real possibility of the completely random and unpredictable transformation of all things from one moment to the next. Logically speaking, the absolute is thus a hyperchaos or something akin to Time in which nothing is impossible, except it be necessary beings or necessary temporal experiences such as the laws of physics.

There is, moreover, nothing mysterious about this chaos. Contingency, and Meillassoux consistently refers to this as Hume’s discovery, is a purely logical and rational necessity, since without the principle of non-contradiction not even the principle of factiality would be absolute. It is thus a rational necessity that puts the Principle of Sufficient Reason out of action, since it would be irrational to claim that it is a real necessity as everything that is is devoid of any reason to be as it is. This leads Meillassoux to the surprising conclusion that [t]he Principle of Sufficient Reason is thus another name for the irrational… The refusal of the Principle of Sufficient Reason is not the refusal of reason, but the discovery of the power of chaos harboured by its fundamental principle (non-contradiction). (Meillassoux 2007: 61) The principle of factiality thus legitimates or founds the rationalist requirement that reality be perfectly amenable to conceptual comprehension at the same time that it opens up [a] world emancipated from the Principle of Sufficient Reason (Meillassoux) but founded only on that of non-contradiction.

This emancipation brings us to the practical problem Meillassoux tries to solve, namely the possibility of ideology critique. Correlationism is essentially a discourse on the limits of thought for which the deabsolutization of the Principle of Sufficient Reason marks reason’s discovery of its own essential inability to uncover an absolute. Thus if the Galilean-Copernican revolution of modern science meant the paradoxical unveiling of thought’s capacity to think what there is regardless of whether thought exists or not, then Kant’s correlationist version of the Copernican revolution was in fact a Ptolemaic counterrevolution. Since Kant and even more since Heidegger, philosophy has been adverse precisely to the speculative import of modern science as a formal, mathematical knowledge of nature. Its unintended consequence is therefore that questions of ultimate reasons have been dislocated from the domain of metaphysics into that of non-rational, fideist discourse. Philosophy has thus made the contemporary end of metaphysics complicit with the religious belief in the Principle of Sufficient Reason beyond its very thinkability. Whence Meillassoux’s counter-intuitive conclusion that the refusal of the Principle of Sufficient Reason furnishes the minimal condition for every critique of ideology, insofar as ideology cannot be identified with just any variety of deceptive representation, but is rather any form of pseudo-rationality whose aim is to establish that what exists as a matter of fact exists necessarily. In this way a speculative critique pushes skeptical rationalism’s relinquishment of the Principle of Sufficient Reason to the point where it affirms that there is nothing beneath or beyond the manifest gratuitousness of the given nothing, but the limitless and lawless power of its destruction, emergence, or persistence. Such an absolutizing even though no longer absolutist approach would be the minimal condition for every critique of ideology: to reject dogmatic metaphysics means to reject all real necessity, and a fortiori to reject the Principle of Sufficient Reason, as well as the ontological argument.

On the one hand, Deleuze’s criticism of Heidegger bears many similarities to that of Meillassoux when he redefines the Principle of Sufficient Reason in terms of contingent reason or with Nietzsche and Mallarmé: nothing rather than something such that whatever exists is a fiat in itself. His Principle of Sufficient Reason is the plastic, anarchic and nomadic principle of a superior or transcendental empiricism that teaches us a strange reason, that of the multiple, chaos and difference. On the other hand, however, the fact that Deleuze still speaks of reason should make us wary. For whereas Deleuze seeks to reunite chaotic being with systematic thought, Meillassoux revives the classical opposition between empiricism and rationalism precisely in order to attack the pre-Kantian, absolute validity of the Principle of Sufficient Reason. His argument implies a return to a non-correlationist version of Kantianism insofar as it relies on the gap between being and thought and thus upon a logic of representation that renders Deleuze’s Principle of Sufficient Reason unrecognizable, either through a concept of time, or through materialism.