The Logic of Gilles Deleuze by Corry Shores;

Author:Corry Shores;
Language: eng
Format: epub, pdf
Publisher: Bloomsbury UK

Figure 6.8 An illustration of Cantor’s diagonalization. Left: A correspondence between real and natural numbers (based on Vergauwen’s illustration).80 Right: The diagonal number escaping those correspondences.

For our purposes here, we arbitrarily consider some place along the series. To keep things simple, we choose the fourth place, but the list always continues after any place you look at. Now, starting with the first decimal of the first real number, we change it so that it is certainly not the same.82 In Vergauwen’s illustration, if it is not 1, then we change it to 1, and if it is 1, we change it to 2. Thus we get the diagonal sequence 3.1211.83

And we next ask ourselves the following questions. Can this new number 3.1211 be the one corresponding to natural number 1 (namely, 3.5827)? No, because we intentionally made the first decimal be different. Can our new number be identical to the second real number in our list? No, because we intentionally made its second decimal be different. This will hold no matter where you go in the sequence of natural numbers that progress downward. We pick any place, obtain the diagonal number, and it will by design not be in the sequence up to that point (Figure 6.8, right). But what if you simply place the diagonalized number as the next real number in the series (thereby “capturing” it among the “regulated flows,” as it were)? That will not solve the problem. We can still find yet another diagonal number that will not be contained in that list, following the same procedure, now applied additionally to the new number in the list. To summarize, the natural numbers are denumerably infinite, and we have attempted to use the naturals to count the reals; yet, because the reals will always in this way exceed the natural numbers, the reals are instead non-denumerably infinite. They (and thus the continuum) have a power that exceeds that of the natural numbers. Again, for Deleuze, this is simply an illustration of how even in mathematical axiomatics, just as with social ones, there will always be something that escapes the axiomatic system even though that something springs from it in accordance with its own axioms.

Deleuze notes that the problematic trend in contemporary mathematics, particularly intuitionism and constructivism, is interested in a calculus of problems, where the Principle of Excluded Middle is rejected as a logical law. Here, mathematical notions are seen as “problems,” because they are understood initially without a truth-value assignment, and in some cases that status of undecidability may hold indefinitely.84 We will turn to these concepts next, but let us first conclude this section about Deleuze’s commentary on the axiomatic and problematic currents in the history of mathematics by showing how the intuitionist notion of problems is continuous with the sort that Deleuze located in previous epochs. Andrei Kolmogorov writes that the “calculus of problems is formally identical with the Brouwerian intuitionistic logic,” and, instead of defining what a problem is, he lists some examples:

To prove the falsity of Fermat’s theorem.

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