From this point of view, computational processing does not mainly guarantee the return to initial conditions, nor does it simply include change derived from an interactive paradigm based on responsive outputs. This is because Chaitin’s conception of incomputability no longer perfectly matches the notion of the limit in computation (i.e., limit for what is calculable). Instead, this limit as the incomputable is transformed: It becomes the addition of new and maximally unknowable algorithmic parts to the present course of computational processing; these parts are algorithmic sequences that tend to become bigger in volume than programmed instruction and to take over, hereby irreversibly transforming the pre-set finality of rules. Chaitin’s re-articulation of the incomputable is at once striking and speculatively productive. What was conceived to be the external limit of computation (i.e., the incomputable) in Turing, has now become internalized in the sequential arrangement of algorithms (randomness works within algorithmic procedures).
At Chaitin’s own admission, it is necessary to see algorithmic randomness as a continuation of Turing’s attempt to account for indeterminacy in computation. Whereas for Turing there are cases in which finality cannot be achieved, and thus computation—qua automation of the finality of reason—stops when the incomputable begins, for Chaitin computation itself has an internal margin of incomputability insofar as rules are always accompanied and infected by randomness. Hence, incomputability is not simply a break from reason, but rather reason has been expanded beyond its limits to involve the processing of maximally unknown parts that have no teleological finality. To put it in other terms, automation is now demarcated by the incomputable, the unconditional of computation. Importantly, however, this challenges the view that computational processing corresponds to calculations leading to pre-programmed and already known outputs. Instead, the limits of automation—that is the incomputable—have become the starting point of a dynamism internal to computation, which exceeds the plan for technocapital’s instrumentalization of reason. From this standpoint, relating Chaitin’s findings to the positioning of critical thought and technocapitalism reveals a new aspect: the incomputable cannot be simply understood as being opposed to reason. In other words, it is not an expression of the end of reason and cannot be explained according to the critical view that argues for the primacy of affective thought.
According to Chaitin, the incomputable demonstrates the shortcomings of the mechanical view of computation, according to which chaos or randomness is an error within the formal logic of calculation. But incomputables do not describe the failure of intelligibility versus the triumph of the incalculable—on the contrary. These limits more subtly suggest the possibility of a dynamic realm of intelligibility, defined by the capacities of incomputable infinities or randomness, to infect any computable or discrete set. In other words, randomness (or the infinite varieties of infinities) is not simply outside the realm of computation, but has more radically become its absolute condition. And when becoming partially intelligible in the algorithmic cipher that Chaitin calls Omega, randomness also enters computational order and provokes an irreversible revision of algorithmic rules and of their teleological finality. It is precisely this new possibility for an indeterminate revision of rules, driven by the inclusion of randomness within computation, that reveals dynamics within automated system and automated thought. This means the following: While Chaitin’s discovery of Omega demonstrates that randomness has become intelligible within computation, incomputables cannot, however, be synthesized by an a priori program or set of procedures that are in size smaller than them. According to Chaitin, Omega corresponds to discrete states that are themselves composed of infinite real numbers that cannot be contained by finite axioms.
What is interesting here is that Chaitin’s Omega is at once intelligible yet nonsynthesizable by universals, or by a subject. I take it to suggest that computation—qua mechanization of thought—is intrinsically populated by incomputable data, or that discrete rules are open to a form of contingency internal to algorithmic processing. This is not simply to be understood as an error within the system, or a glitch within the coding structure, but rather as a part of computation. Far from dismissing computation as the evil incarnation of technocapitalist instrumentalization of reason, one realizes that incomputable algorithms emerge to defy the superiority of the teleological finality of reason, but also of sensible and affective thought.
