Mathematics and Nihilism, Diagonalization and the Good: Response to Pagano on Bacon and Spengler

by metalogike

  1. I claim that diagonalization teaches us that, contrary to our architectonic expectations, cosmic nihilism and local teleology (in the sense of a strengthening of a non-instrumental will to truth) go together. The prototype, for me, is the Gödel sentence: transcendence without a transcendent. It draws a picture for us, a picture into which we are drawn, of the mathematical self-overcoming of nihilism. In the experience of coming to understand diagonalization, our philosophical possibility-space gets reorganized: we learn, to our surprise, that the form that makes local transcendence possible is the same as the form that shows why there is not a global transcendence. The form that makes truths possible is the same as the form that shows why truth cannot be identified with system, even at the limit, even in a passage to the infinite.

  1. Many twentieth-century thinkers have seen the necessity of some sort of negative dialectic – Kojève, Bataille, Sartre, Adorno, to name a few –  but the formal point has almost always been totally muddled, and consequently has not repaired the breach between philosophical and mathematical thought symptomatic (or even partly constitutive?) of nihilism. Badiou (Being and Event, 1988) is the figure who has made the philosophical world slightly familiar with the kind of thinking about these issues I’m suggesting.

  1. Diagonalization, the quasi-formalism in question, borrows unexpectedly both the reflexivity of nous/Geist and the utter irreflexivity of exteriority (space / entropy / nihilism / death. If the latter is lethal, the former is false (and therefore eventually if indirectly just as lethal). Spirit as substance, reflexivity on the cheap, inevitably poisons culture after a certain vaguely defined but inevitable point, viz. the point at which culture develops to a level at which the consequences of the falsity of this picture of reflexivity start to have practical consequences, a metalogical threshold for civilizations. Nietzsche showed us many of the details of the poisoning, though missing the stratum in Plato’s mathematical dialectic that provides a clue for taking the next step. Diagonalization shows us that reflexivity for-itself depends not on substance but on the irreflexive in-itself (though not vice versa) to the very extent that it cannot be reduced to it. (Cf. Davidson’s anomalous monism.)  This curious relation can be maintained because the “mediating” terms are not in conflict with one another, do not even inhabit the same logical space. Dependence shows itself to be a relation (“ontological”) on the side of the in-itself, and irreduction a relation (“epistemological”) of consciousness and knowledge, on the side of the for-itself. Thus the duality has to reassert itself on each side of a fantasized and contradictory mediation. (Cf. Sartre: The problem with the for-itself is that it seeks a mediation with the in-itself, but it already is the only mediation it’s ever going to get. Consciousness already is a relation to being, and there won’t ever be a further mediation of consciousness and being; for that, one may not hope.)

  1. Equivalently, one can say that diagonalization finally provides a solution of the problem of intentionality (which is at once theoretical and existential) which manages to avoid both ontotheological reification and nihilism (pragmatism, constructivism etc.) which had seemed mutually exclusive and exhaustive.

  1. Provisional and abrupt conclusion: Only a second-order Enlightenment, a metalogical Enlightenment (containing what’s true of both the Enlightenment and its critique, unfolding under the cold sun of diagonalization which provides vigor but not cosmic comfort) can save us.