meditation on the arithmetical hierarchy
Let us speak of the kind of hierarchy that actually exists, and whose concept is metalogically valid. Of course, it is not a hierarchy of persons but of propositions.
It’s easy to see that a chain of n identically-quantified variables, for instance, (∀x)(∀y)(∀z) amounts to a single quantified n-tuple (here, ∀(x, y, z)). Also, that it doesn’t matter whether the quantifier is universal or existential (whether the principle in view is unity or totality, to hen or to pan); so long as as only one of them is in view, only what is thought (and not the thinking of it) increases in complexity. (A special sort of complexity: the banality of the procession of dimension.) It’s not just any multiplicity, then, which makes matters difficult, as we recall. As long as the thinker is careful to avoid the slightest iterability in the concept (e.g., Aristotle, Descartes), or the slightest appeal to both principles (e.g., trivializing the Socratic question by reducing it to the ti esti), so long thinking seems to possess an unlimited “formal” power over multiplicity.
What must it be, then, which, even in the sandbox of a first-order speaking of number, records the refusal of being to cede to thinking an immediate and sovereign power to make one-over-many, and indexes an increasing resistance on the part of the content to being counted formally as mere content? I.e., what must be the measure of the difficult, not only insofar as the latter is counted, making one out of the many all-too-easily, but in its rebound upon counting? What must be counted of the language of counting, in order to measure the multiple at the level of the concept, or the difficult proper? (By this third formulation, we’re able to confirm that we’re on the terrain of metamathematics – the mathematics of the language of mathematics.) The question appears hopelessly general, but on the basis of the opening distinction of dimension from difficulty the hypothesis is actually immediate: let us count the number of times that the difference between the principles makes its appearance in the chain of quantifiers, and record this number as an index of the difficulty of speaking about number. Remarkably, this simple inductive leap gives us a full first-order categorization scheme, known as the arithmetical hierarchy.
That the duality of the principles shows itself in a particularly clear and simple way to be the formal cause of complexity, in beings and in thinking, is caviar to the metalogical Platonist. But one is immediately suspicious about reducing the trace of duality to something that can be unambiguously counted. Does this degree of clarity make the difficult too easy to localize, and thus to avoid, removing it across the boundary of thought’s necessary experience, per the wishes of Meno, Critias, Protagoras…? No. Again for a reason which we can reach a priori and confirm rigorously. Suppose there is a simple, expressively-unambiguous metric of computability. Can its effective application march in step with its clarity? No, for in this way, it would solve too much of the halting problem at a single stroke, and contradict the consequences of diagonalization; we know that something must interrupt its straightforward application. Thus the essential irony of this definitive and ineffective system of measurement: we can state, through mere syntactic inspection, an upper bound on complexity, and I suspect we can even state that normally the upper and lower bounds coincide, meaning that the chance of a collapse of complexity leading to the possibility of insight is, in a way, infinitely remote. (This needs to be verified. What is its relation to Chaitin’s “halting probability”?) Yet the interest of thinking is in nothing but this possibility of fulfillment, of the sudden collapse in which an articulable logos compresses the multiple beyond previous bounds. Insight. Thinking must bet against insight, and thus against itself globally. Must – can – thinking then bet against itself locally? Does the truth of global, cosmic pessimism license taking this pessimism as the maxim of thinking’s local act? Can it abandon the tension by which it maintains itself in the still pursuit of insight? Can thinking place itself on the side of the hopeless complexity of being? Can it stop waiting for those events of insight in which the One is momentarily effective? Once again, the consequences of diagonalization’s reconfiguration of local and global would be misinterpreted, I think, in this natural application. At least, it is provably impossible for thinking to be justified in taking this last step of wagering on the normal coincidence of upper and lower bounds of complexity. (In our time – 2002 or 2003 – the discovery that Primes is in P.)
Where we are. The arithmetical hierarchy speaks of the question of how to orient oneself in thinking, or better, of how thinking can orient itself in being.
The locality of thinking is not normal.
A thinking being is both totally exposed to chance and totally incapable of a genuinely random selection, at least when it comes to speaking of number. Unless absolute idealism is true (in which case the problem vanishes, and presumably the temptation to a practical nihilism), thinking operates in a tiny bubble of exception. Were it elsewhere, it would be destroyed. (Will to truth as death drive?) We can know this, and knowing it does not amount to relocating. We can not relocate. No more than we can step out of the indexical expressions of the exception: be there rather than here, then rather than now, the other rather than I. Thinking can’t put itself in advance on the side of the event or in the place of the Other, especially not through an embrace of “paradox”. (The forms of violence and obscurity which arise in this attempt.)
The torsion that the existence of thinking beings introduces into the computational hierarchy, not by an overarching paradoxical view, but by being there. What I rightly know is never realized normally in the place where I know, where I am. Further, the (global) law of this distorting exceptionality is knowable. The existence of my knowledge (whose proper global object is ignorance), is the material bar to its ordinary application, as well as to its paradoxical supplement.
The being of knowledge has an additional meaning which can be interpreted within knowledge’s own scope: not as paradoxical transcendence, but as bar to a global power of judgment. (Completed, the analytic of Dasein purifies Platonism to the extent that it is purified by it; the being of Dasein is indeed care. About what? “About itself”, has not been clarified in the interval between Critias and Heidegger. It cannot be clarified except in relation to number and the Good. (This is one of the many senses in which adequate responses to the problems posed by Plato’s Socrates lie entirely ahead of us, and not in the history of philosophy.))