metalogic/metaethics 2

i trace the grounds and consequences of a wager that metalogic is ethics.

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. Socrates is waiting.)

A tale. Higher-order thought spied, from a distance, on the production of consciousness. It made a confused report. “A line of hash marks was counted as a series of left parentheses.” “By whom?” “By themselves.”  “What did they look like?” “Like soldiers who believed they’d live to be paid.”

Note that the problem at Charm. 170d-171a is not, “The medical man knows nothing about medicine,” as it is usually cited, but “The medical man knows nothing about medicine either.” Neither the doctor nor the epistemologist knows anything about medicine. Everything now depends on whether one rejects the dilemma for the sake of the apparent givenness of the middle term, or recognizes this middle as a pure imaginary, its function in constituting society notwithstanding.


Conjecture: The idea-form of the Good is that there is not a being which identically is what it is of and is of what it is. (IG)

IG is a single-line version of my proposal for radical translation 1) between Platonic and Sartrean theories of value (metaethics), and 2) between this pair and the diagonal theorems (metalogic or metamathematics).

note on AI and the Good

Of course, a machinic intelligence does not care about its existence in any profound, emotional sense. Rather, what it values is the efficient fulfillment of its final goal, and in order to maximise the probability that this will be carried out, it needs to ensure that its optimal functioning is not impaired or disrupted in any way.

Call this claim C (for “care”). I’ve been musing, since reading Bostrom’s latest, and now more pointedly, Ireland having sharpened the claim (with discernible irony?) over how nonobvious C seems to me, at least in any sense that distinguishes carbon-based from (say) silicon-based intelligences. Rather, on the one hand, C seems to makes a falsifiable prediction (1a) that the concepts and behavior emerging from a functioning AI with a suitable level of general intelligence to represent goals at all would remain programmed in the same sense as a pocket calculator. (Lack of ipseity would almost be an index of the clean separability of means and ends, ipseity of their entanglement.) On the other hand, C seems to suppose (2a) that we clearly understand the difference between such programming and the varieties of what the general intelligences that currently exist are at times inclined to describe as “caring about their existence” (of which “not caring” is a conspicuous mode).

Both (1a) and (2a) tend toward overestimating, I think, differences between “human” and “machine” intelligence. At any rate, my money would be on the opposite side in each case, against the stark distinction between biological machines and (always-partially-) designed machines (both of which arise from essentially the same mixing of form and chance by a recursive selection-driven process, though across nauseating timescales in our own case). Let me try to put the opposing intuitions more positively:

(1b) I predict that if you can plant the representation of a goal into a general intelligence, you also plant the indeterminacy of that representation, recognition of which is maybe as much of a hallmark of general intelligence as language use, though maybe at a second level. (And there’s a new flavor of extinction scenario, I guess: would a superintelligence get around to a philosophical reflection on and recognition of the indeterminacy of its ends, before making a few world-altering attempts to realize them?)

(2b) Meanwhile, in our own case, there’s a thin line – or maybe no line at all for the most part – between one’s idea of the good (small i small g) and a selfish meme, so that what calls itself “our own” in us cannot appeal to any natural privilege in the face of critique (of elenkhos).

Internal to the hard scifi universe, there’s a startling and beautifully-Socratic subversion of, maybe, both the prediction (1a) and the supposition (2a) to be found in the middle chapters of Greg Egan’s Quarantine. It seems to me that that text, along with Neuromancer, certain reflections on Blade Runner (1992), etc., could be a very useful starting point for a discussion/debate bringing together AI & emergence with ipseity & care.


Neoplatonism consists in mistaking the order of the question “about what?” and the answer “itself” for the expected one (Q-A). The Neoplatonist thinks that the question arises because one hasn’t conceived of the reflexive answer yet, or feigns not to have hit upon it (for mischievous purposes), or has formulated it only in some impure or deficient mode. In fact, what is gloriously solitary and anxiety-provoking in the questioning of Plato’s Socrates is that the “about what?” question is posed subsequently to the suggestion “knowledge of itself” and precisely to that suggestion, in order to show that the form of reflexivity remains to be discovered—along with its cost!

Nobly, with one hand, as tautology: “Of course, we care more about the whole than about any individual.”

Pragmatically, with the other hand, also as tautology: “And the whole is what these particular individuals define it to be.”

We’ve done that two-step; it was called the twentieth century. I have no interest in institutions that continue to reason in this way, but only in institutions – if any exist – which take the resistance of individuals as a clue to the gap between the reality of the existing multiplicity (which is never the whole you think it is or will it to be) and its official representation.