Gödelizing Gödel (and random thoughts)

It strikes me that Gödel's incompleteness theorem bears a bit of a resemblance to the types of skepticism I keep criticizing, insofar as it makes a universal statement about formal systems which seems to limit or hinder their power. Yet I wonder--could it be that to make this kind of statement successfully requires an implicit perspective subject to the same critique the incoimpletness theorem makes in the first place?

This is an idle thought, and it does not seem to me that Gödel's theorem could be undermined or subverted by its own conclusion, and nor am I qualified to investigate much further.

Analogously, that might have implications for the halting problem.

Just wondering.


And on a completely (?) unrelated note, I feel obliged to mention how Jorge Luis Borges' story "Funes the Memorious" features a man with complete eidetic/photographic memory, who experiences and recalls the "sensory manifold" in toto, rather than as we do in bits and pieces. (Here roughly meaning Kant's "sensory manifold", or whatever modern analogy there is). As a consequence, the man sees little point in abstraction and in fact has difficulty recognizing the similarities behind different species of dogs, or even different objects seen from different angles. I'm reminded next (through reading Gregory Chaitin) of information theory and Kolmogorov Complexity, where we judge the complexity of algorithms or objects based on the smallest instruction set necessary to recreate them.

...

On the other hand

Just when you think there's hope...

And sometimes...

[And sometimes, perhaps, desires can be trusted. Or, miraculously and all the more mysteriously, they can shape.]

I wonder then, were perfection nearly within grasp, how terrifying might that be? And too, how exhilarating, giddying! For Tantalus's hand to brush the grapes, his lips to graze the waters.

Imagine the moment following this unprecedented anomaly: his pulse races, a hitherto dulled and pessimistic mind comes alight, afire! Certainty (and its incumbent predictability) had blunted and wearied his existence. For surely Tantalus had realized – real-ized! – the despotic futility that ruled and overruled his every action, that denied him possibility of relief. Surely, he at last reached a point where unrelenting failure wore away the last of his persistence, leaving him stupefied, resigned, and stultified.

Imagine the moment prior: the cusp of despair, his head bows, and he begs penitently to the gods, as he has done countless times before. He knows well the gesture's uselessness: they will not heed him--and is it that they do not hear or do not care? Or is there anyone to hear? So long has been his imprisonment, that who can say which beings existed, or did fancy alone conjure up his divine imprisoners? Did he really host that profane and awful banquet? Or did he imagine his feat of hubris merely as to justify his own torment? But watch now, as his head bows, his parched and broken lips touch for one instant the inconceivable--the impossible, the unreal--and shatter his dreadful certainty.

How this moisture? How this incomprehensible moisture, there long enough to shock yet gone before it can be tasted? What does it mean, what can it mean? That the gods have heard, relented? Or their powers wane and may now be circumvented?

Imagine his reckless rejuvenation as thoughts careen throughout his jolted mind. An intoxicating force invigorates his ailing hopes and catapults him beyond Reason; and with that same resurgence comes a creeping, dawning horror: he is poised now at the brink of lunatic conclusions, and if he stretches just a little further, will his lips find long-sought respite? Or will the conscious act of striving revoke his supposed progress and rebuke him all the more?

Be this redemption, or a god's mocking laughter?

Is it chance that furnishes him with hope, or insidious design?

Is this opportunity, or is it a desire's wishful mirage? Is there any action he can possibly take to sway the outcome either way? If he chooses wrongly, will he ever get this chance again? And so horrible, if not, to live on knowing that he'd once come so near perfection, but failed and damned himself.

Hegelian Precursor to Derrida

If we cannot indeed make true statements about the whole (because they inevitably lead to an incomplete thesis-antithesis-synthesis triad, which in turn becomes a leg of a new triad...), then yes, I think Derrida's theory of deconstruction--wherein all statements undermine themselves--follows more naturally.

Note, by the way, that I know next to nothing about both Hegel and Derrida here. But anyway.

My objection to deconstructionism hitherto has been similar to my complaints about extreme skepticism generally: by attempting to demolish all foundations, one must necessarily presuppose a new foundation from which to do that demolishing. In other words, the statement "all propositions undermine themselves" necessarily undermines itself, rendering it false. Discussing that with a lit-theory friend of mine, he laughed and called that a beautiful part of the theory; to him, it "shows" the theory in operation on itself. It bothers me, however, since it leaves us left with a paradox and/or a jumble of contradiction.


Parallels

We can view the deconstruction of "All propositions undermine themselves" as analogous to the eternal war between a skeptic and a dogmatist, or an unraveling of the Liar's paradox. To wit, each valid step may be succeeded by a contradictory, equally valid step in the argument. E.g.,

(1) All propositions undermine themselves.

(2) "All propositions undermine themselves" undermines itself, by (1). Ergo, (1) is false.

(3) If (1) is false, then there can be propositions which do not undermine themselves after all. In which case, (2)'s reasoning is incorrect, because (1) might be one of those propositions, and so (1) might be true. Since (1) was given as an initial premise, we should then consider (1) true.

(4) (3)'s conclusion is a proposition which, by (1), undermines itself. Hence (3) must be false, and if (3) is false, then (1) is not true after all.

...etc, etc. This may be argued back and forth as long as we like without resolution.

[Note again: as said before, I know crap-all about Derrida. I have no idea if genuine deconstructions follow the form I just gave, and at the moment am too lazy to verify. Hooray, I'm a bad scholar. You caught me, want a prize?]

Just as with the Liar's Paradox,

(5) This statement (5) is false.

(6) Because (5) is false, (5)'s negation, "This statement (5) is true", must be true. Hence (5) is true.

(7) Since (5) is true, we know that the proposition "(5) is false" is true. Thus, (6)'s conclusion is false, because (5) is not true after all.

(8) Yet, if (5) is false, then its proposition "(5) is false" is false itself, meaning that (5) is really true. That means (7)'s conclusion is falses.

... etc. Clearly, an infinite succession of licitly derived contradictions. I'm trying to make a point about steps directly contradicting the directly previous step, but that's probably confusing, and under an ordinary analysis it is not necessary, so let me show a more intuitive route which is equivalent:

(9) (9) is false.

(10) Because (9) is false, "(9) is false" is false. Thus (9) is true.

(11) But if (9) is true, then "(9) is false" is true. And that means (9) is really false.

(12) If (9) is false, the same reasoning as from (10) shows that (9) is true.

(13) But if (9) is true, then the same reasoning as from (11) shows that (9) is false.

(14) The same reasoning from (10) and (12) shows that (9) is true.

(15) The same reasoning from (11) and (13) shows that...

The main difference is that I'm not spelling out the contradiction of the last step so much as just reasserting either (10) or (11) to refute the last conclusion. Which is really the same thing, so why am I making a fuss about it? The Lord only knows. Really, the smart thing to do is to stop as soon as you've found a contradiction in the argument (since otherwise we run into problems with explosion), but I'm trying to make the analogy to Hegel more explicit.


And speaking of whom, back to Hegel.

Suppose we call the assertion that "The Liar's Paradox statement is true" our thesis, and "The Liar's Paradox statement is false" our antithesis. Clearly, we can always reason toward thesis or antithesis, successfully proving or disproving each conclusion however many times we like, without ever reaching a final resolution. To Hegel, I believe, we should then realize the futility of this exercise, at which point we need to step outside of the system and create a synthesis between thesis and antithesis. Being a Hegelian neophyte, I don't know what the synthesis should be in this case, but it might be something like, "The Liar's paradox is both true and false" or it is"partially true, partially false," or "true at one time, false at another," or some other means of effecting reconciliation.

Now, the fun part about Hegel is that he says the new synthesis, whatever it is, now becomes the thesis or antithesis of a new thesis-antithesis-synthesis triad, which will need its own extra-dichotomous resolution. And we approach (but never reach?) truth through an infinite chain of these triads--reminiscent of Kant's moral progression, where practical reason must postulate an infinity of time (or lifespans?) through which we imperfect beings aspire toward perfection.

Now, where this is relevant more specifically to my thoughts, is that an infinite chain of dialectical syntheses reminds me very strongly of the warring double-helix Ouruborus I mentioned earlier. It seems to me wishful thinking on Hegel's part to claim that the addition of a synthesis makes a new kind of "progression" or development. Rather, the chain of triads fighting with each other is precisely ismorphic to the chain of contradicting (10)s and (11)s I outlined above, the simply unending contradiction. So, either the Liar's Paradox and Derrida's deconstruction already exemplify Hegel's described growth, or there exists simply no progression to speak of either way.

Hegel's syntheses are attempts to establish a new "groundless ground" or self-supporting justification in each controversy. By my thinking, however, this is not progression, since the new synthesis remains just as much a part of the very system it attempted to escape. This is just like trying to defuse Gödel's incompleteness theorem by adding axioms to a formal system; it doesn't matter how many or what axioms you introduce: by the very nature of the system at hand, you will leave yourself open to a new version of the incompleteness theorem. (Unless you reduce your system's axioms to a point where they express less than you originally wanted.)

Now, we might be able to argue that there's a kind of progression/development/growth/whatever here anyway, but I'm not going to investigate further at this point. It may explain (in part) the impossibility of halting philosophical inquiry.

By now, it's beginning to seem to me, naively, that the history of thought is little more than a horribly convoluted deception, an intriciate illusion, contrived to hide the fact that all we've been doing is saying "Nuh-uh!" and "Yeah-huh!" to each other for the last two thousand years.

Silly children.

Volition

Why is it that dreams do not permit free manipulation to suit our own ends (save in the case of the comparatively rare lucid dreams)? Since they are created entirely by our minds, and since we can exert some measure of control over them through thought and will, what is it that prevents a more thorough control/influence?

How does action differ from willing...?

Skepticism and the Root of All Things

Vancouver Philosopher at The Chasm makes the following point about Heidegger:

Heidegger's denial about a fundamental unifying characteristic to philosophy is foolish. When we employ such skepticism about the reality of the ground/central concept, we only wind up grounding such skepticism in denial. The very act of denial becomes its own ground, and this strategy winds up being self-defeating in the end. Instead, we shouldn't think of various philosophical systems as themselves foolhardy in establishing a ground or framework. We should interrogate the framework or ground on its own merits.

This follows my own thoughts about skepticism regarding truth: we cannot sensibly make a statement like "There is no truth" without simultaneously undermining that statement by implying that it itself is true. For skepticism to assert a meaningful proposition (that is, a proposition which supplies us with genuine information), the skepticism itself must be grounded in something. And if the brand of skepticism at hand denies the existence or reliability of all grounds, it necessarily denies its own conclusions. Hence, as Vancouver Philosopher says, denial becomes its own ground, and it invariably ends up defeating itself.

Similarly, skepticism about our epistemic relation to truth undermines itself as well. Suppose our skeptic offers us "an irrefutable argument" that, whether there exist truths or not, we simply cannot possess knowledge of them. E.g., all claims must be justified by other claims, and there are no unjustified, self-supported claims; ergo all claims are unjustified/unacceptable. And yet, if we accept the skeptic's conclusion, we (presumably) have now acquired a new bit of knowledge, viz., the knowledge that "All knowledge is impossible", or "We possess no knowledge"! Wait, how did that happen? How do we know this when we can't know anything? Again the skeptical conclusion undermines itself; it presupposes its own new ground from which to criticize the whole of another position, yet in doing so creates and relies upon that which it seeks to demolish.

Similar arguments apply to skepticism toward the legitimacy of reason generally.


Roots

The essential problem, as I see it, is that the skeptic must engage the opponent on her own territory, so to speak, and using her own tools. This makes strategic sense, since otherwise why should the anti-skeptic (let's say "dogmatist") accept whatever point the skeptic tries to make? Unfortunately, this is a rigged game for the extreme skeptic: when playing by the dogmatist's rules, there is simply no way to "win" or "break outside" the system, because every attempt to do so places you right back into a new system. (This all ties into the Liar's Paradox, Gödel's Incompleteness Theorem, and the Halting Problem; I don't know how to make this more explicit yet, but see Gödel, Escher, Bach by Douglas Hofstadter for related discussion).

Now, life isn't exactly a bed of roses for the dogmatist, either, since she can never really answer the skeptic's demands for an "unmoved mover" in the realm of logic, so to speak. But it's fascinating to imagine the two positions, intertwined together as an infinity of recursive, dialectic contradiction. The skeptic asserts, "You know nothing"; the dogmatist responds, "Then I must know that I know nothing"; the skeptic rejoins, "But to know that, you must presuppose other principles for which you have no justification either!"; and the dogmatist counters, "But in order to know that I need those principles, surely I must know that I need them, so I do know something after all!", and so on ad infinitum. Or ad nauseam, take your pick. The two warring sides endlessly wrap around each other, neither overcoming the other--perhaps like twin strands of intertwined DNA which end up consuming their own tails as an involuted Ouroborus?

But let's not get ahead of ourselves, nor lost in mystical/metaphorical speculation. The key here is that we could apparently resolve the tension if we could ever find a groundless ground. What I call a "groundless ground" shows up in many places: as Aristotle's "prime mover," as Aquinas' "uncaused cause," as the idea of a necessary entity or fact in general, as a self-caused being, as that which needs no justification, as a "primitive fact," as Kant's description of "the unconditioned." To find such a ground and look out from it would yield the fabled view from nowhere, the view sub specie aeternitatis, the God's-eye view. This perspective, and none other, would be satisfactorily "outside of the system" to satisfy both skeptic and dogmatist. (Hopefully.)

It's no surprise, then, that my description of the clash above mirrors Kant's Antinomies of Pure Reason, where my "dogmatism" would map on to the rationalist understanding of metaphysics, and skepticism to the empiricist understanding. Roughly, anyhow. To be more precise, Kant thought that both rationalist and empiricist sought a "groundless ground" (the "unconditioned"), but they sought it from different starting premises; whereas my dogmatist/skeptic divide does not show both sides seeking an unconditioned ground so much as the denial that there is such a thing on the skeptical side. So take the comparison with a grain of salt--I'd have more to say about justifying my analogy/mapping, but I'm running out of motivation at this point.

...

.. to desire a thing so much that one is paralyzed by the very possibility of it being unattainable...