Thursday, January 29, 2009

Leibnizesque Proclivities

I recently borrowed Bertrand Russell's A Critical Exposition of the Philosophy of Leibniz, and I'm rather enjoying what little I've read of it so far.

One thing that struck me is the description (in the beginning) of Leibniz's disinclination toward publishing a fully developed system. Rather, it seems Leibniz tended to craft arguments in response to private correspondence, or to issues which he bore some personal connection to. Unfortunately, this makes giving a comprehensive take on his views a bit difficult, since most if it is fragmentary.

The amusing thing is that I think I find myself doing something similar. I rarely write for the sake of writing--I will always be most motivated to give an in-depth argument when it's directly in response to someone else, peculiarly enough.

Maybe I should work on that.

Sunday, January 25, 2009


Is it satisfying to create a world in which your creatures achieve their desires, so that you may vicariously effect wish fulfillment? To craft a fantasy where your own dreams (and perhaps others') come true?

Is it satisfying to create a world in which your creatures suffer through a morass of confusion, despair, and ever-frustrated longing, so that you spitefully ensure they are never happier than yourself? To craft a fantasy where no one's dreams come true?

Thursday, January 22, 2009

Vacuous Truth

From my post in a topic in the xkcd forums:

"Every woman currently living on Pluto is male."

"Every true falsity is false."
"Every false truth is true."

"Every circular square is neither circular nor square." (This one almost makes an intuitive sense... but then, it really doesn't, when we consider that every circular square is also circular and square.)

"Every present King of France is simultaneously bald and not bald."

By the mathematical/logical principle that propositions are vacuously true when their subject does not exist, no matter what is predicated of that subject, the above statements should be considered true.

This relates to the problem of negative existentials in the philosophy of language, but I don't have more to say about it at the moment. May investigate further.

Must Perfection Equal Stagnation?

At various times I have encountered or myself espoused the view that anything which is perfect must be, in some sense, static. Religious critics might apply it to the idea of heaven, to show that any possible heaven must be a boring/stagnant place; similarly, we might argue that we're better off not being able to reach a state of complete perfection, because perfection would require no change, and therefore what would we do?

The reasoning goes something like this: for change to occur, an entity or circumstance must become other than it is currently. That is to say, at one time we may say of x that Px, while at another time we may say ~Px. So, if x is currently in a state of perfection but that state changes, surely that change now negates the previous, perfect state of x, thus rendering x imperfect.

An analogy with which we should all be familiar is grades: suppose a student has the perfect grade of 100% in the first week of a course. Suppose further that the teacher never allows extra credit, so at this moment the student possesses the highest grade attainable. From here on out, the only possible change to her grade would be to a lower value; so if at any point during the remainder of the semester, her grade experiences any form of change, it must be to a state of imperfection (< s100%).

So far, so good. If perfection be defined as having a grade equal to 100%, this is all true. Any change to that number necessarily yields an imperfect grade. The problem comes when we assume that, because the state of being perfect must not change, therefore nothing else pertaining to the object or circumstance in question can change. And that assumption is simply unwarranted.

To continue with the grade analogy, we should recognize that even while the total grade percentage does not change, a number of other factors do: as the school term progresses, the student continues to do assignments and turn them in. The teacher then grades these new assignments, sums the student's earned points so far, and divides that number into the highest possible point total. So while the grade percentage does not change, the student's earned sum continually rises. And of course, throughout the term, the course progresses, the student strives and sweats over homework, studies for tests, etc.

To me, this makes an obvious case that perfection can occur even while the qualities being judged for perfection change. In my example, the percentage remains at 100% even while the sums increase. This suggests the more general point that "perfection" can describe processes, not merely singular, unchanging states. Which really ought to be obvious, since, after all, we can easily set up (artificial) criteria for a perfect performance or the perfect execution of a technique in music, dance, etc. Yet clearly change does occur in these cases, for they are activities, not frozen states, after all.

Now, the point about lack of change is still true insofar as a perfect performance or process must never deviate from its perfect criteria on pain of falling from perfection. That is to say, the perfect student must continue to score 100% for as long as we judge her, and in this way her score will be predictable and therefore stagnant. However, we can hardly say that the student's homework and grade as a whole are stagnant, since she continues to accumulate new points and produce new work.

Similarly, I say that heaven would be static only insofar as its residents and contents would not deviate from a particular type of perfection; but their behavior could easily be a process or performance, or at least analogous to such. In simplistic terms, if perfection in heaven could be gauged as a percentage of points achieved out of points possible, the points could rise (or hey, they could fall too) without becoming imperfect so long as the total points possible matches them. It could be like a series of games in which one continually succeeds.

It is also important to note that, every time we talk about "perfection," we must qualify ourselves by answering the question, "Perfect according to what criteria?" Would it not be possible to have a constantly changing criteria set for perfection? And could there not easily be criteria which require change of some sort, as perhaps to run a perfect marathon requires that one change one's location and move one's limbs?

Monday, January 19, 2009

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...

Sunday, January 18, 2009

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.

Saturday, January 17, 2009

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.


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.

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

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

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.

Friday, January 16, 2009


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...?

Thursday, January 15, 2009

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.


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.

Tuesday, January 13, 2009


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

Friday, January 2, 2009

Silly details about numbers

The ancient Greeks found numbers fascinating in an unprecedented way (or so Morris Kline's Mathematics for the Nonmathematician informs me). Rather than adopting the more practical attitude of the Egyptians and Mesopotamians toward numbers, the Greeks recognized a conceptual beauty in the ability use the self-same methods on any possible collection of objects. That is to say, abstraction, and the universality thereof.

To assign a single, unique label to every quantity, and then to perform mental operations upon them which, amazingly, correctly modeled comparable situations in reality--astonishing. And I do mean that without irony.

Take a pie or any appropriately divisible object. Cut it into halves, then cut each half into thirds. We take it for granted today that the answer may be computed easily through a mechanical procedure: (1 * 1/2) * 1/3 = 1/6. This is to say that, without having made any cuts or further measurements, we already know with certainty what size the smaller pieces will be! (Or the size that they will approximate, because it is not an ideal world.)

But this is a fantastic discovery, for by beginning from only a few known facts, we discover what must be the case for physical objects after manipulating them--all without having left our armchairs.