Monday, March 19, 2007

Lightdark

Axiomization is the monologic dictation of the infinite toward the finite. Paradoxically, the correct relationship between the infinite and the finite is dialogic, an infinitely modulating feedback loop (i.e. basins of attraction -- attractors -- are places where potentially-causative infinities reside, waiting until the finite is pushed across the event horizon by its own momentum). This is the implication of Goedel, I think. Cantor, with his theory of sets, showed different types of infinities. G showed that no fixed system, no matter how complicated, could represent the complexity of the whole numbers. Geometrization is the codification of B-theoretical truth at the expense of A-theoretical truth. But it is at the point of A-theoretical being, the nunc fluens, where one finds the dialogic strange loop between the infinite and the finite -- where one finds the melody of ontology. It is precisely the fact of ontological melody that Einstein's space-time geometry disallows -- the Now where actualization actualizes, where becoming becomes , where existence exists.

The implication is that no infinites are infinitely actual; the "flowing Now" is always already path-dependent (i.e. subject to nested loops). Standing on the moon was impossible until it wasn't. Time-travel is impossible until it isn't. Heavy elements, organic life, sentience -- all these were once eternally impossible; now they are extant features of our temporal universe. Rule-based impossibilities are inherently suspect.

The work of Escher: true Chiaroscuro. Light only exists where Darkness is absent, and objective existence emerges from the dialogic light-dark imagination.

Jackson and Escher

Many filmmakers borrow from preceding art when composing their pieces, and I get a special joy when I uncover an artist's inspiration. So here's my argument for M.C. Escher as Peter Jackson's muse:

Bilbo's Birthday Fireworks:

Escher:


Passing of the Elves:

Escher:


Rivendell:

Escher:


Mines of Moria:

Escher (yes, again):


Saruman Holding Palantir in Return of the King (couldn't find movie image):


Construction of Barad-dur (couldn't find film image):


"Things that shouldn't be forgotten, were lost." (couldn't find movie image):

Sunday, March 18, 2007

Timaeus

Yourgrau's book "Disappearance of Time," notes and excerpts:

Firstly, on Goedel's Incompleteness theorem: Hilbert's math assumed a fundamental equivalence of Form and Content, so that all Content, all meaning (semantics) could be consistently and Completely represented as pure structure, pure Form, pure Syntax. G's breakthrough was to use Form to elucidate Form's limit in expressing Content. G's theorem, then, is a Formalization of the dialectic of Form and Content, and the inability of the latter to complete reduce to the former. G's complaint of the "formalists" was that they "considered formal demonstrability to be an analysis of the concept of mathematical truth." Instead, formalisms cannot "speak for themselves" about their own significance; they need something more to be complete, something informal or "intuitive."

"Formal questions asks us to supply content of the answer, whereas informal questions demand that we supply both form and content." G's genius was converting the informal into the formal, then introducing constructions that limit the allowable "contentual interpretations" of the Form's domain.

Because of this, one must make a sharp distinction between epistemology and ontology, because the latter cannot be reduced to the former. This is the essence of G's philosophy: the distinction between proof and truth.

G, along with Plato, regarded the world of time as "ontologically suspect," and his philosophical strivings were efforts to merge the realms of the eternal and temporal.

For Kant, the proper image of time was a line generating itself: "by this mode of depicting it alone could we know the singleness of its dimension." For Aristotle this dynamic model of time leaves the past "fixed and continuously supplemented" and the future at least partially indeterminate; the place where they meet, the point of Chiaroscuro, is the Now. But these ideas of direction, movement, trajectory, of flux and points and instants -- these ideas are at the least metaphorically geometric, and at most literally geometric. This "geometrization of physics" reached an apotheosis with Einstein's theory of relativity, which represents a complete "mathematization of time" (geometrization of the temporal). Thereafter, the concept of time was completely absorbed -- completely reduced -- to structure, geometry, Form.

The spatialization of time has a serious implication: if time is susceptible to a purely geometric treatment, then all coordinates of temporal universe are ontologically neutral. Ontological-neutrality is the essence of "space", the essence of extension; therefore, if Einstein was right, an event's location in time has no effect whatsoever on the status of that event's existence. To spatialize time is to collapse the temporal mode of being -- the mode of becoming, of actualization out of the potential -- into platonic objects that, instead of coming into being successively, exist eternally.

But as we've seen, as G showed us, one cannot replace content with form. And this was the meaning of G's solution to E's relativity theory. In his solution G proved the possibility of nonstandard "unlucky universes", the so-called rotating or R-universes, where "objective lapse of time is an illusion" ( i.e. all world-lines loop back to connect to themselves, and time-travel through space is possible) and "even though it is an empirical question whether or not ours is an R-universe (it's not), the collapse of the objectivity of genuine temporal succession in a universe differing from ours only in certain non-lawlike features concerning the cosmic distribution of matter and motion shows that in our universe, too, time is merely 'ideal.' "

So G illuminated an inconsistency. Wittgenstein once said, "The world is all that is the case. It is the totality of facts, not of things." But if time and change are real, Reality grows by accretion of facts, and this would mean that the position in time does carry ontological weight. Tensed existence "entails that we can, at different times, rightly assert about the same thing that it exists and that it does not exist." ("Reality consists of an infinity of layers of "the now" which come into existence successively.") "What is temporally, both is and is not." By proving the possibility of R-universes within Einstein's theoretical domain, G was not trying to prove the non-existence of "objective becoming." Instead, he was demonstrating that Einstein's geometric formalism of time could not possibly be contentually complete: time really is "branchwise asymmetric", time-lapse really is objectively true, and E's theory disallowed it. G believes that time is not a mere illusion, and since E's construction of relativity admitted universes where time-lapse was "provably meaningless", there must be more to time than geometry.

The "direction of time" cannot be captured in purely spatio-geometrical terms. In space motion is relative and reversible. In time movement is unidirectional and irreversible. Events cannot be de-actualized or re-potentialized.

additional notes:

This full Meaning of time is better heard than seen. One needs a different explanation for the "unity of melody, successive in time, defined by time" from the "nonsuccessive unity of chords played spatially." One can grasp a melody structurally (tenselessly) by studying the score, but a piece of music must be performed to be completely comprehended: music is essentially a temporal entity. Music is not music unless it is being performed in real time.

"Eternity for Plato is not temporal duration; it is not a measure but a mode of being." Eternity is not "everlastingness" but "genuineness." The fundamental analogy for time is "a moving image of eternity." So temporal being is not complete being.

Time separates the potential qua potential from its realization qua actual, expressed in the form of motion. Being eternal means being motionless. [me: but even 2+2=4 is not eternal. in the empty set, in the Nothing, even that ceases to exist (though such truths will be the last to decay.]

There are two types of truths. There is the A-series truth, which is constantly in flux due to the nunc fluens -- the flowing now -- and the B-series truths, in which a truth about an event is fixed. A B-theoretic truth is "Kennedy was shot in 1963". An A-theoretic truth is "It is now 2007." The former's truth is fixed. The latter's truth is unable to be fixed permanently. For time to be real, both truth-series have to exist -- have to have ontological weight.

Random thought: memory multiplies the dimensionality of the Past's causality; without memory, causation is temporally linear only. With memory, the past can communicate -- and affect -- the present multi-dimensionally. If the present is the coded past, memory expands and complicates the code.

Sunday, March 11, 2007

Decline of the West, Vol. I

By Oswald Spengler. Notes and excerpts:

Language, even mathematical language, fails when dealing with something eternally inaccessible. Words for these things "are symbols, sounds, not descriptive of the indescribable but indicative of it for him who hath ears to hear. They evoke images, likenesses -- the only language of spiritual intercourse that man has discovered."

There are prime-words, like Will, Space, Eternity, and God, for which we have no notions but only names. Of these things we have an immediate certainty, in ineffable, incommunicable feeling.

The point at which "criticism falls silent and faith begins" is the moment that "analysis is confronted with itself." -- Goedel, i.e.

"The physicist of the inner world" tries to elucidate fictions by more fictions, notions by more notions -- to put a mechanism in place of an organism.

They felt themselves, with evident self-satisfaction, to be "dark and deep." But they understood their subject but partially, and hoped for a similar quasi-incomprehension in their audiences. "The only noteworthy thing they proved was the attractiveness of obscurity."

Schopenhauer reduced the world to Will and Idea, and it is only his ethic and not his metaphysic that decides against the Will. Nietzsche returned to the stronger formula Voluntas superior intellectu.

The primacy of Will or Reason is the basic problem of the Faustian soul.

Since its inception, the Faustian soul has been trying "in labour of many centuries to paint a self-portrait."

Will and Thought correspond to Direction and Extension, History and Nature, Destiny and Causality -- distance-becoming vs. distance-become, direction-feeling vs space-feeling, will vs. reason.

The language of the Faustian soul is spoken in a "dynamic syntax." And therefore the "entire Faustian ethic, from Thomas Aquinas to Kant, is an excelsior -- fulfillment of an 'I,' ethical work upon an 'I,' justification of an 'I' by faith and works; respect of the neighbor 'Thou' for the sake of one's "I" and its happiness; and, lastly and supremely, immortality for the 'I.'"

It is not the notion of Will, but the circumstance that we possess it while the Greeks were entirely ignorant of it, that gives it high symbolic import. Kant's formula, "Space as a priori form of perception," implies an assertion of supremacy of the soul over the alien, of tireless striving; the ego, through the form, is to rule the world. This is expressed in the mathematico-physical concept of force and function, and ultimately in the concept of vector. Plato never felt, as Kant was driven to feel, the ego as centre of a transcendent sphere of effect. The captives in his celebrated cave are really captives, the slaves and not the masters of outer impression -- recipients of light from the common sun and not themselves stars which irradiate the universe. For the Faustian soul, and never for the Classical soul, even spatial interval figures as form, and indeed as prime form thereof, for the notions of capacity and intensity rest upon it.

The search for completeness is the search for God -- the search for the original language (in the beginning was the Word). Goedel proved that we will never find it, never find Him.

"And therefore it is that about 1700 painting had to yield to instrumental music -- the only art that in the end is capable of clearly expressing what we feel about God."

In the Apollonian world, even the Gods are subordinate to blind necessity. In Homer even Zeus must take up the scales of destiny, not to settle, but to learn, the fate of Hector. But for the Faustian God is Will.

"Time is the passion of the third dimension." There is an inward connection between the invisible operations of nature and the unlimited range of the Order, between the arts of Calculus and Fugue.

Can it be that a Culture's prime-phenomenon is valid only for a season? For the Western man, activity, determination, self-control -- these are postulates that did not exist for the Stoics and Epicureans, for whom abstention from these things was an ideal. The Greeks were far from making "struggle" -- Kampf -- an ethical principle. To battle against the comfortable foregrounds of life, against the impressions of the moment, against what is near, tangible, easy; to win through to that which has generality and duration and links past and future -- these are the sums of all Faustian imperatives from earliest Gothic to Kant and Fichte: the causa sui of all our philosophy and science.

Carpe Diem is for the saturated being. For the insatiable soul, Victory is a dream, and the struggle is all.

Psychologists are merely fine students (hardly nowadays to be distinguished from the poet) of spiritual turning-points.

For Michelangelo the question of form was a religious matter, and this explains his sadness: for him it was all or nothing, and his incompletenesses left him fractured, tortured and unsatisfied. And it was this terrible vision that frightened his contemporaries. By mastering stone he was striving to master Death; never before has there been a more open expression of dread in the presence of the "become." And never has there been a comparable effort to tame it with violence.

Cultures are merely organisms which are born, ripen, age, and for ever die.

Brown is the one truly metaphysical color.

From the spheres of Beethoven and the stellar expanses of Kant, Impressionism has come down again to the crust of the earth. Its space is cognized, not experienced, seen, not contemplated; there is tunedness in it, but not Destiny. Modern art is a risky art, meticulous, cold, diseased -- an art for over-developed nerves, and scientific to the last degree. It is natural science as opposed to nature experience, head against heart, knowledge in contrast to faith.

As soon as the word comes to be used as the expression-agent of art, the waking consciousness ceases to express or to take in a thing integrally. The spoken word, when used in any artificial sense, separates hearing from understanding. Soon, motives in art are joined to word-meaning. Thus arrives allegory, or motive that signifies a word.

Thursday, March 01, 2007

Civilization and Its Discontents

By Sigmund Freud. Notes and excerpts:

"Man has, as it were, become a kind of prosthetic God. When he puts on all his auxiliary organs he is truly magnificent; but those organs have not grown on to him and they still give him much trouble at times."

"Writing was in its origin the voice of an absent person."

"Happiness is essentially subjective." me: a complex arrangement of subjective units, each with its own unique "qualia" (and by qualia I mean the product of genotype, phenotype, memory and position). happiness for all is impossible. some happinesses will preclude others. happinesses can be mutually exclusive. "Our possibilities for happiness are restricted by our constitution." me: and by the environment. "We derive intense enjoyment only from contrast, never from the state of things."

"The narcissistic man, who inclines to be self-sufficient, will seek his main satisfactions in his internal mental processes. As a last technique of living...he is offered that of a flight into neurotic illness."

"Religion restricts this play of choice and adaptation -- depressing the value of life, distorting the picture of the real world, and intimidating the intelligence. At this price, by forcibly fixing them in a state of psychical infantilism and by drawing them into a mass-delusion, religion succeeds in sparing many people an individual neurosis. But hardly anything more."

Goethe:
Wer Wissenschaft und kunst besitzt, hat auch Religion;
Wer jene beide nicht besitzt, der have Religion!
(He who possesses science and art also has religion; but he who possesses neither of those two, let him have religion!)

Wilhelm Busch: "He who has cares has brandy too."

Todestrieb -- "death-drive", Trieb - drive, the instinct of destruction, of mastery, the will to power.
das Ich - "the I" or ego,
Eros -- "life-drive" the drive to combine organic substances into ever larger unities
Two Heavenly Powers -- centrifugal and centripetal, entropy and negentropy, disorder and order, yang and yin, distinction and unity, discord and harmony.

"Aggressiveness was not created by property." me: no, it was created by organic chemistry and biological distinction, as a strategy for 'das Ich.' The will to life of das Ich also created Eros as a strategy. Distinction and unity were combined and recombined forever after by the selfish gene, and the play goes on. When balance is lost, a rip-current forms -- of one kind or the other. A man appears to play the deuce.

Wherever we go, this "indestructible feature of human nature will follow." The reign of Todestrieb begets an Eros triumphant. But from the shell of victory Nemesis is hatched...harmony is the foreshadowing leitmotif of a new kind of master.

Trilling: "The Opposing Self" -- A firewall against cultural control it is not -- a too romantic phrase. Rather, it is an inevitability, a necessity; a cure, as it were.

The "oceanic" feeling of eternity, of oneness with the universe.

Christian Dietrich Grabbe, Hannibal: "Indeed, we shall not fall out of this world. We are in it once and for all."

What do men demand of life? "They strive after happiness; they want to become happy and remain so." Almost -- they strive after satisfaction; they are moved to quench desire, to satiate appetite, to accomplish self-elected goals and affirm self-elected identities. Happiness is an empty word. It is hard to bear a succession of fair days. (Goethe). Desires, always and forever, move on. (Heidegger, man as the caring animal).

The drive to become independent of Fate -- the will to power. Fate as Nature, Fate as environment, Fate as other people. Distinction is the essence of the will to power.

Beauty is health, the sublime pinnacle of existence. ("The pleasure before the fall." -- Goethe).

The pursuit of pleasure is ipso facto a drive of distinction, a centrifugal drive of atomizing "I's". As such it needs parameters, channels, conduits, restraints -- it needs these things to ward off distinction's dreaded descendants: dissolution and decay (I am an alliterative bastard!).

Does P = NP?

Notes on computational complexity, from P, NP and Mathematics by Avi Wigderson (excerpts and notes):

"The language of algorithms is slowly becoming competitive with the language of equations and formulas (which are special cases of algorithms) for explaining complex mathematical structures."

In computational complexity, time is measured as the number of elementary operations performed for a given computation. Time is the primary "resource" of algorithms when studying their efficiency.

The equivalence notion in computational complexity -- any understanding we have of one problem can be simply translated into a similar understanding of the other. "The translating functions 'f' and 'h' are called reductions. We capture the simplicity of a reduction in computational terms."

"The asymptotic viewpoint is inherent to computational complexity theory...it reveals structure which would be obscured by finite, precise analysis."

"Efficient computation (for a given problem) will be taken to be one whose runtime on any input of length 'n' is bounded by a polynomial function in 'n'." This is class 'p', at most {An^c}. Polynomial functions typify "slowly growing" functions. Many important natural problems cannot at present be solved faster than in exponential time. Reducing their complexity to (any) polynomial will be a huge conceptual improvement.

The class 'np' contains all properties 'C' for which membership have short, efficiently verifiable proofs. The verification must be checked in polynomial time. Intuitively, this class is of the type that a successful completion can be easily recognized. This class is extremely rich, thousands upon thousands of problems in which arise naturally out of different necessities. 'np' always have trivial exponential algorithms, which is the brute-force way of searching all possible answers and verifying them.

If P = NP, then the implications are colossal: every instance of these tasks can be solved, optimally and efficiently. Is it possible that for every task for which verification is easy, finding a solution is not that much harder? What about the leap of creativity it takes to find a solution? how do you automate that?

The definition of class P is symmetric, the definition of class NP is asymmetric -- i.e. it is not always the case that it is easy to both certify that an object does have a property C, and certify that an object does not have property C (coNP -- if complement is also in NP).

If C can be reduced into D efficiently, then C<- D, and if D is in P, then C is in P. This means that for this case, solving the classification problem C is not computationally that much harder than solving the classification problem D. In some cases this means the importability of techniques from one area to another. The power of reductions is to relate seemingly unrelated notions. Efficient reductions are the backbone of computational complexity.

Hardness and completeness -- a problem D is called c-hard if for every 'C is in c' we have C <- D. If we further have that 'D is in c', then D is called c-complete. In other words, if D is c-complete, it is the hardest problem in class 'c': if we manage to solve D efficiently, we have done so for all other problems in 'c'. If a class has many complete problems, by definition they all have essentially the same complexity. If we manage to prove that any of them cannot be efficiently solved, we have managed to do so for all of them.

NP-complete problems are everywhere, and permeate all branches of science. Finding out a problem is NP-complete usually leads to leaving it alone and searching elsewhere for algorithms. The implication, however, is huge for anybody who finds an algorithm that solves just one NP-complete problem, because that algorithm can be translated to solve all others. Also, there is a near lack of natural objects in the huge void of NP problems that are neither P nor NP complete. This raises wonder about dark matter.

NP-complete is a formal stamp of difficult rather than a mere general feeling. It seems, then, we have two equivalent classes, P for those problems that can be efficiently solved, and NP-complete for those that can't (if P=NP then they are the same). If P is not equal to NP, though, then NP-complete has infinite levels of difficulty. If a problem is both NP and coNP, then it cannot be NP-complete.

Algorithms for natural problems under natural input distributions are almost all NP-complete.

The thrust of lower-bound arguments is that more time buys more computational power, such that there are functions computable in time n^3 that are not for time n^2. The implication is a "universal algorithm" which can simulate every other algorithm with only a small loss in efficiency.

Any natural proof of a lower bound implies subexponential algorithms for inverting one-way functions. The ability to describe a property both universally and existentially constitutes necessary and sufficient conditions -- a holy grail of mathematical understanding.

The artificial upper bound for algorithm is exponential, but this is only because we are inherently bounded by time.

Almost all natural phenomena, to be accurately represented by an algorithm (abstract process), have exponential curves of time as functions of linear increases in inputs.

Finding is hard, verifying is easy: natural selection rule?