"The poet only asks to get his head into the heavens. It is the logician who seeks to get the heavens into his head. And it is his head that splits." G.K. Chesterton

Saturday, March 24, 2012

Eternal Return implies a Finite World, No Beginning of Time, and a single unique Eternity



Eternal Return implies 
a Finite World, No Beginning of Time, 
and a single, unique Eternity.



The construction involves taking into account the set of natural laws NL that govern our world - I assume there may be such NL. Define the set W:=set of all possible states of affairs of the world at any given instant consistent with NL”.
Next we look at the "set of all possible sequences* with no repeats  consistent wit NL (these are our circuits/loops) of the elements of W" - call it Ws. The elements of Ws look like this for any element x of W: (x, y, z, r,…, s, t, x) if W is countable, or the union of [x, r] and [t, x], if W is uncountable, and none of y, z, r,…, s, t  are equal to x (*). Eternal Return demands that Ws is non empty.
It also insists that there are no infinitely long elements in Ws, since if it was otherwise it would be possible to enter a non-terminating sequence with no repeats (in it the ER view would be false) - contradiction. Hence all elements of Ws are finite
  • Ws is finite: there are only finitely many distinct world histories with no return.
    • Hence W is finite or some NL consistent states of affairs never obtain, which raises the question: what makes those states NL consistent? Wouldn’t NL consistency of a state w(a) demand that it can obtain? But then if w(a) can’t obtain, it is no longer NL consistent – contradiction.
    • Hence W is finite.

Next, we model Eternity.
Also ER implies that either, there is no beginning of time, or if there was one we get an immediate contradiction, or if we’re lenient and look the other way we still will be left with an eternity consisting of an impoverished oscillation between states, which also could be considered empirically falsified since we perceive change. A detailed proof could be provided, but just consider the fact that the first instant of time is nor a return point of any predecessor except itself (contrary to ER), and any future instant of time will always lack sufficient predecessors to satisfy both requirements - change and the ER requirement. So we model Eternity to be isomorphic with the integers i.e. it extends into infinity both into the past and to the future.

The construction proceeds as follows - we take some element of W, w(k) i.e. w(k) is some state of affairs at a given instant. Next we construct the “an infinite concatenation* with neither beginning nor end, of all elements of Ws beginning with w(k)”, denoted ?E.w(k). We remove each of the doubling w(k)’s where each sequence joins (*).

However it may be the case that not all concatenations* of elements of Ws are consistent with NL, since there may be “jumps” (discontinuities) in some of them at the joins. So we define E.w(k) identically to ?E.w(k) with the proviso that all concatenations* therein are consistent with NL.
This construction yields an Eternity conditioned on Eternal Return to w(k), denoted E.w(k).

E.w(k) = … w(i) w(k) w(j) w(x)… w(y) w(z) w(k) w(p)… w(q) w(k) w(r)…

If we assume the Eternal Return view as true, then for all w(j), w(k) in W, E.w(j) = E.w(k), since if that wasn’t the case, some instants would not return to themselves contrary to the assumption – contradiction. 

  • For all w(j), w(k) in W, E.w(j) = E.w(k) 
-       Hence there’s only one unique Eternity.





Saturday, March 10, 2012

Some thoughts on William Blake's ..... AUGURIES OF INNOCENCE



‎To see a World in a Grain of Sand 
And a Heaven in a Wild Flower, 
Hold Infinity in the palm of your hand 
And Eternity in an hour.
[...]

Upon some reflection, one may start developing a sense of empathy for the somewhat
romantic and existential character of that perennial plea uttered by humanity, ringing out with a tone of longing, it’s echo reverberating through the Platonic realm, and reflecting back here so softly that only the hearts of poets can hear it.

[...]

Every night and every morn
Some to misery are born,
Every morn and every night
Some are born to sweet delight.

Some are born to sweet delight,

Some are born to endless night.
[...]

God appears, and God is light,
To those poor souls who dwell in night;
But does a human form display
To those who dwell in realms of day.

I'm sure there may be numerous ways of interpreting this poem, and maybe even some of them could be considered as better than others? But it seems that the content of the poem itself opposes the existence of any hierarchy which should serve as the absolute measure of any interpretation's alleged objective adequacy. This, perhaps could be an interesting direction of inquiry in itself, but not one which I presently wish to pursue. Instead I will share some of my reflections inspired by the poem.
.
What I've noticed, prompted by reflections meandering toward the blurry boundaries of
 the poem's subject matter, is the apparent content-relational continuity between the first, and the last few stanzas, and the message their interplay carries. It's plausible that the author intended this symmetry, if not throughout the entire poem then at least between its first and the last stanzas.
.
What it reveals, if one entertains this observation as one not entirely devoid of merit, is
 the conditioning of one's conceptual repertoire to the attributes (nature) of the "realm" they inhabit and experience.
.
Those who are "doomed" - "to endless night", which is some otherworldly-Heavenly domain - and united with
 God's immanent presence, are perchance privileged having such direct and immediate access to the divine and transcendental nature of reality. Being familiar with that realm, they know too well how it permeates everything, and that it should be evident - "seen" - even in a wildflower - a worldly entity. Sadly, such a particular form of being is absent from the realm of the universal which they dwell in. (Note the mysterious and uncrystallized nature of that which cannot be "seen" i.e. during the "night", and consider the magnitude of that mystery with "endless" predicated of it.)
.
As they dream of seeing (a blank concept, although entertainable, yet devoid of any experiencial referent), they imagine seeing that which they only know - "heaven" - in an object (the wild flower) endowed with a particular, grounded yet unique nature. If only they were granted that possibility - to experience, if only an infinitesimal fraction of the "realm of day" - no larger than a grain of sand would suffice.
.
But alas! - their gaze invariably transcends the flower's earthly, finite form, failing to appreciate its pure and simple beauty.
.
Those who dwell in the "sweet delight", are "blessed" with the clarity offered by the world bathed in the "light of day", and unperturbed by that which cannot be seen, can only relate to the divine in terms of the spatiotemporal boundaries, imposed on them by the "realm" they inhabit, and cannot help but reduce the transcendental to such a conceptual confinement: “Hold Infinity in the palm of your hand, and Eternity in an hour”.
.
This is also why, despite persistently harbouring hopes and dreams - which resemble restless chimeras that rummage anxiously through the most elusive depths of the soul, where mystery, doubt and wonder seek refuge from the syrupy tentacles of the hedonistic bliss - they occasionally experience fleeting suspicions with regard to the "all revealing" "sweet delight" - whether there may not be some paradoxical, and double edged nature to it.
.
Being conditioned to the "explicated", worldly realm, which overwhelms the eyes with demonstrative, verging on obtrusive, clarity (hence a presumed "absolute clarity"), and crystallized, unambiguously delineated finite forms, one experiences God accordingly - in a familiar and non hesitant manner.
.
A still further stretched interpretation, on a somewhat relatively deeper reading, would allow seeing
those diverse realms, presented in the poem, as analogues of certain attitudes which we're all free to adopt - or maybe not entirely free, hence the somber tone and categorical emphasis on the phrases "born to ..."
.
Upon adopting, or tentatively veering toward some preferred
 set of attitudes, we progressively develop, and subsequently reinforce, the positive feedback driven interdependence of our experiential and conceptual repertoire - their character in turn, ultimately presents us with a corresponding manifestation of Being.