"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

Thursday, December 27, 2012

Mad Hatter Paradox.

DEFINITION: Mad as a Hatter  – someone is said to be mad as a hatter iff there exists a mental illness (madness) from which they suffer, and they’re ignorant of its existence.


Now, consider someone asserting “I know that I’m mad as a hatter”. Call it the Mad Hatter sentence, and denote it with MH.

Is MH true or false?

Suppose MH is true. So it’s true that the person uttering it knows that they’re mad as a hatter. But by knowing that they are mad as a hatter they’re aware of the illness they purport to be suffering from, i.e. being mad as a hatter and so by definition fail to satisfy the conditions for being mad as a hatter. Hence MH is false, it seems.

But if they are not mad as a hatter, and assert a knowledge of being such, the person asserting MH is oblivious of that ignorance (them in fact not being mad as a hatter), a delusion of sorts, and hence qualifies them for being mad as a hatter, thus rendering MH true - but we know where that leads.

My independent discovery of the sequence A007526

I came across a rather surprising property of the above recurrence relation which I defined when counting the number of models of conditional logics with Lewis-Stanlaker type semantics, i.e. similarity spheres. Because the structure of nested sphere assignments happens to correspond in its form to permutations of non-empty subsets of some set, the formula, known to combinatorialists since Bernoulli, had emerged.

Below is an example taken from The Encyclopedia of Integer Sequences: sequence A007526 entry.


I say those properties were surprising given the context, for in and of themselves they’re not surprising at all. I simply didn’t expect such interesting properties to be inherent in the structure of the systems I was studying.

The sequence s is related to the number of models, given certain parameters (more precisely, the parameter is the size of the filter which is the set of all subformulae of a formula whose satisfiability is being tested - hence the requirement to generate all and only those models which are sufficient and necessary for that task), and given by the following recursive definition:
And this is the "curious" result, which links the above recursively defined function with Euler's Constant e:
It's easy to show that the explicit solution to the above recursive definition of s is given by the formula:
And considering the well known Taylor series, it's easy to show that the result stated above follows.
It turns out that this is a known sequence A007526., whose earliest references are Izquierdo (1659), Caramuel de Lobkowitz (1670), Prestet (1675) and Bernoulli (1713). So it appears that I've discovered/defined this sequence independently of those great mathematicians.

Wednesday, December 26, 2012

Combinations, permutation matrices, and a solution to n-valued valuations (truth value assignments for many-valued logics).

This result can be interpreted as a solution to n-valued valuations (truth value assignments for many-valued logics). A detailed proof is given in a later post.

First we define and construct the extensional valuation function , (intuitively, it can be thought of as the generating function for the classical truth table). The valuation function θ will have dimensions (k = 2i rows by i columns), and the value of each binary entry is given by the following formula, as can easily be checked.
Example θ(7,2), represented on a truth table exhausting all valuations (truth value assignments) for 3 propositional variables (atomic formulae). CORRECTION --- the formula works for the table generated such that zeros come before ones, i.e. with the ones and zeros inverted, so the table below is generated incorrectly for this formula, as we should start with zeros first. (Consequently all the m-valued generalizations should be generated analogously --- starting with the smallest value.)
This map, i.e. θ can be generalized to m-ary truth values. The explicit formula is given below:
I've skipped examples here, naturally since the truth matrices get very large, very quick.

Those explicit solutions to the recursive definitions (truth tables) make computation significantly more efficient. They also turn out to be handy (they're both elegant and compact) when writing algorithms for k combinations on n sets, or k partitions on n sets. They should prove useful to anyone working in writing algorithms for systems based on multiple value assignments.

Proof (sketch) --- a detailed proof is given in a later post.
First we express the truth tables as recurrence relations of congruences modulo the number of truth values in that given logic, then solve them by iteration, i.e. see the pattern. Finally give the solutions an elegant form (ceiling notation) using some useful number theoretic identities. QED

Saturday, September 8, 2012

Zen Koan Therapy

A careless monk experienced a fall,
Thus dislocating his fragile soul.

Reflecting on the apparent pain,
He puzzled over the insight that came.

For there was nothing beneath at all,
upon which one may even fall.

Reason then led him to finally state:
"There is no soul to dislocate."




This poem has been published in the New Lotus magazine.

Wednesday, September 5, 2012

Olympian Spring Song


There is a place one ought to know,
Since reason’s envoy had told me so.

One may rest also firmly assured,
That no virtue’s boundaries here will be blurred.
For I would never dare to reveal,
That which is lacking heart’s precious seal.

The rays of Spring in sunrise borne,
Whilst mending clouds that have been torn,
Decided finally to bring the song,
One which Olympians have known for long.

The world Olympus which bears their name,
Witnesses wonders our minds can't tame,
Among their many abilities,
Olympians utter infinities!

This springtime favorite, Olympian rhyme,
Has been composed to last all time,
Hush little human! and lend an ear,
Is that spring's anthem that you can hear?



Music is currently being written for this piece,

Thursday, July 19, 2012

Notions

...notions, as it happens to be the case, obtain, period. However, there are those that half obtain, obtain only in part, or don't obtain at all. Some even believe, that those that do both, that is to say obtain and don't obtain - all at once - are not that uncommon at all! And who could forget those that obtain only when others don't, and vice versa. Not to mention those that passively obtain - by default, as it were - while those who wish nothing else but to obtain, attempt to comfort themselves with the belief that, surely were they to obtain at all, they would be stuck with it - "it", being that which one presumably discovers once one obtains. I wish to say only this: it appears to me, that the most curious of all, are those which I shall call notions. For...


à la Carroll, Lem, Russell, Dr. Seuss.

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.