"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

Wednesday, November 23, 2011

The scientific method algorithm, computational complexity and TOE's.

I see science, or more precisely the method by which science proceeds (the scientific method) as an algorithm which asks progressively more complex questions, and subsequently receives answers of proportional complexity. Roughly, the science (theory) algorithm goes something like this:

Here we take the chain "(large)-(large with branches)-(large with leaves on some branches)-..." as an analogue for a chain of concepts of increasing complexity.

Science: Is the object General Sherman (largest tree in the world) larger (taller) than 10 metres?
Nature: Yes.
Science: Hoorray!

Science: Does General Sherman have branches (defined in one way or another)?
Nature: Yes.
Science: Hoorray!

Science: Do some of General Sherman's branches have feathers (defined in one way or another)?
Nature: No.
Science: Damn! Back to the drawing board.

Science: Do some of General Sherman's branches have leaves (defined in one way or another)?
Nature: Yes.
Science: Hoorray!

Science: ...?
Nature: ...
Science: ...!
     :
     :
     :
Possibly ad infinitum.

A corollary of this model is, that it may be the case that our theories, as they evolve in discrete steps, one superseeding the previous one, will grow to arbitrary large complexity as they accommodate phenomena of gradually increasing complexity.

It is a theorem of Algorithmic Information Theory that the complexity of certain stings/objects/concepts put a lower bound on the complexity of the theory/program that generates them. Thus one cannot hope for a complete theory of everything (TOE) without some harboured assumptions about the complexity of the universe and the phenomena within it. It would certainly be presumptuous to harbour such assumptions since as far as I know the question whether the universe (multiverse) is finite or infinite, and hence it's complexity, remains an open question.

For it may be the case that our theories can only approach a limit which would be a TOE, but necessarily never reach it.

The above ruminations have been primarily inspired by the work of Gregory Chaitin.

Tuesday, November 22, 2011

Constructing a set disjoint from, and equinumerous with any other set.

A while ago I needed (a lemma in a larger proof) to construct for any arbitrary set A, another set that is disjoint from A and of the same cardinality as A (without the use of higher machinery of ordinals). The task is harder than it may seem. One may choose to attempt a proof by contradiction. One of my friends has in fact proceeded that way. I came up with the following construction which I’m quite fond of.

First, let me state a theorem (which is a direct consequence of Russell’s paradox) that I helped myself to.

Theorem. For each set S, the set
Is not a member of S.
For if we assume otherwise, an immediate contradiction ensues, for
Now for the construction. Let’s take some set A, and construct a set B that is disjoint from it, and equinumerous with it. Consider the set
Theorem: For any set A the following set B is disjoint from A and equinumerous with it:
Less formally
It is clear that B has the same cardinality as A. (post-Clarification: just to be pedantic, it's not just the triple x, y, z that is in A, but rather I should have written "for x, y, z,... in A". I forgot the ellipsis.)

It is also clear that,
since,
which is impossible by construction of RA . Hence B is equinumerous to, and disjoint from A. QED

Friday, November 18, 2011

Thoughts

I don’t think that it is too much of a presumption to claim that our conceptual repertoire developed thought human intellectual history is largely an outgrowth (and conditioned by) of the world we happen to occupy and the form we happen to posses.
We have had some success and seen a development of intellectual brevity which in effect made it possible to surpass our perceptual and rational limitations, leaving them behind as hindering properties of our human form (example: existence of the electromagnetic spectrum - our senses do not perceive X rays for instance, and peculiar existential nature of quantum entities - it seems that the distributivity theorem of propositional calculus fails at the quantum level ) . Nevertheless it is that very form that was the starting point, and persistently forces it’s limitations on our creativity and ingenuity.
Furthermore it should be noted that the universe we happen to occupy is a rather narrow experimental and severely conditioning ground – why, it just happens to be a world where sentient beings are possible. Empiria seems to suggest that other worlds are physically possible, and logic certainly doesn’t demand that possible worlds necessarily contain sentient beings.
One may find oneself wondering whether the universe (or the multiverse in its entire spatio- temporal and/or Hugh Everett’ian sense, or modal realism sense after David Lewis) is finite or infinite. If the latter than we should , with mature humility, accept epistemic defeat, in the “let’s ask nature what she is” game. If the former, then the problem doesn’t disappear but becomes more interesting when we consider questions of information, self-reference and computational complexity. 
But notice that I have introduced a rather presumptuous classical dichotomy which in itself is an effect of intellectual conditioning to particular type of logic. Why not finite and infinite simultaneously (?), or neither finite nor infinite? And if simultaneously finite and infinite, then it appears that, again, I’m  introducing crude notions concerning what we call time, yet have no conclusive answer to what it is or even whether it is a fundamentally coherent notion at all. What we refer to as time may just be a phenomenal manifestation of  some more fundamental properties (Sean Carroll), peculiarities if you will, of the universe we happen to find ourselves in. So again one is reasoning from possibly perceptual limitations, making any inquiry fundamentally unsound.
So we continue on what we supposedly do best – conjure up new concepts about what is, and what could be, with more lucid moments realizing that this is just a peculiar property of some peculiar (what we call, sentient and intelligent) entity occupying some peculiar universe – a property motivated by the desire and hope of finding an alibi for our existence, or maybe not even motivated at all, but it sure sounds more reassuring that such behavior is intended.

Wednesday, November 9, 2011

Super Intelligent Beings.

Imagine yourself, perceiving a plant, say a tree - its incredibly slow metabolic rate and predictable "behaviour" i.e growth. Also in some sense the plant is defenceless against your whim, whatever it may be - to cut it down, burn it, or eat (parts of) it. Better still imagine a stone - its practically unchanging crystal structure, and unique combination of minerals shaped by geological processes over billions of years. Again - the stone is a passive, and for all practical purposes (by definition!), a lifeless object. Now imagine, that a super-intelligent being would perceive YOU in a manner analogous to your perception of that tree, or rock.     

In the time that you would take one breath, it would seem to such a being that eons have passed during which its leaps and strides of insight would dwarf the totality of our (human) current intellectual achievements.