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.

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 R_A . Hence B is equinumerous to, and disjoint from A. QED

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.

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.

A FALSE DICHOTOMY: Science vs Intelligent Design

I realised today something quite obvious, after reading a rather staunchly dismissive response to a book review of The Fallacy of Fine Tuning (Victor Stenger). The response was decisve in its dismissal but superficial if not outright vacuous in justifications for that attitude (I know the person hasn't read the book).

Intelligent and curious individuals with a desire to genuinely understand themselves and the world they live in will search with an opend mind. This philosophical inquiry will attract them to any information that brings them close to revealing yet another mystery, or a new conceptual approach which may make the search more efficient, or even revise and inform the previous approaches. Dubious and self contradicory views will naturally be either subjected to second rate or outhright rejected depending on the degree of internal inconsistencies or dissonances with common sense and direct observation. For example it is this sort of attitude that abandoned a once popular Flat Earth belief. This kind of apprach is a humbling attitude for it soberly aknowledges our epistemological limitations, but makes do with what it can, improving where it can, revising where possible.

The false dichotomy is between scientific thinking, which is highly correlated with the attitude described above, and just one theory/belief - namely Intelligent Design (clearly a subversive oxymoron) that deems itself special for all the wrong reasons (there's nothing genuine about it's pursuits for truth and understanding).

The point is that we don't choose between science and the latter, the distinction (and the correct dichotomy) is more along the lines of  being genuinely curious about the world, and hence searching and discovering its wonderful mysteries or happy to conform to a simplistic set of views that quenches ones inner fears and desires.

Another way of putting this is considering Intelligent Design as one of a plethora of theories and philosophies. Succumbing to the false dichotomy creates an illusion of some special status for ID theory - a status it doesn't deserve.

Can't help but subject "Being" to logic

I certainly endorse an analytical approach to philosophy, but this is not the same as declaring oneself a rationalist. Recently however I've discovered that I may after all be falling within that category.

I have always endowed Being with a supreme status. That is, minds and their way of cognizing, perceiving and talking about Being and its manifestations have a somewhat secondary status. Gottfried Leibniz had introduced the principle of sufficient reason in aiding answering the question "Why is there something rather than nothing?". That principle is related to the anthropic principle which conditions the answer on the existence of intelligent life - us humans in particular . A related anthropic approach would be what I call the logical rather than the standard one with physical attributes criterion - there is something rather than nothing because we happen to occupy a possible world that actually exists - a necessary condition for the existence of beings that ask metaphysical questions.
This would be impossible in a possible world that happened to not be (physically non existent worlds are possible!). That is to say, out of the possible worlds there are those that are and those that are not. Those that happen to exist do not so necessarily i.e. it was possible for them not to be - a subtelty that Parmenides failed to see when he claimed "That which is, is and it is impossible for it not to be" - a modal fallacy identified by Aristotle.

Now that we have the sufficient logical layout, my problem can be clearly presented. It appears that although the question "Why is there something than nothing" has a partially satisfactory answer (in the negative sense, for it does not posit a positive cause to Being) in the sense that it points to the fact that we (physical beings) could not ask that question if nothing existed. Nevertheless, it follows (Alas! the point of my discovery that I'm a metaphysical rationalist) that in those non existent worlds it would be true to say that being is possible, since there is at least one possible world (namely the one we happen to inhabit - proof by inspection;) where being obtains.

The point is that it appears that ontological existence is a subset of modal (logical) existence. That being the case, plenty of worlds are not endowed with (physical) existence --- they don't acually exist, yet they posess modal existence and that is why we can talk about them - moreover some truths hold in those worlds despite them actually existing - "Being is possible" is one of such truths.

"Everyone knows that dragons don't exist. But while this simplistic formulation may satisfy the
layman, it does not suffice for the scientific mind. [...] Indeed, the banality of existence has been so
amply demonstrated, there is no need for us to discuss it any further here. The brilliant Cerebron, attacking the problem analytically, discovered three distinct kinds of dragon: the mythical, the chimerical, and the purely hypothetical. They were all, one might say, nonexistent, but each non-existed in an entirely different way." (Stanislaw Lem, The Cyberiad - Dragons of Probability)

Sensless chatter analysis

Average teenager-mid twenties person wpm (words per minute - spoken) = 150.

Average teenager-mid twenties person wpm (words per minute - spoken) with the phrases (and the variations of) "so/and I/he/she/it/we/they am/is/are/was/were like..." = 45.