Maxwell’s demon and the physical nature of space

Maxwell’s demon is a thought experiment to examine the second law of thermodynamics. But is not only about a special part of physics, but about some unspoken features of our Universe as well.
It is a small, intelligent entity who can detect the speed of individual molecules and opening or closing a door that divides a box into two parts, he/she is able to separate the fast molecules into one while the slow ones into the other part of the box. The result is the decrease of entropy in a closed system – that is contradicts the second law.
Or not, since, according to an argument based on information theory, the main problem is that if Maxwell’s demon’s memory isn’t infinitely large, then sooner or later information should be erased form it.This process emits heat into the box, because information erasing necessarily causes heat [Charles Seife: Decoding the Universe, p. 85.]. So the second law remains valid, since the entropy grows in the closed system.
This answer is partly based on the presupposition that only finitely large memories are possible and includes another presupposition, too, about the nature of space–after all, if you would be able to divide the space into infinitely small amounts, then it would be possible to store an infinitely large amount of information in a finite storage (at least, theoretically). I.e. If you have a two square cm surface of data storage, then you could use the first square cm to store the first piece of data; a half square cm to store the second piece of data; etc. ad infinitum.
An infinitely large memory perhaps not as unreal as it seems to be for the first sight, since there are plausible theories about hypercomputing based on relativistic spacetime of blackhole physics (see the details here). In this case manipulating an infinitely huge amount of data in an finite period of time is possible thanks to the nature: since it is presupposed that time is divisible infinitely many pieces, the result is that we have enough time to perform infinitely many operations in a finite period of time.
Ad analogiam: We can hypothesize that the space’s nature is similar and it is divisible infinitely, so it is possible to build a spatially finite storage to store an infinite amount of information. So we never should erase a bit of information–and the entropy wouldn’t rise in the box.
Obviously, nobody knows whether the space continuous, but it has been shown by this example how the laws of thermodynamics is embedded into the "environment" of the existing physical laws. And we cannot exclude the existence of Maxwell’s demon if the space we live in is not discrete, but continuous.

Thomson’s infinite lamp as a mathematical monster

Imagine that we have a lamp – it is switched off at its initial state and this state can be changed by pressing a button. Having been an hour to play with this lamp, we switch the lamp on after 30 minutes. Then after waiting for 15 minutes, we switch it off – and then we switch it on again exactly 7.5 minutes later – and so on. I think that the end of the story is self-evident: after one hour (and neglecting that it is physically impossible) we pressed the button infinitely many times.
But what will be the result? Will the lamp light? Or will not?
It seems to be an unanswerable question – after all, we can regard the switch off state as an “odd” and the switch on as an "even" number (or vice versa). The source of this problem is that only a natural number is either odd or even – but infinite is not a number in a traditional way.
But there is another analogy and it can help. Nobody knows PI’s exact value since it is an irrational number. What is more, according to our actual knowledge, its digits are randomly distributed. But if we would be able to compute all of its digits, would the last digit be an even number?
Perhaps it seems to be an acceptable answer that there is no a last digit of PI, so it is neither odd nor even. But PI is nothing more than the ratio of the circumference of a circle to its diameter and although we do not know exactly the numerical value of this ratio, it is a certain, existing value. Computing more and more digits of PI, we’ll know it more and more accurately – and computing it to the infinity, we’ll know it exactly.
Ad analogiam: if we press the button of the lamp infinitely many times, then the lamp will be necessarily either switched on or switched off – although we cannot predict the lamp’s state.
At this point we can distinguish to different types in math: random and compressible strings. The previous one means that we cannot find a representation of the given string which is shorter than the original one. Heads and tails is a good example for it: you won't know the result without tossing the coin in reality.
Or onecould mention the cellular automatons (CAs). The state of their cells depend on the neighboring cells’ states and a CA changes in discrete steps. The result is that although the system is absolute deterministic (certain starting configurations always results the same next phases), cellular automation is an incompressible process. We cannot compute the next phase without executing the program itself.
Opposite to these above mentioned examples, a compressible string can be regarded to be “regular” in a sense that if we know the rule, then we can find the nth digit without computing others.
Our lamp represents a totally different solution. We can compute its every stage and its algorithm is ridiculously simple, so it is compressible - except for its endpoint. We cannot answer whether the lamp is switched on at its final stage – unless we de facto pressed that button for infinitely many times.
I wonder whether there are other, strange categories – for example, who could imagine a string which is compressible only at its endpoint? Perhaps other mathematical monsters lurking somewhere.

Super fine-tuned worlds and infinitely many universes

The typical “our Universe is fine-tuned for life” argument states that small changes in certain constants result a lifeless Universe. For example, a 2% percent stronger strong nuclear force’s consequence would be a lifeless Universe, since the formation of heavier elements never would happen under such conditions. “Although our situation is not central, it is inevitably privileged to some extent” says Anthropic Principle’s advocate Brandon Carter [In Modern Cosmology and Philosophy (ed. by John Leslie), 1998, p. 132.].
But as Stephen Jay Gould pointed out “any complex historical outcome – intelligent life on Earth, for example – represent a summation of improbabilities and becomes thereby absurdly unlikely” [ibid, p. 187]. So our special situation is not necessarily a consequence of the fine-tuning.
Similarly, Feynman joked on his luckiness, since he had observed accidentally the ARW357 license plate although the probability of it was practically zero.
And there are other problems with arguments about fine-tuning.
First of all, life is sensitive to some physical parameters. Opposite to it, computing is insensitive to the changes of almost all of them. But it doesn’t mean that our biology is extremely fine-tuned (or extremely matches) to our Universe’s conditions. Probably either a higher or a lower speed of light would not prevent the rise of life. We can play with the idea of a universe where the result of a minor modification of any parameter would result a dead world. It is not the case in our Universe, so we do not live in a so “super fine-tuned” world.
Second of all, “to be privileged” means that the given situation is not average but applying this to the biofil universe concept, there is an unspoken presupposition behind it. If we assume that there are some other universes then uniqueness has a meaning. In other words: if we hypothesize that only a finite number of other universes exist then the “privileged position” is interpretable. But if we are suppose that infinite other worlds (no matter how we define them) exist then we have to believe that infinitely many universes of them are identical with our one. What is more, in this case the number of lifeless; the biolfil and the computable universes are equal – after all, any kind of them have infinitely many identical copies. Thus the meaning of “privileged to some extent” is uninterpretable.