Disclaimer: The following document fragment is presented from an in-character perspective, it should not be taken as the truth of the setting.
Context: Continuing with the theme of last week’s lore post, the concept discussed here are not fictional. At the very least in the sense that they are theoretical concepts in the fields of math and physics. While there is a light touch of fantasy thrown in, by including the Noosphere as one of Earth’s sphere’s, were you to ignore this, the whole excerpt would remain coherent and now relate purely to our own reality, and not the fictional reality of the game’s setting.
Let us consider the Poincaré Recurrence Theorem.
For those unfamiliar with the theorem in question, allow me to briefly explain. Let us consider an arbitrary dynamic system. We postulate that for any state of the system, and for any metric of similarity for the system’s state, there exists a time τ, in which the system will enter a state with similarity lesser or equal to δ.
Speaking otherwise, for any system, there is a finite time before its state repeats itself with a given degree of similarity. At the very least, assuming that the system itself can exist for long enough.
This can also be extended towards instances of a system. That is to say, given two (or more) identical systems within a different state, for any state of any of the systems, there exists a time τ’ until one of the systems in the set repeats the system’s state down to a similarity of at most δ.
This can then be further extended, given a set of systems, there is a time τ” until the state of a chosen segment of the system (for example, a spatial partition) repeats itself within an equivalent segment within any of the instances of a system down to a similarity of at most δ.
Given the fractal nature of reality and the possibility of other universes existing in parallel outside of the constraints given by fractal space, how likely is it for a second identical Earth to exist?
Well, assuming infinite time, infinite space, or an infinite number of existing universes, it appears to be given that, indeed, Earth must repeat itself an infinite number of times.
Or is it?
Or maybe, perhaps, our very notion of this issue finds our minds lost in the magnitude of the problem. What is the state of the Earth? Let’s think of it as such, there exists an arbitrary division of the Earth into 5 spheres:
- the lithosphere is the land, as well as the molten core of the Earth,
- the hydrosphere is the water,
- the atmosphere is the air,
- the biosphere is the sum total of all life on Earth
- the noosphere is the sum total of every thought on Earth.
For a copy of the Earth (with a given degree of similarity, of course) to exist elsewhere in a meaningful way, it would require all five of these spheres to be replicated. After all, if we only take into account an arbitrary arrangement of atoms in space which resembles that of Earth, would it be at all meaningful? Would that Earth be a copy of our Earth if all life upon its surface instantly dropped dead in the next Plank moment of time? After all, we have cared only for the arrangement of atoms in space, not for the systems of the planet itself to have continuity before and after that moment.
So how complex is the state of the Earth: what is the space of all possibilities?
I think we can agree that it is infinite; we’ve used this world a lot already. But not all infinities are equal.
Let’s take the game of Go. On a 19 by 19 board, there are 2e107 possible legal positions of the board. As of 2015, the population of the Earth is approximately 7 billion. If every person on Earth sat down to play a game of Go with someone else, the total state space for this collective Go playoff would be approximately 2e400000000000. Both of these numbers are equally significant when compared to the cardinality of whole numbers, which we denote as א₀ or ב₀.
Were we to theorise if a system of 3.5 billion randomly arranging Go boards would eventually start repeating the positions of all boards at once, the answer would be a firm yes.
But that question doesn’t even compare to the complexity of the lithosphere and the number of possible arrangements of landmass upon a potential alternate Earth.
So what does the complexity of the Earth compare to? ב₁, also known as 𝔠, or continuum, the cardinality of real numbers? ב₂, which is the power set of ב₁ and the cardinality of the set of all deterministic fractals? More?
What does the sum total complexity of life and the mind amount to?
Is it בω?
While it’s impossible to precisely estimate the actual order of ב representing the space of Earth’s state, it’s clearly unimaginably large. And in its unimaginability, it tricks us into thinking many incorrect things.
That is to say, I firmly believe that for any δ of existential importance τ”≅∞, to symbolically express that the value of τ” is too great to be existentially important.
Esoteric Musings... 