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How do we resolve the fact that our finite brains can conceive of mental spaces far more vast than the known physical universe and more numerous than all of the atoms?
For example, the total possible state-space of a game of chess is well defined, finite, but much larger than the number of atoms in the universe (http://en.wikipedia.org/wiki/Shannon_number). Obviously, all of these states "exist" in some nebulous sense insofar as the rules of chess describe the boundaries of the possible space, and any particular instance within that space we conceive of is instantly manifest as soon as we think of it. But what is the nature of this existence, since it is equally obvious that the entire state-space can never actually be manifest simultaneously in our universe, as even the idea of a board position requires more than one atom to manifest that mental event? Yet through abstraction, we can casually refer to many such hyper-huge spaces. We can talk of infinite number ranges like the integers, and "bigger"...

Consider a small finite set--say, the set of members of your immediate family. You understand this set by knowing a list of all the members of the set, and being familiar with all of them. Now, the point of your question is that that approach clearly won't work for infinite, or even very large finite, sets. My grasp of the state-space for chess, or the collection of all integers, cannot involve being familiar with all the elements of the collection. So what does it involve? You hint at a couple of possibilities in your question: In the case of the state-space for chess, you say that "the rules of chess describe the boundaries" of the state-space. So your understanding of that set consists in knowing the rules that determine what is in the set and what isn't, even though you don't have a complete list of the elements. You also refer to the fact that "we can so easily instantiate well-formed members" of these collections. For example, I can generate as many positive integers as I want by counting,...

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