Selected Topics in the Philosophy of Science, Part II: Limited Space, Limited Entities, Infinity

In the lecture on mathematics, Dr. Binswanger considers the concept of infinity and asks whether it is valid, even as a concept of method or potentiality. He answers (tentatively) in the negative, for two essential reasons, the metaphysical and the psycho-epistemological.

The metaphysical argument hinges on the idea that there are a finite number of entities and thus a finite number of things to which we can assign a number, and therefore the concept of infinity as "always being able to add one more" isn't true: we reach a point (albeit a very large point) where our next number doesn't stand for anything.

The psycho-epistemological argument also depends on the finitude of entities: making a symbol to represent the next number takes up space, and even though the human consciousness can make mental condensations, those condensations themselves take up space and therefore there will be a point, even with a computer-augmented mind, where we can no longer hold the next symbol in our head as anything relevant.

In the Q&A period of the mathematics lecture, Dr. Binswanger is asked a question about whether there has been an infinite amount of time, and he answers this to the effect of "infinite time between now and... when? Between now and the beginning of the Universe? Well, there was no beginning. Between now and any actual point in the past? Well, no, that's a finite amount of time."

Dr. Binswanger's response indirectly invalidates the finitude of space and, therefore, the finitude of entities (and therefore the metaphysical and psycho-epistemological arguments as well). Consider the question "Isn't there an infinite amount of space?" Well, we could just say "no, it's finite and limited", or, following Dr. Binswanger's argument in regard to time, we could respond "space between here and... where? Between here and the edge of the universe? Well, there is no edge. Space between here and any given entity? Well, no, that's a finite amount of space."

In this way, there is a metaphysical possibility (Possibility here meaning that the idea does not contradict anything in metaphysics. It is not a claim to any amount of evidence supporting that it's actually correct) of unlimited space which is nevertheless always finite. With the possibility of unlimited space comes the possibility of unlimited entities, by the same argument: withing any given amount of space, there is a finite number of entities, but the size of that given amount of space can increase without limit, and with that the number of entities in that space.

If there is no limit to the number of entities, then any number we come up with has something to refer to, and thus the metaphysical argument is invalidated. Similarly, if there is no limit to the number of entities and if we augmented our brains with computers, it is possible that we would always be able to find enough raw material to continue to increase the number of symbols we can store in our computer-brains, thus invalidating the psycho-epistemological argument against infinity. Infinity is thus restored as a valid term.

The primary issue in this discussion is the idea of applying properties to the universe as a whole that do not apply. To apply "time" to something, you need to have two endpoints, so unless you postulate a beginning and/or an end of the universe, you cannot apply time to the universe as a whole. To apply "distance" to something, you need two endpoints, so unless you postulate an edge of the universe you cannot apply distance to the universe as a whole. To count the number of anything, you must know the limits of the space you're counting. You ask "how many books are there on that shelf," or "how much water is there in this glass," etc. Thus, unless you postulate a limited space, you cannot apply amount of anything to the universe as a whole.

A question that occurred to me based upon this thought process is whether or not "universe" is a proper term at all (note that even if "universe" is proper, it is not a concept but rather an instance of the concept "collection"). The issue I'm having is that, since so many properties like "time", "space", "number of entities" do not apply to the universe, which properties do (i.e. what is its identity)? You could say "the universe is that collection which contains everything that exists", but attributes must be finite (the omitted measurements may be any quantity, but they must be some quantity). So, what exactly is the universe? I don't have an answer, and comments are greatly appreciated.

I should note here that I have been approaching this issue metaphysically. The essence of my argument is that nothing in metaphysics necessitates the existence of an edge of the universe (and therefore nothing necessitates space being limited). At the same time, however, I'm not sure that anything in metaphysics necessitates the absence of such an edge (though I would be open to such an argument). Therefore, as of now the question remains in my mind one for the physical sciences (though I do not count the current discussions of the size of the universe within the scientific community as at all valid).

That was a hefty post, so to sum up: Dr. B claims that infinity is invalid on metaphysical and psycho-epistemological grounds because the number of entities is limited, and then goes on to claim that time is finite but unlimited. I claim that Dr. B's claim of the finite-but-unlimited nature of time applies equally well to space, and therefore to the number of entities, and therefore the metaphysical and psycho-epistemological arguments are invalidated, and infinity is restored as a proper term. I end with some speculation on the propriety of the term "universe", and specify that my argument is purely metaphysical, and that the issue might have to be settled by the physical sciences.

Part I Part II Part III

P.S. Even if you don't accept the finite-but-unlimited nature of space, the finite-but-unlimited nature of time invalidates at least the metaphysical argument against infinity. We can talk about what happened n seconds ago, and we can also talk about what happened n+1 seconds ago, no matter what the value of n is, by Dr. Binswanger's own admission. Therefore, every number we could come up with has something to refer to.