The Unbounded, Finite Universe
Since the infinite is the impossible, the latter choice is rightly dismissed. However, a finite, but unbounded universe can still leave one looking for a satisfactory answer. A spherical analogy is usually offered to explain what seems to be one's only option, but bringing this proposed solution into literal, three-dimensional reality presents itself as being an insurmountable task.
I submit that both sides of this alternative appear contradictory, because they are. Since the question above is inexactly formulated for the given context, what is actually at work here is a false dichotomy.
In order to understand why, let us differentiate. Ask whether the Milky Way is finite or infinite. Needless to say, one would find no unresolved issues with answering that it is finite - on account of the Milky Way having spatial boundaries.
The existence or nonexistence of spatial boundaries is what makes the difference between an easy answer with the Milky Way, and an enigma with the universe. This reveals that it is a certain alleged characteristic of the universe - one whose finiteness would be threatened if boundaries were not present - that is actually being called into question.
In this light, one can readily see that "Is the universe finite or infinite in size?" is the actual question being asked. As such, it is not the universe, but its extent (i.e., size), that is contradictory, whether posited as finite or infinite.
Why, then, must one posit a size of the universe at all? Objectivism does not posit a temporal extension (age) of the universe, and then ask whether it is finite or infinite, so why do the same with spatial extension (size)?
This analogy between the spatial and the temporal is actually very exact. The same problems that arise in a finite or infinite size dichotomy appear in the same exact manner as they do in a finite or infinite time dichotomy: both dichotomies force one to accept that the respective characteristic does in fact exist, when both in fact do not. And, since infinities are impossible, this means: the universe is not finite in size anymore than it is finite in time.
After all: would a theory of "circular time" [1], attempting to reconcile a finite duration that does not possess temporal boundaries, be tenable or necessary?
Objectivism says "No," and thus does away with the temporal false dichotomy of this analogy masterfully. It recognizes that "Is the universe finite or infinite in time?" is a complex question; it assumes that the concept of time is applicable to existence. But, it isn't applicable; so, neither is the question. The universe isn't "in time," so it therefore isn't "finite in time." ("Finite in time" and "in time," like "finite in size" and "in size," are equivalent statements.)
And, this viewpoint can be validly translated into physical reality, too. If one had a time machine that was able to go into the future indefinitely at the rate of octillion years a picosecond, one would never get to an "end" of the universe. One would see the occurrence of event after event, without end. Of course, this is not because the universe is "infinite in time," but because it isn't "in time" at all. Needless to say, this is no threat to identity, and no reification of infinity.
Similarly, if one had a spaceship that was able to travel an octillion light-years a picosecond, one would never get to any "end" (or "edge") of existence. One would fly by existent after existent, and never hit some sort of wall, barrier, or edge of universe. (If there is no edge, then one cannot reach it.) Just like with time, this isn't because the universe is "infinite in size," and it is not because "circular space" (anymore than "circular time") is needed or appropriate. It is because existence isn't "in size" at all.
Just as Dr. Binswanger pointed out that "time is in the universe, the universe is not in time": sizes are in the universe, but the universe is not itself "in size." [2]
However, it is important to remember that - while the universe is not (finite) in time or (finite) in size - it is certainly finite. The reason for this is that "finite" qua adjective is not very descriptive. To say that something is finite is merely to say that it is, i.e., that it possesses a specific identity. The universe is, therefore it is finite. If everything that exists must be finite, then everything that exists (i.e., existence) must be finite. Existence exists – finitely.
Or, as Ayn Rand puts this point:
[D]o you know what we can ascribe to the universe as such, apart from scientific discovery? Only those fundamentals that we can grasp about existence. Not in the sense of switching contexts and ascribing particular characteristics to the universe, but we can say: since everything possesses identity, the universe possesses identity. Since everything is finite, the universe is finite. But we can't ascribe space or time or a lot of other things to the universe as a whole. [3]
Hence, there is nothing in the Law of Identity that mandates every existent possess a (finite) size, anymore than the Law of Identity mandates every existent be (finite) in time. The concepts of size and time apply to certain existents, which have specific natures that allow for such applicability.
But, the very nature of the universe precludes it from having a size, i.e., a three-dimensional extension. Since to be finite is to possess a specific identity, the identity of any extension must be the actual extension. (Existence is identity.) A specific extension is a limited extension that is bound by its nature. Which means: for any extension to exist, i.e., for it to be finite, it must possess bounds.
But, unlike the Milky Way, the universe has no bounds, whether temporal or spatial. Which means: unlike the Milky Way, the universe as a whole has no extension, whether temporal or spatial.
Just as one can integrate the inapplicability of a temporal extension and finiteness, one can do the same with the inapplicability of a three-dimensional extension (i.e. size) and finiteness as well.
Objectivism accepts the former. It is just as logical to accept the latter.
Thus, the one-line answer to the unbounded, finite universe is as follows: in the same exact way that the universe is eternal, it is also "asizal."
It is undeniably true that there cannot be an infinite number of entities in the universe. But, how can that be integrated with the above?
I submit that the answer lies in the same way the universe not having an infinite size (or age) is integrated; i.e., by recognizing that, just as with the concepts of size and time, the alternative to a finite number is not an infinite number, but the concept of number as not applicable to describe the totality of that which exists.
In order to illustrate my point, let's suppose that someone asked an Objectivist the following question: have a finite number of events transpired throughout the entire history of the universe? (I use "event" to refer to any causal sequence, i.e., any instance of motion.) What would an Objectivist properly say to such an inquiry?
There is no reason to believe that all action began at a certain point in the past, and that "before then" everything was motionless. And, since the universe has always existed, the Law of Identity certainly does not mandate an answer of "Yes" to the question above.
But then, is the alternative that an infinite number of events have ever transpired? Is that an Objectivist's only option? No. In short, the above question steals the concept of number from the concept of quantity.
When Dr. Binswanger was asked a similar type of question - wouldn't the universe have to be infinite in time? - part of his response included the following:
An infinite amount of time has passed between now...and when?...To talk about the amount of time that has passed you have to say 'between now, and some other point.' Any point you take (and the same applies into the future)...there's only a finite amount of time between those two points. [4]
This is a very astute observation. When someone asks if a finite number of events have transpired throughout the entire history of the universe (to go back to my hypothetical question above), he is not providing those "two points" that serve as bounds of the alleged duration. As Dr. Binswanger said: "To talk about the amount of time that has passed you have to say 'between now, and some other point.'" Every duration must have a beginning and an end; if it does not, then one cannot properly call it a "duration," and certainly cannot talk about any number of things within the duration. If there are no boundaries, then there is no "within the duration."
As Dr. Binswanger also discussed in the same lecture, a number stands for an amount, or a quantity. (I, like Dr. Binswanger, will take the latter two terms to be synonymous.) "Amount" and "quantity" are metaphysical concepts, and require ostensive definitions; "number," by contrast, is an epistemological concept, and thus is hierarchically dependent on the existence of a quantity. Put negatively: if one does not have a quantity, then one cannot have a number.
So, when someone asks "Have a finite number of events transpired throughout the entire history of the universe?" - since they are not providing two end points of when all these events took place - are not talking about an amount of events, and hence the concept of number is meaningless. "Throughout the entire history of the universe" has no temporal referent, because the universe is not "in time." "Entire history" presupposes that the universe as a whole has a history, i.e., a (finite) duration. But, it doesn't. So, one can't ask how many events have transpired throughout the entire duration of the universe unless there first is a duration of the universe.
But, once again, the alternative to this is not an infinite number of events, amount of time, or duration; the answer is to point out that "time," and that therefore any number referring to or requiring time is meaningless in such a context.
This is my essential argument for why to posit a finite number of entities "in the universe" is contradictory.
Are all the entities in the universe a quantity? This is the criterion that would have to be filled for one to validly state that there are a finite number of entities in the universe. But, what meaning does a quantity have, if one does not provide any boundaries? After all, the universe must be unbounded.
All quantities must be able to be described by a (finite) number, because all quantities are finite in scope (i.e., extension), whether temporal or spatial. But, the universe as a whole does not encompass a (finite) spatial scope. Does this alleged quantity span out 15 billion light-years? 100 octillion light-years? This amount squared? There can be no answer to this question, because the universe has no bounds, and therefore no extension. To extend is to extend finitely, i.e., in a bounded fashion.
Just as in order to talk about an amount of time one has to say, "between now, and some other point," so with an amount of entities one has to say, "between here, and some other point." If one doesn't specify, (i.e., quantify) what one is talking about, "amount," and therefore "number" has no meaning.
So, literally speaking, there is no "in the universe" in this context, because it smuggles in "within the spatial boundaries of the universe," just as "in the entire history of the universe" smuggles in "within the temporal boundaries of the universe."
This is why the universe cannot have a finite number of entities. [5] There are no boundaries to speak of to bound this all-encompassing quantity; hence, it is not a quantity, so "number" is therefore inapplicable to it. But, at this point, there should be no temptation to jump over to the demon of infinity, which in this context means: infinite number.
The dichotomy is not: finite or infinite. But, rather: existence or nonexistence. If it has been shown that the concept of number is inapplicable to describe all the entities "in the universe," then one has blanked out the possibility of any number (including an infinite number, which is a contradiction of the Law of Identity anyway). Either a number exists in this context, or it does not. It is my belief that it does not.
Why does this not reify infinity, you ask? Because it is not what infinity qua violation of identity refers to. The concept of infinity is metaphysically invalid because it attempts to describe an existent (e.g., an attribute) as existing, but as nothing in particular. But, since to be is to be something, infinity is at war with the Law of Identity and reality, and is therefore metaphysically impossible.
However, to declare existence as not possessing spatial boundaries is not to reify infinity. It is not suggesting that a size, extension or number of entities does in fact exist; to the contrary, it is suggesting that it does not. Such a claim upholds the fact that the universe possesses no size, which is fundamentally different than an infinite size – just as the universe not being in time (i.e., being eternal) is fundamentally different than the universe being infinite in time.
Thus, the two questions, "Have a finite number of events transpired throughout the entire history of the universe?" and "Are there a finite number of entities in the universe?" are essentially the same: they both steal the concept of number from the concept of quantity.
Hence, there is still no infinite number of anything. There is only: here we have boundaries, and therefore have a (finite) quantity and a (finite) number; and: here we don't have boundaries, and therefore have no quantity and no number.
The universe doesn't have an infinite number of entities, anymore than it has had an infinite number of events. Once again, the universe is both eternal and "asizal."
[1] Dr. Harry Binswanger mentions in the first question period of his lecture Selected Topics in the Philosophy of Science that Miss Rand had a theory that she called "circular time," which Dr. Binswanger said he unfortunately did not learn very much about. Therefore, my above reference to circular time should not be misconstrued as a reference to Miss Rand's theory of circular time. What I am referring to are theories that speak of "time" as applicable to existence, and that the same events deterministically cycle over and over again within it, like a cosmic movie stuck on "repeat." [Back]
[4] Selected Topics in the Philosophy of Science, first question period. [Back]
[5] It is interesting to note that this point reveals the opposite of what Dr. Binswanger was presenting in his aforementioned lecture: that in reality one never can "run out" of entities to be stood for by a number. As such, this rebuts Dr. Binswanger's metaphysical argument against the mathematical concept of infinity. Whether his psycho-epistemological argument still holds or not is unknown to the author. [Back]