Chapter 1: Spacetime
Whether or not the tension between Euclidean geometry and infinite geometry is real, as opposed to entirely conceptual, we can, nonetheless, speculate about what might happen if this tension tried to work itself out. In general, any two discretely defined points within an infinite space tend toward the same point. That is, however distant from one another two points are when viewed from a finite perspective, they are the same point when viewed from an infinite perspective. That disparity is the essence of the geometric tension between them. Even so, it is certainly beyond debate that this tension, the tendency of points to merge, is merely a figure of speech. The void, after all, is absolute nothingness. And, in any case, points have no physical extent. They are nothing but mathematical abstractions, infinitesimals. It is meaningless to ascribe to them any characteristics whatsoever, particularly anything as definite as a tendency to merge with other points. Or is it?
One outstanding question from cosmology concerns the ultimate fate of our universe. Right now it is expanding, and there is some doubt about whether it will continue to do so, or will instead reverse course one day and begin contracting. I will address this question in Chapter Four. For now, we can treat it as simply a thought experiment. In particular, what will happen if the cosmos goes on expanding forever? The void provides an infinite degree of spatio-temporal freedom to anything that exists. Therefore, if the momentum of expansion exceeds any force of contraction, there will be, literally, nothing out there to get in its way. So where does it go?
Mathematically, if we divide any quantity, however large, by infinity, we get zero, expressed by the equation,
x/∞ = 0.
Put simply, if we distribute any finite amount of stuff over an infinite expanse (fig. 3) it will eventually cease to exist altogether; becoming infinitely diffuse is identical to disappearing. If our cosmos does not reverse course, it has no other choice but to succumb to this strange equation. But because it is expanding at a finite rate, it will require an eternity to undergo this transformation. Now, as I discussed earlier, infinite time (eternity) is, like infinite space, an infinite degree of freedom. Eternity says, “Take all the time you need,” not “This is never going to end.” This infinite degree of temporal freedom offers no resistance to any process that occurs within it, but it is not something over and above that process. Time does not flow; it is not a force that acts on things as if from outside. Physical phenomena tend to evolve in a specific way, from more to less orderly (increasing entropy), but that fact reflects only the phenomena themselves, not the temporal degree of freedom that permits them to occur. If no phenomena are occurring, time, like space, appears as nothingness, with no beginning and no end. However, as merely a facet of nothingness, we are under no obligation to explain how it has no beginning. It is not as though time, qua nothingness, has always been flowing at some finite rate and couldn’t possibly have gotten here had it not started at some particular time.
Once our cosmos has “taken all the time it needs” in order to blink out of existence according to x/∞ = 0, we are confronted with the same sort of paradox, the same sort of tension, between Euclidean and infinite geometry that I introduced above. In particular, it now makes sense to solve the equation for x, revealing that any given infinite expanse of nothingness is equivalent to some certain quantity of something, given by:
x = 0 • ∞.
That is, if we gather up an infinite quantity of nothing, we don’t have nothing anymore, but instead we have some particular amount of something. I’ll grant you, it is much easier to swallow this idea when we imagine something (e.g., our cosmos) ceasing to exist after eternal expansion, than it is when we try to imagine something coming into existence after, presumably, an eternal collapse of nothingness itself. And yet, theoretically, there is no difference. All that distinguishes the two cases is the physical mechanism. We already know our cosmos is expanding, so it requires little to imagine it expanding forever. On the other hand, it borders on the absurd that an infinite expanse of infinitesimal points, nothingness itself, might somehow coalesce into our entire universe.publ...@gadflyllc.com.