August 15, 2010

Finitude of Space

Dear Pr. Craig, I am very found of and convinced by your argument against the possibility of an actually infinite number of past (or future) events. The conclusion that the universe must have begun sounds to me unescapable. But what about space? I guess the argument runs as follow : an actually infinite number of equal segments of space can't exist. Therefore space is limited (I underscore "actually" because of course space is potentially infinite, as it expends indefinitely). However, while the beginning of time is quite easy to admit, the limitation of space is hard to conceive (as well, indeed, as its infinity, like in Kant's first antinomy). . . unless we assume something like "finite-without-boundaries space" (a sort of sphere I suppose?). But, if we do so, I'm tempted to do the same for time: after all, why the time wouldn't be "finite-without-boundaries" like space? It looks like the Hawking's solution to the problem.

So my questions are :

1° how do you conceive the finitude of space? (what is like to be "on the edge of space", seated on the edge of the last galaxy?)

2° do you think the solution (if it is one) for space ("finite-without-boundaries") can be applied to time ?

I add to these questions a simple demand : please forgive my English !

In Christo,


- country not specified

Je suis si content de recevoir une question de quelqu’un en France! Votre anglais est sans doute meilleur que mon francais! C’est formidable, Frederic, que vous vous occuper de ces questions si importantes et interessantes! Que le Seigneur vous benisse dans vos efforts de rendre temoinage de Lui en France d’une maniere intelligente et compatissante! Alors, a vos questions. . . .

I’d say your thinking is basically on target with respect to space. Space is at every point in time finite but expanding and therefore potentially infinite with respect to the future. It will never become actually infinitely large but will forever expand toward infinity as a limit.

This answer implies that space has a closed geometry, like that of the surface of a sphere. Just as you never come to the edge of the surface of a sphere where space ends and you wonder what lies beyond, so one can never come to the edge of space. There is no “edge of space” where the last galaxy resides, anymore than there is on Earth an end of the world where the farthest island lies. Of course, we can no more visualize a curved three-dimensional space than the flatlanders on the surface of a sphere could visualize a curved two-dimensional space. But it is mathematically conceivable, and even testable, although not visualizable for creatures like us.

A closed geometry for space was once thought to be incompatible with space’s being potentially infinite to the future because a closed geometry would necessitate a density so high that the universe would eventually recollapse. But with the discovery of a positive cosmological constant, which causes the expansion of the universe to actually accelerate over time, even high density universes can go on forever.

Now whether you think that time can also have a closed geometry, that is, go in a circle, depends, I believe, on your theory of time. If you adopt what is called a tenseless view of time—I know this doesn’t translate into French, since “time” and “tense” are both translated by the same word “temps,” so that to speak of tenseless time becomes oxymoronic in French (an interesting illustration of how language can actually limit philosophizing!)—a view, that is, in which time is essentially spacelike and there is no objective “now,” then there’s no reason time can’t have the geometry of a circle.

But on a tensed view of time, that is to say, a view of time according to which temporal becoming is real and an objective “now” exists, it seems impossible for time to be circular. For if the same event were to recur, it would occur for a second time, and then a third time, etc., in which case time itself is linear, even if the cycle of events were repetitive. They wouldn’t be literally the same events but new, similar events. For time to be literally circular, you’d have to say that every event is both earlier and later than itself, which seems crazy on a tensed theory of time. On a view in which temporal becoming is objective, it seems metaphysically necessary that no event precedes itself.

The kalam cosmological argument presupposes a tensed theory of time and therefore the linearity of time. For a defense of such a view of time see my Time and Eternity.