General Relativity Proves What, You Say?: A Response to L. A. Mitchell

By Miles Donahue

I. Introduction

Mr. L. A. Mitchell has recently argued [1] that Einstein’s theory of General Relativity proves that the universe has no external cause, and by extension, that God does not exist. In my response, I will first define key terms used in the argument, boil the argument down to its essentials and address what Mitchell’s argument actually proves, and finally, provide a substantial criticism of that argument. I will not try to show that God exists or even formulate an argument for the existence of God. Because Mitchell’s contention is that General Relativity disproves the existence of God, he has the burden of proof to maintain that contention; I have no contentions to defend.

II. Clarifications of Terminology

In section II of his article, Mitchell provides definitions for some key terms used in his argument. But it seems to me that a reader new to the subject might not understand what he’s talking about, so I’d like to provide some further clarifications. First, a closed state of time is an interval of time (that is, a non-zero, finite period of time) which includes its endpoints (e.g., the interval of time from and including 12:00pm to and including 1:00pm). Algebraically, a closed state of time looks like this: 12:00pm ≤ t ≤1:00pm.  Second, a half-open state of time is an interval of time which includes one of its endpoints but not the other. For example, all times later than 12:00pm, up to and including 1:00pm, is a half-open interval of time because it includes one of its endpoints (1:00pm) but not that other (12:00pm). We could explicate a half-open state of time algebraically like this: 12:00pm < t ≤ 1:00pm. Notice that while t includes 1:00pm, it does not include 12:00pm. Third, a first state of time is simply the very first interval of time, whether a closed interval or a half-open interval. This point needs to be underscored: a first state/interval of time doesn’t need to be a closed state of time. We can still talk about the first second of time, even if that second is open in the earlier than direction (that is, there is no first instant bounding that second). So I simply disagree with Mitchell’s definition of “first state of time.” Fourth, an instant is a point in time of zero duration.

I think it’s clearer if we talk about half-open or closed intervals of time, rather than states of time. We can talk about states of the universe (that is, a particular configuration of the universe at some point in time), but the term “state of time” doesn’t have much meaning, because a state of something is a configuration of that thing at a particular time, and you can’t have a configuration of time itself at some time! So let’s stick with “interval” rather than “state” when talking about time.

III. Trimming the Entailment Argument

First, I think we can trim the entailment argument down a bit. Mitchell provides that following formulation of the entailment argument:

1. Every state of the universe is sufficiently caused by a previous state.
2. If every state of the universe is sufficiently caused by a previous state, then every state of the universe has a causal explanation with reference to another state.
3. If every state of the universe has a causal explanation with reference to another state, then there is no state of the universe that was not caused by a previous state of the universe.
4. An external cause of the universe can only exist if there is a state of the universe that was not caused by a previous state of the universe.
5. Therefore, the universe has no external cause (by 3 and 4).
6. Therefore, God does not exist.

The first thing that needs to be pointed out is that the argument is overly complicated. That is, it can still reach (5) with less premises. Premises (2) and (3) superfluous because (5) follows from (1) and (4) alone, and thus (2) and (3) really don’t need to be included in the argument. The second thing that should be said is that (6) doesn’t follow from (1) – (5) by any rule of logical inference. Rather, you need an additional premise to show that God does not exist. Thus, I propose the following reworking of the entailment argument:

1. Every state of the universe is sufficiently caused by a previous state. 

2’. An external cause of the universe can only exist if there is a state of the universe that was not sufficiently caused by a previous state of the universe. 

3’. God, if He exists, is the external cause of the universe. 

4’. Therefore, the universe has no external cause (from 1 and 2’) 

5’. Therefore, God does not exist (from 3’ and 4’).

Notice that this form of the argument is logically valid, by the rule of logical inference known as modus ponens. Because an external cause of the universe requires there to be a state of the universe which is not sufficiently caused by a previous state, and every state of the universe is sufficiently caused by a previous state, there cannot be an external cause of the universe. Because God by definition is the external cause of the universe, He cannot exist. This reformulation makes the logical structure of the argument much clearer.

IV. What the Entailment Argument Actually Proves

I’d like to address the significance of the entailment argument. The theist could agree with the conditional statement, “if General Relativity is true, then there cannot be an external cause of the universe”, but deny that the “if-clause” is true. That is, they would argue that General Relativity does not actually tell us about the nature of time. This is because General Relativity is an incomplete theory of the world; it needs to be married to quantum mechanics, and no one’s sure how that’s going to be done. It might be the case that once we have a quantum theory of gravity, a first closed temporal interval of time will reappear, and by extension a first state of the universe, and premise (1) will be shown to be false. Indeed, this seems to be the case! William Lane Craig outlines this point as follows:

“A typical approach to marrying quantum theory to General Relativity involves describing the evolution of spacetime as a path integral (a sum over all possible paths) in superspace, which is a space of points respresenting three-dimensional configurations of the universe. The points of this configuration space can be regarded as instantaneous states or even as instants, but the fact that in the quantum theory one has a path integral, rather than a single path, makes it impossible to “stack” these instants into a unique history constituting a spacetime. So eventually the dividing process [of states of the universe caused by previous states]…becomes ill-defined in the theory. The fact that the first split-second of the universe’s existence as measured in cosmic time is not resolvable into a unique sequence of ever briefer states is not inconsistent with there being a first second of its existence.” [2]

So on certain theories of quantum gravity, you don’t have a nice splicing of the universe into every briefer intervals of time that converge towards T=0 as a limit; there aren’t states of the universe which are all causally explained as you go backward in time. At the very least, we just don’t have enough information to conclude that premise (1) is true. But let this second point pass. Is the argument, on its own terms, sound? I think not.

V. Criticisms of the Entailment Argument

Let’s turn to premise (1) – every state of the universe is sufficiently caused by a previous state.

First, the careful reader will notice that all of section IV of Mitchell’s article argues that there cannot be a first closed interval of time. But as I already pointed out, there’s no contradiction in asserting that the first interval of time is half-open; indeed, that seems to be what General Relativity implies! The first interval of time can then be expressed as follows: 0 < t ≤ 1, and you can adopt any unit of measure you wish, whether seconds, minutes, hours, etc. Of course, there is no first state of time in the sense of a first instant of time, but that’s not how Mitchell or I define “first state of time.”

Second, we’ve been given no reason to accept premise (1). I can’t garner a single line of evidence for (1) from Mitchell’s paper. All he’s really argued for is that there can’t be a first closed state of time. He hasn’t shown us how this fact implies the truth of (1). Therefore, he’s failed to shoulder his burden of proof. I could stop here and be completely justified, but I think we can be more charitable and reconstruct the argument Mitchell means to give. The idea here is that if the first interval of the history of the universe is half-open, then there are an infinite number of states of the universe prior to any state one picks. The state of the universe at T=1 is proceeded and caused by the state of the universe at T=0.5, which is caused by the state of the universe at T=0.25, which is caused by the state of the universe at T=0.125, ad infinitum. So if the first interval of time in the history of the universe is half-open in the earlier-than direction, then every state of the universe is sufficiently caused by a previous state. Thus, there is no first state of the universe left unexplained. But as I said, this reasoning was left out of Mitchell’s paper.

Let’s now turn to premise (2’) – an external cause of the universe can only exist if there is a state of the universe that was not caused by a previous state of the universe.

Here it’s even more obvious that Mitchell has failed to shoulder his burden of proof. With regard to premise (2’), all Mitchell has to say is, “The first premise is warranted, and I believe that the other premises logically follow from its truth.” Well, I’m glad that Mitchell has such confidence, but no theist will simply trust him when he says that (2’) is true. He needs to provide some argument for its truth; it can’t simply be asserted or assumed. But again, I will be charitable and try to reconstruct the argument for (2’). The idea seems to be that in order for something to have an external cause, there must be a first instant of its existence; it must have a beginning point. In other words, the first temporal interval of a caused thing’s history cannot be half-open in the earlier-than direction. As we’ve seen, if the first temporal interval of the universe of the universe is half-open, then every state of the universe is caused by a previous state, so where, so to speak, does God cause the universe to come into being? Which state of the universe does God cause? We can agree that the universe began to exist, but in a very real sense it would appear to be self-caused. Let me give two responses to this argument:

First, there are parallel cases of things coming into being in half-open intervals of time, and yet having external causes. Thus, we can give a reduction ad absurdum of the argument. Consider first Mitchell’s lifespan. Let’s say he began to exist at T=5 and has lived ‘till T=10. It’s clear that his lifespan forms a closed interval; in other words, T=5 is the instant at which he began to exist. [3] But now consider Mitchell’s truncated life; let us delete the instant T=5 from the temporal series. That is, consider the half-open interval of time 5 < t ≤ 10. Because this interval is half-open, every instantaneous state of Mitchell’s life is caused by a previous state. The state of Mitchell’s life at T=10 is caused by the state at T=7.5, which is caused by the state at T=6.25, ad infinitum. Thus, there is no state of Mitchell’s life in this interval lacking an explanation in terms of a previous state, and there is no first state of Mitchell’s life. And yet, Mitchell’s life during this interval was caused by his life in a previous interval.

Consider next the universe in the interval of time between T=10 and T=20. Again, we shall delete the first instant from the interval [4], forming the half-open interval of time 10 < t ≤ 20. The reasoning will be the same as the above example. Every instantaneous state of the universe has an explanation in terms of a previous state, there is no first state of the universe in this interval, and thus there cannot be an external cause of the universe in this period of time. And yet we know there is, namely the universe as it was in a previous interval of time.

Consider finally the motion of a tennis ball. Let’s say a tennis ball is in motion from 5:00 to 5:01. Next, let’s stipulate that 5:00 is the last instant at which the ball is at rest. Of course, we could consider 5:00 to be the first instant at which the ball moves, but the first characterization is more plausible. As William Vallicella explains:

“For one may reckon the moment of the ball’s transition from rest to motion as the last moment of the ball’s being at rest, which then entails, given the continuity of time, that there is no first state of the ball’s being in motion. Arguably, we must view the matter in this way: otherwise the ball would suddenly be moving at some positive velocity without having accelerated to this velocity from velocity zero through continuum-many instants.” [5]

Now if 5:00 is the last instant at which the ball is at rest, it follows that there is no first instant at which the ball is in motion. This is because instants cannot be immediately next to each other, so to speak, because an instant is by definition a point in time of zero duration, and two zeros is still just zero. But if there is no first instant at which the ball is in motion, then the interval of time at which the ball moves (5:00 to 5:01) is half-open. It then follows, as I’ve explained above, that every state of the ball’s motion during this interval has an explanation in a previous state, and thus it cannot have an external cause. And yet, it does; a tennis player hitting the ball with his racket.

Again, William Vallicella explains:

“And yet there is an external cause of motion, [the person hitting the tennis ball]. Somehow the external cause has to ‘hook onto’ the [moving]-ball process. How? The last instantaneous state of the [hitting] cannot hook onto the first state of the [moving]-ball process, for there is no first state. This suggests that the last state of the [hitting] connects to an entire causal sequence, namely the sequence of [moving]-ball states. Thus a whole sequence would somehow be the effect of a cause.” [6]

William Lane Craig provides a helpful evaluation:

“One thinks in this connection of the ancient sorites-style problems of starting and stopping…If there is a last instant at which some object is at rest, then when does it begin to move? The answer can only be that there is no first instant of its motion. Nonetheless, the object does begin to move and plausibly requires a cause to set it in motion. Similarly, if something begins to exist, it is plausible that it requires a cause to bring it into being whether or not there is a first instant at which it exists.” [7]

Notice that the above three arguments and the entailment argument are parallel, so that if one of them is valid, the other three are valid. But here’s the rub: if one is invalid, the other three are invalid as well. Clearly, the “Mitchell’s Life Entailment Argument”, “The Truncated Universe Entailment Argument”, and “The Tennis Ball Entailment Argument” are all invalid, and thus the original entailment argument is invalid as well. Vallicella, discussing a parody argument very similar to the “Mitchell’s Life Entailment Argument” (except this time concerning Quentin Smith), says this:

“…if the original argument is valid, the parody argument is also valid: after all, the two arguments have the same form, and validity is a matter of form. And if the original argument is sound, then so is the parody. The only difference between the two is in the first premise of each. But it seems that the initial premises are either both true, or else both false. If it is true that the universe is a continuum of successive, instantaneous states, then it is also true that Smith’s life is a continuum of such states. But it is as obvious as anything that the parody argument is unsound, issuing as it does in a false conclusion: we know that the beginning of Smith’s life has an external cause in the conjugal activities of his parents, and since his life’s beginning has an external cause, it cannot have an internal cause or causes on pain of causal over determination. I conclude that the original argument is also unsound, which is to say that it is either invalid in point of logical form, or possesses one or more false premises, or both.” [8]

Second, we can show that the universe not only could have an external cause if its earliest interval is half-open, but needs an external cause [9], because the question, “Why did the universe begin to exist?” is still left unanswered, even if the universe’s earliest interval is half-open. Pointing out that each state of the universe is caused by a previous state might show give an explanation as to why the universe exists, but not as to why the universe began to exist. Vallicella concludes by saying:

“Hence each state’s being caused by earlier states cannot be invoked to explain why the universe began to exist. The question, Why did the universe begin to exist? Cannot be answered by saying that each state of the universe is caused by preceding states; for this could be true even if the universe always existed.” [10]

If the universe were eternal, then every state of the universe would have an explanation in terms of a previous state as well, so merely showing that every state of the universe has an explanation in terms of a previous state wouldn’t go one inch in explaining why the universe began to exist, instead of existing eternally. Something outside or external to the universe is needed to answer that question.

VI. Conclusion

To conclude, I’ve shown that Mitchell’s argument is overly complicated, and given a logically valid form of the entailment argument. I’ve shown that each premise of that argument is problematic. With regard to (1), we’ve seen that Mitchell has given no argument for its truth, and that on certain theories of quantum gravity, the beginning of the universe cannot by “spliced up” into an infinite number of states converging towards T=0. Concerning (2’), we’ve seen three parallel cases of things coming into being lacking a first instant at which they came into being, and yet having external cause. I’ve also shown that the question “Why did the universe begin to exist?” cannot be answered by giving reference to the various states of the universe. Thus, Mitchell’s entailment argument does not show that the universe has no external cause or that God does not exist.

Notes

1. Mitchell, L. A., Doubting Dave. 12 September 2013. Web. 28 September 2013. (http://doubtingdave.com/general-relativity-entails-that-the-universe-has-no-external-cause/).
2. Craig, William Lane, "J. Howard Sobel on the Kalam Cosmological Argument." Canadian Journal of Philosophy 36 (2006). Note 34.
3. Alternatively, we could say that T=5 is that last instant at which he does not exist, but a proponent of the entailment argument would not except this understanding, for, as shall be shown, it would completely undermine the entailment argument.
4. In fact, no deleting is actually required, for as Mitchell says, “every state of time on a FRW model is half-open.” So in the above example, we are talking about the way intervals of time actually are.
5. Vallicella, William. “Could the Universe Cause Itself to Exist?” Philosophy 75 (2000), 609.
6. Ibid.
7. Craig, William Lane, "J. Howard Sobel on the Kalam Cosmological Argument." Canadian Journal of Philosophy 36 (2006).
8. Vallicella, William. “Could the Universe Cause Itself to Exist?” Philosophy 75 (2000), 607.
9. Operating under the assumption that whatever begins to exist has a cause, of course.
10. Ibid, 608.