On the most recent episode of The Atheist Experience Podcast, which I was listening to earlier, one caller asked about the hosts’ feelings on infinite regress. It was a pretty open ended question, but it’s a subject that comes up in a variety of different contexts when we’re arguing about religion. I’ve been thinking about this for a while, because I’ve been listening to old episodes of the show recently and plenty of callers on previous episodes have asked questions or made arguments that boiled down to a problem of infinite regress. I have a few comments and observations of my own.
In religious arguments, infinite regress most often occurs in the First Cause argument, of which I’ll state a somewhat sloppy version in my own words:
Everything that exists is caused by something that came before it. The universe exists; therefore it must have been caused by something. But that something must have been caused by something else, and that must have been caused by something else, and so on. We have an infinite regress, which is impossible, so therefore there must have been some first cause that wasn’t caused by anything else, and this first cause is God.
This argument is not hard to refute by pointing out that if you accept that everything must have a cause, then something had to cause God, and something had to cause that, and so on. There’s no reason why God gets to be exempt from the “everything has a cause” claim, and to suddenly introduce a “first cause” that doesn’t have a cause of its own is to directly contradict the initial claim. There are a number of other issues with the first cause argument, most importantly the fact that there is no reason to think the first cause, if there must be one, is anything like the God most people believe in. But I’m primarily interested in the specfic issue of infinite regress, and the somewhat subtle flaw with using it in this way.
First, you have to be careful exactly how you word your counterargument. Simply demonstrating the failure of the argument to prove the existence of God is not the same as making an argument that disproves God (or at least casts doubt on his existence). To disprove God takes quite a bit more work. Richard Dawkins attempts to adapt the refutation of the first cause argument for this stronger purpose in The God Delusion, and Miller calls him on it here. In the comments there I originally attempted to defend Dawkins’ version of what he calls the Ultimate 747 argument, which is close to what I said above: If everything must be created by something more complex than it, then the universe must have been created by something more complex, namely God. But then God must have been created by a god even more complex, and so on, so we have an infinite hierarchy of gods, which is clearly absurd.
Now, if Dawkins were merely trying to show the flaw in the first cause argument, there is no fallacy. It’s a simple (appropriate) use of reductio ad absurdum: Starting with the assumptions that all things must have a cause and an infinite hierarchy of causes is impossible, we have demonstrated a contradiction, and so one of the initial assumptions is false and the original argument is unsound. But the way Dawkins states his Ultimate 747 argument is somewhat ambiguous, and I agree with Miller that it sounds as though he is making the stronger claim that God is unlikely or impossible as a result of his argument. Unfortunately, one argument that hinges on an infinite regress is as bad as another–you can no more use infinite regress to show that God doesn’t exist than that it does, and now I’ll get to the precise reason it doesn’t work.
Both of these arguments (first cause and Ultimate 747) take for granted that an infinite regress is an absurdity. Even if we ignore the idea of God, it’s still a hard thing to accept. I remember a previous episode of The Atheist Experience in which a caller attempted to set up a hypothetical scenario in which there was no Big Bang (before which there may have been no time, and cause-and-effect arguments get awkward) and the universe had simply existed forever. In that case, he said, we would never be able to make it to the present because an infinite amount of time would have to pass before we could get to this point.
Most people do have a hard time wrapping their minds around the logical potholes that occur with infinity. Zeno’s paradoxes (you can’t walk from point A to point B because first you have to walk halfway there, and then you have to walk halfway from that point to the finish, and then you have to walk halfway from that point, and so on) all deal with infinite regresses, and it wasn’t until a few centuries ago that we finally developed the tools to address them when we formalized the idea of limits. Then there’s the surprising fact that there are exactly as many even numbers as integers, and even worse, as many rational numbers as integers (and once you’ve finally been convinced that all infinities are the same, along come the reals which are provably larger than the set of integers, but let’s not get started on that). The point is, infinity is confusing, and leads to a lot of things that look like paradoxes.
Emphasis on the words look like. None of these are actually paradoxical, in that once we sit down and think about them formally, we don’t reach any logical contradictions. The point I want to make is that, although it makes our brains hurt, there is nothing inherently contradictory about an infinite regress. Nobody has shown that, if an infinite heirarchy of causes occurred, then both A and not-A would be true. We haven’t demonstrated that, with an infinite regress, some basic piece of knowledge about the world would be contradicted. Perhaps someone will produce such a contradiction in the future, but until then we cannot make a sound argument by saying “…which leads to an infinite regress, and is therefore logically impossible”.
Now, there are arguments in which an infinite regress IS a true contradiction. Here’s an example:
Proposition: The square root of 2 is irrational.
Proof: Assume for the sake of contradiction that sqrt(2) is rational. Then it can be written in the form a/b, where a and b are positive integers. Since sqrt(2) = a/b, we can square both sides, to get 2 = a^2/b^2, and we can multiply by b^2 to get 2(b^2) = a^2. Therefore a^2 is even, and so a must be even, and we can write it as a = 2c. Substituting in, we have 2(b^2) = a^2 = (2c)^2 = 4(c^2), or 2(b^2) = 4(c^2), and dividing by 2 we get b^2 = 2(c^2). Therefore b^2 is even, which means b is even and can be written b = 2d. Thus sqrt(2) = a/b = 2c/2d = c/d. But we can do the same thing with c and d, and so on. We have an infinite regress, with the numerator and denominator getting smaller and smaller, which contradicts the fact that every fraction can be written in simplest terms. Therefore there is no fraction that can express sqrt(2).
The reason the use of infinite regress to produce a contradiction is valid in this case hinges on the fact that every fraction can be simplified, which in turn relies on the fact that the natural numbers are well-ordered; that is, every subset of the natural numbers has a least element. This well-ordering principle is extremely useful for proving all sorts of things, but the important thing I want to note is that the well-ordering principle, by its very nature, effectively says that no infinite regresses are allowed. You can count up as high as you want, but you can’t count down forever; you’ll eventually reach a first number. In effect, these arguments work because one of their implied starting assumptions is precisely that an infinite regress is an absurdity.
But nobody said that causes, or moments in time, were well-ordered. In fact (quantum theory aside), they’re very distinctly NOT well-ordered under most circumstances, and so we can’t produce a sound argument by contradiction simply by pointing out an infinite regression. Infinity may be confusing to think about, but that’s not enough to prove (or disprove) God.
[Incidentally, my preferred response to the caller who claimed that if the universe had existed forever we could never reach the present day would be to say, You've got the wrong intuition here. You're thinking of time as something that passes, that it had to start somewhere, and so you're begging the question. A better intuition is to think of the present day as a point on a continuous timeline stretching out to infinity in both directions. After all, we have no trouble visualizing an infinite plane or number line in space, at least in principle. There is no inherent contradiction in the idea of a timeline stretching out to infinity in the past, unless you assume the thing you're trying to prove.]