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.]
I usually think of the first cause argument for a god to be like taking the worst from two worlds. It lacks the parsimony of an uncaused universe, and it doesn’t consistently apply the assumption that everything has a cause as an infinite regress would.
How does something that explains nothing, uncaused universe, a simple explanation? It appears to be no explanation at all, and would seem to violate Occam’s Razor. Worst part is parsimony says nothing about reality or truth, it only talks about what you value. What you value appears to have nothing to do with reality but what you seek of reality or what you seek for an explanation.
Several comments.
The sqrt(2) proof works because you can prove that for any finite number k, you can prove that a and b are greater than k. This contradicts the assumption that a and b are finite numbers. I think this part of the proof requires that natural numbers are ordered, but not necessarily well-ordered.
No such proof works with time, because even in an infinite timeline, there exist no infinitely long intervals; infinity is not a member of the set of real numbers. There just isn’t any paradox to speak of.
It’s also worth noting that there are plenty of religions out there that believe in an infinite timeline (Jainism comes to mind).
As for the “What caused God?” refutation to the cosmological argument, I feel it is flawed because it is quite easy to do some special pleading to prevent any paradoxes. Special pleading is indicative of bias, but it is not fallacious in itself. God, after all, is super-special (for any finitely special object n, God is more special than n
)
However, if you are using the “What caused God?” question to attack the assumption that finite regress is impossible, then I approve!
Perhaps I was misinterpreting your disagreement with Dawkins’ argument. You’re right that special pleading in itself isn’t necessarily a fallacy–you could have started with “Almost everything has a cause” instead of “Everything has a cause”, and then claim without contradiction that God can be the first cause. The problem then becomes, if something doesn’t have a cause, why can’t that something be the universe itself?
The sqrt(2) proof can be formulated in many different ways based on choice of words. You can use the well-ordering of the natural numbers to show that every fraction can be reduced to simplest form. If your ordering is not a well-ordering, I’m pretty sure you could show that “simplest form” is not well-defined.
I had to really contort the proof to get an actual mention of infinite regress in there, though–usually I’d just say from the beginning that a and b have no common factors (so a/b is in simplest form) and then I wouldn’t explicitly mention infinity at all.
My sources tell me that the sqrt(2) proof is what we call a proof by infinite descent. I think I understand why it requires well-ordering now. For every solution, there must exist a still smaller solution, which is impossible, because a well-ordered set must have a least element. Come to think of it, you already said that. How did I miss it?
I had several disagreements with Dawkins’ 747 argument. For one thing, he has all this evo/creo baggage, like the idea of “complexity”. And though you say he was just using this as a reductio ad absurdum, he acts like it’s his primary positive argument, acts like it applies to all theists, not just the ID/creationists. Additionally, it echoes the “What caused God?” question and the Occam’s Razor argument, both of which I feel are poor arguments.
But that’s all past now–I’ve already wrote on my gripes about the philosophical details, and now I can forget about them.
Your caller, like many other people, believes (implicitly or explicitly) that time is like a filmstrip passing through a projector: each frame traveling from future to past, momentarily becoming the present as it’s illuminated by the projector’s light. This is a very common view of time, and if that’s how the universe works, then the idea of an infinite regress does seem problematic: the moving light of the present would have to start somewhere, at “infinity past”, and traverse an infinite number of frames to reach us.
But that view is false. There is no moving light of time. Like you said, Susan, there are just the moments, each one existing eternally like integers on a number line. There’s no problem with the past being infinite, because nothing has to “traverse” that infinite past to reach us in the present, any more than you have to traverse the infinite totality of negative numbers to reach 0.
I don’t think that any one of these metaphors is an accurate depiction of what time is; they all have their advantages when describing various aspects of time, but no one of them is perfect. They’re all ways of capturing an intuition we have about time, and in this case the “infinite timeline” intuition is better because it resolves the problem of having to start. There may be other problems for which the filmstrip metaphor is the better one, but the point is that if your intuition for time indicates a problem, it’s a sign that you need to change your intuition, not insist that everybody else is wrong.
Miller, in my original comments to your post, I was defending Dawkins’ argument as a correct rebuttal to the first cause argument. But later I saw that he was in fact attempting to make a positive argument for the nonexistence of God. You’re absolutely right that as a positive argument it fails in a number of ways.
Susan, you’re right. None of these metaphors is accurate. The closest to describing the flow of time (and it does flow, for any given observer) is Zeno’s paradoxes that you mentioned earlier.
The universe did have a beginning, but there are no “moments” in the sense of integers. The argument ebonmuse links to falls all over the mathematics and the physics in its zeal to deal the death-blow to theism.
Instead, there are paths consisting of a continuum of spacetime points. That is, an uncountably infinite number. And the worldline unquestionably does traverse them. Removing the motion as ebonmuse and the linked argument try to do just buys into Zeno’s paradox.
The answer is not to deny motion, but to accept that the naïve sense of continua as analogous to sequences is false. Just as it was the answer to the paradox.
an infinite series is a complete series, and is, by definition bound by no limit, meaning that you cannot add 1 to the series of time.
so we call such a series a complete series, as opposed to a series that is progressive, that gradually increases, or that perpetuates a moment longer than before, making the series greater than it was before.
time is a collection of events in sequence, whereby one “unit” of time is constantly being added to the next.
the contradiction in your assertion that time is infinite is this:
time is infinite, yet is progressive.
in other words,
time is a complete series, yet is a not complete
how can you call time infinite when it is always becoming longer than it was before? you are saying that time is complete, yet is in progress. this is the contradiction.
time cannot possibly be thought of as an infinite series. the concept of a past, in which things have “passed”, and a future, in which things “have yet to pass”, is a clearly limited system. if you dont see this, then just try and concentrate and maybe youll see.
Agree with Didactic. Also, “But nobody said that…moments in time, were well-ordered.” Actually, one of the most fundamental “laws” of physics says otherwise: entropy always increases over time; therefore, time is in effect well ordered.
If an unending series of events (my predecessors for example ) had to have existed beforehand, this piece could not have been written nor read because neither the writer nor the reader would yet exist.
Time as a concept is unfathomable, as the other (dare I say earlier?) responses show. But no one has any trouble dealing with it in the particular.
HH
Time must move in both directions as its own causality : if time is moving forward it is undoubtedly causing the past as it’s exhaust . when causing something we are in fact moving toward it. the alpha and omega
oh yea, if it’s forward motion is causing the past and the past is a precursor to the future…well you get the example
I fail to see how a infinite regression of gods is illogical if you believe in infinite regression. I have noticed that having to have a ‘beginning’ or a first cause is primarily a western thing. I do believe in God, but I think most people make the mistake of defining God in terms of categories that are passed down through western philosophy. God has to be ontologically unique. God is impassable. God is not contingent, etc. I think that Whitehead was right in he attempted to move the discussion away from such categories and to seek to understand reality as a process, including his understanding of God. If you read the 45th psalm it states that God was anointed God by his God. Most people take a firm stance of the Bible teaching Monotheism. Nothing could be further from the truth. In fact I would say that isn’t a single verse in the entire book that teaches monotheism if you read the verses within its historical context. I believe that these writers used God as an exemption because they were trying to protect monotheism. Though it is a notion that I don’t believe is supported by the Bible.
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