Eternal Recurrence*

In an earlier post—

—we established that time is infinite in both past and future directions.

What are the implications for the overall story of the universe? One implication is, roughly, that everything that happens has happened before infinitely many times, and will happen again infinitely many times. Let’s see why.

1. Simple Philosopher’s Argument

The theory of eternal recurrence has probably been accepted by many people throughout history. In our tradition, it goes back at least to Seneca. Nietzsche later introduced it as a thought experiment in The Gay Science, where he suggests that you should live in such a way that you would be happy to have this life repeat infinitely many times. In later writings, it emerges that Nietzsche thought eternal recurrence was guaranteed by physics.

A simple argument: There are only so many states the world can be in. Given infinite time, the universe will eventually run out of new states to be in and thus will have to repeat one of the earlier states. Supposing determinism is true, then everything after that point will repeat exactly as it did the previous time that state occurred. So there must be some fixed (presumably extremely long) cycle that the universe repeats over and over again. Perhaps this is what Seneca and Nietzsche were thinking.

Problems:

a. The number of possible states of the universe is not limited. E.g., if you just take two particles, there are infinitely many (continuum many) possible distances between them. If you have continuum many possible states, it is possible to go through different states without repeating, forever.

b. Also, the universe may be infinitely large, containing infinitely many particles. This would again give you infinitely many possible states, even if each finite part had only finitely many states.

c. Determinism is probably false, so even if the universe repeats an earlier state, it need not then repeat the whole history following that state.

2. Revised Argument

a. Revised conclusion

First, the revised version of eternal recurrence:

Take the state of any finite part of the universe. There will in the future be states arbitrarily similar to that state. I think this is true.

Take the state the Earth is in now. Some time in the future, there will be a planet that looks very much like the Earth, with beings on it who resemble you and me, with a blog called “Fake Nous”, with the readers of that blog reading a post about eternal recurrence that says all the things this post says. This other earth will not be an exact duplicate of our Earth, mind you. It will merely be close to a duplicate. How close? Well, there will be planets as close as you like—if you specify some (non-exact) degree of similarity, there will be a planet at least that similar to Earth at some time in the future. It will of course take a ridiculously long time for these sorts of things to happen (you know, like a googolplex years).

What happens after that point need not repeat what will be happening now. I.e., suppose state S at time t approximately repeats as state S' at time t'. What happens after t' can diverge from what happens after t. The universe is chaotic, so small changes in initial conditions tend to lead to increasingly large changes in effects as time goes on.

b. Revised argument

Why think that’s true?

We can think of the state of a physical system as a point in a many-dimensional space of possibilities, where the dimensions correspond to each of the fundamental physical quantities that characterize the system’s state. Changes in the system can be thought of as the system’s moving through that space. As the system evolves over the infinite expanse of time, it wanders from one region to another. If the system has fixed total energy, then there is going to be a finite region of the state space to which it is confined.

Aside: Distinguish between the claim that there is a finite range of possibilities and the (false) claim that there is a finite number of possibilities. Illustration: between 2 and 5, there are infinitely many real numbers. However, the range from 2 to 5 is a finite range. Likewise, a physical system with fixed, finite energy has a finite range of possible configurations, even though there are infinitely many possible configurations.

For any sub-region of the state space, there’s going to be some probability of finding the system in that region, if you look at it at a random time. This corresponds to the percentage of the time the system spends in that region, over the long term.

That probability is either zero or nonzero. If it is zero, then you should not expect to ever see the system in that region; you should have credence zero that the system is in that region. If it is nonzero, then the system goes through that region infinitely many times, given infinite time.

So, if you rationally take a physical system to be in a certain region of the state space, then you should expect that the system will be in that region infinitely many times. That’s the eternal recurrence.

Digression about zero-probability events: Any perfectly precise specification of the state of a system is going to have probability zero, given that you have at least one continuous physical property. So any exact state of the system will never repeat. But if you consider some non-exact degree of similarity to a given state, that defines a region of the state space surrounding that particular state, where that region has a nonzero probability. So the system will be wandering through that region for some nonzero percentage of its history.

Q1: If any exact state of the system has probability zero, doesn’t that mean it never happens?

A1: No, but it means it happens 0% of the time, which is compatible with its happening once during infinite time. (Roughly, 1/infinity = 0.)

Q2: But doesn’t your argument imply that, for any exact physical state, you can never rationally believe that that state obtains, since it has probability 0?

A2: Yes. And that implication is correct. We can never identify the exact physical state of any physical system that has at least one continuous variable. No one (sensible) has ever claimed to do so, nor can you think of any way of doing so.

Go back to my description in sec. 2a of the future twin earth, where there is a person who looks like you reading a blog called “Fake Nous,” which describes the arguments for eternal recurrence. That’s a pretty specific qualitative description of a state of (part of) the world, but it isn’t an exact specification (it doesn’t entail the exact value of any continuous variable). So all of that describes a nonzero region of the space of possible configurations for an earth-sized quantity of matter. Given infinite time, and given a sufficient quantity of matter with sufficient energy, that bunch of matter will periodically wander through that region.

Notes:

• You don’t need to have the same bits of matter. If you have different collections of matter (with enough matter and enough energy present), you can apply the same probabilistic argument, as long as you have infinite time during which an appropriate amount of matter exists.

• You also don’t have to look at different times. If the universe is infinitely large (or if there is an infinite multiverse), you can apply the same reasoning to show that there are infinitely many approximate copies of the Earth with Fake Nous blogs, etc., right now. However, I doubt that the universe is infinite in that way.

c. The unbounded case

Technical semi-digression:

So far, we’ve been talking about a system with fixed, finite energy. But the universe might be infinite, in which case it has infinite total energy. Also, there may be no in-principle limit to how much energy can be packed into a given spatial region. So the range of possible states of the universe is unlimited. So, do my above arguments really apply?

In response,

a) If you take parts of the universe of a given, finite size, these parts are going to have finite energy. And for any such part, there is going to be an infinite range of future time during which parts of that size (e.g., with that number of particles) exist.

b) If you have an unbounded, continuous variable, it still must be the case that, at least after some point, higher values of the variable are less probable.

E.g., say you have an earth-sized mass of matter. Maybe there is no in-principle maximum energy that this mass of matter can have. But, at least after some point, it must be that higher energy levels are less probable. As energy level increases, the probability of finding an earth-sized system at that energy level approaches zero.

The current energy level of the Earth has a nonzero probability (density). So, given infinite time and infinite earth-sized physical systems, there will be infinitely many times when an earth-sized system is at approximately the current total energy level of our Earth.

That’s enough for the argument of sec. 2b to work, such that we can conclude that there will be infinitely many occasions when an earth-like planet is in about the state we’re now in.

3. Incredulous Stare

Some people find the eternal recurrence bizarre and unbelievable. It seems that way when you start describing very specific details that are going to repeat.

If you feel this way, think about what you might object to about the argument. Does the recurrence of a state like this one seem impossible, or does it merely seem extremely improbable?

I suggest that it merely seems extremely improbable. If that’s right, then the only other question to ask is: Do you agree that, given infinite time, even the most improbable things eventually happen?

If you keep flipping a coin, eventually, you’ll get 100 heads in a row. That will happen, on average, on one out of every 2^100 sequences of 100 flips. If you accept that, I think you should accept that other extremely improbable (but not probability-zero) things happen some nonzero proportion of the time.

4. Implications

(a)

Nietzsche thought that, whenever you do anything, the fact that you’re doing that means that you’ll do that over and over again, every time the universe repeats. So choose well!

But that’s wrong. Because the recurrences are only approximate, and because the universe is chaotic (and also because we have free will), what you do now does not necessarily tell you what will happen on future occasions when you face similar circumstances.

(b)

The total future utility of the universe is either +infinity or -infinity. This creates theoretical puzzles for utilitarians, who think we should act so as to maximize future utility.

(c)

Nietzsche assumed that during the next cycle of the universe, when there is a person who looks just like you, that person will literally be you. He didn’t argue for this, though.

I have elsewhere suggested that these people who are very similar to you are in fact you—

. More precisely, there is some degree of similarity that a future person can have to you, in some respects, such that they will literally count as an incarnation of you. If so, then we have a great case for immortality. We all live infinitely many total years, though sadly, we lose our memories at the end of each lifetime.

By the way, almost everyone misunderstands the argument of that earlier post (which was based on an academic paper). Almost everyone thinks that the case for reincarnation depends on recurrence of states extremely similar to our current state. It doesn’t. That is merely one way in which reincarnation might work. The argument of the earlier post/paper shows that you get reincarnation merely from the fact that you exist now, plus the fact that time is infinite in both directions. It could be that you get reincarnated in very different physical bodies.

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[ *This is an expanded version of this earlier post. ]