Tuesday, October 26, 2010

Entropy and God

Following my discussion of Boltzmann Brains, I wanted to mention its relation to yet another argument for God.

The argument goes that a low-entropy initial condition is extremely unlikely unless God exists.  Therefore, God probably exists.

(This argument is not to be confused with the argument that evolution contradicts the Second Law, a mistake once made by PZ Myers.)

My response is that we don't know that the initial condition is extremely unlikely.  It's only extremely unlikely if we assume all microstates are equally likely initial conditions.  Also, saying that we do not have an explanation is not the same as saying we do have an explanation and we call that explanation God.

Let's say that I have a differential equation that I can't solve (a common scenario in physics).  A common practice is to posit the existence of a solution and call that solution something like the Bessel Function or the Legendre Function.  But I can't posit a solution and call that solution God.  Got the distinction? The analogy isn't perfect though; in differential equations we have existence theorems, in metaphysics we do not.

Rationalist atheism does not claim that science has all the answers, but rather, that religion has none of the answers.

14 comments:

Larry, a.k.a. The Barefoot Bum said...

More responses:

1) Unlikely is not the same as impossible.

2) Because all conditions are equally unlikely, no particular condition is more unlikely than any other, and both sides must posit there was some initial condition.

3) There's also the weak anthropic condition: if the initial conditions were not such that they produced us (and what we actually observe), we would not be here to observe what we do.

4) If we are going to talk about god in probabilistic terms and since an omnipotent god can by definition create any initial conditions as it pleases, there is a range of gods commensurate with the range of possible initial conditions: there is a measure of the set of all possible gods commensurate with the measure of the set of all possible initial conditions. Therefore it is just as unlikely that a god exists who wanted to create any definable set of initial condition as those initial conditions are without any god.

Of course, whether it's even meaningful to talk about gods in probabilistic terms is a dodgy assumption, but if you're going to talk about the problem in those terms, it seems a fallacy of special pleading to exempt god.

Larry, a.k.a. The Barefoot Bum said...

And not only is there an equivalence between the set of all possible gods and the set of all possible initial conditions plus physical laws, there is also the set of gods who create initial conditions and/or physical laws incompatible with the natural existence and/or evolution of life. Therefore given that we do in fact exist, the probability that a god exists who created conditions & laws compatible with naturalism is smaller than the probability that naturalism is true. (If naturalism were true, we could not exist in a universe with laws & conditions incompatible with the natural existence of life.)

Larry, a.k.a. The Barefoot Bum said...

... there is also the set of gods who create initial conditions and/or physical laws incompatible with the natural existence and/or evolution of life and who create and sustain life anyway.

miller said...

I would be careful with 2) and 3).

2) The Ergodic hypothesis does not say that all conditions are equally likely, but that all initial microstates are equally likely. This means that conditions with high entropy (corresponding to exponentially more microstates) are vastly more likely.

3) The anthropic principle does not resolve this problem. While we would only be around to observe the universe if it had low entropy, the universe is much lower entropy than is necessary.

Larry, a.k.a. The Barefoot Bum said...

The Ergodic hypothesis [says] that all initial microstates are equally likely.

Indeed. But regardless of what macrostate was created, it was created with exactly one microstate. Just a matter of choosing your metric. ;)

The anthropic principle does not resolve this problem. While we would only be around to observe the universe if it had low entropy, the universe is much lower entropy than is necessary.

Hence my qualification about what we actually observe.

And I'm not so sure the universe really did have that much lower entropy than is necessary to account for the existence of human intelligence: there's as much or more cosmological as biological evolution that has to precede the actual development of human beings. Just the formation and coalescence of elements beyond helium (and a bit of lithium IIRC) is a multi-billion year process and requires a great deal of negative entropy. I can't see that we're all that many orders of magnitude away: The universe certainly could plausibly be neither 10s of millions or 10s of trillions of years old (using time as a proxy for entropy).

miller said...

No, I still disagree with you on the Anthropic principle.

As you observed earlier, there are clues in the present to a low-entropy past. But they are only clues in the sense that they confirm our hypothesis. This is what we'd expect to see if the universe started with low entropy.

But they are not clues in the sense of disconfirming every other hypothesis. There are plenty of high-entropy initial conditions which might have led to what we see now. It would have been a pretty amazing coincidence to get from any given high-entropy state to where we are now, but there are plenty of high-entropy states to go around.

The reason we reject these other hypotheses is not because of empirical observations, but because we just have low prior belief that they are true, lower than the Ergodic Hypothesis would suggest.

Larry, a.k.a. The Barefoot Bum said...

I'm not convinced you're correct.

There are plenty of high-entropy initial conditions which might have led to what we see now. It would have been a pretty amazing coincidence to get from any given high-entropy state to where we are now, but there are plenty of high-entropy states to go around.

How does the math work out?

If I understand you correctly (and keep in mind your understanding of thermodynamics and statistical mechanics vastly exceeds my own), that the given the prior knowledge of our own existence probability of high entropy initial conditions is much less than the probability of low entropy conditions.

The number of high entropy initial conditions relative to low entropy initial conditions is, however, much greater than the difference in probability. Therefore although

(1) P (any specific high entropy state | humans) << [much less than] P (any specific low entropy | humans)

it is still the case that

(2a) N (all low entropy states) << N (all high entropy states)

and

(2b) N(low entropy) * P (low entropy | humans) << N(high entropy) * P (high entropy | humans)

When you plug in actual numbers, does the math work out?

Larry, a.k.a. The Barefoot Bum said...

If it does work out, why should "we just have low prior belief that they are true, lower than the Ergodic Hypothesis would suggest"?

Larry, a.k.a. The Barefoot Bum said...

I'm curious to know if in some sense statistical mechanics suggests "last Thursdayism", the idea that there are many more microstates that constitute the physical universe popping up last Thursday — including our memories of what happened before last Thursday — then there are microstates popping up ~15 billion years ago and evolving to our present circumstances.

I'm not at all convinced — at least not yet — that the probability calculations actually suggest that result.

miller said...

Yes, the probability calculations really do suggest Last Thursdayism. That's what Boltzmann Brains are really about!

Here's what the math calculation would look like:

We both agree that:
P(future is higher entropy than present) >> P(future is lower entropy than present)

Which is to say:
N(accessible high-entropy microstates) >> N(accessible low-entropy microstates)

But we can make a one-to-one correspondence between microstates. Just leave all particles in the exact same positions, but reverse their velocities. This corresponding microstate will behave the same way, only with time reversed.

N(accessible microstates) = N(microstates that will lead to present conditions)

Therefore:
N(high-entropy microstates leading to present conditions) >> N(low-entropy microstates leading to present conditions)

P(past was high entropy) >> P(past was low entropy)

If we assign equal prior probability to each microstate, then this leads to Last Thursdayism. But we reject Last Thursdayism, which means that we must be using different prior probabilities.

Larry, a.k.a. The Barefoot Bum said...

Right... we know that at the microlevel, thermodynamics and statistical mechanics are time-symmetric. With time symmetry, then yes, we have as much reason to believe the past was higher entropy than the present as we we have reason to believe the future will be higher entropy. I don't think that's really controversial.

Why specifically we believe the past had lower entropy than the present is certainly an interesting question.

I want to ask a slightly different question, though, related to the anthropic principle: If we add in some (perhaps mysterious for now) thermodynamic arrow of time, i.e. time asymmetry, how would we compare higher- and lower-entropy states microstates that would evolve by the ordinary long-term workings of thermodynamics to present circumstances? Can we start with a universe with significantly higher entropy than we presently believe existed at the big bang and still evolve humans?

miller said...

Given the Second Law (entropy increases forwards in time, not backwards), could the universe have started with higher entropy and still have created humans?

That's a good question, and I don't know the answer.

However, if we allow the universe to be non-uniform, the answer is yes. We could have a universe where just our local galaxy cluster starts with low entropy, and the rest of the universe does not.

Larry, a.k.a. The Barefoot Bum said...

If we discard uniformity, we also have to count possible universes all sorts of different kinds of uniformity, do we not (multiple pockets at different levels of entropy)? That would seem to affect the hypothesis. Also, if there's any sort of convergence effect, to actually observe non-uniformity, we'd have to evolve at a very specific time... and rather quickly, no?

miller said...

In principle, we could look out towards far away galaxies and observe that they're all very high entropy compared to ours. This would have virtually no effect on life here, because that just affects the kind of starlight we'd see.

But there's probably a good cosmological reason for why the universe is uniform.