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People mean different things by “the anthropic principle” so let me clarify what you have in mind. Sometimes the idea is supposed to challenge the fine tuning argument by pointing to the multiverse and invoking the law of large numbers. That argument works if you have some reason to think the multiverse exists and is more probable and theoretically virtuous than theism.
But I also hear much more naive and confused sounding appeals to the anthropic principle. For example, sometimes people seem to be suggesting that because a phenomenon involves the creation of observers, the phenomenon requires no explanation, which is silly. Imagine if I prayed for a parachute while falling from a plane, one spontaneously manifested out of thin air and deployed to save my life, and I reflected afterwards about why that happened. I conclude, “well, I wouldn’t be here to ask the question in the first place if that didn’t happen, so there must be no explanation needed.”
Or imagine saying the theory of evolution is dispensable because “if it didn’t happen we wouldn’t exist. We wouldn’t be here to wonder about it if not, so what is there to explain?”
I am not saying that a phenomenon that involves the creation of observers requires no explanation, I am saying that observing such a phenomenon exists provides no information because such a phenomenon is 100% guaranteed to exist in any universe with observers. So I am saying we cannot draw any useful conclusions from observing such a phenomenon exists.
For example, every person I meet who has ever been skydiving tells me that their parachute has opened every time they've gone skydiving. This fact tells me no information about how likely parachutes are to open (except that the probability is not 0%). It provides no information about parachutes one way or the other.
The surprising part isn't what the phenomenon looks like. It's that there are observers at all in the one-shot universe. It does provide information.
It doesn't provide information, because the only possible one-shot universe we can observe is one with observers. Maybe there was a 0.000...01% chance that the one-shot universe would be fine-tuned for observers, or maybe there was a 99.999...% chance that the one-shot universe would be fine-tuned for observers. Either of these possibilities is consistent with the fact that we, as observers, see a one-shot universe with observers in it.
So imagine we have two hypotheses, to simplify.
Hypothesis 1 is that there's a 95% chance of a rational being coming into existence.
Hypothesis 2 is that there's a 0.001% chance of a rational being coming into existence.
You, the observer, notice, hey, I'm a rational being who came into existence!
I'm saying that it's rational to think that this should update your priors towards hypothesis 1 over hypothesis 2. But only if there's a one-shot or few-shot universe.
It's 95000 times more likely for a rational being to come into existence under hypothesis 1 than under hypothesis 2. So this piece of evidence should update our beliefs by 95000 times using the odds form of Bayes' theorem, I believe.
For a comparison (and I'm not certain that this is a perfect analogy, but I think it works), let's say you are told that there's a 50/50 chance a surgery succeeds. Under the 50% chance you survive, you always wake up. Under the 50% where it fails, they freeze your body, and you estimate that there's a 1/50 chance that they manage to revive you in the distant future.
You wake up. Assuming there's not going to be any distinguishing sensation between the two ways you could wake up, which should you think is more likely? I would think you should think that there's a 50:1 chance that it worked.
No, it shouldn't change your priors, irrespective of whether there's a one-shot or multi-shot universe.
Given that you are a rational being, the odds that you will observe a universe where a rational being came into existence are exactly 100%. This is true regardless of whether hypothesis 1 or hypothesis 2 is true, and therefore it tells you no additional information about which hypothesis is correct.
That's true, because this situation is materially different from the one we are talking about above. Here, we know two sets of probabilities ex ante (both of which can occur), and are now trying to decide, ex post, which is more likely to have occurred. Given two different possibilites, the one with the higher probability is, by definition, the one that was more likely to occur (this is true whether the surgery is one-shot or many-shot, by the way).
In the situation we are discussing, we don't know anything about the probabilities ex ante, and we are trying to derive those probabilities based on our ex post observations.
A better analogy would be: you go into a surgery and nobody knows your odds of survival. You wake up after the surgery. What, if anything, does this tell you about your ex ante odds of surviving the surgery? The answer is, it tells you nothing about those odds. It just tells you that you survived. Your odds of survival could have been 0.001% or 99.9%, but since you can only observe outcomes in which you survive, that fact that you observe your own survival gives you no additional information about the ex ante likelihood of that outcome.
Another example to illustrate the point. Suppose an alien hands you a black box with a screen and a button on it. You push the button, and the number "21" appears on the screen. Pushing the button again does nothing and you cannot disassemble the box to learn how it works. What are the odds that the box was going to display the number "21" when you pushed the button? The answer is, you have no idea (except you know the odds are not 0%). It might have been a 100% chance, it might have been a 0.000000001% chance. You have no way of knowing based on your single ex ante observation.
By starting with "given that you exist," you're assuming away the part you can learn from. Starting from an objective not-taking-into-account-yet-that-you-exist estimate of likelihood that a random universe would be able to contain life, you should have some estimate of a likelihood that a universe could contain life.
But now that there's some agent, that's data! That's information! You can use that to shift your beliefs. Now you're about to respond that there will always be an agent whenever you have the opportunity to think about these things. Maybe so. But there won't always be an agent in the universes in the world-model we were just talking about. And those worlds where there is one look different from the worlds where there isn't one. They were created by mechanisms that result in a higher probability that such agents come to exist. And so you should assume that you, in actuality, are more likely to be in the higher-probability portions, than you'd think on a basis that ignores your coming to exist.
Per your surgery case, I don't think your conclusion is true? Most people who survive a surgery do it in cases where survival is likely, so I would think that you should think that you're among that pool. That seems like another case where the bare fact of your existence provides evidence about the world.
Well, that's not quite just your existence, since what I just said requires knowing how often other people survive surgery, but it would still seem in the general case that you should update to some extent (it may be a small one, depending on your prior knowledge) towards your surgery having odds of survival by waking up.
See Joe Carlsmith's account of SIA here for one analysis that would incline one towards being in universes where more people exist. It's fairly rigorous, but not without difficulties. I don't remember it addressing the question exactly as we've posed it, but it's definitely very relevant.
I'm not convinced that it's quite the same framing as I had used above for the first two paragraphs, so maybe I'll have to think about whether any of that needs revising. But I'm still pretty confident that you coming to exist provides you information that you can use to shift your beliefs.
If we could start there then we would gain some information from the later observation that the universe contains life. But we cannot start there. We start in a universe where the existence of life is a given, with 100% probability.
It's not. You can only observe a universe with agents because you are an agent. That statement is always true, no matter the prior probabilities, so you cannot draw any conclusion about prior probabilities from the fact that it happened.
Right, if we could bring in outside information about how likely a universe is to contain life and what factors influence that, then we might be able to draw some conclusions from the fact of our existence, but we have no such outside information, so we cannot draw such conclusions.
Why can't we start there? Isn't that equivalent to stating that you can't think about universes where life doesn't exist? That's transparently false. Working out what our beliefs should be if we ignore a piece of information is something we're allowed to do.
No, that statement isn't always true. It's only always true for observers. That means that you should shift your probability mass from what they would be if you ignored which worlds you're more likely to exist in as an observer, to what they should be after taking that into account.
Assuming our thinking is at all Bayesian, shouldn't we have some sort of probability distribution? Not sure exactly what one should look like, but that should exist. In any case, did you miss what I said about how it should always be the case, whatever that probability distribution is, that you should update (ignoring other post-waking-up information, on your existence alone) towards the chance of your survival having been higher than you thought it was before your surgery?
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