And the obvious reason why that some superintelligence would be rolling a realistic simulation, is that this is a simulation meant to gain information about a lower-level universe that is very similar to this one.
My bullshit detector is triggered by this.
Yes, sure, this is a hypothesis. A plausible-sounding just-so story. But why are we supposed to privilege it in particular, compared to all the other alternatives? What evidence do we actually have in favor of it?
Probability is a measure function, representing how often events are realized among iterations of probability experiment. The latter is a certain approximation of some real world scenario to the best of your knowledge. A map to a territory, if you will. Probabilities are “subjective” in a sense that they are properties of the map. But they are “objective” in a sense that this map represents the territory. I’m leaving you a link to my sequence on probability theory. It’s unfinished, but I believe that even in its current state it can be quite helpful for some of the questings you are raising here.
I’d recommend to be careful with invoking betting odds. Yes, it is a great validator fo correctness of probabilistic estimates, but it requires to invoke an additional measure function—the whole mathematical apparatus for utilities which is an extra complication and therefore an extra opportunity to get confused. Probability is one thing, utility is the other. If you already feel confused with the former, adding the latter probably isn’t going to make you less confused. It’s better to go back to the basics.
Even more so with “anthropic” scenarious. Appealing to Sleeping Beauty problem while trying to resolve a general confusion about probabilities is like trying to use a metaphor from quantuum mechanics while discussing philosophy. Unless you and the audience are experts in the field, most likely you are going to make yourself and everyone else only more confused.
Cudos for not wanting to pick side in the SSA vs SIA debate. It is a false dylema between two terrible options. You can do so much better than either of them.
We already did it.
We started from considering two types of probability experiments:
Sample between all the observer moments
Sample between being and not being BB
P(BB|1) ~ 1
P(BB|2) = 1⁄2
By the principle of uncertainty we are indifferent between the two.
P(1) = P(2) = 1⁄2
Therefore
P(BB) = P(BB|1)P(1) + P(BB|2)P(2) ~ 3⁄4
And only then we account for the evidence, reflecting on our stable memories. Which simultaneously update us in favor of 2. and against being BB in general
P(SM|2) >>P(SM|1)
P(SM|BB) << P(SM|not BB)
Therefore
P(1|SM) ~ 0
P(2|SM) ~ 1
Which very quickly updates us to near certainty that we are not BBs.
P(BB|SM) ~ 0
That is my argument.
You reflect on your memories. You notice that they point to living in a consistent universe. This is expected on living in a consistent universe. This is not expected on being a BB. Therefore you update in favor of not being a BB.
And the same applies for whether you should be indifferent between observer moments or theories.
That’s not how evidence works, you have two theories, and you have some observation. Is this observation expected on H1? Yes. On H2? Yes. So, it’s not evidence either way.
How does reflecting on your memories and noticing that they correspond to living in consistent and lawful universe is expected on BB?
This is tangential, but there are situation in which something like precommitments are epistemic concepts.
For example, without any precommitments, you do not get to be surprised if you generated a random number between 0 and 100 and it so happened to be 29. In this case it’s just some non-specific number.
However, if you’ve specifically precommited to 29 in particular beforehand, you can be lawfully surprised. This time you are observing a more specific event because of your precomitment.
Or consider Technicolor Sleeping Beauty problem:
Sleeping Beauty experiment, but every day the room that the Beauty is in changes its color from Red to Blue or vise versa. The initial color of the Room is determined randomly with equal probability for Red and Blue
A regular person without any epistemic precomitments upon observing any color of the walls doesn’t get any new information about the state of the coin.
P(Tails|Blue or Red) = P(Tails)
But with an epistemic precomitment to one particular color, a person can correctly guess that the coin is Tails in 2⁄3 of iterations of probability experiment—not mere awakenings! - when the walls of the room are this color. Doesn’t matter which color the walls are, the core part it’s the same color the person has precommited to:
P(Tails|Blue) = P(Tails|Red) = 2⁄3
Once again, the act of precommitment itself is what allows the person to observe a more specific event, and in this case, update their credence.
Well, like, the whole point of Boltzman brain possibility is that your memory is unreliable, and could be created wholesale. You just remember it, there is no like verified history you can rely on
That’s beside the point, really.
Whether you are created wholesale with all your memories or you accumulate experiences during you life, the fact that these memories appear to be ordered, lawful and from the same body—unlike being Canaletto at moment t1 and Jeff Bezos at monent t2, for example - is evidence against you being a boltzman brain and evidence against being randomly sampled among people.
Well, it kinda makes sense? Like, if you can’t distinguish when and where are you, you should have uniform ish distributed credences over those indistinguishable circumstances?
You are supposed to apply principle of indifference representing your lack of knowledge—that’s true. The tricky part here which systematically leads people astray in anthropic scenarious is this:
What are these “indistinguishable circumstances” which we are supposed to be indifferent between?
Are those the observer moments, most of which alledgedly belong to boltzman brains?
Or are those the two hypothesises: “I’m a boltzman brain” and “I’m not a boltzman brain”?
In other words, what is the kind of probability experiment that we are dealing with? The one where my observer moment is randomply sampled between all the observer moment or the one where it’s randomly sampled between two categories of such moments.
And in case we know nothing about it, we once again have to be indifferent between these possibilities. Half the credence goes to 1. and half to 2. So at first there is 50% chance that I’m almostly certainly boltzman brain and 50% chance that I have only 50% chance of being a boltzman brain.
But then, as we accumulate evidence that my observations are stable and ordered it simultaneously priviledges 2. over 1. and makes us being boltzman brain conditionally on 2. less likely. As a result we lawfully become pretty confident that we are not randomly sampled throughout all the observer moments and that we are not boltzman brains. And so mere fact of our existence doesn’t give us any precognitive powers to know with high confidence what happens to the universe in billions of years.
I believe the same line of reasoning that I explore in Doomsday Argument and the False Dilemma of Anthropic Reasoning resolves this problem as well.
Our current experience—your own experience, at this very moment, of seeing ordered letters on a screen—therefore seems to provide overwhelming anthropic evidence against any model of reality or physics which would imply that most brains are Boltzmann brains.
This is true, conditionally on an assumption that my current observer moment is a random sample from all observer moments throughout time. But this doesn’t have to be the case. In fact, to the best of my knowledge, this is most likely not the case. So we can just drop this assumption and then everything is adding to normality.
If a bag is filled with pieces of paper with numbers from 1 to N, a piece is picked randomly and the first one to be picked just so happened to be piece with number 1 - this is quite surprising. But if the algorithm is not a random sample, but instead such that pieces of paper are intentionally picked in order from 1 to N, there is no surprise in it.
Notice that this reasoning isn’t vulnerable to:
“But I don’t believe in anthropic reasoning, so there”—makes the prediction that your experience will dissolve into chaos in the next moment.
As this prediction is also conditioned on the assumption that my experience is the random sample, which is exactly what this reasoning denies.
In other words, just like with Doomsday Argument, Boltzman Brain worries are based on the inaccurate approximation of the causal process of my existence—a map that doesn’t correspond to the territory.
The word real is rather important and I don’t think the conversation can proceed without it unless it is replaced with something else. I would suggest ‘existent’ as such a replacement.
You are failing at the rationalist taboo game. Read this, please, and try again.
The word itself is just a label. The meaning is important. Try explaining the gears behind it, instead of comming up with a synonym.
this implies that mathematics itself is not real, which has not been demonstrated at all.
Taboo the word “real”. What exactly do you mean here?
it is unconditionally true that, for example, the inscribed angle theorem follows from the axioms of Euclidean geometry
What is meant by “follows” is not unconditional. Implication is itself conditional on logical axioms.
However, empiricism doesn’t explain the relative simplicity of the mathematical axioms required to describe reality.
Our brains evolved in the reality and developped the kind of reasoning that is useful for it. No more explanation is required.
i’m just saying that “mathematics is a language, so of course it can be used to describe reality” is arguing against something the surprising-effectiveness crowd does not claim. they do not say “it is surprising that mathematical language can be used to describe reality”, but rather that “it is surprising that specific mathematical objects of interest to mathematicians often end up useful in physics or other empirical sciences.”[1]
Do I understand correctly that you claim that the surprising part is that mathematical objects are “discovered” first for their own sake and only later are revealed to be physically relevant? Because if this is the case, I address this here:
We can still be somewhat surprised if the language has necessary words to describe certain parts of reality before they were encountered. But this is an improbability of a much smaller degree, reduced by the fact that creating new words in this language is an interesting and rewarded activity, which people tend to do for its own sake, as exactly is the case with mathematics.
I’ve met people just generally confused why math can describe reality at all. And math being a language is a part of the answer here. I think Lorenzo Elijah, whose post I’m answering to, is confused in a similar manner. He acknowledges that math is a language but then thinks this is problematic and uses the middle-earth-planaet analogy which would make sense only if math was a story. So I think it’s totally appropriate to point that math is a language and not a story and that this resolves the huge part of the apparent improbability, before addressing the smaller improbability related to the order in which mathematical objects are invented/discovered.
i feel the original essay—in this section—elides the difference between “mathematical language” and “mathematical object”, and thereby argues against a strawman.
Mathematical objects are parts of the language. Like words pr other semantic constructions. I think it’s pretty straightforward.
Can you come up with an example of a mathematical statement that describes reality while not being physically relevant?
Okay, but I don’t think you’ve clarified how to make that choice yet then.
Which choice? What are the possible outcomes of a given experiment? Check this post of mine, I explain the application of the framework of probability experiments in more details.
Like, what if I decide my outcomes are logically possible worlds?
Can you give me an example? I’m not sure I understand your question.
My own view is that you formally need a semantic theory, e.g. model theory. “All logically consistent possible worlds” is often not well-founded as a set (depending on how unconstrained “world” is), so you’d need to clarify this first before trying to do anything with it (which no one does, so no wonder it’s confusing).
Well, yes. That’s why the framework is bad and we should use a better one.
But it doesn’t seem inherently wrong, conceptually, to me.
I don’t understand what you mean by “inherently”. I’m not trying to make any metaphysical claims about essence of the idea of possible worlds or some such. I just notice that it works poorly for what we need from a framework for probability theory and propose a better one.
Interpreted as “all philosophers who talk about possible worlds believe they are real”, that isn’t true. The belief that possible world are real, modal realism, is contentious.
Yes, you shouldn’t interpret it this way. Modal realism is not the point of contention of this post. The whole framework of “possible worlds” is, regardless of whether one thinks they are somehow real or not.
I don’t know how you manage to keep misinterpreting me about it, even though I’ve already explained it to you under a different post.
Talk of possible worlds is common, because of the popularity of modal logic, and modal logic is …old … the sort of thing rationalists should approve of?
Why? Are we suppose to endource everything that’s old? Somehow I didn’t get the memo.
It allows you to put the notions of necessity and possibility on a formal basis, instead of leaving them as vague metaphysical notions.
The sentiment is good the implementation, is not. To be useful such formalization should be based on the framework of probability experiments and not possible worlds, for the reasons described in the post
My issue is with the interpretation of Ω as a set of “all logically consistent possible worlds” instead of set of “all possible elementary outcomes of a probability experiment”.
The difference is that the former idea is untenable, doesn’t work for logical uncertainty and confused everybody about self-location, while the latter provides a clear way to know, what should our Ω be for every specific case.
The issue with the Sleeping Beauty Problem, that makes it unique, is that each running of the experiment produces two outcomes. One on Monday, and another on Tuesday. These are different, and independent, outcomes because an outcome is defined by SB’s ability to experience it
That’s not what an “outcome” means. From Wikipedia:
In probability theory, an outcome is a possible result of an experiment or trial.[1] Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space.[2]
We are, of course, free to talk about something else, that isn’t outcomes, and even build an alternative mathematical model with a different measure function, that isn’t probability, using them. I explicitly do it here. It’s not really necessary, there is no advantages in doing it, but it can be done.
When you are logically pinpointing Y instead of X, you are, quite obvioiusly, talking about Y and not X anymore. This doesn’t change even if some philosophers, in attempt to confuse everyone, insist on calling both X and Y the same name.
According to this modified framework the answer is 1⁄3. But according to probability theory the answer is 1⁄2. And we are interested in probability due to the fact that it correspond to the rational credences of an agent.
And it doesn’t matter how many more alternative problems about memory loss you can come up with. They are all solved in a similar manner that I describe in this post.
Sure.
There is an extra potential complication with information about the setting arriving in a probabilistic manner. For the sake of simplicity, let’s assume that the driver always gets a pamphlet on the intersection.
Yes, and after getting the pamphlet the driver is supposed to realize that they were using an incorrect mathematical model, that fails to capture the specifics of the real-world situation they are in. So, the driver says: “my map was wrong” and starts using the map correctly approximating the setting, instead.
This works exactly the same way as in any other case where you initially had incorrect model of the setting and then learned the truth. For instance, if you though that a D6 die is rolled, but in fact it’s D8. No superpowers involved.