Episode 141: Probability and Reality (with Sam Kuypers)

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Unofficial, reviewed transcript. It may still contain mistakes; check the podcast when wording matters.

Speakers are labeled by first name where identified. Unidentified-speaker labels may refer to different people. Bracketed question marks indicate uncertain wording.

Transcript

[00:00:00]  Peter: Hello out there. This week on The Theory of Anything Podcast, we are joined by Sam Kuypers, I think for the fourth time. Sam does so many cool stuff, it's hard to know where to begin, but he told us that he is a, I think a full-time fellow at the Conjecture Institute right now, which is the organization founded to promote critical rationalism in science. To just read his bio there, Sam is a quantum physicist interested in foundational questions. In his research published in reputable scientific journals, he explores the quantum theory of time, the many worlds interpretation and locality. His work connects these areas under a broad commitment to realism. The view that a real world exists and that science is a quest for good explanations describing this world. And I also might add, he is a super, super nice and humble guy.

[00:01:00]  Unidentified speaker: I really have enjoyed all my interactions with him over the years.

[00:01:05]  Peter: On the podcast this time, we discuss, is probability theory true?

[00:01:12]  Unidentified speaker: Is it useful?

[00:01:14]  Peter: What is the connection between probability theory and physics? Does the many worlds interpretation make probability theory more or less true? It's weird to be on a podcast where I am myself barely hanging on to the trajectory of the conversation at times. Bruce and Sam get deep into the weeds here, but I enjoyed listening to them a lot and I hope someone else gets something out of this too.

[00:01:49]  Unidentified speaker: Welcome to The Theory of Anything podcast.

[00:01:51]  Bruce: We got Peter, and today we've got Sam Kuypers here.

[00:01:54]  Unidentified speaker: Hey, Sam.

[00:01:56]  Sam: Hey, guys, how are you doing? Thanks so much for having me. I'm looking forward to the discussion.

[00:02:00]  Bruce: Oh, total pleasure to have you on.

[00:02:03]  Unidentified speaker: It's such an honor.

[00:02:05]  Unidentified speaker: What an honor, yes.

[00:02:07]  Bruce: Sam is one of my favorite guests on the show. I need to have, we should make Sam a regular or something. He's so fun to talk to. Sam listened to my episode on the Deutsch slot machine, which was mostly me being largely confused, and offered to come on the show, talk with me. Sometimes, I think he thinks he's confused too. One of the things that maybe we can talk about is just how hard it is to wrap our minds around some of these big subjects. Even when we've got degrees in them, and even when we think we know something, it's always not so clear how much we know. So, Sam is going to kind of allow me to ask questions, try to make sense of his understanding of probability, and his understanding as much as possible. Obviously, Sam's not David Deutsch, but as much as possible, what is the point David Deutsch is making, and what can we take away from it? Peter, why don't you go ahead and start with your question, though? I'd actually like to hear what your question was that you wanted to ask.

[00:03:09]  Peter: Oh, well, this was about what I thought was called the Popper Miller conjecture or something close to that. So in the 80s, I think they got together and tried to mathematically work out one of the central tenets behind critical rationalism, at least as I understand it, is that you can only move essentially one direction, I guess, in science or I guess, in life. So the idea behind this Popper Miller, I think it's from a paper from 83. Bruce threw me a curve ball here. I was not expecting to ask this question straight off the bat here.

[00:04:00]  Unidentified speaker: But the idea is that science essentially moves in one direction to refute theories rather than confirm them.

[00:04:11]  Peter: And I think this has to do maybe with the idea that verisimilitude might not be quite right. It sounds like Sam knows a bit about this.

[00:04:21]  Unidentified speaker: Maybe you could explain it.

[00:04:26]  Sam: I haven't worked on it.

[00:04:27]  Unidentified speaker: I know that David and Matjaž are writing a paper on the topic.

[00:04:34]  Sam: Matjaž gave a talk for the Popper Society back when I was still doing my Ph.D. So I have my own understanding of it. His will be, their understanding will probably be richer than mine. Certainly will be richer than mine. But the way I understand it is, so it's a theorem, it's the Popper–Miller theorem. And what it's trying to do is it's trying to say that if you're using the probability calculus to basically perform induction for you, you'll find that you'll [fail?]. So I think specifically, I might be wrong about this, but specifically what it's trying to do is to say, with probability theory, you're not going to get beyond just deductive logic. So if some theory is, you have theories, you have credences for theories, some of them get, or maybe one of them gets refuted or confirmed. Let's say it gets confirmed, so its probability goes to one. Then certain theories will be disconfirmed, and others will also have their probability go up. So some of them, their probability will go to zero, say, and others will become more likely. Then again, the punch line, I'm not sure I'm remembering it right, but I think it's something like all of them basically get raised by the same factor. So if you think that you can now see, oh, well, we've made a measurement, we've done an experiment, the experiment showed that that particular theory is true, and now we can kind of pick that one out. But all the other theories also increase the credence with the same factor. So it can't be that you can choose, it doesn't allow you to choose between theories.

[00:06:36]  Sam: Even theories which initially had a lower credence might now have a higher one, and you're kind of in the same situation.

[00:06:43]  Unidentified speaker: Other than that, you know certain things are now false.

[00:06:45]  Sam: So you can use deduction to say, given that this experimental result is true, certain theories are now refuted, and it doesn't really get you anything beyond that. So when people are trying to use Bayesianism to solve the problem of induction, this theorem says that they will fail to do that. That's basically my understanding of it.

[00:07:07]  Unidentified speaker: It's an interesting result.

[00:07:10]  Sam: My experience with all probability results that I'm familiar with is that they are seemingly simple, but then turn out to be quite deep, especially if you have to construct an explanation of what the result is saying. It's counterintuitive.

[00:07:30]  Sam: But as I said, I haven't worked on it specifically, and I think I'm forgetting some of the details. But I recommend speaking with Matjaž about it. He knows a lot on the topic.

[00:07:43]  Unidentified speaker: Okay.

[00:07:44]  Unidentified speaker: Thank you.

[00:07:44]  Peter: Yeah, it looks like it's from A Proof of the Impossibility of Inductive Probability is from 1983.

[00:07:53]  Unidentified speaker: That's what the paper is called.

[00:07:55]  Unidentified speaker: Right.

[00:07:55]  Sam: Yeah, that sounds about right.

[00:07:57]  Unidentified speaker: Yeah, I think I've also tried to read the original paper and I could not get through it.

[00:08:02]  Sam: That's why there needs to be-

[00:08:04]  Unidentified speaker: I'm doomed then.

[00:08:05]  Unidentified speaker: I'm doomed.

[00:08:06]  Bruce: If Sam can't get through the paper, I'm totally doomed.

[00:08:09]  Peter: Well, that's the other thing that I heard about it, which made me afraid to read it, is that it's very- the math is very mathematical and very complicated.

[00:08:22]  Sam: Yeah, I also just remember it not being very clearly written. So the weird thing is-

[00:08:27]  Unidentified speaker: Yeah.

[00:08:29]  Unidentified speaker: Again, it's been a while since I looked at it.

[00:08:30]  Sam: I only looked at it because, you know, this paper by David and Matjaž is being written. And I thought I’d have a look and I saw the math.

[00:08:41]  Unidentified speaker: The math itself is seemingly not very complicated, although the notation that they use is kind of strange.

[00:08:49]  Sam: Again, just going off my memory now, but then the actual explanation that they give is really hard to follow. So yeah, I don't think- But there's probably people who read it and really got something out of it. And the result itself is genuinely interesting. It's just that I found that paper fairly impenetrable.

[00:09:10]  Bruce: And I ask you something on this. Did you by any chance, Sam, read my blog post that I put out where I was going over Tom Mitchell's proof that there's the futility of unbiased learning? Did you by any chance read that?

[00:09:25]  Sam: I don't think so.

[00:09:27]  Unidentified speaker: Maybe.

[00:09:27]  Sam: I'm always bad at remembering these things.

[00:09:29]  Unidentified speaker: I'm wondering if it's the same thing.

[00:09:33]  Bruce: So Mitchell's a machine learning guy, and he quote, believes in induction. But what he really shows is that when you're doing induction, what you're really doing is you're making a set of assumptions that if you were to put those assumptions explicitly into a deductive logic prover, that it would turn out that the answer coming out of machine learning is precisely deductive, right? So there is no actual induction. Induction is just a certain way of thinking about deduction where often you're using, not always, sometimes using explicit assumptions, but sometimes you're just using implicit assumptions. And it's a really interesting, I call it a proof, but it's not truly a proof. It probably falls short of being a proof, but it's a convincing argument, is what I would say. And it's one that I think Popperians would like because it basically shows that the only kind of [reduction/deduction?] that exists is the deductive kind.

[00:10:31]  Unidentified speaker: Right, yeah.

[00:10:32]  Sam: Yeah, that is just, I wouldn't be surprised if that result kept popping up in different disciplines, which that is essentially, again, my understanding of the Popper Miller Theorem. Also sounds like from the title of the paper, that's what they were arguing.

[00:10:48]  Unidentified speaker: It was quite an explicit title.

[00:10:50]  Sam: So, yeah, I wouldn't be surprised. Probably you could even, if that was not a proper proof, in the case of the machine learning issue, I think you could probably even use the Popper Miller Theorem to show that that is just not possible. Like you're always just ending up with deductive logic.

[00:11:10]  Unidentified speaker: That's right.

[00:11:13]  Peter: There's a one-sentence summary from You Know Where. The Popper Miller argument claims that evidence can eliminate false theories, but cannot logically increase the probability that a theory is true, because any confirmed part is already contained in the evidence itself.

[00:11:37]  Sam: Yeah, that sounds, yeah, so the technical part is...

[00:11:39]  Unidentified speaker: That's the summary I've heard before, yeah.

[00:11:41]  Unidentified speaker: Okay.

[00:11:42]  Sam: It's, yeah, I guess, like, it does increase, I would think that it would increase the credence, but not—it would increase the credence of all of them.

[00:11:55]  Unidentified speaker: So...

[00:11:55]  Bruce: So when you say increase the credence with all of them, I know that you're going off of memory here, right?

[00:12:00]  Unidentified speaker: Yeah.

[00:12:00]  Bruce: There's no way it can increase the credence of all of them because you would normalize at the end. Presumably, you mean before normalization, it would increase the credence.

[00:12:08]  Sam: I think it's the normalization that increases the credence of all of them. So it's like some of them go to zero and some of them increase in credence.

[00:12:15]  Unidentified speaker: But the normalization is what does it and it's kind of uniform.

[00:12:18]  Bruce: It is always the normalization that does it. That is absolutely my understanding. Not of Popper's paper, which I haven't read, but just as a machine learning guide, that is exactly how it works.

[00:12:29]  Sam: I think it's the normalization which raises the probability of all the remaining theories that are non-zero.

[00:12:35]  Unidentified speaker: Yeah.

[00:12:35]  Bruce: So Tom Mitchell, in his book, and I've been wanting to put together a machine learning class for critical rationalists and show some of the interesting overlay between critical rationalism and machine learning, what most people are missing because the language of machine learning is inductive.

[00:12:52]  Unidentified speaker: And I don't think it should be.

[00:12:53]  Unidentified speaker: I think it should be critical rationalist.

[00:12:57]  Bruce: One of the things that he shows is if you were to take something like a Bayesian reasoner for machine learning. So what he does is he starts with an algorithm, the candidate elimination algorithm, which is exactly what it sounds like. You falsify candidates until you get to the only remaining candidate, and that's the correct one. What he shows is that there is a probabilistic interpretation of that. So essentially, you assign all the candidates that aren't yet refuted, equal probability, and then as you eliminate and refute them, it reduces, you normalize out the ones that are refuted, and eventually the ones remaining get higher and higher probability, which is really just a fancy way of saying that they share the version space is what it's called. They are all possible candidates still. And really, that's all probability is in that case. But there is this probabilistic interpretation, but it's really just conjecture refutation, right? Like it's exactly conjecture refutation.

[00:14:08]  Unidentified speaker: Yeah.

[00:14:08]  Sam: No, that makes sense to me. Yeah, you're always going to end up with deductive logic. Like probability theory is not going to allow you to do anything but deduction at the end of the day.

[00:14:22]  Unidentified speaker: That's right.

[00:14:22]  Unidentified speaker: So it's not going to allow you to, for example, say, okay, well, that theory is now more likely than its rivals.

[00:14:27]  Sam: It's just, it's more likely than the rivals which haven't been refuted. But that is not getting you beyond deduction, because deduction can tell you that that theory, a bunch of theories become impossible in the light of the experimental results.

[00:14:42]  Bruce: So I think it's a little more complicated than that. Maybe we can revisit this topic closer to the end of the show, because I kind of want to get into some of the basics that we're probably losing our audience, and I probably need to go over some of the more basic questions first. Let me just say though that there are interesting things around Bayesian use of probability theory that I think critical rationalists would actually like if they understood it. Doing machine learning where you have to touch on this a little, and I've started to study it directly because I became curious. I constantly find overlap between these different ideas in critical rationalism, and not everyone is going to agree with me, but to me it seems pretty obvious that at its root Bayesianism, Bayesian reasoning is critical rationalist. Until you misunderstand it, like a Bayesian epistemologist may not understand Bayesian reasoning. They may think they do, but they may be misapplying it. And then at that point, it's no longer critical rationalist. Now they're doing something different, right? And that's why it's wrong, right? So anyhow, maybe we can dig into that, some of the specifics later. I kind of wanted to ask some of my basic questions. So the main thing that kind of led to this is, I came across critical rationalists on X, who were making these wide-ranging claims about probability that seemed wrong to me, right? I documented this on my show. I don't know if you've seen those episodes.

[00:16:09]  Bruce: They were kind of some shorter episodes where I actually would read what they would say and I would explain why it's not correct, right?

[00:16:15]  Sam: I think I have, yeah, yeah.

[00:16:16]  Unidentified speaker: Okay.

[00:16:17]  Bruce: So I couldn't figure out what was going on, and I would ask questions. Now, most of the critical rationalists on X are not going to be experts in this subject. So they're going to struggle to explain to someone like me, who maybe has actually studied it, what it is that they're trying to get at, right? So they would try to explain it. I would go, no, that doesn't seem right.

[00:16:38]  Unidentified speaker: Here's why.

[00:16:39]  Bruce: They wouldn't necessarily understand what I was saying. They would point me to David Deutsch's talk, Physics Without Probability, and they'd say, you need to go listen to this.

[00:16:48]  Unidentified speaker: This is actually going to set you straight.

[00:16:51]  Bruce: So I actually went and listened to it finally, and it came across and came away still confused. It did not really answer any of my, if anything, it increased my confusion over what was going on.

[00:17:07]  Unidentified speaker: Yeah, I remember you [saying/talking?] like this, yeah.

[00:17:10]  Unidentified speaker: Yeah.

[00:17:10]  Bruce: So having said that, it's not that I necessarily disagree with what he's saying. A lot of times I just can't tell what he's saying. He does make at least one statement that I'm convinced is completely wrong, and that is he says that probability is like a flat earth approximation. Now, I need to be—even here, I need to offer some nuance. If we're talking about subjective probabilities, we can just talk about what those are and if they even exist, since a lot of frequentists deny that they exist. But if we're talking about subjective probabilities, I do think he's right. Probability is like a flat earth approximation. I don't really deny his central thesis, which is you don't need probability theory in quantum physics, if you're looking at the level of the multiverse that is, because it's an entirely deterministic process. The Schrödinger equation is entirely deterministic. There's no probability involved.

[00:18:09]  Unidentified speaker: The probability is emergent, right?

[00:18:11]  Bruce: It's something that you experience at the level of a universe. I don't deny any of that. But when it comes to quantum probabilities, they are not in any way like a flat earth approximation, as far as I can tell. Like they really are probabilities and they match the probability calculus perfectly, even though they are emergent and they're not fundamental if we want to say—want to put it that way. And when I—when I think about that analogy, flat earth approximation, and David even says this in the video. He says, if you're just farming your lawn, you know, farming your land or something, earth is approximately flat, so you can treat the earth like it's flat. But that's going to come back and bite you in some other scenario, because it was just an approximation. And you'll actually get a wrong answer. If I take a catapult and I launch it, and it's really, really powerful, so I have to take the curvature of the earth into consideration, if I try to treat the earth as flat, I'm literally going to get the wrong physical answer as to where the boulder is going to land, because it's going to go further than my calculations say, if I'm assuming it's flat, because it's actually curved, and it's going to go a little further than that because of the curvature, right? So when you use the Flat Earth Approximation analogy, I'm inherently understanding that as meaning there are cases where probability theory is going to give you a wrong answer, and you would need some other theory to get the right answer. And I don't think that's ever actually true for quantum probabilities.

[00:19:49]  Bruce: I think that it is a precise answer for anywhere where we're talking about quantum probabilities. Again, I'm eliminating subjective probabilities from the equation. We're just talking about you're setting up quantum experiments and you're working out which direction the photon is going to go or which slit it's going to go through, or is it spin up or is it spin down? Any of the things that you would do in quantum physics, where you want to take an observation, you need to use the Born rule and you end up with the probability. I believe that those probabilities are not approximate, they are exactly probabilities. Now, am I right about that? Am I wrong about that? And again, giving ground, the moment you talk about subjective probabilities, the moment you talk about life insurance, then I would agree with David that it really is just an approximation.

[00:20:41]  Unidentified speaker: Yeah.

[00:20:41]  Sam: So several things to disentangle.

[00:20:45]  Unidentified speaker: I agree with you about the subjective probabilities, we can leave that aside for the moment.

[00:20:53]  Sam: But my understanding of what David is trying to say there, and maybe regardless of what he's trying to say there, the thing that you might want to distinguish is that there are certain processes which you might call stochastic, where, for example, with some probability in a kind of classical pre-quantum theory sense, for example, a semi-silvered mirror will deflect or transmit a photon. And that theory is wrong in the sense that you can perform interference. So there's interference experiments where if you think that the thing is randomly deflecting the photon, then you're going to run into trouble. And my guess would be that that's what the talk is specifically about.

[00:21:53]  Bruce: Okay, and I would agree with that. We probably need to explain to a lay audience what we're talking about, but keep going for my answer and then back up and try to detail it more.

[00:22:03]  Sam: Yes, so I guess the intuitive explanation of this would be, if you assume that a photon hits this thing called a semi-silvered mirror or a beam splitter, then what happens in practice is it can be transmitted, so it goes straight through or it's deflected, maybe it's deflected upwards, and it does that with certain situations at 50-50 probability. So you can put a detector on either of these paths that the photon takes after it interacts with the beam splitter, and then you will find one of them clicks 50% of the time and the other one clicks the other 50% of the time. And then it seems kind of reasonable to assume, okay, the thing that it's doing is just like there is some microscopic physics going on that we don't really know. But what it's really telling us is, you know, maybe the photon hits a piece of the apparatus of the beam splitter, and then it is deflected upwards. And sometimes it doesn't hit anything and then it goes straight through. And that's just like by chance it does that. Then what happens if you have these things in sequence is that it doesn't keep randomly deflecting them. So you can have one beam splitter, have a photon go through, have them bounce on some mirrors so that they come together again, and then have another beam splitter. Now, if that were random, stochastic, or we have to have like a name for that, if that, I think stochastic is probably the best word.

[00:23:38]  Bruce: Let's call it classical stochasticity, because I would argue that quantum is stochasticity but it's clearly at least different than what you're talking about because of this interference pattern. So, yeah, if it were classically stochastic, you would expect that final beam splitter to, I can't remember the exact numbers, but doesn't it work out to just be 50-50 at the end?

[00:24:00]  Sam: Yeah, if it's classically stochastic, then again, like adding things in sequence which randomly transmit or reflect, we'll just give you a thing which randomly transmits or reflects again. So in the classical case, it's 50-50 again, but quantum mechanically, what happens is that they never go up. They never go up or you never see them going along one path. They always end up going straight through. They always end up being-

[00:24:32]  Unidentified speaker: Right.

[00:24:33]  Bruce: So let's call one path A and let's call one path B. In the classical case, you would get 50% class A path A and 50% path B, and in the quantum case, you get 100% path B and 0% path A. That's correct, right?

[00:24:52]  Unidentified speaker: Yeah.

[00:24:52]  Unidentified speaker: Due to that, that is the interference pattern.

[00:24:55]  Unidentified speaker: Yes, exactly.

[00:24:57]  Sam: So, what quantum mechanics adds is that the quantum mechanical explanation of what's going on during the experiment is not that there's one photon that's being randomly deflected. There is one photon instead, which turns into different instances, and they interact through interference. And it looks and behaves like classical probability when you're just looking at one of these mirrors. So, again, if you have to detect her, if you have one beam splitter, you put the photon through and you see whether it's deflected or transmitted, then that behaves randomly with the usual probability of 50-50 or whatever. And it's just that if you have them re-interfere, if you have a second beam splitter, then it doesn't behave like it would according to the classical theory. And that is because there are multiple photons, like there's one photon branches in two instances and travels along both paths and then re-interfers with itself at the end.

[00:26:08]  Bruce: So my introduction to quantum physics, I went through a textbook on quantum computation and I'm nowhere near Sam's level, which is why Sam's on the show and is answering my questions. Quantum computation for computer scientists, that was the name of the textbook that I went through.

[00:26:26]  Unidentified speaker: I don't remember who the author was.

[00:26:29]  Unidentified speaker: It was an excellent book.

[00:26:30]  Bruce: It's funny, I did it when I was younger. I was, I mean, yes, we're all universal explainers, but I was quicker back then. And you can see in the notes, I'm correcting, like they would have problems with answers in the back of the book. And I'm correcting the problems and the answers where they've done them wrong, right? Because I've understood it really well, and I'm starting to say, oh, they made a mistake here.

[00:26:55]  Unidentified speaker: This is a typo.

[00:26:56]  Bruce: And I go back as an older guy looking at that and I go, I don't think I would ever be able to do this now. Like my focus alone would be hard.

[00:27:04]  Unidentified speaker: It would be hard to even get through this book.

[00:27:07]  Bruce: So it's funny looking back over the margin notes in that book, because I feel like I'm like way stupider than I used to be when I look at it. Maybe I was just way more motivated back then to try to understand it. Despite that fact, I came away from, and I also studied some quantum physics reading Penrose's books. And in fact, the beam splitter example that you use is in his book. And I worked out the mathematics of it and the implications, particularly of the Elitzur–Vaidman bomb experiment, which he brings up in his books. And I put a blog post up, showing it and going through the math in detail so that I could discuss it with other people. It's still available out on millennialstar.org.

[00:27:53]  Unidentified speaker: So Penrose is great in that way.

[00:27:57]  Bruce: Like Penrose, I don't agree with anything he says ultimately, but he really gets into the weeds on mathematics. And he teaches you the math and teaches you the physics so that you will understand his argument. And well enough that I was able to say, oh, he's got his argument wrong, and here's where he's right wrong, right? Despite this, when you came on our show and you explained what C numbers were, I had never heard of them. Like I had made it all the way through quantum computation, and I had made it all the way through Penrose's books, and I had never heard of C numbers, which is basic to the way a quantum physicist would think. And that kind of makes sense. They don't want to introduce that to us computer scientists. They want to just give us a simplified version.

[00:28:46]  Unidentified speaker: They showed us how to put it into matrices and things like that instead, right?

[00:28:50]  Bruce: But that experiment you just described is one of the most basic experiments, the whole beam splitter with the silvered mirror. And absolutely the way you described it is completely correct, right? That prior to the actual observation, it does not follow a classical stochastic, what you would expect at all, which is actually meaningful because I have a physics textbook, or at least I used to have it, I don't know where it is now. I used to occasionally buy them on when they were used or whatever. One of them actually talks about how quantum physics is really just you lack information and it even gives the example of you hit something and it causes the atoms to change and that's why it seems stochastic to us. That's like literally wrong. That was what they were teaching people in college back whenever that textbook era came from. It is like literally wrong. It's known to be wrong. You absolutely cannot explain quantum physics in terms of stochastic probabilities every step of the way. Now, having said that, even that experiment ultimately obeys the probability calculus. It's true that everything prior to the observation follows the Schrödinger equation, which is where the interference pattern takes place. But at whatever point in that process, you decide to measure an observation, it will exactly... The Born Rule will determine what the probabilities are, and then you will observe it exactly according to those probabilities.

[00:30:30]  Bruce: If it's after the interference, where it comes out path B, the Born Rule is going to tell you, it's going to come out with path B with 100% probability, and it will exactly obey that. It will exactly obey that it comes out path B with 100% probability. Maybe a little bit of room for error if the experiment was set up wrong or something like that. One of the things is we sometimes do oversimplify this. Like if you're doing the double slit experiment, it could be that one of the slits is slightly larger than the other, so it's going to be an approximation in that sense, maybe, where we don't quite know the size of the slits. But in principle, we can make those slits finitely as close to each other as we want. It's just an engineering problem to be able to do it, right?

[00:31:16]  Sam: Yeah, that's not really anything that we should be concerned about. For example, again, I can't speak for David, but I assume that when he says that it is an approximation, he doesn't mean that it's an approximation because of the size of the slits or anything like that, that you could have. I think the real issue would be the contrast with the classical stochastic theory, which is, my reading of that talk is that you can use classical stochastic theories as an approximation to what quantum theory does. It just gets you into trouble during these kinds of experiments. I do agree with you that you can still, so also for the listeners, we should probably say what the Born Rule is. The Born Rule is a method of obtaining probabilities from the quantum description. And then also sometimes it is assumed as an axiom of quantum theory, where you say there are probabilities, and when you do a measurement, and then you don't specify what a measurement is in the classical and in the conventional way of explaining this, you say when you do a measurement, then there are probabilities and they adhere to this Born Rule, which is then like the Born Rule itself would be some piece of mathematics for how to obtain those probabilities from the quantum description. And the thing which David and others, David Wallace as well, who worked on this, have done is to say that, well, if you take the many-worlds view of quantum mechanics, you don't actually need that axiom. So you can scrap the axiom and supplement it instead with other things. We can get into that in a moment, I suppose.

[00:33:13]  Sam: But I think some of our disagreement is about this, what is sometimes called the decision-theoretic explanation of probability within the many-worlds interpretation. So to your point, I actually agree that at any particular instance, you can talk about probabilities in the sense of you can have a probability space and that comes from the Born rule. So the Born rule as a piece of mathematics is as a way of getting numbers from the quantum description. And I may even be getting ahead of myself now, but the interesting thing is that, again, in the conventional view, it's not clear what those, like why should those numbers which obey the probability calculus be anything of interest?

[00:34:08]  Unidentified speaker: Why should they really correspond to probabilities?

[00:34:10]  Sam: Why should we make choices with them?

[00:34:12]  Unidentified speaker: That kind of thing.

[00:34:14]  Sam: Anyway, I've said a whole bunch of stuff now. I don't know if this is helping you.

[00:34:19]  Bruce: This is a tough subject, and I know we're probably losing a lot of our audience members. That's okay. This podcast is intended to be a deep dive into a lot of these subjects, and we just trust our audience to come along with us for the ride to whatever degree that they care to. Don't worry too much that we're maybe under-explaining some of this stuff. Let me focus on one thing though, so my main concern is that once you do get to the Born Rule, it is not a flat earth approximation, at least not in the sense that at that point, the numbers are actually going to be quite precise. They're not an approximate number and then some other theory gives you the correct number like a flat earth projection of a catapult. It's literally once you do have an observation, however you define that, once you do have an observation, the Born Rule is going to give you an actual probability of what you're going to say and it's not an approximation.

[00:35:22]  Unidentified speaker: Do you agree with me on that much?

[00:35:25]  Unidentified speaker: Yes.

[00:35:26]  Unidentified speaker: Yeah.

[00:35:26]  Unidentified speaker: Yes.

[00:35:27]  Unidentified speaker: Okay.

[00:35:28]  Unidentified speaker: Yeah.

[00:35:29]  Sam: I mean, I would maybe even put more strongly. I would say that our whole understanding of probability derives from that.

[00:35:37]  Unidentified speaker: So you say we have a probability.

[00:35:40]  Unidentified speaker: I agree.

[00:35:40]  Unidentified speaker: Can you explain that?

[00:35:41]  Bruce: Because I tried to explain this in the podcast, where I went over the Deutsch slot machine. But maybe you can say it better than I can. It seems to me that quantum physics physically explains what probability really is.

[00:35:59]  Sam: So I think so as well. The way Paul Raymond-Robichaud had a nice explanation of this at some point. So he's a mathematician who also publishes on quantum theory, but he's trained in pure mathematics and computer science. And he said when he was doing probability theory in university, one of the things he wondered about is what does it mean for something to have probability half of what's happening? So, by the way, usually some people have got 50-50 probabilities, in which case they're expressed as percentages, but in math, typically we go from 0 to 1. So you say probability half of something happening is comparable to 50-50 in that case. But anyway, what does it mean for something to have probability half of happening? It doesn't say it definitely happens. It doesn't say it doesn't happen. It says it has the probability half of happening. In the real world, it seems like what we see around us is things either happening or not happening. That is also what we get from our classical description of the world. So you can have, for example, a description of the solar system and then either the sun on the Earth and the moon will be in particular positions at particular times or they won't be. Like they're either at position A or they're not at position A.

[00:37:30]  Unidentified speaker: They're either however many meters away from the sun, or the Earth is however many meters or kilometers away from the sun, or it is not.

[00:37:44]  Sam: There is no probability that it will be at some distribution or something, that there's only factual things, things that definitely happen. And the solution to that in quantum mechanics is that there is such a thing as something happening with indeterminacy. So, for example, you have cases where you perform a measurement of a particle and it's in a superposition of two states. So either the particle, for example, is...

[00:38:21]  Unidentified speaker: is not just...

[00:38:22]  Unidentified speaker: sorry.

[00:38:23]  Sam: There are cases where a particle is both at position A and at position B at the same time. Then you measure it, and you find that some of the time it's at position A, and some of the time it seems to be fully at position B. And in the Many Worlds interpretation of that means is just that you branched, and you saw the particle at position A in some branches, and at position B in the other branch. And it turns out that that is... You can use that to explain what probabilities are. So you can say, in the Many Worlds interpretation, in the Multiverse, the probabilities are related to these branches. They're not just related to the number of branches, but also to their thickness. So there is a kind of like a thickness of a branch, which the quantum description gives you. And the thickness of a branch determines how, quote unquote, likely it is that you end up there, you end up seeing that result. And that is what David Deutsch and David Wallace and others have proven. There's actually many different probability proofs. I would be probably aware of this. Like I am partial to the decision theoretic approach, but there are others. The nice thing about the decision theoretic approach is that, although the other, there are other people who have other derivations of probability in quantum mechanics.

[00:39:57]  Unidentified speaker: It's hard.

[00:39:57]  Unidentified speaker: You can kind of...

[00:39:59]  Sam: What's weird about those is that you can get a number out of quantum theory that obeys the probability calculus, but then it doesn't show you that you have to use that to, for example, make decisions with. So it's kind of like the decision theoretic approach has that nice feature where you can say, we don't need to assume, first of all, the Born rule.

[00:40:21]  Unidentified speaker: We can derive that as a result.

[00:40:22]  Sam: We can show that if you use the quantum description, then the thickness of branches basically corresponds to what you would think of as probabilities. And also, if you place bets and things and you have to make choices, that's the number you would use to, that's the number that a rational person would use.

[00:40:42]  Bruce: Because it represents the thickness of basically the proportions of the multiverse, right? So, let's go ahead.

[00:40:52]  Peter: Can I just ask maybe more of a big picture question? If that won't ruin the trajectory of this?

[00:41:00]  Unidentified speaker: No, it's probably good.

[00:41:03]  Peter: So, I've digested, obviously, I'm not in the weeds with this stuff like you guys are, but I've listened to Deutsch quite a bit on this. And I think he pokes some holes in probability theory in a really interesting way. It seems to be most of his comments are based, are from more of a physicist's perspective, and are poking holes more at the level of many worlds. The many worlds interpretation is what I think I get from his comments, but Bruce seems to be saying, what I get from Bruce's perspective is that he's saying, yes, this is true on one level of abstraction, but critical rationalism doesn't just deal with one level of abstraction, it's, you know, and one of the things that I like about critical rationalism is that it seems to really support folk epistemology or sort of more of a common sense way of viewing the world as well. And to me, a common sense way of viewing the world indicates that probability has a lot of uses all the time, that it's a really, it's an important part of, of logical thinking.

[00:42:31]  Unidentified speaker: So I guess, I guess my first, my question is first, does, is, is the arguments against probability all based on many worlds in your view, Sam, and does, doesn't probability have other uses, that other levels of abstraction that are worth considering?

[00:42:54]  Unidentified speaker: Yeah.

[00:42:54]  Unidentified speaker: So I, I think that probability as we use it day to day makes sense, but it makes sense because of the many worlds interpretation.

[00:43:04]  Sam: So for example-

[00:43:06]  Unidentified speaker: Yeah, you can mark that.

[00:43:07]  Bruce: I'm going to ask you more questions about that, but keep going. I don't want to break your train of thought here.

[00:43:10]  Sam: So it's, so say, for example, you know, there's a certain probability that lightning will strike me. It's going to be very small. It doesn't happen a lot. And in fact, lightning has never struck me in my life. But that is true. The reason that the statement, there is a probability that lightning will strike me or could strike me, is true, is because there are other branches where that happens. So that in the multiverse, there is a probability there is, there are branches with a certain thickness where things like that occur. So that kind of gets you around. So it's weird to think about this, because we have like an intuitive notion of what probability means. And that as soon as you start to unpack it in a classical worldview, which most people have, it doesn't really make sense. Like in the classical worldview, there is only, does or does not happen. There isn't, there's no, it will probably happen. All the cases where something will probably be happen are cases where you're forgetting the details of your description. So you're just leaving out that, for example, you can have stochastic processes in classical physics, but they're all based on saying, well, there is some underlying mechanism which is in fact not probabilistic, but we don't know exactly what that is, so we're just going to model it as being about probabilities. That's unsatisfying from a philosophical standpoint. So quantum theory gives you is, you can have a full description of a system, so quantum description is complete, and you still have probability.

[00:44:58]  Sam: So you still have an explanation of what probabilities are because of the multiverse. So that's, I think, one part of the answer. The other thing is, so that's basically saying, it actually fits in with common sense. We have common sense notions, which when we really think about it, don't make sense in our classical world view, and quantum mechanics offers a solution to that. The other point, I think, is that where Bruce might be, but I'm not even sure if we disagree about this. I think I probably agree with Bruce, in fact. What Bruce is saying is that there are still probabilities in quantum physics. It's just that they have an explanation. So you have an interference experiment. Okay, if you do the experiment, then yeah, if you don't measure it and you let the interference experiment do its thing, it's going to violate what you would expect to be the result if you believe in classical stochastic theories.

[00:45:58]  Unidentified speaker: But that doesn't matter.

[00:45:59]  Sam: There are still probabilities.

[00:46:00]  Unidentified speaker: It's not a classical stochastic theory.

[00:46:03]  Sam: But interference happens. And then after that, you still, you can still use the Born Rule to, for example, obtain the result that in the Mach–Zehnder that we discussed, all the photons go straight through instead of being deflected, it's going straight with 50-50 probabilities.

[00:46:25]  Unidentified speaker: So, I agree that there's still probabilities there.

[00:46:29]  Sam: And you can still use the Born Rule in that sense after measurement. So, I don't know, can I?

[00:46:38]  Peter: So, if say like, I don't know, one of these other theories, hidden variables or quantum Darwinism or other, I mean, maybe it's hard for you to imagine, and it's hard for any of us to imagine who are immersed in these theories. But say that many worlds does not pan out, suddenly it becomes clear that this isn't the right theory. Does probability then become more valid than you think about?

[00:47:14]  Unidentified speaker: I would say in that case probability becomes problematic again.

[00:47:17]  Unidentified speaker: So in my view, probability makes sense.

[00:47:20]  Unidentified speaker: I agree.

[00:47:21]  Unidentified speaker: Yeah, I agree with what you're saying here.

[00:47:24]  Sam: Yeah, so again, my view is roughly that we can understand probability. I would go so far as to say that our only understanding of probability really means physically is that there are multiple events being realized at once. So classical probability, even classical probability theory is already describing something like a multiverse really in hindsight.

[00:47:53]  Unidentified speaker: Yes, it really is.

[00:47:55]  Unidentified speaker: That's how I would view it.

[00:47:58]  Sam: That's how I would solve the problem of what classical probabilities theory really means. If it turns out that the quantum description is not correct, or that the many worlds interpretation more specifically, I mean, that's really what you're asking. If it turns out that the many worlds interpretation is not correct, and that this approach to probability, which is really saying probability corresponds to the thickness of branches in the multiverse is false, well, then which could happen?

[00:48:26]  Unidentified speaker: Logically, that's possible.

[00:48:28]  Sam: They don't expect it to be the case, but that could happen. Then what that would do for me is it would make problematic again, our entire understanding of what probabilities are to begin with.

[00:48:41]  Bruce: So let's put this in more explicit terms. Let's say that we discovered that, so we just heard Sam's explanation of what probability physically is. Probability is actually corresponds to the proportions of the thickness of the branches in the multiverse, okay?

[00:48:58]  Unidentified speaker: So that explains probability.

[00:49:00]  Bruce: Probability is now a physically real thing that has nice understandable mathematical explanation, okay? Suddenly we discover, oh my gosh, many worlds is wrong. Copenhagen is the correct interpretation of quantum physics. We have to amend quantum physics and Copenhagen turns out to be correct. Copenhagen would be collapse, where when you make an observation, all the different super positions disappear and only one remains at the end. What Sam is, if I'm understanding you correctly Sam, is saying is, I just solved a problem using many worlds. Throwing Copenhagen on and saying, well, no, actually, classical stochasticity is actually real, reopens the problem.

[00:49:45]  Unidentified speaker: It doesn't solve the problem.

[00:49:46]  Bruce: It makes the problem problematic again, right? Because now you no longer have an explanation of what probability actually is. Presumably, Copenhagen could never turn out to be right because it was never an explanation. You might more realistically imagine something that does do collapses, but unlike Copenhagen, explains what the collapse is.

[00:50:07]  Unidentified speaker: Now, that explanation-

[00:50:08]  Sam: objective collapse theories.

[00:50:09]  Bruce: Yeah. Part of that explanation would have to be a physical explanation of what probability is. Since we don't have such an explanation, I can't give you an example of what that would look like. You may know of some nascent theories, Sam, that you could explain.

[00:50:24]  Unidentified speaker: Maybe not.

[00:50:25]  Bruce: Maybe there aren't any. Maybe many worlds is really just the only explanation at this point for what probability physically is.

[00:50:32]  Sam: Yeah. Well, Penrose thinks that there could be things like objective collapse theories where things like gravity might make the wave function collapse dynamically. And then there's cute little models of that, which are not necessarily-

[00:50:50]  Bruce: That wouldn't really explain what probability is. We're just simply making an axiom of his theory. Is that correct?

[00:50:55]  Sam: Yeah, it would just mean that, like, he has it. So in his view, so again, if Penrose is right, that collapse can happen dynamically, that you can have an account of collapse where the other possible outcomes really disappear. So again, in the many-worlds view, they always remain there. They get hidden from view within a branch, but the other branch just exists simultaneously with their own. So the alternatives exist simultaneously. In these dynamical collapse models, what happens is the alternatives really disappear, but there is an underlying explanation for that, which is not just put in by hand, which again, I'm not persuaded by that, but you can make the argument. In that view, you would again have the issue of, okay, but then what are probabilities? Because previously, we had it as these outcomes of the measurement, and the thickness of branches specifically, and now that account disappears, so we're kind of left with the same question again.

[00:52:06]  Unidentified speaker: That's not to say that you couldn't have an answer, by the way.

[00:52:08]  Sam: Like, it's conceivable, I don't have an argument that you couldn't have an account, it just would make it problematic again.

[00:52:18]  Bruce: Yeah, you'd have to have a separate explanation, part of your explanation, you either would have to have a different explanation for what probability is, and we don't know what that would look like, because we don't have it today. Or you would have to have it as just an unexplained postulate or axiom of your system, basically.

[00:52:37]  Unidentified speaker: So that just doesn't have an explanation, which is satisfying.

[00:52:42]  Unidentified speaker: Yeah.

[00:52:43]  Unidentified speaker: Again, I think in the dynamical collapse models, that might not happen, it would not be a postulate necessarily.

[00:52:49]  Sam: I think the whole premise of those models would be to derive something like collapse from other assumptions, like, okay, we really have the force of gravity and that's really what's doing the collapsing of the wave function. So there would be an underlying reason, but you wouldn't like, again, if you, those models presuppose that there isn't, that the alternatives of the superposition disappear. And any model in which...

[00:53:15]  Bruce: But according to the Born rule's probabilities, right? So like if I've got...

[00:53:19]  Sam: I think so, yeah.

[00:53:21]  Unidentified speaker: Yeah.

[00:53:22]  Bruce: So we typically do 50-50, but let's say we do spin up, like one of Penrose's examples that comes from years ago, and I can't remember it very well. You've got like the spin up and spin down of a photon or an atom or something like that, and then you measure it at an angle. And that's actually going to now give you, let's say, 33% chance of spin up and 66% chance, that adds up to 99, but bear with me, of spin down, right? So I don't understand his collapse model, but under his collapse model, it's not the measurement that causes the collapse, it's that you reach some level of gravity that causes it to, the system has grown to the point where gravity causes it to collapse, but it's still probabilistic, right? Like it still says, and there's going to be a, because you measured it in this direction, it's going to be 30% this way and 70% this way, right? And it's the probability still isn't explained. Like there is an explanation for why the collapse happened, but why it happens to be 70, 30. Like, that's just one of the axioms of, I would think, I don't know this. I would think that would have to just be one of the axioms of his system, of his theory, that it just so happens that when the collapse happens, it's going to obey the following rules of the quantum state that when a collapse happens, it's with this probability or that probability. Am I wrong about that or is that a correct understanding of how-

[00:54:54]  Sam: I'm actually not entirely sure myself. Yeah, I'm not entirely sure myself. I would think that either you put it in by hand or what might happen is that it becomes deterministic again, so that you, like, gravity makes something collapse and makes the other options disappear, but then it does so deterministically so that you always end up, like basically the probability evolves to be one.

[00:55:20]  Unidentified speaker: I think that's what I would expect.

[00:55:23]  Bruce: No, no, okay, so in that case, probability goes back to being subjective, that there is actually a deterministic. If we had access to what was happening behind the scenes, there would be no probability, but because we don't have access to that information, as the gravity, whatever, evolves, it would eventually, that information would lead us to a probability of one at some point, it would be fully determined at that point.

[00:55:47]  Unidentified speaker: I think so, yes.

[00:55:48]  Unidentified speaker: That would be my guess.

[00:55:49]  Sam: I could be wrong about that, by the way. That would be my guess for what happens, because it has to be some kind of dynamical process which picks out, by virtue of being a collapse model, it has to pick out the results, and the system has to evolve to that particular result in an irreversible way.

[00:56:09]  Unidentified speaker: Yeah, so it's a little...

[00:56:11]  Sam: The result would be, yeah, what you just said.

[00:56:13]  Bruce: It's a little like Bohm, where he claims that... Certain aspects of Bohm claim that there is no actual probability, it's actually just that there's hidden information about the position of the atom. I mean, Bohm's weird, like this is... I'm probably doing a poor job of explaining it. I would require a whole podcast just to try to explain it. But certain aspects of Bohm, you observe as probability, but they're in fact just based on the fact that there's slight differences in where the atom is positioned. And under Bohm, it is impossible to know the information that you would need to be able to make the prediction, and therefore we treat it as probability. It's a subjective probability at that point.

[00:57:01]  Sam: Yeah, I think it mixes stuff as well.

[00:57:03]  Unidentified speaker: It does mix stuff.

[00:57:05]  Sam: Yeah, I think there's both—what you said is true, that they have the kind of the hidden variables which determine the position of the particle, which are not known to any observer. You can't know them in enough detail.

[00:57:15]  Bruce: It can't be known, right.

[00:57:17]  Sam: Yeah, and then, but also there is the wave function, which is still doing a lot of the, like basically all the heavy lifting.

[00:57:26]  Unidentified speaker: That's right.

[00:57:27]  Bruce: That's exactly how I understand Bohm, yeah.

[00:57:30]  Sam: Yeah, that's my understanding as well. So you still get some of the quantumness from the wave function and then you kind of insert by hand that there is a real particle and its position is not known and it interacts with the wave function and gives you some of those quantum results. And then, you know, some of the quantum probability stuff mixes with the subjective probability stuff and it gets to be a mess and...

[00:57:55]  Bruce: Yes, there's no explanation for why it does.

[00:57:57]  Unidentified speaker: It just does, right?

[00:57:58]  Unidentified speaker: Yeah.

[00:57:59]  Unidentified speaker: Yeah.

[00:58:00]  Unidentified speaker: Yeah.

[00:58:03]  Unidentified speaker: I haven't really thought about Bohm and probability much, but that is also my understanding.

[00:58:06]  Sam: I find Bohm unsatisfying for other reasons. But yeah, that is also...

[00:58:13]  Unidentified speaker: It's a strange issue.

[00:58:16]  Bruce: Let's go back to Peter's question because I feel like you very adequately answered Peter's question for me, but I'll bet you Peter doesn't know what the answer was. So, Peter, based on our discussion that we just had, can we try to answer your question again?

[00:58:31]  Unidentified speaker: Well, I mean, big picture.

[00:58:33]  Peter: What I'm getting, Sam, is that you actually do find probability theory, a meaningful and useful concept. So maybe there's some disagreement. I don't want to say disagreement, but a different emphasis than the way that David Deutsch talks about it.

[00:58:55]  Sam: Yeah, I think there might be disagreement between David and me. I'm not entirely sure.

[00:58:59]  Unidentified speaker: Part of it is that there's many things.

[00:59:01]  Sam: So in our way of talking about probability, we mix various different things. So we as physicists would talk about probability. I think David does mean stochastic theory. So classical stochastic theories in which what you're really doing is you're forgetting microscopic physics, you're kind of coarse graining and you're saying we don't care about that. And then instead we're going to model that as randomness. And that theory of probability is inadequate.

[00:59:36]  Unidentified speaker: I agree by the way.

[00:59:37]  Bruce: Just to be clear, I agree with Sam what Sam is saying. I don't think I've disagreed with anything Sam has said so far, but continue.

[00:59:44]  Sam: Cool, that's good.

[00:59:47]  Unidentified speaker: We'll disagree on this one too, of course.

[00:59:48]  Unidentified speaker: We've got to get some disagreement going.

[00:59:51]  Unidentified speaker: I'm going to.

[00:59:52]  Bruce: I'm going to bring up subjective probabilities and we'll likely disagree over that.

[00:59:55]  Unidentified speaker: But let's continue.

[00:59:57]  Sam: So I think one of the things to do in the discussion is to distinguish these different notions of probability. So there's probability just in the mathematical sense, that there's certain numbers where you have a probability space, you have an event space, you can have a measure which tells you, well, which is called the probability measure. And there's a mathematical formalism that exists kind of independently of physics, which mathematicians can speak about, and statisticians can speak about when they talk about probability. Then physicists will talk about actual models. So again, there's classical stochastic theories, which are kind of, those are approximative, they aren't literally true. There are quantum models, or rather there is just quantum mechanics, and it has a notion of probability, which is different from that of the classical stochastic models, and is related, again, to the mathematicians' models, which are generally abstract notions that come into play sometimes in physical systems, but might not. So, what I'm saying is these different ways of talking about probability, and depending on your angle, you have to kind of clarify what specifically you mean. And I think some of the disagreement might be coming from that, although I also expect there to be, still, like, residual actual disagreement. So, again, my view is roughly the classical stochastic models are genuinely approximative. They are not true. They can be useful, but the real thing that's doing all the work is quantum mechanics.

[01:01:47]  Sam: So quantum mechanics gives us a genuinely satisfying explanation of what probabilities are supposed to be or are. And they also show why the stochastic, the classical stochastic models are approximative. And then I don't know if I would still have a disagreement with David about that.

[01:02:13]  Unidentified speaker: It might be.

[01:02:15]  Sam: I think he might go further. But I'm not entirely sure.

[01:02:18]  Unidentified speaker: It might be.

[01:02:18]  Peter: That's kind of the tone I get from his comments, at least, is that he's not like, well, probability is valid, except when you're looking at the deepest theories of reality, it breaks down.

[01:02:34]  Unidentified speaker: Which honestly, I think that probably most people who believe in probability theory would, would actually agree with at some level, even if they haven't heard of many worlds.

[01:02:49]  Peter: So if you were to take a statistics class, it's not like you would be smirking in the back of the class, oh, none of this is really true or something like that, right?

[01:03:00]  Unidentified speaker: Yeah, it is.

[01:03:01]  Sam: What I would think is that there is a very strange connection between, well, a statistics class and what they teach and the physics that underlies it. And that was, I took a statistics class at some point, and I think I only attended a couple of lectures and I dropped it. I wasn't so interested in it. But one of the things I found interesting was that very often, they would mix physical models in with the mathematics.

[01:03:34]  Unidentified speaker: Again, you can have a completely abstract mathematical notion of what probabilities are, and it doesn't have to map onto anything physical.

[01:03:41]  Sam: It's just treated as a bit of logic. Like you can say, well, there's mathematical spaces, there's functions on those spaces, and we call a collection of those things we call a probability space.

[01:03:55]  Unidentified speaker: And then they would talk about models.

[01:03:58]  Sam: They would try to make the thing intuitive and say, well, assume that we're talking about a hospital, we're doing medical experiments and things like that. And let's derive some statistical results which are used in real life. I thought at that point, I did have this kind of gut feeling of you're actually doing physics now. Like you're no longer doing just pure math, but you are making substantial assumptions and it's confusing because you're not really, you've crossed over into the real world now. We're no longer just doing abstract mathematics, we're trying to model real things. And you haven't told us this and I find that slightly annoying.

[01:04:44]  Unidentified speaker: So because-

[01:04:44]  Unidentified speaker: I agree by the way.

[01:04:45]  Unidentified speaker: I agree.

[01:04:46]  Bruce: I think that's exactly how they teach it and I think it's confusing.

[01:04:49]  Sam: Yeah.

[01:04:50]  Bruce: I think it's confusing because they're confused. Like they don't understand this either. So they're trying to explain it the way it's been explained to them. But like, there's long been mysteries as to what is probability. And I really feel like there is such a thing as probability. It's physically real. It's explained by the multiverse. It's explained by branches of the multiverse. And it makes it so much easier to wrap your mind around what a probability is. At least in those cases, there's the subjective cases that are harder. But in those cases, if I have a single beam splitter, and I have a half silvered mirror, and then I put a detector on one of the two paths, or both paths, I am with 50% probability, I'm going to have it go down either path A or path B. And that's a physically real probability, because I have set up the experiment to force the branches to be equal thickness, right?

[01:05:49]  Peter: So it sounds like what you're saying is that many worlds actually makes probability more true rather than less.

[01:05:55]  Unidentified speaker: It does, that's—

[01:05:56]  Unidentified speaker: Well, that's what I...

[01:05:57]  Bruce: So this is why I've kind of gotten into this and why Sam's here to help me out. When I go through David Deutsch, what he says, what he says in this video, he's taking a tone that's negative towards probability. But I'm coming away with it feeling like, actually, this explains what probability is and why it's real, right? And so I'm coming away with this positive view of probability, oh, it's physically real, I need to take it seriously, based on the exact same argument, where clearly he's intending a more negative tone. And it's not just that he's intending the tone, like I can't mind read, maybe we're reading too much in. But I know for a fact that his followers that have pointed me to that video are taking it to extremes that are not true, right? Where they're making just wild claims about probability that just are not true at all. And based on the fact that they're clearly reading him as taking a negative tone and they're kind of grabbing the negative vibe and they're kind of running with it, right? And trying to do with it what they think he intends. And yet they're coming away with something like the opposite of what I think the truth actually is.

[01:07:11]  Unidentified speaker: Maybe an analogy will help, which is that we also have it in physics and science generally, that you have these old concepts that you work with, and at some point they get replaced by a new thing.

[01:07:20]  Sam: And so intuitive example is Newtonian gravity. So we all get told there is a force of gravity, it's pulling us, for example, towards the center of the earth, and it's what keeps the earth in its orbit around the sun and all those things. And then at some point, you'll see a lecture about general relativity, and say. There is no gravity, there's no force of gravity, and that is literally true. It turns out gravity isn't a force. What we're experiencing is the curvature of space-time. It's the curvature of space-time. And you might say, well, there is no gravity, but that's not true. But it's true that there is no force of gravity, and we have to distinguish these things. So it turns out this label gravity, which we think is a force, is not a force at all. Gravity is real, but the force of gravity is a fiction. And I think that's somewhat similar here, where we have our initial explanation, where the probabilities are kind of like the force of gravity.

[01:08:25]  Unidentified speaker: So the stochastic theory is like this idea that there is this thing in the world, which our new explanation says doesn't exist, and then explains more deeply.

[01:08:37]  Sam: So just as the force of gravity has been gotten rid of and is replaced by spacetime curvature, so too the original kind of classical view of probability has been superseded by the quantum one.

[01:08:53]  Unidentified speaker: And then we have to be careful about what we mean.

[01:08:55]  Unidentified speaker: I like the way you just said that. I like the way you just said that, by the way, Sam.

[01:08:58]  Unidentified speaker: So thank you.

[01:08:59]  Unidentified speaker: Continue.

[01:08:59]  Unidentified speaker: Sorry.

[01:09:00]  Unidentified speaker: Cool.

[01:09:00]  Unidentified speaker: Yeah.

[01:09:02]  Unidentified speaker: Thanks.

[01:09:02]  Unidentified speaker: Cheers.

[01:09:03]  Unidentified speaker: Yeah.

[01:09:04]  Unidentified speaker: So I hope it's, I know, I hope it's kind of useful.

[01:09:07]  Unidentified speaker: Yeah.

[01:09:08]  Sam: Maybe it's an intuition pump or something like that, where you can say, there is this thing which we thought existed, and it turns out it doesn't. But again, one example being the force of gravity. And then we have an explanation that supersedes the original one, which still explains what we want to explain. So we still call it gravity, but it's no longer a force.

[01:09:35]  Unidentified speaker: It's something entirely different.

[01:09:36]  Sam: And so too, probability is not, like our classical understanding of probability is false. We have to get rid of it. It's, it is true that that is just approximate. And it's superseded by the quantum explanation of probability. And I think Bruce and I can agree that that is like a fully satisfying, philosophically satisfying view of what probability is. And now we have to distinguish these two things, like the thing that replaced our original view and our original view.

[01:10:04]  Unidentified speaker: And the evolving view, right?

[01:10:06]  Bruce: Like, yeah, we have this evolving view of gravity. And you could think of it as gravity is false. And now we have this new thing that we call gravity. Or you could think of it as our view of gravity has changed, right? We've understand it more deeply than we did before. Let me actually throw in a monkey wrench to what you just said, though. Just, I'm not actually disagreeing with you. I'm trying to show the difficulty of speaking about these things. I would argue that there is still, under Einstein's theory, a force of gravity, right? So it's true that our concept of a force evolved over time, to where we had this idea of a force field, something like electromagnetic field. And gravity is not a force in the sense of a force field in Einstein's theory. It's still a force in the casual sense of the word force, right? It's a common sense of it's something that's pulling you towards it, and that's how you experience it.

[01:11:07]  Unidentified speaker: It's still that.

[01:11:08]  Bruce: It's still a force that draws you towards the planet, right? It depends even on what you mean by the word force. Now, you could argue that the original meaning of the word force was more like the force field, and when you say, I mean gravity is not a force, I'm striking down the original meaning of the word force. I would go one step further and say, well, actually, when Newton came up with the word force, there was no force fields, like they didn't exist yet. That was later physics that came along and changed the way we thought of the word force. So I'm not even sure if it's strictly true that we've proven that gravity was not a force in the original Newtonian sense. More likely, nobody gave it much thought. That would be the more likely explanation as to what a force was. It just wasn't part of Newton's thinking just yet. We hadn't gone that deep yet.

[01:12:03]  Unidentified speaker: Thoughts on that?

[01:12:04]  Bruce: Am I like totally off base there?

[01:12:06]  Unidentified speaker: Or can you—[unclear]?

[01:12:07]  Unidentified speaker: [Yeah, I think they all disagree?]

[01:12:09]  Unidentified speaker: So there, I think the way to think about that is that it's a force in the same way that if you're in a car and the car suddenly turns and you feel yourself being pressed against.

[01:12:22]  Unidentified speaker: Yep.

[01:12:23]  Unidentified speaker: It's a force in exactly that sense.

[01:12:25]  Unidentified speaker: Yeah.

[01:12:26]  Sam: So it's a pseudo force which is frame dependent. So if you're in the frame of the car, which as a passenger, you tend to be, then it looks like what's happening is you're being pushed against. So the car is turning, what is it, right? It feels like you're being pushed against the car door when you're sitting on the left and the car is turning. And it turns out the only thing that's really happening is that it's in an inertial frame. So in that case, because the car is turning and you're inside of it, the frame is not inertial. In an inertial frame, there is no force. It's just you're trying to go ahead in a straight line and the car is turning, so it has to force you to go with it. And so you can say it's a pseudo force. I think that would be true. But it's not like a fundamental force of nature, where even if you're in an inertial frame...

[01:13:27]  Bruce: It's not a fundamental force of nature, yeah. So what I'm really getting at here, though, is that the word force doesn't necessarily come prepackaged in some original form that had to be the physics force of nature, fundamental force of nature force. The word pre-exists any of our understandings of physics, right? And so when we talk about a centrifugal force, every physics high school teacher says, there's no such thing as a centrifugal force. We all know there is. We're just talking about force in a much looser sense, right? If you're trying to think about fundamental forces of nature, you're right, there's no such thing as a centrifugal force. If you're trying to talk about it in the loose way that the word force gets used by real people, centrifugal force is a real thing, right? And you can use it and you can use it to do things like it exists.

[01:14:22]  Unidentified speaker: It's an emergent property, though.

[01:14:24]  Bruce: And I think the force of gravity is the same way. When somebody says, you know, there is no force of gravity, I always read them charitably. I read them as meaning there's no fundamental force of gravity. But if somebody says there is a force of gravity, I also read them charitably. And I think, I assume they mean I experience a force of gravity, right, and in the same way I can experience a centrifugal force. I think both are true statements insofar as they go, right? It's not that there's a definitive, you must speak of the word force in this way, and that's the only way you can speak of it.

[01:14:56]  Unidentified speaker: It's a looser term.

[01:14:58]  Bruce: It means different things in different circumstances, depending on how you're using the term.

[01:15:04]  Unidentified speaker: Yeah.

[01:15:04]  Sam: Well, that's also, I mean, that's kind of why we're getting confused partly about the probability issue as well, is we have terms which apply in both cases. So for example, if I say force, I generally mean like a fundamental force of nature, something like a force that isn't just transformed away when you're going to an inertial frame. You can clarify, you can say, well, as we just did, you can say, well, I'm really—the thing I call a force, the thing I want to talk about when I say gravity is a force, is it a fundamental force of nature, like Newton has to suppose it is, or is it something that you can explain as a pseudo force that arises from being in a non-inertial frame due to space-time curvature. So you can disambiguate that way. Similarly, there are several things which we can call probability which are not the same thing. When we say, well, at least when I say, the stochastic, the classical stochastic model is false.

[01:16:21]  Unidentified speaker: Rather, I was going to say, if I say probability is false, I mean specifically, I would have in mind the classical stochastic model.

[01:16:29]  Unidentified speaker: That's a good clarification.

[01:16:30]  Unidentified speaker: Yeah.

[01:16:32]  Sam: So yeah, there are issues around the way we talk about this, and they exist in other cases in physics as well, where you have new theories that superseded old ones, and we still have these remnants of concepts in old ones that we have to disambiguate in the current ones.

[01:16:47]  Bruce: So I used the example of gravity, which is probably not a perfect example of what I have in mind. I think with probability, it's far less clear that the word probability ever intended what you're calling the classical statistical theory. The word probability has this long history, which I'm actually planning to cover in an upcoming podcast, because I read a book about its history, which I thought was really interesting. It was a muddle. Like, it was truly— David, in the video, he points out that they originally were just trying to do mathematics on what they thought were underlying deterministic processes and they had no concept of physical stochasticity in mind. That's true, although you're jumping into the middle of a history that predates it by a thousand years or something. He's picking a certain point in history. If you were to go back further than that, it gets even more muddled. At one point, probable meant it came from somebody I can trust. It literally had this meaning that's so divorced from the way we're now using it today. And the word really just has evolved over time and branched, and different people use it in different ways. And when you say, like, if David said what you just said, and he said, probability is not real, and what I mean is that the kind of classical statistical idea of stochasticity at a classical level is not real. Like, yeah, I totally agree that's true. Like, that doesn't even begin to bounce my radar at all as problematic. It was really the flat earth approximation.

[01:18:28]  Bruce: And here, I think, probably, especially based on what you're saying, but I do mention this in my video before I even talked with you. Likely what he has in mind is the kind of classical stochastic idea that there is a case where it gets you a wrong thing. If you were to assume an atom bounces in a classical way. And to me, that's physics, not probability, but like I can see what he's talking about there. And in that case, yes, it's more like a flat earth approximation.

[01:19:02]  Sam: Again, I can't speak for David, but that would be my reading of it.

[01:19:05]  Bruce: That would be the reading too.

[01:19:06]  Unidentified speaker: Okay.

[01:19:08]  Bruce: That makes sense to me, right? Like it is a flat earth approximation in that case. The thing that really threw me off was that the numbers are exact, exact as they can be to the degree to which we can measure them anyhow. Like when I do the beam splitter and I'm going to do 50 percent path A and 50 percent path B, it will be exactly to whatever degree we have the ability to engineer it, it will be exactly 50-50, right? If we want to change that to be 90-30 or something like that, what does that add up to?

[01:19:41]  Unidentified speaker: 90-10?

[01:19:43]  Unidentified speaker: We can do that.

[01:19:44]  Bruce: We can get whatever probability we want, and it's going to be an exact probability, and it's going to fit the probability calculus exactly because it's a physically real probability at this point.

[01:19:55]  Sam: People actually use it.

[01:19:56]  Unidentified speaker: Yes.

[01:19:57]  Sam: I was listening to Peter and you talk on a podcast that you released I think earlier today, and Peter asked the question to Bayesian, which I've also often wondered about, how often do you actually use Bayesianism? And actually go through the effort of calculating numbers. And here, in quantum mechanics, you actually use the numbers. You actually generate specific numbers, which will be predictions for measurement outcomes and things. Unlike when people try to use subjective theories of probability, in which case the numbers are kind of like window-dressing. And you say, well, there is a formalism that we can refer to. This is maybe a segue into our understanding of subjective theories of probability, which I think are very problematic.

[01:20:47]  Bruce: Before we move on to that, let me actually... I came across a tweet from David Deutsch. This was after I made the videos. And in many ways, it clarifies what he's saying. Although it also clarifies why I'm confused by what he's saying, okay? So he says, more on the Monty Hall problem, and he gives a link.

[01:21:08]  Unidentified speaker: I'll read the...

[01:21:09]  Bruce: it's short, I'll read what the link says. And he says, I still disagree. The problem, Monty Hall problem, the problem has nothing to do with probability nor, therefore, with Bayes' theorem.

[01:21:18]  Unidentified speaker: Okay.

[01:21:19]  Bruce: So the link that he then links to at the end, it's a way of understanding... So the Monty Hall problem is this idea that you have Monty Hall from Let's Make a Deal, and it's an old game show, and there would be like a bunch of curtains, and if you pick the right curtain, you would get a car. Otherwise, you might end up getting something silly like a goat, right? So in the Monty Hall problem, you've got three curtains or three boxes, let's say, and you get to pick one. And then Monty Hall, he's the game show host, he opens one of the boxes you didn't pick. And then the question is, do you stick with your original guess, or do you switch to the remaining box? And people intuitively seem to think that there's no, it makes no difference, right? Because you had a one in three chance and that other box still has a one in three chance. When you actually work this out using probability theory or decision theory, whichever one you're going to use here, it turns out that you want to switch, right? Like it is always to your advantage to switch to the remaining box.

[01:22:32]  Unidentified speaker: And people just, it blows people's minds.

[01:22:33]  Unidentified speaker: There's statistician.

[01:22:34]  Bruce: So what was the lady's name who kind of popularized this, the highest IQ in the world lady, Marilyn?

[01:22:42]  Unidentified speaker: I don't know.

[01:22:44]  Unidentified speaker: I can't remember her name either.

[01:22:45]  Sam: I know who you're talking about.

[01:22:46]  Bruce: Marilyn is the name, but it's like Savant or something like that. Like she probably changed her last name. But she was Guinness Book of World Records' highest IQ. Someone asked her this. She got the right answer, and then she got hate mail for years afterwards, with statisticians telling her that she was stupid and she didn't understand. It popularized the Monty Hall problem.

[01:23:07]  Peter: The key thing that people don't hone in on with that is that the Monty Hall knows where the goat is. If he didn't know where it was, then it wouldn't affect the probability, but he can, you know, it's going, if he knows that there's going to be, there's a 100% chance that the goat is in one of the two remaining doors. So if he opens up a door and shows a goat, then it doesn't affect, it doesn't change the probability at all.

[01:23:40]  Unidentified speaker: Right?

[01:23:41]  Unidentified speaker: That's how I explain it to people, at least.

[01:23:43]  Unidentified speaker: Yeah.

[01:23:44]  Sam: Yeah, it's crucial that he has information.

[01:23:46]  Unidentified speaker: Yeah.

[01:23:47]  Unidentified speaker: So here's Deutsch's version of this.

[01:23:50]  Bruce: Consider a different problem first. You're faced with the same three boxes, but now you can choose any one box or any two boxes. And in the later case, receive the better of the two contents. It's always better to choose two boxes, right? But the rules of the original game allow you to choose two.

[01:24:07]  Unidentified speaker: Here's how.

[01:24:08]  Bruce: First point to the remaining box, the one you're not going to choose, then Monty will open the worse of the two boxes you choose and you take the better one.

[01:24:17]  Unidentified speaker: Okay, that's true.

[01:24:18]  Bruce: That's a correct understanding of Monty Hall problem. I should probably note that the Monty Hall problem is open to interpretation. And those that were arguing with Marilyn over this, you could argue some of them were simply interpreting the problem differently. There are scenarios in which switching makes no sense because they're starting with the assumption that Monty doesn't know anything or something to that effect, right? It's actually a little bit complicated because of that. But most people seem to agree that Marilyn's answer, based on the way it was framed, hers was the right answer.

[01:24:51]  Unidentified speaker: Now, here's the thing.

[01:24:54]  Bruce: I understand exactly what Deutsch is saying here, like I do. Like this makes perfect sense, you're picking two boxes instead of one. Okay, so of course, that's the better answer. I'm having a problem with the way he worded it. The problem has nothing to do with probability.

[01:25:11]  Unidentified speaker: Like, I don't think that's a true statement.

[01:25:14]  Bruce: So I put up on X that I'm very confused by the way he's saying this.

[01:25:19]  Unidentified speaker: Can someone please explain this to me?

[01:25:21]  Bruce: I have a number of knowledgeable crit rats show up on X and explain it to me. Here is part of my explanation to them why I'm confused. I say, in his talk, Physics Without Probability, Deutsch concedes that probability theory can still function as a useful approximation. Why then is the Monty Hall problem not counted as one of these cases? Probability theory describes a problem correctly and yields a precise solution in this case. It is not merely heuristic here, it works exactly. So even on Deutsch's own view, shouldn't this be admitted as a situation in which probabilistic reasoning is at least approximately useful? Given this, it's puzzling to claim that the Monty Hall problem has, quote, nothing to do with probability. Ordinarily, that phrase would be taken to mean that the problem cannot be solved using probability theory at all, which is plainly false. Presumably, what Deutsch really means is something weaker, that the problem can be solved without appealing to probability calculus, a fact that is true but largely unremarkable since all probability problems admit multiple formalisms.

[01:26:25]  Unidentified speaker: So that was what I said at this point.

[01:26:28]  Bruce: I then get pushback from a number of crit rats, we go back and forth. It seems like after a considerable discussion, we did arrive at a consensus that when Deutsch says, the problem has nothing to do with probability, he really means it's something like this. Yes, the problem requires knowledge of probability in the setup, and yes, this problem could also be solved as a probabilistic decision problem. But it can also, after this initial setup, at least, be solved as a non-probabilistic decision problem. Which, by the way, go look this up, it can be solved as a non-probabilistic decision problem. We won't get into what the difference is, but what a decision problem is, and what the difference is between a probabilistic one and a non-probabilistic one. But based on my research from talking with these crit rats, yes, you can solve this problem as a non-probabilistic decision problem.

[01:27:24]  Unidentified speaker: Yeah.

[01:27:25]  Unidentified speaker: So how do you solve it using?

[01:27:26]  Sam: I'm kind of confused both ways.

[01:27:28]  Unidentified speaker: I agree that I can imagine a quantum way of thinking about this, but that feels somewhat artificial.

[01:27:37]  Sam: So how do you do it using just the non-probabilistic decision theory?

[01:27:44]  Bruce: So Deutsch's answer is the non-probabilistic decision theory.

[01:27:48]  Sam: So is that the quantum version? Is that saying you can?

[01:27:51]  Bruce: Yeah, so none of this has to do with quantum, and the quantum complicates it. So let's come back to that as a transition to the discussion on subjective probabilities, because we are probably talking subjective probabilities here, not quantum probabilities here. But this is his, this is Deutsch's example, right? So basically, and this is what all the crit rats would say to me, the knowledgeable ones, like obviously, a lot of them had no clue what was being said here. But there were two in particular, who I won't name, but they're like super smart and they knew what they were talking about. They pointed out that the way Deutsch is framing this, there's no probability. You pick two boxes or you pick one box.

[01:28:31]  Unidentified speaker: Which do you pick?

[01:28:33]  Bruce: That's a fair decision problem. And because it's a fair decision problem, and you didn't have to use probability calculus, you didn't have to use expected values to be able to make the decision, that is the framing of the problem as a non-probabilistic decision problem.

[01:28:49]  Unidentified speaker: Yeah, okay.

[01:28:50]  Sam: So, yeah, I see what you mean.

[01:28:52]  Unidentified speaker: Okay.

[01:28:53]  Bruce: Can you do this as a probabilistic decision problem?

[01:28:55]  Unidentified speaker: Absolutely.

[01:28:57]  Bruce: You could just say expected value of one-third for each, and then you could work out the expected value of winning based on that, using a normal probability theory, and you would get exactly the same answer.

[01:29:08]  Unidentified speaker: Okay.

[01:29:10]  Bruce: After going back and forth, everyone eventually admitted to me, yes, when you see Deutsch say, it has nothing to do with probability, you have to assume that what he really means is, you don't need to make it a probabilistic decision problem. And if that's what you mean, then I agree with him. If the phrase nothing to do with probability only really means not a probabilistic decision problem, then with that interpretive guide, I can agree with what he originally said. I would assume most of them did admit they can see why I was confused though, because that is a weird way to say that. Furthermore, there's still probabilities involved. Like even as the non-decision, non-probabilistic decision problem, that really just means we remove the expected values. There's still an assumption of equal probability here. If you were to take a child, and the child goes, I don't get it, why should you pick two boxes over one? They just don't know, because they're just a child. You would have to explain to them at this point, well, there's an equal chance that it's behind any of the boxes, and you would then be introducing probability even into the non-probabilistic decision problem, because it's at the heart of the initial setup of the problem. To see that this has to be true, the easiest way to do it is to imagine that you happen to know that they don't tend to put the car behind box C, that they only put it behind box C one out of a thousand episodes, or something like that, right? Then Deutsch's formulation breaks down, because it was assuming an equal probability.

[01:30:48]  Bruce: It was implicit, he didn't state it, but it was there. It was still part of the formulation. The reason why I was getting confused then is because I could immediately see that probability played a part of the explanation behind how we were doing this, even though we're now formulating it as a non-probabilistic decision problem, based on certain starting assumptions that are probabilistic. And I think that may explain a lot. Like, when I read that, I thought, oh, that's probably like, I can see how in his talk, maybe he's saying, we don't, you know, physics without probability, we don't need probability, but he may really just be meaning, we don't need a probabilistic decision theory. Because all problems, at least in principle, all quantum probabilities, if you were God and you could see the thickness of the branches, you would never need to work out expected values. You could just simply look at which branch was the thickest and go with that one as a non-probabilistic decision problem. I've wondered if that's really what he meant. You're kind of interpreting—the way you're interpreting it is slightly different, but it's very similar, that he's really just saying, we don't need probability to make decisions, at least in principle, if you had all the information available to a God's-eye view of the multiverse.

[01:32:04]  Sam: Yeah, or even if you have a theory of the branch weights, so if you have...

[01:32:12]  Unidentified speaker: Yes, the quantum state, yeah.

[01:32:15]  Sam: Yeah, you could say it is available to... What you say about the God's-eye point of view is true, but even for us, we can have theories about the branch weights, and they can be very accurate, and we can use those for making both predictions and also we can incorporate those into decision making. So I agree with you that the basic result of the decision theoretic approach is you do classical decision theory, you scrap the assumption that there are classical probabilities, and then instead you insert quantum mechanics into the formalism, and classical decision theory plus probabilities gives the same results as classical decision theory without probabilities, but assuming quantum mechanics is true.

[01:33:07]  Unidentified speaker: Okay.

[01:33:08]  Bruce: This then does lead us to the discussion on subjective probabilities, which I should probably warn the audience is a wild and woolly subject that people much, much smarter than me, maybe as smart as Sam, but definitely smarter than me, have been debating for centuries and have come to no conclusions. We aren't probably going to solve the problem of subjective probabilities today, but I have been studying up on this, and I do have an opinion on the subject that I would like to run past Sam, get his feedback on. If we disagree, which we may or we may not, I would like to maybe argue that back and forth with you a little bit.

[01:33:52]  Sam: I should say, I appreciate [the flattery?], of course. I've not worked that much on probability, but really, I've not published on that, so I'm also just kind of thinking. I have ideas about this and I want to discuss it.

[01:34:07]  Unidentified speaker: I don't want to give people the impression that I am—

[01:34:10]  Unidentified speaker: You're [a world-leading expert on probabilities?]

[01:34:13]  Unidentified speaker: Yeah, yeah. I'm not a world-leading expert on probabilities either, but I do have thoughts.

[01:34:17]  Unidentified speaker: Fair enough.

[01:34:19]  Bruce: So let me just say that this is something that has cropped up from different paths. So you have the divide between frequentists and Bayesians. Frequentists claim they believe in objective probabilities. Bayesians say they believe in subjective probabilities. You've also got this from the discussion we just had. We had this idea of quantum physics being a physically real probability. And then you've got pseudo-randomness, which does not split the multiverse. If I'm running a computer and I'm doing a, let's say, poker or something like that, but it's doing it based on a pseudo-random number generator, there are no branches of the multiverse, even though I'm still using probability theory. So you've got a similar split. It's not the same split, by the way. Like I don't mean to imply one's frequentist and one's Bayesian, but you can see how there's this philosophical divide between two things, and it's really unclear what the two things are.

[01:35:19]  Unidentified speaker: One of them is more clear.

[01:35:20]  Bruce: I feel like, particularly if you buy quantum physics and you buy the many-worlds interpretation, I feel like the idea of a physically real objective probability gets real meaning at this point, for the reasons that we've outlined. In a quantum experiment in particular, it's a really well-defined meaning as to what probability is. But in real life, when we use probability theory, we probably don't in general use it on quantum experiments. We probably use it on things like die rolls or coin flips. It's not even clear to me if those are quantum or not or to what degree they are quantum. I asked David Deutsch this in my interview with him, and he kind of admitted he didn't know either. He thought they were probably quantum. But then he later sent me a paper about how coin flips were deterministic.

[01:36:11]  Sam: Yeah, I also read that.

[01:36:13]  Unidentified speaker: Yeah.

[01:36:13]  Unidentified speaker: Okay.

[01:36:14]  Unidentified speaker: Interesting.

[01:36:15]  Unidentified speaker: Yeah.

[01:36:16]  Sam: Coin flips are just, sorry, so coin flips are just, we don't know the initial conditions and the process is deterministic. But the answer seems like it's all the randomness that we apparently see in the coin flip is just due to us not knowing the initial conditions of the coin. That's my understanding as well.

[01:36:36]  Unidentified speaker: That's from the paper.

[01:36:37]  Bruce: That's my understanding also.

[01:36:39]  Unidentified speaker: Okay.

[01:36:40]  Bruce: So I can set up the idea here is, is that I can set up cases where I would use probability theory. In fact, I have to use probability theory to make a valid rational decision. And yet, I know for sure, I either know for sure, there is no branching in the multiverse in the case of the computer, or I've got good reason to believe there is no branching in the multiverse in the case of the coin flip, or in the case of the die roll, there may be branching in the multiverse because there's little quantum effects that affect where the die is going to land, but it may not be equal probability. Like, I would use equal probability because there's also an issue of initial conditions with a die roll, right? And then we could even go further. We could talk about, this is one I bring up on my podcast, the probability of an asteroid hitting the Earth. Presumably, that's a meaningful number that I could give you on that. I think we worked it out to points, bunch of zeros too or something like that. It was one in 50,000 per year that is the probability being hit by an asteroid. We're basing that solely on past frequencies and on an assumption that may not be true, that forward frequencies are going to be equivalent to past frequencies, okay? That number still strikes me as meaningful, but there's no way it splits the multiverse, right? Because the quantum effects on an asteroid should be so minimal. Maybe I'm wrong about that.

[01:38:06]  Unidentified speaker: If I am, that's fine.

[01:38:08]  Bruce: So we do seem to have two concepts of probability here. For the sake of just wanting to discuss them, but not wanting to confuse things too much, let's call one of them real probabilities, that's the quantum stuff, and let's call the other one subjective probabilities, which may or may not be the same as the Bayesian concept of subjective probabilities. We can discuss separately whether it's the same as the Bayesian concept.

[01:38:32]  Unidentified speaker: Yeah.

[01:38:32]  Unidentified speaker: So one thing to flag immediately is that even the sub, but we're now calling, how do you call them?

[01:38:41]  Unidentified speaker: They're not real ones, we're calling subjective.

[01:38:44]  Unidentified speaker: Subjective, yeah.

[01:38:45]  Bruce: We'll call them subjective.

[01:38:46]  Unidentified speaker: Yeah.

[01:38:47]  Sam: So even those are still based on physical theories. So we're still assuming, so one of the problems I have with the Bayesian approach which we may or may not get into is that we, but for me, probabilities make sense if you have a physical theory, which you can then use to explain where the numbers are coming from, essentially. Like you can say, well, okay, for example, there's asteroids. Well, there is some distribution of asteroids that we don't know, and we can suppose that their rate of hitting the Earth is constant over time because probably the distribution will be the same roughly over time. So the number of them that are hitting us will be similar.

[01:39:38]  Unidentified speaker: Things like that, things where you can say, well, we have like an underlying explanatory framework for why we make those assumptions.

[01:39:43]  Sam: And they could be wrong, and it could be that. There, for some reason, maybe there's going to be lots of asteroids or there's going to be very few of them just because of the distribution of asteroids being not entirely random, things like that. So, so still, that's just the thing to flag, that we're still within the realm of having underlying physical assumptions about the world, which then determine the kind of probability theory that we're setting up, which then brings us on to the next issue of like, what do those numbers actually mean? Because now, like in the case of the asteroids, in the case of the coin flip, there doesn't seem to be an underlying quantum theory of it. And I don't have, like, so for me, this is a genuine problem.

[01:40:34]  Unidentified speaker: I have guesses as to what it might mean.

[01:40:36]  Sam: For example, the best explanation I've heard in terms of the decision theoretic approach is that a rational agent would be indifferent between a quantum game and a classical game in which they don't know these, the distribution of, for example, or if you're playing cards, they don't know how the cards are going to end up, or if they're doing a coin flip, they don't know the initial conditions of the coin. So it might, like, they would be indifferent between using a coin flip and having a, observing a quantum particle in one of two positions and using that instead. So I can imagine that there would be a way of getting the classical probability, so the subjective probabilities from an argument that uses the quantum case, by basically saying, there is a quantum case, there's a quantum analog of what you're talking about. So there's a coin flip, which is deterministic. It's all about us not knowing the initial conditions. There's a quantum analog, which would be, say, a particle would spin up and down, which is then maybe has 50-50 chance of being up or down. And we can say a rational agent would be indifferent between those two games. And then you're still kind of leveraging the results that you know from quantum theory. The other maybe less satisfying move is to say, well, there could be an initial distribution. So, for example, it could be in the multiverse, the initial conditions really are different, right? So you don't know exactly what branch you're in. And then under that assumption, you can again use the quantum results to determine what the probability should be.

[01:42:23]  Unidentified speaker: I'm not sure that is true in all cases.

[01:42:25]  Sam: Like in the asteroid case, I think that actually would make sense because you can say, like, we don't know which branch we're in with regard to how the asteroids formed.

[01:42:34]  Unidentified speaker: That makes sense.

[01:42:35]  Unidentified speaker: Okay.

[01:42:36]  Sam: So for me, that would be, like, at least in certain cases, that seems satisfying. I'm not sure it's satisfying in all cases. Like, it could be like if you're shuffling a deck of cards. That, you know, shuffling is entirely, assuming the shuffling is entirely deterministic, and it's all just due to the algorithm and the initial distribution of cards that you're shuffling, that causes it to be randomly output a particular deck of cards afterwards, so that when you draw a card from the deck, you have whatever the probabilities are there, [they’re massive, aren’t they?] for a deck of cards to have a particular one. I'm thinking, anyway, so if you, in cases where there is a deterministic process and there isn't really any randomness in the initial distribution of the thing that is random across the multiverse, I think you have to leverage the quantum decision theory in the way that I said initially, where you say a rational agent would be indifferent between that game and a quantum game with a similar distribution. So it's kind of, I kind of have a hybrid view, I guess, where sometimes the quantum sense still makes case—but I’m sorry, sometimes the quantum case still makes sense and that would be maybe the asteroids, and other times, I think it is more problematic. I don't know entirely how to solve that.

[01:44:04]  Bruce: So let's take maybe the case that Deutsch brings up that I think is a difficult case, which would be like life insurance. So we're going to use actuarial science, we're going to use probability theories. When we do that, I don't think the actuaries believe that we're dealing with real stochastic events at all, right? They've come to the realization that we can treat a person's death based on our knowledge of their age and whatever information we have on them, that we can treat it like it's a stochastic event. But I would suspect that the majority of them know that this is not a stochastic event.

[01:44:49]  Unidentified speaker: I say that and now I have to retract it slightly.

[01:44:51]  Bruce: It may be partially stochastic, like whether you get cancer might be stochastic, right?

[01:44:59]  Unidentified speaker: But in general, I would not assume that you would try to figure out what age I'm going to die out probabilistically based on trying to look at thickness of branches of the multiverse, right?

[01:45:14]  Unidentified speaker: Like, that wouldn’t—first of all, you couldn't do it because there'd be no way to know the quantum state, so it would be pointless to try.

[01:45:21]  Bruce: Then even if you could, that would probably give you information that's not available, that would allow you to immediately say, oh, this person's long lived, or something along those lines, right? That being able to see the multiverse in the quantum state would probably give you information that would spoil this as a probabilistic problem, okay?

[01:45:42]  Unidentified speaker: So, try to work through [out?], you know, think out loud here if you need to.

[01:45:47]  Bruce: Try to work through, what is it that we're doing? Because obviously actuarial science at least works well enough that insurance companies can stay in business, right? Like you might argue that there are problems with it, that people often live longer than they expected because technology changes, things like that. That's how they have to keep re-looking at it every year and look at new statistics for how long people live, things like that. What is it that we're doing here? Because we're clearly doing something very useful with probability theory, but this is a super messy view that maybe ties a little to the quantum view of probability, but like probably not that much.

[01:46:29]  Unidentified speaker: Cool.

[01:46:29]  Unidentified speaker: Yeah.

[01:46:30]  Sam: Yeah, so my guess would be that actuarial science is trying to model a person that they don't know. So they don't have any specific details about the person, and they're trying to think like how, what would their lifespan be? And then the lifespan issue, I do think, could be modelled through quantum mechanics.

[01:46:54]  Unidentified speaker: So it's probably not what they're actually thinking about because they're just using probability theory, but then I think you can have like a physical interpretation of what that means through the multiverse, where if you think about the lifespan of a particular person, well, that number, the lifespan will vary across the multiverse, so there might be branches in the multiverse where a particular person turns 85, there might be branches where he turns 30 only, and or anything between, and there's a certain probability of each of those branches occurring.

[01:47:30]  Sam: So it may be, if things go really well, then there's very few branches in which you die young and there's only the ones in which you, or this, it’s not—sorry, it’s not even that there’s few, it's that the branch weights of the branches in which you die young is very small, and the ones in which you live long are larger, they're significantly larger. And that does seem to map onto how they're modeling the situation. I mean, obviously, we don't know for particular people how that would look, like we don't have a God's-eye view of a person's life, but we're trying to kind of approximate that with the information that we do have. And then, when you're using lots of different people, it turns out that the averages work out pretty well. But again, I think that's not dissimilar from how that works in actual quantum experiments, where in quantum experiments, you actually know what the numbers are supposed to be, but you still get the frequencies from doing many repeat experiments and tallying them up and seeing what they result in. So I think you can make an analogy there. Definitely, you would expect that someone's lifespan is a variable that differs across the multiverse.

[01:48:48]  Unidentified speaker: Interesting.

[01:48:49]  Bruce: By the way, I think that was a good answer. That gave me a lot to think about there. Let's maybe talk a little bit about the Bayesian view of subjective probability. And because I've just recently studied this, I can talk somewhat authoritatively on what E. T.

[01:49:10]  Unidentified speaker: Jaynes said about it.

[01:49:11]  Bruce: I should note that I'm fairly convinced that E. T. Jaynes is at odds with, say, effective altruism Bayesians. I don't think they know that. I think that the difference is, of course, that he knows what he's talking about, and they're kind of just grabbing ideas and they have a kind of approximate understanding of it.

[01:49:29]  Unidentified speaker: And so they kind of mess them up sometimes, right?

[01:49:33]  Bruce: So let's be a little careful here. I want to try to adhere mostly to what someone knowledgeable like Jaynes would think, but let's acknowledge that the term has come to have other meanings and that we may need to criticize those other meanings where they're being misinterpreted. So when I used to think of the term subjective probability, I sort of had in mind that it was a probability that you just made up, right?

[01:50:05]  Unidentified speaker: It's just your subjective feeling, okay?

[01:50:08]  Bruce: And I think that's actually a pretty good description of how the effective altruists use subjective probability, that they literally say, oh, there's a 95 percent chance that AI is going to destroy us and will go extinct. It's literally just making crap up, right? That doesn't seem to be what Jaynes had in mind. So for those who don't know, E. T. Jaynes wrote a book, The Probability and Logic of Science. And he is, in many ways, that book is the Bible of Bayesians, of modern Bayesianism. Bayesianism pre-exists that book. But he took a lot of ideas that had come out of the Bayesians, pulled them together, and made a case for a unified theory of using probability in science. And it's actually a very compelling argument, I have to tell you. Far more compelling that if you're just a crit-rat talking about these things in X, you have probably missed just how compelling the Bayesian argument when coming from someone knowledgeable like Jaynes, how compelling it actually is. The reason, his point of view is, is that all probabilities are subjective. Now, that may sound like crap at first, but you have to understand what he means. In the case of a physical probability, like quantum physics, first of all, he denied there were physical probabilities.

[01:51:30]  Unidentified speaker: He was a physicist.

[01:51:31]  Bruce: He thought that quantum physics was going to turn out to be some deterministic process and that it was just a subjective probability also. But even if you accept the many worlds version and it's a physical probability that's real, what he would still say is something like this, I think. He would say, you know, ultimately, what a probability calculus is, is it's a plausibility calculus. You've got a set of theories. You're trying to choose between them. You're trying to rank them based on your criticisms or evidence that you've got available. Ultimately, the probability isn't telling you something physically real about the world. Instead, it's telling you something about what you know about these theories as of the day and how good they, what the state of the critical discussion is, and how good one theory is relative to another. It's not really telling you what's the probability this theory is correct. It's really just telling you relative to the other theories that you currently have available. How good is this one compared to its competitors? Because he sees probability is as a plausibility calculus, he thinks that even in a case where there would be a physical probability, ultimately, if I'm rolling a quantum die that splits the multiverse six equally thick branches, [I wouldn't go to bet?] based on that physical probability. But what really matters to me is that I don't know what the outcome is going to be for me. And so in his mind, that's still a subjective probability.

[01:53:08]  Bruce: And that he says, objective probabilities can be rolled into the concept of subjective probabilities. Subjective probabilities is the broader view that includes objective probabilities, but also includes other things that have no actual stochastic event behind them at all, such as the coin flip. So again, I want to emphasize, I'm not, I can't do just, in a future podcast, I'll go through his arguments, I'll try to do better justice to his arguments. I do want to emphasize, this is a smart guy who knows what he's talking about for the most part. And he's making compelling arguments that I don't think critical rationalists have met so far and have understood well enough to criticize. Having said that, there's still something that doesn't sit well with me, with this. And I can see how it then quickly led to the kinds of misunderstandings that I've seen Bayesians fall into, where they just start making up crappy probabilities and there's numerous errors I've seen them fall into. I could give you some others if you were curious, where I've seen them make mistakes that just would not fit with Bayesian reasoning, but they think it does. Even like Sean Carroll, I've seen fall into some of these, right? Where he'll say, you can't set any theory to a probability of zero, because then you can't reconsider it. It's like, no, no, no, you have thoroughly misunderstood good Bayesian reasoning at this point. You absolutely can set the probability to zero, because you're always still able to challenge the model and rewrite the model. It doesn't mean you would never reconsider it.

[01:54:48]  Bruce: It might mean you just have to reconsider the model. But if I have three black balls in a dish, and I say, okay, what's the probability of drawing a black ball out of this dish? The probability is 100. The idea that I have to set it to 99.99999, because otherwise, that is silly. It's a total misunderstanding of what we're even trying to do here. So I've got mixed feelings over this. On the one hand, this is somewhat of an inversion of what you just said. You're trying to look at subjective prob—what we're calling subjective probabilities as an approximation, approximation of objective probabilities, which I kind of like, and that kind of sits well with me. But Jaynes’s argument is that really you should see objective probabilities as being a special case of subjective probabilities. I have to admit his argument makes some sense to me too. Thoughts on this so far before I go on to any specific examples?

[01:55:47]  Unidentified speaker: No.

[01:55:48]  Sam: The only thing that comes to mind is I think this relates to an issue which the philosophers call the principle principle, which is that, as I understand it, you should put the objective probabilities as being equal to the subjective ones if you're doing some kind of Bayesian reasoning, and that there's not really, I mean, maybe you're going to get into this. I think they often struggle to think of reasons why that should be the case. I just thought about that as you talked about Jaynes explaining objective probabilities as being a subset of the subjective ones, because that’s—I could be wrong, but I think that's something that people often struggle with. They have to find some kind of argument for why that actually, those two should be equal in the cases where you're dealing with objective probabilities.

[01:56:47]  Unidentified speaker: I don't know if, yeah.

[01:56:49]  Unidentified speaker: Okay.

[01:56:50]  Bruce: So let's take the best case we can think of, of a subjective probability, which sounds like the one that we agree on would be the coin flip, that it appears based on our current best understandings of a coin flip, coin flips appear to be deterministic, and that what actually brings the quote unquote random element in, is just that we don't happen to know what the initial conditions are. We say initial conditions, but it might include things like, is the coin going to be caught or is it going to fall on the floor? It might be conditions at the end of the flip too, but the conditions around the flip is what we're calling the initial conditions. In a case like this, isn't this a case of a true subjective probability? By the way, does the term subjective probability here rankle you and would you call it something else if you don't like that term?

[01:57:40]  Sam: No, I think it makes sense. As I said before, it could be that you can still model it using quantum theory, just because you can have a distribution of initial conditions. I think, again, with cards, I think that would be harder if you...

[01:57:55]  Unidentified speaker: Yeah, you're right.

[01:57:56]  Bruce: Maybe we should use cards.

[01:57:57]  Unidentified speaker: Okay.

[01:57:58]  Bruce: Let's clarify that for the audience though. So at the moment of the coin flip, it's deterministic. But if you back up, let's say one second, there's going to be a quantum, maybe there's going to be a quantum distribution of initial conditions, which we could then turn it into a quantum problem.

[01:58:16]  Unidentified speaker: That's really what you're arguing, right?

[01:58:18]  Sam: Yeah, basically.

[01:58:19]  Unidentified speaker: Yeah.

[01:58:19]  Sam: That would make sense, as I said, in the cases of the meteors, because in that case, you really would expect there to have been a distribution of universes where meteors form in different ways, and you don't know in which branch.

[01:58:34]  Bruce: It's the same thing, but in this case, you're backing up billions of years instead of backing up one second.

[01:58:40]  Unidentified speaker: Exactly.

[01:58:41]  Unidentified speaker: Yeah.

[01:58:41]  Unidentified speaker: Okay.

[01:58:42]  Unidentified speaker: I'm with you.

[01:58:43]  Bruce: Actually, that's not a bad argument at all in my mind. Let's either use sorting of cards if that works better, or let's even just use pseudorandom numbers on a computer.

[01:58:58]  Sam: Yeah, sorting of cards, they seem similar to that.

[01:59:01]  Unidentified speaker: Okay.

[01:59:02]  Bruce: Is this a case of a true subjective probability that we found?

[01:59:06]  Sam: I think so, yeah. I think you have to think about it. There, the best argument I can think of, although it's not necessarily one I'm entirely sure of, is that you can leverage the decision theoretic approach indirectly. You can say, for example, we have a pseudorandom number generator and we're going to use it in some kind of gambling. I would be indifferent between that situation, provided I don't know anything about how the algorithm works and I don't have any information I can leverage to bet differently than other people would. I would be indifferent between that setup and one where it's a genuine random number generator using quantum effects. But anyway, to answer your question more directly.

[01:59:59]  Bruce: You would still see it. You definitely are upending this compared to Jaynes’s. You would still, even in a case like that, you would see it as it happens to be modeled similar to an objective probability. And that's why you think it still works.

[02:00:15]  Sam: Well, I think that as a rational decision maker, you might be indifferent between the two games.

[02:00:20]  Unidentified speaker: It's not necessarily, I think, that the underlying mechanics are the same.

[02:00:22]  Sam: I just think that you would be indifferent from... So the decision theoretic approach is all about being indifferent between different kinds of games that you play with quantum systems.

[02:00:34]  Unidentified speaker: And here you might say, well, I could be indifferent between the quantum game and the genuine quantum game and the pseudo random number generator game.

[02:00:46]  Sam: And that's not to say that the one is, that the pseudo random number generator has quantum effects. It's just that from the point of view of someone who doesn't know anything about the algorithm or what it's going to produce, it might as well be a quantum game.

[02:01:07]  Unidentified speaker: Okay. Okay.

[02:01:09]  Sam: But it's not saying that the mechanism is similar. And I agree that there is a reason. It seems like a genuine case of subjective probability. What I'm saying instead is the way I would—the way in which I would leverage our understanding of quantum probability is to say, you might be indifferent between these two games, and then you can kind of indirectly use the quantum setup again.

[02:01:35]  Unidentified speaker: All right.

[02:01:36]  Unidentified speaker: I'm with you.

[02:01:37]  Unidentified speaker: OK.

[02:01:38]  Bruce: Let me move to now one that's even more questionable. I'm trying to do this one step at a time here. So I mentioned in my podcast, Cox's theorem. From what I understand, that was actually the first time you had heard of Cox's Theorem. You've had a chance to since look up Cox's Theorem.

[02:01:57]  Unidentified speaker: Yes.

[02:01:58]  Unidentified speaker: So, what are your initial...

[02:01:59]  Bruce: Before I go any further, what are your initial thoughts on Cox's Theorem?

[02:02:04]  Unidentified speaker: Seems interesting.

[02:02:06]  Sam: I think I would call into question some of the assumptions that go into the theorem, that as a...

[02:02:12]  Unidentified speaker: Yeah.

[02:02:14]  Sam: [As far as I can tell, the theorem holds?]

[02:02:16]  Unidentified speaker: Yeah.

[02:02:17]  Bruce: What would you call into question? It's based on three Desiderata. If you question any of those Desiderata, then it doesn't hold. So, it's absolutely true that any circumstance where those three Desiderata do not hold, Cox's Theorem doesn't apply anymore. What would you question on those? You probably don't remember them off the top of your head, but it would probably be the Credence Assumption.

[02:02:42]  Unidentified speaker: Well, maybe.

[02:02:43]  Sam: So, you can tell me, I can tell you what I think is problematic. Essentially, what you get is a, I think it's a pre-order or something.

[02:02:57]  Unidentified speaker: So, you have different kinds of order in mathematics.

[02:02:59]  Sam: One of them is called the total order, where you can assign, you can say, for all elements in a set, you can compare them. So, you can say, as soon as two elements are not the same, either, for example, one of them is larger than the other, or the other is larger than the initial one. And that comes into play with Cox's Theorem, because it says that you can assign probabilities to theories. And that means that you can either, like two theories can have the same probability, or that one of them is larger than the other, or vice versa. And that's always the case. And that makes sense to me in cases where you're talking about rival theories. So if you have theories that are trying to solve the same problem, then I think you can compare them, because you're looking at a narrow set of theories. But what I find kind of strange is the idea that you can have a comparison between, for example, Newtonian mechanics and, say, evolutionary theory like neo-Darwinism. And so this idea that you can completely order all known theories that way seems problematic. I would have thought that you maybe get like a partial order where some theories just they don't relate to other theories. You can't really say, well, they're smaller or larger. We have more or less credence for them. And I don't know what it means.

[02:04:39]  Unidentified speaker: Again, it's one of those cases where I would say it's an open problem.

[02:04:43]  Sam: And maybe the solution to the problem is to just drop it and think about other stuff. Maybe the solution is that there is a genuinely satisfying answer and I don't know. But it strikes me as a problem that you can have a direct comparison between Newtonian mechanics or say Einsteinian relativity, like Einsteinian gravity, and neo-Darwinism, because they're not really competing with each other in any obvious way. I mean, they're not, yeah. So that would be, that was really the main issue I had with it.

[02:05:21]  Bruce: So let me respond how I think Jaynes would respond here, having just read the first, the basic section of his book. So these are the desiderata, right? The things we desire, they're not axioms. So you would not apply Cox's theorem to a case. This is what I think Jaynes would say, but I don't know this for sure. You would not apply Cox's theorem to Newtonian, sorry, Newtonian physics versus neo-Darwinian evolution. Those are not mutually exclusive theories. There's no reason to order them. One of the assumptions you're starting with if you're using Cox's theorem is that you have a situation where these are mutually exclusive theories and that you can, and that they need to be weighed against each other to make a decision of some sort. Now, you're bringing up still a problem, namely that there is an ambiguity here. So for example, what if two theories are competitors, but they disagree over things, and the experiments go one way in one case and a different way in a different case? Like that would be a fair thing to bring up, right? The Newtonian physics versus Einstein's physics is such a self-contained case, and we use it so much, you miss how wild and woolly science actually is, right? In real life, where a lot of times it's just very hard to compare theories. But I do think what you're raising could just roughly be solved as, okay, think about in terms of critical rationalism, the idea of a crucial test.

[02:07:05]  Bruce: There's already, as part of your model, as part of your background knowledge, you're starting with the assumption that only one of these theories could be correct or of the theories you're considering, usually two, but maybe more than two. Only one of them can be correct, and you're going to do an experiment, and you're going to use that experiment to differentiate between these theories. That is the case where I think Cox's theorem does apply, and I actually think it applies correctly in that case. Whether you're actually in that case may be an open question. There's no way to prove you're in that case. It's always partially an assumption of your model and your background knowledge.

[02:07:44]  Unidentified speaker: Right. So, like, if I were to use—

[02:07:49]  Unidentified speaker: Go ahead.

[02:07:50]  Sam: Well, I think to someone who is a Bayesian, at least in the modern sense, that would be like my understanding of Bayesianism is that they do want to have this kind of total ordering of theories using probabilities.

[02:08:06]  Unidentified speaker: Yeah.

[02:08:08]  Sam: And I guess the reason that feels pure is because then you don't need to have auxiliary theories to determine when you should not do that. So now we have to add additional structure to the theory to say when we can use it. We have theories that we make because we attempt to solve a particular problem and they can compete in a niche. So that we can say, given this particular problem situation, we have a bunch of theories and they compete on testability, on ad hocness, on universality, on simplicity of assumptions that go into the model. And narrowly that all makes sense. And yet if you think about, as I said, separate theories, which are not directly competing, although it may turn out that they do at some later stage, compete, such as Newtonian gravity and, or it doesn't really matter which theory of gravity you take, but let's say Einsteinian gravity, because you think it's true, and neo-Darwinism. It's [odd?] that you still want to have, like numbers, which somehow rank those.

[02:09:24]  Unidentified speaker: That's right.

[02:09:25]  Sam: Yeah, so that strikes me as a bit strange.

[02:09:29]  Bruce: And the reason why is because of deductive logic, you often make your assumptions very explicit, and in probability theory, they're often implicit instead. And so it's harder to criticize them, right? So you can imagine that there might be an infinity of ways in which to model this crucial test that you're doing. Probability theory would be one of them, and it would be valid, but it may not be the one that's the most enlightening on terms of the problem that you're currently working on. All Cox's Theorem really says is whatever you do come up with, there will be a probabilistic mapping. You could map it to probability theory, and that's true. As far as I can tell, that part's true, that you will always be able to map it to probability theory at the end.

[02:10:13]  Unidentified speaker: Yeah.

[02:10:13]  Bruce: Now, if that's true, which I think it is, it does make subjective probabilities real in a sense. Like, let's take an example. Let's take one that I still think is valid, but that is really iffy, so that we're making this as difficult as possible. So, you're trying to... There was an example similar to that. I don't have it off top of my head, but like the Increments guys on the Increments podcast asked David Deutsch about how to apply theories in a case where you're trying to remember if you saw somebody at the airport or what time was the flight.

[02:10:53]  Unidentified speaker: Do you remember what it was?

[02:10:54]  Bruce: Like it's something that's very real, like you would do it in real life. And I don't remember the exact specific, so I won't get into that. But let's pick something like, you know, what time is your flight? How well do you remember your flight?

[02:11:06]  Unidentified speaker: Okay.

[02:11:07]  Bruce: You have some level of confidence in what time your flight was, but it's not 100 percent. You know, you might be misremembering something along those lines. That whole ability to say, okay, look, I think it's either 305 or it's 503, because I might be, you know, swapping the five and the three.

[02:11:29]  Unidentified speaker: They look similar, but backwards.

[02:11:32]  Bruce: You have an explanation for what your confidence levels are.

[02:11:35]  Unidentified speaker: You have an explanation for why you might be wrong.

[02:11:37]  Bruce: You have an explanation for why you think you are right.

[02:11:40]  Unidentified speaker: Okay.

[02:11:41]  Bruce: Cox's theorem says that calculus that you're doing in your head, however you're doing it, doesn't matter how you're doing it, that it is isomorphic to probability theory.

[02:11:52]  Unidentified speaker: Give me your thoughts on that.

[02:11:54]  Unidentified speaker: Yeah, I think that for me, that would be a bit of an open problem.

[02:11:59]  Sam: Again, I've said that a couple of times.

[02:12:00]  Unidentified speaker: I see it as an open problem.

[02:12:01]  Unidentified speaker: Okay.

[02:12:02]  Sam: Yeah, so I still find it hard to determine exactly the credences mean in that case. Like you set up a framework using Cox's Theory to talk about credences and intuitively it has a meaning. I think I would still side with the idea that you are, you have explanations for what might be wrong with your memory. So you could say, well, I think I saw a particular time and I often confuse, I don't know, like in my head, I confuse 6's as 9's or something and I rotate them or whatever. And so I know that that's the thing that I do, so maybe I do that in this case.

[02:12:57]  Sam: So there would be some kind of underlying explanation for the reason why you think that you might need to use credences in this case. Conceivably, you could, as you said, think it's like the mathematics shows that you can set up a model. The question—so one other question is, of course, Cox's theorem says that there are numbers that you can use. I guess you could answer this in the form of a question. It's not immediately clear to me that the numbers are something you need to use in thinking about stuff either. Just like, so let me explain it through an analogy. Sometimes in quantum theory, people do is they think of these probability measures. And so they just like David and Wallace, Deutsch and Wallace have the decision theoretic approach. Other people have different approaches. And they say, well, we can put in some basic assumptions and then derive probability as well. Why do we need the decision-theoretic approach? And whenever I speak to these people, they're very nice, very smart people. My question is usually, like, how do you know that I need to use your numbers for making bets with? Because they're numbers and they abide by the probability calculus, but that doesn't mean they should factor into my decision making at all, or into my thinking really about the world. Like, it's a cute fact, but how can you argue that you need to use those? I think here, I also have a similar hesitation, which may or may not be warranted. So there are numbers.

[02:14:40]  Sam: Mathematically, we know that there is a probability space that we can assign to the set of theories and use probability theory to derive certain results. But it's not clear that that maps on to what I'm thinking about. I think you're claiming it must be the case, like the theorem shows that you must be using something that is at least isomorphic to...

[02:15:12]  Bruce: If you're making a decision between specific finite number of theories, then the theorem does say it must be the case. But that is a big assumption. And what you're really kind of arguing is that that assumption may not hold in some cases.

[02:15:26]  Unidentified speaker: I'd actually agree with that.

[02:15:28]  Unidentified speaker: Yeah, well, I guess I'm saying it's not clear...

[02:15:33]  Sam: For example, then you're assuming decision theory, I would say. Then you're assuming classical decision theory and Cox's part...

[02:15:40]  Unidentified speaker: Is that correct?

[02:15:40]  Sam: Cox is then part of a result from classical decision theory or not?

[02:15:43]  Bruce: It's not a result from, but you can use it in... It ties into decision theory, right?

[02:15:51]  Sam: Right, right, right.

[02:15:52]  Unidentified speaker: So, but like...

[02:15:53]  Sam: I guess that's kind of the limits of my knowledge.

[02:15:56]  Unidentified speaker: Right.

[02:15:56]  Sam: I guess that's kind of the limit of my knowledge of decision theory.

[02:15:59]  Unidentified speaker: Yeah, I don't immediately have anything, other than what I just said, I think for me, either I would have to delve into the details more.

[02:16:07]  Sam: My basic concern would be like, I still don't quite know what the credences are.

[02:16:13]  Unidentified speaker: Right.

[02:16:15]  Sam: I don't know what. In all the other cases that we talked about so far, I think I can always either turn it back into a story about physical probabilities in the sense that we talked about before, where it's branch weights in the multiverse, or we can some other way we can use the decision theoretic approach on quantum theory because we can say, a rational agent would not see the difference between these two setups.

[02:16:46]  Unidentified speaker: One of them is classical, one of them is quantum.

[02:16:49]  Sam: It would be indifferent between playing the classical and the quantum game, but the classical game has the, for example, the pseudo-random number generator, or the non-genuine, the subjective probabilities.

[02:17:00]  Unidentified speaker: In this case, this is kind of like our third category of—

[02:17:07]  Unidentified speaker: It is, yeah.

[02:17:09]  Unidentified speaker: I'm trying to stretch this as far as I can to make it as messy as I can, so, yeah.

[02:17:13]  Sam: Yeah, so for me, assuming, I'll go along with you, because you seem to know more about the theory than I do, that all of this is correct, that would raise a question for me about what the credences actually represent, because it's not like the theory is going to be true in some universes and not in others.

[02:17:32]  Unidentified speaker: That's right.

[02:17:34]  Sam: So, yeah, I don't have another response to that.

[02:17:39]  Bruce: So, let me actually try to answer the question myself as best I can. Keep in mind that I don't know that I've got the answer either.

[02:17:46]  Unidentified speaker: I'm just trying to think this through out loud too.

[02:17:51]  Bruce: So, I think that I completely agree with you that it boils down to your explanation. So, let's say that I'm thinking it's either 305 or it is 503, because I swear I saw one of those two, right? And so, in my mind, I then criticize that and I say, you know, I don't think I've ever seen a plane flight leave at 03, but I have seen them leave at 05 before. You know what, maybe once I did see it leave on 03, but you know, or some weird number like that. But like usually it's a more round number. So, I go, you know what, I'm pretty sure it's 305. Okay, so I've now worked through a set of criticisms. Now, does this match Cox's Theorem? Yes, it does, because I had to choose between a finite number of mutually exclusive theories. Does this mean that what I did is isomorphic to probability?

[02:18:46]  Unidentified speaker: Yes, it does.

[02:18:48]  Bruce: What is it that I'm actually doing here though? And could I, maybe could I even tie it back to an objective probability in some way, right? So, I don't know if I could tie it back exactly to an objective quantum probability. Maybe I could, but let's just tie it to frequencies because frequencies might be easier to talk about. In there was a criticism. It's really rare that you have a plane leave at a weird number that's not round. Okay, I've only seen that once out of how many number of times that I've done this. You've now introduced a frequency, right? Where you know sometimes it's one way, but it's usually the other way. Even though you don't know the exact frequency, it's not like you've actually stopped and counted it up so that you could turn it into what a frequent twist would consider an objective frequency. You've probably got a pretty strong gut feel.

[02:19:42]  Unidentified speaker: You know what?

[02:19:42]  Bruce: I've done this probably a hundred times, and I've only seen it once go the other way. So I'm going to assign a frequency of one out of 100 to this. Based on that, I'm 99 now percent sure that it was 305, not 503.

[02:19:57]  Unidentified speaker: Okay.

[02:19:58]  Bruce: So you can see how, in this case, there is a probability, and it kind of ties to a physical frequency, which you might even argue has some sort of quantum branching if you were to go back far enough in time to look at it. How often do they do this, and what are the reasons why they would do an odd number, and things like that?

[02:20:20]  Sam: Yeah, well, I would have said actually that maybe the mistake itself is something which you've realized you make with a certain frequency. Error rates and things also have a nice interpretation in the multiverse.

[02:20:33]  Unidentified speaker: Yeah.

[02:20:34]  Unidentified speaker: Yes.

[02:20:35]  Sam: Because errors happen with a particular frequency, so it could be that you know that you mess up certain things, that like in this case, you know that you sometimes misremember the order of the numbers, say.

[02:20:50]  Unidentified speaker: Right.

[02:20:52]  Unidentified speaker: So yes, that would be part of the model.

[02:20:54]  Bruce: You've got a gut feeling of how often you mess up the order based on your past history. So I don't think there's anything here that is the thing that's really problematic here. First of all, there's nothing problematic here. If all you're trying to do is make a decision right now, and this is the only information you have available to you, I can completely see why this is your best theory, that it's actually 305. So we're going to go with that because the odds are in your favor, what we really mean is that theory has survived criticism better relative to the only other theory that we had at the moment. Based on frequencies, based on your—under certain assumptions about how often you misremember things, based on a number of different things. The real problem I see here is that this subjective probability, this is an extreme example of a subjective probability, is it's not rooted in much. It's rooted, it's not, to put this in Popperian terms, it's not severely tested, right? You have a gut feeling that you mess up an error rate of one out of 10 maybe, you have got a gut feeling that usually you've only once seen an odd number, you might be misremembering that by the way. And so you've created this probabilistic model in your head, something that's, it's really not a probabilistic model, it's isomorphic to a probabilistic model. You've created this model in your head and you're gonna go with it because it's the best you've got at the moment. But really most of your assumptions are pretty untested.

[02:22:28]  Bruce: And so I think this is where I would actually criticize Bayesians, is that they will take a subjective probability that is rooted in physics, where we've got a really strong explanation as to what the frequencies are gonna be because it's a quantum physically real probability. And then they'll take something like 95% chance that, you know, AGI is gonna wipe us out based on nothing more than a gut feeling that's, in this case, not—not even rooted in past experience, right? And they will try to, even though, even though all of these are probabilities, because a probability is just a mathematical formalization, they're not, they're not equivalent probabilities anymore, right? Like one of those is a severely tested probability, and one of those is something that you made the crap up on, right? And I think the real problem that Bayesian, this is my opinion, that Bayesian's bump into, is that they come, just because the mathematical formalization is the same for both of these as per Cox's Theorem, they think they can mingle them without having to severely test them. And I actually think that is the problem here. That's my guess, is that they're lacking the Popperian concept of severe testing in how they try to do their probabilities.

[02:23:42]  Unidentified speaker: Thoughts on that?

[02:23:43]  Bruce: This is my initial thought. I'm going to tease this out in future podcasts, try to write it out. I may change my mind as I try to write it down. Give me your initial thoughts on what I'm saying, and do you think I'm headed in a productive direction, or do you think I'm off base here?

[02:23:59]  Unidentified speaker: Sounds productive.

[02:24:00]  Sam: I think also, the other thought I had was that we kind of managed to squeeze it back into some kind of physical model of where the credences are coming from. I'm wondering, I mean, presumably, it's definitely the Bayesians would say, well, we can have cases where maybe we don't have any physical basis for thinking that there's an error, and we could still use the credences like in more severely tested scientific cases where we have several theories which are not, like we don't think that we're misremembering bits of string theory, say, but we might still have a credence for string theory, which is higher or lower. That's what the Bayesians would argue again.

[02:24:45]  Bruce: There may be cases where they're right to do that. I suspect that the vast majority of cases where they do this, they're just wrong.

[02:24:52]  Unidentified speaker: Yeah.

[02:24:53]  Unidentified speaker: Again, I don't know what it would mean.

[02:24:55]  Unidentified speaker: It remains an issue of having more confidence in the theory is not a thing that I can make sense of except to say, well, there are certain attributes of physical theory, just theories in general that we can compare, but it's kind of multi-dimensional.

[02:25:15]  Sam: It's [testability?], there is ad hocness, there is simplicity. If you have two things which make the same predictions, but one of them has [fewer/two?] assumptions, that is one axis in which you can compare the two theories with one another. So the issue I have with the Bayesians is when they talk about credence, I think you can compare theories, but there's many different things going on when we are comparing them, and they try to condense it down to these single numbers. So it's like a one-dimensional thing that they used to say, well, this theory is better than the other one. Definitely when you're talking about theories which are actively competing with one another without there being a single consensus on which one is better, usually it is that some of them are better in certain regards, and the other ones are better in other ways, and you don't know, you're conflicted about which one will solve the problem.

[02:26:28]  Sam: The short answer would be our comparisons are multidimensional, but it's a single dimension.

[02:26:33]  Bruce: If the comparisons are multidimensional, they don't fit Cox's theorem. You're absolutely correct about that. I could even give a real-life case of that. I've been sending this for a future podcast, but light wave theory and light particle theory would be an example of where there was actually multidimensional, because of course, quantum theory was the more correct example. So you cannot fit light wave theory and light particle theory into a Bayesian Cox's theorem type analysis.

[02:26:59]  Unidentified speaker: Interesting.

[02:27:00]  Unidentified speaker: Yeah.

[02:27:01]  Bruce: I've now given away the answer to a problem I'm going to pose in a future podcast, so now I've given the answer in advance.

[02:27:08]  Unidentified speaker: Nice.

[02:27:09]  Unidentified speaker: Okay.

[02:27:10]  Unidentified speaker: Well, Bruce, we want Sam to come back on the podcast many times.

[02:27:16]  Peter: So we have to be respectful of his time and wrapping up.

[02:27:21]  Unidentified speaker: This was supposed to be a shorter episode, right?

[02:27:24]  Unidentified speaker: What did we do?

[02:27:25]  Bruce: Three hours or something?

[02:27:27]  Unidentified speaker: I think we did that.

[02:27:27]  Unidentified speaker: All right.

[02:27:28]  Bruce: Thank you very much for your talking with me, Sam. This has been awesome. I think that this has been very helpful to me at least, to try to make sense of this. I think you've given a lot of clarity, brought a lot of clarity that I wasn't able to get elsewhere when I was trying to understand what we were talking about here.

[02:27:48]  Unidentified speaker: I really appreciate that.

[02:27:50]  Unidentified speaker: Yeah.

[02:27:50]  Sam: It was great talking with both of you. Definitely open to coming back, of course.

[02:27:56]  Unidentified speaker: Thanks so much for having me.

[02:27:56]  Sam: It was really fun.

[02:27:58]  Peter: Thank you, Sam.

[02:27:59]  Unidentified speaker: I got a lot out of listening to you and I enjoyed it so much.

[02:28:02]  Unidentified speaker: Thank you.

[02:28:03]  Unidentified speaker: Great.

[02:28:04]  Sam: Glad to hear it.

[02:28:12]  Unidentified speaker: Hello again.

[02:28:13]  Peter: If you've made it this far, please consider giving us a nice rating on whatever platform you use, or even making a financial contribution through the link provided in the show notes. As you probably know, we are a podcast loosely tied together by the Popper Deutsch Theory of Knowledge. We believe David Deutsch's Four Strands tie everything together. So we discuss science, knowledge, computation, politics, art, and especially the search for artificial general intelligence. Also, please consider connecting with Bruce on X at bnielson01. Also, please consider joining the Facebook group, The Many Worlds of David Deutsch, where Bruce and I first started connecting.

[02:28:56]  Unidentified speaker: Thank you.