Spacetime Can Solve the Halting Problem

Hinzugefügt: 2026-06-01

[00:00:00]So something you can do if a Malament-Hogarth  spacetime exists is run a machine, run a program, which you don't know if it's going to halt or  not. And in this universe, in a Minkowski universe purely, you would not be able to tell  if your Turing machine is going to halt.

[00:00:15]But it's said that you could take  a laptop, send it to some point in an MH spacetime, and you'll be able to  receive whether the laptop halts or not.

[00:00:29]But then there's issues of  so-called blue-shifted photons. So what is a blue-shifted photon?  Is it actually a problem?

[00:00:39]Yeah, yeah. The blue shift happens because,  well, so you're absolutely right in your characterization. What a Malament-Hogarth  spacetime is, is you identify a point, an event in the spacetime, and you look at its past  light cone. And in the past light cone, you have an observer with an infinite future that's  contained in your past light cone. So for you, you're making an observation at a  particular finite point in space and time, but part of what you're observing is an  infinite future of some other observer.

[00:01:17]And so that other observer could be a computer  running some kind of program or something.

[00:01:22]What happens with the infinite blue shift situation is when the computer  is sending signals back to that point, they can start to bunch up. And so the signals  can get very unreliable and they can also become very close together. So the wavelength  will go towards the blue side of things.

[00:01:45]And so some people have said, well, you know,  this really isn't physically reasonable because of blue shift situations like that. And I think I  agree with that. What I've tried to do in some of my work has been to collect up a bunch of reasons  that people want to exclude these things for.

[00:02:06]So blue shift would be one of them. Another one  would be whether the standard energy conditions are satisfied. Another would be, well, we don't  want to have this computer having to accelerate really wildly and have infinite fuel to traverse  whatever world line it's supposed to traverse.

[00:02:28]So you can collect up a bunch of these things.  And what I do is I say, look, here's a model.

[00:02:37]I go to work in my little workshop  building a little model, and the model is a Malament-Hogarth spacetime that has none of those unphysical properties. The  irony is that this model is pretty unphysical.

[00:02:53]Even by your standards?

[00:02:54]Even by my standards, yeah.

[00:02:56]So there's an old technique introduced by  Penrose, and Geroch used it a lot in his work. Hawking used it with Ellis in his textbook.  You might term it the cut-and-paste construction.

[00:03:11]It's where you take models of spacetime, like  relatively well-behaved models, like something like a Minkowski spacetime. And what you're  going to do is you're going to cut slits here, and you're going to glue things together.  You're going to create wildly crazy topologies in this way. They're going to be standard models  of GR. They're going to count as models of GR, but they're going to be wildly unphysical.

[00:03:40]Now, those folks — Penrose, Geroch,  others — they emphasized early on that the point of these examples was not to show,  you know, it wasn't to exhibit a physically reasonable model of our universe. But rather,  I mean, if you're studying something like logic or something like this, counterexamples  are just so important. And so they serve the purpose of being counterexamples  to certain lines of reasoning, right?

[00:04:05]And so, in this case, I would say the example  that I'm coming up with that's very unphysical is a counterexample to a certain line of reasoning  that goes like this: Malament-Hogarth spacetimes can't possibly happen. Why? Because  they have unphysical property X, Y, Z.

[00:04:24]I'm going to say, well, I have a counterexample to  that line of reasoning because I've now exhibited a spacetime which is Malament-Hogarth  that doesn't have property X, Y, or Z.

[00:04:36]And so what it does is it puts the ball in  the other person's court again and says, if there's something else you're thinking  of that makes it unphysical, then speak up.

[00:04:46]Come up with some other criticism here, but the  criticisms you have so far aren't quite enough.

[00:04:52]Uh-huh.

[00:04:53]You know, the hawk-eared — or hawk-eared,  the analogy for hawk-eyed, but for listeners, the eagle-eared, I'm not sure — listener may  have noticed that you used the term "obtain."

[00:05:04]So people who study philosophy are  distinguished by their use of the word "obtain." Physicists who haven't studied  philosophy never use the word "obtain."

[00:05:12]Really? Okay.

[00:05:13]So "obtain" just means that  something becomes the case.

[00:05:17]I want to know — you mentioned earlier  that there's a difference between not only the way that Malament does the  philosophy of physics compared to other philosophers of physics, but you also mentioned  that there's a difference between the way philosophy of physics is done and philosophy  per se as such. So what is that difference?

[00:05:36]Well, I mean, I think one thing to say is that  philosophy of physics even is a very diverse thing. So there are all sorts of styles of  philosophy of physics. You encountered some philosophers of physics, right? You had David  Wallace on your program, Tim Maudlin, I think.

[00:05:59]So you're familiar with — I mean, both of those  folks have a very different style than I do.

[00:06:07]I'm in a style which is on the more mathematical,  logical, rigorous side of things, I would say.

[00:06:15]The way I attack a problem is  to try to formalize that problem and come up with a proof or a counterexample  for a given philosophical claim.

[00:06:26]But if you look at the work of other philosophers,  it's going to be very different. They're going to not necessarily be so rigorous. Here's an  example. This is not my field at all, but something like quantum field  theory. There's going to be some philosophers of physics who want to really make  that into a really rigorous type thing. That's difficult. What they might do is they might  go to an algebraic approach to that, maybe look at C*-algebras or something like that,  whereas someone like David Wallace or others, they might want to dive into the practice of  how folks are trying to make sense of quantum field theory within a physics  department or something like that.

[00:07:17]So these styles can clash, and these clashes can  pop up at a conference. There are more styles than that, too. I mean, there are styles where it's  very disconnected from the actual practice of physics, where you're looking at something like  time travel from... you're starting in a different place. You might bring in physics here or there,  but you're not starting with, okay, let's assume we're working within the context of GR, and  let's ask this question. Rather, you're saying, I'm not going to start from GR. I'm going to  start from my own — I've got my own system.

[00:07:58]And so, yeah, there's a lot of different ways  to do philosophy. And looking back on my own trajectory, when I was a youngster, I just  wasn't aware of all of these ways. I mean, for me, it was just like, there's physics,  there's philosophy. Bringing them together, that's got to be like this one  thing, but it's not one thing.

[00:08:19]Right. There are a couple levels here. So number  one, when you tell people you go into math — if you tell lay people who haven't studied any math,  they may think that what a mathematician does is something like an accountant, where you're  multiplying just larger and larger numbers together, and maybe something like the Pythagorean  theorem. And they don't realize that it's much more than that. And then when you  realize that it's much more than that, you don't think in terms of there are styles to  that much more than that. That Grothendieck style is different than Terry Tao style. And something  that didn't occur to me is that there are different styles of philosophy of physics until  just speaking with you now and thinking about it.

[00:08:57]Is there a style of philosophy of physics that you  envy? That you think, oh, I wish I could do that?

[00:09:07]Yeah, I mean, I certainly envy —  David sort of invented this style of doing philosophy of physics.

[00:09:15]Which David?

[00:09:17]David Malament.

[00:09:18]I would say, yeah, I'm very envious. I try in my  own way to work in his style. I have my own style, though. My style is one where I don't put a  sentence in a paper, I don't put a word in a paper, unless it's necessary. And so my papers  can be two, three pages. That's very unusual in philosophy. It's maybe more common in physics,  although that's even pretty short for physics.

[00:09:47]But I like to let the results speak  for themselves. I think a lot of philosophers — what they'll want  to do is they might have a result.

[00:09:55]Most philosophers don't prove theorems. So they're  not actually making progress on the mathematical, physical side of things. What they're going  to do is they're going to take existing statements and they're going to  philosophize about them in some way.

[00:10:13]For me, my theorem is my philosophy. So  I let the results speak for themselves, and I try not to add on top of the  results some flowery prose about whatever, or what the meaning of this is, or something.  The result is what it is. That's the philosophy.

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