I see you continuously blending computation (the mechanical enumeration of finite steps) with provability (what a formal theory can or cannot prove in principle, given its axioms) all the time. Everybody with an eye for it can. You want an example: reread yourself. As long as you keep conflating those two notions, we’ll never find common ground on a ToE or Gödel’s incompleteness applying or not because of lack of clarity. Gödel’s results hinge on provability in a system strong enough to represent arithmetic, not on whether you can simulate 𝑁 steps of a cellular automaton. Invoking uncomputable inference rules (like the ω-rule) further muddles the line between informal physics-adjacent and logical methods, without clarifying which framework you’re using + to which end. You admit yourself that you enjoy playing teacher. Thank you for that information and best wishes for your "ToE" and "teaching".
On Saturday, January 18, 2025 at 4:53:57 PM UTC+1 Jesse Mazer wrote: > On Sat, Jan 18, 2025 at 5:29 AM PGC <multipl...@gmail.com> wrote: > >> >> >> On Tuesday, January 14, 2025 at 9:06:29 PM UTC+1 Jesse Mazer wrote: >> >> On Tue, Jan 14, 2025 at 4:56 AM PGC <multipl...@gmail.com> wrote: >> >> >> >> On Monday, January 13, 2025 at 11:58:38 PM UTC+1 Jesse Mazer wrote: >> >> >> Doesn't Godel's theorem only apply to systems whose output can be mapped >> to judgments about the truth-value of propositions in first-order >> arithmetic? A cellular automaton would seem to have "evolving quantities >> and/or qualities through numerical or some other equivalent formalism's >> means", but Godel's theorem places no limitations on our ability to compute >> the behavior of the cellular automaton for N time-increments, for any >> finite value of N, so I would think Godel's theorem would likewise place no >> limitations on our ability to compute the physical evolution of the >> universe's state for any finite time interval. For some cellular automata >> it may be possible to set up the initial state so that the question of >> whether some theorem is ever proved true or false by the Peano axioms (or >> other axioms for arithmetic) is equivalent to a question about whether the >> automaton ever arrives at a certain configuration of cells, so Godel's >> theorem may imply limits on our ability to answer such questions, but this >> is a question about whether something happens in an infinite time period. I >> assume there are similar limitations on our ability to determine whether >> certain physical states will ever occur in an infinite future >> (straightforwardly if we build a physical machine that derives theorems >> from the Peano axioms, or a machine that derives conclusions about whether >> various Turing programs halt), but most of what physicists do is concerned >> with predictions over finite time intervals, I don't see how Godel's >> theorem would pose any fundamental obstacles to doing that. >> >> Jesse >> >> >> Gödel’s first incompleteness theorem states that any sufficiently strong >> formal system (capable of arithmetic) contains statements that are >> undecidable—neither provable nor disprovable within that system. The second >> theorem says such a system cannot prove its own consistency from its own >> axioms. These are statements about provability in formal theories. They are >> not directly about whether you can compute a finite number of steps in a >> system like a cellular automaton. >> >> You can absolutely compute, step by step, the evolving states of a >> (finite) cellular automaton for N time steps, and Gödel’s theorems do not >> say you can’t. They say something deeper: if your formal axioms are strong >> enough to represent integer arithmetic (like Peano Arithmetic or any >> Turing-complete formulation), there will be statements expressible within >> that framework which it cannot resolve. That’s a statement about what can >> or cannot be proven within the system, not about whether a machine can run >> a simulation for some finite time. >> >> You also assume that Gödel’s incompleteness only restricts what can >> happen in an “infinite time” scenario. For example: “Well, sure, there >> might be some question about whether a certain configuration arises >> eventually, but for finite intervals we have no Gödel-limited obstacles.” >> This misreads Gödel: Gödel’s first theorem does not hinge on infinite time >> steps; it is about the intrinsic logical structure of the formal system. >> Even for trivial seeming statements involving finite objects (e.g., “this >> specific integer has property P”), the theorem shows there can be >> statements that the system cannot prove or disprove. It’s not that you >> can’t “run the simulation long enough,” but rather that the theory itself >> cannot settle certain propositions at all. >> >> >> I think you are misunderstanding my claim, I didn't say that Godel's >> theorem itself is directly stated in terms of number of time steps of some >> computation, only that if we look for applications of Godel to >> computational dynamical systems like cellular automata or computable >> physics (and Deutsch's result on p. 11-13 at >> https://www.daviddeutsch.org.uk/wp-content/deutsch85.pdf suggests the >> evolution of any finite quantum system is computable), the only >> applications will be to questions about whether the dynamical system ever >> reaches a certain state in an unlimited time. You said yourself that >> Godel's theorem places no limits on our ability to compute the behavior of >> such a system for N time steps given any specific value of N, so I don't >> think you disagree with this. If you do disagree, i.e. you think there is a >> way of applying Godel's theorem to a finite computable system that places >> limits on our ability to deduce something about its behavior over a finite >> series of time steps, please give an example. >> >> >> First I'll address the rest of your post as there's not really much to >> talk about: Uncomputable inference rules (like the ω-rule) aren’t used in >> standard physical theories much, so invoking them misses the core point >> about Gödel’s incompleteness unless you have some incredible non-standard >> result to show; in which case, prove/show it. >> > > *What* core point do you think I'm missing by invoking them, can you be > more specific? I only invoked them to make a simple point about your own > statement "Gödel’s point is more general: the existence of some undecidable > statements is guaranteed"--my point that this is only true if by "general" > you are talking about the general case of *computable* systems whose output > can be mapped to statements about arithmetic. If you agree with the point > that Godel's proof only applies to computable systems (i.e. that his own > definition of 'formal systems' is now understood to be equivalent to > computable systems), then I don't think you have any reason to dispute what > I said there, I brought up the case of it not applying to non-computable > systems for no other reason besides making that point. In general it seems > like you have a rather uncharitable way of reading what I write, jumping to > conclusions that I am talking nonsense without reading very carefully, as > in this case or the earlier case (which you haven't tried to defend) where > you decided I was claiming that 'the existence of straightforward > computable truths (e.g., “2 + 2 = 4” or “state s follows from state s_0 > after N steps”) somehow negates Gödel’s theorems', something I definitely > was not doing. > > Assuming you agree with the point that Godel's theorem applies > specifically to computable arithmetic-provers, I wonder what you think of > my followup point that the theorem can be trivially translated into a > statement about a certain kind of halting problem (i.e. a statement about > whether something will occur in an unlimited number of computational steps): > > 'And given the understanding that the theorem is specifically about > computational systems, we can think in terms of a program that takes any > given set of axioms, like the Peano axioms, and methodically finds all the > possible propositions that can be derived from them; then for any specific > WFF, one can modify the program so it will halt if it ever derives either > the WFF or its negation. In that case, the claim that there are certain > statements that are undecidable is equivalent to the claim that this > program will never halt on such a statement, meaning that when translated > into these terms it does become another "will something ever happen in an > infinite time" question, even if that wasn't Godel's original formulation.' > > > >> Again, we can simulate a finite system for N time steps, but Gödel’s >> result is not about whether you can brute-force a finite trajectory—it’s >> about the existence of statements in a sufficiently strong formal framework >> (one that encodes arithmetic) which no consistent axiom system can decide. >> Consequently, if a “theory of everything” in physics is robust enough to >> interpret integer arithmetic, then Gödel’s incompleteness theorems apply. >> > > I have never disputed that they apply to certain physical questions, I've > just said that they apply only to questions about whether some physical > state will occur in unlimited time, not about the local dynamics which is > what physicists usually mean when they talk about looking for a "theory of > everything". Are you actually disagreeing, and making a positive claim they > limit our ability to answer questions about dynamics over a finite time? > > >> There's really nothing more I can say regarding all the vagueness in your >> reply, because you clearly are performing rhetorical moves to limit the >> generality of Gödel's contributions and separating "modern physics" from it. >> >> And doing the same while demanding “an example” for finite steps is >> meaningless unless we specify exactly which formal system’s provability >> we’re talking about >> > > OK, pick any specific cellular automaton which is known to have the > property of computational universality (and thus one can design an initial > state such that later states can be mapped to statements about arithmetic > generated by an algorithm), like Conway's Game of Life. Do you claim that > Godel's theorem would place any limits on our ability to predict the > behavior of a finite collection of cells for a finite time? Do you claim it > would place any limits on whether any intelligent beings that might exist > within such a cellular automaton would be able to discover the basic cell > transition rules, which can be thought of as analogous to physicists in the > real world finding a final physical ToE? > > > >> That omission reveals a misunderstanding of Gödel: he never claimed you >> can’t compute discrete steps in a small system, >> > > Again you are leaping to uncharitable conclusions--I never said or implied > that Godel claimed you can't compute discrete steps in a small system. > Since you seem to keep misreading me as arguing there is something wrong > with Godel's proof, I want to be clear on the point that I have never > claimed there is any problem with Godel's theorem as *he* stated it, I've > only made some points about *your* claim that Godel has application to > physics or other computational dynamical systems. And my point is not even > that this claim is wrong in itself, just that the application is limited to > questions about whether certain states will occur in an unlimited series of > time-steps, that it does not imply any limits on our ability to find a > final dynamical ToE or to answer questions about behavior over a finite > time interval, i.e. the types of questions physicists are actually > concerned with in practice (do you dispute that these are the types of > questions physicists are usually concerned with in practice, not questions > about whether arbitrary physical states will arise in unlimited time?) > > > >> but rather that any consistent, arithmetic-level theory remains >> incomplete about some statements. This conflation of “finite-step >> computation” with “formal provability” underscores why your rhetorical >> moves lack any sort of precision >> > > Saying I am conflating "finite-step computation" with "formal provability" > is itself a rhetorical move lacking precision--I'm not clear on what you > mean by conflating the two, but I don't think I have done that, I've just > said that when we apply Godel's theorem (a theorem about formal > provability) to computational systems, the application is only about the > behavior of some such computational systems in the limit, Godel's theorem > does *not* imply any problem with answering questions about their behavior > over a finite series of steps (which I suppose is what you mean by > 'finite-step computation'). > > > >> and ultimately misrepresent/misunderstand not only Gödel’s theorems but >> the nuanced notion of provability in basic terms. >> > > Again a rather vague rhetorical move--what specific claim do you think I > have made (that I would recognize as something I have genuinely claimed and > not a misreading) that misrepresents/misunderstands Godel's theorem and > provability? > > >> Because provability is relative and computability is not. >> > > I don't think I have said or implied otherwise--by "relative" I assume you > just mean relative to the choice of axiomatic system? Consider the > paragraph of mine about translating Godel's theorem into explicitly > computational terms, the one starting with the words "And given the > understanding that the theorem is specifically about computational systems, > we can think in terms of a program that takes any given set of axioms, like > the Peano axioms"--obviously the notion of what theorems would be proven by > such a computational program would be relative to which program we choose, > we could have such a program for generating all statements provable using > the Peano axioms, or we could have a *different* program for generating all > statements provable with some other axiomatic system. > > > >> Your stock fell for me with this reply. Please don't pretend to spoon >> feed me. You can play teacher with AG, which is out-of-topic regarding ToE. >> AG can pay for lessons somewhere and play "not convinced" twirling his >> moustache and adjusting his monocle elsewhere and the >> trolling/grandstanding, playing god is so abundant... moderate it guys. >> Passivity kills freedom and discourse. >> > > I don't think I am trying to spoon feed you, just defend my view from your > accusations that I am getting something wrong about Godel's theorem, and > assert my own view about what physical questions Godel's theorem applies to > (is there any way I could try to make the same substantive points such that > you would *not* see it as 'pretending to spoon feed you' or 'playing > teacher', even if you continued to think I was incorrect? In other words, > is it something about my language or tone that's bothering you, so that if > I had written things differently you'd have found my posts less > objectionable even if I had made exactly the same counterarguments?) Sorry > if my discussion with AG bothers you, I agree it's off-topic but others > were already responding so I thought I'd try to weigh in as well, like a > lot of nerds I admit I have a bit of the syndrome at https://xkcd.com/386/ > > Jesse > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/4215858a-68b3-4943-9cc2-976edc0791d7n%40googlegroups.com.
