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 <[email protected]> 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 <[email protected]> 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 [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/4215858a-68b3-4943-9cc2-976edc0791d7n%40googlegroups.com.
