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You're overlooking something basic and uncontroversial that may prevent such an analysis from being deterministic. Gödel's incompleteness theorems say that, for

by lutusp 13y ago

You're overlooking something basic and uncontroversial that may prevent such an analysis from being deterministic. Gödel's incompleteness theorems say that, for a sufficiently complex system, there are true statements that cannot be proven from within that system, using the system's methods. This means that, from a perspective within the universe, we may be constrained by the incompleteness theorems from making any kind of conclusive determination as to its true nature. Gödel's incompleteness theorems have a wide reach. They prevent a solution to the Turing Halting problem, and they placed a firm limit on projects such as Russell and Whitehead's tendentiously named "Principia Marthematica" (http://en.wikipedia.org/wiki/Principia_Mathematica http://en.wikipedia.org/wiki/Principia_Mathematica), a very ambitious project that was fatally undermined by Gödel's work. The incompleteness theorems need to be considered in the present question -- we're inside the "system" we're trying to analyze, consequently there are questions we can't meaningfully resolve. I think the question about the universe being a simulation meets this classic criterion for indeterminacy.

evanb 13y ago
I would love a way, and have indeed tried (in my ambitious, misspent youth) to formulate a way, to make this observation precise for quantum field theories. I am sure there is a way, but it's tough. Only the Gödel sentences are true but not provable. Maybe the sentence "the lattice spacing is finite" is not a Gödel sentence. Indeed, the thrust of the paper seems to be exactly that, and I agree. When I think too hard about apply Gödel to physics in my night thoughts, it makes me wonder if there is really a "final theory of everything" or if every theory is really just an effective field theory [0] with another deeper theory underneath, and it's turtles all the way down. [0] https://en.wikipedia.org/wiki/Effective_field_theory https://en.wikipedia.org/wiki/Effective_field_theory
- lutusp 13y ago
  > Only the Gödel sentences are true but not provable. An important aspect of the incompleteness theorems, relevant to the present topic, is that one cannot deterministically locate those particular true claims that cannot be proven true, to which the theorems apply. If this were not so, it would undermine the very notion of indeterminacy. But the subjective nature of our view of the universe extends beyond the incompleteness theorems into more pedestrian issues like how we can claim to objectively define a property of physical reality, using tools and methods embedded in that same reality. > When I think too hard about apply Gödel to physics in my night thoughts, it makes me wonder if there is really a "final theory of everything" or if every theory is really just an effective field theory [0] with another deeper theory underneath, and it's turtles all the way down. "Turtles all the way down" -- I love that expression -- it aptly describes the problem faced by creatures trying to fully define their domain from within. From time to time I picture the day this expression came into being ... "'You're very clever, young man, very clever,' said the old lady. 'But it's tortoises all the way down!'" http://en.wikipedia.org/wiki/Turtles_all_the_way_down#Origin http://en.wikipedia.org/wiki/Turtles_all_the_way_down#Origin
  - evanb 13y ago
    > An important aspect of the incompleteness theorems, relevant to the present topic, is that one cannot deterministically locate those particular true claims that cannot be proven true, to which the theorems apply. If this were not so, it would undermine the very notion of indeterminacy. Right, I agree, but that doesn't necessarily mean that we cannot prove any true claims. Unless I misunderstand, it just means that true claims whose proof we do not know cannot be distinguished from true claims whose proof does not exist. We can search for proofs to both, and sometimes we will find a proof, and sometimes we won't. But not finding a proof doesn't mean that the claim is a Gödel sentence, it could also mean that we just didn't look hard enough. Also: I should really... proof read my posts better. "too hard about apply..." I cringe at the fact that my carelessness will be preserved for posterity.
    - lutusp 13y ago
      > Right, I agree, but that doesn't necessarily mean that we cannot prove any true claims. That's certainly true, but the fact that we can't locate a demarcation between provable and unprovable makes all analysis inside system X problematical as to the true nature of X, with an obvious connection to the present topic. > Unless I misunderstand, it just means that true claims whose proof we do not know cannot be distinguished from true claims whose proof does not exist. Not exactly. The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. The second category is a problem that undermines any global analysis of logical systems. Remember that Russell and Whitehead's famous tome (http://en.wikipedia.org/wiki/Principia_Mathematica http://en.wikipedia.org/wiki/Principia_Mathematica) wasn't edited after Gödel's work, to simply remove those parts that were unsupportable, it was instead abandoned in its entirety. Gödel's work demonstrated that any such deterministic analysis of logical systems wasn't possible, in whole or in part. But again, this doesn't argue that a given logical system can't be used to prove theorems. It only argues against the claim that such a system can be proven to be both complete and internally consistent.
    - evanb 13y ago
      >The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. Can you clarify that a little bit? It seems like you define the demarcation in terms of itself.
evanb 13y ago
Also, Seth Lloyd turns the question [0] on its head and suggests that it's silly to call the universe anything but a computer, since it contains systems that are Turing Complete. What is it computing? Its own history / dynamics. That said, it is different to ask if we're in a simulation vs. in a computer---the word simulation makes one think of some programmer compiling some source code and executing some program: it entails agency. [0] http://www.randomhouse.com/kvpa/lloyd/ http://www.randomhouse.com/kvpa/lloyd/
- lutusp 13y ago
  Good point. When we ask whether this is all a simulation, I think most people assume an agency, a wizard behind a curtain. But the question doesn't necessarily imply that form. But think -- until about 50 years ago we wouldn't have necessarily thought of the universe as a computer, because we didn't think of things in terms of computers until recently. My point is this conversation and its terminology is only possible because of the present state of technical evolution -- in us and in our machines. > What is it computing? Its own history / dynamics. Yes, agreed. On a scale smaller than the universe, we can examine evolution by natural selection as an example of an algorithm that works in a reliable, predictable way (predictable only in a statistical sense). I prefer evolution as an example of a natural algorithmic model because it's easy to express, but produces so much complexity.
- lotsofmangos 13y ago
  By that logic, my house is a computer as it contains many systems that are Turing complete.
  - evanb 13y ago
    The logic should more precisely be: the dynamics of the universe, ie. the things that physics allows, includes the operation of Turing machines, so that the physical laws are at least Turing complete. Thus, the universe, from a computational point of view, a computer.
    - dinkumthinkum 13y ago
improv32 13y ago
I'm not sure the GIT applies here, it is my understanding that the GIT says that for any formal system (i.e. sentences derived by repeated application of rules of derivation to axioms) that is able to represent the Peano arithmetic, has no deterministic decision procedure, due to the ability of a sentence to self-reference. Is that accurate, or is GIT broader than I thought?
- lutusp 13y ago
  I mentioned the GIT as an example of the risk of invalidation by self-reference, but whether they formally apply depends on how we state our hypotheses. They certainly apply in some degree to any effort to define strict deterministic criteria by which one might conclude that the universe is a simulation, carried out from within that same universe. The more formal and detailed the criteria, the more likely we are to be trapped by self-reference. > Is that accurate, or is GIT broader than I thought? I think the ambition of the simulation question offers enough similarity to a complex logical system that one needn't assume any greater scope for the GIT than they're known to have. I'm not claiming that the GIT obviously applies, only that the possibility needs to be taken into account on a list of issues that attend this question.
eli_gottlieb 13y ago
A) Goedel sentences involve some form of mathematical self-reference/recursion within the proof system. B) Empirical/physical science doesn't deal with conclusive proof from axioms, anyway: it deals mainly in probability and evidential falsification. Ergo, Goedel's Theorem doesn't really tell us there's some great, unlearnable Fact About The Universe we can never figure out, since science isn't a proof system subject to the Theorem in the first place. We can probably put very good probability bounds (so to speak) on anything about real life that we actually need to know.
- lutusp 13y ago
  > Empirical/physical science doesn't deal with conclusive proof from axioms, anyway: it deals mainly in probability and evidential falsification. Of course it does. Modern physics requires rigorous mathematics, the latter certainly are influenced by the incompleteness theorems. If I shape a theory about the universe in modern physics, if my theory doesn't have a mathematical form it won't be taken seriously for very good reasons. Therefore modern physics, the most scientific of sciences, requires a very high level of mathematical reasoning. Therefore the incompleteness theorems need to be taken into account. Even something as trivial as the Turing halting problem is known to be insoluble because of its connection to the incompleteness theorems. Link: http://en.wikipedia.org/wiki/Halting_problem http://en.wikipedia.org/wiki/Halting_problem Quote: "The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem." This is just one of many examples in which the incompleteness theorems affect the outcome of a pedestrian scientific issue. > ... since science isn't a proof system subject to the Theorem in the first place. This is false. Beyond the examples given above, many aspects of present scientific theories are influenced, directly or indirectly, by the incompleteness theorems.
  - DerpDerpDerp 13y ago
    Your reversing of the relationship between the Halting Problem and the Incompleteness theorem (based on the quote you provided) makes me doubt the rest of your conclusions. It's not that the Incompleteness Theorem tells us about the Halting Problem, but rather, that the Halting Problem tells us about the Incompleteness theorem. Also, we have no reason to think that the universe satisfies some of the requirements of the Incompleteness Theorem (or even things like the Halting Problem). Both require some kind of unbounded or self-referential behavior, which we might not have in the universe.

by lutusp 13y ago

evanb 13y ago

I would love a way, and have indeed tried (in my ambitious, misspent youth) to formulate a way, to make this observation precise for quantum field theories. I am sure there is a way, but it's tough. Only the Gödel sentences are true but not provable. Maybe the sentence "the lattice spacing is finite" is not a Gödel sentence. Indeed, the thrust of the paper seems to be exactly that, and I agree. When I think too hard about apply Gödel to physics in my night thoughts, it makes me wonder if there is really a "final theory of everything" or if every theory is really just an effective field theory [0] with another deeper theory underneath, and it's turtles all the way down. [0] https://en.wikipedia.org/wiki/Effective_field_theory https://en.wikipedia.org/wiki/Effective_field_theory

lutusp 13y ago
> Only the Gödel sentences are true but not provable. An important aspect of the incompleteness theorems, relevant to the present topic, is that one cannot deterministically locate those particular true claims that cannot be proven true, to which the theorems apply. If this were not so, it would undermine the very notion of indeterminacy. But the subjective nature of our view of the universe extends beyond the incompleteness theorems into more pedestrian issues like how we can claim to objectively define a property of physical reality, using tools and methods embedded in that same reality. > When I think too hard about apply Gödel to physics in my night thoughts, it makes me wonder if there is really a "final theory of everything" or if every theory is really just an effective field theory [0] with another deeper theory underneath, and it's turtles all the way down. "Turtles all the way down" -- I love that expression -- it aptly describes the problem faced by creatures trying to fully define their domain from within. From time to time I picture the day this expression came into being ... "'You're very clever, young man, very clever,' said the old lady. 'But it's tortoises all the way down!'" http://en.wikipedia.org/wiki/Turtles_all_the_way_down#Origin http://en.wikipedia.org/wiki/Turtles_all_the_way_down#Origin
- evanb 13y ago
  > An important aspect of the incompleteness theorems, relevant to the present topic, is that one cannot deterministically locate those particular true claims that cannot be proven true, to which the theorems apply. If this were not so, it would undermine the very notion of indeterminacy. Right, I agree, but that doesn't necessarily mean that we cannot prove any true claims. Unless I misunderstand, it just means that true claims whose proof we do not know cannot be distinguished from true claims whose proof does not exist. We can search for proofs to both, and sometimes we will find a proof, and sometimes we won't. But not finding a proof doesn't mean that the claim is a Gödel sentence, it could also mean that we just didn't look hard enough. Also: I should really... proof read my posts better. "too hard about apply..." I cringe at the fact that my carelessness will be preserved for posterity.
  - lutusp 13y ago
    > Right, I agree, but that doesn't necessarily mean that we cannot prove any true claims. That's certainly true, but the fact that we can't locate a demarcation between provable and unprovable makes all analysis inside system X problematical as to the true nature of X, with an obvious connection to the present topic. > Unless I misunderstand, it just means that true claims whose proof we do not know cannot be distinguished from true claims whose proof does not exist. Not exactly. The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. The second category is a problem that undermines any global analysis of logical systems. Remember that Russell and Whitehead's famous tome (http://en.wikipedia.org/wiki/Principia_Mathematica http://en.wikipedia.org/wiki/Principia_Mathematica) wasn't edited after Gödel's work, to simply remove those parts that were unsupportable, it was instead abandoned in its entirety. Gödel's work demonstrated that any such deterministic analysis of logical systems wasn't possible, in whole or in part. But again, this doesn't argue that a given logical system can't be used to prove theorems. It only argues against the claim that such a system can be proven to be both complete and internally consistent.
  - evanb 13y ago
    >The demarcation is between theorems whose truth we can prove (and can prove that we can prove), and theorems we're not sure about but that may fall on the far side of the demarcation line. Can you clarify that a little bit? It seems like you define the demarcation in terms of itself.

evanb 13y ago

Also, Seth Lloyd turns the question [0] on its head and suggests that it's silly to call the universe anything but a computer, since it contains systems that are Turing Complete. What is it computing? Its own history / dynamics. That said, it is different to ask if we're in a simulation vs. in a computer---the word simulation makes one think of some programmer compiling some source code and executing some program: it entails agency. [0] http://www.randomhouse.com/kvpa/lloyd/ http://www.randomhouse.com/kvpa/lloyd/

lutusp 13y ago
Good point. When we ask whether this is all a simulation, I think most people assume an agency, a wizard behind a curtain. But the question doesn't necessarily imply that form. But think -- until about 50 years ago we wouldn't have necessarily thought of the universe as a computer, because we didn't think of things in terms of computers until recently. My point is this conversation and its terminology is only possible because of the present state of technical evolution -- in us and in our machines. > What is it computing? Its own history / dynamics. Yes, agreed. On a scale smaller than the universe, we can examine evolution by natural selection as an example of an algorithm that works in a reliable, predictable way (predictable only in a statistical sense). I prefer evolution as an example of a natural algorithmic model because it's easy to express, but produces so much complexity.
lotsofmangos 13y ago
By that logic, my house is a computer as it contains many systems that are Turing complete.
- evanb 13y ago
  The logic should more precisely be: the dynamics of the universe, ie. the things that physics allows, includes the operation of Turing machines, so that the physical laws are at least Turing complete. Thus, the universe, from a computational point of view, a computer.
  - dinkumthinkum 13y ago

improv32 13y ago

I'm not sure the GIT applies here, it is my understanding that the GIT says that for any formal system (i.e. sentences derived by repeated application of rules of derivation to axioms) that is able to represent the Peano arithmetic, has no deterministic decision procedure, due to the ability of a sentence to self-reference. Is that accurate, or is GIT broader than I thought?

lutusp 13y ago
I mentioned the GIT as an example of the risk of invalidation by self-reference, but whether they formally apply depends on how we state our hypotheses. They certainly apply in some degree to any effort to define strict deterministic criteria by which one might conclude that the universe is a simulation, carried out from within that same universe. The more formal and detailed the criteria, the more likely we are to be trapped by self-reference. > Is that accurate, or is GIT broader than I thought? I think the ambition of the simulation question offers enough similarity to a complex logical system that one needn't assume any greater scope for the GIT than they're known to have. I'm not claiming that the GIT obviously applies, only that the possibility needs to be taken into account on a list of issues that attend this question.

eli_gottlieb 13y ago

A) Goedel sentences involve some form of mathematical self-reference/recursion within the proof system. B) Empirical/physical science doesn't deal with conclusive proof from axioms, anyway: it deals mainly in probability and evidential falsification. Ergo, Goedel's Theorem doesn't really tell us there's some great, unlearnable Fact About The Universe we can never figure out, since science isn't a proof system subject to the Theorem in the first place. We can probably put very good probability bounds (so to speak) on anything about real life that we actually need to know.

lutusp 13y ago
> Empirical/physical science doesn't deal with conclusive proof from axioms, anyway: it deals mainly in probability and evidential falsification. Of course it does. Modern physics requires rigorous mathematics, the latter certainly are influenced by the incompleteness theorems. If I shape a theory about the universe in modern physics, if my theory doesn't have a mathematical form it won't be taken seriously for very good reasons. Therefore modern physics, the most scientific of sciences, requires a very high level of mathematical reasoning. Therefore the incompleteness theorems need to be taken into account. Even something as trivial as the Turing halting problem is known to be insoluble because of its connection to the incompleteness theorems. Link: http://en.wikipedia.org/wiki/Halting_problem http://en.wikipedia.org/wiki/Halting_problem Quote: "The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem." This is just one of many examples in which the incompleteness theorems affect the outcome of a pedestrian scientific issue. > ... since science isn't a proof system subject to the Theorem in the first place. This is false. Beyond the examples given above, many aspects of present scientific theories are influenced, directly or indirectly, by the incompleteness theorems.
- DerpDerpDerp 13y ago
  Your reversing of the relationship between the Halting Problem and the Incompleteness theorem (based on the quote you provided) makes me doubt the rest of your conclusions. It's not that the Incompleteness Theorem tells us about the Halting Problem, but rather, that the Halting Problem tells us about the Incompleteness theorem. Also, we have no reason to think that the universe satisfies some of the requirements of the Incompleteness Theorem (or even things like the Halting Problem). Both require some kind of unbounded or self-referential behavior, which we might not have in the universe.