4 ms·

That's circular logic. Under the mathematical universe hypotheses, a map, being isomorphic to the territory, would be equivalent.

pron 9y ago
Right, but the claim that mathematical isomorphism is the same thing as physical equivalence is a non-obvious claim at all.
- deleted 9y ago
  [deleted]
TheOtherHobbes 9y ago
The original claim is circular. Physics may be mathematical, but that doesn't mean you can start from the Big Bang and use mathematical physics to predict the contents of this particular thread on Hacker News. So you immediately have the problem that mathematical physics cannot possibly be a complete description of the universe. Of course you can argue that we simply don't know the math yet, and if we did we'd be able to start from the primordial singularity and predict the existence of Max Tegmark, Y Combinator, and me and you. But that's a very strong claim, and I'm not aware of any evidence that comes close to supporting it. Even if you make a much looser claim - that mathematical physics can predict classes of phenomena like Max Tegmark, etc - there's still no math around at the moment that even begins to pretend it can formally prove that assertion. Worse - strong claims about mathematical provability immediately fall apart because of Godel. How can you have an abstract formal system running the universe if that system can't possibly be complete, provable, and formally consistent in its own terms?
- akvadrako 9y ago
  Which has nothing to do with my point, which was that the grandparent was assuming the MUH wasn't true as evidence that it isn't.
  - pron 9y ago
    I didn't assume it is not true nor claimed it is not true, simply pointed out that it is not obviously justified as the simplest explanation to the fact that that math can explain the universe.
- posterboy 9y ago
  > Worse - strong claims about mathematical provability immediately fall apart because of Godel incompleteness means true but unprovable propositions in specific forms of higher order logic, but Kurt also proved the completeness of first order logic. However, the scheme of induction used to lower peano arithmetic to first order logic (Presburger Arithmetic e.g.). Likewise, an incomplete higher order logic can (ostensibly - I couldn't) be proved in a logic of even higher degree, which is again incomplete, so again that's an argument ad infinitum. Obviously that's abstract nonsense, but the question shouldn't be whether the universe can be predicted up to infinity. The question at face value is rather, whether quantitative analysis can solve any given qualitative problem. It's the syntax semantics duality - what comes first?
  - pron 9y ago
    For someone who's on a first-name basis with Gödel, you should review your logic. The incompleteness theorems hold for any formal system whatsoever, regardless of order. They are simple corrolaries of the halting theorem. First order logic is indeed complete in the following sense: A sentence of FOL is provable iff it's valid, i.e. true in all interpretations. But according to the incompleteness theorem, logical theories (sets of axioms) of a certain richness would always yield sentences that are true in some interpretations and false in others, or, alternatively, that are neither provable nor disprovable (and first-order Pano arithmetics and obviously all formal set theories are among them). Order and other particulars of a formal system are irrelevant, and indeed Gödel was only fully satisfied with his incompleteness theorems after he read Turing's famous paper on computability, and regarded it a "miracle" (as it applies to any formal system, and depends on none). Obviously, many quantitative problems are unsolvable as a direct result of the halting theorem, and more generally the time-hierarchy theorem (which states that there are problems of any arbitrary difficulty), as many decidable problems are still unsolvable due to intractibility.
    - posterboy 9y ago
      see, the halting problem is much more appropriate for this audience :) However, there is the calculus of constructions, which, as far as I can tell, tries to be provable in all admissible formulas. It's a totally functional language. So, is the universe totally functional, will it halt? I'm not sure that's the actual question, neither whether the universe is beyond our understanding, that's out of the question, only, does the universe follow rules or is it deterministic and not only in the quantum sense. > all computable mathematical structures (in Gödel's sense) exist - en.wikipedia/wiki/Mathematical_universe_hypothesis That's trivially true, because they exist in the heads of the mathematicians constructing them.
- kazinator 9y ago
  > Physics may be mathematical, but that doesn't mean you can start from the Big Bang and use mathematical physics to predict the contents of this particular thread on Hacker News. Only because the only available approaches for even beginning to do anything of that sort are constrained by being embedded in that same universe. It's unsurprising that a system cannot exist, and at the same time embed a calculation which calculates all of itself. (We cannot even use mathematics to predict simple N-body problems; we have to switch to numerical techniques which have limited precision. Once you have floating-point in there, it's no longer the math, but an approximation.)

pron 9y ago

Right, but the claim that mathematical isomorphism is the same thing as physical equivalence is a non-obvious claim at all.

deleted 9y ago
[deleted]

TheOtherHobbes 9y ago

The original claim is circular. Physics may be mathematical, but that doesn't mean you can start from the Big Bang and use mathematical physics to predict the contents of this particular thread on Hacker News. So you immediately have the problem that mathematical physics cannot possibly be a complete description of the universe. Of course you can argue that we simply don't know the math yet, and if we did we'd be able to start from the primordial singularity and predict the existence of Max Tegmark, Y Combinator, and me and you. But that's a very strong claim, and I'm not aware of any evidence that comes close to supporting it. Even if you make a much looser claim - that mathematical physics can predict classes of phenomena like Max Tegmark, etc - there's still no math around at the moment that even begins to pretend it can formally prove that assertion. Worse - strong claims about mathematical provability immediately fall apart because of Godel. How can you have an abstract formal system running the universe if that system can't possibly be complete, provable, and formally consistent in its own terms?

akvadrako 9y ago
Which has nothing to do with my point, which was that the grandparent was assuming the MUH wasn't true as evidence that it isn't.
- pron 9y ago
  I didn't assume it is not true nor claimed it is not true, simply pointed out that it is not obviously justified as the simplest explanation to the fact that that math can explain the universe.
posterboy 9y ago
> Worse - strong claims about mathematical provability immediately fall apart because of Godel incompleteness means true but unprovable propositions in specific forms of higher order logic, but Kurt also proved the completeness of first order logic. However, the scheme of induction used to lower peano arithmetic to first order logic (Presburger Arithmetic e.g.). Likewise, an incomplete higher order logic can (ostensibly - I couldn't) be proved in a logic of even higher degree, which is again incomplete, so again that's an argument ad infinitum. Obviously that's abstract nonsense, but the question shouldn't be whether the universe can be predicted up to infinity. The question at face value is rather, whether quantitative analysis can solve any given qualitative problem. It's the syntax semantics duality - what comes first?
- pron 9y ago
  For someone who's on a first-name basis with Gödel, you should review your logic. The incompleteness theorems hold for any formal system whatsoever, regardless of order. They are simple corrolaries of the halting theorem. First order logic is indeed complete in the following sense: A sentence of FOL is provable iff it's valid, i.e. true in all interpretations. But according to the incompleteness theorem, logical theories (sets of axioms) of a certain richness would always yield sentences that are true in some interpretations and false in others, or, alternatively, that are neither provable nor disprovable (and first-order Pano arithmetics and obviously all formal set theories are among them). Order and other particulars of a formal system are irrelevant, and indeed Gödel was only fully satisfied with his incompleteness theorems after he read Turing's famous paper on computability, and regarded it a "miracle" (as it applies to any formal system, and depends on none). Obviously, many quantitative problems are unsolvable as a direct result of the halting theorem, and more generally the time-hierarchy theorem (which states that there are problems of any arbitrary difficulty), as many decidable problems are still unsolvable due to intractibility.
  - posterboy 9y ago
    see, the halting problem is much more appropriate for this audience :) However, there is the calculus of constructions, which, as far as I can tell, tries to be provable in all admissible formulas. It's a totally functional language. So, is the universe totally functional, will it halt? I'm not sure that's the actual question, neither whether the universe is beyond our understanding, that's out of the question, only, does the universe follow rules or is it deterministic and not only in the quantum sense. > all computable mathematical structures (in Gödel's sense) exist - en.wikipedia/wiki/Mathematical_universe_hypothesis That's trivially true, because they exist in the heads of the mathematicians constructing them.
kazinator 9y ago
> Physics may be mathematical, but that doesn't mean you can start from the Big Bang and use mathematical physics to predict the contents of this particular thread on Hacker News. Only because the only available approaches for even beginning to do anything of that sort are constrained by being embedded in that same universe. It's unsurprising that a system cannot exist, and at the same time embed a calculation which calculates all of itself. (We cannot even use mathematics to predict simple N-body problems; we have to switch to numerical techniques which have limited precision. Once you have floating-point in there, it's no longer the math, but an approximation.)