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There is no objective mathematical reality, because it cannot include a statements about its own consistency I'm assuming we're talking about Gödel's incomplet

by cygx 10y ago

There is no objective mathematical reality, because it cannot include a statements about its own consistency I'm assuming we're talking about Gödel's incompleteness theorems? Don't they just say that if there's such a thing as objective mathematical reality, it can't be effectively axiomatized?

candiodari 10y ago
> Don't they just say that if there's such a thing as objective mathematical reality, it can't be effectively axiomatized? No they don't. Real space doesn't appear to be infinite, and Zn is not subject to Godel's incompleteness theorem. If you drop the requirement of infinite numbers and "recursive" infinites (e.g. real numbers), as reality appears to do, there is no problem.
- cygx 10y ago
  Ok. Not sure I buy that (eg right now, we have no reason to believe that the lifetime of the universe is finite), but that's not what I was getting at: My point is that at worst, the incompleteness theorems only imply that we won't be able to write down all the rules that govern the universe.
  - Tloewald 10y ago
    No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Not being able to write all the rules is a completely different and unrelated thing.
    - wbhart 10y ago
      If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. So I can assume them to be true or false, and develop perfectly consistent mathematics. In this way, I can always add more rules to the system (albeit a nonstandard one).
      - lurker83256 10y ago
        This is a misunderstanding. Statement A is actually true in the system, you just cannot prove that it is true. Adding an axiom specifying its falsity would be a contradiction (although you could not prove this).
        baddox 10y ago
        Adding either the Godel sentence or its complement would ruin the consistency of the axiomatic system, because the whole point of the Godel sentence is that it claims that itself cannot be proven to be true in its axiomatic system. But you don't get to add axioms to an axiomatic system anyway, because doing so yields a new axiomatic system to which the original Godel sentence does not refer.
        drdeca 10y ago
        I think by adding the axiom, they mean considering the axiom system with the old axioms, and also the one being added? So, assuming the initial system is consistent, adding the godel sentence, or its negation, produces a new consistent system, iirc. But in the second case, it would be omega-inconsistent ?
      - baddox 10y ago
        A Godel sentence claims that itself cannot be proven in its axiomatic system. If you add another axiom, you have a new axiomatic system that can prove the Godel sentence in the original axiomatic system, but the new axiomatic system will have its own new Godel sentence.
      - lmm 10y ago
        > If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. Assuming the original system was consistent (which you can never prove), yes. But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).
    - cygx 10y ago
      No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Careful there: Gödels first incompleteness theorem proves that in a consistent effectively axiomatized formal system, there will be statements that are true or false that cannot be proven to be true or false. So let's assume there were such a thing as objective mathematical reality and that it could be formalized as The System. As far as I'm aware, all the incompleteness theorem implies is that The System's set of theorems is not recursively enumerable (ie we will never be able to 'write them down'). All we can do is craft necessarily incomplete subsystems applicable to areas of our interest. It's a bit sad that humans will never know the Whole Truth, but such is life...
      - Tloewald 10y ago
        I'm being quite careful: 'consistent effectively axiomatized formal system' is implied by my use of the word 'proves'.
- Jweb_Guru 10y ago
  > Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.
  - naasking 10y ago
    Google "Bekenstein Bound".
    - Jweb_Guru 10y ago
      That doesn't prove anything about the continuity (or lack thereof) of spacetime, just indicates that energy is discretized. Also has no bearing on the original statement about infinity. It's also worth pointing out that the original statement is nonsensical because the theory of real closed fields doesn't suffer from incompleteness in the first place, so there's no need to "save" it from paradox the way you have to save Peano arithmetic.
      - naasking 10y ago
        The Bekenstein bound, as a starting point, addresses the more general point that all of known physics needs only computable abstractions. Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen, which does address your specific question about whether space is truly continuous.
        cygx 10y ago
        Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.
        naasking 10y ago
        We simulate quantum mechanics on classical computers all the time. Superposition requires exponentially more resources in some cases, but it doesn't somehow make QM non-discrete.
- pron 10y ago
  That doesn't really matter. If all you care about is physical reality, you have to apply its implications on everything, and Gödel's incompleteness reappears in a different form. First, consider that undecidability is more general -- i.e., stronger -- than incompleteness, as undecidability implies incompleteness but not vice-versa. If you admit infinity, then you have undecidability and therefore incompleteness. If you don't have infinity because you decide to only care about physical reality, then you don't have undecidability, but then you must treat time as physical (and therefore finite) as well. While you no longer have undecidability (and therefore no incompleteness, either), you do have its finite form, namely infeasibility (due to computational complexity), which, given an assumption that time is finite, amounts to precisely the same result. Almost every theorem about undecidability can be generalized to infeasibility. For example, instead of undecidability of the halting problem, we get infeasibility of the bounded halting problem (that's the basis for the hierarchy theorems, which gave birth to the notion of computational complexity, just as the halting problem gave birth to the notion of computability). Similarly, by bounded halting, instead of a consequence of incompleteness, i.e. "there are statements that can neither be proven nor falsified", you get "there are statements that can be neither feasibly proven nor falsified" -- this is a direct corollary of the hierarchy theorems -- and the problem remains the same. The only question is whether you're willing to admit infinity (or what kinds of infinity you're willing to admit) as a convenience, or as an approximation of large finite quantities. But choosing not to admit infinity doesn't really make anything simpler (on the contrary, things may get more tedious).
akud 10y ago
Was going to say the same thing. Most of my professors were secretly platonists, though they had to pass as formalists to get respect in polite society. It's hard to make absolute claims about the nature of mathematical reality; the formalists need to explain why math is so successful in the real world, and the platonists need to give an account of the ontological nature of mathematical objects.
- breuleux 10y ago
  > why math is so successful in the real world Well, math is a universal approximator -- if there are patterns in what we observe of reality, math can fit them, but that's a far cry from them "being" math. Other formal systems like Turing machines or Lambda calculus can also approximate anything (including each other, naturally) with different primitives... and they have an easier ontology (as far as I can tell). I mean, if you posit that fundamental reality is a computer of sorts, you get your ontology, and you get math as a very good formalism to describe the particular program we're in.
fmap 10y ago
Yes, and indeed Gödel himself believed in an objective mathematical reality. What I meant to say is that the commonly accepted basis for mathematics (first order logic and ZFC) was first justified using arguments which later turned out to be false. Logically, we are not finding better and better approximations to some mathematical laws of nature, but rather making a series of arbitrary choices (you can assert the truth or falsity of any independent statement). Philosophically, we can argue about the sense in which the continuum hypothesis ought to be true or false (that is what Gödel did), but practically it makes more sense to think of logic as a tool that you build to describe and solve specific problems.

by cygx 10y ago

candiodari 10y ago

> Don't they just say that if there's such a thing as objective mathematical reality, it can't be effectively axiomatized? No they don't. Real space doesn't appear to be infinite, and Zn is not subject to Godel's incompleteness theorem. If you drop the requirement of infinite numbers and "recursive" infinites (e.g. real numbers), as reality appears to do, there is no problem.

cygx 10y ago
Ok. Not sure I buy that (eg right now, we have no reason to believe that the lifetime of the universe is finite), but that's not what I was getting at: My point is that at worst, the incompleteness theorems only imply that we won't be able to write down all the rules that govern the universe.
- Tloewald 10y ago
  No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Not being able to write all the rules is a completely different and unrelated thing.
  - wbhart 10y ago
    If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. So I can assume them to be true or false, and develop perfectly consistent mathematics. In this way, I can always add more rules to the system (albeit a nonstandard one).
    - lurker83256 10y ago
      This is a misunderstanding. Statement A is actually true in the system, you just cannot prove that it is true. Adding an axiom specifying its falsity would be a contradiction (although you could not prove this).
      - baddox 10y ago
        Adding either the Godel sentence or its complement would ruin the consistency of the axiomatic system, because the whole point of the Godel sentence is that it claims that itself cannot be proven to be true in its axiomatic system. But you don't get to add axioms to an axiomatic system anyway, because doing so yields a new axiomatic system to which the original Godel sentence does not refer.
        drdeca 10y ago
        I think by adding the axiom, they mean considering the axiom system with the old axioms, and also the one being added? So, assuming the initial system is consistent, adding the godel sentence, or its negation, produces a new consistent system, iirc. But in the second case, it would be omega-inconsistent ?
    - baddox 10y ago
      A Godel sentence claims that itself cannot be proven in its axiomatic system. If you add another axiom, you have a new axiomatic system that can prove the Godel sentence in the original axiomatic system, but the new axiomatic system will have its own new Godel sentence.
    - lmm 10y ago
      > If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. Assuming the original system was consistent (which you can never prove), yes. But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).
  - cygx 10y ago
    No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Careful there: Gödels first incompleteness theorem proves that in a consistent effectively axiomatized formal system, there will be statements that are true or false that cannot be proven to be true or false. So let's assume there were such a thing as objective mathematical reality and that it could be formalized as The System. As far as I'm aware, all the incompleteness theorem implies is that The System's set of theorems is not recursively enumerable (ie we will never be able to 'write them down'). All we can do is craft necessarily incomplete subsystems applicable to areas of our interest. It's a bit sad that humans will never know the Whole Truth, but such is life...
    - Tloewald 10y ago
      I'm being quite careful: 'consistent effectively axiomatized formal system' is implied by my use of the word 'proves'.
Jweb_Guru 10y ago
> Real space doesn't appear to be infinite Citation needed. People keep saying things like this ("real space isn't infinite" or "real spacetime must be discrete") but never show their work.
- naasking 10y ago
  Google "Bekenstein Bound".
  - Jweb_Guru 10y ago
    That doesn't prove anything about the continuity (or lack thereof) of spacetime, just indicates that energy is discretized. Also has no bearing on the original statement about infinity. It's also worth pointing out that the original statement is nonsensical because the theory of real closed fields doesn't suffer from incompleteness in the first place, so there's no need to "save" it from paradox the way you have to save Peano arithmetic.
    - naasking 10y ago
      The Bekenstein bound, as a starting point, addresses the more general point that all of known physics needs only computable abstractions. Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen, which does address your specific question about whether space is truly continuous.
      - cygx 10y ago
        Continuous abstractions are merely for our convenience, and not fundamental as far as we've seen I do not see how that's true for quantum mechanics (take linear superposition as just one example). People may speculate about discrete underpinnings, but at this time, it's merely speculation.
        naasking 10y ago
        We simulate quantum mechanics on classical computers all the time. Superposition requires exponentially more resources in some cases, but it doesn't somehow make QM non-discrete.
pron 10y ago
That doesn't really matter. If all you care about is physical reality, you have to apply its implications on everything, and Gödel's incompleteness reappears in a different form. First, consider that undecidability is more general -- i.e., stronger -- than incompleteness, as undecidability implies incompleteness but not vice-versa. If you admit infinity, then you have undecidability and therefore incompleteness. If you don't have infinity because you decide to only care about physical reality, then you don't have undecidability, but then you must treat time as physical (and therefore finite) as well. While you no longer have undecidability (and therefore no incompleteness, either), you do have its finite form, namely infeasibility (due to computational complexity), which, given an assumption that time is finite, amounts to precisely the same result. Almost every theorem about undecidability can be generalized to infeasibility. For example, instead of undecidability of the halting problem, we get infeasibility of the bounded halting problem (that's the basis for the hierarchy theorems, which gave birth to the notion of computational complexity, just as the halting problem gave birth to the notion of computability). Similarly, by bounded halting, instead of a consequence of incompleteness, i.e. "there are statements that can neither be proven nor falsified", you get "there are statements that can be neither feasibly proven nor falsified" -- this is a direct corollary of the hierarchy theorems -- and the problem remains the same. The only question is whether you're willing to admit infinity (or what kinds of infinity you're willing to admit) as a convenience, or as an approximation of large finite quantities. But choosing not to admit infinity doesn't really make anything simpler (on the contrary, things may get more tedious).

akud 10y ago

Was going to say the same thing. Most of my professors were secretly platonists, though they had to pass as formalists to get respect in polite society. It's hard to make absolute claims about the nature of mathematical reality; the formalists need to explain why math is so successful in the real world, and the platonists need to give an account of the ontological nature of mathematical objects.

breuleux 10y ago
> why math is so successful in the real world Well, math is a universal approximator -- if there are patterns in what we observe of reality, math can fit them, but that's a far cry from them "being" math. Other formal systems like Turing machines or Lambda calculus can also approximate anything (including each other, naturally) with different primitives... and they have an easier ontology (as far as I can tell). I mean, if you posit that fundamental reality is a computer of sorts, you get your ontology, and you get math as a very good formalism to describe the particular program we're in.

fmap 10y ago

Yes, and indeed Gödel himself believed in an objective mathematical reality. What I meant to say is that the commonly accepted basis for mathematics (first order logic and ZFC) was first justified using arguments which later turned out to be false. Logically, we are not finding better and better approximations to some mathematical laws of nature, but rather making a series of arbitrary choices (you can assert the truth or falsity of any independent statement). Philosophically, we can argue about the sense in which the continuum hypothesis ought to be true or false (that is what Gödel did), but practically it makes more sense to think of logic as a tool that you build to describe and solve specific problems.