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All formal systems are incomplete or inconsistent. There is zero evidence that the universe is either.

drdeca 6y ago
There is a difference between a mathematical object and a formal system. Surely you would not call a differential equation with a given initial condition describing a simple harmonic oscillator along with an initial condition “inconsistent”. Would you call it incomplete? Doing so would seem rather odd seeing as there is exactly one solution (given the initial condition). And the sense of “incomplete” that the incompleteness theorems are about doesn’t apply to this. I don’t just mean that it isn’t true of it. I mean that it is not an applicable idea. It is like asking what the northernmost point of the color blue is. People should be careful when trying to apply Gödel’s incompleteness theorems to philosophical questions, as people often do so without really understanding what they say, and, importantly, what they do not say. Something that may help is to familiarize oneself with Gödel’s _completeness_ theorem.
- guscost 6y ago
  > Surely you would not call a differential equation with a given initial condition describing a simple harmonic oscillator along with an initial condition “inconsistent”. No, certainly not. > Would you call it incomplete? Yes, absolutely. There is no such thing as a harmonic oscillator that exactly follows an equation like that. Show me one. Either the formal system has nothing to do with the real world, or it is an incomplete approximation of it. All other known physical theories are the same way. Can even a part of the universe be described completely by any (incomplete) formal system? Is it even meaningful to talk about a “part” of the universe in this sense? Well, there’s exactly zero evidence either way. That’s what I’m implying when I say “the universe may not be incomplete.”
  - drdeca 6y ago
    > Either the formal system has nothing to do with the real world, or it is an incomplete approximation of it. Ok, but this is a totally separate meaning of “incomplete” than what the incompleteness theorems are talking about. So, if that is what you meant by “incomplete” in your original comment, you should know that your comment looked like it was appealing to Gödel’s incompleteness theorems, which do not deal with that sense of “completeness”.
    - guscost 6y ago
      Added some clarification - Yes, the incompleteness theorems only concern the behavior of mathematical/formal systems themselves. But to me it seems perfectly reasonable that correspondingly, there may be features of the universe ("true statements") that cannot be modeled ("proven") by any given physical theory ("formal system"). I cannot prove this, but nobody can prove the opposite belief, either.

There is a difference between a mathematical object and a formal system. Surely you would not call a differential equation with a given initial condition describing a simple harmonic oscillator along with an initial condition “inconsistent”. Would you call it incomplete? Doing so would seem rather odd seeing as there is exactly one solution (given the initial condition). And the sense of “incomplete” that the incompleteness theorems are about doesn’t apply to this. I don’t just mean that it isn’t true of it. I mean that it is not an applicable idea. It is like asking what the northernmost point of the color blue is. People should be careful when trying to apply Gödel’s incompleteness theorems to philosophical questions, as people often do so without really understanding what they say, and, importantly, what they do not say. Something that may help is to familiarize oneself with Gödel’s _completeness_ theorem.

guscost 6y ago
> Surely you would not call a differential equation with a given initial condition describing a simple harmonic oscillator along with an initial condition “inconsistent”. No, certainly not. > Would you call it incomplete? Yes, absolutely. There is no such thing as a harmonic oscillator that exactly follows an equation like that. Show me one. Either the formal system has nothing to do with the real world, or it is an incomplete approximation of it. All other known physical theories are the same way. Can even a part of the universe be described completely by any (incomplete) formal system? Is it even meaningful to talk about a “part” of the universe in this sense? Well, there’s exactly zero evidence either way. That’s what I’m implying when I say “the universe may not be incomplete.”
- drdeca 6y ago
  > Either the formal system has nothing to do with the real world, or it is an incomplete approximation of it. Ok, but this is a totally separate meaning of “incomplete” than what the incompleteness theorems are talking about. So, if that is what you meant by “incomplete” in your original comment, you should know that your comment looked like it was appealing to Gödel’s incompleteness theorems, which do not deal with that sense of “completeness”.
  - guscost 6y ago
    Added some clarification - Yes, the incompleteness theorems only concern the behavior of mathematical/formal systems themselves. But to me it seems perfectly reasonable that correspondingly, there may be features of the universe ("true statements") that cannot be modeled ("proven") by any given physical theory ("formal system"). I cannot prove this, but nobody can prove the opposite belief, either.