- #36
Grinkle
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WWGD said:(except maybe theoretical physics)
Per my own lay understanding, except for theoretical physics, the rest is experimental physics.
WWGD said:(except maybe theoretical physics)
WWGD said:But physics is not developed/modeled axiomatically but instead experimentally, isn't it (except maybe theoretical physics)?
I guess a sort of T.O.T.E https://en.wikipedia.org/wiki/T.O.T.E . ?stevendaryl said:Well, the way that physics can be described is an iteration of:
- Do experiments.
- Make up a mathematical model that would allow you to predict the results.
- Do other experiments to test that model.
- If it fails, go back to 2 and repeat.
The trailing dot was chopped on your link.WWGD said:https://en.wikipedia.org/wiki/T.O.T.E .
Grinkle said:I may be missing something (I am certainly missing many somethings!) but I don't understand why this question should be a concern. We have many models that describe reality, and the reason they are seen to be incomplete is because of observations about reality. I think that it may be impossible to prove mathematically that 1+1 is 2. That is interesting, but it has no bearing on why F=MA is known to not be accurate as velocities approach c, and why it is known that relativity cannot make predictions when the force of gravity becomes very large.
If we ever formulate a mathematical model of everything that is consistent across all known and hypothesized physical situations, and every prediction that model makes is verified by experiment and observation, I expect few will be concerned that we cannot prove from a finite list of axioms that 1+1 is 2, and so by implication we cannot establish any mathematical proof the model from a purely mathematical perspective. (I have probably not phrased that lat part correctly at all, but I hope that is the gist of it).
stevendaryl said:Well, it is certainly not true that "it is impossible to prove that 1+1=2". That's about the easiest proof there is.
If one is ok to unbound the life of the solar system ignoring that the sun will explode sometime, then what you argue makes sense to me. I can see how even with a solid theory of orbital mechanics it might be impossible to use that theory to prove that orbits are stable for an infinite time. Is that truly a consequence of Godel's theorem? (not implying rhetorically that I doubt it, I am really asking)
The "laws of physics" only predict what kinds of events will happen, not that any particular event will happen. That's why they are called general laws. I don't think GIT has anything to do with the accuracy of predictions based on initial conditions (such as whether the solar system is stable or not). Just because we don't have enough information to be completely accurate does not mean that the rules we use are inconsistent or incomplete.stevendaryl said:Even if we have the complete theory of everything, there would still be questions that we couldn't answer about that theory.
Sure, Whitehead and Russell needed only a few hundred pages to prove that.stevendaryl said:Well, it is certainly not true that "it is impossible to prove that 1+1=2". That's about the easiest proof there is.
Has anyone proved that (sq root of 2)^{2}=2?Demystifier said:Sure, Whitehead and Russell needed only a few hundred pages to prove that.
friend said:I don't think GIT has anything to do with the accuracy of predictions based on initial conditions (such as whether the solar system is stable or not).
If Godel's Incompleteness Theorem is the reason of unpredictability in the laws of physics, then there MUST be some underlying system of logic from which those law are derived. What do you suppose that could be? And how does self-reference enter the picture to produce said incompleteness?stevendaryl said:I would say that it does. Certain types of questions, of the form "Will state S ever evolve into state S'?" are not solvable, even with perfect knowledge of the initial conditions and perfect knowledge of the laws of motion by which one state evolves into another. Whether chaotic natural systems are an example or not is not clear (to me), but I don't know of any reason to think that they are not subject to Godel's incompleteness.
friend said:In order for Godel's incompleteness theorem to apply to physics, you'd have to prove that the mathematical laws of physics were derived from some system of logic.
The GIT ONLY applies if you require every mathematical formula provable in math. The laws of physics do not require every possible equation of math provable in math. It only requires a subset of that math. Math is not equivalent to physics.stevendaryl said:Once you have a theory of physics, that question is now a completely mathematical question, and such mathematical questions are subject to Godel's incompleteness theorem. There may be questions of that type that cannot be answered.
friend said:If Godel's Incompleteness Theorem is the reason of unpredictability in the laws of physics, then there MUST be some underlying system of logic from which those law are derived. What do you suppose that could be?
friend said:The GIT ONLY applies if you require every mathematical formula provable in math. The laws of physics do not require every possible equation of math provable in math. It only requires a subset of that math. Math is not equivalent to physics.
stevendaryl said:Well, our theories of physics depend at the very least on arithmetic. Here are some arithmetic facts:
Any system which is capable of deriving the above general facts is subject to Godelian incompleteness. And if it isn't capable of deriving the above facts, then I would say it was too weak to use for physics.
- For any integer [itex]n[/itex], [itex]n+0 = n[/itex]
- For any integer [itex]n[/itex], [itex]n*0 = 0[/itex]
- For any two integers [itex]i[/itex] and [itex]j[/itex], [itex](i+1) + j = (i+j)+1[/itex]
- For any two integers [itex]i[/itex] and [itex]j[/itex], [itex](i+1) * j = (i*j)+j[/itex]
I think you're confusing two different things: (1) What is the world really doing? (2) What can we predict about the world?
The first is not subject to Godel's theorem--the world does whatever it does, and doesn't care about anybody's theorems. The second is about our ability to reason about the world using mathematics. Godel's theorem certainly applies to that.
So when you talk about the "laws of physics", I'm not sure whether you talking about #1 or #2. I'm not claiming that Godel has anything to do with what the world actually does, but it certainly is relevant to our ability to predict what will happen in the far future.
friend said:What are you saying, that 1+1=2 is incomplete?
Of course I'm talking about epistemology and not ontology. By laws of physics one is inherently talking about the language used to describe reality. In this case the language used is math.
Then what is the physical mechanism that implements the self-reference used in GIT? Godel was able to construct a metalanguage about mathematics using mathematics itself. This enabled him to construct the self-referential statement "This statement is not provable" - all within math itself. Now, if the laws of physics are incomplete by GIT, then where does that self-reference occur in the subset of the math used for physics? How is the metalanguage constructed within the math for physics? The math used in physics is used to refer to objects of physics, and it does not refer to other mathematical objects.stevendaryl said:Math is not equivalent to physics, but predicting the future based on the past certainly involves mathematics. If there are certain mathematical questions that are unsolvable, and those mathematical questions are involved in predicting the future, then those predictions are impossible to make.
Some physical problems have been shown to be unsolvable. Making predictions about chaotic systems may be an example, or it may not be.
friend said:Then what is the physical mechanism that implements the self-reference used in GIT?
stevendaryl said:Can every even integer greater than 2 be written as the sum of two prime numbers?
That question doesn't involve self-reference. But we don't know how to solve it. (For that particular question, there is no proof one way or the other as to whether it is solvable using standard mathematics, or not.)
martinbn said:Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true. So is your example an example that shows incompleteness or is it irrelevent?
martinbn said:Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true.
Your examples of unprovability would have been well known before Godel so that if they were definitively shown to be precises unprovable, then Godel would have never had need to construct the GIT. For there would be the examples of unprovability without Godel, and everyone would have know it. So your examples don't mean a thing. They only show that we don't know how... yet.stevendaryl said:It's an example of the type of problem that is unsolvable: Do all elements of this infinite set have property P?
Every specific yes/no question is potentially solvable someday. But there are sets of yes/no questions such that it is provable that it is impossible to solve every problem in the set.
I've already given one example: Does such and such polynomial equation have an integer solution? For any method you choose to answer such questions, there will be questions that your method gets wrong.
Mathematically, there is no function which given an equation (written in ascii, for definiteness) will return True or False depending on whether the equation has an integral solution.
friend said:Your examples of unprovability would have been well known before Godel so that if they were definitively shown to be precises unprovable.