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A single toss of a fair coin cannot be predicted. But if we observe a large number of tosses, we can prove mathematically the law that roughly half of them will show up heads.

The movements of individual molecules in a gas cannot be predicted and can be assumed to be random. But if we observe some macroscopic phenomena such as temperature or pressure, we can prove mathematically that some laws are satisfied.

Individual quantum events are random. But if we observe a large number of such events, we discover experimentally that they satisfy the laws of quantum mechanics. Could the laws of quantum mechanics be proved mathematically as in the examples above?

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    $\begingroup$ It's unclear to me what you mean by prove. What, on your view, would constitute proof of a physical law? $\endgroup$ – Alfred Centauri Dec 24 '14 at 14:20
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    $\begingroup$ Ugh. There are a lot of statements in this question which, if I correctly guess what's in your head, are not correct. That makes answering the question kind of hard. $\endgroup$ – DanielSank Dec 24 '14 at 14:55
  • $\begingroup$ @DanielSank: Could you please point at least one of my statements above that is not correct? $\endgroup$ – Bob Dec 26 '14 at 14:59
  • $\begingroup$ @AlfredCentauri: for instance, the kind of proofs there are in Einstein's theory of Brownian motion $\endgroup$ – Bob Dec 26 '14 at 15:07
  • $\begingroup$ Let's start with "Individual quantum events are random". I just don't know what you mean there. $\endgroup$ – DanielSank Dec 26 '14 at 15:10
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No, you cannot prove that mathematical law. All you can say is that your experiments are consistent with your premise.

I recommend you take the time to read some introductory books on physics, statistics, and the scientific method (which is to say, don't just take my word that my initial statement is valid).

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    $\begingroup$ Mathematical laws that cannot be proven are not laws. Do you mean physical law? $\endgroup$ – ACuriousMind Dec 26 '14 at 17:20
  • $\begingroup$ @ACuriousMind I meant what I wrote. Go read some math books: in math, a law, or axiom, is a starting point from which you derive a system. The "parallel postulate" is a well-known example of such a law. $\endgroup$ – Carl Witthoft Dec 26 '14 at 19:22
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    $\begingroup$ I've never heard the usage of "law" for "axiom". To me, mathematical laws are what is derived from the axioms, and perhaps including the axioms. Regardless, while your suggestion to read about the scientific method is surely right, I think your answer is a bit unclear, as it should rather state that physical truth is not about mathematical proofs than leave OP's suggestion that "the laws of quantum mechanics" are "mathematical laws" unchallenged. $\endgroup$ – ACuriousMind Dec 27 '14 at 18:03
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Physics is not mathematics.

Physical laws are not axioms.

A physical theory is not the derivation of all possible hypotheses from a set of axioms.

Please repeat this like a mantra a hundred times a day for the next three weeks.

Instead a physical theory is patterned as a set of naive ontological assumptions about the approximate usefulness of a set of observational terms like "mass", "force", "gravity" etc.. "Laws of nature" are shorthand descriptions for a wide range of observations that these terms satisfy within a more or less well defined domain of application of the theory.

The theory is both expected and accepted to fail outside of this domain and some of the most interesting aspects of it is how it fails along the boundaries of this domain, where one or several different theories with different sets of observational assumptions and laws of nature take over. That failure boundary is where, at any given time, the active scientific work takes place.

One of the important questions of physics is the question how far one can push that boundary and what the relation between the terms on both sides of it are. Sometimes we don't know, yet, what lies behind the range of application of a theory, sometimes we have important hints from multiple theories of what that could be, but we haven't found the right language, yet, to express those ideas concisely.

I know that this must be awfully frustrating for someone who thinks about the world in terms of axiomatic set theory and existence proofs. The world just isn't like that and physics is not that neatly organized. That, however, is what makes physics endless intellectual fun!

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