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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?

closed as off-topic by Dave, James Kingsbery, user2953, Swami Vishwananda, Nick May 25 '16 at 21:00

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  • We do not prove anything mathematically in your examples, we justify abductive claims (hypotheses) empirically and by this make them inductive ones with more certainty. Mathematical proves are apodictic. Empirical, i.e. inductive laws are falsifiable. You could start looking these terms up in the SEP and IEP to clarify your misunderstanding. – Philip Klöcking May 3 '16 at 15:32
  • @alanf Referring to the very specific framework of Popper may help to clarify some misunderstandings after already having had an idea of what is going on. But it is a very specific discourse with specific terms and not at all uncontested or 'eternal truth' without knowing what one may think under the terms used first. Therefore your link is highly related, but not really appropriate to the question imho. – Philip Klöcking May 3 '16 at 16:24
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    Here's my charitable reading of the question: Is there an axiomatic formulation of QM in which "empirical" laws are mathematically proven in the axiomatic framework? Is this what @Bob is looking for? – DBK May 3 '16 at 16:28
  • @PhilipKlöcking You're wrong, as illustrated by the arguments given at the link, which you haven't replied to. – alanf May 3 '16 at 21:05
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I would like to emphasize the differences between the three cases:

1) The fair coin: We know the probability distribution of the single experiment - both results have the same probability p = 1/2. Using the law of large numbers we can mathematically derive how the result of many tosses approaches this distribution.

2) Statistical mechanics: We hypothesize that the molecules obey the lays of Newton mechanics. Then we can mathematically derive that the mean values of the observables satisfies the laws of thermodynamics, e.g., pV = RT (state equation of the ideal gas)

3) The fundamental laws are the laws of quantum mechanics itself, e.g., the Schroedinger equation. We do not know other laws from which the laws of quantum mechanics can be derived mathematically.

Hence: The laws of quantum mechanics cannot be derived mathematically as in the example 1) and 2)?

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You are not using the term "prove mathematically" accurately. Mathematical theorems are in the realm of pure mathematics, and have conclusions that can be demonstrated to be logical consequences of the initial definitions.

The probability of a coin toss, and the movement of molecules in a gas, are part of the material world. We can demonstrate empirically, through experiment, that they appear to match the predictions of a mathematical model, and we can do so with enough reliability that we take it as an established fact that they will always match the predictions of the model, but that is not the same thing, or even the same kind of thing, as proving a mathematical theorem.

With that said, there's nothing intrinsically different about establishing results in quantum mechanics as in the macroscopic world (except that we have less of an understanding of the underlying mechanisms, and the results are more counterintuitive).

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