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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?

Bob
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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 03 '16 at 15:32
  • No. http://philosophy.stackexchange.com/questions/4028/how-does-induction-relate-to-falsifiability/12661#12661 – alanf May 03 '16 at 15:54
  • @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 03 '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 03 '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 03 '16 at 21:05
  • @alanf I simply have other intentions. Like with other models, theories have explanatory force for different problems. And if the problem is rather basic, I will not use the most sophisticated theory. I will use a simple model first, even if it may be totally 'wrong'. Because it helps to get the idea. You do not begin to use orbital equations for explaining Rutherford's gold foil experiment for beginners in physics either, but e.g. Bohr's 'wrong' one. Popper really isn't the gold standard either, but it is rather sophisticated. My aim is the conciousness of destinctions first. – Philip Klöcking May 03 '16 at 22:24
  • IMHO this is a physics question. It is an axiom of the mathematical treatment of of QM that the magnitude squared of the relevant quantities maps onto conventional probabilities of different events occurring. – Dave May 03 '16 at 22:38
  • We can not predict mathematically what a "fair coin" will do unless we assume that: a) it stays "fair" forever; b) the tosses are independent of each other forever. Neither a) nor b) can be proved mathematically, they are extrapolations from observations, but probabilistic conclusions can be proved from them. Similarly, quantum mechanical laws can be derived from a couple of basic axioms, but the axioms themselves are taken from observations. – Conifold May 03 '16 at 23:46

2 Answers2

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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)?

Jo Wehler
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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).

Chris Sunami
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