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I am really struggling with this question.

Do you have an example which is not statistically describable? If not, is everything statistically describable? if yes, why is everything statistically describable?

My final goal is to understand, whether it is possible or not that there are things or processes in our universe that are not statistically describable.

edit: To clarify:

  • Let every process in the universe be in Set A.
  • Let B be the best possible finite set of laws of nature.
  • Let C be the currently know set of laws of nature. Let
  • Let f(X) be all processes described by the laws in set X
  • We know C ⊆ B, but is f(B) = A or f(B) ⊂ A

If the universe is inherently nondeterministic (like quantum physics), the laws are also allowed to be statistical, since they describe non-determinism perfectly.

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    Possible counter-examples: 1) What is the propositional content of the following sentence: "Snow is white"? 2) What is it like to be a bat? 3) What is "Of Mice and Men" supposed to teach us?
    – Einer
    Sep 1, 2014 at 13:14
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    The right question to ask is when are Statistical techniques are appropriate to use; and what kind of additional information do they give. Sep 1, 2014 at 13:51
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    Well, I care most about fundamental laws of nature. I'am wondering whether laws cover everything or not. But if you could discribe everything statistically, there couldn't be a undescribable thing in the universe.
    – Odin
    Sep 1, 2014 at 13:55
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    Well, the 'laws of nature' aren't 'everything'; as Einer helpfully points out; even in physics, fields such as statistical mechanics complement Newtonian Mechanics; Chomsky points out science is generally not done by statistical means - even when statistics are being used. Sep 1, 2014 at 13:59
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    'The whole of nature can be covered by laws of nature' isn't a scientific proposition; but a philosophical assumption; generally we say in short that the universe is 'intelligible'; however its generally understood to be circumscribed in many different ways; a physicist such as Verlinde suggested it would be suprising that the universe in its totality is intelligible; but not suprising that it must be to some extent; after all we couldn't operate in this universe if that was not the case. Sep 1, 2014 at 14:10

5 Answers 5

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Maybe you're using statistics in a nuanced way I'm not quite understanding. If you switched it to empirical data, the critique I suggest below fails.

Statistics by themselves cannot be knowledge, because they are probabilistic and predict not what it is but the probability of what will be. Once it is, it's no longer statistics. Thus, working from QM with Heisenberg uncertainty, we can narrow down the area in which the particle exists, but that is not identical to knowing either where the particle is or what its velocity is.

Or to give a more mundane example, you can state definitively that 103 of 203 human pregnancies will be a male. You cannot using just statistics state whether the child is a male or female. You would need to gather more direct empirical data.

Clearly, the mode of description that statistics affords is invaluable for doing science, but it is incomplete precisely because the strength of its predictive power is precisely in that it does not make mistakes about specific empirical facts, but this is precisely because such facts are statistically unavailable.

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Let's assume A is the set of laws all physical processes can be described with. Now why are those laws the way they are? Is that a) arbitrary (it is just the way things are) or is there b) a set of rules that governs the laws of A (maybe in a progression of generalization like Kepler->Newton->Einstein)?

If a) is the case there is nothing won if you apply statistics (or any other mathematical apparatus) to it. You will find nothing new since those laws are arbitrary. In this scenario you have described everything in the realm of physics except for the laws of physics. If you think, the question why A is the way A is, would not belong to the realm of physics, bear in mind, that it Einstein explained why Newton is the way it is. That is part of the job of physics. So in this case, there is something left in the physical world, that cannot be statistically described.

if b) is the case, there will be a set A' that governs A. But A' is no different in nature than A, so here it will again be the question if A' is just arbitrary or if it is governed by a set A''. So this whole process just iterates on and on.

In The Life of the Cosmos* Lee Smolin tries to offer a (partial) solution. He proposes, that univeres are created by collapsing black holes. Only if a universe has a special set of laws, it is able to create a black hole, which in turn is able to spawn a new universe with a similar set of laws. If the laws are unsuitable to create such a black whole, it will not spawn a new universe and hence it cannot be a universe we live in, asking why things are the way they are. Statistically it would be highly unlikely that of all universes in existence we turn up in one that is unable to create black wholes: The universes with black-wholy laws have just the greater reproductive rate (i.e. > 0). The whole process is therefore governed by a variant of evolution. And evolution can indeed be described statistically. So this might be a way out, if you subscribe to his theory.


*which is the book this answer is heavily inspired by.

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  • +1 because it is interesting, but it's not what I am looking for, I think.
    – Odin
    Sep 1, 2014 at 14:59
  • @Odin Makes me glad and sad at the same time to hear that ;-) What are you missing?
    – Einer
    Sep 1, 2014 at 15:07
  • I edited some text to the question, but MathJax does not seem to show it properly...
    – Odin
    Sep 1, 2014 at 15:21
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It seems you can't use statistics to describe something that has only happened once. There would be no further information to draw a general case.

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You seem to be including quantum mechanics in your definition of a "statistical theory". But in fact, as @alanf states in his answer, quantum phenomena cannot be described by classical statistics.

In a sense, scientists developed quantum mechanics precisely because, as you phrased it, "there are things or processes in our universe that are not statistically describable".

Scott Aaronson, a professor who studies quantum computing at MIT, put it this way:

[I]f you're lucky, after years of study [of quantum mechanics] you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.

Quantum-mechanical theories all deal with amplitudes, not probabilities.

Of course, then you can ask the same question again: are there natural phenomena that cannot be described with quantum-mechanical amplitudes?

The answer to that is: we don't know.

Gravity, one of the first phenomena to be studied in the modern scientific era, does not yet have a quantum-mechanical description. It may turn out that quantum-mechanical amplitudes can describe gravity, or it may turn out that a new mathematical formalism is needed.

EDIT: I read more of Aaronson's lecture (linked to above), and he actually argues that quantum-mechanical amplitudes are probably expressive enough to describe everything.

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Events to which probabilities can be assigned can be described statistically, but there are classes of events for which this is not possible. In quantum mechanics if you arrange an interaction in which information about an observable is copied from one system to another, after the interaction the square amplitudes of the relevant outcomes obey the calculus of probabilities. So after the interaction the outcomes can be described statistically. But before such an interaction the square amplitudes of those outcomes may not obey the calculus of probabilities, e.g. - they don't obey the calculus of probabilities during an interference experiment. So during an interference experiment the events taking place can't be described statistically. See

http://arxiv.org/abs/math/9911150.

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