TITLE: Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?
QUESTION [12 upvotes]: It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite number of trials or that it "relates the axiomatic concept of probability to the statistical concept of frequency". Isn't this is a methodological mistake of ascribing an interpretation to a mathematical term, perhaps relying too much on the colorful language, that does not at all follow from how this term is mathematically defined? Recall the typical derivation of the WLLN:

Let $X_1, X_2, ..., X_n$ be a sequence of n independent and identically distributed random variables with the same finite mean $\mu$, and with variance $\sigma^2$ and let:
$\overline{X}=\tfrac1n(X_1+\cdots+X_n)$
We have:
$E[\overline{X}] = \frac{E[X_1+...+X_n]}{n} = \frac{E[X_1]+...+E[X_n]}{n} =
 \frac{n\mu}{n} = \mu$ $Var[\overline{X}] = \frac{Var[X_1+...+X_n]}{n^2} = \frac{Var[X_1]+...+Var[X_n]}{n^2} = \frac{n\sigma^2}{n^2} =
\frac{\sigma^2}{n}$
And from Chebyshev's inequality:
$P(|\overline{X}-\mu|>\epsilon) \le \frac{\sigma^2}{n\epsilon^2}$
And so X is said to converge in probability to $\mu$.

Now consider what is strictly speaking the meaning of this expression in the axiomatic framework it is derived in:

$P(|\overline{X}-\mu|>\epsilon) \le \frac{\sigma^2}{n\epsilon^2}$

$P()$, everywhere it occurs in the derivation, is known only to be a number satisfying Kolmogorov's axioms, so a number between 0 and 1, and so forth, but none of the axioms introduce any theoretical equivalent of the intuitive notion of frequency. If additional assumptions about $P()$ are not made, the sentence can obviously not be interpreted at all, but what is also important the theoretical mean $\mu$ is not necessarily the mean value in an infinite number of trials, $\overline{X}$ is not necessarily the mean value from n trials, and so forth. Consider an experiment of tossing a fair coin repeatedly - quite obviously, nothing in Kolmogorov's axioms enforces using 1/2 for the probability of heads, you could just as well use $1/\sqrt{\pi}$, yet the derivation continues to "work", except the meaning of the various variables is not in agreement with their intuitive interpretations. The $P()$ might still mean something, it might be a quantification of an absurd belief of mine, the mathematical derivation continues be true regardless, in the sense that as long as the initial $P()'s$ satisfy axioms, theorems about other $P()'s$ follow, and with Kolmogorov's axioms providing only weak constraints on and not a definition of $P()$, it's basically only symbol manipulation.
This "relative frequency" interpretation frequently given seems to rest on an additional assumption, and this assumption seems to be a form of the law of large numbers itself. Consider this fragment from Kolmogorov's Grundbegriffe on applying the results of probability theory to the real world:

We apply the theory of probability to the actual world of experiment
  in the following manner:
...
4) Under certain conditions, which we shall not discuss here, we may
  assume that the event A which may or may not occur under conditions S,
  is assigned a real number P(A) which has the following
  characteristics:
a) One can be practically certain that if the complex of conditions S
  is repeated a large number of times, n, then if m be the number of
  occurrences of event A, the ratio m/n will differ very slightly from
  P(A).

Which seems equivalent to introducing the weak law of large numbers in a particular, slightly different form, as an additional axiom.
Meanwhile, many reputable sources contain statements that seem completely in opposition to the above reasoning, for example Wikipedia:

It follows from the law of large numbers that the empirical
  probability of success in a series of Bernoulli trials will converge
  to the theoretical probability. For a Bernoulli random variable, the
  expected value is the theoretical probability of success, and the
  average of n such variables (assuming they are independent and
  identically distributed (i.i.d.)) is precisely the relative frequency.

This seem to be mistaken already in claiming that from a mathematical theorem anything can follow about empirical probability (the page on which defines it as the relative frequency in actual experiment), but there are many more subtle claims that technically also seem erroneous from the above considerations:

The LLN is important because it "guarantees" stable long-term results for the averages of random events. 

Note that the Wikipedia article about LLN claims to be about the mathematical theorem, not about the empirical observation, which was also historically sometimes been called the LLN. It seems to me that LLN does nothing to "guarantee stable long-term results", for as stated above those stable long-term results have to be assumed in the first place for the terms occuring in the derivation to have the intuitive meaning we typically ascribe to them, not to mention something has to be done to at all interpret $P()$ in the first place. Another instance from Wikipedia:

According to the law of large numbers, if a large number of six-sided die are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the precision increasing as more dice are rolled.

Does this really follow from the mathematical theorem? In my opinion, the interpretation of the theorem that is used here, rests on assuming this fact. There is a particularly vivid example in the "Treatise on probability" by Keynes of what happens when one follows the WLLN with even a slight deviation from this initial assumptions of p's being the relative frequencies in the limit of an infinite number of trials:

The following example from Czuber will be
  sufficient for the purpose of illustration. Czuber’s argument is as
  follows: In the period 1866–1877 there were registered in Austria
m = 4,311,076 male births
n = 4,052,193 female births
s = 8,363,269
for the succeeding period, 1877–1899, we are given only
m' = 6,533,961 male births;
what conclusion can we draw as to the number n of female births? We
  can conclude, according to Czuber, that the most probable value
n' = nm'/m = 6,141,587
and that there is a probability P = .9999779 that n will lie between
  the limits 6,118,361 and 6,164,813. It seems in plain opposition to
  good sense that on such evidence we should be able with practical
  certainty P = .9999779 = 1 − 1/45250 to estimate the number of female
  births within such narrow limits. And we see that the conditions laid
  down in § 11 have been flagrantly neglected. The number of cases, over
  which the prediction based on Bernoulli’s Theorem is to extend,
  actually exceeds the number of cases upon which the à priori
  probability has been based. It may be added that for the period,
  1877–1894, the actual value of n did lie between the estimated limits,
  but that for the period, 1895–1905, it lay outside limits to which the
  same method had attributed practical certainty.

Am I mistaken in my reasoning above, or are all those really mistakes in the Wikipedia? I have seen similar statements all over the place in textbooks, and I am honestly wondering what I am missing.

REPLY [3 votes]: I understand OP's concern and I want to illustrate it by an example from geometry.
Pythagorean theorem. Let $V$ be a two-dimensional real vector space with inner product $\langle \cdot, \cdot \rangle$ and induced norm $\| \cdot \|$. In linear algebra we learn that $\langle a,b \rangle = 0$ implies $\| a \|^2 + \| b \|^2 = \| a + b \|^2$, and this result is called Pythagorean theorem. This is Step 2 of the routine in Torsten's answer [1]. Clearly, we would like to know what the connection of this Pythagorean theorem is with right-angled triangles drawn on a real sheet of paper. So we need to think about Steps 1 and 3 of the routine. Generations of students have drawn right-angled triangles and measured the lengths of the legs $a,b$ and the hypothenuse $c$, arriving at the identity $a^2+b^2=c^2$ to some degree of precision. Based on the amount of data, it is plausible to assume that there exists an empirical Pythagorean theorem in the real world. Now we can use the empirical Pythagorean theorem and the standard interpretation (using a rectangular coordinate system) to 'identify' $\| a \|$ with the length of $a$. In this way we obtain an interpretation of the Pythagorean theorem (in $V$) in terms of lengths. Doesn't it then feel wrong to say that the empirical Pythagorean theorem is a consequence of the Pythagorean theorem (in $V$) under the above interpretation?
There is an empirical result often called the empirical law of large numbers or the stability of frequencies, which states, for example, that the relative frequencies of heads in a long sequence of coin tosses 'converge' to some value $p$. In my opinion it is this empirical law which Kolmogorov refers to in the excerpt cited by OP. Afterwards OP argues that since we are using stability of frequencies to interpret probabilities and thereby the law of large numbers (LLN), it feels wrong to say that LLN 'guarantees' stability of frequencies.
I agree that it seems unnecessary to say that LLN is responsible for stability of frequencies whenever stability of frequencies is used to provide an interpretation.

However, stability of frequencies only looks good on paper. Since we cannot make infinitely many observations, this empirical law isn't of much help for determining probabilities in practice. On top of that many problems of practical interest are not as reproducible as a coin toss. I am not a probabilist or statistician, so from here on I have to rely on the opinion of other people.
First of all, let me quote Mark Kac. OP has provided a short excerpt in [4]. Here is how Kac continues.

The applicability of such a theory [probability theory] to natural sciences must ultimately be tested by an experiment. But this is true of all mathematical theories when applied outside the realm of mathematics, and the vague feeling of discomfort one encounters (mostly among philosophers!) when first subjected to statistical reasoning must be attributed to the relative novelty of the ideas.
To me there is no methodological distinction between the applicability of differential equations to astronomy and of probability theory to thermodynamics or quantum mechanics.
It works! And brutally pragmatic as this point of view is, no better substitute has been found. ([2], p.5)

What Kac suggests is a more pragmatic point of view, that of a physicist. Let me quote Krzysztof Burdzy.

[Compared with mathematicians,] physicists have a different idea of a 'proof' – you start with a large number of unrelated assumptions, you combine them into a single prediction, and you check if the prediction agrees with the observed data. If the agreement is within 20%, you call the assumptions proved. ([3], p.41)

I think that Burdzy has made up the figure of 20%, but I am not a physicist. More importantly, we can apply probability theory (including LLN) with any assumption that we deem fit, which makes stability of frequencies in some sense obsolete. As long as we can produce predictions that can be tested, we don't have to worry about the 'vague' link between the model and the real world. Over time and by doing a lot of experiments, we acquire a certain confidence in our claims (if they agree with the observations) and then they become accepted by the math/science community.
All of this is rather difficult to comprehend for a beginner in probability/statistics. In a first probability course, the statistical tools that are needed for the predictions only enter very late or not at all, which may be the reason why students don't see this 'pragmatic' approach to applied probability/statistics. On the other hand, stability of frequencies may still be useful for gaining intuition.
A layman (like myself) gets lost easily in the big philosophical frequentist vs. Bayesian debate(s). As a mathematician, I can accept that the only definition of probability appears in the Kolmogorov axioms and I don't need to know its 'true' meaning in order to learn and apply the theory. My goal in writing this was to provide some consolation for a specific group of people (including myself), i.e. those who have gone through a similar thought process as OP.

[1] Torsten Schoeneberg (Aug 19, 2021)
[2] Mark Kac, "Probability and related topics in physical sciences"
[3] Krzysztof Burdzy, "The search for certainty: on the clash of science and philosophy of probability" (suggested by Bjørn Kjos-Hanssen in [4])
[4] Logical issues with the weak law of large numbers and its interpretation
[5] Is probability and the Law of Large Numbers a huge circular argument?