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Hypothetical frequentists defined probability as limiting long run frequencies of hypothetical random infinite sequences, such as an infinite sequence of coin tosses. One argument against this definition is that these infinite sequences do not exist.

Richard von Mises, one of the main proponents of frequentism in the 20th century argued against this objection that e.g. instantaneous velocity is also defined as a limit, which does really not exist, namely as ds/dt for dt → 0. Or chalk lines on a black board are represented by lines without any width in geometry even though this is of course incorrect. But it is nevertheless helpful as an idealization and the question of whether it really exists is simply irrelevant, we shall judge the use of the concept by how it helps us do describe reality, and which predictions it allows.

Unfortunately I do not know too much about physics, so my question is whether you think that this comparison is valid? Just because there are other idealizations does of course not mean that everything goes. So is it a problem for hypothetical infinite frequentist that the infinite sequence does nor actually exist?

Furthermore do you have any reference to philosophical papers that deal with such idealized concepts in physics?

  • The tile of the questions doesn't seem to match the body, i.e. you don't ask about the consequences of the 'existence' of infinite sequences for Frequentism. – Discrete lizard Apr 1 '18 at 7:06
  • You still have to be more precise. If Mises said “probability just means the limiting long run frequencies of hypothetical random infinite sequences (if such existed in the real world)” this is very different from coming up with a formalism in which probability is defined on the basis of random infinite sequences. – wolf-revo-cats Apr 1 '18 at 9:56
  • But my question is not about the exact formalism but more about the fact that it is based on something that does not exist. I am not sure whether I underdtand your comment though – Sebastian Apr 2 '18 at 7:31
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Most mathematicians today would say that the "real" definition of a limit comes from epsilon-N proofs: the limit of the sequence is x if and only if, for any given epsilon, there is some N such that all elements after the first N are within epsilon of x. In other words, the sequence gets arbitrarily close to x and, at some point, it never gets further away from x. This definition doesn't require any appeal to infinitesimals or an actual infinity; just the potential behavior of the sequence indefinitely into the future.

I don't know the work by von Mises you're referring to, so I can't comment on whether that was what he had in mind. But it does seem that textbook frequentism uses exactly the same notion of the potential behavior of a sequence indefinitely into the future.

On idealization in physics, there's some discussion in the SEPh article on models in science (scattered, so search in the page text). And Angela Potochnik has a new book on the topic.

  • Thank you for your comment! But then still without knowing the whole sequence i can never be sure that my sequence stays within [x-eps, x +eps] for all n >N. – Sebastian Apr 1 '18 at 7:06
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The argument claiming the complete existence of infinitely many terms is false. This concerns the sequence of elements of {0, 1} in case of von Mises as well as the sequence of all rational numbers (q_n) in case of Cantor. It is obvious that infinitely many terms do not and cannot exist in the finite accessible universe.

The analogy to the limit in mathematics is invalid. The analytical limit a of a potentially infinite sequence (a_n) is defined as follows:

For every eps > 0 there exists a natural number m such that for n > m: |a_n - a| < eps.

This limit a does not require the existence of the complete infinite sequence; it requires only that the above condition is satisfied for every term a_n that can be dealt with (can be imagined or written).

The ideas of von Mises and Cantor are rather naive: They presuppose that the limit is a term following upon all finite terms. Unfortunately many contemporary mathematicians adhere to this delusion too and claim that they can count to infinity.

But the non-existence of complete infinite sequences is not a problem in frequentism (contrary to set theory) because arbitrarily long sequences exist, and therefore the error can be made as small as possible.

  • With "can be dealt with" you mean all the measurements that can be actually made? – Sebastian Apr 1 '18 at 7:21
  • If i want to know whether it converges i better know all the terms – Sebastian Apr 1 '18 at 7:29
  • @0rangetree: There is nothing like "all the terms". Every term belongs to a finite initial sequence whereupon infinitely many are following ("infinitely many" only means: There is no upper bount of the number of terms following - no magical aleph or omega). To know all the terms means to count to infinity since no term remains. That is nonsense as every sober mind can confirm you. – Wilhelm Apr 1 '18 at 8:32
  • But you never know how small the error is. I can make 10^100 measurements and be practically confident that it is close to the limit. But theoretically it can easily be that i am not even close. What i am lacking is information about the behaviour of the terms that follow which are possibly infinite – Sebastian Apr 2 '18 at 7:40
  • @0rangetree: Yes, in principle you could have all oxygen molecules of a room in one of its corners. But the probability for ten of them to be in that corner is already very small. This can be checked by experiment. Absolute certainty is not available. But is a certainty restricted to say 10^(-100) really a problem? – Wilhelm Apr 2 '18 at 8:50
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If we throw a coin and win if we get heads, Mises would say that the ratio

number-of-wins : trials

would approach 0.5 for an infinite sequence. This is the case for the following very regular sequence

W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,W,L,…

But now, if the rules were different and we would only win if we get heads twice in a row, we should assume a ratio approaching 0.25. But a regrouping of the sequence above would not work. With H = heads (formerly W), T = tails (formerly L) written grouped:

HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,HT,…

we get only losses. Never two heads in a row.

In a realistic example, we can intuit that it would work. A “truly” random sequence would more start like this:

W,L,W,W,L,W,L,L,W,L,W,L,W,W,W,W,L,L,W,W,W,L,L,L,L,W,L,W,L,L,W,L,L,W,W,W,W,L,L,L,L,W,L,L,L,L,L,W,W,W,L,W,W,W,L,L,W,W,W,L,…

grouped:

HT,HH,TH,TT,HT,HT,HH,HH,TT,HH,HT,TT,TH,TH,TT,HT,TH,HH,HT,TT,TH,TT,TT,TH,HH,TH,HH,TT,HH,HT,…

and so in wins and losses if the new rules are applied (two times heads in a row):

L,W,L,L,L,L,W,W,L,W,L,L,L,L,L,L,L,W,L,L,L,L,L,L,W,L,W,L,W,L,…

– roughly one quarter are wins and it seems intuitively the ratio would converge to 0.25.

So obviously, just any and all infinite sequences, which converge “correctly” (like the first very regular one), will not do. They may in some sense not be “random” and so not admissible.

But if we want to base our concept of probability on infinite sequences, we would at least need an exact grasp what infinite sequences are random and therefore admissible. Mises failed to give such a criterion. And so his analogy with the concept of velocity in physics breaks down, similar problems don't occur in the calculus machinery used there.

It's still, by itself, a non-argument to just say that infinite sequences do not exist in observable reality.

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