Is it problematic to use indeterministic models for deterministic processes

Question

The title basically says it all. My question is whether it is problematic that deterministic processes such as coin tosses are modelled as indeterministic random processes by Frequentist statisticians.

More specifically I am interested which positions in the philosophy of science would object to this step and which position would consider this unproblematic.

Personally I always considered the argument against Frequentism made e.g. by Jaynes quite convincing. One passage from "The logic of science" is given here:

"The writer has never thought of a biased coin ‘as if it had a physical probability’ because, being a professional physicist, I know that it does not have a physical probability. From the fact that we have seen a strong preponderance of heads, we cannot conclude legitimately that the coin is biased; it may be biased, or it may have been tossed in a way that systematically favors heads. Likewise, from the fact that we have seen equal numbers of heads and tails, we cannot conclude legitimately that the coin is ‘honest’. It may be honest, or it may have been tossed in a way that nullifies the effect of its bias."

Answer 1

First, why think that a coin flip is deterministic? In the nineteenth and, especially, early twentieth century, quite a few scientists and philosophers of science argued that the notion that the world is "governed" by "laws" is a holdover from Christianity. Without that notion of laws, determinism falls away as empty. (What's doing the determining?) And even if we accept the notion of laws, they needn't be deterministic. Ironically, Newtonian mechanics isn't deterministic. Starting with the development of population statistics and statistical mechanics in the 19th century, a number of philosophers and scientists were proponents of "real chance" theories, on which chances or probabilities are real, irreducible features of the world. Peirce and Fischer are notable examples here; I think possibly also Maxwell, although I'm less confident about that. Note that other frequentists — such as the senior Pearson (I always forget whether that's Egon or Karl) — were very strong empiricists, and argued that we should replace the notion of laws with something like "concise descriptions of observations."

There is a sense in which quantum mechanics is deterministic in a way that Newtonian mechanics is not. But even then the observations that I make of random variables or processes (e.g., which path a single photon travels in a double slit experiment) are, in a deep sense, probabilistic. The Stanford Encyclopedia entry on determinism (linked above) is a good place to start reading about determinism.

Second, there's a line of thought in philosophy of science arguing that there are cases in which determinism and indeterminism are empirically equivalent. That is, we would see exactly the same thing if determinism were true as if it were false. There's further controversy over whether, in these cases, indeterminism might be preferable. One good entry into this literature is Charlotte Werndl's "Evidence for the Deterministic or the Indeterministic Description?"

Third, if we abandon the idea that science is about discovering laws, then many philosophers suggest that we think of science as developing maps or models. But maps and models don't need to be perfectly accurate representations of the thing they're representing. They are typically selective — they're only intended to represent certain aspects of their targets — and distorted — introducing simplifications or errors. Maps and models are selective and distorted because doing so makes them more useful as representations. Think of the way the Mercator map projection distorts the polar regions. Doing so allows the map projection to both represent courses of constant bearing as straight lines and use a roughly constant scale over moderately-sized areas. So if you want to know how to get from England to Barbados in the 17th century, you can draw a straight line, read off the course reading, and use a single scale to calculate the distance.

More generally, when we move from laws to models, we arguably need to replace the notion of truth with a notion like "adequacy for purpose." Wendy Parker talks about this move in the context of climate modeling, and Angela Potochnik has a good discussion of the implications of this shift for notions such as objectivity. In this light, your question can be paraphrased as something like "are indeterministic models more adequate for some purposes than deterministic models?" The answer is evidently yes! In statistical mechanics, epidemiology, or quantitative social science, we frequently use indeterministic models to represent systems that include large numbers of individual parts (gas molecules; human beings in a large society) when we don't have a good predictive model of their individual behavior (human beings) or when modeling all of those parts at an individual level would be computationally intractable (gas molecules, especially in Maxwell's day).

Answer 2

I would say it is not problematic, because we need not think of probabilistic models as being indeterministic. Even if we agree for the sake of argument that the universe is deterministic, at least at the macro level, there is still plenty of scope for using probabilities to describe the fact that we have limited information about the universe. In fact, I share your distrust of frequentism and would say that fundamentally probability is a calculus of partial information. Information is often inaccurate, always imprecise and always incomplete. But it is useful to quantify information and perform calculations with it, and this is what the probability calculus does. The fact that probability is capable of performing this role can be argued from two different sources. One comes from the work of Richard Cox, which Edwin Jaynes also uses, which makes use of how information is preserved under deductive relations. The other approach is due to Bruno de Finetti and appeals to how our degree of certainty about propositions must be constrained in order to avoid making irrational decisions, such as getting 'Dutch booked'.

Since you give coin-tossing as an example, consider an experiment involving tossing a coin repeatedly. A freqentist might say that since the coin has a head and a tail (we shall ignore other possibilities, like landing on its edge) there is a long run frequency of its landing heads and this might be 0.5 if the coin is unbiased. I agree with you and Jaynes that this is an unsatisfactory thing to say: the frequency is not a property of the coin. If we tossed the coin in exactly the same way every time, we would get the same result every time. (Again, I'm assuming determinism for the sake of argument, but I'm fairly sure that quantum effects would be tiny at the scale of a coin toss.) The reason the coin toss has a probability at all is because we have underspecified the description of the experiment. We have not stated the starting position of the coin, the magnitude and direction of the impulse applied to it, the distance to the surface on which it will land, etc. The coin will land sometimes heads and sometimes tails, not because there is some probabilistic disposition of the coin, but because there are variables in the specification of the experiment that have been left unstated. When we have an underspecified problem but we need an answer, the best we can do is work with the partial information we have. We could assume some particular values for the unknowns, but this is not helpful here. So instead we take a range of possible values for the unknowns and integrate them out. When we perform a (possibly weighted) sum or integral over all the reasonable values for the starting position, impulse, etc., the best answer we can come up with is a probability of 0.5, assuming no bias. This is not a statement about the coin, but a statement about the partial information we have of an underspecified experiment.

The upshot is: we don't have to be concerned with whether the universe is actually deterministic or not, because we still need models that take into account our partial information.