This is a question I asked on Physics Stack Exchange a while back, and I thought it would be interesting to hear people's thoughts on it here. You can find the original question here.
What do we mean when we say that we have a probabilistic theory of some phenomenon?
Of course, we know from experience that probabilistic theories "work", in the sense that they can (somehow) be used to make predictions about the world, they can be considered to be refuted under appropriate circumstances and they generally appear to be subject to the same kinds of principles that govern other kinds of explanations of the world. The Ising model predicts the ferromagnetic phase transition, scattering amplitude computations of quantum field theories predict the rates of transition between different quantum states, and I can make impressively sharp predictions of the ensemble properties of a long sequence of coin tosses by using results such as the central limit theorem. Regardless, there seem to be a foundational problem at the center of the whole enterprise of probabilistic theorizing - the construction of what is sometimes called "an interpretation of the probability calculus" in the philosophical literature, which to me seems to be an insurmountable problem.
A probabilistic theory comes equipped with an event space and a probability measure attached to it, both of which are fixed by the theory in some manner. However, the probability measure occupies a strictly epiphenomenal position relative to what actually happens. Deterministic theories have the feature that they forbid some class of events from happening - for instance, the second law of thermodynamics forbids the flow of heat from a cold object to a hot object in an isolated system. The probabilistic component in a theory has no such character, even in principle. Even if we observed an event of zero probability, formally this would not be enough to reject the theory; since a set of zero probability measure need not be empty. (This raises the question of, for instance, whether a pure quantum state in some energy eigenstate could ever be measured to be outside of that eigenstate - is this merely an event of probability , or is it in fact forbidden?)
The legitimacy of using probabilistic theories then rests on the implicit assumption that events of zero (or sufficiently small) probability are in some sense negligible. However, it's not clear why we should believe this as a prior axiom. There are certainly other types of sets we might consider to be "negligible" - for instance, if we are doing probability theory on a Polish space, the collection of meager sets and the collection of null measure sets are both in some sense "negligible", but these notions are in fact perpendicular to each other: can be written as the union of a meager set and a set of null measure. This result forces us to make a choice as to which class of sets we will neglect, or otherwise we will end up neglecting the whole space !
Moreover, ergodic theorems (such as the law of large numbers) which link spatial averages to temporal averages don't help us here, even if we use versions of them with explicit estimates of errors (like the central limit theorem), because these estimates only hold with a probability for some small , and even in the infinite limit they hold with probability , and we're back to the problems I discussed above. So while these theorems can allow one to use some hypothesis test to reject the theory as per the frequentist approach, for the theory to have any predictive power at all this hypothesis test has to be put inside the theory.
The alternative is to adopt a Bayesian approach, in which case the function of a probabilistic theory becomes purely normative - it informs us about how some agent with a given expected utility should act. I certainly don't conceive of the theory of quantum mechanics as fundamentally being a prescription for how humans should act, so this approach seems to simply define the problem out of existence and is wholly unsatisfying. Why should we even accept this view of decision theory when we have given no fundamental justification for the use of probabilities to start with?
As I see it, probability is essentially just a measure of our ignorance, or the ignorance of any model that's used to make predictions. An event with a probability of 0.5 implies that in half of all situations where I have information indistinguishable from the information I have now, this event will occur; in the other half of all such indistinguishable situations, it won't happen.
For example, all I know is that I have a coin with two sides of equal weight that I plan to flip carelessly through the air until it lands on a flat surface. I'm not tracking how all the action potentials in the neurons of my motor cortex, cerebellum, and spinal cord will affect the precise twitches of individual muscle fibers as I execute the flip, nor the precise orientation of the coin prior to the flip, nor the position of every bone and muscle in my body, nor the minute air currents that might interact differently with the textures on the heads versus tails side, nor any variations in the texture of the landing surface, nor that sniper across the street who's secretly planning to shoot the coin once it's in the air, nor etc., etc., etc. Under the simplified model, where that's all you know, it really will land heads half the time and tails half the time across all possible instantiations of the situation where you can't tell any difference in the relevant initial conditions. In the reality of a deterministic universe, however, the coin (of any particular Everett branch of the multiverse) will either land heads-up or it won't, with no in-between state that could be called "probability".
Similarly, temperature also measures our ignorance, or rather lack of control, of the trajectories of a large number of particles. There are countless microstates that produce identical macrostates. We don't know which microstate is currently happening, how fast and in what direction each atom is moving. We just know that the molecules in the fluid in the calorimeter are bouncing around fast enough to cause the mercury atoms in the thermometer to bounce against each other hard enough to cause the mercury to expand out to the 300K mark. But there are vigintillions of distinct ways this could be accomplished at the subatomic level, which are nevertheless indistinguishable to us at the macroscopic level. You could shoot cold water through a large pipe at 100 mph and we would still call it cold, even though the average kinetic energy of the water molecules is now equivalent to a significantly higher temperature. This is because we have control over the largest component of their motion, because we can describe it with a simple model.
To a God-level being that actually does track the universal wave function and knows (and has the ability to control) the trajectories of every particle everywhere, there is no such thing as temperature, no such thing as probability. Particles just have whatever positions and momenta they have, and events either happen or they don't (neglecting extra nuances from QM). For those of us bound by thermodynamics, however, these same systems of particles and events are far less predictable. We can't see all the lowest-level details, much less model them with the same precision as reality itself, much less control them with God-level orchestration. Thus, probability, temperature, etc. become necessary tools for predicting and controlling reality at the level of rational agents embedded in the physical universe, with all the ignorance and impotence that comes along with it.
Here I think you're mixing two different approaches. One is the Bayesian apporach: it comes down to saying probabilistic theories are normative. The question is how to reconc... (read more)