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We know how to handle the mathematics of probability, thanks to the work of the Russian mathematician Andrey Kolmogorov and others. Probabilities are real numbers between zero and one, inclusive; the probabilities of all independent events add up to one; and so on. But that’s not the same as deciding what probability actually is.
There are numerous approaches to defining probability, but we can distinguish between two broad classes. The “objective” or “physical” view treats probability as a fundamental feature of a system, the best way we have to characterize physical behavior. An example of an objective approach to probability is frequentism, which defines probability as the frequency with which things happen over many trials, as in our coin-tossing example.
Alternatively,
there are
“subjec-
tive” or
“evid-
enti
-al”
views
which
treat
proba-
bility
as personal, a reflection of an individual’s credence,
or degree of belief, about what is true or what will happen. An example is Bayesian probability, which emphasizes Bayes’ law, a mathematical theorem that tells us how to update our credences as we obtain new information. Bayesians imagine that rational creatures in states of incomplete information walk around with credences for every proposition you can imagine, updating them continually as new data comes in. In contrast with frequentism, in Bayesianism it makes perfect sense to attach probabilities to one-shot events, such as who will win the next election, or even past events that we’re unsure about.
Interestingly, different approaches to quantum mechanics invoke different meanings of probability in central ways. Thinking about quantum mechanics helps illuminate probability, and vice versa.
Or, to put it more
pessimistically:
Quantum mechanics
as it is currently
understood
doesn’t really
help us choose
between
competing
conceptions of probability, as every
conception has a
home in some quantum
formulation or other.
An illustration of a dice showing numbers 3, 5 and 6..
Let’s consider three of the leading approaches to quantum theory. There are “dynamical collapse” theories, such as the GRW model proposed in 1985 by Giancarlo Ghirardi, Alberto Rimini and Tullio Weber. There are “pilot wave” or “hidden variable” approaches, most notably the de Broglie-Bohm theory, invented by David Bohm in 1952 based on earlier ideas from Louis de Broglie. And there is the “many worlds” formulation suggested by Hugh Everett in 1957.
Each of these represents a way of solving the measurement problem of quantum mechanics. The problem is that conventional quantum theory describes the state of a system in terms of a wave function, which evolves smoothly and deterministically according to the Schrödinger equation. At least, it does unless the system is being observed; in that case, according to the textbook presentation, the wave function suddenly “collapses” into some particular observational outcome. The collapse itself is unpredictable; the wave function assigns a number to each possible outcome, and the probability of observing that outcome is equal to the value of the wave function squared. The measurement problem is simply: What constitutes a “measurement”? When exactly does it occur? Why are measurements seemingly different from ordinary evolution?