> Another example of misleading mathematical terminology is "random variable", which are not random or variable but instead are well-defined mappings.
Indeed. Kolmogorov's random variables are a special case of observables in mathematical physics.
In classical physics, the state space for a free particle is S = R^3 x R^3. As an observable we might take the particle's velocity in some particular direction, which defines a mapping f : S -> R. We want such mappings to respect the relevant structure of the state space. A classical state space usually has the structure of a smooth manifold, so the mappings should be smooth as well.
In classical stochastic physics, the states now have the structure of a probability space. We want the mappings to be measurable so we can take the preimage of a measurable set of observable values to calculate its probability. This is exactly the Kolmogorov definition of a random variable.
Quantum theory doesn't quite fit into the above scheme, but there are several ways these three cases can be unified. For example, from the viewpoint of C* algebras of observables, classical systems have commutative algebras and quantum systems have noncommutative algebras.
By the way, there is a connection to monads that I find enlightening. With monads, there is both an internal semantics and an external semantics. From the internal point of view, random variables are indeed random and variable. From the external point of view, they are neither. (This can be given a Bayesian spin by thinking of the internal and external observers as someone with respectively imperfect and perfect knowledge.) This is analogous to someone's perspective on the state monad from the internal and external points of view. From the viewpoint of someone who lives in the state monad, the same expression can return different values depending on context. But from an outsider's explicit state-passing point of view, everything is referentially transparent.
Indeed. Kolmogorov's random variables are a special case of observables in mathematical physics.
In classical physics, the state space for a free particle is S = R^3 x R^3. As an observable we might take the particle's velocity in some particular direction, which defines a mapping f : S -> R. We want such mappings to respect the relevant structure of the state space. A classical state space usually has the structure of a smooth manifold, so the mappings should be smooth as well.
In classical stochastic physics, the states now have the structure of a probability space. We want the mappings to be measurable so we can take the preimage of a measurable set of observable values to calculate its probability. This is exactly the Kolmogorov definition of a random variable.
Quantum theory doesn't quite fit into the above scheme, but there are several ways these three cases can be unified. For example, from the viewpoint of C* algebras of observables, classical systems have commutative algebras and quantum systems have noncommutative algebras.
By the way, there is a connection to monads that I find enlightening. With monads, there is both an internal semantics and an external semantics. From the internal point of view, random variables are indeed random and variable. From the external point of view, they are neither. (This can be given a Bayesian spin by thinking of the internal and external observers as someone with respectively imperfect and perfect knowledge.) This is analogous to someone's perspective on the state monad from the internal and external points of view. From the viewpoint of someone who lives in the state monad, the same expression can return different values depending on context. But from an outsider's explicit state-passing point of view, everything is referentially transparent.