“Controlled by abstract fact” as in Controlling Constant Programs idea?
That post gives a toy model, but the objects of acausal control can be more general: they are not necessarily natural numbers (computed by some program), they can be other kinds of mathematical structures; their definitions don’t have to come in the form of programs that compute them, or syntactic axioms; we might have no formal definitions that are themselves understood, so that we can only use the definitions, without knowing how they work; and we might even have no ability to refer to the exact object of control, and instead work with an approximation.
Also, consider that even when you talk of “explicit” tools for accessing abstract facts, such as apples or calculators or theorems written of a sheet of paper, these tools are still immensely complicated physical objects, and what you mean by saying that they are simple is that you understand how they are related to simple abstract descriptions, which you understand. So the only difference between understanding what a “real number” is “directly”, and pointing to an axiomatic definition, is that you are comfortable with intuitive understanding of “axiomatic definition”, but it too is an abstract idea that you could further describe using another level of axiomatic definition. And in fact considering syntactic tools as mathematical structures led to generalizing from finite logical statements to infinite ones, see infinitary logic.
That post gives a toy model, but the objects of acausal control can be more general: they are not necessarily natural numbers (computed by some program), they can be other kinds of mathematical structures; their definitions don’t have to come in the form of programs that compute them, or syntactic axioms; we might have no formal definitions that are themselves understood, so that we can only use the definitions, without knowing how they work; and we might even have no ability to refer to the exact object of control, and instead work with an approximation.
Also, consider that even when you talk of “explicit” tools for accessing abstract facts, such as apples or calculators or theorems written of a sheet of paper, these tools are still immensely complicated physical objects, and what you mean by saying that they are simple is that you understand how they are related to simple abstract descriptions, which you understand. So the only difference between understanding what a “real number” is “directly”, and pointing to an axiomatic definition, is that you are comfortable with intuitive understanding of “axiomatic definition”, but it too is an abstract idea that you could further describe using another level of axiomatic definition. And in fact considering syntactic tools as mathematical structures led to generalizing from finite logical statements to infinite ones, see infinitary logic.