Consider the function f on the real numbers such that for any real number x, f(x) is 1 if x is irrational and f(x) is 0 otherwise.
We have an axiomatic measure theory, which tells us that the integral over values of x from 0 to 1 of f(x) is 1.
I don’t believe there is any physical system, in our brains or otherwise, that represents this function or its integral. There is a physical system in our brains that represents the rules of logical inference, and it is through this physical system that we can form beliefs about mathematical facts that do not model any known physical system. That is, we can make statements of the form: if there were a physical system that satisfied this proposition, it would also satisfy this other proposition, regardless of whether that physical system actually exists.
The facts that we call mathematical and logical truths can be represented in any universe capable of representing the rules of inference (and having sufficient memory and processing power). In this sense, they are independent of our particular physical universe.
Not that I disagree with your conclusion (or agree – mostly I’m just confused), but:
I don’t believe there is any physical system, in our brains or otherwise, that represents this function or its integral.
Including the representation in your computer, or your brain, of the phrase “the function f on the real numbers such that for any real number x, f(x) is 1 if x is irrational and f(x) is 0 otherwise”?
Ah, I should clarify that point, it is confusing as I wrote it.
I meant that there is no physical domain over which some point wise property varies discontinuously as a function of whether the point, in some measure, has a rational or irrational distance from some reference, and that the only physical systems that in any sense represent the function do so indirectly, by representing propositions about it (such as the examples you gave).
Consider the function f on the real numbers such that for any real number x, f(x) is 1 if x is irrational and f(x) is 0 otherwise.
We have an axiomatic measure theory, which tells us that the integral over values of x from 0 to 1 of f(x) is 1.
I don’t believe there is any physical system, in our brains or otherwise, that represents this function or its integral. There is a physical system in our brains that represents the rules of logical inference, and it is through this physical system that we can form beliefs about mathematical facts that do not model any known physical system. That is, we can make statements of the form: if there were a physical system that satisfied this proposition, it would also satisfy this other proposition, regardless of whether that physical system actually exists.
The facts that we call mathematical and logical truths can be represented in any universe capable of representing the rules of inference (and having sufficient memory and processing power). In this sense, they are independent of our particular physical universe.
Not that I disagree with your conclusion (or agree – mostly I’m just confused), but:
Including the representation in your computer, or your brain, of the phrase “the function f on the real numbers such that for any real number x, f(x) is 1 if x is irrational and f(x) is 0 otherwise”?
Ah, I should clarify that point, it is confusing as I wrote it.
I meant that there is no physical domain over which some point wise property varies discontinuously as a function of whether the point, in some measure, has a rational or irrational distance from some reference, and that the only physical systems that in any sense represent the function do so indirectly, by representing propositions about it (such as the examples you gave).