It might be informative to note that the ratio of the square of a circle’s area to its radius, or of its circumference to its diameter, is only necessarily what it is in Euclidean space because of certain axioms and postulates; in Hyperbolic space it is variable. This means it is logically possible for the number defined in this way to vary but with implications concerning other properties of geometry. In this sense, a (large) disagreement about the value of π could be cordoned off by information about the curvature of spacetime. On the other hand, if you were to define π in a way which included all of the axioms and postulates of Euclidean geometry then imagining it to have a different value would become logically impossible and I don’t know how to make a decision here beyond deriving a contradiction. If you actually believed this but were otherwise logical, you could believe anything because of the principle of explosion.
When I attempt to imagine myself operating in a universe in which π has slightly different digits, and conclude it wouldn’t affect my decisions , beliefs or behaviors very much, I think what I am doing is not actually conditioning on π having those digits directly, but rather making the observation that my model of the universe is of a kind of ‘equivalence class’ of all the different universes in which the first few digits of π are those which I know, which I can infer has certain properties which all of them share from the knowledge that π is between 3 and 4. As a mind, I consider myself to be living in this ‘equivalence class’ of universes in logical space when I can’t distinguish between them by observation. If I had to consider a situation in which π(in the Euclidean sense) was equal to 2, I would necessarily “expand” in mathematical/logical space to a much broader type of universe where my system of axioms would not be sufficiently powerful to prove theorems like that 3<π<4, which would be a weirdly constrained place to live (and of course I can’t actually do this).
Conversely, if I was a much more computationally powerful and intelligent but similarly inflexible mind, I wouldn’t be able to imagine even the 2^2^2nd digit of π being anything other than what it is.
It might be informative to note that the ratio of the square of a circle’s area to its radius, or of its circumference to its diameter, is only necessarily what it is in Euclidean space because of certain axioms and postulates; in Hyperbolic space it is variable. This means it is logically possible for the number defined in this way to vary but with implications concerning other properties of geometry. In this sense, a (large) disagreement about
the value of π could be cordoned off by information about the curvature of spacetime. On the other hand, if you were to define π in a way which included all of the axioms and postulates of Euclidean geometry then imagining it to have a different value would become logically impossible and I don’t know how to make a decision here beyond deriving a contradiction. If you actually believed this but were otherwise logical, you could believe anything because of the principle of explosion.
When I attempt to imagine myself operating in a universe in which π has slightly different digits, and conclude it wouldn’t affect my decisions , beliefs or behaviors very much, I think what I am doing is not actually conditioning on π having those digits directly, but rather making the observation that my model of the universe is of a kind of ‘equivalence class’ of all the different universes in which the first few digits of π are those which I know, which I can infer has certain properties which all of them share from the knowledge that
π is between 3 and 4. As a mind, I consider myself to be living in this ‘equivalence class’ of universes in logical space when I can’t distinguish between them by observation. If I had to consider a situation in which π(in the Euclidean sense) was equal to 2, I would necessarily “expand” in mathematical/logical space to a much broader type of universe where my system of axioms would not be sufficiently powerful to prove theorems like that
3<π<4, which would be a weirdly constrained place to live (and of course I can’t actually do this).
Conversely, if I was a much more computationally powerful and intelligent but similarly inflexible mind, I wouldn’t be able to imagine even the 2^2^2nd digit of π being anything other than what it is.