I like your attempt to see where the focus of our statements shifts back onto the territory, but in the crucial place I’m not sure how an IsRock predicate may or may not “obey counting axioms”—not sure what this means.
My take on this: a “rock” is a a concept that exists in the map, not in the territory. “Is a rock” (and, much more importantly and centrally, “Is an object”) is a predicate that exists within the map. However, “this is a rock” is a statement about the territory, not the map. How’s that work? When I state “this is a rock”, what I’m saying is that “out there” in reality there’s Stuff which, when mediated through my senses and subjected to the discretization process my map comes equipped with, will yield an Object that my map will recognize as a Rock. Not all possible configurations of Stuff in reality are such that this process will finish with a Rock in my map. By saying “this is a rock”, I claim a constraint on the Stuff in reality that ensures the successful completion of such a process (that I may or may not actually carry out).
Suppose the state of the universe at a particular time can be considered as a set of atoms in R^3.
Now let’s suppose we choose IsIsSomeRocksPredicate such that IsIsSomeRocksPredicate(P) implies:
P(r,u) is a binary predicate (the first argument being our candidate bunch of rocks and the second argument being the state of the universe)
Let T be an isometry. Then P(r,u) implies P(T(r), T(u))
P(r,u) implies r is a subset of u
P(r,u) implies that r is a finite set of atoms
Let u and u’ be identical in some neighborhood of r. Then P(r,u) implies P(r,u’)
P(r,u) implies that r is not solidly connected to anything within u (this stops you defining half a rock as a rock)
P(r,u) probably implies some other stuff about the essential rockiness of r.
Then we can pick a P that satisfies this, and come up with an IsSameNumberOfRocks equivalence function. We can also come up with a successor function (adding an extra rock) that is unique up to IsSameNumberOfRocks. A big problem is that the successor function isn’t always defined if there’s a finite number of rocks in the universe, so I guess rocks really don’t behave like numbers.
(I hadn’t realized just how far you need to unpack “2 rocks + 2 rocks = 4 rocks”… I’m interpreting it to mean something like “if you have 2 rocks, and then include another 2 rocks which don’t include any from the first set of 2, then you end up with the same number of rocks as if you picked an entirely different set of 4 rocks”
I’m not sure you’ve specified your definition fully.
Imagine there’s a biscuit that looks just like a rock. When filtered through your senses, you’ll recognize it as a rock and so by your definition it actually is a rock. Statements such as “this looks like a rock to me but actually it’s something else” become always false by definition.
I like your attempt to see where the focus of our statements shifts back onto the territory, but in the crucial place I’m not sure how an IsRock predicate may or may not “obey counting axioms”—not sure what this means.
My take on this: a “rock” is a a concept that exists in the map, not in the territory. “Is a rock” (and, much more importantly and centrally, “Is an object”) is a predicate that exists within the map. However, “this is a rock” is a statement about the territory, not the map. How’s that work? When I state “this is a rock”, what I’m saying is that “out there” in reality there’s Stuff which, when mediated through my senses and subjected to the discretization process my map comes equipped with, will yield an Object that my map will recognize as a Rock. Not all possible configurations of Stuff in reality are such that this process will finish with a Rock in my map. By saying “this is a rock”, I claim a constraint on the Stuff in reality that ensures the successful completion of such a process (that I may or may not actually carry out).
Suppose the state of the universe at a particular time can be considered as a set of atoms in R^3.
Now let’s suppose we choose IsIsSomeRocksPredicate such that IsIsSomeRocksPredicate(P) implies:
P(r,u) is a binary predicate (the first argument being our candidate bunch of rocks and the second argument being the state of the universe)
Let T be an isometry. Then P(r,u) implies P(T(r), T(u))
P(r,u) implies r is a subset of u
P(r,u) implies that r is a finite set of atoms
Let u and u’ be identical in some neighborhood of r. Then P(r,u) implies P(r,u’)
P(r,u) implies that r is not solidly connected to anything within u (this stops you defining half a rock as a rock)
P(r,u) probably implies some other stuff about the essential rockiness of r.
Then we can pick a P that satisfies this, and come up with an IsSameNumberOfRocks equivalence function. We can also come up with a successor function (adding an extra rock) that is unique up to IsSameNumberOfRocks. A big problem is that the successor function isn’t always defined if there’s a finite number of rocks in the universe, so I guess rocks really don’t behave like numbers.
(I hadn’t realized just how far you need to unpack “2 rocks + 2 rocks = 4 rocks”… I’m interpreting it to mean something like “if you have 2 rocks, and then include another 2 rocks which don’t include any from the first set of 2, then you end up with the same number of rocks as if you picked an entirely different set of 4 rocks”
I’m not sure you’ve specified your definition fully.
Imagine there’s a biscuit that looks just like a rock. When filtered through your senses, you’ll recognize it as a rock and so by your definition it actually is a rock. Statements such as “this looks like a rock to me but actually it’s something else” become always false by definition.
(I’ll address the counting question separately)