I pretty much agree with this analysis. I have one comment on 2 though.
Our minds have opinions on the following topics:
Whether or not something is a rock
Whether or not something is a valid candidate for the IsRock predicate
It’s worth pointing out that these are both somewhat arbitrary, with no right or wrong answer. If we say “yes this is a rock” or “no this doesn’t look like an IsRock predicate” then we’re expressing statements about our map/mind rather than about the territory itself.
That said, it feels like there is more than one predicate which:
You would feel is a valid IsRock candidate
Is talking about the territory rather than anyone’s map
Obeys counting axioms (subject to some complications such as already having a UnionOfTwoPortionsOfSpace predicate)
If we say “yes this is a rock” … then we’re expressing statements about our map/mind rather than about the territory itself.
Just realised that I didn’t quite mean this.
If we say “this is/isn’t a valid IsRock predicate” then we’re expressing a statement about the map
Under one semantics, “this is a rock” means “I believe this is a rock” and is a statement about the map
By picking a predicate P which satisfies the IsIsRockCandidate metapredicate, we can give “this is a rock” the semantics P(This), which is a statement about the territory.
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 pretty much agree with this analysis. I have one comment on 2 though.
Our minds have opinions on the following topics:
Whether or not something is a rock
Whether or not something is a valid candidate for the IsRock predicate
It’s worth pointing out that these are both somewhat arbitrary, with no right or wrong answer. If we say “yes this is a rock” or “no this doesn’t look like an IsRock predicate” then we’re expressing statements about our map/mind rather than about the territory itself.
That said, it feels like there is more than one predicate which:
You would feel is a valid IsRock candidate
Is talking about the territory rather than anyone’s map
Obeys counting axioms (subject to some complications such as already having a UnionOfTwoPortionsOfSpace predicate)
Just realised that I didn’t quite mean this.
If we say “this is/isn’t a valid IsRock predicate” then we’re expressing a statement about the map
Under one semantics, “this is a rock” means “I believe this is a rock” and is a statement about the map
By picking a predicate P which satisfies the IsIsRockCandidate metapredicate, we can give “this is a rock” the semantics P(This), which is a statement about the territory.
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)