How do you determine whether a physical process “behaves like integers”? The second-order axiom of induction sounds complicated, I cannot easily check that it’s satisfied by apples. If you use some sort of Bayesian reasoning to figure out which axioms work on apples, can you describe it in more detail?
I don’t have an answer to the specific question, only to the class of questions. To approach understanding this, we need to distinguish between reality and what points to reality, i.e, symbols. Our skill as humans is in the manipulation of symbols, as a kind of simulation of reality, with greater or lesser workability for prediction, based in prior observation, of new observations.
“Apples” refers, internally, to a set of responses we created through our experience. We respond to reality as an “apple” or as a “set of apples,” only out of our history. It’s arbitrary. Counting, and thus “behavior like integers” applies to the simplified, arbitrary constructs we call “apples.” Reality is not divided into separate objects, but we have organized our perceptions into named objects.
Examples. If an “apple” is a unique discriminable object, say all apples have had a unique code applied to them, then what can be counted is the codes. Integer behavior is a behavior of codes.
Unique applies can be picked up one at a time, being transferred to one basket or another. However, real apples are not a constant. Apples grow and apples rot. Is a pile of rotten apple an “apple”? Is an apple seed an apple? These are questions with no “true” answer, rather we choose answers. We end up with a binary state for each possible object: “yes, apple,” or “no, not apple.” We can count these states, they exist in our mind.
If “apple” refers to a variety, we may have Macintosh, Fuji, Golden delicious, etc.
So I have a basket with two apples in it. That is, five pieces of fruit that are Macintosh and three that are Fuji.
I have another basket with two apples in it. That is, one Fuji and one Golden Delicious.
I put them all into one basket. How many apples are in the basket? 2 + 2 = 3.
The question about integer behavior is about how categories have been assembled. If “apple” refers to an individual piece of intact fruit, we can pick it up, move it around, and it remains the same object, it’s unique and there is no other the same in the universe, and it belongs to a class of objects that is, again, unique as a class, the class is countable and classes will display integer behavior.
That’s as far as I’ve gotten with this. “Integer behavior” is not a property of reality, per se, but of our perceptions of reality.
Well, it comes from the fact that apples in a bowl is Exclusively just that, as verified by your Bayesian reasoning. There are no other “chains” of successors (shadow apples? I can’t even imagine a good metaphor).
So, now you in fact have that bowl of apples narrowed down to {0, S0, SS0, SSS0, …} which is isomorphic to the natural numbers, so all other natural number properties will be reflected there.
How do you determine whether a physical process “behaves like integers”? The second-order axiom of induction sounds complicated, I cannot easily check that it’s satisfied by apples. If you use some sort of Bayesian reasoning to figure out which axioms work on apples, can you describe it in more detail?
I don’t have an answer to the specific question, only to the class of questions. To approach understanding this, we need to distinguish between reality and what points to reality, i.e, symbols. Our skill as humans is in the manipulation of symbols, as a kind of simulation of reality, with greater or lesser workability for prediction, based in prior observation, of new observations.
“Apples” refers, internally, to a set of responses we created through our experience. We respond to reality as an “apple” or as a “set of apples,” only out of our history. It’s arbitrary. Counting, and thus “behavior like integers” applies to the simplified, arbitrary constructs we call “apples.” Reality is not divided into separate objects, but we have organized our perceptions into named objects.
Examples. If an “apple” is a unique discriminable object, say all apples have had a unique code applied to them, then what can be counted is the codes. Integer behavior is a behavior of codes.
Unique applies can be picked up one at a time, being transferred to one basket or another. However, real apples are not a constant. Apples grow and apples rot. Is a pile of rotten apple an “apple”? Is an apple seed an apple? These are questions with no “true” answer, rather we choose answers. We end up with a binary state for each possible object: “yes, apple,” or “no, not apple.” We can count these states, they exist in our mind.
If “apple” refers to a variety, we may have Macintosh, Fuji, Golden delicious, etc.
So I have a basket with two apples in it. That is, five pieces of fruit that are Macintosh and three that are Fuji.
I have another basket with two apples in it. That is, one Fuji and one Golden Delicious.
I put them all into one basket. How many apples are in the basket? 2 + 2 = 3.
The question about integer behavior is about how categories have been assembled. If “apple” refers to an individual piece of intact fruit, we can pick it up, move it around, and it remains the same object, it’s unique and there is no other the same in the universe, and it belongs to a class of objects that is, again, unique as a class, the class is countable and classes will display integer behavior.
That’s as far as I’ve gotten with this. “Integer behavior” is not a property of reality, per se, but of our perceptions of reality.
Well, it comes from the fact that apples in a bowl is Exclusively just that, as verified by your Bayesian reasoning. There are no other “chains” of successors (shadow apples? I can’t even imagine a good metaphor).
So, now you in fact have that bowl of apples narrowed down to {0, S0, SS0, SSS0, …} which is isomorphic to the natural numbers, so all other natural number properties will be reflected there.