I’m not sure what the etiquette is of responding to retracted comments, but I’ll have a go at this one.
Why is the same identical thing the same?
That’s what I mean when I say they are identical. It’s not another, separate thing, existing on a separate occasion, distinct from the first but standing in the relation of identity to it. In mathematics, you can step into the same river twice. Even aliens in distant galaxies step into the same river.
However, there is something else involved with the stability, which exists in time, and which is capable of being imperfectly stable: oneself. 2+2=4 is immutable, but my judgement that 2+2 equals 4 is mutable, because I change over time. If it seems impossible to become confused about 2+2=4, just think of degenerative brain diseases. Or being asleep and dreaming that 2+2 made 5.
So the question becomes, “If “2+2” is just another way of saying “4″, what is the point of having two expressions for it?”
My answer: As humans, we often desire to split a group of large, distinct objects into smaller groups of large, distinct objects, or to put two smaller groups of large, distinct, objects, together. So, when we say “2 + 2 = 4”, what we are really expressing is that a group of 4 objects can be transformed into a group of 2 objects and another group of 2 objects, by moving the objects apart (and vice versa). Sharing resources with fellow humans is fundamental to human interaction. The reason I say, “large, distinct objects” is that the rules of addition do not hold for everything. For example, when you add “1” particle of matter to “1″ particle of antimatter, you get “0” particles of both matter and antimatter.
Numbers, and, yes, even logic, only exist fundamentally in the mind. They are good descriptions that correspond to reality. The soundness theorem for logic (which is not provable in the same logic it is describing) is what really begins to hint at logic’s correspondence to the real world. The soundness theorem relies on the fact that all of the axioms are true and that inference rules are truth-preserving. The Peano axioms and logic are useful because, given the commonly known meaning we assign to the symbols of those systems, the axioms do properly describe our observations of reality and the inference rules do lead to conclusions that continue to correspond to our observations of reality (in (one of) the correct domain(s), groups of large, distinct, objects). We observe that quantity is preserved regardless of grouping; this is the associative property (here’s another way of looking at it).
The mathematical proof of the soundness theorem is useless for convincing the hard skeptic, because it uses mathematical induction itself! The principle of mathematical induction is called such because it was formulated inductively. When it comes to the large numbers, no one has observed these quantities. But, for all quantities we have observed so far, mathematical induction has held. We use deduction to apply induction, but that doesn’t make the induction any less inductive to begin with. We use the real number system to make predictions in physics. If we have the luxury of making an observation, we should go ahead and update. For companies with limited resources that are trying to develop a useful product to sell to make money, and even more so for Friendly AI (a mistake could end human civilization), it’s nice to have a good idea of what an outcome will be before it happens. Bayes’ rule provides a systematic way of working with this uncertainty. Maybe, one day, when I put two apples next to two apples on my kitchen table, there will be five (the order in which I move the apples around will affect their quantity), but, if I had to bet one way or the other, I assure you that my money is on this not happening.
I’m not sure what the etiquette is of responding to retracted comments, but I’ll have a go at this one.
That’s what I mean when I say they are identical. It’s not another, separate thing, existing on a separate occasion, distinct from the first but standing in the relation of identity to it. In mathematics, you can step into the same river twice. Even aliens in distant galaxies step into the same river.
However, there is something else involved with the stability, which exists in time, and which is capable of being imperfectly stable: oneself. 2+2=4 is immutable, but my judgement that 2+2 equals 4 is mutable, because I change over time. If it seems impossible to become confused about 2+2=4, just think of degenerative brain diseases. Or being asleep and dreaming that 2+2 made 5.
So the question becomes, “If “2+2” is just another way of saying “4″, what is the point of having two expressions for it?”
My answer: As humans, we often desire to split a group of large, distinct objects into smaller groups of large, distinct objects, or to put two smaller groups of large, distinct, objects, together. So, when we say “2 + 2 = 4”, what we are really expressing is that a group of 4 objects can be transformed into a group of 2 objects and another group of 2 objects, by moving the objects apart (and vice versa). Sharing resources with fellow humans is fundamental to human interaction. The reason I say, “large, distinct objects” is that the rules of addition do not hold for everything. For example, when you add “1” particle of matter to “1″ particle of antimatter, you get “0” particles of both matter and antimatter.
Numbers, and, yes, even logic, only exist fundamentally in the mind. They are good descriptions that correspond to reality. The soundness theorem for logic (which is not provable in the same logic it is describing) is what really begins to hint at logic’s correspondence to the real world. The soundness theorem relies on the fact that all of the axioms are true and that inference rules are truth-preserving. The Peano axioms and logic are useful because, given the commonly known meaning we assign to the symbols of those systems, the axioms do properly describe our observations of reality and the inference rules do lead to conclusions that continue to correspond to our observations of reality (in (one of) the correct domain(s), groups of large, distinct, objects). We observe that quantity is preserved regardless of grouping; this is the associative property (here’s another way of looking at it).
The mathematical proof of the soundness theorem is useless for convincing the hard skeptic, because it uses mathematical induction itself! The principle of mathematical induction is called such because it was formulated inductively. When it comes to the large numbers, no one has observed these quantities. But, for all quantities we have observed so far, mathematical induction has held. We use deduction to apply induction, but that doesn’t make the induction any less inductive to begin with. We use the real number system to make predictions in physics. If we have the luxury of making an observation, we should go ahead and update. For companies with limited resources that are trying to develop a useful product to sell to make money, and even more so for Friendly AI (a mistake could end human civilization), it’s nice to have a good idea of what an outcome will be before it happens. Bayes’ rule provides a systematic way of working with this uncertainty. Maybe, one day, when I put two apples next to two apples on my kitchen table, there will be five (the order in which I move the apples around will affect their quantity), but, if I had to bet one way or the other, I assure you that my money is on this not happening.