I see a lot of people arguing that “2”, “3″, “+”, and “=” are defined in terms of the Peano axioms, and as such, aren’t actually relevant to the behavior of physical objects. They say that the axioms pin down the numbers, regardless of how physical objects behave or start behaving.
But the Peano axioms use something called a “successor” to generate the natural numbers. And how do we figure out what the successors are? Well, one notation is to append an “S” to the previous number to indicate that number’s successor, such that the successor of “SS0” is “SSS0″. Then “+” would mean “repeatedly apply the successor operation to the number to the right of this operator for however many times indicated by the number to the left of this operator”, and “=” would indicate “these two numbers, when written out using successor-notation, have the character ‘S’ occur the same number of times”.
So how do we calculate “2 + 2”? Well, we take the number of occurrences of the character “S”, count them up, then put the total number of occurrences in front of a single “0”: “SSSS0“, which we interpret as “4”. “2 + 2 = 4”.
But now let’s imagine your brain’s visual cortex (or whatever it uses to visualize things) being messed up somehow. Actually, let’s go further than that, and suppose everyone’s visual cortexes got messed up in the same way, so that now, when we visualize two objects and two more objects, and put them together, we see, not four objects, but three. So if you were to try and visualize a group of two dots coming together with another group of two dots, you would end up seeing, in your mind’s eye, a final group of three dots.
Now imagine counting the number of times “S” occurs in “SS0″, then putting two of those groups of “S” characters together. How many “S” character are in the final group? Visualize it, now...
Why, three. Of course there’s three. If you take “SS” and put it with “SS”, of course you get “SSS”. What’s that? You say you’re getting “SSSS”? Where are you getting that extra “S” from? What? No—can’t you see it? It’s obviously three—how can you put those two groups together and get four?
So, “SS0 + SS0 = SSS0”.
My point is this: you can’t just point to the Peano axioms and say, “Ha! Your hypothetical situation involving the behavior of mere physical objects is meaningless before the might of my absolute mathematical assumptions!” Remember, when you try to perform “logic” on those axioms, you’re still using your brain to do it. And however ivory-tower untouchable you imagine your axioms to be, your brain is a real, physical object performing real physical computations. If we lived in a slightly different universe, your brain would take one look at “SS0” and “SS0“, visually add together the number of “S” characters, and see “SSS0”. Anyone who saw “SSSS0” would be seen as crazy, or brain-damaged, or something.
They say that the axioms pin down the numbers, regardless of how physical objects behave or start behaving.
Not sure who “they” are. May I suggest reading up on model theory? Maybe Enderton’s “mathematical logic” (?).
It’s important to think separately about the “model” (the thing we are studying), the “language” (where we make statements about the model), and the “theory” (a set of statements in a language). The same theory in a particular language may and in fact generally does apply to multiple distinct models. The same model (object) may be described by theories in different languages of different strengths.
So in one sense Peano axioms “pin down” the natural numbers, but in another sense they don’t because we can invent crazy objects that contain a lot more than just the natural numbers to which Peano axioms also apply (this is the content of the Lowenheim-Skolem theorem).
We can use a more powerful language than first order logic to describe the natural numbers, and that would rule out some of the crazy models. That will capture more properties of the natural number line, but not everything.
Physicists play a similar game to logicians, except perhaps a bit less formally. But their models (in the sense of ‘object of study’) “bite back.”
It’s confusing that to a model theorist “the model” refers to the territory, while to a statistician “the model” refers to the map.
My bad; I wasn’t clear. “They” refers, not to any person or group of people in academia, but some of the commenters on this LW post. As an example: this comment.
Um, but the (+) operator in peano arithmetic is actually defined in terms of Sx + y = x + Sy. It would be somewhat circular to “suggest counting the S’s up” in a method of defining numbers, after all. So the way you calculate 2 + 2 is more like
SS0 + SS0
(thing on the left starts with an S)
S(S0) + SS0
(move the S to the right)
S0 + SSS0
(thing on the left starts with an S)
S(0) + SSS0
(move the S to the right)
0 + SSSS0
(eliminate "0 + " with axiom)
SSSS0
I imagine you’d need a rather more devious brain modification to prevent one from carrying out these steps correctly, in such a way that the result is SSS0.
I imagine you’d need a rather more devious brain modification to prevent one from carrying out these steps correctly, in such a way that the result is SSS0.
All right. Let me take a stab at it.
S(0) + SSS0
(move the S to the right)
Okay. Following you so far...
0 + SSSS0
Eh? Where’d you get the extra “S” from?
(This hack would have the unfortunate side effect of making every addition with at least one term less than or equal to 3 and a result greater than 3 come out to 1 less than it’s supposed to, however. If you wanted to only make 2 + 2 = 3, and preserve all other additions as-is, I can’t think of any brain hack that could do that. That’s not to say no such hack is possible; I’m sure one is, but I just can’t think of one.)
This hack would have the unfortunate side effect of making every addition with at least one term less than or equal to 3 and a result greater than 3 come out to 1 less than it’s supposed to, however.
I think it would even result in any addition with a term ≤ 3 and result > 3 come out to exactly 3, unless you have some sort of rule for S + SSS0 sometimes becoming SSSS0 instead of SSS0.
Note also that an enterprising soul can line up the two steps:
S0 + SSS0
0 + SSS0
And notice that they are confused, because the SSS0′s are identical, even though they shouldn’t be, because Sx + y = x + Sy was the rule applied and Sy ≠ y.
A brain hack that made all of this work is surely possible, of course, but it seems like it would have to be a bit more systematic.
I see a lot of people arguing that “2”, “3″, “+”, and “=” are defined in terms of the Peano axioms, and as such, aren’t actually relevant to the behavior of physical objects. They say that the axioms pin down the numbers, regardless of how physical objects behave or start behaving.
But the Peano axioms use something called a “successor” to generate the natural numbers. And how do we figure out what the successors are? Well, one notation is to append an “S” to the previous number to indicate that number’s successor, such that the successor of “SS0” is “SSS0″. Then “+” would mean “repeatedly apply the successor operation to the number to the right of this operator for however many times indicated by the number to the left of this operator”, and “=” would indicate “these two numbers, when written out using successor-notation, have the character ‘S’ occur the same number of times”.
So how do we calculate “2 + 2”? Well, we take the number of occurrences of the character “S”, count them up, then put the total number of occurrences in front of a single “0”: “SSSS0“, which we interpret as “4”. “2 + 2 = 4”.
But now let’s imagine your brain’s visual cortex (or whatever it uses to visualize things) being messed up somehow. Actually, let’s go further than that, and suppose everyone’s visual cortexes got messed up in the same way, so that now, when we visualize two objects and two more objects, and put them together, we see, not four objects, but three. So if you were to try and visualize a group of two dots coming together with another group of two dots, you would end up seeing, in your mind’s eye, a final group of three dots.
Now imagine counting the number of times “S” occurs in “SS0″, then putting two of those groups of “S” characters together. How many “S” character are in the final group? Visualize it, now...
Why, three. Of course there’s three. If you take “SS” and put it with “SS”, of course you get “SSS”. What’s that? You say you’re getting “SSSS”? Where are you getting that extra “S” from? What? No—can’t you see it? It’s obviously three—how can you put those two groups together and get four?
So, “SS0 + SS0 = SSS0”.
My point is this: you can’t just point to the Peano axioms and say, “Ha! Your hypothetical situation involving the behavior of mere physical objects is meaningless before the might of my absolute mathematical assumptions!” Remember, when you try to perform “logic” on those axioms, you’re still using your brain to do it. And however ivory-tower untouchable you imagine your axioms to be, your brain is a real, physical object performing real physical computations. If we lived in a slightly different universe, your brain would take one look at “SS0” and “SS0“, visually add together the number of “S” characters, and see “SSS0”. Anyone who saw “SSSS0” would be seen as crazy, or brain-damaged, or something.
Physics trumps math and logic.
Not sure who “they” are. May I suggest reading up on model theory? Maybe Enderton’s “mathematical logic” (?).
It’s important to think separately about the “model” (the thing we are studying), the “language” (where we make statements about the model), and the “theory” (a set of statements in a language). The same theory in a particular language may and in fact generally does apply to multiple distinct models. The same model (object) may be described by theories in different languages of different strengths.
So in one sense Peano axioms “pin down” the natural numbers, but in another sense they don’t because we can invent crazy objects that contain a lot more than just the natural numbers to which Peano axioms also apply (this is the content of the Lowenheim-Skolem theorem).
http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem
We can use a more powerful language than first order logic to describe the natural numbers, and that would rule out some of the crazy models. That will capture more properties of the natural number line, but not everything.
Physicists play a similar game to logicians, except perhaps a bit less formally. But their models (in the sense of ‘object of study’) “bite back.”
It’s confusing that to a model theorist “the model” refers to the territory, while to a statistician “the model” refers to the map.
My bad; I wasn’t clear. “They” refers, not to any person or group of people in academia, but some of the commenters on this LW post. As an example: this comment.
Um, but the (+) operator in peano arithmetic is actually defined in terms of Sx + y = x + Sy. It would be somewhat circular to “suggest counting the S’s up” in a method of defining numbers, after all. So the way you calculate 2 + 2 is more like
SS0 + SS0
(thing on the left starts with an S)
S(S0) + SS0
(move the S to the right)
S0 + SSS0
(thing on the left starts with an S)
S(0) + SSS0
(move the S to the right)
0 + SSSS0
(eliminate
"0 + "with axiom)SSSS0
I imagine you’d need a rather more devious brain modification to prevent one from carrying out these steps correctly, in such a way that the result is SSS0.
All right. Let me take a stab at it.
Okay. Following you so far...
Eh? Where’d you get the extra “S” from?
(This hack would have the unfortunate side effect of making every addition with at least one term less than or equal to 3 and a result greater than 3 come out to 1 less than it’s supposed to, however. If you wanted to only make 2 + 2 = 3, and preserve all other additions as-is, I can’t think of any brain hack that could do that. That’s not to say no such hack is possible; I’m sure one is, but I just can’t think of one.)
I think it would even result in any addition with a term ≤ 3 and result > 3 come out to exactly 3, unless you have some sort of rule for S + SSS0 sometimes becoming SSSS0 instead of SSS0.
Note also that an enterprising soul can line up the two steps:
And notice that they are confused, because the SSS0′s are identical, even though they shouldn’t be, because
Sx + y = x + Sywas the rule applied andSy ≠ y.A brain hack that made all of this work is surely possible, of course, but it seems like it would have to be a bit more systematic.
That seems fair. Would you agree that my original point (that your grasp of logic stems from a physical brain and can be muddled) stands, though?