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.
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?