If, whenever we took 2 bananas and stuck them together with 2 more bananas, we ended up with 3 bananas, 2+2=4 would still be ‘true’ in the abstract sense that it proceeds naturally from the axioms[.]
I’m not so sure of that. If putting 2 S’s next to 2 S’s got us 3 S’s, we could prove 2+2=3 in PA with the usual definition of addition:
(dfn) \a. 0 + a = a
(dfn) \ab. Sb + a = b + Sa
\a. SS0 + a = S0 + Sa = 0 + SSa = SSa
SS0 + SS0 = SSS0
Depending on the universe’s other rules for putting n things next to m things, we might also be able to derive “2+2=4”. In this case, we would decide that PA is inconsistent! Whatever the other rules are, this already shows that the “abstract” conclusions we can draw from a set of axioms depend on the way symbol manipulation works in our world.
I don’t think this is really a problem for your argument, but it’s an interesting complication. Many (most?) physical facts seem to have no influence on the symbolic manipulations we can use to derive them. For instance, symbolically computing a series for pi doesn’t seem to involve any actual circles the way shuffling symbols to add 2 and 2 in PA involves putting SS next to SS.
If putting 2 S’s next to 2 S’s got us 3 S’s, we could prove 2+2=3 in PA with the usual definition of addition
Nitpick but important: we couldn’t actually prove it, just produce a convincing (in that world) false proof (that is actually a proof of a theorem in some other, inconsistent, system with slightly different inference rules).
We could figure out that our symbolic manipulation is inconsistent with the axioms based on the quirk you consider.
There are more axioms needed to define Peano Arithmetic. Taking axioms 7 and 8 from Wikipedia, translated into your notation:
\a. Sa != 0
\ab. Sa = Sb → a = b
(I also use the symmetry and transitivity of equality.)
Note, from the axioms you stated:
\a. S0 + a = 0 + Sa = Sa
So, axiom 8 can be restated as:
\ab. S0 + a = S0 + b → a = b
So, starting with your result:
SSS0 = SS0 + SS0 = S0 + SSS0
But also,
S0 + SS0 = SSS0 = S0 + SSS0
So, by the restatement of Axiom 8:
SS0 = SSS0
And then using the original form of Axiom 8 twice:
S0 = SS0
0 = S0
We have a contradiction of Axiom 7.
Thus, it is proven that our symbol manipulation does not follow the Peano Axioms. This does not invalidate the Peano Axioms. It simply means that a given physical system does not follow them. Of course, it would be difficult for people living in an alternate universe where symbols really behaved this way to notice the distinction. And they likely would not have to, if all objects behaved that way; they would figure out some other math to represent their situation. And at some point, mathematicians in their ivory towers would develop this weird math that is really hard to write down and has no known application in the real world (both properties bringing great joy to these academicians).
You’re right. PA is still consistent (i.e. has a model) even if
N = the set of strings of the form S*0
0 = the string "0"
S = the function that prepends "S" to its argument
fails to be one because of the way string concatenation works. There’s nothing mathematically special about theories that can use physical objects as a model.
(Minor quibble: the definition of addition isn’t an axiom. It’s just a relation definable in the first-order theory of arithmetic.)
I’m not so sure of that. If putting 2 S’s next to 2 S’s got us 3 S’s, we could prove 2+2=3 in PA with the usual definition of addition:
Depending on the universe’s other rules for putting n things next to m things, we might also be able to derive “2+2=4”. In this case, we would decide that PA is inconsistent! Whatever the other rules are, this already shows that the “abstract” conclusions we can draw from a set of axioms depend on the way symbol manipulation works in our world.
I don’t think this is really a problem for your argument, but it’s an interesting complication. Many (most?) physical facts seem to have no influence on the symbolic manipulations we can use to derive them. For instance, symbolically computing a series for pi doesn’t seem to involve any actual circles the way shuffling symbols to add 2 and 2 in PA involves putting SS next to SS.
Nitpick but important: we couldn’t actually prove it, just produce a convincing (in that world) false proof (that is actually a proof of a theorem in some other, inconsistent, system with slightly different inference rules).
We could figure out that our symbolic manipulation is inconsistent with the axioms based on the quirk you consider.
There are more axioms needed to define Peano Arithmetic. Taking axioms 7 and 8 from Wikipedia, translated into your notation:
\a. Sa != 0 \ab. Sa = Sb → a = b
(I also use the symmetry and transitivity of equality.)
Note, from the axioms you stated:
\a. S0 + a = 0 + Sa = Sa
So, axiom 8 can be restated as:
\ab. S0 + a = S0 + b → a = b
So, starting with your result:
SSS0 = SS0 + SS0 = S0 + SSS0
But also,
S0 + SS0 = SSS0 = S0 + SSS0
So, by the restatement of Axiom 8:
SS0 = SSS0
And then using the original form of Axiom 8 twice:
S0 = SS0 0 = S0
We have a contradiction of Axiom 7.
Thus, it is proven that our symbol manipulation does not follow the Peano Axioms. This does not invalidate the Peano Axioms. It simply means that a given physical system does not follow them. Of course, it would be difficult for people living in an alternate universe where symbols really behaved this way to notice the distinction. And they likely would not have to, if all objects behaved that way; they would figure out some other math to represent their situation. And at some point, mathematicians in their ivory towers would develop this weird math that is really hard to write down and has no known application in the real world (both properties bringing great joy to these academicians).
You’re right. PA is still consistent (i.e. has a model) even if
fails to be one because of the way string concatenation works. There’s nothing mathematically special about theories that can use physical objects as a model.
(Minor quibble: the definition of addition isn’t an axiom. It’s just a relation definable in the first-order theory of arithmetic.)