Until the time of writing this comment, I did believe that there would be a meta-logic that governs the math or logic that “can exist”. However, I concede that such a statement requires proof and I do not have enough (read: any) background in logic to know if such a proof is forthcoming.
This is the question: if there were two physical universes realizing two distinct sets of mathematical ideas such that at least one mathematical idea is realized in one set and not the other, then would mathematics still provide a meta-framework for both sets of mathematical ideas?
More succinctly: Given any two models, can you always find a meta-model that includes them both? Or maybe it can be proven that this is unprovable within a given model?
Given any two models, can you always find a meta-model that includes them both?
I think this is a confused question. The answer is yes, but when I tell you what the meta-model is, I don’t think your underlying curiosity will be satisfied.
Given 2 models A and B, we can construct a model C. The set of objects in C will be the union of the set of elements of the form (A, a) for all objects a in the model A, and the set of elements of the form (B, b) for all objects b in the model B. Then every proposition about A can can correspond to a proposition about C, particularly about its objects of the form (A, a), by simple substitution of (A, a) for a; and similarly propositions about B correspond to propositions about the objects of the form (B, b). This is, of course, a trivial combination in which the parts that correspond to the original models do not interact with each other at all, hence my belief that you will not be satisfied.
Perhaps a better question to ask would be: is there a meta framework that distinguishes some mathematical systems as “good” and others as “bad”? I think the answer to this question is the criteria that a mathematical system be self-consistent, that is, it does not produce contradictions. Of course, this criteria is not always possible to verify.
Yes, the question was confused. I got distracted thinking about stuff I don’t know about (asking “how” instead of “whether”).
You don’t need “meta-logic”, whatever that might be, to know that 2+2=3 cannot be consistent with Peano arithmetic.
Here’s the “proof”. We know that 2+2=4 in “our” Peano Arithmetic. Suppose that 2+2=3 in “another” Peano Arithmetic in an alternate reality. Then the two Peano Arithemetics are actually different because they have a different set of “trues”. When we say that 2+2=4 in “our” Peano Arithmetic, we mean “our” PA and not the other. Whatever distinguishes the two arithemetics can be incorporated in our PA as an axiom – indeed it was included implicitly in what we meant by PA all along even if we lacked the imagination to explicitly identify it.
Until the time of writing this comment, I did believe that there would be a meta-logic that governs the math or logic that “can exist”. However, I concede that such a statement requires proof and I do not have enough (read: any) background in logic to know if such a proof is forthcoming.
This is the question: if there were two physical universes realizing two distinct sets of mathematical ideas such that at least one mathematical idea is realized in one set and not the other, then would mathematics still provide a meta-framework for both sets of mathematical ideas?
More succinctly: Given any two models, can you always find a meta-model that includes them both? Or maybe it can be proven that this is unprovable within a given model?
A logician’s input on this would be helpful.
I think this is a confused question. The answer is yes, but when I tell you what the meta-model is, I don’t think your underlying curiosity will be satisfied.
Given 2 models A and B, we can construct a model C. The set of objects in C will be the union of the set of elements of the form (A, a) for all objects a in the model A, and the set of elements of the form (B, b) for all objects b in the model B. Then every proposition about A can can correspond to a proposition about C, particularly about its objects of the form (A, a), by simple substitution of (A, a) for a; and similarly propositions about B correspond to propositions about the objects of the form (B, b). This is, of course, a trivial combination in which the parts that correspond to the original models do not interact with each other at all, hence my belief that you will not be satisfied.
Perhaps a better question to ask would be: is there a meta framework that distinguishes some mathematical systems as “good” and others as “bad”? I think the answer to this question is the criteria that a mathematical system be self-consistent, that is, it does not produce contradictions. Of course, this criteria is not always possible to verify.
Yes, the question was confused. I got distracted thinking about stuff I don’t know about (asking “how” instead of “whether”).
You don’t need “meta-logic”, whatever that might be, to know that 2+2=3 cannot be consistent with Peano arithmetic.
Here’s the “proof”. We know that 2+2=4 in “our” Peano Arithmetic. Suppose that 2+2=3 in “another” Peano Arithmetic in an alternate reality. Then the two Peano Arithemetics are actually different because they have a different set of “trues”. When we say that 2+2=4 in “our” Peano Arithmetic, we mean “our” PA and not the other. Whatever distinguishes the two arithemetics can be incorporated in our PA as an axiom – indeed it was included implicitly in what we meant by PA all along even if we lacked the imagination to explicitly identify it.
But still, just to be clear, with a meta-logic or not, you can’t have 2+2=3 be consistent with Peano arithmetic in a different physical reality.