Physics doesn’t guarantee that mathematical reasoning works.
All of math can be built on top of first-order logic. In the sub-case of propositional logic, it’s easy to see entirely within physics that if I observe that “A AND B” corresponds to reality, then when I check if “A” corresponds to reality, I will also find that it does. Every such deduction in propositional logic corresponds to something you can check in the real physical world.
The only infinity in first-order logic are quantifiers, of which only one is needed: FORALL, which is basically just an infinite AND. I don’t think it’s too surprising that a logical deduction from an infinite AND will hold in every finite case that we can check, for similar reasons to why logical deductions hold for the finite case.
It is mysterious that physics is ordered in a way that this works out, but pending the answer you say exists, it’s not any more mysterious than asking why math is ordered that way.
Again: every mathematical error is a real physical even in someone’s brain, so , again, physics guarantees nothing.
The only infinity in first-order logic are quantifiers,
There are of course, lots of infinities in maths. Our ability to reason about them means mathematical reasoning includes symbolic reasoning, not just direct calculation. That’s a computation or pyschology-level observation—it doesn’t add anything to point out that brains are made of quarks.
Again: every mathematical error is a real physical even in someone’s brain, so , again, physics guarantees nothing.
I don’t get what you’re trying to show with this. If I mistakenly derive in Peano Arithmetic that 2 + 2 = 3, I will find myself shocked when I put 2 apples inside a bag that already contains 2 apples and find that there are now 4 apples in that bag. Incorrect mathematical reasoning is physically distinguishible from correct mathematical reasoning.
There are of course, lots of infinities in maths.
Everything we know about all other infinities can be built on top of just FORALL in first-order logic.
In ZFC, the Axiom of Infinity can be written entirely in terms of ∈, ∧, ¬, and ∀. Since all of math can be encoded in ZFC (plus large cardinal axioms as necessary), all our knowledge about infinity can be described with ∀ as our only source of infinity.
Only for the subset of maths that’s also physical. You can’t resolve the Axiom of Choice problem that way.
You can’t resolve the Axiom of Choice problem in any way. Both it and its negation are consistent.
All of math can be built on top of first-order logic. In the sub-case of propositional logic, it’s easy to see entirely within physics that if I observe that “A AND B” corresponds to reality, then when I check if “A” corresponds to reality, I will also find that it does. Every such deduction in propositional logic corresponds to something you can check in the real physical world.
The only infinity in first-order logic are quantifiers, of which only one is needed: FORALL, which is basically just an infinite AND. I don’t think it’s too surprising that a logical deduction from an infinite AND will hold in every finite case that we can check, for similar reasons to why logical deductions hold for the finite case.
It is mysterious that physics is ordered in a way that this works out, but pending the answer you say exists, it’s not any more mysterious than asking why math is ordered that way.
Again: every mathematical error is a real physical even in someone’s brain, so , again, physics guarantees nothing.
There are of course, lots of infinities in maths. Our ability to reason about them means mathematical reasoning includes symbolic reasoning, not just direct calculation. That’s a computation or pyschology-level observation—it doesn’t add anything to point out that brains are made of quarks.
I don’t get what you’re trying to show with this. If I mistakenly derive in Peano Arithmetic that 2 + 2 = 3, I will find myself shocked when I put 2 apples inside a bag that already contains 2 apples and find that there are now 4 apples in that bag. Incorrect mathematical reasoning is physically distinguishible from correct mathematical reasoning.
Everything we know about all other infinities can be built on top of just FORALL in first-order logic.
Only for the subset of maths that’s also physical. You can’t resolve the Axiom of Choice problem that way.
Whatever “built on top of” means. Clearly, we can intend transfinite models.
In ZFC, the Axiom of Infinity can be written entirely in terms of ∈, ∧, ¬, and ∀. Since all of math can be encoded in ZFC (plus large cardinal axioms as necessary), all our knowledge about infinity can be described with ∀ as our only source of infinity.
You can’t resolve the Axiom of Choice problem in any way. Both it and its negation are consistent.