I agree with this statement—and yet you did not contradict my statement that second order logic is also not part of mainstream mathematics.
A topologist might care about manifolds or homeomorphisms—they do not care about foundations of mathematics—and it is not the case that only one foundation of mathematics can support topology. The weaker foundation is preferable.
The last sentence is not obvious at all. The goal of mathematics is not to be correct a lot. The goal of mathematics is to promote human understanding. Strong axioms help with that by simplifying reasoning.
If you assume A and derive B you have not proven B but rather A implies B. If you can instead assume a weaker axiom Aprime, and still derive B, then you have proven Aprime implies B, which is stronger because it will be applicable in more circumstances.
If you were writing software for something intended to traverse the Interplanetary transfer network then you would probably use charts and atlases and transition functions, and you would study (symplectic) manifolds and homeomorphisms in order to understand those more-applied concepts.
If an otherwise useful theorem assumes that the manifold is orientable, then you need to show that your practical manifold is orientable before you can use it—and if it turns out not to be orientable, then you can’t use it at all. If instead you had an analogous theorem that applied to all manifolds, then you could use it immediately.
There’s a difference between assuming that a manifold is orientable and assuming something about set theory. The phase space is, of course, only approximately a manifold. On a very small level it’s—well, something we’re not very sure of. But all the math you’ll be doing is an approximation of reality.
So some big macroscopic feature like orientability would be a problem to assume. Orientability corresponds to something in physical reality, and something that clearly matters for your calculation.
The axiom of choice or whatever set-theoretic assumption corresponds to nothing in physical reality. It doesn’t matter if the theorems you are using are right for the situation, because they are obviously all wrong, because they are about symplectic dynamics on a manifold, and physics isn’t actually symplectic dynamics on a manifold! The only thing that matters is how easily you can find a good-enough approximation to reality. More foundational assumptions make this easier, and do not impede one’s approximation of reality.
Note that physicists frequently make arguments that are just plain unambiguously wrong from a mathematical perspective.
I understand your point—it’s akin to the Box quote “all models are wrong but some are useful”—when choosing among (false) models, choose the most useful one. However, it is not the case that stronger assumptions are more useful—of course stronger assumptions make the task of proving easier, but the task as a whole includes both proving and also building a system based on the theorems proven.
My primary point is that EY is implying that second-order logic is necessary to work with the integers. People work with the integers without using second-order logic all the time. If he said that he is only introducing second-order logic for convenience in proving and there are certainly other ways of doing it, and that some people (intuitionists and finitists) think that introducing second-order logic is a dubious move, I’d be happy.
My other claim that second-order logic is unphysical and that its unphysicality probably does ripple out to make practical tasks more difficult, is a secondary one. I’m happy to agree that this secondary claim is not mainstream.
My primary point is actually that I don’t care if math is useful. Math is awesome. This is obviously an extremely rare viewpoint, but very common among.
But I do agree with that quote, more or less. I think that potentially some models are true, but those models are almost certainly less useful for most purposes than the crude and easy to work with approximations.
I agree that second-order logic is not necessary to work with the integers. Second-order logic is necessary to work with the integers and only the integers, however. Somewhat problematically, it’s not actually possible to work with second-order logic.
I agree with this statement—and yet you did not contradict my statement that second order logic is also not part of mainstream mathematics.
A topologist might care about manifolds or homeomorphisms—they do not care about foundations of mathematics—and it is not the case that only one foundation of mathematics can support topology. The weaker foundation is preferable.
The last sentence is not obvious at all. The goal of mathematics is not to be correct a lot. The goal of mathematics is to promote human understanding. Strong axioms help with that by simplifying reasoning.
If you assume A and derive B you have not proven B but rather A implies B. If you can instead assume a weaker axiom Aprime, and still derive B, then you have proven Aprime implies B, which is stronger because it will be applicable in more circumstances.
In what “circumstances” are manifolds and homeomorphisms useful?
If you were writing software for something intended to traverse the Interplanetary transfer network then you would probably use charts and atlases and transition functions, and you would study (symplectic) manifolds and homeomorphisms in order to understand those more-applied concepts.
If an otherwise useful theorem assumes that the manifold is orientable, then you need to show that your practical manifold is orientable before you can use it—and if it turns out not to be orientable, then you can’t use it at all. If instead you had an analogous theorem that applied to all manifolds, then you could use it immediately.
There’s a difference between assuming that a manifold is orientable and assuming something about set theory. The phase space is, of course, only approximately a manifold. On a very small level it’s—well, something we’re not very sure of. But all the math you’ll be doing is an approximation of reality.
So some big macroscopic feature like orientability would be a problem to assume. Orientability corresponds to something in physical reality, and something that clearly matters for your calculation.
The axiom of choice or whatever set-theoretic assumption corresponds to nothing in physical reality. It doesn’t matter if the theorems you are using are right for the situation, because they are obviously all wrong, because they are about symplectic dynamics on a manifold, and physics isn’t actually symplectic dynamics on a manifold! The only thing that matters is how easily you can find a good-enough approximation to reality. More foundational assumptions make this easier, and do not impede one’s approximation of reality.
Note that physicists frequently make arguments that are just plain unambiguously wrong from a mathematical perspective.
I understand your point—it’s akin to the Box quote “all models are wrong but some are useful”—when choosing among (false) models, choose the most useful one. However, it is not the case that stronger assumptions are more useful—of course stronger assumptions make the task of proving easier, but the task as a whole includes both proving and also building a system based on the theorems proven.
My primary point is that EY is implying that second-order logic is necessary to work with the integers. People work with the integers without using second-order logic all the time. If he said that he is only introducing second-order logic for convenience in proving and there are certainly other ways of doing it, and that some people (intuitionists and finitists) think that introducing second-order logic is a dubious move, I’d be happy.
My other claim that second-order logic is unphysical and that its unphysicality probably does ripple out to make practical tasks more difficult, is a secondary one. I’m happy to agree that this secondary claim is not mainstream.
My primary point is actually that I don’t care if math is useful. Math is awesome. This is obviously an extremely rare viewpoint, but very common among.
But I do agree with that quote, more or less. I think that potentially some models are true, but those models are almost certainly less useful for most purposes than the crude and easy to work with approximations.
I agree that second-order logic is not necessary to work with the integers. Second-order logic is necessary to work with the integers and only the integers, however. Somewhat problematically, it’s not actually possible to work with second-order logic.
What sort of practical tasks are you thinking of?