Ah okay, I think we have different understandings of the motivation for axioms. If I understand correctly you are saying axioms should be motivated by intuition. I disagree. Axioms should be motivated for their generative capacity, i.e. how much fun math we can squeeze out of them. Axioms don’t need to be “true”, rather they need to be interestingly generative.
When I encounter math I don’t immediately have an intuition for, I remind myself what Von Neumann said:
“In mathematics you don’t understand things. You just get used to them.”
For me at least, I have a hard time grasping the intuition behind a lot of math so I cling to this quote like dogma.
I see. And I agree that there’s a place for such an attitude, that it can be interesting to explore the formal consequences of a system such as ZFC even if you don’t necessarily think it’s “true” in any Platonic sense. (Although maybe you want to at least be able to hope that the system is consistent; no one proves things in naive set theory anymore even though it is an extremely generative system!)
Anyway, part of my point was that systems like PA that are weaker and more intuitive than ZFC can be nearly equally “interestingly generative”, with some narrow exceptions. So even if that is your primary criterion for choosing an axiom system, you could still come to the conclusion that PA is a better foundational system if you still care at all about an abstract notion of truth.
Regarding the von Neumann quote, presumably the math you are encountering has already been vetted by other mathematicians, so you can assume it is likely to be true even without understanding it. In such a situation, it makes sense to allow yourself to get used to it. It is different if you want to study the foundations of math; since people disagree about it you will need to form your own opinion.
Ah okay, I think we have different understandings of the motivation for axioms. If I understand correctly you are saying axioms should be motivated by intuition. I disagree. Axioms should be motivated for their generative capacity, i.e. how much fun math we can squeeze out of them. Axioms don’t need to be “true”, rather they need to be interestingly generative.
When I encounter math I don’t immediately have an intuition for, I remind myself what Von Neumann said:
“In mathematics you don’t understand things. You just get used to them.”
For me at least, I have a hard time grasping the intuition behind a lot of math so I cling to this quote like dogma.
I see. And I agree that there’s a place for such an attitude, that it can be interesting to explore the formal consequences of a system such as ZFC even if you don’t necessarily think it’s “true” in any Platonic sense. (Although maybe you want to at least be able to hope that the system is consistent; no one proves things in naive set theory anymore even though it is an extremely generative system!)
Anyway, part of my point was that systems like PA that are weaker and more intuitive than ZFC can be nearly equally “interestingly generative”, with some narrow exceptions. So even if that is your primary criterion for choosing an axiom system, you could still come to the conclusion that PA is a better foundational system if you still care at all about an abstract notion of truth.
Regarding the von Neumann quote, presumably the math you are encountering has already been vetted by other mathematicians, so you can assume it is likely to be true even without understanding it. In such a situation, it makes sense to allow yourself to get used to it. It is different if you want to study the foundations of math; since people disagree about it you will need to form your own opinion.