Multisets are defined as a tuple of a set and a function . They are not a new construction which are somehow orthogonally different to sets.
ZFC was not built to be intuitive. It was built to unify and formalize all of mathematics. It is not supposed to be some kind of universal human way of seeing the world through a mathematical lens. It’s just a set of axioms we build on, and we can choose different sets of axioms at will.
Your example of a water molecule assumes we do not distinguish between the hydrogen atoms with indexing and . I’d argue sets are more natural as we as humans naturally are able to distiguish separate objects even if there exists an equivalence relation between them e.g. both and are of the class “hydrogen”.
In short:
You seem to be conflating the foundations of mathematics with the foundations of mathematical thinking. These are not the same thing. The former is how we justify doing math, the latter is how we map abstract concepts to intuitive patterns us meat-brained apes can understand.
Multisets are defined as a tuple of a set and a function . They are not a new construction which are somehow orthogonally different to sets.
Multisets can be defined in terms of sets, and conversely sets can be defined in terms of multisets. I don’t see either of them as “more fundamental” than the other necessarily, except in terms of which one we’ve chosen to give a more prominent place to.
ZFC was not built to be intuitive. It was built to unify and formalize all of mathematics. It is not supposed to be some kind of universal human way of seeing the world through a mathematical lens. It’s just a set of axioms we build on, and we can choose different sets of axioms at will.
If ZFC is not intuitive (and my opinion is that it is not), then we shouldn’t assume that any theorems it proves are actually true, and we should if possible rely on a weaker system which is intuitive, such as the Peano axioms. Peano arithmetic is in fact powerful enough to deal with almost all of modern mathematics (except for abstract set theory), it’s just that people are so used to using ZFC as the foundation that they rarely check to see whether their theorems can be proven in PA. See for example this book where the author develops a “strong undergraduate curriculum” using a system conservative over PA as the foundation.
Your example of a water molecule assumes we do not distinguish between the hydrogen atoms with indexing and .
On the atomic level, quantum effects prevent us from distinguishing and . Eliezer wrote about this in the Sequences.
You’re missing my point, possibly intentionally. Lets go through your points in reverse order:
Yes I know hydrogen atoms are indistinguishable. The point was about objects in general, the atomic example was a holdover from the original post. If you need help abstracting think of two apples and an orange instead.
Truth intuitiveness. Assuming as much would mean throwing out a lot of the most beautiful parts of intellectual thought which has occurred over human history. This feels like a plug or advertisement for PA not an actual point.
Yes of course. The point was that using multisets as a foundation isn’t significantly different from using sets as a foundation. It was a critique of the original post, not an assertion about the best foundation of numbers.
I’m getting the feeling you want to make a point but you’re not looking at the context of my comment.
I’m getting the feeling you want to make a point but you’re not looking at the context of my comment.
Actually I was trying to make three separate unrelated points. I think we agree sufficiently on the first and third points that it would be unproductive to discuss them further. Regarding my second point:
This feels like a plug or advertisement for PA not an actual point.
Well I do like to plug for PA when I can (it’s a good foundation for math!), but I think I do have an actual point as well.
Truth intuitiveness.
I mean, sure, there are a lot of unintuitive things that you can prove from true axioms. But with the axioms themselves, the only basis we could have for asserting that the axioms are true is that they are intuitive. I mean, why else would you think they are true? (In mathematics at least, of course in other fields you could have empirical evidence of something unintuitive.)
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.
ZFC is the kind of framework that kind of just asserts itself as true and then still leaves huge holes open in mathematics. The complex plane and quaternions dropping communicative properties leaves my intuition to say there’s fundamental issues in mathematics that haven’t been resolved.
A few problems I have with this post:
Multisets are defined as a tuple of a set
ZFC was not built to be intuitive. It was built to unify and formalize all of mathematics. It is not supposed to be some kind of universal human way of seeing the world through a mathematical lens. It’s just a set of axioms we build on, and we can choose different sets of axioms at will.
Your example of a water molecule assumes we do not distinguish between the hydrogen atoms with indexing
In short:
You seem to be conflating the foundations of mathematics with the foundations of mathematical thinking. These are not the same thing. The former is how we justify doing math, the latter is how we map abstract concepts to intuitive patterns us meat-brained apes can understand.
Multisets can be defined in terms of sets, and conversely sets can be defined in terms of multisets. I don’t see either of them as “more fundamental” than the other necessarily, except in terms of which one we’ve chosen to give a more prominent place to.
If ZFC is not intuitive (and my opinion is that it is not), then we shouldn’t assume that any theorems it proves are actually true, and we should if possible rely on a weaker system which is intuitive, such as the Peano axioms. Peano arithmetic is in fact powerful enough to deal with almost all of modern mathematics (except for abstract set theory), it’s just that people are so used to using ZFC as the foundation that they rarely check to see whether their theorems can be proven in PA. See for example this book where the author develops a “strong undergraduate curriculum” using a system conservative over PA as the foundation.
On the atomic level, quantum effects prevent us from distinguishing
You’re missing my point, possibly intentionally. Lets go through your points in reverse order:
Yes I know hydrogen atoms are indistinguishable. The point was about objects in general, the atomic example was a holdover from the original post. If you need help abstracting think of two apples and an orange instead.
Truth
Yes of course. The point was that using multisets as a foundation isn’t significantly different from using sets as a foundation. It was a critique of the original post, not an assertion about the best foundation of numbers.
I’m getting the feeling you want to make a point but you’re not looking at the context of my comment.
Actually I was trying to make three separate unrelated points. I think we agree sufficiently on the first and third points that it would be unproductive to discuss them further. Regarding my second point:
Well I do like to plug for PA when I can (it’s a good foundation for math!), but I think I do have an actual point as well.
I mean, sure, there are a lot of unintuitive things that you can prove from true axioms. But with the axioms themselves, the only basis we could have for asserting that the axioms are true is that they are intuitive. I mean, why else would you think they are true? (In mathematics at least, of course in other fields you could have empirical evidence of something unintuitive.)
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.
ZFC is the kind of framework that kind of just asserts itself as true and then still leaves huge holes open in mathematics. The complex plane and quaternions dropping communicative properties leaves my intuition to say there’s fundamental issues in mathematics that haven’t been resolved.