An idea which has picked up some traction in some circles of pure mathematicians is that numbers should be viewed as the “shadow” of finite sets, which is a more fundamental notion.
You start with the notion of finite set, and functions between them. Then you “forget” the difference between two finite sets if you can match the elements up to each other (i.e if there exists a bijection). This seems to be vaguely related to your thing about being invariant under permutation—if a property of a subset of positions (i.e those positions that are sent to 1), is invariant under bijections (i.e permutations) of the set of positions, it can only depend on the size/number of the subset.
See e.g the first ~2 minutes of this lecture by Lars Hesselholt (after that it gets very technical)
Equivalence classes of sets under directions is the standard way of defining “size” (cardinality), so unless you are talking about some more specific idea this is not just something in some circles of math.
There are also ordinals, which are ordered sets under order isomorphism. You use cardinals when you only care about size (meaning each element doesn’t have an individual identity), and ordinals when you also care about order (so each apple still doesn’t have an individual identity, but the way they are lined up on the plate matters)
Honestly neither are useful to most mathematicians, beyond distinguishing countably infinite from uncountably infinite.
This is great!
An idea which has picked up some traction in some circles of pure mathematicians is that numbers should be viewed as the “shadow” of finite sets, which is a more fundamental notion.
You start with the notion of finite set, and functions between them. Then you “forget” the difference between two finite sets if you can match the elements up to each other (i.e if there exists a bijection). This seems to be vaguely related to your thing about being invariant under permutation—if a property of a subset of positions (i.e those positions that are sent to 1), is invariant under bijections (i.e permutations) of the set of positions, it can only depend on the size/number of the subset.
See e.g the first ~2 minutes of this lecture by Lars Hesselholt (after that it gets very technical)
[Deleted]
Equivalence classes of sets under directions is the standard way of defining “size” (cardinality), so unless you are talking about some more specific idea this is not just something in some circles of math.
There are also ordinals, which are ordered sets under order isomorphism. You use cardinals when you only care about size (meaning each element doesn’t have an individual identity), and ordinals when you also care about order (so each apple still doesn’t have an individual identity, but the way they are lined up on the plate matters)
Honestly neither are useful to most mathematicians, beyond distinguishing countably infinite from uncountably infinite.
Ooh, I like that formulation. It’s cleaner—it jumps straight to numbers rather than having to extract them from counts.