>>14400262That's strictly cardinal numbers, that is, the numbers of size. The cardinal number '9' is the size of a set with nine elements, as you rightly point out.
But numbers can also be ordinals, that is, the numbers of sequences. Within the set of nine elements, we can count them (in different orderings) as ('1', first); ('2', second); etc. There are distinct functions of ordinal and cardinal numbers that are blurred if only considered naively. In fact set theory constructs ordinals and cardinals as entirely distinct objects.
In fact, the ordinals are arguably much simpler than the cardinals since ordinals are only loaded with the information about succession (e.g., '4' is the successor to '3') while cardinals are loaded with a concept of cardinality (e.g., '4' corresponds to a set with '4'-many elements [i.e., cardinality 4]).
So it's disingenuous to say that "that is all numbers are". They are functionally at least two types of numbers, and there's a lot more to say about them philosophically. Do they exist beyond their physical correspondents? And are they invented or discovered? And so on.
