2 ms·
It is so “philosophically” because the larger your set, the less inner structures it has (the larger your set the less you can say). Akin to “you cannot say mu
by pfortuny 5y ago
It is so “philosophically” because the larger your set, the less inner structures it has (the larger your set the less you can say).
Akin to “you cannot say much about a genus but you can say a huge lot about a species”.
Of course this is a very rough explanation if at all.
Edit: there is much more to it, obviously, than my lame comparison.
- autopilot23 5y agoThat's not nessecarily true. Bigger (cardinality?) doesn't mean more or less inner structures. If anything, it's the assumptions you choose to make that determine how much you can say about an object. A reason why the complex numbers is tame in certain contexts is that it's algebraically closed. Discrete theory also tends to be more difficult imo since by taking its limit, you should still recover the continuous theory.
- feoren 5y agoIt doesn't have much to do with how "large" the set is. The complex numbers have the same cardinality as the reals; the rational numbers have the same cardinality as the integers and whole numbers. Even talking about finite sets, the group of integers modulo 13 is a lot less "interesting" than the group of integers modulo 12, because how such a group acts depends heavily on the division structure of the modulus: the more divisors there are in the modulus, the more interesting things get. So the sets of integers modulo any prime are all basically the same (and boring), while the integers modulo highly composite numbers like 60 are wild, complex structures.