>>13818176>Understanding goes from examples and then generalizations. Mathematicians write down the generalization first and then go like "heres some examples" if ever.It depends on how you think, different people think in different terms. You can think in incredibly abstract terms from the get-go and have that come naturally - grothendieck style. Your personality is important, everyone does things a bit differently.
It is also really difficult to by hand work with the examples of a lot of theorems - for instance the banach fixed point theorem is easy and very general - but the examples of application of it can be arbitrarily hard. Or in group theory you absolutely need to employ general arguments about groups, normal subgroups, sylow subgroups, commutator subgroups, transfer homomorphisms, etc before you are actually able to do practical things with them like enumerate them up to isomorphism, because the things get out of hand extremely quickly when you allow the order to grow.
> Its like studying law in law school without ever reading any law, just the meta concept of law.Well, the subject itself is different from law. In law you would not be able to and are never expected to walk into a courtroom on an alien planet inhabited by sentient molluscs, take position of the judge, reproduce their entire legal code yourself on the spot without reading it and apply it to the case in question. Meta-law is different from law no matter how you spin it, but meta-mathematics is the same as mathematics when done right. Mathematics is closed under applying the meta-functor, you could say.
