> If you don't come from a math-y background and are trying to "get into serious mathematics", the only way to do this is to get a PhD.
> I realize this might come across as gatekeeping, but the reality is that each subfield of math (or any scientific discipline) has its own set of tactics for approaching problems, which have been developed over the years by the people actually in the trenches.
There are degrees of seriousness. Sure, even a very good foundation is not enough to do novel work in say, algebraic geometry (but then again, sometimes it is enough to make progress in combinatorics) - but the strongest undergrads are still much closer to freshly minted PhDs than they are to laymen.
Mathematical maturity is the first and hardest step; after that, people will know where to go for the folklore if they want it.
> I realize this might come across as gatekeeping, but the reality is that each subfield of math (or any scientific discipline) has its own set of tactics for approaching problems, which have been developed over the years by the people actually in the trenches.
There are degrees of seriousness. Sure, even a very good foundation is not enough to do novel work in say, algebraic geometry (but then again, sometimes it is enough to make progress in combinatorics) - but the strongest undergrads are still much closer to freshly minted PhDs than they are to laymen.
Mathematical maturity is the first and hardest step; after that, people will know where to go for the folklore if they want it.