Very much appreciate this! I've been interested in studying higher level mathematics mostly as... well, just an interest and also to brush up on things that now feel long forgotten but didn't know where to start and wasn't really looking to attend school for it.
Terence Tao wrote that mathematical maturity is developed in three stages: the pre-rigor stage, the rigor stage, and the post-rigor stage.
The rigor stage usually begins during the second or third year of university, and ends about half way through one's PhD. Although in reality it's more of a very gradual tapering off, and is something one likely never fully exits.
By my estimation, you need to do between 1000-2000 proofs to complete the rigorous stage of your math education. Doing these proofs forces you to slowly "ablate away" your bad mathematical intuition, replacing it with correct intuition, informed by the rigor super-structure within which you're required to operate.
Assuming one does on average 50 proofs per course, that's, at a minimum, twenty proof-based math courses you need to take.
The hard part about learning advanced mathematics on your own is that you need an expert to read your proofs and provide feedback. There is no compiler for pure mathematics which can take the place of a human. I would advise anyone who is serious about learning advanced mathematics to enroll in courses at a university. There's no need to matriculate, most universities offer open university on a pay-per-course basis.
One only needs to visit the viXra to see what happens when people try to skip the hard work and go directly from the pre to the post-rigor stage.
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[ 3.1 ms ] story [ 13.3 ms ] threadNow to see if I have the discipline...
The rigor stage usually begins during the second or third year of university, and ends about half way through one's PhD. Although in reality it's more of a very gradual tapering off, and is something one likely never fully exits.
By my estimation, you need to do between 1000-2000 proofs to complete the rigorous stage of your math education. Doing these proofs forces you to slowly "ablate away" your bad mathematical intuition, replacing it with correct intuition, informed by the rigor super-structure within which you're required to operate.
Assuming one does on average 50 proofs per course, that's, at a minimum, twenty proof-based math courses you need to take.
The hard part about learning advanced mathematics on your own is that you need an expert to read your proofs and provide feedback. There is no compiler for pure mathematics which can take the place of a human. I would advise anyone who is serious about learning advanced mathematics to enroll in courses at a university. There's no need to matriculate, most universities offer open university on a pay-per-course basis.
One only needs to visit the viXra to see what happens when people try to skip the hard work and go directly from the pre to the post-rigor stage.