Based on the commentary I’ve heard from grad students, getting truly good at math is on par with bringing the ring to Mt. Doom in terms of difficulty, if not danger.
Oh, I at least somewhat disagree with that. The form of mathematics is extremely difficult and even somewhat obscurantist. The content is, while not easy, no harder than any of the other difficult intellectual skills we regularly teach to intelligent people, and gets easier with practice just like everything else.
No, the miserable thing is when someone starts out into “Let there be a Blah from the set Herp, then define its Bibbity to be the Derp such that De Hurr...” when what they really mean is, “I will prove a theorem quantified over all Herp’s by pretending I have an arbitrary Herp called Blah, and then using Blah to build Bibbity, which is a Derp, and due to the way I built it, it has this property De Hurr that I wanted.” While all that English is obviously quite verbose for everyday work, computer theorem-proving languages are usually about as terse as “real” mathematics while being much clearer and easier to understand, especially because in programming, giving things descriptive names is considered good form, whereas in mathematics the custom is to use single letters, ideally Greek ones or in funny fonts, for absolutely everything, the better to prove the theorem without ever explaining what it means or why you built it that way.
If we bothered to treat mathematics as a form of communication (specifically: communication of rigidly defined, highly specific computational structures) rather than as a personal exploration of Platonic Higher Realms, we could definitely all get much better at it.
(Why yes I am a massive fanboy of Bob Harper… Why did you ask?)
Based on the commentary I’ve heard from grad students, getting truly good at math is on par with bringing the ring to Mt. Doom in terms of difficulty, if not danger.
Oh, I at least somewhat disagree with that. The form of mathematics is extremely difficult and even somewhat obscurantist. The content is, while not easy, no harder than any of the other difficult intellectual skills we regularly teach to intelligent people, and gets easier with practice just like everything else.
No, the miserable thing is when someone starts out into “Let there be a Blah from the set Herp, then define its Bibbity to be the Derp such that De Hurr...” when what they really mean is, “I will prove a theorem quantified over all Herp’s by pretending I have an arbitrary Herp called Blah, and then using Blah to build Bibbity, which is a Derp, and due to the way I built it, it has this property De Hurr that I wanted.” While all that English is obviously quite verbose for everyday work, computer theorem-proving languages are usually about as terse as “real” mathematics while being much clearer and easier to understand, especially because in programming, giving things descriptive names is considered good form, whereas in mathematics the custom is to use single letters, ideally Greek ones or in funny fonts, for absolutely everything, the better to prove the theorem without ever explaining what it means or why you built it that way.
If we bothered to treat mathematics as a form of communication (specifically: communication of rigidly defined, highly specific computational structures) rather than as a personal exploration of Platonic Higher Realms, we could definitely all get much better at it.
(Why yes I am a massive fanboy of Bob Harper… Why did you ask?)
And you get to keep the ring afterwards! :-)