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For sheer beauty, study math. But economically, I think it's much like any advanced level STEM course. In such courses you come across a huge range of things,

For sheer beauty, study math. But economically, I think it's much like any advanced level STEM course. In such courses you come across a huge range of things, but the main benefit is that you now know that X is a thing that people have investigated. If you know that X is a field of some depth, that completely changes your own course of action. The thing is I'm not sure the actual meat is necessary. Yes it sounds criminal, but what you just indexed everything and came back to learn it properly when needed? When I look at HN type people that seems to be the actual way things happen, rather than learn X -> apply X. It's more like people kinda sorta remember there was something that sounded like it might be useful in this other area, then they circled back, read up on the details, and applied it.

zasdffaa 4y ago
> For sheer beauty, study math. I wish. I wonder what proportion of people reading your post don't see or feel the beauty in it. I think you need a special and probably quite rare 'sixth sense' to perceive it.
trashtester 4y ago
> But economically, I think it's much like any advanced level STEM course. I agree. Almost all of those would be some kind of applied mathematics, though. I was not arguing above for the importance of studying pure math exclusively. For most people, math is a force multiplier when combined with some other knowledge/skill. > It's more like people kinda sorta remember there was something that sounded like it might be useful in this other area, then they circled back, read up on the details, and applied it. Absolutely. By having a good and broad generic understanding of a field, you can circle back to it. Understanding something in a more abstract way makes that easier, as it promotes a deeper understanding rather than just memorizing facts. For instance, if you think of euler's identity in terms of a rotation group it becomes very easy to prove it (at least for someone with a mostly visual understanding of math an physics). But just as important, the rotation group gives a much better understanding of many cases where complex numbers are useful, ranging from physics to electronics to electrical engineering. Quantum wave functions, capacitors/inductors and electrical motors all have this kind of rotational symmetry in common. So while some colleges may teach quantum mechanics, electronics or electrical engineering with a shallower math curriculum as the base, some abstract algebra can help gain deeper understanding of each, compared to memorizing the formulaes without fully understanding them. Also, even if many with a STEM degree often end up in a different field than what they studied in collage, the ability to apply math and statistics can often be useful way outside of the field studied initially. (At least, that's my experience as a former physicist. While I rarely need quantum mechanics, the ability to manipulate probability density function, find and describe symmetries, and have developed a quantitative intuition that spans a 50 orders of magnitude remains useful, both professionally and even as a citizen. )
ZeroGravitas 4y ago
I feel like that's my kind of approach, but I guess you also need another kind of person to just keep drilling away at some weird little math problem that has absolutely no practical use for decades until it suddenly does.

zasdffaa 4y ago

> For sheer beauty, study math. I wish. I wonder what proportion of people reading your post don't see or feel the beauty in it. I think you need a special and probably quite rare 'sixth sense' to perceive it.

trashtester 4y ago

> But economically, I think it's much like any advanced level STEM course. I agree. Almost all of those would be some kind of applied mathematics, though. I was not arguing above for the importance of studying pure math exclusively. For most people, math is a force multiplier when combined with some other knowledge/skill. > It's more like people kinda sorta remember there was something that sounded like it might be useful in this other area, then they circled back, read up on the details, and applied it. Absolutely. By having a good and broad generic understanding of a field, you can circle back to it. Understanding something in a more abstract way makes that easier, as it promotes a deeper understanding rather than just memorizing facts. For instance, if you think of euler's identity in terms of a rotation group it becomes very easy to prove it (at least for someone with a mostly visual understanding of math an physics). But just as important, the rotation group gives a much better understanding of many cases where complex numbers are useful, ranging from physics to electronics to electrical engineering. Quantum wave functions, capacitors/inductors and electrical motors all have this kind of rotational symmetry in common. So while some colleges may teach quantum mechanics, electronics or electrical engineering with a shallower math curriculum as the base, some abstract algebra can help gain deeper understanding of each, compared to memorizing the formulaes without fully understanding them. Also, even if many with a STEM degree often end up in a different field than what they studied in collage, the ability to apply math and statistics can often be useful way outside of the field studied initially. (At least, that's my experience as a former physicist. While I rarely need quantum mechanics, the ability to manipulate probability density function, find and describe symmetries, and have developed a quantitative intuition that spans a 50 orders of magnitude remains useful, both professionally and even as a citizen. )

ZeroGravitas 4y ago

I feel like that's my kind of approach, but I guess you also need another kind of person to just keep drilling away at some weird little math problem that has absolutely no practical use for decades until it suddenly does.