# [Question] Learning Abstract Math from First Principles?

I find that I am usu­ally quite good at ap­plied math, and en­joy it. I am tak­ing a course cur­rently that is split into two parts, Vec­tor Calcu­lus and Com­plex Anal­y­sis. The vec­tor calcu­lus makes sense to me and I can see how and why it works, and I find it in­ter­est­ing and en­joy­able to learn.

On the other hand, I spend quite a bit of men­tal en­ergy wrap­ping my head around the hows and whys of the more ab­stract com­plex anal­y­sis. I am not sure if I en­joy ab­stract math or not in gen­eral be­cause I do not un­der­stand it as well. So, my ques­tion: Does any­one have any recom­mended re­sources for learn­ing (any) ab­stract math­e­mat­i­cal topic from first prin­ci­ples, that ex­plains rea­son­ably well what’s go­ing on with the math, rather than just how to do it?

• For your spe­cific situ­a­tion, may I recom­mend curl­ing up with Vi­sual Com­plex Anal­y­sis for a few hours? 😊 http://​​pi­pad.org/​​tmp/​​Need­ham.vi­sual-com­plex-anal­y­sis.pdf

On a more gen­eral note, I find that any­one who says they “learned it from first prin­ci­ples” is usu­ally putting on airs. It’s an odd in­tel­lec­tual pu­rity norm that I think is un­for­tu­nately very com­mon among the math­e­mat­i­cally- and philo­soph­i­cally-minded.

As evolved chim­panzees, we are ex­cel­lent at see­ing a few ex­am­ples of some­thing and then un­der­stand­ing the more gen­eral ab­strac­tions that guide it on a gut level; we have an amaz­ing abil­ity to ar­rest form from thing, but our abil­ity to go the other way around is a lot more limited.

I think most of your in­tel­lec­tual idols would agree that while even­tu­ally be­ing able to build up “from first prin­ci­ples” is a great goal to shoot for, it’s ac­tu­ally not the ped­a­gogy you want. It’s okay to start con­crete and just prac­tice and grind un­til the more ab­stract stuff be­comes ob­vi­ous!

Take it from a guy who leapt off the deep end this quar­ter into ab­stract alge­bra, real anal­y­sis, sig­nal pro­cess­ing and prob­a­bil­ity the­ory at the same time—there is no way I would be perform­ing at the level I am in these classes if I didn’t force my ab­strac­tion-lov­ing ass down to ground level and ac­tu­ally just crank out prob­lem sets un­til the ab­strac­tions fi­nally started to make sense.

• I don’t re­ally un­der­stand what you mean by “from first prin­ci­ples” here. Do you mean in a way that’s in­tu­itive to you? Or in a way that in­cludes all the proofs?

Any field of Math is typ­i­cally more gen­eral than any one in­tu­ition al­lows, so it’s a lit­tle dan­ger­ous to think in terms of what it’s “re­ally” do­ing. I find the way most peo­ple learn best is by start­ing with a small num­ber of con­crete in­tu­itions – e.g., groups of sym­me­tries for group the­ory, or posets for cat­e­gory the­ory – and grad­u­ally ex­pand­ing.

In the case of Com­plex Anal­y­sis, I find the in­tu­ition of the Rie­mann Sphere to be par­tic­u­larly use­ful, though I don’t have a good book recom­men­da­tion.