Hetware wrote: > I'm working through a 1953 edition of Thomas's _Calculus And Analytic > Geometry_. When I work problems, I use Mathematica to type my > transformations, and to check my results. I use it for far more, as > well; graphing, numerical solutions, etc. > > Many years ago I found computers to be a nuisance when it came to math, > and more importantly physics. I was contented to have a piece of chalk > or a pencil and an eraser, than to have all the computing power in (the) > Universe. Time was the only resource I found in short supply. > > Now that I have used them for years, I realize that computers can do a > whole lot. They can find integrals for equations which I cannot > integrate by hand. They can produce graphics which a human could never > produce, etc. > > I've used a pocket calculator since the 1970's. But, I feel as if I > should have learned to work the same problems on my own. I feel > somewhat crippled by using it as a crutch. > > I'm in a conundrum twixt the use of computers to do my thinking for me, > and learning to think for myself. Should a child learn his times > tables, or learn to use a computer to do it for him?
The question 'What math should I learn if computers can do it all?' in your title suggests to me that you might want to look into what computers can't do (even idealized ones as fast as you like and with limitless memory). Look up "limitative theorems in logic."
-- Madam Life's a piece in bloom, Death goes dogging everywhere: She's the tenant of the room, He's the ruffian on the stair.