In article <Pine.NEB.firstname.lastname@example.org>, William Elliot <email@example.com> wrote:
> On Mon, 11 Nov 2013, 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? > > Computers don't think, they calculate. Use computers for calculations. > To understand mathematics, don't use a computer; use a text book. > If you don't understand mathematics, then you can't be thinking but > only guessing. That's what I've seen students do, to use a calculator > as a guessing machine, checking their guesses until they get an answer > they guess is correct.
A computer is a tool--no, it's a collection of tools, a toolbox. There is software that can do the things you mention, but you need to know enough mathematics to know WHICH tool to choose from the toolbox. Also, if you CAN solve a problem analytically it is better to do that and (where appropriate) then use the computer to calculate numerical values.
A simple example: I can solve the two-body central force problem numerically, and get very accurate values for position as a function of time. But if I solve the problem analytically I find that the solution is a conic section' plus all the other general results about angular momentum, etc., that you learn in physics class. I've learned something, which may help me solve a different problem.
Example of a different problem (artificial example, but it illustrates the principle): Suppose the Earth suddenly stops at the apogee its orbit (instantaneous velocity zero relative to the Sun). How long will it take to fall into the Sun? You can actually integrate the equation of motion; it's a bitch but it's possible. Doing it numerically is probably easier. But there's a much easier method. Approximate the Earth's motion by an extremely elongated (high eccentricity) orbit, and remember that the period only depends on the semimajor axis. The answer falls out.
The modern computational tools available on computers are wonderful, but they don't do much to help you understand the physics.