TITLE: Why learn to solve differential equations when computers can do it?
QUESTION [62 upvotes]: I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background.
One question is haunting me. 
Why do I need to learn to do complex math operations on paper when most can be done automatically in software like Maple. For instance, as long as I learn the concept and application for how aspects of linear algebra and differential equations work, won't I be able to enter the appropriate info into such a software program and not have to manually do the calculations?
Is the point of math and math classes to learn the big-picture concepts of how to apply mathematical tools or is the point to learn the details to the ground level?
Just to clarify, I'm not trying to offend any mathematicians or to belittle the importance of math. From CS I recognize that knowing the deep details of an algorithm can be useful, but that is equally important to be able to work abstractly. Just trying to get some perspective on how to approach the next few years of study.

REPLY [3 votes]: One issue is that you have to communicate your problem to the computer, before it can "automatically" solve it. Even if the computer understands the communicated problem, the actual form of the reported solution or whether any solution at all is reported can depend on "irrelevant" minor details. Here is an excerpt from a question about my most recent issue of this form:

I "suspect" that $\mathcal{F}\{(1-r^2)^n\operatorname{circ}(r)\}(\rho)=\frac{n!J_{n+1}(2\pi\rho)}{\pi^n\rho^{n+1}}$.
...
I tried to tell Wolfram Alpha that I want to know the 2d radially symmetric Fourier transform of $\operatorname{circ}(r)$, but it neither understands "$\operatorname{circ}$" nor "2d Fourier transform". Finally I wrote

integral of 2 pi x besselj(0,2 pi r x) from 0 to 1
integral of 2 pi x (1-x^2) besselj(0,2 pi r x) from 0 to 1

The first query gave the correct answer $\frac{J_1(2\pi r)}{r}$, but the second gave a cryptic $\pi{}_0\tilde{F}_1(;3;-\pi^2r^2)$.  But

integral of x (1-x^2)^2 besselj(0,r x) from 0 to 1

where I omitted the $2\pi$ factors, gave the helpful answer $\frac{8J_3(r)}{r^3}$.

Note that the only answer to this question proposes a way to communicate this problem to the computer in a "more direct less preprocessed" way, with the result that the computer doesn't generate any answer at all.