I understand that computer-algebra systems are useful for physists, engineers, or other users of mathematics. But are they useful in mathematics itself?
- Are they usually taught in undergraduate or graduate education in (pure) mathematics?
- Do the majority of professional mathematicians use them?
- Can they be replaced with free and open source ones like Maxima, PARI/GP and Sage?
- Would there be a serious problem if a student or a resercher in mathematics did not use them?
My main concern is that if mathematics students or reserchers have to use blackbox software like Mathematica, it seems to me against the spirit of mathematics: never treat results as truth until their proof is provided.