Are CASs useful in mathematics? I understand that computer-algebra systems are useful for physists, engineers, or other users of mathematics.  But are they useful in mathematics itself?
Specifically,


*

*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.

Perhaps you can watch this and see.
http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html
Note: there is research into this and many classes are using such tools and I think it can be a very enriching approach, but there are certainly issues as you mention. 
I believe some of the comments are fair and there is no doubt so much waste in the things we are teaching our kids (akin to working out log tables).
This 'new' area seems to be gaining some ground and is called Experimental Mathematics: see: http://mathworld.wolfram.com/ExperimentalMathematics.html
You might also want to look at: http://en.wikipedia.org/wiki/List_of_computer_algebra_systems
I wish that teaching would invest more time and figure out how to teach budding mathematicians how to think about mathematics and proofs, and also share with them the usefulness of these math packages and programming skills to do computations.
This will better prepare them for real-world jobs while not sacrificing the theoretical disciplines or tract.
-A
