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.
 A: The answer is a big YES. Computers and CAS have brought experimentation to Mathematics. Here are some links that you may find interesting:


*

*Wikipedia page on Experimental Mathematics.

*Link to the Experimental Mathematics journal.

*Experimental Mathematics: Computing Power Leads to Insights.

*Ten Problems in Experimental Mathematics, by David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor and Eric W. Weisstein, in the Mnhly.


Or just Google experimental mathematics.
A: 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
A: 0.Yes.


*

*Yes.

*Yes. (I hope.)

*More or less.

*Yes. 


They are not blackboxes, and not against the spirit of mathematics. 
(In everyday life you use a lot of blackboxes.) In math if a researcher want to prove a statement he/she uses other statements but not verifies each of them. They hope it was checked others and results are correct. In many cases checking is almost impossible because the university isn't subscribed for many journals or the researcher does not understand that language. 
As I know (i'm not a specialist of this topic, sorry if I say wrong things) computer scientists develop a method or a software that in principle could verify other source code. So I encourage you to use CAS, but "be prepare".
A: If I want to look up a summation or an integral or other similar task, Mathematica is usually much more convenient than consulting a book of tables.
I also take advantage of Mathematica to manipulate symbolic expressions far more complex than I would be willing to do by hand. And even to manipulate ones that I am willing to do by hand, if I can do it faster in Mathematica (or I want to double check my results).
A: Even outside of experimental mathematics a CAS can be useful for proving routine problems that come up in the course of more complicated problems. For example:


*

*Quadratic Diophantine equations

*Hypergeometric identities (see A = B)

*Factorization of integers and polynomials

*Integer relations (LLL, PSLQ, etc.)

