One of the frequent concerns I remember from my days in physics (where Mathematica was frequently and heavily used) was the question of how scientists could trust the results of the program. To the best of my knowledge, (although I must concede my knowledge is some years out of date) no computer algebra system is currently regarded as being bug free. There is always the question: "how do I know that I got the right answer *this* time?"
While auditing the logic of one of the open source systems (Axiom, Maxima, Reduce, etc.) is at least theoretically possible, in practice the challenge is sufficiently difficult that anyone other than a dedicated specialist is unlikely to tackle it. The problem is compounded in cases like Mathematica and Maple, which are proprietary and closed source - even the theoretical opportunity for an in-depth audit is lacking.
The most promising approaches I have head put forth to address these sorts of problems are to add the ability to generate a “formal proof” transcript (targeting something like ACL2 or COQ) as a byproduct of the standard computer algebra system calculations, which can in turn be verified by a much simpler proof checker (or a very determined human.) What are your thoughts on the question of producing trustworthy answers to complex mathematical problems via computer? Is leveraging formal proof techniques a good way forward, or should we be looking in other directions?