The early development of "computer algebra" systems in the 1960s was prompted in part by the belief that human mathematical reasoning represented a plausible domain for demonstrating artificial intelligence. In the 70s and 80s computer algebra system builders and researchers turned away from the field’s AI roots and toward basic representation issues, algorithm development, programming language design, applications and user interfaces. Advances in programming technology and formal algebraic descriptions provided further paths for development. With today’s improved computational speed and low costs it seems worthwhile to examine those unattained goals of the 60s. From this ambitious perspective, the current commercial systems do not provide sufficient scope for encoding mathematical knowledge. Success in building new, more ambitious, systems seems vital to the prospect of using them as building blocks for general mathematical reasoning in AI. These new systems must have far more effective methods (and heuristics) for computing with domains and functions including intervals, inequalities, singularities, special functions, and complex variables. This paper suggest directions and challenges for these new systems to re-address the frontiers for progress in AI and computer algebra.