The original authors of a lot of Maple were computer scientists and not pure mathematicians. They believed that mathematics, as taught to them in their undergraduate mathematics courses, could be implemented as a set of direct rules (because it feels like that in undergraduate mathematics). Unfortunately, with enough mathematical education, you realize that that is quite wrong, and that even simple mathematics is much more complicated than that. Unfortunately a lot of the incorrect design decisions that were taken in the early days by these (otherwise brilliant!) computer scientists were so deeply buried in the fabric of Maple that many of them took years of hard work to 'undo'. I know, I was one of the people who spent years repairing Maple after the over-simplification of (x^2)^(1/2) was removed from Maple's kernel. Very slowly, over the past 15 years, more of these have been 'fixed'. But there are some over-simplifications which remain, and they may never be fixed, not without creating a whole new system.

Other parts of Maple are 'broken' because there are huge holes in the theory. For example, Calculus is all about functions, while Maple is not at all about functions, it is about expressions. It other words, it is about syntactic expressions which are supposed to **denote **functions. Unfortunately, there is no denotational semantics for open terms! [defn: open term = term with free variables]. Just as bad, if you use the naive interpretation of open terms as being implicitly quantified (or more precisely, an implicit lambda), that still doesn't help much because:

- The lambda calculus is usually typed -- but what is supposed to be the domain of definition?
- Not all Maple expressions are functions, some are meant to be formal objects (like polynomials or series, for example)
- The denotational semantics of the lambda calculus is via pointwise evaluation, and functions are all assumed to be total [partial functions are totalized by adding an explicit 'bottom' element]. This is not standard mathematics!

I could go on, but that is probably enough.

I know that many of the current Maple developers are indeed mathematicians -- and I know several who would really love to fix several of those early mistakes, but who also know that it may never happen.

I should point out that, without some serious theoretical work, I don't think the SAGE developers have much chance to really improve things either. The problem is quite deep. If it was possible to 'algebraize' calculus, I am quite sure that the Axiom people would have done so long long ago -- but they didn't. There is a good reason for that!