>> Most CAS systems I've seen are hard-wired for algebra, and many of them
>> have the transformation rules hard-wired as well. Mathematica is more
>> general than that.
>
> Underneath it all, Maxima is similar. Its all symbolic. Certain things
> have certain meanings. All functionality and rules are defined in
> libraries, it has a standard set that it loads when started that gives
> pretty standard algebra and calculus. Depending on how far down the
> rabbit hole you want to go, you can even from the Maxima command line
> give it instructions in Lisp.

Many of these CAS programs have a parser written in C that snips up text 
and turns it into internal data structures. Then other C code does 
algebraic transformations such as collecting like terms and so forth. If 
you want to input something that isn't algebra, or expressions in a 
non-associative algebra... sorry, you can't do that.

Mathematica doesn't work this way. The core implements a transformation 
engine, and arbitrary precision math. Beyond that, the parser, the 
printer, the simplification rules, *everything* is Mathematica source 
code, which can be altered at will (if you're so-inclined).

I have no idea about the architecture of Maxima, but it wouldn't 
surprise me. There are advantages to doing it this way. (The main 
disadvantage is performance...)

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