Some pure maths and algebraic manipulation apps I've found:
Applications:
- Mathematica - the best (and most expensive) thing out there
- Maxima, a very interesting analog, available on Fink. Sadly, it runs under X, which rather limits its appeal for Mac OS X users.
- MuPad - for Linux and Windows
- Mathomatic - command-line only.
- Octave
- Scilab
- R