15Feb08: DanielSoto told me about SAGE today. On checking it out, there are some interesting links:

• Octave is an open-source environment for Math with Linux, Windows and Mac versions. It is sort of an alternative to Matlab. Syntax is similar to Matlab. How to install on Os X http://island94.org/setting-octave-and-gnuplot-osx
  • Octave manual with examples
  • Octplot for plotting routines (similar to Matlab)
• SAGE open source symbolic math (sort of similar to MAXIMA or Macsyma).
• R is for statistical computing
• Plot Nice 2D graphics...

A comment on Sage and Octave on a blog site:
I use closed source applications: Mathematica, MATLAB and open source ones: Sage, Octave, Maxima. Now, Mathematica is an extremely high quality program with brilliant visualisations. Nothing comes close to the polish of it in open source. It's like damnsmalllinux vs Mac OSX Leopard. Now, MATLAB sucks and may be easily replaced with Octave. MathWorks is a really poorly polished product which looks like Windows 95, and IMHO Octave beats it. So, if I'm a pro, my choice would be use the most efficient product: Mac OSX, Mathematica, AND Octave and R. Not MATLAB. Sage is absolutely necessary to bring many open source products together. Including Python.

Sage notes

Sage is a very large agglomeration of stuff in Python. It is being put together by and for mathematicians. If you check out the SAGE tutorial you'll find some stuff is easily accessible, some less so.

Like Octave it runs using the Terminal utility. Read the README instructions to follow the slightly non-intuitive installation and operation.

You can do symbolic math via MAXIMA, which is included in SAGE (no wonder it's big). For example:

sage: a = x*(x+3)*(x+2)
sage: a
x*(x + 2)*(x + 3)
sage: maxima.ratsimp(a)
x^3+5*x^2+6*x

Note the object referencing scheme. The Maxima functionality in SAGE is not especially well documented, but you can read the MAXIMA documents and more or less figure it out.