I've been using Maple and numpy for a few years. I stopped upgrading my Maple at version 13 because improvements didn't compel me to pay for a newer version for my use. I chose Maple as a happy medium between the numerical and matrix-focused MATLAB and the symbolically-focused Mathematica. They can all do symbolics and matrix operations but they differ in how elegantly they're performed. But, as I gave Maple more complex mathematical tasks such as the calculus involved with Fourier transforms, and symbolic tensor & matrix operations, it became clear that it wasn't up to the task. I knew Mathematica was superior for those tasks but I chugged along with Maple until I got to George Mason University where they had a site license for Mathematica.

Right off the bat I could see a clear superiority in the user experience (mind that I didn't use Maple beyond ver 13). Vectors and matrices are just lists. In Maple there are objects called vectors, lists, tables, and matrices and you sometimes had to import them from a module. Clearly, in just this area, Mathematica is superior. Mathematica includes alot of functionality out of the box. In contrast, in Maple, I had to import modules for what I considered basic functionality which makes the experience not as consistent.

In comparison to other math environments:

- SAGE: SAGE uses a web interface which can't come close to a desktop application's functionality. The UI does not come close to Mathematica. Also, the open-source symbolic packages it uses are primitive compared to Maple let alone Mathematica. Oh and good luck installing it on Windows.
- numpy/scipy: Numpy and scipy give mathematical functionality as opposed to being a math environment; basic ones at that. But, its indexing coolness does exist in Mathematica.
- Mathematica has pattern-matching..others don't enough said.

For math-centric programming, Mathematica is a high bar to get to in terms of consistency, documentation, functionality, and general experience. My general gripe about open-source is the lack of an umbrella vision that moves a project in a certain direction. SAGE is the best (the only) hope but it has a long way to go. Even in the best scenario, getting different python packages seamlessly working together is doubtful (eg. look at my sym2num function).

Conclusion: If all you need is basic functionality, Octave, SAGE, scipy, sympy, and maxima could suit you. For more advanced tasks, be prepared to pay up.

PS: This post will change as I learn more.

Support for my opinion.

Support for my opinion.