Personally, I would love to see Mathematica's simplification and integration routines some day and discuss them with everyone. That's one of the "basic" things that the free CASes still get so woefully wrong, and for me is one of the most interesting secrets.
So you guys (?) released publicly some boring code about linking Mathematica with Hadoop. Big whoop, it's just a wrapper. Where's the real interesting stuff?
Which means that open-sourcing our projects during their actual development isn't worth the significant development cost (an entire toolchain would need to be open-sourced, and documented). And it also is somewhat antithetical to our design process, which is kinda centralized around Stephen himself, who for the most part really does have a good sense of design.
I think that could change, though, and then I think there is a really good opportunity to open source more of our stuff and start building stuff collaboratively with a wider community. Especially when our Cloud stuff comes online.
Also, I regularly bug Wolfram about open-sourcing some core part of the kernel. He's not in principal against it, but points out that it would cost a lot, buy us nothing in the short term, and fork the language. But I think it'll happen eventually, hopefully sooner rather than later.
Collaborations in Wolfram language will be a pretty unique experiment, for they are likely to unite people from very different fields (including, of course, lots of non-programmers). It's going to be great.
There is likely a lot left to be done in the core language, but encourage folks to start with v1 of the language: http://reference.wolfram.com/legacy/v1/
Once the core of the language is working, implementing things like simplification and integration become much easier: http://webcache.googleusercontent.com/search?q=cache%3Arefer...
It seems unlikely that such a movement would grow to be the size of Mathematica or the Wolfram language, but then again Linux and Wikipedia also seemed unlikely at the time.
Should I work on Mathics, Sympy, Maxima, or Axiom? Or is Sage itself growing its own routines?
I suggested Mathics because you appear to be interested in what Mathematica can do but resent that it closed source. Since it does not appear that it will be open sourced soon (Taliesin discussed this elsewhere and I got a similar answer from Stephen in an AMA http://www.reddit.com/r/IAmA/comments/tmutz/stephen_wolfram_...), and because duplicating even a portion of a system with a 25-year head start would require a massive amount of effort, it seems the best choice would be to work on a program that would best organize the efforts of the other programs. To agree with that statement one probably has to agree that the language is the best one to build a system around.
I currently work on Octave because there's so much Matlab code out there that needs to be liberated. Is it the same case with Mathematica code? Are people able to easily take their code from Mathematica and run it in Mathics?
I'd really like to see Mathics become the Octave of Mathematica.
)set output tex on
)set output mathml on
If you're familiar with / interested in Mathematica, I'd suggest helping Mathics out. The Mathematica language is massive, and there's heaps of low hanging fruit left in Mathics.
(Mathics core developer here btw)
We used to integrate with Sage, but no longer. The main reason is that it wasn't really used for anything, but made installing Mathics much more complicated