Python is an alternative, especially within the Sage package .
Then there is also Maxima .
Will check out Sage, haven't used Maxima since 2004. Thanks!
Edit: Sage looks nifty, Maxima looks unchanged, at least at screenshot-level.
Octave is more of an open source Matlab equivalent. That is, mostly linear algebra.
Mathematica _has_ open source equivalents. It has equivalents that predate Mathematica. Their libraries just aren't as extensive.
But I could be wrong; I used mathematica and R for very different things so I may just be unaware of their similarities.
Of course this won't ever happen because of the paradigm -- R is for data crunching and does this way better than Mathematica.
Seriously though, he did seem open to the idea of at least partially open-sourcing Mathematica during a recent AmA: