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Common Lisp has a robust and comprehensive numerics facility. Poking around the CLHS and reading about ratios, fixnums, bignums, etc., will lead you to the conclusion that you can get your numeric work done (albeit in many cases much more slowly than you would like).

Add one of the pattern matching and term rewriting packages and you'll be very close to the foundation of the core of Mathematica. It will be a SMOP and adding to the library of facts enabling recognition of reducible terms.

There are a number of these. The one I have been using lately is sympy - a pure python symbolic math library.

You may also be interested in the open source "sage mathematics", which is a frontend/interface to many different open-source symbolic solvers such as sympy, maxima, GAP, PARI, Singular, and octave.

They are not as good as mathematica currently, but can already do quite a lot.

http://stackoverflow.com/questions/33550/best-open-source-ma...

1) Mathics 2) SAGE 3) Sympy 4) Maxima

well, off the top of my head, there's GNU Octave.

that'd be matlab not mathematica