More than any other product, Mathematica embodies symbolic computing, and this recent post on the Wolfram blog, Symbolic Programming Visualized hints why. The Mathematica language is based entirely on repeated transformations on symbolic expressions through pattern-matched rules.
I have never seen any mention of symbolic computing in anyone's dreams for future programming languages. Instead, I see a laundry list of incremental features for the next "big" programming language. Why can't the language be small, incorporating only the most expressive ideas.
Symbolic computing is the blind spot of the technology community. Even Turing thought of both brains and computers as manipulating symbols. Methinks, several years from now, we will probably see again the next revenge of LISP, which are operations on symbols.
According to Microsoft researchers in their roadmap "towards 2020 science," symbolic computation will not be integrated into programming languages until 2012. (We do see a precursor in LINQ expression trees, being introduced in Visual Studio Orcas. There are also glimpses in some of the newer functional languages.) This gives me five years headstart.
If we want intelligent software, symbolic computing is the way to get there.