My view in brief: in Symbolic Computation computers shall be able to manipulate and operate symbols, like mathematical expressions, geometrical objects, molecule structures or even programs. To manipulate mathematical symbols you speak in the language of mathematics. Simplified, one could say symbolic problem solving has to do with the elimination of Quantifiers in this language. You are happy, if you get a closed form solution (quantifiers are eliminated, all variables are bound). Closed form solutions are "economic" in the sense that they are easy to interprete and compute. BUT, they often approximate only "small worlds", as the Black Scholes Formula, a solution of the BS Model under constraints.
Approximating the real world of financial deal types numerical schemes need to be applied. Dealing with numerical analysis in quantitative finance you have to cover, model analysis, find numerical solvers, which are robust related to noisy input, inverse problems, model calibration and adaptivity, ..
In this label, dominantly Andreas and Michael will give full explanation on the mathematical background, in a series of posts.