I start this Blog with a reference to Pau Wilmott's Blog Numbers People and Symbols People. Paul is one of the most influencing quantitative analysts, in his blog he discusses the ability of people to handle abstractions.
But let me go back. When I studied Math in the early 70ies, I learned already about Symbolic Computation. In SC computers shall be able to manipulate and operate symbols, like mathematical expressions, geometrical objects, molecule structures or even programs. To manipulate and operate symbols needs a special language, the language of Mathematics. If you think of questions like "for all N, is there a K, so that sum (i, 1, 2, ..., N) = K?", the answer is "yes, K=N*(N+1)/2". If you do not want to verify this across infinite many cases you ned a proof. This part of mathematics has to do with Quantifier Elimination. Solutions are in close form, if all quantifiers are eliminated. Close form solutions are exact and convenient. BUT, they usually only describe a small world.
SC systems I used at that time: Mathematica, Maple, Macsyma, Derive, ... I was responsible for factory-automation software at Austria's largest industry enterprise and prototypically applied them in robot and CNC programming and simulation.
1990, I met Stephen Wolfram, when he was on a world tour and I co-organized a talk in front of 400 people at the University of Linz. Fortunately I found out quickly that Stephen's Mathematica is the (only) adequate implementation of the language of mathematics with a declarative layer for programming and a representation of all symbols in unified expressions for unambiguous evaluation.
At this time, I launched my own company with the objective of creating innovative solutions on top of Mathematica.
But as mentioned, many problems cannot be transformed into closed form solutions. They need to be treated numerically. Good numerical schemes fit well along domains where close form solutions are available and do not lose this accuracy and robustness "in between". MathConsult (Andreas Binder, CEO) is renowned for accurate and robust numerical solutions of the most complex technical problems.
When such numerical schemes are integrated into Mathematica, the solution inherits from Matematica its full declarative environment, visualisation, link technologies, ... and Mathematica's functionality is extended. UnRisk is such a solution.
So, it is Numbers AND Symbols. I will come back to this AND-effect.