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### Symbols or Numbers - Pure vs Applied Maths?

When in London, we enjoyed a meeting with Paul Wilmott. Among other things we talked about mathematical creativity and whether computer based learning can create the creativity. And about the creativity in applied mathematics, that is often questioned by pure mathematicians.

From Paul's blog: Numbers People or Symbols People

IMO, it starts with the question: axiomatic or algorithmic mathematics? In axiomatic maths two functions are defined "identical" if they have the identical I/O relation. In algorithmic maths you need to take care of the economics (resources usage, performance, ..).

As a former algebraist I worked in axiomatic maths. A free algebra is the language interface of algebraic structures, say, a polynomial ring. It needs operational semantics to calculate, say, polynomial functions. And when I studied in the 70s I learned already about symbolic computation.

Dance with symbols

In SC computers shall be able to manipulate symbols that can be mathematical expressions, geometrical objects, molecule structures … or even programs.

In my free Rings manipulation is restricted to Ring properties, in the Ring class all Ring instances inherit the Ring properties but more properties are hidden in the concrete implementation (expansion and simplifications rules if special functions are involved).

We describe in the language of mathematics and try to solve with SC methods (exact, closed form solutions). But the world of closed form solutions is usually a small world. Just think of the Black Scholes formula.

It's all numbers

Many problems cannot be transformed into closed form solutions. You need numerical schemes (in most of the real world cases). You need them to solve models of material flows, chemical reactors, conservation of energy in a blast furnace,  adaptive optics, …

Good numerical schemes fit perfect along sub-domains where closed form solutions are available and do not lose accuracy or robustness "in between".