Yes, problem solving principles are different in different fields.
The steps (workflow) of computational problem solving
- Transform the problem description into a model
- Transform the model into a form that makes it of good nature for calculation
- Calculate
- Interprete the results
In general step one is the most difficult one. It is the most important work of quants. The work to add value.
Pure or applied (calculational) mathematics?
In mathematical problem solving, in step one a linguistic description of a problem is transformed into expressions, in the language of mathematics (mathematical notation). And now the difference between mathematical and other problem solving comes into play: mathematicians do not want to test their solvers (the operational semantics) in finite many cases but proof that they are correct in infinite many cases. And those researching completeness and consistency do the theoretical work that makes mathematics an unprecedented powerful problem solving universe - they work with mathematical objects. It shows its power in step two - especially when closed form solutions can be achieved.
To achieve closed form solutions you apply a kind of "quantifier elimination" task - for all n is there a k so that k=sum(i, i=1, …n)? Yes, k=n*(n+1)/2 .. simple proof by induction.
To achieve closed form solutions you apply a kind of "quantifier elimination" task - for all n is there a k so that k=sum(i, i=1, …n)? Yes, k=n*(n+1)/2 .. simple proof by induction.
But this has its traps. The proof-for-infinite-many-cases technique may make the problem domain smaller - often too small.
Symbolic computation is the computer technique that supports step two in the mathematical workflow. But is goes far beyond, it understands symbols not only in mathematical notion … symbols are graphs, geometric elements, .. even programs. All can be represented in a unified expression language.
The calculation step can become difficult itself, if the model is complex (for a wide domain) and cannot be simplified much. This is where numerical schemes are indispensable.
Asymptotic mathematics
It suggests that striving for exact solutions, one could decompose a domain so that exact solutions are possible in the sub-domains, when impossible in the domain. The total solution comes from an asymptotic recomposition of sub-results.
Adaptive Integration is such a technique in UnRisk. It integrates step two and three.
What if a domain is influenced by dogmatic thinking?
There are complaints that in economics and finance rationale debates of ideas are often replaced by dogma - a set of principles that are defined by an "authority". To change this was a matter of education, … but also technology can help.
Critical thinking requires not only research, but doubt and questioning. From a system development point of view it requires a bottom up approach, evolutionary prototyping, constructive learning …
A model is a model is a model
But what if they do not fit to real world conditions? Engineers are really good at manipulating models contextually - they usually rely on computer-aided-engineering technologies that have a wide implementation of (mechanical, electrical, micro-electronical …) engineering languages.
Its easy to simulate-for-insight in such languages. And their algorithmic implementation covers a wide range of engineering objects, function complexes and systems.
This is why we have unleashed UnRisk Financial Language and its wide algorithmic implementation - the UnRisk Engines. To enable quants manipulating financial objects and models contextually.
p.s. I know that theorem proving is more than a "test avoider" - a constructive proof is an algorithm that solves the problem described in the theorem. A proof might even tell a story of the transformation.
p.s. I know that theorem proving is more than a "test avoider" - a constructive proof is an algorithm that solves the problem described in the theorem. A proof might even tell a story of the transformation.
