Beyond The Beyond

It is a role-playing video game, a blog in WIRED about future trends like augmented reality ...

I often ask myself, what is beyond mathematics, beyond simulation, ... ?

Some take an extreme position: the scientific approach will become outdated. 2009 I wrote about this in The Petabyte Age - linking to the WIRD article: The End Of Theory: The Data Deluge Makes The Scientific Method Obsolete.

And we have the The Exabyte Revolution now - the age of Big Data .... (Googling Big Data leads to over a million results - the industry is quite creative with new names).

I still believe in intelligent combinations of modeling and data-driven methods.  I share the enthusiasm about the possibilities of data science but there are some barriers and limitations - I wrote about them here.

At the other hand it is not always easy to propagate the effectiveness of mathematics in computational science. Much of computational science is devoted to simulation, especially as computing muscles become really strong. The starting point of simulation is frequently expressed as the solution of systems of PDEs. One of the mathematical problems is to find adequate solvers (methods). Problems include model calibration that is an (ill-posed) inverse problem.

One of the questions that arrive are: continuous versus discrete models?
In Asymptotic Mathematics we try to decompose a domain so that exact solutions are possible in the sub-domains and get the total solutions from an asymptotic recomposition. At the other "end" we need to try to  discretize the problem at the earliest possible stage - like the Discrete Mechanics and Optimal Control method

Summarizing, numerical implementations of dynamic systems models depend on approximations - and those are subtle .... and simulation is often unlikely to answer all questions .... Mathematical theory can guide us to create clever algorithms enabling us to get better insight.

So, the interplay between symbolic and numerical computation and robust implementations will continue to make simulation a powerful scientific tool for more complex problems.

I really recommend  to organize objects, models and methods orthogonally.

Consequently beyond mathematics and simulation, we find mathematics and simulation in interplay.