I have studied technical mathematics and applied it for so many years. And with more experience, I got better and more competent. And I liked my competence and I worked hard to maintain it. And I did things I were good at.
I still do, but with a little change.
I realized that you can get a degree of competence where your processes and solutions become absolutely predictable and reliable - in a particular domain or problem-set. You solve problems by reuse. Industrial.
In math you have the chance to enter a next abstraction layer, but why would you change approaches that are so successful? Cross-sectoral math? It makes you even more competent. If you are an artist in solving convection-reaction-diffusion PDEs in chemical reactors or finance applying a few dirty tricks, to finite element solvers, like up-winding techniques, you are really competent.
Competence can become a danger, if it erects a barrier to change. Change brings us into the situation to become less competent for a while.
Aren't robots and computer systems the most competent problem solvers?
Math is not longer math, it is computer math - it is plugged. Mathematica is not only a math system, it is a programming system, a platform, ... for a vast variety of problems - it supports sophisticated approaches as well as brute force algorithms. It depends on you....
Is it possible to become competent in speaking? In programming?
So what has changed in my/our problem solving life? We disrupt ourselves by trying new things, do explorative and experimental prototyping, try the hard parts of projects with new approaches - like combining math with data-driven methods, use Montecarlo techniques instead of instant solvers, ... and we even reinvent our systems - as we do with newUnRisk.
Replacing sophisticated PDE solvers, by modified and optimized Montecarlo techniques, will reduce the code size from about a million lines to less than 400.000. And it is easy to make this code inherently parallel.
Competence is two-sided - it has an obvious benefit, but also a danger. Consequently, there should be optimal competence... and no, I am not (longer) an industrialist, maximizing productivity. I like dynamics, creativity, ... the uncertainty and difficulties ahead.
Generic technologies, like Mathematica help us to entering the new without the risky horror of an icy-and-storm-lashed-night.