decoupling intuitive relations from complex calculations in mathematics education

Got my hair cut by a european lady with an advanced degree in mathematics this morning, and we managed to find a lot to agree about but it seems I need to work on this idea about a new conception of what math is and how to teach it. On the one hand, as she pointed out, you can't enjoy the power and fluency of a skill unless you practice it, repeating over and over, training yourself and ingraining it where you no longer struggle with the basic parts. On the other hand, there are lots of ways to use logic inside a structured rubric which don't involve numbers, such as the design methods I'm learning about at ID.
Between those two extremes of mathy versus non-mathy logical tools, there are many useful frameworks that are already part of the standard conception of mathematics, yet which don't require full fluency with numeric calculations to understand. As evidence I submit the popularizations of quantum and theoretical physics by many practicing physicists like Stephen Hawking, Murray Gell-Man and Michio Kaku, as well as accessible (if heavy) explanations of underlying/overarching themes in other mathematics-heavy disciplines like Hofstaedter's 'Godel, Escher, Bach' in computer science and 'The World is Flat' in economics.
Yet to my knowledge, no one has popularized any pieces of advanced mathematics so. Doing so could facilitate the reader's ability to see the world in such a way that quadratic equations and derivatives are relevant. I'm currently reading 'fooled by randomness' by Nicolas Tassim Taleb, which comes fairly close to doing this by relating a lot of powerful vignettes where reality fails to conform to some of the basic tenets of statistics and economics. I think he doesn't achieve what I'm describing only because the book is focused so heavily on the financial industry that the broader application is only vaguely suggested. It's also not a great introduction to the principles involved because it is essentially demonstrating Godel's assertion (perhaps when I finish reading I will be able to say whether it's a derivative or variation thereof?), which is all about what you can't do with mathematics. Perhaps there's a way to turn Godel's theorem inside out so that it's about what you *can* do with mathematics?

Here are the parts of mathematics that seem to me to have relevance outside of numbers-intensive disciplines:
- set theory, which is about groups and their relationships (especially container/contained and identity/collection). Has applications in systems theory, could be exemplified through stories about societies and looking at emergence.
- Boolean and Bayesian logic, which are about alternatives, decision making, or branching structures. Could be exemplified through stories about management, flow of goods, and control structures (which is another way to say flow of information?).