Recently I listened a lecture by Stephen Wolfram where he talks about math and how badly it’s taught nowadays. Especially there’s a lot of attention devoted to low level mechanical stuff (which is done by computer mostly) and much less attention to big ideas. Interesting part is that big ideas could be used like building blocks to approach high level problems without gaining an expert knowledge in low level details.

I was wondering, what are those big ideas? For me good examples are differentiation (for optimizing), integration (for aggregating), dimensionally reduction (for finding hidden structure). I could use them like simple functions in a whole range of interesting problems. What are big ideas for you?

To be clear, the idea is not to avoid low level calculation, but to familiarize with high level concepts. Analogy from programming: of course it’s good to know assembler and low level details of concurrency primitives, but you won’t use it too much in daily problem solving, you will build upon existing libraries and frameworks. I want to get sort of hierarchy of mathematical concepts and taught myself broadly of what can I do with them. Sort of top down approach.