Comment by aljgz

As a programmer who uses math heavily, I can answer this. As a programmer, your intuitive understanding of the world is infinitely important. You use it to formulate ideas, rule out solutions that would not be feasible, have an estimate of the expected cost and quality of solutions, etc.

Being able to dig deeper is important, but what's more important is to have an intuitive understanding of many many things: psychology, economy, finance, physics, art, etc. It's important to know the limits of your familiarity with any of these. For instance, my understanding of the fundamental practices in Accounting is really good (I've led a budget aggregation software for a huge conglomerate), but details is bad (tax rules for each industry, etc).

When I needed to create a software for optimizing stone cutting, I needed to know enough from computer vision, computational geometry, and optimization to know that our solution is feasible, task team members to learn what they need to do, and get into implementation, debugging and optimizing with them when needed.

After that, I still can't write code in computational geometry that handles all corner cases.

It's really good if we know everything with infinite precision, but for a programmer it's not efficient. We need to know where to stop.