> Just as you don't really understand how a CPU works until you've programmed in assembly, you don't really understand math if you haven't done hand-computation, at least up to a point.

Conrad Wolfram addresses this very point in the video (11:30). He says that the most valid point that hand-calculating helps understanding is that teaching processes themselves helps understanding, but that there's a better way to do that: programming. In other words, programming a math solution is better for teaching math understanding that hand solving it.

In the end, I think the idea he's trying to impart is that the core of math is not calculation at all, it just seems that way because they've been intertwined for so long. But now that we have computers to do the tedious calculations, we can work on defining and exploring what math really is: how different conceptual intuitions are related