> Except memorizing multiplied numbers is not foundational knowledge for any higher math
But my point wasn't that it's foundation in general, category theory, set theory, etc are foundational, but that it's important to start from it. Just because something is foundational doesn't mean it's a good starting point.
> it doesn't mean you need to know an entire table.
Kids at an young age are usually very good at memorizing things and once they have memorized then they can build on that and learn new tricks, relationships, abstract and foundational principles.
> Not knowing 7x8 never held me back in higher math/math-relevant classes like Calculus I-III,
Neither did me either. Embarrassed to say, I had forgotten a lot of it. But I would have been struggling early on in the 4-9th grades if I didn't know how to add or multiply numbers then. I would have been wasting time on tests doing it the slow way, instead of thinking of more interesting problems.
So to summarized, yes, it is not foundational but it all depends on age, it's better to take advantage of what the brain already knows how to do well at that particular state in time.
> In fact I'm pretty sure mathematicians are notorious for poor arithmetic.
Could be but if they are poor they would be left behind in early grades and might never becomes the mathematicians they are. There is also a bit survivorship bias in the sense that if there mathematicians who are poor ar arithmetic, they'd stand out and become memorable, but those that are good are not noticeable because they are expected to be good with numbers.