This is a really important point on this topic. I don't believe (nor do I think you were claiming) that this means math cannot be taught more intuitively. An e

This is a really important point on this topic. I don't believe (nor do I think you were claiming) that this means math cannot be taught more intuitively. An even broader example to your point is that Calculus itself was invented without foundation, at least without foundation modern mathematicians would find satisfactory; intuition patched some holes around 'infinity.' But Calculus could none the less be developed by blackboxing its inner workings and justifying its existence by just how spectacularly useful and predictive it was. Likewise with arithmetic and algebra itself: ancient civilizations who factored numbers or solved equations did not require Peano's work to justify inventing some math. Calculus and all the math before it would indeed rest on foundations absent any Set Theory. The wider perspective is that discovery starts in the middle of a concept, and works out towards its implications and inwards towards its axioms. The justification taught to those learning math (or anything) need not start from its axioms, but it should start from its history in context to how humans found use for the concept - that is justifying enough and likely more satisfying than learning axioms developed after the fact when the question is 'why?'