Even Kant acknowledged that although what is known a priori, say for example that visually detected objects will always have size and shape, does not depend on verification by subsequent experience, our knowing of it certainly depends on having previously experienced all such objects as being extended and shaped. In that sense it is a kind of a posteriori knowledge, just of the most universal kind.

We don't get numbers, operations or equations from observing nature. Two, plus and equals don't exist as objects anymore than a perfect circle or PI does. — Marchesk

So, I would say we do ultimately get all those things from observing 'external' nature; and this of course in conjunction with observing the nature of our mental processes; which are also surely part of nature, properly conceived.

It's undoubtedly true that there is, as you say, a "conceptual apparatus", but what is in question is the nature and origin of that conceptual apparatus. Must it have supernatural origins?

So, I would say we do ultimately get all those things from observing 'external' nature; and this of course in conjunction with observing the nature of our mental processes; which are also surely part of nature, properly conceived. — Janus

Sure, mental process are part of nature. The problem here is that mental processes can be about things that are not part of nature. Our understanding of the world can have mistaken notions. Our experience of the world can be mistaken. It's clear that problems arise between language and the world. Simon Blackburn called it a "loose fit" between mind and world. What constitutes identity? When is something a pile? Can you have non-simple objects? Are forms required for knowledge? Does time flow? Do we look out on the world perceiving things as they are? And so on.

Language does seem to bestow the ability to reify what are merely concepts, leading us to imagine that they are concrete entities.