I think you are both getting lost in the weeds trying to make this metaphor work, or not work.

Math is simply the logical conclusion of a set of conditions someone accepts as inherently true. If this, then that. Follow this logic far enough and you end up where we are today.

I think this conflates logic and mathematics. Some would dispute that logic underpins mathematics. Counting can be analyzed logically, but it does not in any meaningful sense seem to depend on logic.

Sure it does. How else would you prove that one number follows or precedes another? I think you are conflating the act of physically counting with the logical foundation of our number systems.

It doesn't seem correct to equate mathematics with proof. If I express a mathematical construction like the whole numbers (let Whole = Zero | Succ Whole), and I build further constructions on that foundation, am I doing mathematics? If so, then it seems mathematics does not depend on logic, as logic depends on propositions and there are no propositions to be seen.

Certainly you can analyze such constructions using logic, but that's again conflating logic with mathematics. There's overlap, but they aren't strictly the same.

I would say that a mathematical construction is based in logic, maybe not in the traditional sense, but there is definitely logic behind the construction itself. I think we are talking past one another, what I mean by logic here is more nebulous than predicate calculus. There is an innate logic behind the philosophy of mathematics and I believe you cannot divorce mathematics from logic.

Show me a cartographic map of a 5-dimensional universe.