Jon A> Had been wracking my brain to imagine a simple reply to Paul Tyson's...

Short answer:   Nobody thinks in formal proofs, not even professional mathematicians.

But everybody -- including their pet cat or dog -- thinks in diagrammatic patterns that could be represented in mathematics.

For details, see "Peirce, Polya, and Euclid -- integrating logic, heuristics, and geometry", http://jfsowa.com/talks/ppe.pdf .

John