But to top it all off, if this was truely a law of mathematics, then show me a proof, theorem, or even a mathematical conjecture, about order of operations.
Our friend doesn't know what a mathematical proof is, and will instead try to give you an example in which he posits a real-world calculation, writes down an arithmetic expression for it according to one convention, interprets it with another, gets a different answer, and tells you this is "proof" that it's wrong.
When I explained to him how you could write down the expression according to a different convention, then interpret it with the same convention and get the same answer, he just denied, denied, denied, with no sign of understanding.