Anyhow, glad we agree :)
> there is a very ancient philosophical debate about math as a "discovery" versus as an "invention."
Yes, and it's a super interesting idea & debate too. Not to get too far off topic, but I'm a little bit more of a mind that representations are invented, while many fundamental truths are discovered (and can only be expressed through representation).
One math example that for me relates to software just a tiny bit, and illustrates why this is a debate is the 0^0 problem. We've mostly figured out how to better classify and represent that, but not before having big arguments where smart people say 1 and other smart people say 0. Mathematicians and programmers have mostly picked a convention, even though both answers are right depending on context. I would call that picking of a convention part of the "invention" side of math, as is calling it indeterminate.
But yeah, agreed, invention and discovery are both weird and problematic when you talk about math.
That said, the line I'm talking about isn't the representational line, it's the physical line. Math can be written down in multiple representations too. The reason that math was deemed not patentable isn't because I can or can't write it multiple ways, nor whether I can solve the math problem by hand.
Patents were just designed around products made out of physical devices and physical processes. If I had to figure out how to build a machine using gears and hydraulics, or wires and chemicals, then it was patentable. If I used math to design a better airplane wing, and built the wing, then the wing was patentable along with the methods to design the wing. If I came up with a better Laplace transform, then it's pure concept and not physical product, so it was decided not patentable. I believe the distinction is mainly about you having to build & sell the product you invented, because the patent is a business protection and not a copyright. If you invent something in theory but don't build it or try to sell it, then you don't get patent protection.