One example is the surprisingly common view that "all mathematical propositions are tautologies," and therefore can’t convey any new informationand of course I can't help but take the bait. As you surely recall from this discussion, I'm a firm Platonist:
Pythagoras's theorem is a statement about objects that have no width, mass, or time duration. It is not a statement about depressions in sand, sticks, or strings. [...] The fact that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse was true long before Pythagoras or even planet Earth was around; that it was discovered by some humans (long before Pythagoras, actually) has no bearing on its validity.However, I had also dug myself into a bit of a hole:
Yes, the boundary between "discovery" and "invention" is indeed blurry; I am not sure I can give a meaningful answer to whether chess was invented or discovered.And now, thanks to Scott, I think I can dig myself out of that hole. We are going to define two realms: E (for Euclid) and B (for Borges). E contains all the mathematical "tautologies". Thus, if you seed E with the definition of a group, E will also contain all the facts about groups, including the theorems we've discovered, ones we've yet to discover, ones we'll never discover, and ones that are true but unprovable. B is a much more boring set -- it is the collection of all possible statements, true and false, about anything. It includes a statement and proof of Pythagoras's theorem (and its negation with a false proof), a description of the game of chess (and its infinite variations), as well as lots of pure gibberish.
Now I can make a meaningful distinction between invention and discovery. We discover elements of E, but invent elements of B. We discover mathematical truths, but invent proof techniques. The game of chess belongs squarely in B, and thus is an invention.
And what bearing does this have on Scott's comment? Well, E consists of self-contained truths, or tautologies. We can only access a tautology via a proof. The heart of math isn't making true statements, it's finding clever proofs!