This is what happens when people take ideas seriously. Ideas like Toby's one, that a proof in maths is a story, and you are free to go wherever you like with it. Here's my comment: Here's a way you can develop your idea of "there's always one more" as a definition of infinity. You could define "infinitely long" as "there''s always one longer" so if you have a line of length N then you can double it and get a line of length 2N, and there's always a longer one. Then you can think of infinitely short as "there's always one shorter" So if you have a line of length N then there's a shorter one of length N/2 which half as long. And then you might think of ratios of lines, and products of lines, which are areas. I think this is a bit closer to the way the ancient Greeks thought about arithmetic, because they were putting everything in terms of geometric constructions. So there is one book of Euclid's Elements whi