One Plus One Equals Two (2006)

Thanks for sharing! I like to look at this example inside the debate of if mathematics are invented or discovered.

> That is how Whitehead and Russell did it in 1910. How would we do it today? A relation between S and T is defined as a subset of S × T and is therefore a set.

> A huge amount of other machinery goes away in 2006, because of the unification of relations and sets.

Relations are a very intuitive thing that I think most people would agree that are not the invention of one person. But the language to describe them and manipulate them mathematically is an invention that can have a dramatic effect on the way they are communicated.

I'd say mathematics is discovered and definitions are invented. E.g. "ordered pair" is not part of set theory, it's an invented name we give to a convenient definition of a set schema.

Even base-N representations are an invention: S() and zero are all you need, but Roman Numerals were an improvement over base-1 representations and base-N is significantly more convenient to work with.

Mathematics is entirely founded on human invention.

When we wrote simple mathematics on the Pioneer and Voyager probes I think it was under the assumption that anyone or anything else finding them would have co-discovered enough mathematics to recognize it on the plaques. That's the sense in which I use the word "discovered" for much of mathematics. Our definitions will differ from aliens but the foundations will be translatable.