This debate is partly a semantics game of the meanings of "discovered" and "invented" If an idea describes reality does that make it a discovery instead of an

This debate is partly a semantics game of the meanings of "discovered" and "invented" If an idea describes reality does that make it a discovery instead of an invention? What if you think the idea describes reality but it really doesn't. For instance, let's use the sense of inventing as pulling something out of thin air, and discovery as growing an understanding of what is in front of you. Math certainly involves elaboration according to fixed rules which could be seen as discovery. It also involves what-if anxioms that redefine the basis for the mathematical thought (say 4-dimensional space), that could be thought of as invention. This article seems to think of invented and discovered in terms of whether it is not based on reality or whether it is based on reality. This is trickier, I think it is fair to say that the needs of reality do drive certain branches of mathematics, but certain branches are proudly not based on reality. The funny thing is that periodically we realize that the math we thought was firmly not based on reality ends up describing reality better than the math we based on reality. So you have math that was spun out of an idle what-if thought that ends up describing with high accuracy an important physical concept. Is that an invention or a discovery?