as a mathematician, are you a mathematical realist? do you think math was discovered or invented? thanks
Well, I'm an applied mathematician, and so generally I don't really care so much. One important idea of proving that something possesses a certain quality is to assume it does not have this quality and then show that this implies something that cannot be true given everything else we assume to be true. If mathematical entities are truly "discovered" in the sense that they exist apart from a human's ability to describe them, then what about the concept of the odd number 4? I can prove that 4 is not odd by assuming is odd and then showing that as an odd number 4=2k+1, where k is an integer. However, this implies k=3/2, which is not an integer, and therefore 4 cannot satisfy the definition of an odd number. Not the most elegant of proofs, but important to it is the notion of the odd number 4. Can this thing exist? Well, I'm talking about it, and so I suppose it does. However, can it be "discovered"? I don't really think so. It was something I had to construct in order to probe the qualities possessed by the even number 4.
I guess I'd say I have a modified version of "God made the integers, but all else is the work of man" idea, but without the intent of besmirching things like irrational numbers or infinity. There are certain concepts, like what the symbol 2 represents, that, well, they're so fundamental I don't think they really are even "discovered." Maybe even like how Chomsky says there's something innate about language in a child's brain. I hadn't really thought of it before, but if math is the language of science, then this has some credence. However, if you look at something like the Cantor set, it seems pretty clear to me that it was invented. Or maybe constructed is a better word. This in no way diminishes its importance or beauty, though.
An equation that I try to solve represents an idealization of physical phenomenon; it came to me because certain simplifying assumptions were made about the shape of the molecules, and certain forces where included whereas others were excluded, and then I chose to apply these specific boundary conditions. Did this equation exist before I wrote it down? I suppose one could argue that it did, and indeed very different physical phenomena are often modeled by the same equations. However, the context that led me to meet this equation is connected to it. Furthermore, just because I can write down the solution to an equation does necessarily mean that I have a good understanding of its behavior. This is something that is definitely discovered through experimentation. And perhaps I will discover that for some parameter values, its prediction doesn't match the physical phenomenon at all, indicating that one of those forces I ignored turned out to be pretty significant after all.