I have recently posted a couple of comments relating to whether mathematics is real. These were blog comments or forum comments. And, to be clear, these posts were elsewhere (not on this blog). This post will expand on those.
Discovery or invention
A blogger asked whether mathematics is a matter of discovery or invention, and I commented. But I’m not sure where that was. So I am reconstructing and expanding my comment.
My first remark was that mathematics is an invention. I’m reminded that Kronecker famously said “God gave us the natural numbers; all else is the work of man”. But it has long seemed to me that Kronecker gave too much credit to God. It is all the work of man.
I did not want to dispirit the blogger, so I tried to connect invention with discovery.
Nobody doubts that Lego blocks were invented. But give children some Lego blocks, and they will quickly discover all kinds of interesting things that they can do with those blocks. You really cannot separate invention and discovery. If you invented something that was useless and boring, you would quickly forget it. Any important invention involves the discovery that what you invented is useful or interesting or both. We might say that “invention or discovery” is a false dichotomy. They go together hand in hand.
Getting back to mathematics, presumably at some time it was discovered that counting was a useful practice. But maybe the practice had to be invented before you could discover that it was useful. Again, discovery and invention are intertwined.
Numbers are not just counting. For counting, you need a sequence of names to use. But numbers themselves are abstract entities, presumably derived from the practice of counting. So the use of numbers in this way was a separate invention. But again, it was an invention that depended on discovering the usefulness of numbers.
The Heretical Philosopher 