(November 6, 2016 at 12:59 pm)Alex K Wrote: In my opinion, ideally, it is a discovery: one sets certain axioms as a starting point and discovers the tree of theorems which results from them. In reality, a lot of developing is going on, namely shaping and formulating the concepts with which one works. For instance, the concept of connectedness or simple connectedness was developed by mathematicians to express certain properties of sets. The rules which sets have this property and what other properties such sets have, is something that is discovered by formulating conjectures and proving them.
Pretty much the way it seems to me too. The discoveries all take place within a domain we've completely contrived. Learning mathematics feels like discovering immutable facts but those facts depend entirely on the definitions we've chosen.
