A lot of mathematics is about exploration. You look at some system and try to figure out how it works. If you have a good idea (conjecture), you try proving or disproving it. The goal is to gain some understanding.
Once in a while, it turns out that exploration hits the gold ore in the mine. You get something that applies to solving some problem we've had for years. As an example, algebra was considered meaningless for years. Then cryptography came along.
There are other good examples. Reed-Solomon coding relies on a basis of finite fields.
The problem is we don't really know when we'll strike gold in many cases. So people go exploring. It also involves humans, and they can disagree on a lot of things. Stephen seems to run into disagreements more often than the average researcher, at least historically.