I'm not sure I understand why you see this as a dichotomy. Sometimes inspiration comes from a weird idea, sometimes it falls out of mathematical analysis.

It's not like it is exclusive, everyone thinks a bit different thankfully. Like your example of the positron and electron seems fine; math and experiment in a cycle of discovery. You wouldn't know to look for a positron if you didn't study the electron experimentally and try to come up with some math for it.