Does research have to be "goal oriented"? Could one spend say three years studying topological braids, find random identities/crap and publish it?

Question

Obviously anyone can publish anything at certain places, so I'm speaking in the purview of writing something that wouldn't make you look bad or is in an academic setting of some sort. Anyway would what I described be a bad idea? Also how would one even do that? Write seperate papers for each individual niche/tidbid/identity they found?

It seems like it would be easier to sort of bumble around in the dark so to speak when trying to find stuff in mathematics, rather then sort of brutishly attacking particular problems head on. Or is this not how mathematical research is performed? Do most people start out with the goal in mind? Can one just sort of bumble around like I described? Or is that "bad" so to speak.

Answer 1

What you describe is just really unlikely to accomplish anything as far as research goes.

If you find a bunch of random identities, why would a journal want to publish it? They are likely to have already been found by someone, and just don't appear in literature because they haven't yet proven useful for anything.

The problems you solve don't need to be huge, Fermat's Last Theorem type of problems. But you want them to be standing open problems that some part of the community has an interest in, or to make progress towards some huge problem, or have some sort of interest for the mathematical community.

Answer 2

Mathematicians do mathematics because they're curious. Some just want to understand how these interesting (to them) abstract objects behave. Some want to see how the abstractions match real world phenomena (that's applied mathematics).

Some mathematicians try to solve known open problems (what you call "brutishly attacking"). Some just explore things they find puzzling (what you call "bumbling around").

When they find new theorems they look for a way to publish them so that other equally curious mathematicians can see what they've done, perhaps extend it. If a journal's editor and referees agree that the new mathematics is correct and potentially interesting to readers of that particular journal then the work is published there.