The Subtle Art of the Mathematical Conjecture

well, one may say your mind is battling rigid logic

If two individuals who had no contact whatsoever reliably came up with the exact same ideas and structures, it is reasonable to view it as something objectively existing.

On the other hand, is this philosophical debate of any relevance?

The vast majority of mathematicians hold platonist views. There are interesting alternative views (e.g. ultra-finitism) but I'm not well-versed enough to tell you if they yielded interesting insights or have a significant audience.