Is there any field of mathematics that deals with the role of the observer? E.g., some formulation in which a set is changed, in some unspecified way, when it is observed? Or maybe some philosophy of mathematics that addresses this? (I am more interested in formal mathematical systems than philosophical expositions, though.)
I was thinking a bit about the Axiom of Choice and the Well Ordering Principle. The Axiom of Choice feels true, and the Well Ordering Principle feels false. However, one might envision mathematics differently as an observer dependent practice, if we posit (as some foundational axiom) that a set exists only if it has been observed, or its definition has been observed, or something (?). In that case, the order of observation might impose an ordering on any set of sets, and this ordering might differ among observers. Though, that is just one way I can imagine a set being affected by observation.
Let me zoom out a bit. In the 20th century, several fields of human endeavor moved in a direction broadly known as postmodernism where subject matter was no longer considered independently of the observer. E.g., postmodern literature, and observer dependent physics of the 20th century. I am wondering if there has been anyone who's considered what an observer dependent mathematics might look like.