I don't have any authoritative references for the following -- so far as I know, it's my own invention, based on some fairly well-established (readily observable) facts. I expect one or more of you can poke holes in it, but it's worth a try...
The very nature of human language defeats the solipsist and eliminates any doubt about the presence of "objects" that exist outside an individual's internal mental state. Language can only be learned through cooperative interaction with other individuals -- it is something that each individual develops through a combination of (a) accepting/mimicking the expressions of others and (b) assembling / inventing "generative" (creative) systems and structures that serve to keep all this linguistic information "sensible", manageable, and adaptable to new experience.
When Decartes said "Cogito ergo sum", a large proportion of the cogitation he referred to was dependent on a vocabulary and grammar that he learned from his parents and community, based on shared experience of objects, events, behaviors and relationships. (Picture Decartes at age 3, as his mother says to him, "Yes, René, that's a dog. The dog just licked your hand. Wasn't that nice? Would you like to pet the dog?" He acquired language the same way we all do.)
The language we use for mathematics is considerably more specialized and constrained, but it operates the same way: as concepts are discerned and understood, symbols and expressions are needed to describe them in a manner that can be shared. Unlike the natural language that we all acquire instinctively and automatically, the symbols and expressions used for math must obey strict rules that disallow ambiguity, emotional affect, etc. As a collection of symbols and grammatical rules, it has the ability to stand up as a coherent system on its own without direct dependence on physical reality, despite being founded on (and drawing its purpose from) reality. The "perfect circle" exists only in the sense that there is an expression to define it: the shape that is built up from all points that lie on a plane at an equal distance from a single, central point. Given such a definition, all sorts of derivative expressions can arise to describe various properties of a circle, including the relation we call pi, and all this comes in very handy as we try to get a more detailed grip on reality itself. (For one thing, we get to see the degree of "error" between an idealized shape and an observed shape, and this in itself can be very informative, in many ways.)
But, returning to the OP question: "objective reality" could be defined as the domain of linguistic expressions that seek to represent, as accurately and consistently as possible, the interaction of objects with the aggregate of human senses.
I'll use the term "objects" to refer to things that reside in the physical world and are accessible to human senses (by whatever physical means, including specialized instruments) -- thus "objects" include humans and their behaviors, as well as everything else from sub-atomic particles to clusters of galaxies. Objects, their actions and their relationships are the primary building blocks on which human languages (including mathematical language) are built -- they are the basis for the success of our languages as tools of communication.
When objects demonstrate properties or actions that are perceived uniformly by humans, such as temperature or gravitational attraction, it is the uniformity of perception that constitutes "objective reality", and our attempts to accurately describe such properties and actions are "objective descriptions". It is the ability to establish an agreed-upon description, based on unambiguously common perceptions, that constitutes "objectivity".
Now, consider objects whose properties or actions are not perceived uniformly by humans. One example might be psycho-active drugs, which tend to induce very different perceptions from one individual to another -- the particular effects can be profoundly subjective. Until more is understood about "normal" perception (and memory and "creativity") in the human brain, as well as what these drugs actually do to brain chemistry, there isn't much we can say in "objective" terms about these objects -- there isn't enough that can be recognized and accurately described as common, shared experience.
Another example, of course, is human behavior (including natural language behavior -- i.e. things people say). As of just the last 100 years or so, we have the ability to create a physical ("objective") record of a given human behavior: movements and voices can be captured on physical media for repeated review and analysis, and there will be consensus among watchers and listeners about the physical details of the recorded event. But (apart from the fact that such recordings may be limited in fidelity or completeness), the unavoidable shortcoming we face is our inability to record a person's intent.
If multiple viewers / listeners of a given recording are asked "what was this person trying to do / say?", we are prone to get multiple (possibly conflicting) answers, especially in the case where the actions or utterances (or persons) that were recorded happen to impinge on any sort of emotional or prejudicial bias held by the viewers / listeners. Even seemingly simple actions can be ambiguous, e.g.: "he's tying his shoe" vs. "he's trying to duck down but doesn't want others to think that he's ducking"; or "he's describing the weather" vs. "he's trying to avoid engaging in substantive discourse", and so on. Describing other people's intent is a notoriously subjective endeavor.
As for anything that would be called "metaphysical"... well, that's a different matter entirely -- neither "objective" nor "subjective", IMHO. Metaphysics is a form of guesswork that explicitly forswears any reliance on sharable physical perception when trying to explain things that have been perceived (or worse, when explaining things that have not been and cannot be perceived). I consider metaphysical descriptions to be a "natural" extension of our innate habit for inducing purpose; they may also have something to do with the kind of thinking that led the famous emperor to command admiration for his "new clothes" {AbE: as well as the kind of thinking that led so many of the emperor's subjects to show admiration}.
The diversity, pervasiveness and persistence of metaphysical belief says a lot about the nature of human cognition, while saying virtually nothing substantive about the nature of reality.
Edited by Otto Tellick, : minor grammar repair
Edited by Otto Tellick, : (addition as indicated in next-to-last paragraph)
autotelic adj. (of an entity or event) having within itself the purpose of its existence or happening.
™ Version 4.2
