Was pi invented or discovered?

[FallentoReason]

Fair enough, but the reason that they aren't perfect representations of said concept is that we have devised said concepts requirements in a way that excluded them. We could just as easily have defined a circle as a concept referring to the exact dimensions of the sun or ripples on a pond (or some equivalent ratio thereof). Had we done so, pi (not our pi, but thi sun pi we've never gone to the truoble of defining) would still exist, there would still be a relationship between the various dimensions ad ways of measuring the sun that yielded a guide as to what a circle was - and it would reside in the real world as filled with substance and material as the dimensions of any circle in our minds - based upon those relationships which exist in the real world and are contingent upon those properties of that world that we can detect - exemplified by the sun, as our prime example.

I think the way in which a circle is currently defined wasn't done like that so as to exclude physical objects from literally being a circle. It's kind of the same thing as the concept of a chair: we have small ones, big ones, wide® ones... but why would we arbitrarily chose one of the many representations of the concept to be the chair, just like you proposed we could have chosen the sun to be the representation of a circle? It seems to me that all of them (chairs & shapes resembling circles) have some intrinsic value that allows us to place them into the category of a chair/circle, and therefore it is correct for us to steer clear of physical manifestations of the concept by understanding the concept to be of a "pure" nature where it only exists in our minds.

Our concept of a circle (and our determinations of what is or is not a cricle) have changed over time...but it's more than fair to describe our initial concept of a circle as something very similar to what a child today might point to when asked to show an example of a circle. The group of things the child singles out will not be circles as we now understand them, nor will the child leverage or likely understand pi, but they will still be circles, and the concept is still very useful in describing those things.

Yeah absolutely.

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With a bit of imagination, I can tell you I just drew a representation of the concept of a circle, and in our everyday understanding of the world, you wouldn't call me out on it, but instead superimpose what you know a circle to be over my obviously imperfect circle. In practice, it will always work whenever we draw a circle in a diagram, blueprints or maths textbook provided that the person who sees it has an understanding of the concept. No, it's not a true circle, but it's a good enough representation of it that the message will get through to the other person that the concept is being dealt with.

It shouldn't be surprising that a system refined for the systems sake begins to become divorced from what it was initially crafted to communicate. All that is required to show where a circle (or pi) resides in the world outside that system is for dimensions to exist, for an area to be singled out and the relationship between the borders of that area quantified in any way, if we single out a circular area, this will plainly demonstrate where pi resides and where it waited to be discovered in hard inches (if you will). Regardless of what we call that area (if not a circle) and regardless of the manner in which it is quantifed (if not the numbers and math you and I are familiar with). Whatever concept devised and whatever system leveraged to study this area will yield a result that is equivalent to our own. IOW, a rose by any other name.

I think the system itself kept improving to the point that it reached the "pure" understanding of a circle. Like the badly drawn circle above, I could try again to improve it. If I were living back in the days when e.g. sketching equipment had just been invented, I could have grabbed a compass and drawn as best I could a circle. Then for a while we could all go believing that that is a perfect circle because all points are at the same distance from the centre, but eventually we would realise that in fact I have drawn an infinite amount of circles because my pencil has a thickness i.e. there is a range of circles with radius r to r + t (where t is the thickness of the pencil). Bring if forward to today and our modern understanding of atoms would lead us to think a circle can never be but then when we bring in our understanding of atoms, we then can see that a circle can never be made in real life because any substance you use will have a thickness.

I don't know if this properly addresses your last paragraph. I wasn't too sure what you meant by the "system"...