I was thinking about this in the shower the other day. Circles don't "exist" because they are a pure abstraction; as an element of mathematics they are simply a bag of conclusions that are inevitable if you are willing to accept other axioms. The language with which we talk about these axioms and conclusions might be fallible, and certainly the conclusions themselves may be fallible (though not likely so for elemental enough mathematics), but the kernel of truth in logic at the middle is, I think, unassailable.
Perhaps so. I trimmed down the parts where I referenced Euclid in a similar vein to “conclusions … inevitable if you … accept other axioms.” I was thinking though of this Veritasium video outlining the development of non-Euclidean geometries:
We never know any data before interpreting it through theories. All observations are, as Popper put it, theory-laden, and hence fallible, as all our theories are. . . . The growth of knowledge consists of correcting misconceptions in our theiories.
Pi is a great example. Did circles exist before minds existed? Or are circles approximations of the observed world?