https://en.wikipedia.org/wiki/Riemannian_geometry

I realized as a child, intuitively, something that’s incredibly difficult to articulate. But it’s something like this:

The credibility or consistency of postulates depends not upon the postulates themselves, or, necessarily on how we relate them. It is derived from elsewhere. This elsewhere is the »reference frame(s) in which they acquire meaning. And the primary determinant of their relationships with the reference frames … is the specific, necessary and intentional way we chose »what to exclude from them.

In simpler terms, the validity is never intrinsic in a postulate. It’s in the relation with the reference frame, which was constructed on purpose, by excluding nearly everything, in order to produce the desired functions within the very narrow and limited properties of the result. We then expand slightly, placing our minds in the position of observers of the qualities evident in the relationships within the given frame.

First, we have to de-infinitize a reference frame. This means a series of intentional exclusions. The ‘toy’ that remains produces the appearance of the consistency of the postulates. This is, in part, why mathematics cannot be placed on a reliable logical foundation.

Reimannian geometry gave us an expansion of the reference frames in which Euclidian geometry had established axioms thought to be nearly inviolable. An expansion of the reference frame to include other-dimensional spaces… returned depth to the study of geometry.

But a similar revolution awaits us… in our relationships with language and meaning.