What Hyperbolic Geometry Actually Is — Viewed Through Systemic Functional Linguistics

Penrose (2004: 40, 43):

The existence of various different models of hyperbolic geometry, expressed in terms of Euclidean space, serves to emphasise the fact that these are, indeed, merely ‘Euclidean models’ of hyperbolic geometry and are not to be taken as telling us what hyperbolic geometry actually is. Hyperbolic geometry has its own ‘Platonic existence’, just as does Euclidean geometry.

… hyperbolic geometry does actually exist, in the mathematical sense that there is such a consistent structure. In the terminology of §1.3, hyperbolic geometry inhabits Plato’s world of mathematical forms.

 

Blogger Comments:

To be clear, in the very simplest of terms, 'hyperbolic' geometry replaces the flat plane of Euclidean geometry with curved plane of the inside of a sphere. (Likewise, 'elliptical' geometry replaces the flat plane of Euclidean geometry with curved plane of the outside of a sphere.)

From the perspective of Systemic Functional Linguistic Theory, hyperbolic geometry, like all mathematics, is second-order meaning (metaphenomenal), and like all meaning, forms part of the content of the collective consciousness of its community.

From this perspective, the expression of hyperbolic geometry in terms of Euclidean space is the use of the expression plane of one geometric system to realise the content plane of another.