From Practical to Pure Geometry and Back

Abstract

The purpose of this work is to address the relation existing between ancient Greek (planar) practical geometry and ancient Greek (planar) pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical geometry, some of which are basically explicit in definitions, like that of segments (straight lines) in Euclid‘s Elements. Then, we will address how in pure geometry we, so tospeak, ―refer back‖ to practical geometry. This occurs in two ways. One, in the propositions of pure geometry (due to the accompanying figures). The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment.

A hub-and-spoke model of geometric concepts

“…In fact, in Plato's Meno there is an example of ancient Greek practical geometry (Plato, 1997, pp. 881-5;Valente, 2020), and the corresponding text can be seen as a written rendering of the oral teaching and discussion of geometry in ancient Greece (Saito, 2018). 8 In the case of ancient Greece, there is also evidence of the division of land in rectangular plots (Lewis, 2004, p. 3;Cuomo, 2001, pp.…”

“…Another motivation is that having models of the neural representation of geometric concepts gives us a new venue to address the issue of the relationship between geometric figures and geometric objects. That has already been addressed from a historical perspective (see, e.g., Valente, 2020), and also cognitive considerations have been taken into account in somewhat related issues (see, e.g., Giaquinto, 2007;Dal Magro & García-Pérez, 2019;and Ferreirós & García-Pérez, 2020). But here, we present specific models developed in the context of a theoretical framework (the hub-and-spoke theory), which provides a more 'tangible' way to address this issue (since we have models of the neural representation of geometric figures and objects).…”

On the Relationship Between Geometric Objects and Figures in Euclidean Geometry

“…This relation of idealization is made clearer by taking into account similar relations for lines and points (Valente 2020a). They are all taken into account, implicitly, in the definition of circle.…”

“…Again, it is due to the relation of idealization that we have between geometric objects and concrete objects. For instance, a geometric segment is in a relation of idealization not only with, e.g., a practically drawn segment but also, e.g., with a rod, a stretched rope, or with the 'visual fire' taken to be a sort of light beam (Valente 2020a).…”