Copyright © Lucas Amiras 2020



The main issue of protogeometry can be conveyed through a question, which emerges right from the beginning at learning geometry, but has never hitherto been answered:

What is the meaning of “straight” (line) or “flat” (surface)?

One can nowhere get a satisfactory reply to this question, neither in geometry lessons at school nor at university. Regarding the information given in schoolbooks and scientific textbooks on geometry and comparing it with the speech about straight lines and flat surfaces in elementary handicraft or technology the impression is created that there is a fundamental problem: It seems paradoxical that these objects, being terminologically undetermined on the one side, are on the other side successfully produced and widely used.

Protogeometry attempts in a philosophical manner, which I call “operational phenomenology”, to solve this paradox. It tries to explicate technical actions and phenomena related to these concepts in elementary technology and to use them as a normative basis for concept and theory formation of geometry.

The table gives an idea about the relationships between these practical domains, the related concepts and the issue here:

After establishing an elementary vocabulary dealing with figures on bodies the issue is to support definitions for “even” and “straight” on this terminological basis. Protogeometry comes to an end with an effort to interrelate it with geometric theory. However, the common geometric theory (or their axioms) cannot simply be derived from it. It is rather a methodical connexion we like to establish between them. Thus, Protogeometry becomes the status of a pre-theory of theoretical geometry. Protogeometry provides so, due to its character, the foundations of theoretical and especially axiomatic endeavours in geometry, so that these can be better understood methodologically.



Touching, Contact


be positioned on, overlapping,
coincidence of markings, inside-outside of figures, borders, ends etc.

Position markings on bodies, drawing borderlines

contact, fitting of figures, Cuts or divisions of figures

Handling of bodies, tool and mould construction, building

movement, movement of figures, guided movements, Spatial arrangement, touch ability, fitting of figures

Congruence, Congruence principle

Shape, reproduction of shape, conncy of shape

Ideal geometrical elementary shapes and their properties

Elementary functional properties of basic geometrical forms: universal fitting, smoothness, slid ability etc.

Technical practice with geometrical shapes

equality of form, similarity

pictures, moulds, models

Drawing, representations

Geometrical concepts, ideas

Practical concepts, elementary phenomena

Practical domains, domains of experience