# The Nature of the Axioms of Geometry

June 28, 2021In the second half of the 19th century and the beginning of the 20th century, a philosophical debate was accompanied by the nature of synthetic a priori judgments.

The discovery of non-Euclidean geometries brought with it the following question: These geometries contain contradictory statements, especially considering the fifth axiom. One geometry speaks of a single parallel, while another mentions that no parallel can be drawn or that more than one (even an infinite number of) parallels can be drawn. Therefore, these geometries contain propositions that are not consistent with each other and cannot be true under the same interpretation.

If the axioms of geometry have universal necessity and objective validity, as Kant claims, all of them must be simultaneously necessary and objectively valid. However, two contradictory propositions cannot be necessarily and objectively valid at the same time. Being aware of this, Kantian philosophers, when they became aware of non-Euclidean geometries, gave two types of reactions: First of all, they accepted the intellectual possibility of these geometries, but they argued that only Euclidean geometry determines the form of intuition and therefore has objective validity. Secondly, they argued that non-Euclidean geometries may not be consistent, that is, it can be shown that they contain contradictions over time.

However, historical developments have revealed the invalidity of these two arguments. First of all, Riemann’s proof of relative consistency removed the second criticism by revealing that Euclidean geometries are as consistent as Euclidean geometry. The elimination of the first criticism took place in two stages. In the first stage, models of non-Euclidean geometries were found and it was seen that these geometries had an objective validity in the mathematical field. However, proponents of Kantian thought still argued that Euclidean geometry was the geometry of space. The second stage was realized with Einstein’s general theory of relativity. Einstein stated that a non-Euclidean geometry is more suitable for describing the laws of physics.

As a result of all these developments, the idea that the axioms of geometry are synthetic a priori judgments has become untenable. So what to say about the semantic and epistemological status of the axioms of geometry? In what philosophical framework can this question be answered?

A discussion on this subject is about the nature of basic terms such as “point”, “straight line” mentioned in the axioms of geometry. On the one hand, Frege and the mathematician and philosopher David Hilbert (1854 – 1912), on the other hand, Russell and the French physicist and philosopher Henry Poincaré (1862 – 1943), engaged in a discussion around this issue and exchanged letters. Since the 1880s, Poincaré has argued that the axioms of geometry neither have a factual content nor express a logical necessity, nor are they based on synthetic a priori judgments, but are “implicit definitions”. Hilbert similarly stated that the axioms of geometry are definitions. Interestingly, two prominent advocates of the reduction of mathematics to logic, Russell and Frege, opposed these approaches and for a long time tried to persuade Poincaré and Russell, respectively, that the axioms of geometry were not descriptive.

In these discussions, Hilbert and Poincaré’s view is that the terms in question are defined within axioms and have no meanings and references outside these definitions. Poincaré expressed this view as the axioms of geometry do not have a propositional content but should be considered as “definitions in disguse”. By being descriptive, the axioms of geometry must be analytical.

Discussions on the nature of the simple and undefined terms mentioned in the axioms of geometry played a key role in understanding the nature of language and logic in a certain way. In a way, the tradition of analytic philosophy has clarified its logic and a priori understanding within these discussions. We can summarize the development of the subject as follows: Kant presented a critique of Platonic ideas and intellectual vision as a means of acquiring knowledge about these ideas in his own thought. At the center of Kant’s critique of the possibility of metaphysics is a critique of Platonism.

By vision is meant a space or interface where we come into contact with objects directly (without the mediation of concepts). If the objects in question (such as Platonic objects or ideas) do not exist in the sensory field, the vision takes the adjective “intellectual” (“rational”).

On the other hand, while trying to justify mathematical judgments, Kant, as opposed to conceptual knowledge,