# Nature of Axioms of Geometry

December 24, 2019 By Felso

In the second half of the 19th century and the early 20th century, a philosophical discussion of the nature of synthetic a priori judgments was accompanied.

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

If, as  Kant argues, the axioms of geometry have universal necessity and objective validity, they must all be obligatory and objective valid at the same time. However, the two contradictory propositions cannot be one and simultaneously obligatory and objective. Aware of this, Kantian philosophers reacted when they became aware of non-Euclidean geometries: Firstly, they accepted the intellectual possibility of these geometries, but claimed that only Euclidean geometry determined the shape of the image and therefore had objective validity. Secondly, they argued that non-Euclidean geometries may not be consistent, that is, it can be shown that there are contradictions over time.

However, historical developments have revealed the invalidity of these two arguments. First of all, Riemann’s relative proof of consistency eliminated the second criticism by suggesting that Euclidean geometries are as consistent as Euklides geometry. The elimination of the first criticism took place in two stages. In the first step, models of non-Euclidean geometries were found and it was found that these geometries had an objective validity in mathematical field. However, advocates of Kantian thought, however, have argued that Euclidean geometry is the geometry of space. The second stage was realized by Einstein’s general theory of relativity. Einstein stated that a non-Euclidean geometry is more appropriate 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 about the semantic and informational status of the axioms of geometry? What kind of philosophical framework can this question be answered?

A discussion on this subject concerns the nature of the basic terms such as “point”, “straight line geçen, which are 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 subject and corresponded. Poincaré argued that from the 1880s onwards, the axioms of geometry had no factual content, expressed no logical necessity, nor were based on synthetic a priori judgments, but were “implicit definitions Po. Hilbert likewise stated that the axioms of geometry are definitions. Interestingly, two important names advocating the reduction of mathematics to logic, Russell and Frege, opposed these approaches and tried to persuade Poincaré and Russell, respectively, that the axioms of geometry would not be descriptive for a long time.

In these discussions Hilbert and Poincaré advocate that these terms are defined in axioms and do not have any meaning and references beyond these definitions. Poincaré expressed this view that the axioms of geometry do not have any propositional content but should be considered as definitions implicit definitions dis. As they are descriptive, the axioms of geometry have to be analytical.

In the axioms of geometry, the debate on the nature of simple and undefined terms, often referred to as the key to understanding the nature of language and logic in a certain way. In a way, the tradition of analytical philosophy has clarified its logic and a priori understanding in these discussions. The development of the subject can be summarized as follows: Kant presented a critique of the intellectual view as a means of Platonic ideals and of our knowledge of these ideals. At the heart of Kant’s critique of the possibility of metaphysics is a critique of Platonicism.

Vision refers to a space or interface with which we come in direct contact with objects (without the mediation of concepts). If these objects (such as Platonic objects or ideals) do not exist in the sensory field, the view is considered to be el intellectual ”(“ intellectual ”).

On the other hand, Kant, while attempting to base mathematical judgments, used visual knowledge as opposed to conceptual knowledge and argued that mathematics was based on pure vision. As a result, synthetic a priori judgments are based on the direct knowledge of objects constructed in pure vision. However, as we have stated above, the view that geometry is based on synthetic a priori judgments has now become untenable with the emergence of Euclidean geometries. Thus, a tendency has arisen to reduce arithmetic in particular, and mathematics in general, to analytic (logical). However, a question to be asked is where the knowledge of the logical originates. Logical forms (forms of logical propositions and valid inferences), if seen as facts, we can say that we have conceptual information about themselves. In this case, however, the logical a would have a posteriori status, which is not an easily defensible view. If our knowledge of the logical is not conceptual, the remaining option is that the information is visual. However, in this way, the intellectual vision is reintroduced as a kind of knowledge that Kant criticizes and contributes to. This is equally unacceptable. In this context, it is very important to show that the axioms of geometry are implicit definitions: This approach is of great importance in terms of reducing mathematics to logical one hand; on the other hand, no reference is made to the visual information during this reduction.

The final point of the process of the withdrawal of philosophy within the language will be put in Wittgenstein Tractatus. The general style of criticism for the understanding of logical forms in a Platonic way is negative. So, it is aimed at revealing what they are not. Wittgenstein tried to reveal what they are within the boundaries of language.

Prepared by:
Sociologist Ömer YILDIRIM
Source:  Ömer YILDIRIM’s Personal Lecture Notes. Atatürk University Department of Sociology 1st Grade Giriş Introduction to Philosophy ”and 2nd, 3rd, 4th Grade Tarihi History of Philosophy” Lecture Notes (Ömer YILDIRIM); Open Education Philosophy Textbook