Demonstration in geometry: historical and philosophical perspectives

Saddo Ag Almouloud

doi:10.33361/rpq.2020.v.8.n.18.344

Abstract

In this article, we weave historical-philosophical reflections about demonstration in mathematics, based on works of researchers that discuss the different philosophical perspectives on the topic, more specifically on geometry. We focus first on demonstration and its relationship with intuition and figural representations. Second, we criticize Poincaré’s conception of mathematical demonstration. Third, we reflect, in a non-exhaustive way, on the philosophy of demonstration in geometry, confronting Kant’s conceptions with the axiomatizations of the non-Euclidean geometries. In this text, we do not adopt a single definition that would cover all modes of scientific validation, since we admit the possibility of an evolution of ideas about the validity of a proposition. Not to fall into the symmetrical flaws of the glorification of the Ancients or even being ungrateful to them, we must start from the naive idea that the demonstration has a historical origin and, therefore, maintains a historical character, but we should be more attentive to what characterizes, in its particularity or even its uniqueness, the productions of past and present centuries. Keywords: Philosophy of demonstration; Axiomatization; Induction; Intuition; Representation.

Highlights

Demonstration occupies a central place in mathematics, as it is the method of proof whose systematic use characterizes this discipline among the sciences
If mathematics had no others, it would be immediately blocked in its development; but it resorts to the same process again, that is, to reason by recurrence and can continue marching forward. - At each step, if we look closely, we find this way of reasoning in the simple form we have just given it, or in a more or less modified way
The mathematical demonstration can be seen as an argument by which someone is convinced or convinces others that something is true; it may seem difficult to go beyond the epistemic conversation about explanatory proof

Summary

Introduction

Demonstration occupies a central place in mathematics, as it is the method of proof whose systematic use characterizes this discipline among the sciences. The demonstration among the Greeks is a consequence of reflective thinking influenced by the political-social and philosophical demands that were established by the need to “convince” the other. Demonstrative speech is demanding, scrupulous, unlike ordinary speech, the vehicle of our conventional ideas. Rigor honours those whose existence is based on principles. We chose to weave historical-philosophical reflections about the demonstration in mathematics. These reflections are supported by the work of researchers that discuss different philosophical perspectives of the demonstration in mathematics, in geometry. We should carefully consider what characterizes the productions of past centuries in their particularity or even their uniqueness

Demonstration: the role of intuition and figural representations

The mathematical demonstration

The philosophy of demonstration in geometry

Conclusions