Foundations of Trigonometry: Conceptual and Logical, Being an Essay Towards a Conceptual Foundations of Mathematics

TL;DR

This paper critiques historical treatments of trigonometry, addressing foundational philosophical and technical questions, and proposes a conceptually sound, logically consistent framework distinguishing geometric and analytic trigonometry based on Eudoxean ratios.

Contribution

It introduces a new foundational approach to trigonometry, defining geometric functions via Eudoxean ratios and establishing a clear relationship with real numbers and measures.

Findings

Eudoxean ratios are suitable as positive elements of real numbers.

A distinction between geometric and analytic trigonometric functions is established.

The measure of an angle is defined as its Eudoxean ratio to the radian.

Abstract

Noticing that all of the 19th, 20th and 21st centuries treatments of trigonometry surveyed in this article are conceptually or logically defective, it is required to seek a conceptually sound and logically correct foundations of the subject. To this end, several questions have to be discussed: Is mathematics arbitrary? What does it have to do with nature? Reality? Applications? What is measure? Direct measure? Indirect measure? What are ratios? Eudoxean ratios? What is their relationship to measure? What are the real numbers? What is their relationship to ratios? To measure? What are the geometric trigonometric functions? The analytic trigonometric functions? What is the relationship between them? What is the measure of an angle? what is its relationship to trigonometry?......? After dealing with these philosophical and technical questions, a treatment of both geometric and analytic trigonometry which would be conceptually sound and logically correct foundations of the subject is proposed. This treatment distinguishes between geometric trigonometric functions (the elements of whose domain are angles) and analytic trigonometric functions (the elements of whose domain are real numbers), the bridge from the former to the latter is the measure of angles. The geometric trigonometric functions are defined to be the Eudoxean ratios between appropriate straight line-segments, and the measure of an angle is defined to be its Eudoxean ratio to the radian, which is to be defined beforehand. Also, the Eudoxean ratios were defined beforehand, and it was advocated that it is conceptually sound to consider them as the positive elements of the field of the real numbers.