Foundations of Trigonometry: Conceptual and Logical, Being an Essay Towards a Conceptual Foundations of Mathematics
Mohamed A. Amer

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…
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Taxonomy
TopicsHistory and Theory of Mathematics · Mathematics and Applications · Mathematical and Theoretical Analysis
