The One and the Dyad: the Foundations of Ancient Mathematics. What Exists Instead of Infinite Space in Euclid’s Elements?

Zbigniew Król

This paper contains a new interpretation of Euclidean geometry. It is argued that ancient Euclidean geometry was created in a quite different intuitive model (or frame), without infinite space, infinite lines and surfaces. This ancient intuitive model of Euclidean geometry is reconstructed in the connection with Plato’s unwritten doctrine. The model creates a kind of “hermeneutical horizon” determining the explicit content and mathematical methods used. In the first section of the paper, it is argued that there are no actually infinite concepts in Euclid’s Elements. In the second section, it is argued that ancient mathematics is based on Plato’s highest principles: the One and the Dyad and the role of agrapha dogmata is unveiled.


Keywords: Euclidean geometry · Euclid’s Elements · ancient mathematics · Plato’s unwritten doctrine · philosophical hermeneutics · history of science and mathematics · philosophy of mathematics · philosophy of science

Zbigniew Król – Associate Professor at the department of the Philosophy of Science, Sociology and Foundations of Technology, Faculty of Administration and Social Science, Warsaw University of Technology, and at the two departments at the Institute of Philosophy and Sociology of the Polish Academy of Sciences (Warsaw, Poland): of the Philosophy and Hermeneutics of Mathematics and of the Inquiries on Ancient Philosophy and the History of Ontology. His research concerns the philosophy of mathematics and science, history of mathematics and science, logic, mathematics, ontology and epistemology.   »  

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