Limitations of the Axiomatic Method in Ancient Greek Mathematical Sciences

Abstract

My thesis in this paper is that the admiration many of us have for the rigor and relentlessness of the axiomatic method in Greek geometry has given us a misleading view of the role of this method in the broader framework of ancient Greek mathematical sciences. By stressing the limitations of the axiomatic method or, more explicitly, by stressing the limitations of the role played by the axiomatic method in Greek mathematical science, I do not mean in any way to denigrate what is conceptually one of the most important and far-reaching aspects of Greek mathematical thinking. I do want to emphasize the point that the use of mathematics in the mathematical sciences and the use in foundational sciences, like astronomy, compare rather closely with the contemporary situation. It has been remarked by many people that modern physics is by and large scarcely a rigorous mathematical subject and, above all, certainly not one that proceeds primarily by extensive use of formal axiomatic methods. It is also often commented upon that the mathematical rigor of contemporary mathematical physics, in relation to the standards of rigor in pure mathematics today, is much lower than was characteristic of the 19th-century. However, my point about the axiomatic method applies also to 19th-century physics.