Axiomatic method

This semester, I decided to take a course on Advanced English Writing (this class) and Introduction to Oriental Philosophy. The philosophy class is taught by the professor Woosuk Park. I enjoyed reading his book "Philosophy of Baduk" in the past.

In the Wednesday class, the professor Park raised the issue that the axiomatic method is apprently lacking in the East. He argued that the axiomatic method, examplified in the Elements by Euclid, formed the basis of the scientific revolution in the West. And the lack of axiomatic method is why the West could overtake the East in science. Then he wanted to discuss what role the philosophy played in creating such difference.

I pointed out that the rigorous axiomatic treatment of mathematics is a modern development. He replied that Grundlagen der Geometrie by David Hilbert, motivated by non-Euclidean geometry, may be considered the beginning of the rigorous axiomatic mathematics, but the germ of the axiomatic idea was already present long before. I wasn't satisfied, but I didn't pursue the argument any further at the time.

So here are some of my thoughts. First, Euclid was not rigorous, at least not in the modern sense. Let's look at his first proposition: To construct an equilateral triangle on a given finite straight line. Given the segment AB, two circles, one with center A and radius AB, and the other with center B and radius BA, are constructed. Let C be the point at which two circles intersect. Then that ABC is equilateral follows from the definition of the circle. Now, why should the point C exist? Two circles may not intersect at all, and the existence of the point C is nowhere justified. In general, the Elements is not very careful about between-ness and inside-ness.

Second, it was the analysis, not the geometry, which enabled the development of Newtonian physics and other sciences. And the early analysis was anything but axiomatic. Newton and Leibniz made heavy uses of infinitesimals, treating an infinitesimal as non-zero in one part of the "proof" (where they divide by it), and as zero in the other part of the "proof" (where it is eliminated). The need to prove convergence was not realized for a long time. It was only by the arithmetization of analysis, the 19th century development to found the analysis on the algebra, typified by epsilon-delta definition of limit by Weierstrass, that the analysis became rigorous.

Therefore it seems clear to me, that the axiomatic method is not what made the scientific revolution possible. At the time, the axiomatic method was not applied to the mathematics used by the science, which was the analysis.