Mathematical Paradoxes and Three Mathematical Crises

Author: Han Xuetao / Country: Mainland China
Publisher:
Publish Date: 2006-05-01
Features: Through several passages by the famous Qing Dynasty mathematician Mei Wending (1633–1721) of China, we can further appreciate this point. In his mathematical work Theory of Equations, he wrote: "Mathematics is one, but when divided, it has both measure and number; measure refers to the method of geometry, and number refers to arithmetic, both of which progress from simple to complex. Therefore, the method of measure begins with surveying fields, advances to lesser breadth and height, and culminates in the Pythagorean theorem." Here, the method of measure refers to geometry. This passage emphasizes the position of properties and algorithms related to right-angled triangles in Chinese-style geometry. In Comprehensive Solutions to Geometry, he wrote: "Geometry does not speak of the Pythagorean theorem, yet its principles are intertwined with it. Therefore, what is difficult to understand is clarified when explained using the Pythagorean theorem. Trusting the meaning of the ancient Nine Chapters, it encompasses all aspects." In Examples of the Pythagorean Theorem, he said: "The application of the Pythagorean theorem is miraculous. Measurement techniques in Western mathematics are detailed, but they cannot be established without the Pythagorean theorem. Thus, trigonometry is a variation of the Pythagorean theorem, and the eight lines are its established formulas." When Western geometry was introduced, Mei Wending mistakenly believed that Western geometry was merely a variation of Chinese Pythagorean mathematics, with nothing new. However, as he pointed out, to understand the original form of ancient Chinese geometry, one must start with the Pythagorean theorem and its related properties, which is correct. Of course, the Pythagorean theorem is not only important to traditional Chinese mathematics. In fact, along with its corollaries and extensions, the Pythagorean theorem has broad applications in the real world and plays an extremely important role in the development of mathematical theory. In plane geometry, this beautiful, famous, and useful theorem shines like a gem. The astronomer Kepler once called it the "golden" theorem in geometry, and it is truly worthy of the title! Moreover, what is even more important is that as a fundamental and renowned mathematical theorem, the Pythagorean theorem deeply permeates many branches of mathematics. Many mathematical formulas and propositions in mathematics are derived from it or are based on its foundation. It can be said that in mathematics, the Pythagorean theorem has been and remains an indispensable tool that spans many fields. If one were to name the most important theorem in mathematics, it would undoubtedly be this one. The following anecdote serves as evidence. In 1955, Greece issued a stamp to commemorate the contribution of ancient Greece to the Pythagorean theorem 2,500 years ago. The design featured three chessboards arranged together. In 1971, the government of Nicaragua issued a set of stamps titled "The 10 Most Important Mathematical Formulas in the World," with each stamp displaying the selected formula and a brief explanation of its significance on the back. The second stamp in this set was the Pythagorean theorem. The renowned and late Chinese mathematician Hua Luogeng once considered using the Pythagorean theorem as a language to communicate with extraterrestrial civilizations. In his article The Uses and Development of Mathematics, he wrote: "If our spacecraft reaches a planet where there are advanced beings like humans, what should we use as a medium of communication? Take a painting—unfortunately, the scenery is unfamiliar. Bring a recording—still no connection. I think it would be best to bring two figures: one 'number' and one 'number-shape relationship' (the Pythagorean theorem)." P10-11