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Was Pythagoras the First to Discover Pythagoras’s Theorem?

(This article was originally published in SIAM News.)

According to Brown University mathematician David Mumford, the answer to the question is an emphatic "No!" On February 27, 2013, in a public lecture at the Institute for Mathematics and its Applications at the University of Minnesota, Mumford showed how ancient cultures, including the Babylonians, Vedic Indians, and Chinese, all proved the beloved formula long before the Greeks. He argued that the theorem is ultimately the rule for measuring distances on the basis of perpendicular coordinates. This comes up naturally in calculations of land area for purposes like taxation and inheritance, as shown in Figure 1. He further suggested that the Greeks' love of formal proof may have contributed to the Western belief that they discovered what Mumford calls the "first nontrivial mathematical fact."

Along with Pythagoras's theorem, Mumford discussed the discovery and use of algebra and calculus in ancient cultures. One of his key points is that deep mathematics was developed for different reasons in different cultures. Whereas in Babylonia algebraic "word" problems were posed seemingly just for fun, the Nine Chapters on Computational Methods, considered the Chinese equivalent of Euclid's Elements, was compiled in about 180 BCE for very practical applications--among them Gauss-ian elimination for solving systems of lin-ear equations, which the Chinese carried out using only counting rods on a board (Figure 2). Riemann sums grew naturally out of the necessity for estimating volume. Mumford suggested that Vedic Indians even pondered problems of limit in integral calculus.

Contrary to Western historical belief, Mumford showed, the West did not always lead in mathematical discovery. Apparently, the origins of calculus sprang up totally independently in Greece, India, and China. Original concepts included area and volume, trigonometry, and astronomy. Mumford considers the year 1650 a turning point, after which mathematical activity shifted to the West.

Mumford's presentation runs counter to current texts on the history of mathematics, which often neglect discoveries occurring outside the West. He showed that purposes for which mathematics is pursued can be very culturally dependent. Nevertheless, his talk points to the fundamental fact that the mathematical experience has no inherent cultural boundaries.

Mumford, a professor emeritus in the Division of Applied Mathematics at Brown University, has worked predominantly in the area of algebraic geometry and is a leading researcher in pattern theory. Mumford received a Fields Medal in 1974; his more recent awards include the Shaw Prize (2006), the Steele Prize for Mathematical Exposition (2007), the Wolf Prize (2008), and the National Medal of Science (2010).

(Anna Barry, a postdoctoral fellow at the Institute for Mathematics and its Applications at the University of Minnesota, followed up on her coverage of David Mumford's IMA lecture with an interview. The full article and interview is available online at SIAM News.)