Errol Morris on Pythagoras, Incommensurability, and Whiggish History

In March 2011 Errol Morris, an award-winning documentary filmmaker and director of TV commercials, published a series of five opinion pieces in the New York Times with the collective title "The Ashtray". Those who know about Morris through his films and ads may not be aware of his philosophical writings about the nature of truth, knowledge, and history. If those topics put you off from reading Morris, perhaps it would help to know that the series title comes from the object thrown at the author by Thomas Kuhn, author of "The Structure of Scientific Revolutions", when Morris was a graduate student under Kuhn at Princeton. (If that's still not enough, then see his description of a Dan-Brownish library encounter, below.) The series as a whole is a critique of Kuhn's notion of "incommensurability" of scientific paradigms, but the part that appealed to me particularly is the third in the series, on the origin of the term "incommensurable" in mathematics and the legends surrounding the Pythagorean proof that the square root of two is irrational.

The term "incommensurable" comes from the proof of the irrationality of √2. To say that √2 is irrational is to say that the ratio of the diagonal of a square to one of its sides cannot be expressed as a ratio of whole numbers; that is, the side of a square and the diagonal are "incommensurable". The proof of this goes back to the Pythagoreans, who according to legend tried to suppress knowledge of the proof and ordered the death of Hippatus of Metapontum for revealing the secret. Morris, puzzled by Kuhn's use of incommensurability as a metaphor for scientific paradigm shifts, decided to dig deeper into the history of the Pythagorean proof and the various legends surrounding it.

The main source for the legends seems to be Iamblichus of Chalcis, writing 800 years after the Pythagoreans. While researching him, Morris relates an encounter in the stacks of Harvard's Widener Library that sounds like it comes out of a Dan Brown novel:

I took the elevator to the fifth floor. Looking for the call number –– WID-LC B243.I2613.1986. I stopped. Turned down an aisle, tripped a motion-sensor, and a light clicked on. An old man – possibly in his 70s – was walking towards me from the other end of the aisle. The gap closed between us. I bent down to reach for a book – Iamblichus’s “Life of Pythagoras, or, Pythagoric Life (De vita pythagorica).” As he passed me, he said, “Be careful. Iamblichus is not to be trusted.”

Apparently Iamblichus gives several conflicting versions of the story. Morris goes on to make the point that the historical evidence is very sketchy, and that the characterization of the proof as a crisis in Greek mathematics is a construction of the 19th century, when mathematicians faced their own foundational crisis. In other words, it is an example of "Whiggish history" (a term coined by Herbert Butterfield): the interpretation of the past not on its own terms but in the context of present concerns.

So what does this have to do with teaching high school mathematics? The proof of the irrationality of √2 is a staple of HS math, and the story of its discovery and legends surrounding it is a sidebar frequently encountered in HS textbooks. I think the legends should continue to be included - it makes a good story, and humanizes the math. But care should be taken to avoid Whiggish history and associate the stories with the proper historical context. As Morris makes clear, it's about the 19th century, not the Greeks. The idea of incommensurability should also be included, and made precise. It does not mean immeasurable, that is, not measurable; nor does it mean incomparable or not comparable. Rather, it describes a specific relation between two things - that they do not have a common unit of measurement. It can only be applied to two or more objects: a number cannot be incommensurable by itself, only in relation to another. Incommensurable magnitudes can still be compared - the diagonal of a square is larger than a side - without necessarily having a common unit of measure. This idea of a common unit that can measure two quantities is one that can be concretely demonstrated in class (and by the way leads to the Euclidean Algorithm for finding the greatest common factor) and showing that the assumption of a common unit leads to a contradiction is the basis for the proof of √2.