wanderings on a sea of non-identity

One of the books I'm currently reading is A history of mathematics: from Mesopotamia to modernity by Luke Hodgkin (Oxford, 2005). The author includes some startling quotes. Thus at lunch today I encountered: "the soul ... wanders hither and thither on the sea of non-identity ... immersed in the storm of the coming-to-be and the passing-away, where there is no standard of measurement." Kind of poetic, isn't it? The discussion is of "incommensurabilities" or irrational numbers, which were said to trouble Pythagoras greatly. The classic incommensurable and the one which supposedly caused the great mathematician such trouble was the ratio of the sides of a square with its diagonal.

Wikipedia explains:

The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction . However Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. He could not disprove their existence through logic, but his beliefs would not accept the existence of irrational numbers and so he sentenced Hippasus to death by drowning.

Here's an explanation, not from Hodgkin, but from this source:

The ratio of the length of the diagonal of a square figure to the length of one of its sides is not “the ratio of a number to a number.” Pappus, in his Commentary on Book X of Euclid’s Elements, says about this pythagorean discovery,

[The pythagoreans] sought to express their conviction that firstly, it is better to conceal (or veil) every surd, or irrational, or inconceivable in the universe, and, secondly, that the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wanders [thereafter] hither and thither on the sea of non-identity (i.e., lacking all similarity of quality or accident), immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement.

A couple of pages further along, Hodgkin -- still discussing incommensurables -- moves on to Samuel Beckett's masterpiece Murphy:

"But betray me," said Neary, "and you go the way of Hippasos."

"The Akousmatic, I presume," said Wylie. "His retribution slips my mind."

"Drowned in a puddle," said Neary, "for having divulged the incommensurability of side and diagonal."

"So perish all babblers," said Wylie.

"And the construction of the regular dodeca - hic- dodecahedreon," said Neary. "Excuse me."

Hodgkin tells us what this is about, if you care to know; see p. 45.

Here's a partial explanation from another source (citation given below):

One of Samuel Beckett's enduring themes was the search for a suitable language to express that mental, inner realm of experience that lay beyond the rational, a language that did for the written text what irrationals did for mathematics. In Beckett's early novel Murphy (1938), we witness the early stages of his struggle with the problem. ... [T]he narrator tells us that Murphy went to sit at the feet of Neary, "a Pythagorean," who had learned the art of stopping his own heart: "For Murphy had such an irrational heart that no physician could get to the root of it." This is signaled most clearly when we find a direct reference to Hippasos who is drowned in a mud puddle for revealing the incommensurability of side and diagonal. (As usual, Beckett has slightly altered the image to suit his own obsessions--in this case, changing the ocean to the recurrent puddle.)

{Source: Configurations 2.3 (1994) 537-571, via jstor:
Hacking the Brainstem: Postmodern Metaphysics and Stephenson's Snow Crash by David Porush}