The Asthray By Errol Morris Part 3

Le penseur par Rodin, via Wikipedia

The third installment of The Ashtray is online over at the NYT. It’s the longest of the series, until now.

Commensurability or incommensurability is a concept in the philosophy of science. Scientific theories are described as commensurable if one can compare them to determine which is more accurate; if theories are incommensurable, there is no way in which one can compare them to each other in order to determine which is more accurate.

Pythagoras and the Pythagoreans were devoted to a higher spookiness. It is their distinction. With his vein-ruined hands describing circles in the smoky air, Pythagoras has come to believe in numbers, their unearthly harmonies and strange symmetries. ‘Number is the first principle,’ he affirmed, ‘a thing which is undefined, incomprehensible, having in itself all numbers…’ Half-mad, I suppose, and ecstatic, Pythagorean thought offers us the chance to peer downward into the deep unconscious place where mathematics has its origins, the natural numbers seen as they must have been seen for the very first time, and that is as some powerful erotic aspect of creation itself…
David Berlinski, “Infinite Ascent”

There is an anomaly — an inability to find a rational fraction that measures the diagonal of a unit-square. This is followed by a mathematical proof that shows conclusively, irrefutably that there is, that there can be, no such fraction.

It’s a very nice essay, presented in an almost academic fashion. A must for any mathematician.

The Ashtray: Paradigm Shifts by Errol Morris

Errol Morris is at it again with a new essay called The Ashtray. The first part was published yesterday. The second appeared today.

The most important and most controversial aspect of Kuhn’s theory involved his use of the terms “paradigm shift” and “incommensurability.” That the scientific terms of one paradigm are incommensurable with the scientific terms of the paradigm that replaces it. A revolution occurs. One paradigm is replaced with another. And the new paradigm is incommensurable with the old one. He made various attempts to define it — changing and modifying his definitions along the way. In the 1962 edition of “Structure” incommensurability was likened to a Gestalt-flip. Presumably, it was about how we see the world.