This morning I was working on a linear algebra problem for a client that I first solved by doing calculations with indices. As I was writing things up I thought of the phrase “the debauch of indices” that mathematicians sometimes use to describe tensor calculations. The idea is that calculations with lots of indices are inelegant and that more abstract arguments are better.

The term “debauch of indices” pejorative, but I’ve usually heard it used tongue-in-cheek. Although some people can be purists, going to great lengths to avoid index manipulation, pragmatic folk move up and down levels of abstraction as necessary to get their work done.

I searched on the term “debauch of indices” to find out who first said it, and found an answer on Stack Exchange that traces it back to Élie Cartan. Cartan said that although “le Calcul différentiel absolu du Ricci et Levi-Civita” (tensor calculus) is useful, “les débauches d’indices” could hide things that are easier to see geometrically.

After solving my problem using indices, I went back and came up with a more abstract solution. Both approaches were useful. The former cut through a complicated problem formulation and made things more tangible. The latter revealed some implicit pieces of the puzzle that needed to be made explicit.

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John,

How do you feel about Penrose’s tensor diagrams? Are they just a different notation for Cartan’s “débauches”, or do they enable more abstract thinking?