Finally I found the source of the dictum "debauches of indices". It is most often used in singular ""debauch", as in Spivaks's Vol.II p.211. The original is from the first preface in E. Cartan's "Lecons sur la Geometrie des Espaces de Riemann". Alas I don't understand enough French. Can somebody please translate it? (Google translator is worthless.)

Les services éminents qu'a rendus et que rendra encore le Calcul différentiel absolu du Ricci et Levi-Civita ne doivent pas nous empécher d'éviter les calculs trop exclusivement formels, ou les débauches d'indices masquent une realité géométrique souvent tres simple. C'est cette realité que j'ai cherché a mettre partout en evidence.

  • $\begingroup$ Presumibley he means a proliferation of complex symbols often difficult to "interpret" geometrically. $\endgroup$ Jan 14 '16 at 19:56
  • $\begingroup$ See here for a comment. $\endgroup$ Jan 14 '16 at 21:04
  • $\begingroup$ @Mauro ALLEGRANZA: your link does not work. $\endgroup$ Jan 14 '16 at 21:41
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    $\begingroup$ Google translates "debauche" as "debauchery", so it is a legitimate English word. When you look for synonyms, one that seems appropriate is "orgy". $\endgroup$ Jan 14 '16 at 21:44
  • $\begingroup$ I don't know if we have an explicit policy, but unless you're asking for the historical content behind the paragraph, and simply a translation, this would seem to be off-topic. $\endgroup$
    – HDE 226868
    Jan 15 '16 at 0:24

superabundance of indexes There is a triple negation "ne doivent pas nous empêcher d'éviter" which makes the text hard to grasp, that is ../they/ should not prevent us from avoiding, but the sense is:

"Notable oppurtunities that we have received and will continue to receive further from le Calcul différentiel absolu de Ricci et Levi-Civita should not prevent us from avoiding purely formal calculations where a superabundance of indexes hides an often simple geometrical reality."

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    $\begingroup$ Thanks a lot! But, is it "purely formal calculations" or is it "purely formal calculi" (i.e. methods of calculation like the one of your untranslated part, "the absolute differential calculus of Ricci and Levi-Civita")? - (P.S.: I guess you're right, will mark as "answered" tomorrow.) -- I'm not surprised of the triple negation: From the little I could grasp of his maths this seems typical. $\endgroup$ Jan 15 '16 at 0:10
  • $\begingroup$ Perhaps he could have written "calculi" :) My hunch is that he was thinking of calculations because he says "often" and "a simple geometry". $\endgroup$
    – sand1
    Jan 15 '16 at 10:39

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