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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.

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  • $\begingroup$ Presumibley he means a proliferation of complex symbols often difficult to "interpret" geometrically. $\endgroup$
    – Mauro ALLEGRANZA
    Jan 14, 2016 at 19:56
  • $\begingroup$ See here for a comment. $\endgroup$
    – Mauro ALLEGRANZA
    Jan 14, 2016 at 21:04
  • $\begingroup$ @Mauro ALLEGRANZA: your link does not work. $\endgroup$
    – Alexandre Eremenko
    Jan 14, 2016 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". $\endgroup$
    – Alexandre Eremenko
    Jan 14, 2016 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  • $\begingroup$ I don't know if we have an explicit policy, but unless you're asking for the historical content behind the paragraph, and simply a translation, this would seem to be off-topic.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, 2016 at 0:24

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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. $\endgroup$
    – Martin Gisser
    Jan 15, 2016 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, 2016 at 10:39

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