Source of a Quote by M. Stone on Poincaré and Bourbaki

Question

The quote in question is the following:

For Bourbaki, Poincaré was the devil incarnate. For students of chaos and fractals, Poincaré is of course God on Earth.

The common reference for this quote seems to be MacHale's Comic Sections: The Book of Mathematical Jokes, Humour, Wit and Wisdom (p.145) (see e.g. https://mathshistory.st-andrews.ac.uk/Biographies/Stone/quotations/ or Eisner, Farkas, Haase & Nagel's Operator Theoretic Aspects Of Ergodic Theory (p.9) available at https://www.math.uni-leipzig.de/~eisner/book-EFHN.pdf).

I would like to know if there is a written reference for this, or else if it can be verified otherwise. MacHale's book doesn't seem to list references.

Thank you for your time.

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I should add that before posting the question I did a fast reference chase which led me to Mandelbrot's Fractals and Chaos: The Mandelbrot Set and Beyond (p.280) (this part seems to be from a reproduction of an earlier Intelligencer article of Mandelbrot, titled "Chaos, Bourbaki, and Poincaré"):

Do not forget that for Bourbaki, Poincaré was the devil incarnate, who had left behind a mess of unproven assertions and loose ends. They boasted that they had cleaned up that mess. Of course, Poincaré has long been a source of concern to French mathematicians. In the 1880s, Hermite kept writing to Mittag-Leffler to complain that young Poincaré never completed a proof. For students of chaos and fractals, Poincaré is, of course, God on Earth.

In light of this the question remains: either the quote is being misattributed to Stone (which seems likely to me, given the bombastic language (I gather from the rest of the book that Mandelbrot really disliked Bourbaki)), or Mandelbrot is quoting Stone. In this book Stone's paper "The Revolution in Mathematics" is listed as a reference; though as far as I can see Stone does not mention Bourbaki nor Poincaré there.

Answer 1

This does not directly answer if the quote "For Bourbaki, Poincaré was the devil incarnate" is from Stone or Mandelbrot (although I would bet on the latter), but only points out the complex relation between Bourbaki (I mean, the young French mathematicians of the 30s) and Poincaré's heritage.

In the the imaginary biography of Bourbaki Notice sur la Vie et l'œuvre de Nicolas Bourbaki (Notice sur la Vie et l'œuvre de Nicolas Bourbaki), probably written under Henri Cartan, we read that in 1906 he gets a scholarship in order to study under Poincaré and Hilbert.

On the other hand, the idea that Bourbaki was born in reaction to the stagnation in French mathematics caused by the First World War and Poincaré, is essentially due to Raymond Queneau (here the original handwritten text of its paper Bourbaki et les mathématiques de demain. Critique 18(176), 3-18 (1962)):

Il ne faut pas croire d’ailleurs que, merveille génétique, Bourbaki se soit engendré lui-même. Son père même, on peut le nommer, c’est Hilbert, et il eut comme nourrice, entre autres, Emmy Noether, et ses parrains furent quasiment tous des étrangers [...] Poincaré a joué un mauvais tour à la mathématique et à la philosophie française en raillant le grand Péano et sa definition "logistique" de zéro [...] son autorité a détourné les mathématiciens français des recherches logique [...] La mathématique fraçaise s'est trouvée, et par le fait de la guerre de 14-18, et par l'influence de Poincaré, en plein crise gérontique. [...] Vers les années 30, quelques jeunes mathématiciens ont pris conscience du retard pris par les mathématiques françaises non seulement dans l’enseignement (il ne cesse de l'être que depuis peu) mais encore dans la recherché; et ils créèrent Bourbaki; et depuis 1945, trois Français ont eu la médaille Fields, sorte de prix Nobel des mathématiques, Schwartz, Serre et Thom.

i.e.

Moreover, one must not believe that, as a genetic marvel, Bourbaki generated himself. His very father, we can name him, was Hilbert, and he had Emmy Noether as his nurse, among others, and his godparents were almost all foreigners [...] Poincaré played a bad trick on French mathematics and philosophy by mocking the great Péano and his "logistic" definition of zero [...] His authority diverted French mathematicians from logical research [...] French mathematics found itself, both because of the 14-18 war and because of Poincaré's influence, in the midst of a gerontological crisis. [...] Towards the 1930s, a few young mathematicians became aware of the backwardness of French mathematics, not only in teaching (which has only recently ceased to be the case) but also in research; they created Bourbaki; and since 1945, three Frenchmen have been awarded the Fields Medal, a kind of Nobel Prize for mathematics, Schwartz, Serre and Thom.