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There's a quote by Poincare on the "new functions", such as continuous functions without derivatives, that were appearing during the second half of the 19th century. The fullest version I've found online is in this comment by John Baez on the n-Category Café. A shorter version is on the MacTutor's function concept page, which attributes the year of the quote to 1899.

It reads:

Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner.

In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.

If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum.

Does anyone know the source?

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    $\begingroup$ It appears on pp.132-133 of Poincare's Science et méthode in the 1909 edition Google Books shows in response to the query "musée tératologique Poincaré ". The local Uni library has a copy of a 1999 reprint, which might be related to MacTutor's date. Or Poincaré might have reused this passage. $\endgroup$ Oct 11, 2023 at 3:17

2 Answers 2

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McTutor most likely took the passage from Kline's Mathematical Thought From Ancient to Modern Times, v.3, p.973, they reproduced his translation verbatim. Kline references Poincare's essay Dans la Science Mathématique published in L'Enseignement Mathématique, 11 (1899) 157-62 (Ouvres, 11, 129-34). Electronic version is available from SwissDML.

In 1908, the essay was folded into Poincare's book Science et Méthode. An English translation of it is accessible on Internet Archive and differs from Kline's:

"Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.

Formerly, when a new function was invented, it was in view of some practical end. Today they are invented on purpose to show our ancestors' reasoning at fault, and we shall never get anything more than that out of them.

If logic were the teacher's only guide, he would have to begin with the most general, that is to say, with the most weird, functions. He would have to set the beginner to wrestle with this collection of monstrosities. If you don't do so, the logicians might say, you will only reach exactness by stages."

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  • $\begingroup$ For completeness, the excerpt in question is found in chapter II of part II of Science and Method. The English translation quoted here is by Francis Maitland. In the 2001 Modern Library edition The Value of Science: Essential Writings of Henri Poincaré, the quote is on page 447. $\endgroup$ Oct 11, 2023 at 15:59
  • $\begingroup$ @Conifold. Beautiful. Thanks! $\endgroup$
    – JMJ
    Oct 11, 2023 at 19:07
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This text appears in Poincaré's 1909 Science et méthode, starting on p.132. I do not know if he published this passage earlier.

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