| Poincare's remark is quoted more fully and the controversy described
| in Morris Kline's _Mathematical Thought_, p1003 (Oxford UP pb, 1990, v3)
|
| "But it has happened that we have encountered certain paradoxes, certain
| apparent contradictions, which would have pleased Zeno of Elea and the School
| of Megara [...] Later generations will regard [Cantor's] Mengenlehre as a
| disease from which one has recovered." (Poincare quoted in Kline, 1003)
Did Poincare state "Later generations will regard [Cantor's] _Mengenlehre_
as a disease from which one has recovered"?
He might have said it... BUT, it is NOT attested that Poincare state such a
thing, and I would love to be proved wrong!
In *my* opinion, we can only promise ourselves the joy of the doctor called
in to follow another beautiful case of sloppy scholarship!
To our knowledge, Poincare just alluded to Cantorian set theory as "beautiful
pathological case" in connection with his proposal to restrict mathematics to
finitely expressible objects.
Le cantorisme.
"J'ai parle plus haut du besoin que nous avons de remonter sans
cesse aux premiers principes de notre science et du profit
qu'en peut tirer l'etude de l'esprit humain. C'est ce besoin qui
a inspire deux tentatives qui ont tenu une tres grande place dans
l'histoire la plus recente des mathematiques. La premiere est le
cantorisme, qui a rendu a la science les services que l'on sait.
Un des traits caracteristiques du cantorisme, c'est qu'au lieu de
s'elever au general en batissant des constructions de plus en plus
compliquees et de definir par construction, il part du _genus
supremum_ et ne definit, comme auraient dit les scholastiques, que
_per genus proximum et differentiam specificam_. De la l'horreur
qu'il a quelque temps inspiree a certains esprits, a HERMITE par
exemple, dont l'idee favorite etait de comparer les sciences
mathematiques aux sciences naturelles. Chez la plupart d'entre
nous ces preventions s'etaient dissipees, mais il est arrive qu'on
s'est heurte a certains paradoxes, a certaines contradictions
apparentes, qui auraient comblé de joie ZENON d'Elee et l'ecole de
Megare. Et alors chacun de chercher le remede. Je pense pour mon
compte, et je ne suis pas seul, que l'important c'est de ne jamais
introduire que des etres que l'on puisse définir completement en un
nombre fini de mots. Quel que soit le remede adopte, nous pouvons
nous promettre la joie du medecin appele a suivre un beau cas
pathologique." [(*) p. 182]
Here's my translation of the last part:
"...but it has happened that one has encountered certain paradoxes,
certain apparent contradictions, which would have pleased ZENO of
Elea and the school of Megara. And then each one must seek the
remedy. I, for one, think that the important issue is never to
introduce objects that one cannot completely define in a finite
number of words -- and I am not alone in this. Whatever the adopted
remedy, we can promise ourselves the joy of the doctor called in to
follow a beautiful pathological case."
(*) Poincare, Henri:
"L'avenir des mathematiques" (pp. 167-182) in _Atti del IV Congresso
Internazionale dei Matematici_ (Roma, 6-11 Aprile 1908), pubblicati per
cura del Segretario Generale G. Castelnuovo (Prof. all'Universita di Roma),
vol. I (Relazione sul Congresso - Discorsi e Conference), Roma: Tipografia
della R. Accademia dei Lincei (Proprieta del Cav. V. Salviucci), 1909.
PS Incidentally, what do you think of Morris Kline as a historian of
mathematics?
--JGC