TITLE: What was the motivation behing Poincaré saying that Topology was a disease?
QUESTION [6 upvotes]: Poincaré once said that:

"Point set topology is a disease from which the human race will soon recover."

Today we know that he was clearly wrong. What was on his mind when he said that?

REPLY [6 votes]: The purported Poincaré's quotation is about set theory [Mengenlehere].
We can see :

Jeremy Gray, Did Poincaré say “set theory is a disease” ? (1991).

According to Gray, the source is James Pierpont's quotation from a footnote into Otto Hölder's Die mathematische Methode, 1924, page 556.
Gray translate it as :

Poincaré at the International Mathematical Congress of 1908 in Rome said that one would later look back on set theory [Mengenlehere] as a disease one has overcome.


Regarding Poincaré point of view against Cantorism, see :

Henri Poincaré, Science and method (original ed, 1908) :
Ch.I : II. The Future of Mathematics
Ch.II : III. Mathematics and Logic.