TITLE: What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
QUESTION [49 upvotes]: I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe that such a fight is winnable?

Question. What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do
  they tend to succeed or fail?

Definitions.

Recent = In the past fifty years.
Highly-entrenched = The literature at the time had the old terminology written all over it.
Succeeded = The new terminology is now used almost exclusively in research papers.

REPLY [13 votes]: From around 1900 to 1970, there was a highly-entrenched practice to write maps as $f:x\to y$ with an arrow between argument and value (e.g. Weyl 1913, p. 54).
Starting in the 1930s, a programme arose to write them instead as $f:X\to Y$, with the arrow now between domain and codomain (e.g. Eilenberg-Steenrod 1952, p. 4).
These conflict: does $f:X\to X\cup\{X\}$ mean von Neumann’s successor map, or an unspecified map between these sets? This can get bad when dealing with sheafs, or localization.
In the late 1950s, some bourbakists (Serre and Grothendieck?) attempted to lift the ambiguity by writing argument-to-value with what they called a tortillon : $x\rightsquigarrow y$. This didn't catch on.
Around early 1964, more bourbakists (Serre and Borel?) tried again, this time with what they called  a poussoir : $x\mapsto y$. This was a hit, and almost entirely displaced the old $\to$ in research papers.
Lesson: enroll Borel?