Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a problem and then ask, ‘What if I generalize this a bit?’
Other times they arise only after struggling with a subject and realizing you were looking at it from the wrong angle. An appropriate definition can then make all the difference, by reorganizing the thought and sheding light into the problems, somehow making them sharper and more focused.
I would like to collect evidences and instances of this idea. An answer should be a story of how someone came up with a good definition and how this was crucial to their understanding of a topic. If you talk about someone else, then ideally provide a reference.
(My interest in this is mainly psychological, namely, how the action of naming something somehow brings it into existence and organizes the world around it.)
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It was suggested that this question is a duplicate of (Examples of advance via good definitions), which asks about good definitions. It was pointed out there, and also here, that basicaly all notions that stood the test of time qualify as good definitions.
I was not asking, although it seems many people construed it this way, for a collection of good definitions, but for a collection of stories that showed how a proper definition actually changed the perception about a field.
A typical such story would have someone saying "Wait a moment! I should not be dealing with [concept A] at all! That's the wrong way to approach this. Instead I should define this other guy, [concept B], and then everything will make a lot more sense!"
I realize this is hard to make precise, so I understand if the question gets closed.
Best Answer
A famous definition which led to a completely new point of view is the definition of Schwartz distribution. It changed the understanding of what a "function" is, even among engineers.
Actually, the definition of function by Dirichlet in 19th century also clarified many things.