TITLE: How to define the disciplines of mathematics
QUESTION [4 upvotes]: Would you say that it is possible to give a unified, general definition of the different structures of mathematics and draw a clear distinction between them? I have been repeatedly trying to come up with such a distinction myself, but they either all fail to cover the entire discipline, or they end up covering something that is also dealt with in other disciplines. The best one I have myself is

Analysis is the study of the general idea of limits (which is perhaps a bit unprecise), of the real numbers (and those number structures that are based on them), and --- perhaps a bit controversially --- the study of the applications of the Axiom of Choice.
Algebra is the general study of operations on sets.
Topology is the general study of the global structures of spaces and maps between spaces.

I only covered three major branches, and it already starts to go wrong, particularly when you want to draw a distinction between algebra and topology. Both deal with abstract structures in sets in one way or another. Perhaps someone has a better distinction?
Another part of what makes this task difficult could perhaps be that the disciplines are more divided with regards to how they operate than what exactly they deal with. Any thoughts of this?

REPLY [1 votes]: I think those first pages of the Princeton Companion to Mathematics are what you're looking for:
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