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There are certain ideas in mathematics which, although very general, may easily be characterized. Some of these ideas are so important that they revolutionize more than one field.

For example:

  1. The idea of looking for conserved quantities. This is used in Hamiltonian dynamics and diffusions/martingales, etc. also we can consider Euler's formula concerning vertices, edges and faces and I'm sure there are many other things.

  2. The idea of creating a new space just so that we can perform an operation on an object which technically doesn't work on the proper object. For example, we couldn't take a square root of -1 so we made the space bigger so that we could do it. Another example is weak derivatives. We made the space bigger so we could differentiate and this idea alone basically created modern pde theory. However it is the same general idea as creating the complex plane.

Obviously there are many other types of ideas in mathematics which may not fit into this context I am trying to describe. My question for this post is what other general ideas like 1 and 2 above can you list.

Joe
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    Mathematics might be said to consist of "very general" (abstract) ideas, some of which are not at all modern (eg. proof by contradiction, using symbols to represent unknowns and "solve" for them) and some which are comparatively recent. I'm not getting a clear notion of what, for your purposes, are "ideas in mathematics which may not fit into this context." If there is a historical distinction to be drawn, you might do better asking about specific ideas at History of Science and Math SE. – hardmath Oct 08 '15 at 23:05

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