Here are some criteria. The first three are extracted from a comment Gowers made on Scott's Blog. I do not know how many a result has to have to be a breakthrough or even if such a threshold makes sense. Perhaps some sort of weighted sum, but would be hard to define.
- The result breaks a long standing barrier.
- The techniques introduce a fundamentally different method.
- The result is practical (or close to it).
- The problem being discussed is important. This may be a bit circular in that it then depends on What is Important?
- The result has to make substantial progress on the problem. This may depend on What is substantial? That notion may depend on how long the problem has been open.
- There has to be a reason why the problem was thought to be hard. E.g., a proof that a new technique is needed, problem has been open for a long time, people in the field say its hard.
- A paper that STARTS an important field could be a breakthrough. Cook's Theorem and Valiant's PAC learning qualify here.
- A paper that FINISHES a field could be a breakthrough if the field is important. Again this may be a bit circular in that it then depends on What is Important?
- The techniques are new. One might end up debating what is new.
- The techniques can be used on other problems.
- The paper inspires other papers. For this criteria you need to wait a few years. Some papers are not appreciated for a while.
Do you have other criteria, examples, counterexamples, constructive criticism of my criteria, or anything that is an intelligent contribution to this topic? Is so, please post a comment!