How does new math get invented/discovered?
In: Mathematics
People solving problems, people making new problems, people forming theories and definition, constructing new objects, and so on.
Where do people get problems from?
1. External motivation. Some real world problems get modeled into a math model. Many questions in differential equation started with this.
2. Intrinsic interest. Sometimes people just get curious. This usually applies to many basic question in number theory.
3. Conjecture. People suspected that something might be true, based on heuristic reasonings, and want to confirm or deny it.
What about theories and definition and new objects?
1. Definitions can be motivated by a problem being solved (gives a simple name to a complicate-to-describe idea, or constructing a new object), or part of a theory. Occasionally it’s also an attempt at solidifying an intuitive but imprecise concept. Definitions might persist beyond the solution of a problem if it turned out to be useful enough, for example it might define a new object with helpful property.
2. Theory comes from attempt at consolidating or generalizing previous math knowledge. They set up a framework, defining new objects, and conjectures about them.
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