At what threshold can a mathematician truly claim they "understand" a branch of mathematics?
I’ve been thinking about the epistemological boundary of what it actually means to know or understand a specific domain of mathematics. Excluding elementary arithmetic (which relies heavily on procedural computation), higher branches like abstract algebra, differential geometry, or algebraic topology demand a completely different level of mental modeling. Even just calculus, when can we truly claim to understand a subject?
At what point do you consider yourself to have crossed into genuine "comprehension" of a subject?
Is it when you:
- Can construct formal proofs from scratch without consulting references?
- Develop an intuitive visual/geometric picture of abstract axioms?
- Reach the point where you can translate problems into that subject's framework naturally?
- Understand the core motivations behind the foundational counterexamples and edge cases?
- Being able to do math mechanically about a specific subject?
For those specializing in pure or applied math whether it's topology, analysis, abstract algebra, or PDE what is your personal threshold for claiming true fluency in a subfield? Where does mechanical competency end and genuine mathematical understanding begin for you?