Somewhat related is the way one tends to learn mathematics, mostly seeing definitions, proofs, and sequences of lemmas that have been improved on over many deca

Somewhat related is the way one tends to learn mathematics, mostly seeing definitions, proofs, and sequences of lemmas that have been improved on over many decades. When you learn real analysis you get Cauchy’s epsilon–delta formulation mixed with quantifiers, set theory, the notion of continuous functions, and Riemann integration. You don’t need to first learn vague notions of infinitesimals and explanations that seem to make sense but fall apart when you try them yourself. When you learn graph theory, you can use the language of set theory with straightforward definitions of what a graph is, and you can see innovations like the probabilistic proof or subjects like Ramsey theory. When you do applied maths you have tools like contour integration and computers and you don’t need to rely on Euler’s technique of applying the ‘universality of analysis’ (at the time analysis meant what we might now call non-abstract algebra; over history it has meant basically every different topic that is not geometry) and expanding and rearranging infinite expressions and somehow getting the right answer. When you learn Galois theory you already know what a group is, and you have lots of examples and tools you can use. You also know about rings and fields and polynomial rings. You don’t need to simultaneously invent the notion of a field extension, a group, a functor, and a normal subgroup. When you learn group theory you seem to find all these easily-proven theorems with grand names like Lagrange’s theorem or Cayley’s theorem but maybe they are only easy in a modern context. When you learn category theory, most of your examples of categories come from things that you learned from materials that post-date category theory and therefore can have a structure that makes the categories more obvious.