Yesterday I posted on @TopologyFact
The uniform limit of continuous functions is continuous.
John Baez replied that this theorem was proved by his “advisor’s advisor’s advisor’s advisor’s advisor’s advisor.” I assume he was referring to Christoph Gudermann.
The impressive thing is not that Gudermann was able to prove this simple theorem. The impressive thing is that he saw the need for the concept of uniform convergence. My impression from reading the Wikipedia article on uniform convergence is that Gudermann alluded to uniform convergence in passing and didn’t explicitly define it or formally prove the theorem above. He had the idea and applied it but didn’t see the need to make a fuss about it. His student Karl Weierstrass formalized the definition and saw how generally useful the concept was.
It’s easy for a student to get a warped perspective of math history. You might implicitly assume that mathematics was developed in the order that you learn it. If as a student you learn about uniform convergence and that the term was coined around 1840, you might reasonably conclude that in 1840 mathematicians were doing what is now sophomore-level math, which is far from true.
Gudermann tossed off the idea of uniform convergence in passing while working on elliptic functions, a topic I wasn’t exposed to until sometime after graduate school. My mathematics education was more nearly reverse-chronological than chronological. I learned 20th century mathematics in school and 19th century mathematics later. Much of the former was a sort of dehydrated abstraction of the latter. Much of my career has been rehydrating, discovering the motivation for and application of ideas I was exposed to as a student.
The book “Calculus Reordered” by D. Bressoud discusses mathematics history around this idea. *Very roughly* the development goes like this :
Integration -> Derivation -> Limits and Continuity
https://press.princeton.edu/books/hardcover/9780691181318/calculus-reordered
I can imagine much of the category theory being developed today will eventually be taught before uniform continuity!
I’m tempted to call this “revisionist history” except that phrase implies dishonest intent. A sort of compaction and refinement takes place over time, sorta like metamorphic rock. This makes progress possible.
But it needs to be complemented with some historical perspective. Otherwise you can get the impression that your predecessors were lightweights, or you can get the impression that they were gods. Neither perspective prepares you to advance what they started.
“I learned 20th century mathematics in school and 19th century mathematics later. Much of the former was a sort of dehydrated abstraction of the latter.”
The opposite happens with quantum mechanics/QFT. We’re taught every single historical step along the way and in a way that makes it “weird” and “paradoxical.” We could teach our modern understanding, as is, as a straightforward theory and do away with all the mysterian fetishism.
Nice turn of phrase, by the way.