TITLE: Can I use Ravi Vakil's way of learning for elementary subjects?
QUESTION [25 upvotes]: I mean you can't learn math in a linear order. Can I just read a paper first on a subject I haven't studied and just work backwards? For example, I have never studied combinatorics but I sort of have a fuzzy idea about the paper on alternating permutations by Stanley.

REPLY [6 votes]: This is too long to be a comment. I address the last paragraph in David Speyer's answer, which has prompted quite a bit of discussion amongst friends in real life recently, mathematical and otherwise.

I think this sort of study requires you to be scrupulously honest with yourself about whether you actually know a complete proof, and not to be satisfied until you do. I've seen a lot of students whose "fuzzy" understanding proved, on examination, to be no understanding at all.

I strongly disagree. I have "faked it until I have made it" quite a bit in my undergraduate career, confused beyond all belief by trying to study mathematics way beyond me that struck me as more interesting than the mathematics I was supposed to learn at the time. It has only been in hindsight I realized I had learned quite a lot, even though at the time I had drawn the very severe criticism and ridicule of elders for:

not "going slow and steady",
not "knowing what I am talking about".

We do not need more students to be repulsed by nor fall in love with the Sylow theorems and very basic results from linear algebra and general topology. This is what has turned off so many of my friends from pursuing mathematics beyond the undergraduate level.
I end with two quotations from some writings about mathematics for a popular level, courtesy of Quanta.
https://www.quantamagazine.org/a-path-less-taken-to-the-peak-of-the-math-world-20170627/

Huh tried to use these lunches to ask Hironaka questions about himself, but the conversation kept coming back to math. When it did, Huh tried not to give away how little he knew. “Somehow I was very good at pretending to understand what he was saying,” Huh said.

https://www.quantamagazine.org/peter-scholze-and-the-future-of-arithmetic-geometry-20160628/

At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which says that the equation xn + yn = zn has no nonzero whole-number solutions if n is greater than two. Scholze was eager to study the proof, but quickly discovered that despite the problem’s simplicity, its solution uses some of the most cutting-edge mathematics around. “I understood nothing, but it was really fascinating,” he said.
So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.”
As Scholze burrowed into the proof, he became captivated by the mathematical objects involved — structures called modular forms and elliptic curves that mysteriously unify disparate areas of number theory, algebra, geometry and analysis. Reading about the kinds of objects involved was perhaps even more fascinating than the problem itself, he said.