I'll accept the premise, but I still wonder if there are things that can be done to make it easier for someone. In my case, I've been trying to learn some more mathematics recently, and one of the most annoying things is coming across notation that isn't defined in a paper, presumably because "everyone" who can read the paper is familiar with the context and knows what the "skinny long arrow" means (good luck with that internet search). I wonder if there could be a wiki-like / forum / stackoverflowish site, which people could use to discuss and provide running commentary on a paper/book. Especially useful would be the ability for people to be able to annotate the paper by translating the formulas in to a formal language where you could track down the definition of the various operators, and try to figure out why the author used both of → and ↦ in the paper, when they both appear to be for functions/maps. (Just to preempt the easy objections, I'm not trying to suggest that each paper be formalized and proven in something like Isabelle/Coq).
In the ideal form, this website would allow you to see the paper or book page in question, and then see all the people who commented or had questions on each particular sentence (in the margin?). There could be filtering and voting so that experts could bypass the newbie commentary, etc..
I suppose part of my problem would be solved by getting a book like:
...(which I just came across when composing this message).
Maybe someone has a other suggestions for something like this? Maybe a site similar to this already exists?
And on a slightly related note to making things easier to learn, I think learning programming is much easier than math, because even though both are abstract, at least with programming you get a tangible, concrete thing (the program) that you can run and modify and extend, and the computer will tell you when you went wrong (e.g. won't compile, output result is unexpected, etc.).
Unlike mathoverflow, it is meant for every kind of math question below research level.
(Regarding $\to$ vs $\mapsto$, I think of it as type-level vs lambda expression. I think you can find it in any introductory abstract algebra book that assumes you still need to learn a thing or two about functions.)
In my experience, it seems the usual way people in the math community resolve these issues is to ask an expert, or at least a knowledgeable grad student.
Forgive me if I'm making incorrect assumptions about your background, but usually you learn math from books of varying degrees of difficulty which naturally force you to become accustomed to various kinds of notational conventions.
You wouldn't try to learn math from papers until you've built that foundation (unless you have access to a tutor/mentor), at which point the notation usually shouldn't be an issue.
That sounds like the traditional method of learning math. I was wondering if we could leverage technology and our experiences with teaching/learning the formal systems of programming languages to make more math more accessable. For instance, I'm thinking this little instance of geometric algebra:
...might be easier for me to understand if I could use Haskell to implement the wedge and geometric product operators on an algebraic data type describing the scalar/vector/bi-vector thingy. There is probably an applied vs. pure thing here as well. My motivations for investigating geometric algebra is to see if geometric algebra makes synthesizing mechanical linkages easier, whereas maybe most expositions on geometric algebra are focused on teaching geometric algebra to advance the state of geometric algebra. That's probably a long winded way of saying that mathematicans are writing for mathematicians (whether by design or accident). I suppose I should re-read Mindstorms again, but this time in the context of adult learning.
I'm not sure if this is what you're looking for, but I've had this book on my wishlist for a quite a while and it seems to fit: http://www.geometricalgebra.net/
I also what a running commentary would do for authors. Would they get ideas for improving their next paper, by looking at what had people confused? Surprised by who is reading their papers (especially those outside of their field)? Would they merely be horrified by YouTube style commenters?
I'll accept the premise, but I still wonder if there are things that can be done to make it easier for someone. In my case, I've been trying to learn some more mathematics recently, and one of the most annoying things is coming across notation that isn't defined in a paper, presumably because "everyone" who can read the paper is familiar with the context and knows what the "skinny long arrow" means (good luck with that internet search). I wonder if there could be a wiki-like / forum / stackoverflowish site, which people could use to discuss and provide running commentary on a paper/book. Especially useful would be the ability for people to be able to annotate the paper by translating the formulas in to a formal language where you could track down the definition of the various operators, and try to figure out why the author used both of → and ↦ in the paper, when they both appear to be for functions/maps. (Just to preempt the easy objections, I'm not trying to suggest that each paper be formalized and proven in something like Isabelle/Coq).
In the ideal form, this website would allow you to see the paper or book page in question, and then see all the people who commented or had questions on each particular sentence (in the margin?). There could be filtering and voting so that experts could bypass the newbie commentary, etc..
I suppose part of my problem would be solved by getting a book like:
https://www.amazon.com/Mathematical-Notation-Guide-Engineers...
...(which I just came across when composing this message).
Maybe someone has a other suggestions for something like this? Maybe a site similar to this already exists?
And on a slightly related note to making things easier to learn, I think learning programming is much easier than math, because even though both are abstract, at least with programming you get a tangible, concrete thing (the program) that you can run and modify and extend, and the computer will tell you when you went wrong (e.g. won't compile, output result is unexpected, etc.).