moofs wrote:Yeah, I know. <continuationrant>As has been shown in lots of other fields though, tradition and historic significance aren't necessarily the best. I suppose there are probably sects within the maths field that use more clarified sets of conventions, just grumpy that it's not taught that way (as I really would have liked to have that bit of understanding under my belt).
You would not get a disagreement from me. Freshmen and Sophomore math text books are published at a far lower quality than authors whom write for junior and senior level math books, and then in freshmen and sophomore grad schools another set of authors arise where I got to editions that haven't changed since the 70s. These graduate student books are hallmarks of the introduction into the trade. The level of clarity that is expounded in these volumes is not even comparable with lower level books, let alone freshmen and sophomore texts that really are designed to teach people who are interested in using math, rather than understanding how math really works.
moofs wrote:I think it also bugs me in that I'm used to programmatic functions with the definition and usage more immediately available, i.e. without the need to look up the source code for each function you're using and reimplement it every time you need to solve something (unless you have something like Mathcad), and that the definitions are written in the mentioned obtuse Greek notations, with several things at any given point (in even summaries) not being defined in any one location.
I think the trade off with naming variables is clarity of label versus length of label. It is similar to the trade off between userfriendly vs beginner friendly. I don't disagree with you in the slightest, math notation can be terse, but it's part of the culture.
In your computing example, different functions have standard naming conventions that are exceptionally clear, e.g. hungarian notation, gnu c/c++ style, etc etc. The reason why code is necessarily labelled like this is because code projects can literally be millions of lines long. Along with cognition limitations, you also have internal namespace pollution that needs to be avoided.
In math, proofs seldom go on for more than 20 pages. Referring back to the definitions is something people often do when they start learning math, but one of the skill sets of reading math is maintaining a short term memory of the definitions whilst going through the exposition. It's not something most other fields care for in developing in their students. There is also a particular trend in math books to say things as tersely rather than being verbose, leaving just enough for the student to play with and work out on their own. This approach really encourages students whom are driven to this stuff, but everyone else finds it impossibly obtuse. This terseness is taken to a whole new level for paper publications. I have profs that stay on the same paragraph for a week for something a colleague wrote, which that paragraph can be expanded to a full dozen pages worth of math writing. But then in math journals, space being relatively confined and the author wanting to show off his math eloquence leaves the proof to be really "unencumbered". I understand your conundrum. I learned Math in Hong Kong chinese schools, English schools, Hong Kong private schools that used a french system and then I came to the United States to study latter. Notations moved on me everytime.
moofs wrote:Not presenting all their formulas in the form of proofs would make them more comprehensible as well
The equivalent in programming is a 10,000 line spaghetti code app with documentation haphazardly interleaved over three 40,000 page documents and with several critical pieces omitted. Not only that, but the API (programming interface - like DLL's - functions that other people built that you can reference so everything doesn't have to be rebuilt from scratch every time) would have 14 unit testing functions built in for each function that actually needed to be interfaced, but with no naming conventions and no clarification as to what needed to be used and why.</continuationrant>
Short Version: I don't like mathematics' style of documenting itself because it artificially makes things far harder than they need to be.
As for talking about formulas in the form of proofs, unfortunately, proofs are what make math internally consistent. You've got no way around it. Formulas are basically constructs that are shown to work in a fashion by its construction, the proof. I understand that proofs are really hard to read, and in freshmen and sophomore math, the students are actually not equipped at all to read the proofs, but I don't think it's the fault of proofing. Rather, I think it is because schools want students to learn how to use math to do things, and that knowing what's really going on has became secondary. I spent a year of graduate school classes thumbing through a 350 page book called Principles of Analysis by Walter Rudin, just to learn how to prove everything in Engineering Calculus 1. It took me 3 months at 5 hours a week of classes plus some hw time when I was 18 to learn to do the calculations. Then it took me a whole year to figure out what the hell was going on when I was 27. That's two semesters of graduate courses, about 20 hours a week on each graduate course and when I got Bs in the class I can still pick up that book three years latter and learn something new from it. The good thing down the road is that the sooner you get use to using proofs to get your point across, the easier it is to read them. And using English to describe math gets to be a real hurdle down the road. Using math is just much much better.
As per the proof reads like spaghetti code with linking issues all over the place, such as definitions of presumed knowledge not immediately available, some authors just assume that you know them, by accident. That's why getting a freshmen/sophomore to read what the hell is going on in a probability book and get it is nigh impossible. You're not suppose to figure out what is going on. You just use the methods prescribed and go home. To get what is going on in a junior level stats course, you need graduate school. You will need some probability, and to understand how that probability works, like how those integrals of a normal distribution really go to 1, you need measure theory, which is the next course after the Principles of Analysis course I talked about before. Notation isn't really the hurdle. Most of the time, this stuff is just hard as hell.
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