mathwonk said:
lighten up. life is long. youll be fine.
Fizix, Mathwonk has just generously given you some of the most profound advice I have ever heard given to another person. Indeed, if it was a theorem, it would produce 1,000 more theorems.
Please let this post and its length serve to you as an abstract template of how mathematics/ physics is truly done. Mathwonk dropped the nugget. Here comes the depth behind it. The same way math is really done.
Lets get some things straight. You're very young and it appears you're very bright and beyond your years. A sincere congratulations. And I definitely applaud your efforts and achievements. But. (Here comes the but)
Life is infinite dimensional young friend, and not discretely either. With that, you may have stumbled upon the interconnectedness of everything. I am not being pretentious, I mean what I say. It is all interrelated.
Within that interelation of all things is
your youth. Yeah yeah, sounds like parental rhetoric, I know. It's not.
My friend, you may have a great superficial understanding of all these subjects and indeed that is a very worth while and meaningful understanding. With it you can go deeper into everything. And you'll find many an academic in college who is sorely lacking that understanding - which often causes all kinds of problems for everyone.
But, with all due respect, it is not likely that you could pass without books any serious college level test administered by Mathwonk or I or any other older adult.
I absolutely don't mean to dampen your spirits, just a caution to the wind and a possible challenge that you may not being going to the depths truly required in understanding these texts.
My advice young friend is to honestly slow down and peer into the depths of what you're learning. I was once told if you spend less than an hour on a page of mathematics you've gone too fast. And with my experience, I'm going to tell you that is absolutely true.
There is a difference between reading a theorem knowing vaguely what it means - vice truly deeply understanding a theorem; what it does, where it goes, came from, and is going to be.
Moreover knowing a theorem implies I could ask you anytime to go the blackboard and proof it - while being able to explain in painful detail the necessity of every line in the proof.
They are different understandings my friend. And my advice is to make sure the later is the priority not the former.
Mathwonk said in that dense nugget these things. Stop and think about it friend. Truly, stop, and walk into the unknown without the aid of a book. That is a real scholar. A real scholar does not consume book after book after book without spending inifinitely more time in between books and pages and theorems looking at the depths of what was just been said.
So here's my 2 advices in short for you.
1) For every theorem you read, spend at a minimal 1 hour trying to do something with that theorem - without the aid of a book. Just a pencil and a blank piece of paper. Do something with it and then it yours for life. But don't just do trivial things or simple things, really truly take it to the depths.
2) Get in touch with the philosophy of math and physics. This is where you should spend your next dollars on dover books. Here you will find true power in mathematics and physics. Not knowing and not being able to communicate and think about the philosophy of these subjects relegates you to a mere foot soldier of those who can. Not the place for a mind like yours.
Answers to your questions.
My advice, take
no classes. That's right, none. You don't need them until you're in college, and you are truly squabbling your intelligence if you do take them. Many many many many students get caught in this "prestige" game. You've been snared in at a young age.
It's been awhile since I was in 10th grade, but I thought I remember getting my driving permit. I speculate that you've skipped a grade or two. Which in of itself is a great ticket.
Here's another reason not to take these classes. No one is impressed. You as a "young'n" are impressed, we who are elder are not. If you wish to impress us, do what I said earlier and take one of these subjects to its depths on
your own time and learn to enjoy your youth. Moreover, it would be more impressive if you studied something else other than hard science and mathematics.
But at the days end, you're a fool to try to impress other fools. Indeed you should seriously only do any of this for yourself. Why not then truly take these subject to their depths?
Question: How many languages do you know
natively?
That is the point of this whole post. It's great you can say "what's up" in 20 languages and can order food in 10. But you still only speak then 1 language. The point is to actually take the time and truly learn, to the native level a language or a single subject of mathematics. Then and only then will you have impressed someone. Otherwise you only fool yourself my friend.
Dover books.
Dover books are great, but they only go so far. They're usually (always) outdated works. Not to mention they are often written in ancient tongues. But they're a great place to start.
Fundamental Concepts of Algebra by Bruce E. Meserve (Not what you think)
Challenging Problems in Algebra by Alfred S. Posamentier and Charles T. Salkind
Problems in Group Theory by John D. Dixon
The next two are highly recommended for you young friend.
Introduction to Mathematical Philosophy by Bertrand Russell (5 stars on a 4 star scale)
Physics and Philosophy by Sir James Jeans
The only one of these I have honestly broached but not finished is Russell's. Some I have tried problems out of, but they are very difficult. But that IS the point. Challenge yourself and forget trying to impress people with what college courses you've taken online early.
Do yourself a huge favor now while it is still easy, and ignore this game of prestige and who has done what - and truly learn something on your own. I believe you can do it. You sound like you believe you can do it. Why aren't you doing it?
My concluding remark is this, and please listen up:
If you, Fizix, are destined
to be a great mind (like Einstein), then the courses you suggest and the books you mentioned are worthless. They are absolute garbage and with bare minimal resources you will find a way to surmount the heights of great mountains on your own.
I say, you sound like you might just be one of those minds, so then, get started with doing this. Take a single subject to its depths and take the long haul up a mountain to your throne.
And a final note. It took Mathwonk one minute to type that dense nugget. It took me over an hour to compose this post. So too is the way of all theorems in mathematics.
] on my own. Specifically, I am using Shilov's Linear Algebra as well as Shilov's Real and Complex Analysis. I'm using Hocking and Young for Topology, and Deskins' Abstract Algebra. Are these good books? What other books would you recommend that are relatively cheap [I'm talkin' Dover price ranges. 
Perhaps there are some good "real" Statistics books out there? 
