Independently Learning Mathematics

[EOB]

I realize it's a very broad statement to say that I'm independently learning mathematics because I am definitely not learning all of mathematics. I am currently studying the calculus and of right now via M. Spivak and have also been consulting R. Courant's Differential and Integral Calculus, vol 1.

I have zero formal education in the calculus or any other higher mathematics for that matter. I am now in my early twenties and beginning a "formal" learning of mathematics on my own. I was planning on finishing at least one of the aforementioned works on the calculus and then finish reading the following works regarding analysis. The books that I have planned on consulting regarding this endeavor are Analysis 1 & 2 by Einar Hille and Foundations of Modern Analysis by Jean Dieudonne (I do have copies of all of the works that are listed in this query.)

When I'm finished with those works I am hoping that this will give me sufficient preparation to finish reading E. Bishop's Foundations of Constructive Mathematics and A.A. Markov's The Theory of Algorithms. Regarding all of the previously mentioned titles, I have read parts of them and I believe that I have been reading them in an "incorrect" order. I'm feeling overwhelmed as to whether or not my approach is "correct" because minus the works on the calculus alone, I comprehend very little of the other works listed. Thus far all of the aforementioned works have taxed my mental faculties to such an extent that I could no longer pursue the reading of some of them because I do not have the required background to understand much of the content contained within them and I do not want to develop a lack of appreciation of the information that these books contain if I were to continue reading them without really understanding what is being discussed. Any advice on how I should learn formal mathematics and whether the order in which I am currently studying in order to ultimately study constructive mathematics is the "right" way/a way that will allow for that goal to actualize will be greatly appreciated. Thank you for your time/thoughts/recommendations. :)

Regards,

EOB

 

[CJ2116]

As someone who self studies math/physics (admittedly, not in a highly disciplined fashion, unfortunately!) I can tell you right off the bat that you are expecting too much of yourself, too quickly. By starting off that advanced, you are building on a shaky foundation and may be liable to get discouraged and give up. Or worse yet, misunderstand topics and have it come back later to haunt you (not good if you're trying to prove something logically!)

Spivak is a great book and it helped me tremendously when I took real analysis in college, but this was only after a few years of building up to it. Start with a basic calculus book used in college calculus (the 9th edition of Finney and Thomas is an excellent and cheap book) and work through it methodically and carefully - the "brute force" computation not only enhances what is going on in proofs, it builds problem solving skills. Then, afterwards, go to the more elegant formulations of calculus!

Hopefully this helps a little!

 

[IGU]

Have you looked at the How to Become a Pure Mathematician (or Statistician) page? As a quick reference, it gives you some idea of what areas of study come before others.

 

[EOB]

CJ2116 and IGU:

Your advice falls on ears that are not deaf and I thank both of your very much for your recommendations. I'm currently perusing the information contained in the redirect link provided by IGU. I thank you both for your time and your recommendations!