Greetings,
First my compliments and thanks to the people who support and participate in this forum. It was down quite a long time and surely it was greatly missed . Thanks!
My studies in the area of business information systems in last years in a mysterious ways have taken a unsuspected course and I found myself trying to read about various aspect of computational economics and data mining. Books like Peter Jackel's Monte Carlo methods in finance and others have led me to the conclusion I need to know a lot more about mathematics to be able to continue in this direction. The problem was that the only academic-level course I've taken that had to do anything with mathematics was elementary linear algebra (operation on matrices) with little bit of operations research thrown in. It was in my freshman's year so it is thoroughly forgotten. So I decided I will educate myself about as many areas of mathematics and the quick way as in "for dummies" books just won't do. Being too self-minded and too self-confident to ask for advice I did quite a wandering around but to that moment I've read and reasonably well mastered the material in some wonderful books (some of which among other things have quite diminished my self-confidence to the level that I'm able to seek advice now ):
1. J. Keisler, Elementary Calculus: An Infinitesimal Approach (Edt. from 1986)
Found it very easy first book, written with the understanding that the readers have little prior knowledge of mathematics.
2. W.Feller, An Introduction to Probability Theory and Its Applications, Vol. 1
Nothing to do with the easiness of the first one. But still readable and well known book. It even has translation in my natural language and every mathematician around has read it. They said it was tough book and recommended some local "for dummies" books. Didn't follow this advice... Great delays when one is trying to get all proofs (more like arguments) and examples right. I must read it again several times in the near future to be able to pretend that I've mastered it fully.
3. Hrbacek, Jech, Introduction To Set Theory (3rd edt.)
Next I tried to read "Probability & Measure Theory, Second Edition" by R.B.Ash. The set-theoretic notation and concepts seriously scared me. Not that they were complex but because I saw things like that for the first time. Was too big step I gess. So I decided to drop it for a while and see what set theory is by reading one of the two recommended books. This one and the "Elements of Set theory" by "Herbert B. Enderton". I've chosen the first at random and because it looked less intimidating in terms of notation. In term of notation it may be less intimidating but I often found myself going to the other book for easier explanations and proofs. Maybe the whole set theory thing was a mistake - this book took me longer than any other. Proofs are just alien... and beautiful, reading of the last 3-4 chapters is postponed until the need arises.
4. J.Fraleigh, A First Course In Abstract Algebra
Choosed this one, because some notes in the previous book about operators and structures as well as the development of the arithmetic from the axioms of set theory looked super interesting. I'm reading it now. I'm up to cosets and I intend to cover at least the basic course up to ideals, rings, fields and factor rings. When looking at the examples I have feeling I'm not familiar with many phenomenas in mathematics which this book is trying to generalize. It's readable though.
5. D.Poole: Linear Algebra -A Modern Introduction (I thought for some reason abstract algebra comes before linear. Wrong I guess. Stopped reading the abstract algebra just after the material about groups and for a month and a half and started this book. It was great discovery. After reading it I suddenly found out something amazing - most of the cryptic Wikipedia articles on various topics in mathematics and general informatics became much more readable. The applications, chosen to demonstrate the subject - great hit. This book helped me understand mathematical texts much more than a single one of the others did. Should have read it much more early.)
So what should I read next? I hope that the description of my problems with the above books give hint about my level. What do I want?
In the beginning I was only wishing to be able to read Wikipedia articles on mathematics and informatics. Little did I know that they are normally written at quite high level. I'm still want to read them and still many elude me. This is a good example: Ideal lattice cryptography - Wikipedia, the free encyclopedia
Now that I've read some books, I worry that although I know the concepts, I have the feeling that there might be a problem with my problem-solving skills. I follow proofs, sometimes I am able to devise my proofs when this is hinted or required, but in general often have to seek advice. I think it will be difficult for me to translate the theoretical knowledge into computing a real-world problems if the need arises. Not that I'm planning ever to design dam walls or shoot rockets in space. Maybe some creative usage of mathematics and statistics on some dataset. That's all. Maybe it is because I didn't do enough extra problems in the books I've read? Should I next read a second book in some of the areas I've already covered? Or try to find some extra book with tasks and problems alone?
Another thing that I want is to get to the level of understanding to be able to read books and articles that deal with stochastic differential equations in economics.
So....
Could You recommend where and on which books should I focus next:
Should I read second book on linear algebra? First payed off very well in terms of conceptual knowledge. Are there "classic" books in the area?
Second on abstract?
Do differential equations come before texts on number theory?
How about complex numbers, should I go to extend my very basic understanding about them first before starting serious number theory at all?