TITLE: Can the order of learning be changed?
QUESTION [12 upvotes]: I have been advised by many people to learn from scratch. So I decided to learn it. But I have the following questions in my mind. 

Can we skip "the Trinity" and learn something else directly? ("the Trinity" refers to analysis, topology and algebra)
Why are these things so important and why are they taught for 3 years? 
Are there any books that give a integrated introduction about this subject and take us to the advanced level?
What are the most important concepts that one needs to master in them? I saw many mathematical problems that are applied, like finding integrals etc. Can they be skipped? 

Thanks a lot.

REPLY [2 votes]: I have come to realize there is a marked difference between a math/scientific education and a liberal arts education which might be applicable to your question. In a liberal arts education, you can hop on the train at any station. There is nothing stopping you from looking at the great problems of Shakespeare long before you confront Chaucer. In many ways this dynamic devalues the liberal education.
Whereas in math, science or technical fields, there is a cumulative progression.This imputes a sense of rigor into the dynamic, as many topics assumes an explicit familiarity with prerequisites - and they mean it. Granted there is something very satisfying in studying, e.g., a number theory text rather than a text with a litany of theorems on commutative algebra. That is, you get to use something you know more so than accumulating theorems. 
But without a sound foundation, you're not in the game. And will at some point wish you had cultivated more capacity when you had the energy and opportunity to do so.