Introduction to Mathematics in Words of a Heretic
The Core of Mathematics
Different people see it from very different angles. I see it from an angle of a weird, unpopular, crazy, software developer, who's views are not recognized even by other software developers. Professional mathematician like Bertrand Russell have a different view to the topic.
As of May 2013 my view is that absolutely all of Mathematics, regardless of theory, is about transformations, i.e. how to transform one thing, let's say, $A$ as $(7+3)$, to another, let's say, $B$ as $(10)$.
Sometimes the transformations can be done only in one direction, e.g. $A -> B$. For example, the reminder of 99 when divided by 5 is 4, where 4 is the $B$.
Sometimes the transformations can be done in both directions. For instance, $(7+3)$ can be transformed to $10$ and $10$ can be transformed to $(7+3)$.
Sometimes there is more than one way to perform the transformation. For example, $10$ can be transformed to $(7+3)$ and $(6+4)$ and $(5+5)$ and $(1+9)$ etc.
Transformations can form series, like $(10)$, $(7+3)$, $((4+3)+3)$, $(((2+2)+3)+3)$, etc.
In terms of mathematics skills the only difference between a 2nd-grader and a mathematics professor is the amount of different transformations that they can perform and the ability to see the greater picture. The 2nd-grader does not have enough pieces to assemble the whole puzzle, whilst the professor can generalize from a lot of pieces.
If the idea that mathematics is nothing but knowledge about transformations seems too oversimplifying, then one additional supporting argument is that digestion with all its complexity is nothing but conversion of food to "useful nutrients" and that holds for lizards, crocodiles, sharks, cows, humans, mollusks, ants, spiders, plants, jelly-fish, crabs, bacteria and so on and so forth. That is to say, the statement about digestion is a correct summary of the process regardless of its simplifying appearance and there might be other statements that look blatantly simple, but are still exact, not oversimplifying. One of those statements might be the statement that mathematics is nothing but knowledge about transformations.
(I admit that as of 2013 even I am missing a lot of the pieces from the puzzle, but this article does reflect only my views and I never promised to present the ultimate truth.)
A few Words About Professional Mathematicians
The main problem with them tends to be that they hardly ever explain, why the things that they present are relevant to You, why You should care about theory X in stead of theory Y or Z.
The other general problem with them tends to be that they totally ignore the fact that IT IS NOT OK TO PRESENT ONE MATHEMATICALLY CORRECT SOLUTION!!! What's needed is to CHOOSE (or even do some original research to figure out, create) mathematically correct solutions that are tailored, optimized, according to learning psychology. If there are no known transformation paths from form A to form B then coming up with any path from form A to form B is a novel and great contribution to science and human knowledge, but TEACHING IS NOT ONLY ABOUT BEING CORRECT. Teaching is about FACILITATING THE INCREASE OF SKILLS OF LEARNERS!!!
I understand that it's not very highly regarded to find new, novel, solutions to classical problems that have been solved ages ago, but then one might ask, what is teaching about? Computers (historically books) can preserve and convey knowledge better than humans, unless the human teachers/scholars add some extra value to the learning process.
It is practical to bear in mind that often, although not always, the reasons, why the professional mathematicians are not able to do all the necessary work is that they are under-financed and/or over-loaded with work. Specially many university lecturers. There are slackers among mathematicians, including university lecturers, but it is a fact that fine people tend to be really busy with various kind of work and there are situations, where no amount of extra money can buy them more time, specially if they are doing research.
How to Learn Mathematics
As of May 2013 I do not have the answer to it. I just write down the few things that I do know and believe to be true.
The main part of the answer is that in order to gain the ability to notice different transformation paths, one has to rigorously do exercises. There's no need to do them on paper, but one definitely has to do them oneself. Just reading the solutions is not sufficient.
As it's not possible to spend hours or days for deriving everything "from scratch", some intermittent solutions, theorems, formulae, etc. must be known by heart. There's no need to intentionally memorize them, because they will transfer to memory without extra effort during the exercises.
Sometimes it helps, if one uses one's own notation in stead of the classical, official, historic notation. Often the classical notation is misleading and not optimal for keyboard based data entry. The use of a computer avoids the drift of attention that happens during the manual drawing of mathematical symbols.
In old-fashioned lectures one should take photos of the blackboard in stead of writing things down manually.
Keep in mind that the theory that mathematicians produce, is often a form of software that is run only by humans and therefore can run with mistakes without crashing. People do make mistakes even, if they are not sloppy, and that includes mathematicians. Given that the percentage of sloppy people amongst mathematicians is roughly the same as in other occupations, expect to run into horrible mistakes that are visible even to physicists, chemists, IT people, practically anyone who has ever learned any university level mathematics.
Expect mathematicians to use new terms before defining them. That is to say, if You're stuck somewhere and wondering, what does something mean, then just read on and You might find the definition. It might take even many pages to grasp, what they really meant.
Expect different mathematicians to use different assumptions. For example, there are cases, where 0 is not considered a natural number or a vector that has the length of 0 is considered to be an exception due to some direction calculation peculiarities.
If a word that mathematicians use is familiar to You, then do not assume that You know, what they are talking/writing/lecturing about, unless You have checked out the precise definition that the particular mathematician uses in that particular lecture. Mathematicians tend to redefine words. An example: series in calculus and series in statistics and sequence in general.
If a mathamatics text contains a reference to a formula, e.g. "according to (number-bla-bla-bla)", then it often helps to understand the text by actually flipping the pages and taking a look at the formula before reading on.