Reference: Chapter 1: Abstract Algebra Third Edition, I. N. Herstein, Prentice Hall International Edition:
For many readers/students of pure mathematics, such a book will be their first contact with abstract mathematics. The subject to be discussed is usually called “abstract algebra,” but the difficulties that the reader may encounter are not so much due to the “algebra” part as they are to the “abstract” part.
On seeing some area of abstract mathematics for the first time,be it in analysis, topology, or what not, there seems to be a common reaction for the novice. This can best be described by a feeling of being adrift, of not having something solid to hang on to. This is not too surprising, for while many of the ideas are fundamentally quite simple, they are subtle and seem to elude one’s grasp the first time around. One way to mitigate this feeling of limbo, or asking oneself “What is the point of all this?” is to take the concept at hand and see what it says in particular concrete cases. In other words, the best road to good understanding of the notions introduced is to look at examples. This is true in all of mathematics.
Can one, with a few strokes, quickly describe the essence, purpose, and background for abstract algebra, for example?
We start with some collection of objects S and endow this collection with an algebraic structure by assuming that we can combine, in one or several ways (usually two) elements of this set S to obtain, once more, elements of this set S. These ways of combining elements of S we call operations on S. Then we try to condition or regulate the nature of S by imposing rules on how these operations behave on S. These rules are usually called axioms defining the particular structure on S. These axioms are for us to define, but the choice made comes, historically in mathematics from noticing that there are many concrete mathematical systems that satisfy these rules or axioms. In algebra, we study algebraic objects or structures called groups, rings, fields.
Of course, one could try many sets of axioms to define new structures. What would we require of such a structure? Certainly we would want that the axioms be consistent, that is, that we should not be led to some nonsensical contradiction computing within the framework of the allowable things the axioms permit us to do. But that is not enough. We can easily set up such algebraic structures by imposing a set of rules on a set S that lead to a pathological or weird system. Furthermore, there may be very few examples of something obeying the rules we have laid down.
Time has shown that certain structures defined by “axioms” play an important role in mathematics (and other areas as well) and that certain others are of no interest. The ones we mentioned earlier, namely, groups, rings, fields, and vector spaces have stood the test of time.
A word about the use of “axioms.” In everyday language, “an axiom means a self-evident truth”. But we are not using every day language; we are dealing with mathematics. An axiom is not a universal truth — but one of several rules spelling out a given mathematical structure. The axiom is true in the system we are studying because we forced it to be true by “force” or “our choice” or “hypothesis”. It is a licence, in that particular structure to do certain things.
We return to something we said earlier about the reaction that many students have on their first encounter with this kind of algebra, namely, a lack of feeling that the material is something they can get their teeth into. Do not be discouraged if the initial exposure leaves you in a bit of a fog.Stick with it, try to understand what a given concept says and most importantly, look at particular, concrete examples of the concept under discussion.
Follow the same approach in linear algebra, analysis and topology.
Cheers, cheers, cheers,