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                      MODERN, ABSTRACT MATHEMATICS

The modern mathematics that has come into vogue in the last hundred years or so is distinctly different from the classical mathematics of previous centuries. It is characterized by a very different approach, a different way of thinking, and by an abstractness that sets it apart from classical mathematics. It is a mathematics based on an axiomatic approach: Concepts such as a group, ring or linear space are defined in terms of a set of axioms or postulates. One then proceeds to figure out just how much follows logically from the stated axioms, just as in Euclidean geometry one starts with a stated set of axioms and logically deduces an entire body of theorems that follow logically from that stated set of axioms. The general procedure is indeed essentially the same as that of Euclidean geometry but there is a big difference. In Euclidean geometry one deals with figures in space that can be visualized. He is dealing with objects that are relatively concrete. Intuition plays a big role. Modern mathematics applies the Euclidean geometry technique of logical deduction from a set of axioms to abstract algebraic structures. As a consequence the reasoning and results tend be distinguished by great abstractness. The beginning set of axioms are generally stated in very abstract terms and on first sight one is likely to have difficulty figuring out what their real meaning might be. On first encounter they generally sound like gobbledegook. Generally they consist of a statement of a list of abstract properties or laws of some concrete system that one may not be familiar with. And the statement of the axioms themselves give no hint of what that concrete system might be. They may seem intellectually impenetrable. Generally there is some concrete model behind them that the creator of the axioms had in mind when he created them. For example, in the case of a linear space it is the vector space of three dimensions. In the case of a group the concrete model is the permutation group. In spite of the great abstractness and abstruseness of modern mathematics it can't be denied that there is a lot of power in the method. It can be surprising how much can be deduced from a small set of axioms. Profound results can be deduced. A good example is group theory. The discovery of quotient groups is a profound result deduced directly from the simple axioms for a group.

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