Matroids: The Value of Abstraction - Introduction
Posted January 2003.
Many lay people find mathematics intimidating because it is abstract. How does this abstraction come about? One way that mathematics grows is by putting into broader perspective lots of examples which have something important in common. This is one source of the abstraction. The advantage of abstraction is that it enables one to show that something is valid for lots of specific things without having to do a separate verification for each specific thing. When a mathematical concept is central it is also rare that there is a single road to the idea. Rather, like all roads leading to Rome, there are many avenues by which to approach the central, important concept. Unfortunately, for reasons of efficiency, it is sometimes the case that one begins by explaining the abstractions first, rather than going back to the root examples that are being generalized.
Matroid Theory is an example of a part of mathematics that was born by abstraction. The way that the theory grew (putting together separate ideas from different areas of mathematics that were also important in their own right) is a good example of the process of how mathematics grows. There are many examples of matroids: binary matroids, transversal matroids, graphic matroids, rigidity matroids, regular matroids, and k-connected matroids. All of these objects are matroids first, and second, they are an attempt to capture what is special about the class of examples that they were designed to abstract or generalize.
We will begin with two different sets of examples (vector spaces and graphs) which show that there are natural sets of axioms which apply to the above examples. The interesting fact is that by making suitable definitions, these two sets of axioms can be shown to be equivalent. Keep in mind that unlike scientists who must live with the world the way it is, mathematicians create the world they work in. Though the concepts they study may be chosen because they are models (representations) of real world objects, mathematicians still get to choose the way they wish to define terms. When definitions are just right they become standardized and then form the platform for the next series of investigations.
York College (CUNY)
- Vector spaces and graphs
- Multiple births
- The development of a theory of matroids
- Applications of matroids
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