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IMPORTANCE OF CONCRETE MODELS IN UNDERSTANDING MANY ABSTRACT CONCEPTS IN MATHEMATICS

Often very dark and obscure topics in mathematics which just make no sense at all suddenly make great sense when you realize what that concrete model is that lies behind them.

Examples:

1. In studying linearly independent and dependent vectors and concepts of space, subspace, basis, etc. in matrix theory they all make sense when you consider the concrete model of the meaning of these ideas. What is the concrete meaning? It is these concepts as applied to vectors in two and three-dimensional space. When the concepts are phrased in general, abstract terms with no reference to the underlying model they are dark, obscure and impenetrable. When one realizes their meaning with respect to the vectors of two and three-dimensional space they make great sense.

2. The theory of quadratic and bilinear forms and point transformations can be difficult, obscure and hard to understand until you focus on a concrete problem of plane and solid analytic geometry that serves as a model and brings into clear light what the real problem and objective is and which gives intuition and insight into how to go about solving it. What is this model? In two and three-dimensional space a second degree surface (such as an ellipse or ellipsoid) has a particularly simple form, called its canonical form, when expressed with respect to a particular coordinate system but has a more complicated form when expressed with respect to any other coordinate system. The problem posed is: Given the equation of a surface in some complicated form (which is complicated because it is not referred to its canonical coordinate system) translate and rotate the coordinate system in which the surface is expressed so as to bring it into coincidence with its canonical coordinate system. The object is to re-express the equation with respect to another coordinate system (or "basis", to use the terminology of matrix theory) which reduces the equation to canonical form. The idea is simple when mentally viewed in these concrete terms of two or three-dimensional space but becomes very dark, obscure and difficult to grasp when couched only in abstract notations, concepts and terminologies and no reference or hint whatsoever is made to the concrete model behind it.

The above sort of thing happens all the time in abstract mathematics. Abstractions are, almost by definition, extensions and generalizations that are made from concrete problems, ideas and concepts. When one explains abstract ideas and concepts one ought to go into the concrete problems and ideas from which they arose. Only then will the ideas fall together and make sense. American mathematicians seem to just stubbornly refuse to do this. It is a matter of dogma or philosophy or something. You can look through dozens of books on Matrix Theory and you will never find in one a hint of the concrete problems from plane and solid analytic geometry that I mentioned in the two examples above that would give the student insight and understanding into the dark, obscure, abstract ideas they are expounding. It is as if they do not want to help the student to understand the subject -- as if imparting honest understanding was not their real objective. It is as if their real purpose was to intimidate, stymie, confuse, put down, and "squash" the student. Or to just awe and impress him with their own intellectual prowess. This seems like a rather harsh suggestion. It suggests real malice. But if it is not true one would have to ask another question: Do the professors who wrote those books have a real, honest understanding of the subject themselves? Because, for a person with a real, honest understanding of the subjects, the importance of the concrete models to providing a real understanding of the abstract ideas is obvious.