Models in the Theory of Several Complex Variables

concept has become familiar, the model is regarded as a "realization" of the concept. The most important disadvantage of diagrams is that they are not able to describe the manifold completely, e.g. to indicate whether a circle is to be taken with its boundary or without it. Because of this during the transition from the diagram to the abstract notion we often commit mistakes by attributing to the manifold properties which are possessed only by the model. A means of checking our conclusions would be to repeat our reasoning but now operating exclusively with abstract symbols, e.g. of the type indicated earlier in this section. This situation necessitates a certain kind of training and particular flexibility of mind, enabling one to pass from an intuitive model to an abstract formulation of notions and back. These difficulties arise naturally in operating with models of four-dimensional space; in addition to this there occur further difficulties, which we shall indicate in what follows. 4. The advantage of models in teaching and research. Despite the difficulties mentioned before it seems that in many cases the use of models for a geometrically minded individual is of great value in research. Two important reasons may perhaps be indicated: First, our interest in a thing is determined to a considerable extent by the amount of mental energy which is required for working with it. We involuntarily prefer to think in terms of familiar things which can be conceived without special effort. In other words, the extent of our interest in any subject depends to a certain degree on our previous associations-and here again we see the advantage of being able to use intuitive geometry to help in reasoning about abstract relations. Second, if a relationship is known, it is not too important how we formulate it, provided only that our formulation is sufficiently precise. Research, on the other hand, is the search for new relations and in that case it becomes highly significant in which way our problems are formulated. If we can succeed in stating our problem in terms of a class of objects with which we have gained great familiarity, then we may be able to exploit our previous experience with these things. The form of the problem in the new representation may suggest comparisons and hypotheses and further questions, and our previous experience will enable us to see new relationships all the more readily. Thus, the moment that complex numbers are regarded as points in a plane, the analogy with integrals of real functions leads to the formulation of curvilinear integrals in the complex plane-the most fruitful approach in the theory of functions. Furthermore, even after some fact has been discovered and it has been reduced to abstract form which completely and rigorously describes it, even then, the intuitive processes by means of which we have worked are of importance in further research because of the ready associations and the many problems which in this way arise almost spontaneously. This content downloaded from 157.55.39.180 on Tue, 27 Sep 2016 05:23:13 UTC All use subject to http://about.jstor.org/terms 1946] MODELS IN THE THEORY OF SEVERAL COMPLEX VARIABLES 499 5. Models of four-dimensional manifolds. It would, perhaps, be of interest to illustrate the foregoing by a description of a type of model used for visualization of four-dimensional space.* These models are moving three-dimensional pictures, that is to say time is chosen as one of the coordinates. A four-dimensional domain is then represented as a moving picture. The continuous process is thus recorded by a finite number of "frames," and this last is in turn often replaced by a number of stills. In order to have a better idea as to how intuitive these models are, let us put ourselves in the position of beings ("Flatlanders") who are able to "see" two-dimensional domains but who do not possess intuition for three-dimensional space. xs