saucer
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A Good Tautology is Hard to Find!
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 Posted 26/07/2007 12:35:01 AM | | -
Topologically speaking there are few possibilities for solid objects; essentially the only variable is the number of holes. Interesting things happen, however, when we work with abstract objects in more than three dimensions. This is not the dry academic game that it might appear at first sight. Consider the ubiquitous use of graphs in science. One often sees graphs of (say) pressure against temperature. Neither variable is naturally a geometric quantity. Nonetheless, they can be represented by distances on a sheet of paper. Moreover, geometric concepts such as slopes of lines and areas of regions can usefully be interpreted in terms of thermodynamics. In principle, none of these geometric ideas are neccessary; all the arguments can be carried through algebraically without any diagrams. Nonetheless, most of us find diagrams helpful and illuminating, because we have a better intuitive feel for geometry than for algebra. Now suppose that instead of two variables (pressure and temperature) we are faced with a problem involving five or six. We might like to draw a six dimensional graph, but we do not have enough space. We can try to imagine what six dimensional geometry might be like, but it is difficult to have confidence in our intuition about such a theoretical construct so far removed from experience. On the other hand, we can try to do the problem by pure algebra, but in doing so we forgo the visual insight which is one of the most powerful faculties of the human mind. In fact, mathematicians have developed a rather strange yet highly fruitful approach to such problems. The questions addressed are officially formulated in terms of logic and algebra rather than geometry, and solutions are required to be expressed in such terms. Where geometrical concepts are used, they must be defined in terms of algebraic ones. All use of such concepts is to be justified purely in terms of such definitions, rather than in terms of geometric intuition. The visual imagination guides the course of the argument, and suggests what one should try to prove, but the argument must stand in its own right.
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