442 VII. THE MECHANISM OF TIME-BINDING |
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line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles'. We should note, in passing, that a straight line is
assumed to be of 'infinite' length, which involves a definite type of structural metaphysics of 'space', common to the A and older systems. This postulate of Euclid can be expressed in one of its equivalent forms, as, for instance, 'Through a point outside a straight line one, and only one, parallel to it can be drawn'. Lobatchevski and others decided to build up a geometry without this postulate, and in this they were successful. Let us consider what Lobatchevski did. For this, we go to a deeper level - otherwise, to a higher order abstraction - where we discover that what on his level had been the dropping of an assumption becomes on our deeper level or higher order abstraction the introduction of an assumption; namely, the assumption that through a point outside a straight line there passes more than one parallel line.Now such a process is
structurally inherent in all human knowledge. More than this, it is a unique characteristic of the structure of human knowledge. We can always do this. If we pass to higher orders of abstractions, situations seemingly 'insoluble', 'matters of fact', quite often become problems of preference. This problem is of extreme semantic importance, and of indefinitely extended consequences for all science, psychiatry, and education in particular.The examples I have given show a most astonishing semantic situation; namely, that one question can sometimes be answered 'yes' or 'no', 'true' or 'false', depending on the order of abstractions the answerer is considering. The above facts alter considerably the former supposedly sharply defined fields of 'yes' and 'no', 'true' and 'false', and,. in general, of all multiordinal terms. Many problems of 'fact' on one level of abstraction become problems of 'preference' on another, thereby helping to diminish the semantic field of disagreement.
It is interesting to throw some light on the problem of 'preference'. Which statement or attitude is preferable? The one claiming that Lobatchevski dropped a postulate, or the one claiming that Lobatchevski introduced a new postulate ? Both are 'facts', but on different levels, or of different orders. The dropping appears as an historical fact; the introducing as a psycho-logical fact inherent in the structure of human knowledge. The preference is fairly indicated; the psycho-logical fact is of the utmost generality (as all psycho-logical facts are) and, therefore, more useful, since it applies to all human endeavours and not merely to what a certain mathematician did under certain circumstances. |
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