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An Introduction To Non-aristotelian Systems And General Semantics.

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504          VII. THE MECHANISM OF TIME-BINDING
'emotional' with the 'intellectual', which was hampered, to say the least, by the old el languages and methods and systems. The organism by structural necessity acts as-a-whole, but the old elementalism with its psychophysiological effects prepared the semantic background for split personalities, which a non-el system helps to re-integrate. We find a rather astonishing result; namely, that the structure of human achievements corresponds to the principle of stratification with the resulting consciousness of abstracting, usually limited to the special field. The majority of individual and group difficulties are found, also, to be due to the very general disregard of this principle.
By the scientific data of 1933 it seems well established that the enlargement of the field of 'consciousness' is extremely desirable. With this aim, a more general enquiry into the character of the 'unconscious' may also be worth while. Let us investigate the structure of science (1933), and see if some 'unconscious' factors cannot be found there. We find a curious fact, that mathematicians, in addition to their other activities, make a business of unravelling hidden unconscious assumptions. Their enquiries have led to a thorough investigation of the structure of their language in two directions: one, investigation of the underlying assumptions; the other, working out the 'implications'.
Let us give a simple structural example of this. Two assumptions are said to be equivalent when each of them can be deduced from the other without the help of additional new assumptions. For instance: (a) The fifth postulate of Euclid - 'If a straight line falling on two straight lines make 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 the two right angles', (b) 'Two straight lines parallel to a third are parallel to each other', (c) 'Through a point outside a straight line one and only one parallel to it can be drawn'. Each assumption silently, unconsciously, presupposes the other, so that they can be deduced from each other. They actually are different forms of the same propositional function.
Another case is equivalence relatively to a fundamental set of assumptions A, B, C,. .. M. It might happen that, in diminishing the fundamental set, two assumptions which were formerly equivalent cease to be so. For instance, the following assumptions are mutually equivalent and also equivalent to the fifth postulate of Euclid, (a) 'The internal angles, which two parallels make with a transversal on the same side, are supplementary.' (Ptolemy.) (b) 'Two parallel straight lines are equidistant.' (c) 'If a straight line intersects one of two parallels, it also intersects the other.' (Proclus.) (tf) 'A triangle being given, another