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triangle can be constructed similar to the given one and of any size whatever.' (Wallis.)
(e) 'Through three points, not lying on a straight line, a sphere can always be drawn.' (W. Bolyai) ,.But the following two assumptions are only equivalent to the
E fifth postulate if we retain the postulate of Archimedes*: (a) 'The locus of the points which are equidistant from a straight line is a straight line'; (b) 'The sum of the angles of a triangle is equal to two right angles.' (Saccheri.)*The crucial point of this discussion is that all that has been said here is
not obvious even to the attentive and intelligent reader, nor to many mathematicians. It took nearly two thousand years and some of the rfforts of the best scientists of the world to discover these connections and implications. The above examples illustrate a general underlying structure of all our languages. They have inherent interconnection, underlying assumptions and implications, the analysis of which, outside of mathematics, is seldom, if ever, carried far enough. Now these structural assumptions and implications are inside of our skin when we accept a language - any language. If unravelled, they become conscious; if not, l hey remain unconscious. In the present work we have already had an opportunity to become acquainted with unconscious implications which are concealed in the structure of any language. We saw that we must start with undefined terms, which represent structural assumptions and postulates, as we have no means to explain them or define them at a Kiven date. We found that these undefined terms represented our unconscious metaphysics, and that the way to make this unconscious metaphysics conscious was to start explicitly with undefined terms, produce a system of postulates., a procedure which is completely fulfilled only in mathematics.It should be noticed (as this is very important) that the undefined terms, being undefined, are overloaded with
'emotional' values. As the higher nervous centres cannot handle them, the lower nerve centres work upon them overtime. If we do not analyse our languages into their undefined terms and structural postulates, our strongest 'emotional' and semantic components, which made these languages, remain hidden and unconscious.♦The Postulate of Archimedes is stated by Hilbert thus: Let
Abe any point upon a straight line between the arbitrarily chosen points _{t}. A and B. Take the points .i,, Aso that _{s}, . . . Alies between _{x} A and A A, between Aand _{x} A. ; moreover, let the segments _{%}AA... be all equal. Then, among this series of points, there always exists a certain point _{lt} A_{X}A_{V} A,A An, such that B lies between A and An.This hypothesis is used by Saccheri in its intuitive forrn; namely, a segment, which passes continuously from the length a to the length b, different from a, takes, during its variation, every length intermediate between o and 6." |
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