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An Introduction To Non-aristotelian Systems And General Semantics.

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CHAPTER XXXIII
ON LINEARITY
The conception of linear transformation thus plays the same part in affine geometry as congruence plays in general geometry; hence its fundamental importance. (547)                                                       HERMANN WEYL
It is instructive to compare the mathematical apparatus of quantum theory with that of the theory of relativity. In both cases there is an application of the theory of linear algebras. (215>                        w. heisenberg
This "perturbation theory" is the complete counterpart of that of classical mechanics, except that it is simpler because in undulatory mechanics we are always in the domain of linear relations. (466)            e. schrodinger
As a result of experimental research on association, in 1904, I was led to show the complexity of the factors governing evocation. . . . And I have often insisted since then on this essential idea, in opposition to the simple schema of linear associative connection. (4ii>                henri pieron
We have already had several occasions to mention the 'plus' or additive issues as connected with linearity. This problem is of structural and linguistic as well as empirical and psycho-logical semantic importance. It is sufficient for our purpose at present that we should notice two facts; namely, (1) That in one dimension, linearity expresses the relation of proportionality; (2) That the problems of linearity are dependent on the relation of additivity.
The structural notion of additivity is of great antiquity. Being the amplest of such notions, it naturally originated very early in our history. The earliest records show *hat the Babylonians and the Egyptians used the additive principle in their notation. Our primitive ancestors, long before any records were written, had similar structural conditions present, open for investigation and reflection, that we have today. That this was the case is not a mere guess. Otherwise we would still be at their stage of development. Some beginning had to be made somewhere. There is little doubt that the men of remote antiquity presented many types of make-up, as we do today. Some, for instance, were more curious than others; some more inventive, some more reflective., which, as we know today, is found even among animals. These more gifted individuals were, as usual, the inventors, discoverers, and builders of systems and language of their period. They could not long fail to recognize the fact that a stone and a stone, or a fruit and a fruit are different from one stone or one fruit. For instance, the two stones might have saved the early observer's life in defence, or the two fruits might have satisfied his hunger or thirst, where one would not have done so. An accumulation of objects was obviously somehow different from a single object. As these problems were often of vital importance to their lives, names for such accumulations of objects began to be invented, and one and one was called two, two and one was called three ,. Number and mathematics were born as a structural semantic life-necessity
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