ON LINEARITY |
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this difficult structural and semantic problem of linearity versus non-linearity, of additivity versus non-additivity?
Indeed the problems demanding our attention are extremely baffling and difficult. Even in such a perfected science as physics, we have great difficulties in using non-linear equations, and are still at the stage where we solve few equations other than linear ones. To make any progress at all we must start with the
simplest available problems in this field; namely, mathematical problems. The main point at this stage is not a solution of the problem but its formulation. When formulated and brought to the attention of mankind, there is no doubt that it will be eventually solved.To better understand the additive principle, let us consider a group of elements, the individuals of which we denote by letters,. Let us take
two or more of these elements and produce a synthesis which results in a third orentity. Let this synthesis be of such a nature that the characteristics
ascribed to the elements are also present in the resultant synthesis, in other words, let them have the so-called group characteristic. If our elements are, for instance, numbers, the new synthesis is also a number and belongs to the original group. We must notice that the problem of
order is important in the formulation of the additive principle. If a and b are the two elements the synthesis of which we define, we must be clear that a first and b second, or b first and a second, must be recognized in the synthesis. Let us assume also that only the two alternative orders a and b, or b and a, are of importance in this case. The commutative law asserts that a plus b is equal to b plus a, which means that the two possible alternative orders give equivalent results. We must notice that this does not mean that order does not enter into this synthesis; in such a case the above mentioned commutative law would make no assertion at all. It is of importance that order should be involved in the synthesis. It is a matter of indifference only as far as equivalence by a commutative law is concerned.We should notice for our purpose that the synthesis has the 'same' characteristics as the elements had. In other words, if we know the characteristics of the elements we know the characteristics of the result. For instance, if the elements were numbers, the result will be a number, and no characteristic absent in the elements will appear in the result. This predictability from the characteristics of the elements to those of the result is perhaps one of the most striking characteristics of additivity. On the one hand, it allows us to foretell the future; on the other hand, it limits considerably the applicability of the additive principle. It is obvious that when we combine elements, and the results have
new characteristics absent in the original elements, the new problems are structurally no more of an additive character, and the synthesis must be different.Only a few of the simplest entities in physics possess additive characteristics. If we take, for instance, 'weight' or 'length' or 'time', we see that these units are additive. One pound, or inch, or second, if added respectively to one pound, or inch, or second, gives us two pounds, or two inches, or two seconds. |
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