640 IX. ON EMPIRICAL AND VERBAL STRUCTURES |
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term 'absolute' has a meaning; namely, relative, no matter to what; all of which leads to the only content of knowledge - structure.
The whole Einstein theory should, in this sense, be called the 'theory of the absolute', and can be expressed as the simple demand that 'universal laws' should be formulated in an invariant form, a most revolutionary demand and yet so
structurally natural that no one can deny it.When we mathematize or speak about potential or actual measurements, we are dealing with
ordered, extensional, actional, behaviouristic, functional and operational entities, and so we build up a language which at least has a similar structure to the external events. Numbers imply units, quanta, but also order. It seems that number is the only abstraction upon which we all must agree. We never doubt that a statement, such as that 'I have in my pocket five pennies', may be perfectly definite and ascertainable for all. The specific and unique relations called numbers seems to have absolute significance. It must be added that the existence of non-quantitative branches of mathematics does not alter what is said here. In these branches, the asymmetrical relation of order remains paramount and we may treat numbers from either of their two aspects, the cardinal or the ordinal.The epoch-making significance of the Einstein-Minkowski work consists precisely in the fact that they were the first to
apply the above, though without, it is true, formulating the general principle. The lack of such a general, j[, epistemological formulation retards considerably the understanding of their work, and so laymen miss the enormous structural, and semantic beneficial effect upon the proper working of our nervous system and our sanity.Before giving a short methodological account of the Einstein theory it will be well to recall some structural and semantic conclusions which the differential calculus suggests.
When we were dealing with the notion of a variable, we saw that the variable might be
any element selected out of an ordered aggregate of elements. We can select elements relatively widely separated from each other, as, for instance, the numbers 1 and 2, or points, let us say, an inch apart. It is obvious that if we choose, we can make the gaps smaller, and postulate an infinity of intermediate steps. When we make our gaps smaller, the elements are ordered more densely and closer together. In the limit, if we choose indefinitely many elements between any two elements, our series become compact, if we still have a possibility of gaps; or they eventually become what we call continuous, when there are no more gaps.Without legislating as to whether the entities we use in physics are 'continuous', 'compact', or 'discontinuous', we may grant that the maximum elucidation of the above terms in mathematics is very useful. We can easily see that in terms of action a continuous series gives us action by contact, since consecutive elements are indefinitely near each other. As the differential and integral calculus were built on the structural assumption of continuity, the use of the calculus brings us in touch not only with our * but also with its indefinitely close neighbour x+dx. We see that the calculus introduces a most important |
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