ON LINEARITY |
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The joint effect of
two causes working together is not the sum of their effects separately^{2}, and we need oo-valued causality.Analytically, if we have
linear differential equations and we have oneLinear problems and linear equations play a very important structural role in science and there is little doubt that linear equations preponderate enormously, although many fundamental events cannot be described by such equations. A universe which can be described by linear differential equations of the second order has definite structural characteristicsin the main in rough accord with observation. As such differential equations give us the tendency of a process, we may use them to describe large-scale phenomena by integration, or the statistical phenomena of great numbers.
Unfortunately, the study of non-linear problems is structurally very difficult and largely a problem of the future.
There is one very important point which we should not miss. We know already that there is a fundamental difference between different orders of abstractions. Physical abstractions have always characteristics left out, and our higher order abstractions are further removed from life, but they have all characteristics included. The problem of sanity being a problem of adjustment, we must somehow correlate these abstractions in which characteristics are
left out with those which include all characteristics, and so must proceed by approximations. Mathematical methods, particularly those of the differential and integral calculus, have evolved the best technique of approximation in existence today, which, as we have seen, is strictly connected with linearity or additivity.A similar urge which prompted us in the expression of our additive tendencies and methods in the structure of language, has led to the production of the calculus. For organisms which abstract in so numerous and such different orders, the methods of the calculus are therefore fundamental psycho-logical devices, conditioning sanity.
In conclusion, we should notice two quite important facts. One of these is that the nervous system, being in a state of nervous tension, cannot structurally be a simple additive affair in all its functions, a fact which every one of us has experienced. Too many stimulations dull, or abolish, or change reaction in an enormous variety of ways. Pieron, as a result of experimenting in association, has not only shown the complexity of these processes, but also reaches the conclusion that the associative connections are non-linear. ^{3} The other most important point is that structurally the term 'and' implies addition. When we confuse orders of abstractions or levels of analysis, the 'and' additive implications falsify the issues. Thus for instance two atoms of hydrogen and one atom of oxygen and a spark produce water. The second 'and', at least, is used illegitimately, as it applies to an entirely different level (the spark) from that of the atoms. Linguistically we introduced additive implications, |
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