ON GEOMETRY |
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One of the main difficulties is that the structure of this world is such, that it is made up of absolute individuals, each with unique relationship toward environment (in the broadest sense); and we have to speak about it in terms of generalities. 'Laws'. , formulated in the
old two-valued ways, can never account adequately for the facts at hand, being only approximations. The mathematical methods which have already been explained give us at once a great advantage. We have seen that if we have a function, y =/(x), let us say, and take the graph of this function, to every point of the graph there corresponds a pair of values x and y. We have seen also that each of the four quadrants I, II, III, IV has a characteristic pair of signs. In quadrant I, both x and y are positive; in II, * is Ynegative and
y positive; in III, both * and y are negative; and finally, in IV, * is positive and y negative. We can easily see that the value of the variables may be thought of as variable conditions different for each individual, and that definite localizations correspond to them. In our example we had to do with a function of one independent variable, and we had a one-dimensional line, curved in two dimensions. When we had a function of two independent variables we had a surface, which in general was curved in a thirddimension. By analogy we may pass to any number of dimensions, where by dimension we do not mean anything mysterious, but roughly the number of variables involved in the problem.
We see that if we think of the activity of the nervous system in terms of a mathematical function with an enormous number of variables, we shall not only have place for the uniqueness of each individual, determined by the value of the variables and the character of the function, but that this would also imply a
localization, which is permanent in a given individual at a given 'time'; which again implies the totality of 'circumstances',. Our function would be N"f(x_{u} x_{2},x_{t}, . . .x_{n}).In fact it is hard to see how it is possible to analyse the activities of the nervous system in any other way. The facts are, that every organism is an individual, distinct and different from others, and so we must have means to take this individuality into account. Different values for different variables take care of this point. Similarities are accounted for by the general structural character of the functions. For instance, any quadratic equation with two unknowns gives us a conic section. An equation of the type
yrepresents a parabola, the graph of any equation of the form ^{i} =ax xy =o represents a hyperbola ,. For every definite set of values of our variables the implied localization is also definite, which corresponds to the fact that in a given individual at a given 'time'. , the localization is definite. One value for the whole function can be |
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