680 IX. ON EMPIRICAL AND VERBAL STRUCTURES
some continuous material present in 'space' and 'time' we speak in terms of density. Density multiplied by a three-dimensional volume of 'space' gives us mass, or what appears as its equivalent - energy. From a four-dimensional, or space-time point of view, density multiplied by a four-dimensional volume of space-time gives us action. We see that the multiplication of density by the three dimensions of 'space' gives us mass or energy. A fourth multiplication by the dimension of 'time' gives us mass or energy multiplied by 'time' which becomes action by definition. It is obvious that, structurally, action must be more fundamental than the older quantities.
In terms of curvature, action represents the curvature of the world, because where we find 'action', we also find 'matter', acceleration, gravitation , .4
'Action' is fundamental, because structurally in a four-dimensional metrical manifold it takes the form of the simplest integral invariant that can exist at all. On this form of action Maxwell's electromagnetic theory is built. The quantity action appears as a pure number,8 a unique, specific relation which conditions structure.
We should expect that the action represented by the number 1 would be most interesting and would eventually represent the indivisible atom of action. The modern quantum theory seems to favour such a point of view.
When we encounter a pure number having such crucial significance in this world we should not wonder that such a number intrigues us. As yet it is impossible to state that action cannot have fractional numbers. What, then, would the action represent?
Eddington suggests that the number may represent a probability or some function of a probability.
We combine probabilities by multiplication, but we combine actions in two regions by addition. We see, therefore, that the logarithm of a probability gives the function indicated and Eddington suggests the provisional equivalence of action with the negative of the logarithm of the statistical probability of the state of the world around us. Such a suggestion is extremely appealing and important because the principle of Least Action can be stated as the principle of greatest probability. The laws of nature appear to be such that the actual state of the world is represented by that which is statistically the most probable!6
That such structural conclusion can be drawn at all is of tremendous semantic importance for us because, as we are abstracting in different orders all through, the only appropriate language in which we can eventually hope to speak correctly, is the language of probabilities, statistical averages , .
Action is one of the terms of pre-einsteinian physics which has survived unmodified, the only other one being entropy. The law of gravitation, the laws of mechanics, and the laws of electromagnetism, can all be, not only summed up, but also deduced, from a single principle of least action. This important structural unification was accomplished even before the advent of the einsteinian theory, and only the addition of gravitation to this list is new.7