13. A deep structure for biology

Our focus, as presented here in biological and ecological matters, is admittedly narrow. In the same way, Bjerknes saw a “deep structure” between meteorology and electromagnetism, and he used those observed correspondences to reform his subject. We are looking for a similar deep structure of biology with both the vector calculus and with Newtonian concepts of mass and inertia. We have just observed that a change in energy density is a change in structure and in configuration. It is a change in deep structure. This does not mean that we deny that biological organisms have any other features. It means only that we are looking for a different way of describing them. We are simply abstracting, from biological and ecological phenomena, those features that might allow the model we have in mind to be constructed. We do this with a view to isolating different features; running experiments; taking measurements; and then making predictions, all while using that observed deep structure.

Figure 13: Deep Structure: Some Examples of Isomorphism
 Deep Structure: Some Examples of Isomorphism

Group theory is the branch of mathematics specializing in deep structures. It draws objects together on the basis of operations held in common (Garling, 1986). A “one-to-one correspondence” exists between all members of the group, under the given operations. Thus even though the square, the cross-head screw, and the units of G, the Gaussian integer field, displayed in Figure 13 look like vastly different entities, they nevertheless hold certain properties in common. Those properties are the deep structure that then define the group.

In the specific case depicted in Figure 13, each of the three objects shares, for their deep structure, an ability to rotate through 90º, after which each is left essentially unchanged. In the first, we pick up a piece of square Origami paper and rotate it clockwise through 90º. In the second, we take a Phillips screwdriver and drive the screw inwards by a quarter turn. In the third we multiply by -i (and since i = √-1 this gives 1 → -i → -1 → i → 1). The Gaussian field of imaginary numbers is an enormously powerful tool that allows far-reaching conclusions about the other members of this group to be drawn, and which probably could not be drawn in any other way. Those conclusions can then be validated by experiment. As long as these operations are respected, all such conclusions will apply equally well to the others, no matter how diverse their other features. These three objects are therefore isomorphic … they have the same deep structure.

The immediate question is: are biological populations truly isomorphic with vector bodies that possess inertia and specified directions of action, and so that behave as a Newtonian mass? If so then all the equations and the powerful mathematical equipment pertaining to those vector fields of study, such as Green’s theorem and planimeters, will immediately become available to biology and ecology, which would surely be an overwhelming benefit.