| Bird Distributions as Surjective and Bijective Chickadee species in North America do not migrate. The four species have different areas where they occur. Whether each species is breeding or is not breeding, each inhabits a unique area. But this area is not adapted to its species, because it is ambiguous whether the same area is to be aligned with the species as a breeding entity or is to be aligned with the species as a non-breeding entity. This ambiguity stands in contrast to the fact that the species must be adapted – must have adaptedness – to its area, otherwise it would not be there. (The reader can see that the convenient nominalistic adapted contrasts with the realist’s adaptedness.) Sparrow species in North America do migrate. Each breeding species has its northern breeding area. And species and area must be adapted to each other. Each species when not breeding has its southern, wintering, non-breeding area, and species and area must be adapted to each other. They must be adapted to each other, otherwise the species would not be in these areas. There are four chickadee species, composing a set of four. Three of the six sparrow species of the genus Spizella form a set of three. But if each species is considered to be a breeding entity or a non-breeding entity then there is different line-up between breeding species and area and non-breeding species and same area in the case of chickadees from the line-up of breeding species and breeding area and non-breeding species and non-breeding different area in sparrows. The line-up is given in Tables 3 and 4, and each line-up affirms the basic structure requirement of bird, separable from the function of adaptedness, separable in turn from inhabited area. The narrow form of function applies: the function of adaptedness binds a breeding species to its area and the function of adaptedness binds a non-breeding species to the same area, in the case of chickadees, Table 3: f(1) = w and f(2) = w, but 1 
2; f(3) = x and f(4) = x, but
34;
f(5) = y and f(6) = y, but 56;
f(7) = z and f(8) = z, but 78.
Thus this is surjective. And this intimates a modified fourth principle
of adaptation, wherein eight entities, four species in breeding and
non-breeding condition, have adaptedness to four different areas.The narrow form of function applies again: the function of adaptedness binds a breeding species to its breeding area and the same but non-breeding species to its different non-breeding area, in the case of sparrows, Table 4: 1 to u, 2 to v, 3 to w, 4 to x, 5 to y, 6 to z – i.e., f(1) = u, f(2) = v, f(3) = w, f(4) = x, f(5) = y, f(6) = z. This is to say that when 1 is bound to, assigned to, aligned with, matched with, paired with u, such functional relationship equates 1 to u and that such relationship is best described as an adaptational relationship. Likewise for 2 and v. And so on. This is bijective, all elements on the left perfectly lined up, and singly so, with all elements on the right. But this perfect lining up makes a reverse line-up obvious, so the functional relationship of adaptation is reversible (invertable) and so there is adaptedness both ways, in accord with the third principle. But paramount is the accord with the fourth principle, six species adapted to six areas, six particulars having adaptedness to six different particulars. For the scarlet tanager, the bobolink, the Atlantic golden plover and the Pacific golden plover (Lincoln and Hines, 1950) the situation is the same as the sparrows (Figure 2), particularly because breeding area and non-breeding areas are so far apart. By the fourth principle each breeding species has adaptedness to its breeding area and each non-breeding species has adaptedness to its non-breeding area. By the third principle each breeding species has adaptedness to its breeding area which is adapted to it; each non-breeding species has adaptedness to its non-breeding area which is adapted to it. For these audaciously migrating species there is the set A of four species and the set B of four areas. There is the function of adaptedness that binds a breeding species to its breeding area in a bijective relation, for example: f(a1) = b1. There is the function of adaptedness that binds a breeding area to its breeding species in a bijective relation, for example: f(b1) = a1. The function of adaptedness that binds a non-breeding species
to its non-breeding area, f(a2) = b2,
binds its non-breeding area to the non-breeding species, f(b2)
= a2, the functions being invertible (reversible). |