Science and the modern world

by Alfred North Whitehead

We can now explain how the analytical character of the realm of eternal objects allows of an analysis of that realm into grades.

In the lowest grade of eternal objects are to be placed those objects whose individual essences are simple. This is the grade of zero complexity. Next consider any set of such objects, finite or infinite as to the number of its members. For example, consider the set of three eternal objects A, B, C, of which none is complex. Let us write R(A, B, C) for some definite possible relatedness of A, B, C. To take a simple example, A, B, C may be three definite colours with the spatio-temporal relatedness to each other of three faces of a regular tetrahedron, anywhere at any time. Then R(A, B, C) is another eternal object of the lowest complex grade. Analogously there are eternal objects of successively higher grades. In respect to any complex eternal object, S(D₁, D₂, ... Dₙ), the eternal objects D₁, ... Dₙ, whose individual essences are constitutive of the individual essence of S(D₁, ... Dₙ), are called the components of S(D₁, ... Dₙ). It is obvious that the grade of complexity to be ascribed to S(D₁, ... Dₙ) is to be taken as one above the highest grade of complexity to be found among its components.

There is thus an analysis of the realm of possibility into simple eternal objects, and into various grades of complex eternal objects. A complex eternal object is an abstract situation. There is a double sense of ‘abstraction,’ in regard to the abstraction of definite eternal objects, i.e., non-mathematical abstraction. There is abstraction from actuality, and abstraction from possibility. For example, A and R(A, B, C) are both abstractions from the realm of possibility. Note that A must mean A in all its possible relationships, and among them R(A, B, C). Also R(A, B, C) means R(A, B, C) in all its relationships. But this meaning of R(A, B, C) excludes other relationships into which A can enter. Hence A as in R(A, B, C) is more abstract than A simpliciter. Thus as we pass from the grade of simple eternal objects to higher and higher grades of complexity, we are indulging in higher grades of abstraction from the realm of possibility.

We can now conceive the successive stages of a definite progress towards some assigned mode of abstraction from the realm of possibility, involving a progress (in thought) through successive grades of increasing complexity. I will call any such route of progress ‘an abstractive hierarchy.’ Any abstractive hierarchy, finite or infinite, is based upon some definite group of simple eternal objects. This group will be called the ‘base’ of the hierarchy. Thus the base of an abstractive hierarchy is a set of objects of zero complexity. The formal definition of an abstractive hierarchy is as follows:

An ‘abstractive hierarchy based upon g,’ where g is a group of simple eternal objects, is a set of eternal objects which satisfy the following conditions,

(i) the members of g belong to it, and are the only simple eternal objects in the hierarchy,

(ii) the components of any complex eternal object in the hierarchy are also members of the hierarchy, and

(iii) any set of eternal objects belonging to the hierarchy, whether all of the same grade or whether differing among themselves as to grade, are jointly among the components or derivative components of at least one eternal object which also belongs to the hierarchy.

It is to be noticed that the components of an eternal object are necessarily of a lower grade of complexity than itself. Accordingly any member of such a hierarchy, which is of the first grade of complexity, can have as components only members of the group g; and any member of the second grade can have as components only members of the first grade, and members of g; and so on for the higher grades.

The third condition to be satisfied by an abstractive hierarchy will be called the condition of connexity. Thus an abstractive hierarchy springs from its base; it includes every successive grade from its base either indefinitely onwards, or to its maximum grade; and it is ‘connected’ by the reappearance (in a higher grade) of any set of its members belonging to lower grades, in the function of a set of components or derivative components of at least one member of the hierarchy.