Analogy

1. A methodological integration of the sciences can be furthered by drawing valid analogies between the methods of the different sciences. Resemblance between different sets of phenomena may be taken as suggesting the existence of some law or principle common to both sets, especially where a comparison can be made between the functions of elements in two systems. The fruitfulness of such analogies for science depends on whether consequences can be deduced from them which can be tested or observed, and this is likely to depend on whether the resemblance selected as analogous is of a fundamental or merely superficial kind. If structural relations can be reproduced in a simplified form in a different medium, a model may be constructed. The relation between model and thing modelled can be said generally to be a relation of analogy of which two kinds may be distinguished: (a) in the case of a logical model of a formal system, there is analogy of structure or isomorphism between model and system, since the same formal axiomatic and deductive relations connect elements and predicates of both system and model; and (b) in a replica model, in which there are material similarities between the parent system and its replica.

2. A member x of a set is analogous to its fellow member y when (a) x and y share several objective properties (or are equal in some respects) or (b) there exists a correspondence between the parts of x or the properties of x and those of y. If x and y satisfy the first condition, they may be said to be substantially analogous (e.g. in the case of any two atoms). If the second condition holds, then x and y are formally analogous irrespective of their constitution. If both conditions hold, the analogy may be called a homology. Homology implies both substantial and formal analogy, and substantial analogy implies formal analogy, but not conversely.

3. If x and y are sets, then correspondence under condition (b) leads to several degrees of formal analogy: (1) plain, or some-some analogy, when some elements of x are paired with some elements of y; (2) injective, or all-some analogy, when every element of x is paired with an element of y; (3) bijective, or all-all analogy, when the preceding relations hold both ways.

4. If there exists a correspondence that maps every element of x onto some element of y and, in addition, preserves the relations and operations in x, then there is a homomorphism of the set x into the set y, which is an all-some (injective) structure-preserving analogy. If there is also a homomorphism from y into x, and in addition the two morphisms compensate each other, the analogy is called an isomorphism. Isomorphism is perfect analogy which implies homomorphism, injective analogy, and plain analogy.