1. A one-to-one correspondence between elements in different systems such that the relationship between the elements is preserved.
2. A perfect analogy. Whatever is in one system and happens in that system has its isomorphic image in a second system and conversely. Isomorphism implies both homomorphism (described separately) and bijective (all-all) analogy; the latter implies injective (all-some) analogy, which in turn implies plain (some-some) analogy.
3. Whenever the symbols of one mathematical model stand in one-to-one correspondence with those of another, including the symbols of relation, and whenever the relation is preserved when a pair of symbols in one model is replaced by their counterparts in the other, the two models are isomorphic. Then the phenomena represented by the models can also be viewed as isomorphic. An isomorphism between two phenomena can be established only if their corresponding models (from which isomorphism is deduced) are sufficiently faithful representations.
4. There are three prerequisites for the existence of isomorphisms in different fields and sciences:
(a) the number of simple mathematical expressions which will be preferably applied to describe natural phenomena is limited; for this reason, laws identical in structure will appear in intrinsically different fields (The same applies to statements in ordinary languages; here, too, the number of intellectual schemes is restricted, and they will be applied in quite different realms).
(b) the structure of reality is such as to permit the application of conceptual constructs and is not too complex to be represented by the relatively simple schemes which can be elaborated.
(c) the parallelism of general conceptions or even special laws in different fields is a consequence of the fact that these are concerned with systems and that certain general principles apply to systems irrespective of their nature.