In mathematics, a one-to-one correspondence between the elements of two or more sets or classes that preserves the structural properties of the domain (e.g., the Arabic numerals 1, 2, 3 and the Roman numerals I, II, III). In biology, it refers to similarity in form or structure between two or more substances or entities. Three sorts of isomorphisms can be drawn between different levels of organisation: 1. analogical isomorphisms: also known as the ‘soft’ systems approach, the aim is to demonstrate similarities in functioning between different levels. However, they say nothing about the causal agents or governing laws involved; 2. homological isomorphisms: also known as the ‘hard’ systems approach, the phenomena under study may differ with regard to causal factors, but they are governed by the same laws or principles based on mathematical isomorphisms. The latter can be derived, for example, from allometry, game theory, and linear or non-linear dynamics as well as from a broad range of frequency distributions (e.g., a Poisson distribution); 3. explanatory isomorphisms: the same causal agents, laws or principles are applicable to each phenomenon being compared. Moving from analogical to homological isomorphisms is a refinement of the question ‘how’, and from homological to explanatory isomorphisms is a change to the question ‘why’.
See Allometry, Analogy, Game(s) theory, Homology, Law, Levels of organization, Linear dynamical systems, Metaphor, Non-linear dynamics, Poisson distribution, Principle, Reductionism