Magic Metaphor and Isomorphism

Towards a 21st Century Structuralism

抄録

The straightforward expressions that compose magical acts have made anthropologists question how two completely different categories can be combined. This paper reinterprets magical acts reported in Papua New Guinean ethnography from the point of view of isomorphism in mathematical category theory. First, it is made clear that the set of elements that make up magic in the realm of living things and the similar set in the realm of the spirits are isomorphic; from this it is demonstrated that the two are so much the same that they are interchangeable.

Furthermore, it necessarily follows from this that operation in the realm of the living corresponds to operation in the realm of the spirits, and conversely, operation in the realm of the spirits also corresponds to operation in the realm of the living. In this way, the concept of isomorphism can succinctly explain the combination of different categories and the pursuit of unique causal relationships, which the study of magic has focused on. All of these are derived from isomorphism and are an inevitable consequence of isomorphism. The limited role of the various characteristics of magic, including metaphorical expressions, becomes clear if one regards them as contrivances and efforts to meet the demands of isomorphism.

However, the condition for isomorphism—that there is a one-to-one correspondence between the elements of the set on the side of the living and the elements of the set on the side of the spirits—is by no means guaranteed. It seems that the people of Papua New Guinea are aware both of this lack of guarantee and of the way in which they should make efforts; and under these conditions they endeavor to actualize isomorphism, which is to say, they continue to practice magic. Here the important nature of magical acts—that they rely on the idea of isomorphism and aim at the realization of isomorphism—becomes clear.

The study of magic in anthropology, which has a history and continues to be important, gains a new perspective from the language that embodies mathematics in this way. This topic, which is universal across space-time and moreover does not fit comfortably within the bounds of modern logic, is given a very succinct explanation through the language of mathematics, which has made remarkable progress in the modern era. In this way mathematics can contribute to anthropology and open up anthropology to other modern disciplines. This does not by any means mean quantification, but rather the re-emergence of an approach that re-understands human thought—which is not easily quantifiable—as structure; an approach which may perhaps call itself the structuralism of the 21st century.