I went looking for Lowie, because I was curious about the introduction of the diffusion concept into cultural anthropology. The mathematical description of diffusion, originally developed in thermodynamics, became important in statistical genetics during the first half of the twentieth century. In particular, R. A. Fisher introduced diffusion methods to examine the effects of natural selection in his 1922 paper, "On the dominance ratio." Diffusion methods made it possible to derive analytical approximations for many interesting biological parameters, and also came to underlie models of population dynamics outside the field of genetics.
So I wondered: How did the use of diffusion in cultural anthropology compare to the introduction of the diffusion concept in genetics? Kroeber's systematization of the concept of cultural diffusion was certainly later than Fisher and Wright's major works on diffusion theory.
It turns out that "diffusion" was initiated into cultural anthropology by E. B. Tylor himself. The OED has the earliest anthropological use of the term in Primitive Culture:
How good a working analogy there really is between the diffusion of plants and animals and the diffusion of civilization, comes well into view when we notice how far the same causes have produced both at once.
Later, Boas also makes extensive use of the concept of diffusion, in essentially the modern sense as an alternative explanation to independent invention for a cultural trait. For example, in his 1891 article, "Dissemination of tales among the natives of North America," he compares the transfer of myths and stories among groups in the New World to that in the Old. He writes:
Then, we may ask, is there no criterion which we may use for deciding the question whether a tale is of independent origin, or whether its occurrence at a certain place is due to diffusion? I believe we may safely assume that, wherever a story which consists of the same combination of several elements is found in two regions, we must conclude that its occurrence in both is due to diffusion. The more complex the story is, which the countries under consideration have in common, the more this conclusion will be justified (Boas 1891:13-14).
So by the time Lowie was writing his Culture and Ethnology, the concept of cultural diffusion was well-established, and the Boasian school was concerned with classifying culture similarity in terms of diffusion.
Cultural problems tended to involve the spread of relatively large quantities of information -- reflected by the Boas quote above and its concern with the "combination of several elements." We can interpret this focus as a statistical consequence of observing culture: With so many potential observations, diffusion is difficult to distinguish from the null hypothesis of parallel development (chance similarity) unless the similarities are sufficiently detailed (involve a threshold of information) to prove otherwise.
I have not yet found anywhere that this concept of assessing diffusion by "information content" was formalized beyond verbal descriptions like Boas' above. I will review Kroeber's contribution later; he does not provide any formalism at all.
What I want to point out here is that this concept of diffusion is delimited quite differently from the use of diffusion models in genetics. Fisher and Wright initially introduced diffusion methods to deal with the effects of random changes of gene frequencies. In the case of Boasian cultural diffusion, random change will almost always fall short of the information content necessary to identify specific resemblances in a cross-cultural context.
I can imagine datasets on cultural traits that would be sufficient to test the hypothesis of undirected (non-selected) diffusion. For example, phonological data on dialects is often detailed and coded to geographic locations. If we approached these data with a diffusion model, they would in many cases be sufficient to demonstrate departures from the null model of undirected diffusion.
But most observations from ethnography are not of this character. For this reason, cultural diffusion is a priori a phenomenon involving direction by some selective mechanism. In genetics, this would be natural selection. In cultural variation, the selective mechanism may be less clear -- some combination of conscious decision, customs concerning borrowed behaviors, and functional efficacy may be involved.
In the case of natural selection driving a gene substitution through space, Fisher's model of a wave of diffusion assumed only a single parameter determining intrinsic increase -- the "reaction" term in a reaction-diffusion equation. This was sufficient because the only relevant difference between the selected and non-selected alleles was fitness.
The case of cultural diffusion, in contrast, makes it tempting to suggest that there might be many independent terms contributing to the spatial dispersion of a cultural trait. Traits might be different in that some are more transmissible than others; thereby spreading more widely. Some might have greater functional advantages than others. Some ideas might be impeded because they conflict with widespread taboos; others might be facilitated by the same factors.
I write all this because I'm curious about why there was not a more formal development of diffusion in the context of cultural theory. There was every potential of it: Boas did not begin with a formalization, but the ideas critical to a formal theory are present in his description. But the development of the field quickly went in the opposite direction -- away from formalization and toward description. This was despite the fact that the concept of diffusion became incredibly important in the conflict between the Boasian school of ethnology and the cultural evolutionism of Leslie White and others.
Others (non-anthropologists) did develop more formal theory, but this had little (if any) impact within anthropology. As I continue, I'll go back to the early cultural evolutionists Tylor and Morgan, and trace their influence on 20th century neo-evolutionists. Additionally, no account of cultural diffusion can omit the importance of the Kulturkreis school and its concept of the culture area.