An Evaluative Framework

North American anthropologists and archaeologists have long confused the Midas-touch-like transformational evolution of Lewis Henry Morgan, Edward B. Tylor, and Herbert Spencer with the variational evolution of Charles Darwin. Following Franz Boas, evolution as a theory of change was allegedly discarded by North American anthropologists and archaeologists at the beginning of the twentieth century. However, they used the term “development” instead of “evolution” and spoke of culture change in evolutionary terms, often mixing elements of the theories and ontologies of transformational and variational evolution. Documenting culture change under variational evolution demands the materialist paradox be circumvented. This paradox highlights the question: How do we measure change in continuously variable phenomena? Paleontologists adopted the approach that each population of organisms is polymorphic; individuals are members of the same species but formally variable. Paleontologists compare central tendencies of temporally sequent populations, or chronospecies. Archaeologists who undertook frequency seriations adopted an approach that focuses on morphospecies, forms or types that occur in two or more temporally sequent populations. The occurrence of multiple types per temporal period highlights the variation upon which a sorting mechanism such as selection works, and the occurrence of one or more types in each of two or more temporally sequent assemblages provides evidence of connection between them required of studies of change. Recognizing that graph types, how phenomena are parsed into types, and theories of change are mutually influential allows evaluation of archaeological graphs of change in terms of their implied theories and ontologies.

Archaeology Spindle Graphs in the 1930s

During the 1930s, archaeological spindle graphs in the form of seriograms (straight-sided spindle graphs) were published. Three of these represent the investigator’s suspicions about culture change rather than being strictly empirical. Stylistically, seriograms were seldom subsequently published, suggesting these graphs minimally influenced later researchers. By the 1920s, based on the basically unimodal frequency distributions observed in frequencies of specimens of various pottery types in the American Southwest, anthropologists had begun to suspect there were so-called stylistic pulses reflecting the vogue or popularity of particular kinds of artifacts. Explanations fell back on probability theory, likely as a result of the influence of Franz Boas’s statistical reasoning; kinds of phenomena simply should display unimodal temporal frequency distributions given probability theory. Although conceptually unsophisticated, graphic models of these stylistic pulses published by anthropologists in the 1920s took the rough form of spindle graphs and represent a then unrecognized nod to the theory of variational evolution. These spindle graph models may be the ultimate source of archaeological spindle graphs, but these models were a bit difficult to decipher. Many graphs of culture change appearing in the 1920s and 1930s imply variational evolution.

Final Thoughts

Graphs are analytical tools and communication tools, and they summarize visually what has been learned. Granting that a major purpose of archaeology is to document and explain culture change, it is odd that the hows and whys of graphing culture change have received minimal attention in the archaeology literature. Spindle graphs will likely continue to be the most frequently used graph type for diagraming change, but continued development of computer software may result in new graph types and styles. Recent modifications to spindle graphs include scaling bar thickness to temporal duration of the represented assemblage. Classic data on temporal change in kaolin pipe stem hole diameters can be graphed using a regression line, a bar graph, and a spindle graph; the different graphs highlight that how phenomena are classified, how data are graphed, and one’s theory of change are mutually influential. Deciding which graph type to use in any particular situation will depend on what the researcher hopes to illustrate, along with the goal to produce a readily deciphered graph. The majority of archaeological graphs that appeared in the twentieth century depict variational evolution. Once developed in the late 1940s, spindle graphs quickly became the graph type preferred by North American archaeologists. There is weak circumstantial evidence archaeologists may have borrowed the idea of spindle graphs from paleontology, but it seems more likely the idea was stumbled upon by early archaeologists who perceived unimodal pulses in artifact frequencies over time and developed general models of those pulses.

Covariant Variational Evolution and Jacobi brackets: Fields

The analysis of the covariant brackets on the space of functions on the solutions to a variational problem in the framework of contact geometry initiated in the companion letter[Formula: see text] is extended to the case of the multisymplectic formulation of the free Klein–Gordon theory and of the free Schrödinger equation.

Covariant variational evolution and Jacobi brackets: Particles

The formulation of covariant brackets on the space of solutions to a variational problem is analyzed in the framework of contact geometry. It is argued that the Poisson algebra on the space of functionals on fields should be read as a Poisson subalgebra within an algebra of functions equipped with a Jacobi bracket on a suitable contact manifold.