Martin H. Krieger’s research while affiliated with University of Southern California and other places

Complex behaviour can occur in any system made up of large numbers of interacting constituents, be they atoms in a solid, cells in a living organism, or consumers in a national economy. Analysis of this behaviour often involves making important assumptions and approximations, the exact nature of which vary from subject to subject. Foundations of Complex-system Theories begins with a description of the general features of complexity and then examines a range of important concepts, such as theories of composite systems, collective phenomena, and stochastic processes. Each topic is discussed with reference to the fields of statistical physics, evolutionary biology, and economics, thereby highlighting recurrent themes in the study of complex systems. This detailed yet nontechnical book will appeal to anyone who wants to know more about complex systems and their behaviour. It will also be of great interest to specialists studying complexity in the physical, biological, and social sciences.

The traditions of design—in natural science, computation, theology, architecture, and literature and art—provide models for some aspects of planning. These models are especially concerned with order, coherence, meaning, structure, and judgment, and parallels or homologies among them. They also show how the making of wholeness (the “manufacture of transcendence”) is in tension with contingency, fetishism, and failure.

The postulate of randomness thus resolves itself into the question, “Of what population is this a random sample?” which must frequently be asked by every practical statistician (R. A. Fisher, 1922).1

The Argument In this paper I argue first that Marx's Critique of Political Economy employs “critique” in the Kantian meaning of the term—i.e., determining the domain of legitimate application of the categories involved and maintaining that outside these borders understanding is led into error and entangled in metaphysics. According to Marx, his predecessors in political economy transgressed these boundaries of application, and therefore conceived of all different modes of production as being essentially similar to commodity production, and thus implied that commodity production and the bourgeois form of life corresponding to it are “natural” not historical and transitory. In Marx's conception there are no super-historical economic categories or laws. I argue moreover that Marx's methodology of reconstructing the “development” of socioeconomic entities and categories from their “germ” or “cell” also serves his critical intention. Whereas social theorists of the time referred with organic metaphors to human collectives (“family,” “community,” etc.), Marx referred with such metaphors to economic entities only (“commodity,”“money,” etc.). The difference is crucial, since the first carries deterministic consequences for the development of society while the latter does not: Social form and historical development in Marx are contingent and not necessary, historical and not natural, transitory and not eternal. I also stress that Marx's procedure of critique is internal. He uses only such assumptions, observations, and arguments as could in principle also be used by the scholars criticized. Nevertheless the outcome of the critique is not merely a new theory but an entirely different one — i.e., a historical conception of the discipline of political economy and of its laws.

Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the arithmetic addition of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such as accelerations, are, for the most, part irrelevant, affirming that, in the end, most of the world's particulars may be averaged over (a very un-Scriptural point of view). (We work out all of this in technical detail, including a nice geometric picture of stochastic integration, and a method of calculating the variance of the sum of dependent random variables using renormalization group ideas.) These fundamental theorems affirm a culture that is additive, ahistorical, Cartesian, and continuist, sharing in what might be called a species of modern culture. We understand mathematical results as useful because, like many other such artifacts, they have been adapted to fit the world, and the world has been adapted to fit their capacities. Such cultural interpretation is in effect motivation for the mathematics, and might well be offered to students as a way of helping them understand what is going on at the blackboard. Philosophy of mathematics might want to pay more attention to the history and detailed technical features of sophisticated mathematics, as a balance to the usual concerns that arise in formalist or even Platonist positions.

This Note points out a homology of structure employed by elementary particle physicists and by social anthropologists, and suggests its origin in a systemic order reflecting a ‘plenitude’ of nature, and a non-degeneracy of names.