The Logic of Emergence
David H. Wolpert, David C. Krakauer
``Emergence'' refers to the fact that different mathematical languages and theories seem to be required to describe scientific theories that operate at different spatial and temporal scales. Highly circumscribed, special cases of emergence have been investigated using many different formal tools, including coarse-graining (over space and / or time), renormalization, dimension reduction, broken symmetry, and various formulations of causality. Here we extend the mathematics of model theory to unify and extend these earlier investigations. As a preliminary application of our framework, we investigate canonical examples of emergence, including the emergence of genetics from chemistry, of psychology from neurobiology, and of different forms of collective behavior from the laws governing their constituent parts. By exploiting the power of model theory we also uncover many aspects of emergence that were previously obscured. In particular, we uncover some of the rich hierarchical structure that relates different scientific fields. For example, we show that scientific fields can be related via a collection of multiple partial orders, all defined over the same set. On the other hand, under an alternative relation among scientific fields, one which allows cycles, we show that it is possible that the set of all of scientific fields must collapse into a single scientific field. Going forward, these investigations point to the formal tools that might need to be developed in order to arrive at a deeper understanding of this emergent hierarchy of scientific reality.
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