The Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics takes place at Chapman University as its main host, and is co-organized together with researchers from UC Riverside, CSU San Bernardino, and Pitzer College. It also occasionally integrates the Chapman University D.Sc. program in Math, Philosophy and Physics as its Graduate Colloquium.

The seminars are held in hybrid format on the Chapman University campus in the Keck Center, home of Schmid College of Science and Technology, or on Zoom. On March 15, 2024, Professor Alan Baker presented his talk: Three Kinds of Topological Explanation in Science.

Abstract: The increasing prevalence of network models in science, especially when dealing with systems of high complexity, has led philosophers to take an interest in the notion of topological explanation. Network models work by mapping a physical phenomenon onto a mathematical graph (network) and then analyzing topological properties of the network such as node connectivity, clustering, and 'small world' features. How does topological explanation relate to scientific explanation more generally? Is topological explanation similar to other types of mathematical explanation in science?

In this talk I propose a classification of topological explanation in science into three basic kinds. The first kind involves cases where the target phenomenon literally possesses the topological properties in question (for example, the early stages of a developing embryo). The second kind comprises the network modeling approach described above, in which the target phenomenon maps onto a mathematical graph, and the topological analysis pertains directly to the graph and only indirectly to the target phenomenon. The third kind involves cases in which data is collected about the target phenomenon, resulting in a multi-dimensional array of data points, and topological analysis is performed on this array. I use this threefold classification to draw some more general philosophical conclusions concerning how applied mathematics works in actual scientific practice.

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