# Abstract

Philippe Huneman has recently questioned the widespread application of mechanistic
models of scientific explanation based on the existence of structural explanations,
i.e. explanations that account for the phenomenon to be explained in virtue of the mathematical
properties of the system where the phenomenon obtains, rather than in terms of
the mechanisms that causally produce the phenomenon. Structural explanations are very diverse, including cases like explanations in terms of bowtie structures, in terms of the topological
properties of the system, or in terms of equilibrium. The role of mathematics in
bowtie structured systems and in topologically constrained systems has recently been examined
in different papers. However, the specific role that mathematical properties play in
equilibrium explanations requires further examination, as different authors defend different
interpretations, some of them closer to the new-mechanistic approach than to the structural
model advocated by Huneman. In this paper, we cover this gap by investigating the explanatory
role that mathematics play in Blaser and Kirschner’s nested equilibrium model of the
stability of persistent long-term human-microbe associations. We argue that their model is
explanatory because: i) it provides a mathematical structure in the form of a set of differential
equations that together satisfy an ESS; ii) that the nested nature of the ESSs makes the
explanation of host-microbe persistent associations robust to any perturbation; iii) that this
is so because the properties of the ESS directly mirror the properties of the biological system
in a non-causal way. The combination of these three theses make equilibrium explanations
look more similar to structural explanations than to causal-mechanistic explanation.