There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. Topology provides an ideal ground for such purported non-causal explanations since topological manifolds, on which the parameters of a dynamical system can be modelled, are typically associated with multiple invariants, which remain unaltered even if the manifold is bent, stretched or twisted, reflecting a change in the parameter modelled on the manifold. Understood in this sense, topological explanations seem to provide modal information about certain constraints on the system that may not be evident in detailed, and often, cumbersome causal explanations.

This thesis examines some foundational issues in the applicability of topology to the natural world and their bearing on the debate on such purported non-causal (mathematical) explanations. More specifically, this thesis looks into various topological and geometrical formulations that essentially exploit the geometry of oscillating and complex systems, as an exercise in ‘geometric mechanics’, to provide a simple explanation of certain constraints imposed on their dynamics by the virtue of their geometry. The central question answered in this thesis is whether topology provide sufficient structure for such non-causal explanations. The answer, as the thesis demonstrates, is negative because topological explanations critically rely on idealisations, such as continuity and smoothness, which are realised only contingently in the natural world (or in mathematical approximations/models of the natural world); these idealisations impose some foundational limitations on the application of topology in modelling such systems ‘non-causally’. Consequently, purported topological explanations fail to fully circumvent the causal dependencies of such systems implying that these are not really ‘non-causal’ explanations.

This thesis also extends the argument to mathematical explanations in general. It argues that purported mathematical explanations are essentially causal explanations in dis- guise and are no different from ordinary applications of mathematics to the natural world. This is because these explanations work not by appealing to what the world must be like as a matter of mathematical necessity, but by appealing to various contingent causal facts. These contingent facts, although assumed away in the why-question pertaining to a physical fact, still participate as causal facts in an explanation of why the fact obtains in the world. That is, the explained physical fact does not obtain because of a mathematical necessity but by appeal to the world’s network of causal relations.