Mathematical explanation, grounding, and mathematical research

The present thesis is focused on mathematical explanation as an integral part of mathematical practice, especially mathematical research. To this aim, I have adopted a philosophical approach with several examples from the actual practice of mathematics at different levels of complexity. First, I offer a philosophical analysis of mathematical explanation based on several constraints that I claim could be taken as intuitive. I advance a specific view of metaphysical grounding (i.e. a separatist theory). Subsequently, I claim that mathematical explanation could be seen as a variety of the latter grounding relation. Afterward, I discuss some possible counter-arguments for the latter view due to Mark Lange, and show that the best attempts at showing that mathematical explanation and metaphysical grounding diverge are unsuccessful. I then proceed to highlight an element which I take as essential to an adequate account of mathematical explanation, i.e. mathematical background knowledge, also through an intermediate case-study from the model theory of ACF. Subsequently, I explore how it is possible to re-prove the elimination of imaginaries for real closed valued fields with a particular method, and introduce ‘conceptuality’ as a variety of explanation in mathematics. Finally, I gesture towards a social-constructive view of mathematics, also sketching some preliminary steps toward the view that mathematical practice could be understood as a theatrical performance. In the end, I discuss further open questions that could be seen as the next steps of the present research.