About.

The event will take place via Zoom on Friday: March 20, 2026

Explanation and understanding are central topics in the philosophy of science. This discipline has made important developments to account for the role explanations play in scientific activities. A current trend is that philosophers emphasize the relation between explanation and understanding, and stress the need for theories of scientific understanding. This events wants to bring together scientists from different fields tackling the question of mathematical explanations both in the sense of explanation within mathematics as well as mathematical explanations in the sciences. 

You can see this event as an informal continuation of Explanation and Understanding in Mathematics which took place in December 2019 in Brussel and lead to the following issue of Axiomathes. It is clearly a continuation of the first iteration happening in 2022 and the second one happening in 2024.

Registration is free, but needs to be done. 

"How Mathematics Applies_From Pragmatic Schemes to Semantic Frames (A Working Approach)." by  Carlos Falcon Machuca

In this work-in-progress, I propose a simple way to understand the applicability of mathematics as a guided passage from everyday experience to formal rules. The central claim is that application is not automatic: we first stage empirical situations into mathematizable scenes through pragmatic schemes (repeatable practices of measuring, comparing, and stabilizing). I then introduce operational iconicity to highlight how diagrams and notations function as manipulable tools that make inferential consequences visible. Finally, semantic frames articulate the staged scene by fixing roles, relations, and idealizations, thereby constraining when a formal rule genuinely applies. The aim is to outline a practical, experience-based pathway for understanding how mathematics connects to the empirical world.

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"A Sosaian virtue-epistemological reconstruction of Hilbert’s conception of activity of understanding" by Zhouwanyue Nata Yang

Abstract: In his virtue epistemology, Ernest Sosa argues for a relational concept of knowledge that acknowledges the highest epistemic value of knowledge and explains it as a relational property embodied by apt truth-aiming performances of believing. With aptness, Sosa means that such performances’ accuracy (i.e., their truth-attainment) is attributed to their adroitness (i.e. their manifesting the agents’ performances of their own intellectual virtues under ordinary circumstances). Using the functional dependence of accuracy on adroitness that Sosa encodes in aptness, I reconstruct Hilbert’s conception of activities of understanding as presented in his finitist program in terms of Sosa’s AAA structure. According to this relational account, Hilbert’s approach to the activity of understanding via surveyable proofs imposes a requirement on the acceptance of mathematical propositions: for such acceptance, the validity of a symbolic expression of a proposition must be traced back to an apt proving procedure such that the procedure’s accuracy is attributed to its adroitness. 

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"Gauge Symmetries as a Distinctively Robust Mathematical Explanation " by Lorenzo Miláns del Bosch and José A. Perez-Escobar

Distinctively mathematical explanations are non-causal explanations in which mathematical facts play an indispensable explanatory role in empirical science. While such explanations have been defended in influential work by Marc Lange and others, they have also been subject to criticism, particularly concerning their explanatory directionality, their alleged lack of modal robustness, and their potential to collapse into merely generic explanations. In this paper, we argue that explanations based on gauge symmetries constitute a genuine and non-trivial instance of distinctively mathematical explanation that avoids many of these standard objections. Gauge symmetries explain physical phenomena not by identifying causal mechanisms, but by imposing mathematically articulated constraints on the space of physically admissible models. We show that these explanations are robust across a wide range of contexts, and that their explanatory force derives from the non-causal, modal necessity encoded in the in a symmetry structure that is representationally redundant. Thus, gauge symmetries constitute the first concrete, non-trivial example of a symmetry acting as a distinctively mathematical explanation of a physical phenomenon, and the first case to which many criticisms of this notion do not apply. 

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"Unification, Generalization, and Explanation in Algebra"  by Marc Lange

I will examine some kinds of explanations in mathematics to which elementary algebra is indispensable. The explanations that will be examined explain why various “word problems” have numerically the same answer, why the solutions to quadratic equations take the same form in terms of the equations’ coefficients, and why certain quadratic equations have only one solution rather than the usual two. Each of these explanations works by subsuming the explanandum under some algebraic generalization. But more than that is required: a unified proof of the generalization or even something more. Algebra unifies various mathematical results, explaining why they hold (not merely proving that they hold) by showing their similarity to be no coincidence or their difference to be, fundamentally, no difference at all.

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