EXUNMA: Explanation and understanding with(in) mathEMATICS
WLD 2022 in Brussel (virtual)

About.

The event will take place via Zoom on Sunday - January 16, 2022

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.

This event is part of the World Logic Day 2022

Registration is free, but needs to be done.

Explanation, Understanding and Definitions Vincent Coumans (Radboud Universiteit)

Joint work with: Joachim Frans, Henk W. de Regt

Following trends in research concerning mathematical explanation, we look at explanatory definitions. In particular, we build on the work of Lehet (2021a, 2021b), who describes explanatory definitions as those definitions that generate understanding. From this starting point, we investigate what types of understanding are associated with definitions, in order to develop a conception of explanatory definitions. Using the work of Baumberger (2014), we show that definitions can be explanatory in two ways: symbolically or factually. Furthermore, we zoom in on the impact of contextual factors on the explanatory value of definitions.

Baumberger C (2014) Types of Understanding: Their Nature and Their Relation to Knowledge. Conceptus: Zeitschrift Fur Philosophie 40 (98):67-88

Lehet E (2021a) Induction and explanatory definitions in mathematics. Synthese 198 (2):1161-1175. doi:10.1007/s11229-019-02095-y

Lehet E (2021b) Mathematical Explanation in Practice. Axiomathes 31 (5):553-574. doi:10.1007/s10516-021-09557-4

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Counter countermathematical explanations by Atoosa Kasirzadeh (University of Edinburgh)
Recently, there have been several attempts to generalize the counterfactual theory of causal explanations to mathematical explanations. The central idea of these attempts is to use conditionals whose antecedents express a mathematical impossibility. Such countermathematical conditionals are plugged into the explanatory scheme of the counterfactual theory and—so is the hope—capture mathematical explanations. Here, I dash the hope that countermathematical explanations simply parallel counterfactual explanations. In particular, I show that explanations based on countermathematicals are susceptible to three problems counterfactual explanations do not face. These problems seriously challenge the prospects for a counterfactual theory of explanation that is meant to cover mathematical explanations.

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Mathematical explanations in biology: a challenge to necessity and unification by José Antonio Pérez Escobar (ETH Zurich / Paris)
In this talk I will challenge the ideas that mathematical explanations in biology provide explanations by showing the necessity of explananda and that they contribute to the unity of science. To this end, I will provide a reconstruction of the explanation of the hexagonal shape of the honeycomb and analogies with other shapes in biology.

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A novel and formal account of mathematical explanations by Francesca Poggiolesi (CNRS, IHPST UMR8590, Université Paris 1 Panthéon-Sorbonne)

In this talk we will show a novel account of mathematical explanations that is mainly based on the notion of complexity and uses logical tools, namely derivations in natural deduction calculi. We will illustrate how the account works with the explanatory proof of Pythagoras theorem.

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Explanatory proofs in the mathematics classroom Keith Weber (Rutgers University)

Many mathematics education researchers call for explanatory proofs to play a more prominent role in mathematics classrooms. The proofs that students see should not only convince them that a theorem is true, but also explain why the theorem is true. Unfortunately, there is no shared criteria in mathematics education for what makes a proof explanatory.

The goal of this talk is twofold. First, I will present considerations that mathematics educators use when deciding if a proof is explanatory which have hitherto been ignored in the philosophy literature. Whether a proof is explanatory may depend on (i) which student is reading the proof, (ii) the activities that the student engages in as they read the proof, and (iii) how mathematical objects are represented, cognitively by the student and in the text of the proof. Second, I will present my own theory about what it means for a proof to be explanatory for a student.