Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free

Abstract

This work gives a new argument for ‘Empirical Philosophy of Mathematical Practice’. It analyses different modalities on how empirical information can influence philosophical endeavours. We evoke the classical dichotomy between “armchair” philosophy and empirical/experimental philosophy, and claim that the latter should in turn be subdivided in three distinct styles: Apostate speculator, Informed analyst, and Freeway explorer. This is a shift of focus from the source of the information towards its use by philosophers. We present several examples from philosophy of mind/science and ethics on one side and a case study from philosophy of mathematics on the other. We argue that empirically informed philosophy of mathematics is different from the rest in a way that encourages a Freeway explorer approach, because intuitions about mathematical objects are often unavailable for non-mathematicians (since they are sometimes hard to grasp even for mathematicians). This consideration is supported by a case study in set theory.

Showing Mathematical Flies the Way Out of Foundational Bottles: The Later Wittgenstein as a Forerunner of Lakatos and the Philosophy of Mathematical Practice

“…For this reason, Wittgenstein's late philosophy is a powerful tool in ethnomathematics (François and Vandendriessche 2016). Perhaps also for this reason the philosophy of mathematical practice requires a strong empirical focus (Kant, Pérez-Escobar, and Sarikaya 2021). Wittgenstein did not think that mathematical foundations should or could determine mathematical practices, and a focus on mathematical foundations may limit philosophical clarifications of actual practices, and perhaps the potential of the practices themselves.…”

Purifying applied mathematics and applying pure mathematics: how a late Wittgensteinian perspective sheds light onto the dichotomy

“…Both directions have been recently explored from different angles. For instance,it has recently been argued that input from the real world (mainly about mathematical practices, including uses of intuitions and metaphors) is essential to philosophize about mathematics (Kant et al, 2021). Moreover, it has been shown that mathematics is connected to intuitions about the real world, and accommodates them via representational ambiguity (Pérez-Escobar, 2020) .…”

“…The question of the prose of mathematics has been systematically studied, for instance, inKant & Sarikaya (2020), in the context of narratives about mathematical practice.…”