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

This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as precursors for the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy.

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

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.

On the centenary of Zygmunt Janiszewski (1888-1920): ideals of mathematical practice and the constitution of continuum theory

This paper shows the importance of applying a certain approach to the history and philosophy of mathematical practice to the study of Zygmunt Janiszewski's contribution to the topological foundations of Continuum theory. In the first part, a biography of Janiszewski is presented. It emphasizes his role as one of the founders of the Polish School of Mathematics, and the social, political and military facets in which his intellectual character was revealed, as well as the values that guided his academic and scientific life. Kitcher's view of mathematical practice is then adopted to examine the philosophical conceptions and epistemological style of Janiszewski in relation to the construction of the formal axiomatic system of knowledge about the continua. Finally, it is shown the convenience of differentiating in Kitcher's approach, the methods, procedures, techniques and strategies of practice, and the aesthetic values of mathematics. Keywords: Zygmunt Janiszewski; Continuum theory; Philosophy of mathematical practice; Polish school of mathematics.