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This article introduces a special issue of Philosophical Investigations focused on Ludwig Wittgenstein's philosophy of mathematics and its connection to the philosophy of mathematical practice (PMP). The author acknowledges that while Wittgenstein is a highly influential philosopher, his work in the philosophy of mathematics is often overshadowed by his other writings. The special issue includes nine articles that cover various topics related to Wittgenstein's philosophy of mathematics and PMP. The articles explore Wittgenstein's practice-based approach to meaning, his views on set theory, his struggles with arithmetic and higher mathematics, and his critique of Cantorian set theory. The volume aims to deepen our understanding of mathematical practice, contribute to the discourse on Wittgenstein's philosophy of mathematics, and introduce Wittgensteinian ideas to the wider community of philosophers of mathematical practice. [Extracted from the article]

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. [ABSTRACT FROM AUTHOR]