Review of M. Wille, Mathematics and the Synthetic A Priori: Epistemological Investigations into the Status of Mathematical Axioms [Book Review]

Philosophia Mathematica 16 (1):130-132 (2008)
Abstract
Kant famously thought that mathematics contains synthetic a priori truths. In his book, Wille defends a version of the Kantian thesis on not-so-Kantian grounds. Wille calls his account neo-Kantian , because it makes sense of Kantian tenets by using a methodology that takes the linguistic and pragmatic turns seriously .Wille's work forms part of a larger project in which the statuses of mathematics and proof theory are investigated . The official purpose of the present book is to answer the question: what is mathematics . Wille sets himself the task of finding a definition that enables him to distinguish between mathematics and proof theory . His solution reads roughly as follows. Mathematics is about how to generate synthetic a priori knowledge by acting within some calculus . This definition does not seem to be a promising starting point, because in this way a very controversial claim becomes true by definition. However, Wille's strategy is to make sense of his definition during the course of his argument and to show that the definition is appropriate regarding research that is commonly taken to be mathematics .The second part of the book explains what ‘acting within some calculus’ amounts to. In Wille's view, mathematics should be characterized in …
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