Making and breaking mathematical sense. Histories and philosophies of mathematical practice (Q2826025)

The book is devoted to problem connected with the new trend in the philosophy of mathematics -- namely with the trend in which the mathematical practice is investigated. The author combines philosophical, historical and cognitive studies to describe mathematics not as an absolute ideal but as a human endeavor taking shape in specific social and institutional contexts.The book consists of seven chapters. In Chapter 1 some histories of classical philosophies of mathematics are introduced. Chapter 2 describes economical-mathematical practice with algebraic signs and subtracted numbers in the abbaco tradition of the Italian late Middle Ages and Renaissance. Chapter 3 provides a general outline of a philosophy of mathematical practice -- this forms the theoretical core of the book. The author reflects on the function of mathematical statements (following Wittgenstein), their epistemological position, mathematical consensus and mathematical interpretation and semiosis. In Chapter 4 one find reflection on some of the ideas of the previous chapter with concrete case studies focusing on problems of mathematical semiosis -- how mathematical sings obtain and change their senses. The preceding cognitive concerns are articulated in a more systematic manner in Chapter 5. The neuro-cognitive debate on the mental representation of numbers is reviewed and the cognitive theory of mathematical metaphor is presented. The cognitive problematic from Chapter 5 is fleshed out in Chapter 6 with case studies of medieval and early modern geometric algebra and of the history of notions of infinity. Those case studies demonstrate the limitations of the cognitive theory of mathematical metaphor. The last Chapter 7 complements the discussion by thinking of mathematics not only as subject to constraints but also as feeding back into the reality that shapes it. A brief narrative follows Fichte, Schelling and Hermann Cohen. A solution to Mark Steiner's formulation of Wigner's problem of ``unreasonable'' applicability of mathematics to the natural sciences is offered. The general purpose of the author is to recall that mathematics was and can be different from the mathematics that we are used to today. The book is accessible to readers with a general interest in philosophy and mathematics.