Philosophy of mathematics: Bolzano's responses to Kant and Lagrange (Q2714826)

Starting from \textit{Bolzano}'s characterization of the philosophy of mathematics as ``an activity aimed at discovering the objective grounds of propositions which we already know with the greatest certainty'' (p. 399), the author deals with Bolzano's discussion of two of his most influential precursors, \textit{Kant} and \textit{Lagrange}. Although Bolzano's method used in both cases was the same, the results were quite different. ``In the case of Kant, there was in Bolzano's view little worth retaining [\dots], what he retained from Lagrange was far more substantial'' (pp. 401-2).NEWLINENEWLINENEWLINEIn particular Bolzano's rejection of Kant's epistemology is discussed, differences concerning the attitude towards analytic judgements, and finally towards the rôle of intuition (\textit{Anschauung}) in mathematics.NEWLINENEWLINENEWLINEIn the section on Bolzano and Lagrange (pp. 416-426), Bolzano's transformation of Lagrange's analysis (not published before 1930) is illustrated by considering the concept of continuity. The author summarizes: ``In Bolzano's understanding, philosophy of mathematics is an active, creative force, one aimed not at understanding a static object from a distance but at changing a dynamic one from within. To be sure, if we follow him in his usage, it will turn out that most philosophy of mathematics has been done by mathematicians rather than philosophers'' (p. 427).