In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books.

This paper consists of three main sections. In the first section, we consider how early attempts at understanding the relationship between mathematics and philosophy in Leibniz’s thought were often made within the framework of grand reconstructions guided by intellectual trends such as the search for “the ideal of system”. In the second section, we proceed to recount Leibniz’s first encounter with contemporary mathematics during his four years of study in Paris presenting some of the earliest mathematical successes which he made (...) there. In particular, we argue that recently published letters and papers reveal how his youthful mathematical reflexions were deeply intertwined with important philosophical insights that, in turn, acted as guiding ideas for his mathematical research. Finally, in the third section, we situate the central themes of the essays of the present volume within the new understanding of the interrelations between philosophy and mathematics in Leibniz’s thought briefly indicated in the opening section. (shrink)