This book provides the first comprehensive account of the relationship between philosophy of mathematics and the mathematical practice of the seventeenth century - the most eventful period of mathematical development in history. Starting with the Renaissance debates on the certainty of mathematics, the author leads the readers through the foundational issues raised by the emergence of new mathematical techniques including the influence of the Aristotelian conception of science in Cavalieri and Guldin. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical developments and philosophical reflection in seventeenth century mathematics.