On the status of proofs by contradiction in the seventeenth century

Abstract

In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations are causal demonstrations influenced the reflection on proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism.