FROM CAUSES TO RELATIONS: THE EMERGENCE OF A NON-ARISTOTELIAN CONCEPT OF GEOMETRICAL PROOF OUT OF THE QUAESTIO DE CERTITUDINE MATHEMATICARUM
Tobias SCHÖTTLER
Abstract. Insofar as many Renaissance thinkers regard Aristotelian philosophy of science as the framework for their understanding of mathematics and its proofs, they consider geometrical proofs as syllogisms using causes.
Furthermore, they identify geometrical proofs as demonstrationes potissimae, which are a kind syllogism that provides both the cause and the effect of an event. By questioning this assumption, Piccolomini initiates the so-called Quaestio de certitudine mathematicarum. Several scholars agreed with him. Others either maintained that mathematical proofs are demonstrationes potissimae or tried to prove that at least some mathematical proofs satisfy the conditions for being demonstrationes potissimae. Despite their differences in detail, all participants in the debate recognized Aristotelian scientific theory as the norm. Yet even traditionally Aristotelian answers take on a new meaning by virtue of a new context. This marks the birth of a genuinely new debate which has unwittingly left its Aristotelian roots behind. As a result, geometrical proofs are not longer
thought of as being based on causes or principles of being, but on the relationship between the different figures. Such a relationalism opens up the possibility of further development of mathematics.
Keywords: mathematics, proof, geometry, Aristotelianism, Renaissance, certainty