Introduction: Mathesis Universalis, Proof and Computation

Abstract

By “mathesis universalis” Descartes and Leibniz understood a most general science built on the model of mathematics. Though the term, along with that of “mathesis universa”, had already been used during the seventeenth century, it was with Descartes and Leibniz that it became customary to designate it as a universal mathematical science that unifies all formal a priori sciences. In his Dissertatio de arte combinatoria (1666), early Leibniz writes that the mathesis is not a discipline, but unifies parts from different disciplines that have quantity as their subject. A while later, in a fragment entitled Elementa Nova Matheseos Universalis (1683?) he writes “the mathesis […] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” The more mature Leibniz considers all mathematical disciplines as branches of the mathesis and designs the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects” (“objets quelconques”). It is an abstract theory of combinations and relations among objects whatever.