Abstract:

Although Leibniz scholars know that Leibniz devised his own set of symbols for geometry, much of this symbolism is still not well understood. For the growing and recent work on Leibniz’s geometry of situations does not treat the notation that he created for this geometry, what he likes to call his “calculus of situations.” As a result, our most detailed and extended commentary on Leibniz’s calculus continues to be a chapter from Louis Couturat’s 1901 book. And Couturat explains Leibniz’s notation by explaining what its symbols mean, translating them into his geometry. While Couturat’s approach illuminates parts of Leibniz’s symbolism, it also leaves some fundamental questions unanswered because there are questions that we can ask about a symbol that go beyond its meaning. In Leibniz’s case, it is appropriate to ask whether any extra-geometrical principles guided him in devising his calculus. For one way to look at Leibniz’s notation for geometry is to see it as an expression of his geometry. Since it is well-known that Leibniz has a theory of expression and that he sees a tight link between his theory of expression and his work on geometry, it is natural to wonder whether any principles from Leibniz’s theory of expression shaped his symbolism for geometry.
It is this question that I will answer in my talk. Whereas other commentators treat Leibniz’s calculus of situations primarily as an outgrowth of his geometry of situations, I argue that Leibniz’s calculus is just as much an outgrowth of his theory of expression. For I claim that Leibniz’s theory of expression convinced him that the best sort of expression for an object is one that represents it in terms of its building blocks, much like how the expression “H2O” represents water with symbols that stand for its ingredients. It is this sort of expression that Leibniz thinks geometers should use to represent the objects of their field and that Leibniz does not believe Descartes’s algebra can ever produce. To make Leibniz’s theorizing concrete, I concentrate on how he represents two fundamental objects in geometry: shapes and magnitudes. I show that Leibniz represents both with situations because he has clear and philosophically interesting reasons for thinking that situations are the building blocks for shapes and magnitudes.
My talk has three sections. The first sets the stage for our inquiry by sketching how Leibniz’s theory of expression informs his notation for geometry. In particular, I demonstrate how, according to Leibniz, his notation for geometry is itself a solution to a problem at the intersection of his theory of expression and his geometry. The problem, in a nutshell, is to find representations for shapes and magnitudes that conform to what Leibniz calls his “law for expressions:” an expression for an object must be fashioned from symbols that stand for the object’s ingredients. Section two presents how Leibniz’s law for expressions meshes with his geometry so as to produce his expressions for shapes, while the third and final section does the same for magnitude.