Geometry and Logic. The Shape of Mathematical Proof from Antiquity to the Early Modern Age

Co-financed by Max-Planck-Institute for the History of Science, Berlin

17 June 2013 - 19 June 2013

Aims

The conference aims to discuss the connections between traditional logic and mathematical practice from Antiquity to the Modern Age, especially concentrating on the shape of deduction in geometry. The main focus will be in the period that precedes the birth of mathematical logic in the 19th century and the explicit attempts to merge together the two traditions of mathematical and logical proof. The conference will discuss, among other things, the role of principles in geometry in different ages and the various classifications of them, the epistemological meaning of definitions of primitive terms, the role of diagrams in geometrical demonstrations, and the development of the logical tools properly used in mathematical proofs. Some possible topics are: the role of geometrical examples in Aristotle's Posterior Analytics and the possibility to apply a theory of syllogism in a mathematical proof; Euclid's system of principles and the role of diagrams in ancient mathematics; the development of demonstrative tools in post-Euclidean classical mathematics, from Archimedes to the late commentators (e.g. Pappus, Simplicius); the late-antiquity debate on genetic definitions and their role in geometry (e.g. Hero, Proclus); the role of rigid motions in ancient mathematics and the new demonstrative means developed in the Islamic Middle Age; the Western quaestiones on mathematics in the medieval period and their relation with the rediscovery of the Aristotelian logical writings; the so-called quaestio de certitudine mathematicarum in the Renaissance and its developments in late Scholasticism; the first attempts of reduction of mathematical proofs to logical inferences (Herlinus, Dasypodius, Clavius); the birth of algebra and its consequences on the relations between geometry and logic (Viète, Descartes, etc.); the discussions on the possibility of a mathesis universalis; the new mathematical epistemologies of the Early Modern Age and the redefinition of the boundaries between logic and mathematics (e.g. Descartes, Hobbes, Locke, Hume); Leibniz, his new logic, and his program in logically proving the whole of mathematics; the many attempts in a logical characteristica; the development of infinitary procedures in mathematics and their fallout on logic in the 18th century; Kant's epistemology of mathematics and the relations between intuition and logic; the birth of non-Euclidean geometries and their consequences on logic and epistemology; the development of manifold deductive systems; the first steps in a mathematical logic at the beginning of the 19th century.