Leibniz on the Parallel Postulate and the Structure of Space

Vincenzo De Risi

Max Planck Institute for the History of Science Berlin

Abstract: 
Gottfried Wilhelm Leibniz (1646-1716) worked at the foundations of geometry during all his life, often labeling his various researches on the topic under the general name of analysis situs. One of the primary aims of Leibniz’ geometrical studies was that of ground and rigorize the classical geometry of Euclid, by means of a throughout examination of the principles and proofs of the Elements in their ancient and modern editions. In fact, he had liked to prove all the axioms of geometry, and ground it as a purely logical and unhypothetical science. One of the main obstacles in the foundations of geometry was represented by the Parallel Postulate: a principle that, since Antiquity, was regarded somehow as misplaced in a system of axioms, and was almost universally considered as provable. Already at the time of Leibniz, one could count several dozens of attempts to prove it, which however resisted untouched to the efforts of the best mathematicians of the Early Modern Age. Leibniz, who read a large number of these commentaries on Euclid, was perfectly informed of the regrettable situation of a simple geometrical statement that seemed impervious to proof, and attempted himself several demonstrations of the Parallel Postulate. It is quite clear that Leibniz was never satisfied with his proofs of the Postulate. He first tried to prove it, probably, in 1679; but in 1712, after more than thirty years of sporadic but untiring attempts, he had to admit to have tried much, but the demonstration still escaped him. In any case, Leibniz never published anything on the topic, and current histories of non-Euclidean geometry don’t mention him among the mathematicians who have worked on the proofs and implications of the Parallel Postulate. In the talk, I will present some of the attempts done by Leibniz in this direction, drawing from unpublished Leibnizian manuscripts found in the Hannover library. I will sketch, moreover, some epistemological views that directed Leibniz’ proofs, and some of the philosophical consequences that he envisaged in his geometrical studies on the Parallel Postulate. In particular, as Leibniz’ analysis situs was defined as a proper science of space (a characterization of geometry that was quite uncommon at the time), I will show how Leibniz’ researches on the theory of parallels may be regarded as an attempt to determine the mathematical structure of space itself. Finally, I will discuss Leibniz’ very last attempt in a proof of the Postulate, in which he tries to ground it on the Principle of Sufficient Reason, a principle that should only apply to physical and contingent truths. This attempt is deeply connected with Leibniz’ late metaphysics of space (as it is expounded, for instance, in his correspondence with Clarke), and seems to show that Leibniz was able to conceive a non-Euclidean space as a consistent (non-contradictory) ideal structure, even though physical space cannot possibly share non-Euclidean features.