Leibniz on the Parallel Postulate and the Foundations of Geometry

This book surveys Leibniz’s frustrated attempts to prove Euclid’s famous fifth postulate, while unable and even unwilling to explore the non-Euclidean geometries that lay at hand. The first of two parts is an overview of Leibniz’s idiosyncratic epistemology of geometry in the context of his time. It also provides a detailed commentary on his writings from correspondence, unpublished pieces, and marginalia on the theory of parallels. The thoughts from antecedents such as Clavius and Proclus, contemporaries such as Borelli and Vitale Giordano, and later thinkers such as Wenceslaus Karsten and Carl Hindenburg figure in significantly. The second part is a collection of Leibniz’s writings on the theory of parallels in the original Latin verso and an English translation recto. Since most of the transcriptions are published here for the first time, this volume adds to the history of mathematics and philosophy as well as non-Euclidean geometry.

Leibniz’s work in this area has not had much attention, as it consisted of flawed efforts toward an unattainable goal without significant influence on the trend of progress. His groping from mechanical analogy to metaphysical reasoning makes for intriguing reading as Leibniz parries and feints with his quarry from a letter here to a jotting in a margin there. A profusion of footnotes amplifies the subtle shadings of argument in a basically chronological exposition. Unlike the concluding excerpts from Leibniz’s writings, most of this source material is untranslated from the original French, Latin, etc. (When translations are available, and this is only for Leibniz, it is spelling out in the footnote which text is relevant to the passage.) I feel translations into English of all quotations would enhance this work. As the author observes, Leibniz’s “position about the use of imagination” is “complex and nuanced”. His vision of the analysis situs and the conviction that “imagination plays an important role in discovering mathematical truths (the ars inveniendi)” are delicately shaded beauties hidden away in the untranslated material.

The author does much in the first part to explore Leibniz’s philosophical approach to mathematics. Sections such as 3.3 Geometry and the Science of Space stand by themselves as essays elucidating such Leibnizian concepts as compossibility, a geometrical situation, and the analysis situs as a framework to completely analyze all geometrical concepts. Similarly, 3.4 Philosophy and the Parallel Postulate explores the metaphysical foundations Leibniz was compelled to bring in and that seem to have done more to cloud the matter than to clarify it, as explored later along with the effect on the foundations of statics in 5.4 The Principle of Reason. This internal struggle brought forward here from Leibniz’s “private and very private notes” never intended for publication makes for fascinating reading. Only in covering this area is it possible to understand how Leibniz’s “last attempt to prove the Parallel Postulate ended up with the implicit admission that non-Euclidean geometry is impossible”.

The presentation and organization with context of these previously unpublished pieces allows us to look over Leibniz’s shoulder as he confronts the Parallel Postulate and is defeated by it. With “Parallelae sunt lineae aequidistantes” in his 1673 reading notes to “Synopsis geometrica…” (Honorato Fabry), we find that in copying this definition, Leibniz had encountered and internalized “the definition of parallel lines that he endorsed throughout his life.” Through five chapters leading up to the reproduced texts, the author delves into even Leibniz’s impressive yet finite mind, encountering the “usual difficulties with infinite measure and cardinality” that when joined with his attempts of “a philosophical or metaphysical flavor” formed his own dead ends in scrutinizing Euclid’s axiom, as happened to others over the millennia. Many of those similar dead ends are documented here. The blind alleys and heroic efforts to craft a solution from a perfectionist God’s best of all possible worlds, or dodge acknowledgement of alternate geometries (even Gauss would not admit to that), or express eccentric notions about infinite extents make this survey of the Parallel Postulate in Leibniz’s world a fascinating read for any one that loves this history of mathematics, indeed the history of understanding the infinite.

Tom Schulte teaches mathematics with frequent historical references to students of Oakland Community College in Auburn Hills, Michigan.