Epistemology of Geometry

by

Joshua Eisenthal, Lydia Patton

• LAST REVIEWED: 19 May 2023
• LAST MODIFIED: 25 March 2020
• DOI: 10.1093/obo/9780195396577-0402

Introduction

From Euclid to Einstein, geometry has continually shown itself to be a remarkably rich area of philosophical inquiry. Major geometrical traditions find their roots in the Āryabhatīya of Āryabhata from the classical age of Indian mathematics, the Nine Chapters on the Mathematical Art from Han dynasty China, and in Euclid’s Elements of ancient Greece, conveyed via the Arabic textual tradition to medieval scholars. Ancient reasoning methods—characteristically relying on pictures or diagrams—have found a renaissance in the scholarship of the last few decades and are now seen as far more rigorous than has often been supposed. Much contemporary geometry traces back to the development of algebraic methods (spurred in the Western tradition through philosophers such as Descartes and Leibniz), and geometry today overlaps with many other areas of mathematics (including set theory, category theory, and homotopy type theory). In the interim, a number of traditions have risen to prominence. Kant regarded geometry as the paradigm example of the synthetic a priori, and Hilbert’s work on geometry set a new standard for the axiomatization of a mathematical theory. Having always been admired as a paradigm of science, there is a long-standing tradition of seeking out “geometrical formulations” of physical theories (including classical mechanics, electromagnetism, and quantum physics). The emergence of a plurality of non-Euclidean geometries—sending shock waves through mathematics, physics and philosophy—led in particular to the 19th-century “problem of space,” which then fed into Einstein’s theories of special and general relativity, radically transforming the physical application of geometrical notions. This article is organized into three major areas of the epistemology of geometry. The first, Geometrical Reasoning, concerns the epistemology of geometrical practice itself. The second, Geometry and Philosophy, concerns the various pathways between geometry and philosophical theorizing more generally, including the historical engagement between the two. And the third, Geometry and Physics, concerns the intimate connections between geometry and physics, especially (of course) the physics of space. The bibliography begins with a selection of general overviews and textbooks that can serve as entry points into more specific areas.

General Overviews

These overviews cover some of the most important periods in the history of geometry. Mueller 2012 discusses the context of Euclid’s Elements. Sarasvati Amma 1999 aims to correct the historical error that Indian mathematicians of the premodern period were brilliant algebraists but did not make major strides in geometry. Klein 1893 and Torretti 1978 both discuss the major advances in geometry during the 19th century, and Friedman 1983 surveys a novel approach to geometry that emerged in the aftermath of relativity theory.

• Friedman, Michael. Foundations of Spacetime Theories: Relativistic Physics and Philosophy of Science. Princeton, NJ: Princeton University Press, 1983.

  A rigorous but accessible account of the overarching conflict between conventionalism and realism concerning spacetime geometry. Friedman applies a four-dimensional approach (embedded in the formal apparatus of differential geometry) to classical physics, special relativity, and general relativity, providing a gateway into the contemporary philosophy of spacetime physics.

• Klein, Felix. “A Comparative Review of Recent Researches in Geometry.” Bulletin of the American Mathematical Society 2.10 (July 1893): 215–249.

  DOI: 10.1090/S0002-9904-1893-00147-X

  A survey of the various branches of geometry in the second half of the 19th century by one of the masters of the period. Topics covered include group theory, projective geometry, line geometry, and analysis situs. Klein’s text offers insights into the evolving subject matter of geometry itself and the relationship between geometry and other branches of mathematics.

• Mueller, Ian. “Greek Mathematics to the Time of Euclid.” In A Companion to Ancient Philosophy. Edited by Mary Louise Gill and Pierre Pellegrin, 686–718. Oxford: Blackwell, 2012.

  DOI: 10.1002/9781444305845.ch35

  Mueller’s paper establishes the sources of Euclid’s Elements in the Greek tradition, including in the traditions of Plato, Aristotle, and the Pythagoreans. The first part of the paper focuses on plane geometry, including the relation between geometry and algebra; the second part deals with the history of Greek arithmetic; and the final part delves into lesser known figures and traditions in Greek geometry.

• Sarasvati Amma, T. A. Geometry in Ancient and Medieval India. Delhi: Indological Publications, 1999.

  A classic text on ancient and medieval Indian geometry. Surveys the Sanskrit and Prakrit traditions, including an early focus on the canonical texts of the Sulbasutras in the Vedic literature. Deals in depth with the intimate connections between geometry and astronomy in those traditions, in particular in the work of well-known Indian geometers such as Aryabhata, Sripati, and Bhaskara. The historical account extends to the early 17th century.

• Torretti, Roberto. Philosophy of Geometry from Riemann to Poincaré. Dordrecht, The Netherlands: Springer, 1978.

  DOI: 10.1007/978-94-009-9909-1

  A landmark review and analysis of the history of the philosophy of geometry, focused on the major developments of the 19th century. Torretti provides a brief background on ancient geometry; a lucid discussion of metric, projective, and axiomatic geometry; and an analysis of empiricist and conventionalist interpretations of geometry.