Nineteenth-Century Geometry

Abstract

In the nineteenth century, geometry, like most academic disciplines,went through a period of growth verging on cataclysm. During thisperiod, the content of geometry and its internal diversity increasedalmost beyond recognition; the axiomatic method, vaunted sinceantiquity by the admirers of geometry, finally attained true logicalsufficiency, and the ground was laid for replacing, in the descriptionof physical phenomena, the standard geometry of Euclid by Riemann'swonderfully pliable system. Modern philosophers of all tendencies— Descartes and Hobbes, Spinoza and Locke, Hume and Kant —had regarded Euclidean geometry as a paradigm of epistemic certainty.The sudden shrinking of Euclidean geometry to a subspecies of the vastfamily of mathematical theories of space shattered some illusions andprompted important changes in our the philosophical conception of humanknowledge. Thus, for instance, after these nineteenth-centurydevelopments, philosophers who dream of a completely certain knowledgeof right and wrong secured by logical inference from self-evidentprinciples can no longer propose Euclidean geometry as an instance inwhich a similar goal has proved attainable. The present article reviewsthe aspects of nineteenth century geometry that are of major interestfor philosophy and hints in passing, at their philosophicalsignificance.