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Nineteenth Century Geometry

Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) First published Mon Jul 26, 1999; substantive revision Wed Jan 13, 2010 In the nineteenth century, geometry, like most academic disciplines, went through a period of growth verging on cataclysm. During this period, the content of geometry and its internal diversity increased almost beyond recognition; the axiomatic method, vaunted since antiquity by the admirers of geometry, finally attained true logical sufficiency, and the ground was laid for replacing, in the description of physical phenomena, the standard geometry of Euclid by Riemann's wonderfully 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 vast family of mathematical theories of space shattered some illusions and prompted important changes in our the philosophical conception of human knowledge. Thus, for instance, after these nineteenth-century developments, philosophers who dream of a completely certain knowledge of right and wrong secured by logical inference from self-evident principles can no longer propose Euclidean geometry as an instance in which a similar goal has proved attainable. The present article reviews the aspects of nineteenth century geometry that are of major interest for philosophy and hints in passing, at their philosophical significance.

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