Hume on space, geometry, and diagrammatic reasoning

Synthese 186 (1):169-189 (2012)
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Abstract

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar

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Graciela De Pierris
Stanford University

Citations of this work

Perceiving Necessity.Catherine Legg & James Franklin - 2017 - Pacific Philosophical Quarterly 98 (3).
Knowledge and Sensory Knowledge in Hume's Treatise.Graham Clay - 2021 - Oxford Studies in Early Modern Philosophy 10:195-229.
Certainly useless: empiricists’ uncomfortable relationship with intuition.Lewis Powell - 2023 - British Journal for the History of Philosophy 31 (4):724-743.

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References found in this work

A treatise of human nature.David Hume & D. G. C. Macnabb (eds.) - 1969 - Harmondsworth,: Penguin Books.
An essay concerning human understanding.John Locke - 1689 - New York: Oxford University Press. Edited by Pauline Phemister.
An enquiry concerning human understanding.David Hume - 2000 - In Steven M. Cahn (ed.), Exploring Philosophy: An Introductory Anthology. New York, NY, United States of America: Oxford University Press USA. pp. 112.

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