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Hume on space, geometry, and diagrammatic reasoning

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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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References

  • Baxter D. (1988) Hume on infinite divisibility. History of Philosophy Quarterly 5: 133–140

    Google Scholar 

  • Bos H. (2009) Descartes’ attempt, in the Regulae, to base the certainty of algebra on mental vision. In: Glymour C., Wei W., Westerståhl D. (Eds.), Logic, methodology and philosophy of science: Proceedings of the thirteenth international congress. College Publishers, London, pp 354–373

    Google Scholar 

  • Euclid. (1926). The thirteen books of the Elements (3 Vols., 2nd ed.) (Sir T. L. Heath, Trans. & Ed.). Cambridge: Cambridge University Press.

  • Falkenstein L. (1997) Hume on manners of disposition and the ideas of space and time. Archiv für Geschichte der Philosophie 79: 179–201

    Article  Google Scholar 

  • Falkenstein L. (2006) Space and time. In: Traiger S. (Ed.), The Blackwell guide to Hume’s Treatise. Blackwell Publishing, Oxford, pp 59–76

    Chapter  Google Scholar 

  • Flew A. (1976) Infinite divisibility in Hume’s Treatise. In: Livingston D., King J. (Eds.), Hume: A re-evaluation. Fordham University Press, New York

    Google Scholar 

  • Fogelin R. (1985) Hume’s skepticism in the Treatise of human nature. Routledge and Kegan Paul, London

    Google Scholar 

  • Franklin J. (1994) Achievements and fallacies in Hume’s account of infinite divisibility. Hume Studies 20: 85–101

    Google Scholar 

  • Frasca-Spada M. (1988) Space and the self in Hume’s Treatise of human nature. Cambridge University Press, Cambridge

    Google Scholar 

  • Holden T. (2002) Infinite divisibility and actual parts in Hume’s Treatise. Hume Studies 28: 3–25

    Article  Google Scholar 

  • Holden T. (2004) The architecture of matter. Oxford University Press, Oxford

    Book  Google Scholar 

  • Hume, D. (1975). Enquiries concerning human understanding and concerning the principles of morals (L. A. Selby-Bigge, Ed., 3rd ed., with text revised and notes by P. H. Nidditch). New York: Oxford University Press.

  • Hume, D. (1978). A treatise of human nature. (L. A. Selby-Bigge, Ed., 2nd ed., with text revised and notes by P. H. Nidditch). Oxford: Oxford University Press.

  • Hume, D. (1999). An enquiry concerning human understanding (T. L. Beauchamp, Ed.). New York: Oxford University Press.

  • Hume, D. (2000). A treatise of human nature (D. F. Norton & M. J. Norton, Eds.). New York: Oxford University Press.

  • Jacquette D. (2001) David Hume’s critique of infinity. Brill, Leiden

    Google Scholar 

  • Kemp Smith N. (1964) The philosophy of David Hume: A critical study of its origins and central doctrines. Macmillan, London

    Google Scholar 

  • Locke, J. (1979). An essay concerning human understanding (P. H. Nidditch, Ed.). Oxford: Oxford University Press.

  • Quine W. V. (2003) 1946 Lectures on David Hume’s philosophy. Eighteenth-Century Thought 1: 171–254

    Google Scholar 

  • Stein H. (1990) Eudoxus and Dedekind: On the Ancient Greek theory of ratios, and its relation to modern mathematics. Synthese 84: 163–211

    Article  Google Scholar 

Download references

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Correspondence to Graciela De Pierris.

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De Pierris, G. Hume on space, geometry, and diagrammatic reasoning. Synthese 186, 169–189 (2012). https://doi.org/10.1007/s11229-012-0071-5

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