Abstract
In this chapter, I consider – from the two perspectives (Idealist and Materialist) – our geometric beliefs: their acquisition, truth and warrant. I present a difficulty attending the attempt to explain our acquisition of geometric beliefs, and consider two suggestions for contending with it. My conclusions will be that the difficulty notwithstanding, Hume’s Idealist can account for our having the geometric beliefs we do, although he can’t adequately explain why we do not have some non-Euclidean ones. Furthermore, many of them are justified, even count as knowledge, albeit “imprecise” (in a sense to be explained). I then consider the questions from the Materialist perspective. I argue that Hume the Materialist can only account for the acquisition of our geometric belief if he thinks they are acquired empirically, and that they are then unjustified, indeed, unjustifiable.
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Notes
- 1.
I argued (Sect. 9.4) that Hume’s nominalism cannot account for the fact that a generalisation based on an exemplar is a priori, rather than inductive. But the a priorists may still be right in ascribing to Hume the a priori view of geometry.
- 2.
There are two pieces of evidence for the (minority) a posteriorist view. First, Hume suggests that geometry’s “first principles are still drawn from the general appearance of the objects” (T 1.3.1.4; SBN 71, italics mine). But as the following sentences show, Hume means here by “objects” – ideas. As Grene (1994) notes, he uses the word ‘object’ to mean several things. Second, Hume alludes to our correcting “our judgments of our senses…by a juxta-position of the objects…or…by the use of some common and invariable measure” (T 1.2.4.23; SBN 47). And since we cannot juxtapose lines in an idea or apply to them a “common measure” (like a ruler), it might seem as if he takes geometry to be based on our experience with physical objects. But I take Hume to be discussing applied geometry in these passages, “the geometry of the surveyor or the carpenter” (Johnson, 1995, p. 97), and its empirical nature is compatible with theoretical geometry being a priori. So much for supporting evidence for the a posteriorist reading. Much stronger reasons can be cited in support of the a priorist reading. First, Hume places geometry in the a priori prong of the fork (T 1.3.1.1; SBN 68). Second, when he says (T 1.2.4.17; SBN 45) that “with regard to such minute objects, they are not properly demonstrations, being built on ideas, which are not exact, and maxims, which are not precisely true”, he is implying that other derivations are demonstrations, so their premises are a priori.
- 3.
Berkeley (N 258), Warnock (1953), Fogelin (1988), Jesseph (1993) and Pressman (1997) note the falsity of much of Euclidean geometry in a finitely divisible space. Elsewhere (PHK, §123), Berkeley sees things less clearly, suggesting that Euclidean geometry is not committed to infinite divisibility: “[t]he infinite divisibility of finite extension, though it is not expressly laid down either as an axiom or theorem in the elements of that science, yet is throughout the same everywhere supposed and thought to have so inseparable and essential a connection with the principles and demonstrations in geometry, that mathematicians never admit it into doubt, or make the least question of it”. And he is similarly mistaken when he suggests (PHK, §131) that “upon a thorough examination it will not be found that in any instance it is necessary to make use of…quantities less than the minimum sensible”. Pythagoras’ theorem is false in a discrete space, but it is certainly useful.
- 4.
The Thirteen Books of Euclid’s Elements, translated with commentary by Sir Thomas L. Heath, 2nd edition revised, New York: Dover Publications, 1956.
- 5.
Berkeley, too, thinks this is the discrete standard for equality of length: “If [with] me you call those lines equal [which] contain an equal number of points, then there will be no difficulty [in suggesting a curved line can be equal to a right line]. That curve is equal to a right line [which] contains as [many] points as the right one doth” (N 516).
- 6.
In denying the existence of a square whose sides are composed of 10 minima, since “the number of points must necessarily be a square number”, Berkeley (N 469) is noting another violation of Euclidean geometry in a discrete space.
- 7.
Gray (1978) argues that Berkeley thinks minima do have shape and extension. And he points out that this supposition engenders difficulties, because each particular candidate shape (a circle, for instance) violates plausible assumptions. Anyway, Hume explicitly denies that minima have shape or extension.
- 8.
In the passage I quoted above (T 1.4.2.7; SBN 190), Hume is less diffident about our knowledge of impressions, suggesting we are infallible with respect to them.
Bibliography
Badici, E. (2008). On the compatibility between Euclidean geometry and Hume’s denial of infinite divisibility. Hume Studies, 34, 231–244.
Batitsky, V. (1998). From inexactness to certainty: The change in Hume’s conception of geometry. Journal for General Philosophy of Science, 29, 1–20.
Bennett, J. (1971). Locke, Berkeley, Hume. Clarendon Press.
Bracken, H. (1987). On some points in Bayle, Berkeley, and Hume. History of Philosophy Quarterly, 4, 435–446.
Falkenstein, L. (2000). Hume’s finite geometry: A reply to mark Pressman. Hume Studies, 26, 183–185.
Fogelin, R. J. (1988). Hume and Berkeley on the proofs of infinite divisibility. Philosophical Review, 97, 47–69.
Gray, R. (1978). Berkeley’s theory of space. Journal of the History of Philosophy, 16, 415–434.
Grene, M. (1994). The objects of Hume’s treatise. Hume Studies, 20, 163–177.
Jesseph, D. (1993). Berkeley’s philosophy of mathematics. University of Chicago Press.
Johnson, O. A. (1995). The mind of David Hume. University of Illinois Press.
Newman, R. (1981). Hume on space and geometry. Hume Studies, 7, 1–31.
Pressman, M. (1997). Hume on geometry and infinite divisibility in the treatise. Hume Studies, 23, 227–244.
Strawson, G. (1989). The secret Connexion: Causation, realism, and David Hume. Clarendon Press.
van Bendegem, J. P. (1987). Zeno’s paradoxes and the Weyl tile argument. Philosophy of Science, 54, 295–302.
Warnock, G. J. (1953). Berkeley. Penguin.
Weyl, H. (1949). Philosophy of mathematics and natural sciences. Princeton University Press.
Wright, J. P. (1983). The Sceptical realism of David Hume. University of Minnesota Press.
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Weintraub, R. (2024). Geometry from the Materialist and Idealist Perspectives. In: Humean Bodies and their Consequences. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-50799-1_10
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