Abstract
I start by considering Mark Steiner’s startling claim that Hume takes geometry to be synthetic a priori, which engenders the Kantian challenge to explain how such knowledge is possible. I argue, in response, that Steiner misinterprets the (deceptive) relevant passage from Hume, and that Hume, as the received view has it, takes geometry to be analytic, although in a more expansive sense of the word than the modern one. I then note a new challenge geometry engenders for Hume. Unlike Euclidean space, Humean space is finitely divisible, for which several Euclidean axioms and theorems do not hold. I argue, in response, that (crucially for his scientific project) Hume can account for our belief in the truth of Euclidean geometry on the basis of non-Euclidean ideas, although (innocuously for him) not all should be true. I conclude by arguing, on a less optimistic note, that Hume cannot point to a geometry that is true of our (discrete) ideas.
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Notes
- 1.
I follow Steiner in restricting my comments to the Treatise. So I remain uncommitted vis-a-vis the interpretative dispute as to whether Hume’s view about geometry in the Enquiry is different.
- 2.
References to the Treatise are to A Treatise of Human Nature, ed. David Fate Norton and Mary J. Norton (New York: Oxford University Press, 2000), and to A Treatise of Human Nature, ed. L.A. Selby-Bigge, 2nd edition, revised by P.H. Nidditch (Oxford: Clarendon Press, 1978), hereafter SBN.
- 3.
This (interpretative) view is predominant. The exceptions are Bennett (1971, p. 241), Newman (1981), Wright (1983), Fogelin (1988), Batitsky (1998) and Badici (2008), who think that at least in the Treatise, Hume takes geometry to be contingent, empirical. I find this (minority) view implausible. There are two pieces of evidence for it. First, Hume’s 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, and its empirical nature is compatible with theoretical geometry being a priori. Two, 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. 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. Despite my interpretative choice, my discussion will be of interest even for those who accept the empirical interpretation of Hume’s geometry.
- 4.
I think Steiner didn’t know of his (interpretative) predecessors, who anticipate him in criticising Kant’s attribution of the analytic view of geometry to Hume.
- 5.
If a Humean “demonstration” is construed liberally, so must be a Humean “contradiction”, since “wherever a demonstration takes place, the contrary is impossible, and implies a contradiction” (T abstract 11; SBN 650). We think the denial of a contradiction is a logical truth. But Hume takes a contradiction to be the denial of a demonstrable proposition. Since the latter needn’t be a logical truth, the former needn’t be logically impossible. In fact, for Hume, contradictions come in degrees. Only the most extreme kind is a logical contradiction, “the flattest of all contradictions, viz. that ‘tis possible for the same thing both to be and not to be” (T 1.1.7.4; SBN 19). So the denial of S is contradictory, albeit not a logical contradiction.
- 6.
References to the Enquiry are to Enquiry Concerning Human Understanding, ed. Tom L. Beauchamp (Oxford: Clarendon Press, 2000), and to Enquiry Concerning Human Understanding, ed. L.A. Selby-Bigge, 3rd edition, revised by P.H. Nidditch (Oxford: Clarendon Press, 1975), hereafter SBN.
- 7.
The term “revival set” is Garrett’s (1997, p. 24).
- 8.
- 9.
The Thirteen Books of Euclid’s Elements, translated with commentary by Sir Thomas L. Heath, 2nd edition revised, New York: Dover Publications, 1956.
- 10.
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” (1707–8, #516).
- 11.
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 (1707–8, #469) is noting another violation of Euclidean geometry in a discrete space.
- 12.
In a similar vein, Berkeley (1710, §25) thinks that our ordinary causal judgements are false, because causation requires agency, and the objects of sense are ideas, which are inactive. The true cause of all events is God.
- 13.
Elsewhere (T 1.4.2.7; SBN 190, original italics), Hume is less diffident about our knowledge of impressions. He says it is not “conceivable that our senses shou’d be more capable of deceiving us in the situation and relations, than in the nature of our impressions. For since all actions and sensations of the mind are known to us by consciousness, they must necessarily appear in every particular what they are, and be what they appear. Every thing that enters the mind, being in reality as the perception, ‘tis impossible any thing shou’d to feeling appear different. This were to suppose, that even where we are most intimately conscious, we might be mistaken”.
- 14.
Thus, the proof of Euclid’s first proposition, according to which an equilateral triangle can be constructed on any given finite line segment, invokes this “notion”, as does the proof of the proposition that any finite line segment can be bisected.
- 15.
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.
- 16.
“And tho’ we must endeavour to render all our principles as universal as possible, by tracing up our experiments to the utmost, and explaining all effects from the simplest and fewest causes, ‘tis still certain we cannot go beyond experience; and any hypothesis, that pretends to discover the ultimate original qualities of human nature, ought at first to be rejected as presumptuous and chimerical.” (T 0.8; SBN xvii, my italics).
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This research was supported by the Israeli Science Foundation, Grant Number 751/20.
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Weintraub, R. (2023). Hume’s View of Geometry. In: Posy, C., Ben-Menahem, Y. (eds) Mathematical Knowledge, Objects and Applications. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-21655-8_14
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