Abstract
David Hume devoted a long section of his Treatise of Human Nature to an attempt to refute the indivisibility of space and time. In his later Enquiry Concerning Human Understanding, he ridiculed the doctrine of infinitesimals and the paradox of the angle of contact between a circle and a tangent.
Following up Hume’s mathematical references reveals the role that a handful of mathematical examples (in Hume’s case, the indivisibility of space and the angle of contact) played in the work of philosophers who (like Hume) were not otherwise interested in mathematics, and who used them to argue for either fideist or sceptical conclusions. Such paradoxes were taken to mark the limit of rational mathematical enquiry, beyond which human thought should either fall silent or surrender to religious faith.
The fideist argument occurs, for example, in Malezieu’s Éléments de Géometrie, to which Hume refers indirectly in the Treatise.
Hume did not seem to appreciate that while bringing rigour to the differential and integral calculus was a central problem for mathematics, the angle of contact was (by his time) a non-problem that arose in the first place only owing to the antique authority of Euclid.
Following Hume’s mathematical sources thereby shows us something about the role and significance of mathematics in the wider intellectual culture of his time. A small number of isolated and fossilized puzzles became emblematic of mathematics as both rational authority and inaccessible mystery.
I am grateful for the comments of two anonymous referees, which helped me to improve this chapter considerably.
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Notes
- 1.
For studies and defences of Hume on mathematics, see: Atkinson (1960), Badici (2008, 2011), Batitsky (1998), Baxter (1993), Coleman (1979), De Pierris (2012), Jacquette (2001), Waxman (1996). Regarding Hume’s mathematical training, it is possible that he studied mathematics amongst his other subjects at Edinburgh (see Butler (2015)). Jacquette (p. 35) quotes Mossner’s speculation that Hume may have studied with James Gregory or Colin Maclaurin, who were teaching at Edinburgh during Hume’s time there. However, Hume never graduated and there is nothing like a transcript to indicate which classes he took or how much he understood. His explicit references to mathematical sources in the Treatise and the Enquiry are all to texts dating from the previous century, and the level of technical mathematical knowledge on display is minimal. Jacquette (p. 22, n29) mentions a letter in which Hume recommends Pierre Bayle’s (1697) article on Zeno of Elea, and erects a reading of the Treatise on the hypothesis that Hume adopted Bayle’s arguments about infinite divisibility but not his conclusion. There is no evidence that Hume’s understanding of the mathematics of his day went beyond the level required to read Bayle.
- 2.
Heath (1956).
- 3.
“Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process is repeated continually, then there will be left some magnitude less than the lesser magnitude set out.”
- 4.
De Olaso (2012, p. 105).
- 5.
The main part of the proposition (that is, that a further line cannot be interposed between the tangent and the circle) does occur in proofs of propositions in books III, IV and XII, but the claim that the angle of contact is a real quantity does not. In his commentary on this proposition, Heath suggests that Euclid dropped this line in just because other writers discussed horn angles and it would have seemed odd not to mention them at all (Euclid also defines rhombus and rhomboid without going on to say anything about them). See Heath (1956, p. 39).
- 6.
See Rashed (2012).
- 7.
See Katz (2016, pp. 174–178).
- 8.
Heath remarks that, “The violence of the controversy between [Peletier and Clavius] will be understood from the fact that the arguments and counter-arguments (which sometimes run into other matters than the particular question at issue) cover, in [Clavius’] book, 26 pages of small print” (1956, p. 41). Axworthy (2018) explains that Peletier and Clavius had a broader disagreement over the interpretation of Euclid that included differing judgements about the legitimacy of superposition in geometrical proof. She argues that superposition was not criticised prior to Peletier’s commentary of 1557, but was regularly condemned as a mechanical procedure thereafter, which suggests something about the changing status of Euclid in the sixteenth century.
- 9.
Rashed (2012, fn 1) quotes Mersenne’s text.
- 10.
- 11.
De Olaso (2012, p. 113). I am not aware of any candidate counterexamples.
- 12.
References here are to the Cambridge edition translated and edited by Jill Vance Buroker.
- 13.
Buroker’s introduction, p. xxiii.
- 14.
Buroker’s introduction, p. xii.
- 15.
As Kotevska (2021) explains, Arnauld and most especially Nicole doubted that mathematics is intrinsically valuable and worried that it was a distraction from worthwhile activities. Arnauld’s Nouveaux Elémens de Géométrie (1667) was written to deploy mathematics for religious edification.
- 16.
Critique of Pure Reason, A421-A460/B449-B488 (1998). In Kant, the restlessness of reason arises from dialectical pairs of proofs of contradictory conclusions, rather than a single proof of something incomprehensible. Note A424/B452, where Kant exempts mathematics from the antinomies, which arise from the transcendental contemplation of various kinds of completeness. However, the second antinomy concerns “The absolute completeness of the division of a given whole in appearance” (A415/B443), which, after translation into Kant’s idiom, is what Arnauld and Nicole, and Hume, understood by the infinite divisibility of space.
- 17.
Quoted from De Olaso (2012) p. 109.
- 18.
Strickland (2011) p. 89. Sophia (1630–1714) was Electress of Hannover, granddaughter of James I of England and VI of Scotland, mother of George I of Great Britain, and sister of the Princess Elizabeth who corresponded with Descartes.
- 19.
Strickland (2011, p. 91). It’s clear from the context that the angle of contact is what he has up his sleeve here.
- 20.
Translation by the author.
- 21.
Treatise (I, ii, ii, ed. Selby-Bigge, p. 30).
- 22.
Most scholars who consider them regard Hume’s arguments about space and time to be untenable and his understanding of mathematical argument to be defective. Fogelin (1988) gives a crisp account of the reasons why. Falkenstein (2015) helps Hume recover some dignity, but only by anachronistically deploying topological notions. See also Garrett (1997). Jacquette (2001) offers a spirited defence of Hume’s position, at the cost of dismissing large swathes of modern mathematics. Baxter (1993) presents Hume’s views on space and time as consequences of Pyrrhonian Scepticism (rather than the early modern Jansenist scepticism discussed in this paper). However, in the course of defending Hume, Baxter recognises that Hume, “would have to say that there is a maximum size beyond which geometry does not apply” (p. 123, n31). That is, effectively, an admission that Hume’s views are untenable.
- 23.
Treatise (I, ii, iv, ed. Selby-Bigge, p. 43).
- 24.
Treatise (I, iv, i, ed. Selby-Bigge, p. 180).
- 25.
Treatise (I, ii, iv, ed. Selby-Bigge, p. 46). Barrow’s Lectiones Mathematicae were published in Latin in 1683. There was an edition in English in 1734, which William Whewell (writing much later) dismissed as “so badly executed that it cannot be of use to any one” (Whewell 1860, p. viii). It is believed that Hume relied on this.
- 26.
Treatise (I, ii, ii, ed. Selby-Bigge, p. 30).
- 27.
Enquiry (book 1, section 12 part ii, Selby Bigge ed. p. 156).
- 28.
Ibid.
- 29.
Ibid.
- 30.
Op. cit. pp. 156–7. Notice Hume’s use of the inner angle theorem an example of a theorem we can easily understand and believe. It is the standard, go-to example of a geometrical theorem in philosophy, going back at least as far as Aristotle (Posterior Analytics 85b4–8). This is further evidence that Hume was relying on mathematical examples picked up from philosophical sources rather than any personal knowledge of mathematics.
- 31.
- 32.
There is some circumstantial evidence that Hume continued to think about mathematics after the publication of the Enquiries. See Gossman (1960). However, none of it suggests that Hume understood that philosophical questions about infinitesimal reasoning had to be answered without depriving mathematics of its greatest prizes.
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Larvor, B. (2023). The Limits of Understanding and the Understanding of Limits: David Hume’s Mathematical Sources. In: Zack, M., Waszek, D. (eds) Research in History and Philosophy of Mathematics. Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-21494-3_7
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