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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
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.
References
Arnauld, A. & Nicole, P. (1662) Logic or the Art of Thinking. Jill Vance Buroker (ed.), Cambridge University Press edition, 1996.
Atkinson, R. F. (1960). Hume on mathematics. Philosophical Quarterly, 10(39).
Axworthy, Angela (2018) The debate between Peletier and Clavius on superposition, Historia Mathematica 45(1) pp. 1–38, ISSN 0315-0860, https://doi.org/10.1016/j.hm.2017.09.004.
Badici, E. (2008). On the Compatibility between Euclidean Geometry and Hume’s Denial of Infinite Divisibility. Hume Studies, 34(2).
Badici, E. (2011) Standards of equality and Hume’s view of geometry. Pacific Philosophical Quarterly, 92(4).
Batitsky, Vadim (1998) From Inexactness to Certainty: The Change in Hume’s Conception of Geometry Journal for General Philosophy of Science / Zeitschrift für allgemeineWissenschaftstheorie, Vol. 29, No. 1, pp. 1–20.
Baxter, Donald L.M. (1993) Hume’s Theory of Space and Time. In Fate Norton & Taylor (eds) The Cambridge Companion to Hume. Cambridge University Press. (2009 edition), pp. 105–146.
Beeley, Philip; Scriba, Christoph J. (2008) Controversy and modernity. John Wallis and the seventeenth-century debate on the nature of the angle of contact. Mathematics celestial and terrestrial, 431–450, Acta Hist. Leopoldina, 54, Dtsch. Akad. Naturforscher Leopold., Halle/Saale.
Butler, Annemarie (2015) Hume’s Early Biography and A Treatise of Human Nature. In Ainslie & Butler (eds) The Cambridge Companion to Hume’s Treatise.
Coleman, Dorothy P. (1979) Is Mathematics for Hume Synthetic a Priori? Southwestern Journal of Philosophy 10 (2):113–126.
De Olaso, E. (2012) Scepticism and the infinite. In Lamarra (ed.) L’Infinito in Leibniz: Problemi e Terminologia. pp. 95–118.
De Pierris, G. (2012) Hume on space, geometry, and diagrammatic reasoning. Synthese, 186(1).
Falkenstein, Lorne (2015) The Ideas of Space and Time and Spatial and Temporal Ideas in Treatise 1.2 in Ainslie, Donald & Butler, Annemarie (eds) (2015). The Cambridge Companion to Hume’s Treatise. Cambridge University Press.
Fogelin, Robert (1988) Hume and Berkeley on the Proofs of Infinite Divisibility The Philosophical Review 97(1) pp. 47–69.
Garrett, Don (1997) Cognition and Commitment in Hume’s Philosophy. Oxford University Press.
Gossman, L. (1960). Two Unpublished Essays on Mathematics in the Hume Papers. Journal of the History of Ideas, 21(1/4).
Heath, Thomas L. (1956) The Thirteen Books of Euclid’s Elements. London: Constable & co.
Hume, David (1739) A Treatise of Human Nature. Reprinted from the Original Edition in three volumes and edited, with an analytical index, by L.A. SelbyBigge, M.A. (Oxford: Clarendon Press, 1896). Penguin classics edition, 1985.
Hume, David (1777) Enquiries concerning the human understanding and concerning the principles of morals. Selby-Bigge (ed.) Oxford: Clarendon press, 1902.
Jacquette, Dale (2001) David Hume’s Critique of Infinity. Leiden, Boston, Köln: Brill’s Studies in Intellectual History.
Jesseph, D. M. (1998) Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes. Perspectives on Science 6(1), 6–40. The MIT Press. Retrieved December 18, 2018, from Project MUSE database.
Katz, V.J. (2016) Sourcebook in the Mathematics of Medieval Europe and North Africa. Princeton University Press.
Gilbert-Charles Le Gendre (1735) Traité de l’opinion ou Memoires pour server à l’histoire de l’esprit humain, 2nd ed., vol. 3. Paris.
Kotevska, Laura (2021) Moral improvement through mathematics: Antoine Arnauld and Pierre Nicole’s Nouveaux éléments de géométrie. Synthese 199:1727–1749
Malézieu, Nicolas de (1729) Elémens de géométrie de Monsieur le Duc de Bourgogne. Avec l’Introduction a l’application de l’albegre a la geometrie (par mr Guisnée) 3. éd., revûë, corrigée & augmentée d’un traité des logarithms. Paris.
Mason, H. T. (1967) The Leibniz-Arnauld Correspondence. Manchester: Manchester University Press.
Popkin, R. (1955) The Skeptical Precursors of David Hume. Philosophy and Phenomenological Research, Sep., 1955, Vol. 16, No. 1 pp. 61–71.
Rashed, R. (2012) L’angle de contingence: Un problème de philosophie des mathématiques. Arabic Sciences and Philosophy, 22(1), 1–50. https://doi.org/10.1017/S0957423911000087
Ryan, T. (2012) Hume’s “Malezieu Argument” Hume Studies 38(1), pp. 105–118.
Strickland, Lloyd (2011) Leibniz and the Two Sophies: The Philosophical Correspondence. Toronto: Centre for Reformation and Renaissance Studies.
Waxman, W. (1996) The Psychologistic Foundations of Hume’s Critique of Mathematical Philosophy. Hume Studies, 22(1).
Whewell, William (1860) (ed.) The Mathematical Works of Isaac Barrow. Cambridge: Cambridge University Press.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-031-21494-3_7
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-21493-6
Online ISBN: 978-3-031-21494-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)