The limits of understanding and the understanding of limits: David Hume’s mathematical sources

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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.
Original languageEnglish
Title of host publicationResearch in History and Philosophy of Mathematics
Subtitle of host publication The CSHPM 2021 Volume
Place of PublicationSwitzerland
PublisherSpringer Nature
ISBN (Electronic)978-3-031-21494-3
ISBN (Print)978-3-031-21493-6
Publication statusPublished - 12 May 2023

Publication series

Name Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques
VolumeCSHPM 2022 Volume
ISSN (Print)2662-8503
ISSN (Electronic)2662-8511


  • mathematics
  • philosophy
  • Hume


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