## 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.

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 language | English |
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Title of host publication | Research in History and Philosophy of Mathematics |

Subtitle of host publication | The CSHPM 2021 Volume |

Place of Publication | Switzerland |

Publisher | Springer Nature Link |

Pages | 115-131 |

ISBN (Electronic) | 978-3-031-21494-3 |

ISBN (Print) | 978-3-031-21493-6 |

DOIs | |

Publication status | Published - 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 |
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Publisher | Springer |

Volume | CSHPM 2022 Volume |

ISSN (Print) | 2662-8503 |

ISSN (Electronic) | 2662-8511 |

## Keywords

- mathematics
- HISTORY
- philosophy
- Hume