Mathematical Memes in the Age of Reason: The limits of understanding and the understanding of limits

David Hume devoted a long section of his Treatise of Human Nature to an attempt to refute the indivisibility of space and time. In the 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 precisely these mathematical examples (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. That is to say, this handful of paradoxes were taken to mark the limit of rational mathematical enquiry, beyond which human thought should either fall silent or surrender to religious faith.

This argument occurs, notably, in Malezieu’s Éléments de Géometrie, to which Hume refers indirectly in the Treatise. This book was an elementary textbook suitable for aristocratic youth, and therefore far from the discourse of the mathematical elite.

The fact that it was the same pair of examples turning up in extra-mathematical books (such as the Port Royal Logic) or elementary mathematical writing (such as Malezieu’s Elements), without ever including obvious alternative candidates such as Torricelli’s horn of plenty, indicates that they constituted a stable unit of discourse that was reproduced without further reference to mathematical literature or expertise—a meme.

Following Hume’s mathematical sources 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.