A Diagram of Choice: The Curious Case of Wallis’s Attempted Proof of the Parallel Postulate and the Axiom of Choice

Abstract

Wallis’s attempted proof of Euclid’s Parallel Postulate is an important but oft neglected event leading to the discovery of non-Euclidean geometries. Our aim here is to show Wallis’s own reliance on three non-constructive diagrammatic inferences that are not (fully) explicit in his own supplement to Euclid’s axioms. Namely, there is i- an implicit assumption concerning the possibility of motion; ii- an implicit assumption about the continuous nature of space and time; and iii- an explicit assumption about the existence of similar triangles which conceals an appeal to a combinatoric principle of reasoning that is tantamount to appealing to the Axiom of Choice.