From the Mathematization of Logic to the "Logicalization" of Mathematics? Imagination and Impossibility Between Late-Medieval Semantics and the Rise Complex Mathematics

Graziana S. Ciola (Radboud University)

Abstract

Why is medieval logic not mathematized? This is a longstanding problem in the historiography of medieval logic. I suggest flipping that question on its head: rather than asking why medieval logic was not mathematized, it is more felicitous to asks how developments in logic shaped contemporaneous and subsequent developments in the philosophy and practice of mathematics.

The case in point is precise and consequential. I argue that the algebraic treatment and philosophical problematisation of complex numbers, emerging in 16th-century mathematics, has its conceptual and historical roots in a decisive shift in 14th-century modal semantics. This shift transformed the absolutely impossible into something imaginable and understandable, and the
imaginable into something mathematically operable.

In ancient and medieval logic and mathematics, necessarily empty terms — i.e., those terms signifying something intrinsically contradictory and therefore absolutely impossible — and the square roots of negative numbers occupied the same conceptual space: both were dismissed as inconceivable, as violations of the boundaries of rational thought itself. The parallel is not incidental. It reflects a shared metaphysical commitment to the limits of the thinkable.

What breaks this impasse is a profound semantic reorientation. In late-14th-century modal logic, most notably in the work of Marsilius of Inghen and his followers, absolute impossibilities are drawn into the logical domain: while not real, there are conceivable; they remain nonexistent but are manipulable.

The reception of this new semantics of imaginable impossibilities across the 15th and 16th centuries was widespread and influential This paper traces a direct line of conceptual continuity —through views, texts, and theories — from Marsilius of Inghen to Girolamo Cardano, arguing that new approach to imaginable impossibilities launched by late-medieval logicians is precisely what made the mathematical imagination of complex numbers possible.