The Impossible and the Imaginable: Late-Medieval Semantics of Impossibility and the Roots of Complex Mathematics

2026-07-22

"A popular view in the history of philosophy and science holds that we only conceive what is in some sense possible and cannot truly understand anything intrinsically contradictory. The Chimera is the paradigmatic example of such an inconceivable absolute impossibility in medieval logic. ""Chimera"" is a necessarily empty term lacking signification and reference. Analogously premodern mathematics dismisses the square root of a negative number as the impossible result of an impossible operation, i.e. something as nonsensical as the Chimera. Yet by the 16th century complex numbers–i.e. those numbers having an imaginary part i = the numerical value of [root–1]–start being used and problematised. Somewhere down the line these ""impossible numbers"" had become conceivable and manipulable. How and why did this shift happen? i2 seeks the answer in the late medieval semantics of necessarily empty terms. In the 14th century, Marsilius of Inghen (ca.1341-96) makes terms like ""Chimera"" properly signifying, understandable and referential in his semantics. Marsilius' account of imaginable absolute impossibilities is widely influential in the 15th and 16th centuries. Girolamo Cardano–credited with the philosophical problematisation of complex numbers–seems at least partly aware of this Marsilian tradition. Marsilius' semantics of imaginable impossibilities, its later reception and its possible influence on Cardano have not yet been explored. This is what i2 is set to do for the first time, through a key-concept analysis and the applications of the method of historical and rational reconstruction to a rich textual corpus. i2 will produce a groundbreaking account of the rise of complex mathematics in relation to the medieval semantics of impossibility and imaginability. Based at Radboud's Center for the History of Philosophy and Science, and advised by an international board of scholars, i2 will deliver high-impact outputs in top journals, two books and knowledge dissemination results."

Netherlands

2025-12-15