Periodic Reporting for period 1 - i2 (The Impossible and the Imaginable: Late-Medieval Semantics of Impossibility and the Roots of Complex Mathematics)
Período documentado: 2023-05-01 hasta 2025-10-31
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 [root1]) 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 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 and knowledge dissemination results.
Yet by the 16th century complex numbers (i.e. those numbers having an imaginary part i = the numerical value of [root1]) 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 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 and knowledge dissemination results.
Two years into the project, i2’s first results shed new light on the history of logic, epistemology, metaphysics, and mathematical thought from the late 14th to the 16th century.
Our first Objective (O1) is to reassess mid-14th century “nominalist” semantics—i.e. theories on the relation between words, thoughts, and things by authors like John Buridan, Albert of Saxony, and Marsilius of Inghen. Unlike Buridan, and contrary to what one might expect from a “nominalist,” Marsilius considers chimeras and absolute impossibilities imaginable, understandable, and even proper referents. But how can we grasp such impossible concepts? If knowledge arises from reality, how can we form a concept of a chimera? While Marsilius omits these issues in his logic, he addresses them in some of his commentaries on Aristotle—e.g. Questions on Metaphysics and Questions on De Anima. Using provisional editions we prepared, we are answering these questions by examining Marsilius’ logic alongside his metaphysics. We found that this new semantics of necessarily empty terms is neither meaningless nor a mere technical tool divorced from our understanding—as shown in specialized communications and forthcoming publications.
O2 aims to reconstruct the “via Marsiliana” in the 15th–16th centuries, supporting our historical hypothesis of a tradition linking Marsilius’ semantics to Cardano’s early thoughts on complex numbers. We are mapping Marsilius’ influence, particularly at the University of Pavia, through texts later familiar to Cardano as a student and young master. As demonstrated in specialist communications and forthcoming works, we identified Giovanni Apollinare Offredi as a key figure in the reception and transmission of Marsilius’ views at the intersection of logic and natural philosophy. We are preparing a critical edition of Offredi’s Treatise on the First and Last Instant, a pivotal yet overlooked link in this chain of ideas.
O3 investigates three seemingly distinct senses of “imagination”: (a) a cognitive step elaborating sense perception; (b) a form of abstraction in mathematical construction; and (c) by the 14th century, the conceiving of things that are more or less—even absolutely—impossible. After two years, we found these senses less distinct than they appear, often merging in late medieval debates. This is exemplified by John Dorp of Leiden, an influential late-14th-century nominalist logician influenced by Buridan and Marsilius. As shown in scientific communications, Dorp connects impossible, perceptual, and mathematical imaginations.
Our first Objective (O1) is to reassess mid-14th century “nominalist” semantics—i.e. theories on the relation between words, thoughts, and things by authors like John Buridan, Albert of Saxony, and Marsilius of Inghen. Unlike Buridan, and contrary to what one might expect from a “nominalist,” Marsilius considers chimeras and absolute impossibilities imaginable, understandable, and even proper referents. But how can we grasp such impossible concepts? If knowledge arises from reality, how can we form a concept of a chimera? While Marsilius omits these issues in his logic, he addresses them in some of his commentaries on Aristotle—e.g. Questions on Metaphysics and Questions on De Anima. Using provisional editions we prepared, we are answering these questions by examining Marsilius’ logic alongside his metaphysics. We found that this new semantics of necessarily empty terms is neither meaningless nor a mere technical tool divorced from our understanding—as shown in specialized communications and forthcoming publications.
O2 aims to reconstruct the “via Marsiliana” in the 15th–16th centuries, supporting our historical hypothesis of a tradition linking Marsilius’ semantics to Cardano’s early thoughts on complex numbers. We are mapping Marsilius’ influence, particularly at the University of Pavia, through texts later familiar to Cardano as a student and young master. As demonstrated in specialist communications and forthcoming works, we identified Giovanni Apollinare Offredi as a key figure in the reception and transmission of Marsilius’ views at the intersection of logic and natural philosophy. We are preparing a critical edition of Offredi’s Treatise on the First and Last Instant, a pivotal yet overlooked link in this chain of ideas.
O3 investigates three seemingly distinct senses of “imagination”: (a) a cognitive step elaborating sense perception; (b) a form of abstraction in mathematical construction; and (c) by the 14th century, the conceiving of things that are more or less—even absolutely—impossible. After two years, we found these senses less distinct than they appear, often merging in late medieval debates. This is exemplified by John Dorp of Leiden, an influential late-14th-century nominalist logician influenced by Buridan and Marsilius. As shown in scientific communications, Dorp connects impossible, perceptual, and mathematical imaginations.
After two years, i2 has significantly expanded our understanding of late-medieval and Renaissance semantics of empty terms within the broader philosophical context of the time. Among other discoveries, our research has undeniably shown Marsilius of Inghen to be a major figure in the history of Western thought. We have identified new key authors who developed and disseminated the semantics of impossibilities beyond the boundaries of logic—for example, John Dorp. We have made significant advancements in our knowledge of Apollinare Offredi’s life and works, and thus in our understanding of the circulation of philosophical ideas in 15th-century Italy. For instance, we discovered or identified manuscripts of Offredi’s work—some of which were previously misattributed to Giovanni Marliani—and confirmed their true authorship (a critical edition is underway). These are just a few examples of the many historical discoveries and theoretical results achieved by members of the i2 team.
Through these advancements, i2 has traced the provable transmission of Marsilius’ semantics to Cardano’s intellectual milieu during his lifetime. Our future research will focus on Cardano’s reception of these texts and ideas, and we will further develop our initial findings on the interconnected meanings of “imagination” in late-medieval and Renaissance philosophy.
Together with other projects and initiatives, i2 is contributing to a renewed interest in premodern logic and mathematics—particularly in the history of semantics grappling with impossibilities. A notable example of this global interest is the 2023 conference "Existence and Nonexistence in the History of Logic", organized by Graziana Ciola (PI of the i2 project), Milo Crimi, and Calvin Normore. On this occasion, several contributions focused on the late-medieval and early modern logic of empty terms, engaging both with historical treatments of these themes in other traditions and with contemporary philosophical approaches to the logic of nonexistence in a mutually enlightening dialogue.
Over the past two years, i2 has actively fostered this broader dialogue through it research, dissemination activities, seminars, and website: https://i2erc.wordpress.com/(se abrirá en una nueva ventana).
Through these advancements, i2 has traced the provable transmission of Marsilius’ semantics to Cardano’s intellectual milieu during his lifetime. Our future research will focus on Cardano’s reception of these texts and ideas, and we will further develop our initial findings on the interconnected meanings of “imagination” in late-medieval and Renaissance philosophy.
Together with other projects and initiatives, i2 is contributing to a renewed interest in premodern logic and mathematics—particularly in the history of semantics grappling with impossibilities. A notable example of this global interest is the 2023 conference "Existence and Nonexistence in the History of Logic", organized by Graziana Ciola (PI of the i2 project), Milo Crimi, and Calvin Normore. On this occasion, several contributions focused on the late-medieval and early modern logic of empty terms, engaging both with historical treatments of these themes in other traditions and with contemporary philosophical approaches to the logic of nonexistence in a mutually enlightening dialogue.
Over the past two years, i2 has actively fostered this broader dialogue through it research, dissemination activities, seminars, and website: https://i2erc.wordpress.com/(se abrirá en una nueva ventana).