Breaking the lines is the research project that I carried out at KU Leuven as an FWO Junior Postdoc (2021-2024).
It is centered on one fundamental issue in the history of philosophy and science, namely, the debate about atomism, particularly as explored by late medieval thinkers. The aim of the project is to shed new light on the mathematical approach employed by medieval scholars in addressing a central problem in natural philosophy: the nature of continuum.
This medieval debate was marked by two opposing stances on the nature of geometrical extension: ‘divisibilism’ (which claims that everything which is in a space – be it a geometric figure or a body – is indefinitely divisible) and ‘indivisibilism’ (which maintains that extended spaces are grounded on elements – atoms – that cannot be divided). These positions were held by Aristotelians and atomists, respectively. 13th- and 14th-century discussions on indivisibles were grounded in the reading of Aristotle’s works on natural philosophy – especially the Physics – but Islamicate sources, such as Avicenna’s and Averroes’s works, enriched this debate substantively.
A distinctive feature of the debate concerning the continuum between Aristotelians and atomists was the use of mathematical arguments to defend their respective positions. Specifically, geometrical reasoning, employing techniques like parallel or radial projection, animated these discussions. In geometry, assuming the existence of indivisibles (points or indivisible lines) led to paradoxical conclusions, such as the diagonal and side of a square, whose ratio is impossible in Euclidean geometry, sharing a common measure, i.e., the indivisible itself. These arguments gained prominence in the realm of natural philosophy, forging a renewed and inseparable link between mathematics and physics.
From a contemporary viewpoint, it’s accurate to assert that Aristotelian philosophy indirectly contributed to the medieval debate on atomism by offering geometrical explanations, as the core Aristotelian writings did not explicitly delve into mathematics. However, medieval thinkers engaging with atomism could refer to a ‘Aristotelian’ mathematical work, the De lineis indivisibilibus. Translated into Latin in the 13th century and possibly authored by Theophrastus, this concise yet pivotal text was ascribed to Aristotle throughout the Middle Ages. Tracing back to Xenocrates, the theory of indivisible lines postulates the existence of a primary and indivisible unit of linear measurement, by which even the immeasurable magnitudes can be related. The De lineis indivisibilibus argued against the theory of indivisible lines, hence informing the research project’s name.
Who translated the De lineis indivisibilibus into Latin? Scholarship credited Robert Grosseteste (ca. 1170 – 1253), 13th-century polymath, to be responsible for the first Latin translation of the text. Although this assertion necessitates further validation, Grosseteste’s familiarity with De lineis indivisibilibus aligns with some of his philosophical principles, notably in his De luce and Glosses to the Physics, where he expounds on his original account of atomism.
The pseudo-Aristotelian text offered no unequivocal interpretation. Did it contribute to delineating a clear boundary between atomists and Aristotelians or foster distinct forms of atomism? Or did it contribute to developing special forms of atomism? How were its geometric reasonings received and exploited? Only a careful consideration of the reception of this fundamental source can unequivocally define the earliest medieval roots of Renaissance and Modern atomism and shed new light on this crucial moment in the history of philosophy and science.
One certainty is that the De lineis indivisibilibus enjoyed significant success in the late Middle Ages, evident from its preservation in over seventy codices housed in libraries across Europe!