Medieval Philosophy of Mathematics

The trends of Platonism which proved to be the most influential throughout the Renaissance were born roughly around the same period as the Greek corpus attributed to the Egyptian sage Hermes Trismegistus. They resulted from the rich intermingling of Greek philosophy with other Near Eastern cultures since the time of Alexander the Great. It is not by chance, then, that their fortunes were bound together until the Early Modern period. Legend has it that Cosimo de’ Medici was highly impressed by (...) the Platonic wisdom of the Greek émigrés visiting Florence in 1439, during the Council of Union between the Eastern and Western Churches, and particularly by the eminent philosopher George Gemistos Plethon. More than twenty years later, Cosimo entrusted a young Marsilio Ficino with the task of translating into Latin a Greek manuscript of Plato’s dialogues, possibly bequeathed by the Byzantine emperor, if not by Plethon himself. Before completing his rendering of the first series of ten dialogues, Ficino presented his elderly patron with the Pimander, a translation of fifteen Greek treatises on theology and occult lore by the “thrice greatest” Mercury or Hermes, believed to be the first in a venerable tradition of ancient sages which culminated in Plato. Certainly, these and similar newly recovered collections helped to shape and enrich the intellectual life of the emergent Renaissance. Their novelty and relevance, however, tended to be overstated in some historiographical perspectives. Fortunately, profound critical studies of the various sources from the Platonic, Neoplatonic, and Hermetic traditions have multiplied since the 19th century, gradually providing a clearer picture of the extent and nature of their influence on Renaissance and Early Modern scholars. Some of the most interesting topics discussed currently regard the lines of continuity between the medieval and Renaissance receptions of Platonism and Hermetism. Indeed, the Latin, Arabic, Hebrew, and Byzantine Middle Ages offer an immense repository of Platonic and Hermetic wisdom to Renaissance humanists and philosophers, which includes new theoretical and practical approaches, interpretative methods, translations, and commentaries. Only after elucidating these elements of continuity and change can one adequately ponder the distinctive character and originality of Renaissance Platonists and Hermeticists. Another hotly debated issue since Lynn Thorndike’s pioneering studies is the role of these ancient and medieval traditions in the development of experimental sciences and the emergence of the scientific revolution around the 16th and 17th centuries. (shrink)

…sit necessaria sciencia mathematice ad bona anime procuranda.Scientific humanism in the 15th and 16th century witnessed the spread of Greek and Arabic mathematics, whose reading was disciplined by philological research, enriched by the practical sense of abacus masters and diffused by the press. This doesn't mean that before this time many of these works were totally unknown. Around the 13th century mathematics scholars were already familiar with the work of Theodosius, Archimedes, Vitruvius, the Banū Mūsā brothers and so on; however, (...) these works were confined to small intellectual circles, and turned into an object for philosophical speculation.This was the case of... (shrink)

Responsione nostra disputationem cum L. Novák prosequimur, qui tractationem nostram, cui titulus “Různá pojetí matematiky u vybraných autorů od antiky po raný novověk”, impugnavit. Impugnatio a L. Novák sub titulo “Tomáš Akvinský instrumentalistou v matematice?” conscripta ansam praebuit nobis ad nonnulla, quae dixeramus, non solum clarius, sed etiam latius ac profundius explananda. Qua in re inprimis ad hoc attendimus, quomodo S. Thomas mathematicam, scientiasque medias necnon philosophiam intellexerit. Adhuc in nostra sententia sistimus, duplicem scil. ac valde diversam interpretationem harum disciplinarum (...) proponi posse. Quamquam multi textus testantur, S. Thomam mathematicam scientiasque medias realistice intellexisse, inveniuntur tamen apud eum dicta nonnulla, quae interpretationi favent instrumentalisticae. Hanc duplicem interpretationem prae oculis ponere multum iuvat, nostro iudicio, ad subsequentem intellectualem historiam, praecipue modernae aetatis, adaequate intelligendam.In our contribution we continue our discussion with L. Novák, who criticised our paper “Různá pojetí matematiky u vybraných autorů od antiky po raný novověk.” Novák’s critique titled “Tomáš Akvinský instrumentalistou v matematice?” served as an incentive for us not only to clarify certain points, but also to deepen our original exposition. We focused on Aquinas’s understanding of mathematics, the middle sciences and philosophy. We still insist that two substantially different interpretations of these disciplines are possible. On the one hand, there is much evidence for Aquinas’s realistic approach to mathematics and the middle sciences. On the other hand, ideas can also be found in Aquinas’s texts supporting an instrumentalist reading. In our opinion, it is important to point out these two approaches to the mathematical sciences in order to adequately understand the subsequent evolution of the history of ideas, especially in the modern period. (shrink)

In his De primo et ultimo instanti, Walter Burley paid careful attention to continuity, assuming that continua included and were limited by indivisibles such as instants, points, ubi, degrees of quality, or mutata esse. In his Tractatus primus, Burley applied the logic of first and last instants to reach novel conclusions about qualities and qualitative change. At the end of his Quaestiones in libros Physicorum Aristotelis, William of Ockham used long passages from Burley’s Tractatus primus, sometimes agreeing with Burley and (...) sometimes disagreeing. How may this interaction between Burley and Ockham be understood within its historical context? (shrink)

This paper focuses on the exegetical proposal of the Tractatus de sex dierum operibus by Thierry of Chartres and it is tasked with analyzing the twofold interpretative framework adopted by the Cancelor: first, the accordance between the narration of Genesis and the heuristic models of physical and cosmological causality; second, the mathematical theology, which revises the work of creation according to an arithmological approach. The study is divided into two parts which follow the structure of the Tractatus. In the first (...) part, I analyse the physical plausibility of Christian cosmogony and subsequently the conception of matter with regard to both William of Conches’ inordinatum and the hermetic locus mundi. In the second part, I examine how the numerical discourse interprets the creatural unfolding and dependence on God, as well as the creation per verbum and the individuality of each created being. My purpose is threefold: first, to test the inner philosophical coherence of the Tractatus; second, to interpret specific theoretical points at the light of Thierry’s commentaries on Boethius’ De Trinitate, and finally, to relate the arithmetical issues of the Tractatus to Thierry's Commentum super Arithmeticam Boethii. (shrink)

The Scientific Revolution was far from the anti-Aristotelian movement traditionally pictured. Its applied mathematics pursued by new means the Aristotelian ideal of science as knowledge by insight into necessary causes. Newton’s derivation of Kepler’s elliptical planetary orbits from the inverse square law of gravity is a central example.

The fundamental idea of a Neoaristotelian inherence ontology of mathematical entities parallels that of an Aristotelian approach to the ontology of universals. It is proposed that mathematical objects are nominalizations especially of dimensional and related structural properties that inhere as formal species and hence as secondary substances of Aristotelian primary substances in the actual world of existent physical spatiotemporal entities. The approach makes it straightforward to understand the distinction between pure and applied mathematics, and the otherwise enigmatic success of applied (...) mathematics in the natural sciences. It also raises an interesting set of challenges for conventional mathematics, and in particular for the ontic status of infinity, infinite sets and series, infinitesimals, and transfinite cardinalities. The final arbiter of all such questions on an Aristotelian inherentist account of the nature of mathematical entities are the requirements of practicing scientists for infinitary versus strictly finite mathematics in describing, explaining, predicting and retrodicting physical spatiotemporal phenomena. Following Quine, we classify all mathematics that falls outside of this sphere of applied scientific need as belonging to pure, and, with no prejudice or downplaying of its importance, ‘recreational’, mathematics. We consider a number of important problems in the philosophy of mathematics, and indicate how a Neoaristotelian inherence metaphysics of mathematical entities provides a plausible answer to Benacerraf’s metaphilosophical dilemma, pitting the semantics of mathematical truth conditions against the epistemic possibilities for justifying an abstract realist ontology of mathematical entities and truth conditions. (shrink)

The creators of modern science, such as Galileo Galilei and Isaac Newton, begun their work based on the principles which created generations before them. Nature Research has its origins back to antiquity. The Middle Ages is in general wrongly referred to as "The Dark Ages". Therefore dominates the incorrect opinion that the science had not been developed. In fact, now forgotten medieval scholars in comparison with antiquity had brought a fundamental change in the view of nature. The Christian doctrine of (...) God, who created the universe out of nothing and gave the world laws, prompted them to generate the principles of empirical research. Foundations of modern science were laid in "Renaissance of 12th Century", when mathematics, physics and engineering has been developed. Subsequently, several medieval natural scientists dedicated their lives to research in various areas of mathematics, physics, astronomy, optics, mechanics and other subjects of science. Renaissance scholars were able to build on this platform and Newton, using the methods of new mathematical formula could establish a modern natural science. (shrink)

Throughout his works, St. Augustine offers at least nine distinct views on the nature of time, at least three of which have remained almost unnoticed in the secondary literature. I first examine each these nine descriptions of time and attempt to diffuse common misinterpretations, especially of the views which seek to identify Augustinian time as consisting of an un-extended point or a distentio animi . Second, I argue that Augustine's primary understanding of time, like that of later medieval scholastics, is (...) that of an accident connected to the changes of created substances. Finally, I show how this interpretation has the benefit of rendering intelligible Augustine's contention that, at the resurrection, motion will still be able to occur, but not time. (shrink)

Summary The practice of the disputatio in the medieval universities gave rise to a particular literary genre, the questio. This genre is caracterised by the production of arguments in favour of or against the thesis submitted for questio, before the author develops his own answer. This genre is common to philosophy and theology. But to present a mathematical problem in the form of the questio may seem paradoxical since it leads to the production of false proofs. We shall examine three (...) texts of the 14th and 15th centuries in which is raised the question of the incommensurability of the diagonal and the side of a square: Nicole Oresme's Questions on the geometry of Euclid, an anonymous question from a 15th century manuscript and Blasius of Parma's Questions on Thomas Bradwardine's treatise on ratios. We shall establish a relationship between these three texts. We will show how the mathematical content is combined with the literary genre of the questio. We shall particularly concentrate on the false proofs and consider their status. Finally, we shall ask ourselves how these texts are linked with a possible didactic purpose. Résumé La pratique de la «dispute» dans les universités médiévales a donné lieu à un genre littéraire particulier, la Question. Ce genre est caractérisé par la production d'arguments pour ou contre la thèse soumise à questionnement, avant que l'auteur ne développe sa propre réponse. Ce genre est courant en philosophie et en théologie. Mais présenter un problème mathématique sous la forme d'une Question peut sembler paradoxal: cela conduit en effet à la production de démonstrations fausses. Nous examinerons trois textes des xiv e et xv e siècles dans lesquels est posée la question de l'incommensurabilité de la diagonale et du côté d'un même carré: Les questions sur la géométrie d'Euclide de Nicole Oresme, une question anonyme dont le manuscrit est daté du xv e siècle et les Questions sur le traité des rapports de Thomas Bradwardine rédigées par Blaise de Parme. Nous établirons une filiation entre ces trois textes. Nous montrerons comment s'articule le contenu mathématique avec le genre littéraire de la Question. Nous porterons une attention particulière aux démonstrations fausses sur le statut desquelles nous nous interrogerons. Et nous nous demanderons quel lien entretiennent ces textes avec un éventuel enseignement. (shrink)

Review essay: Les mathématiques infinitésimales du IXe au XIe siècle. Volume 4: Ibn al-Hatham, méthodes géométriques, transformations ponctuelles, et philosophie des mathématiques (London: Al-Furq¸n Islamic Heritage Foundation, 2002), pp. xiii+1064+vi ¤ 106.71 ISBN 1 87399 260 2.

One of the basic elements of Nicholas of Cusa's philosophy of mathematics is his theory of mathematical objects as “entities-of-reason” (entia rationis). He refers to these as being “abstracted from sensible things”. That is why it is possible to assume that Nicholas bases his theory of mathematics on Aristotle's philosophy of mathematics. Aristotle too describes mathematical objects as coming into being through abstraction (ex aphaireseos). The author analyses Cusa's understanding of abstraction in De docta ignorantia and De mente and tries (...) to show that – according to Nicholas of Cusa – the abstraction, which is ens rationis, simultaneously stimulates the human mind to produce mathematical objects from within itself. The author attempts to show how Cusa's philosophy of mathematics is not directly based on Aristotle's philosophy of mathematics – Aristotle is not an abstractionist and does not ascribe the existence of mathematical objects to the mind of the mathematician – but on the abstractionist interpretation of Aristotle by Alexander of Aphodisias, who was followed by the predominantly neoplatonist commentators of Aristotle. These commentators did not see any important differences in the metaphysical or epistemological underpinnings of abstractionism and the so-called projectionism, i.e. the theory according to which mathematical objects pre-exist in the soul. (shrink)

The ArgumentThe major part of the mathematical “classics” in Hebrew were translated from Arabic between the second third of the thirteenth century and the first third of the fourteenth century, within the northern littoral of the western Mediterranean. This movement occurred after the original works by Abraham bar Hiyya and Abraham ibn Ezra became available to a wide readership. The translations were intended for a restricted audience — the scholarly readership involved in and dealing with the theoretical sciences. In some (...) cases the translators themselves were professional scientists (e.g., Jacob ben Makhir); in other cases they were, so to speak, professional translators, dealing as well with philosophy, medicine, and other works in Arabic.In aketshing this portrait of the beginning of Herbrew scholarly mathematics, my aim has been to contribute to a better understanding of mathematical activity as such among Jewish communities during this period. (shrink)

SOME MODERN THOMISTS claiming to follow the lead of Thomas Aquinas, hold that the objects of the types of mathematics known in the thirteenth century, such as the arithmetic of whole numbers and Euclidean geometry, are real entities. In scholastic terms they are not beings of reason but real beings. In his once-popular scholastic manual, Elementa Philosophiae Aristotelico-Thomisticae, Joseph Gredt maintains that, according to Aristotle and Thomas Aquinas, the object of mathematics is real quantity, either discrete quantity in arithmetic or (...) continuous quantity in geometry. The mathematician considers the essence of quantity in abstraction from its relation to real existence in bodily substance. "When quantity is considered in this way," he writes, "it is not a being of reason but a real being. Nevertheless it is so abstractly considered that it leaves out of account both real and conceptual existence." Recent mathematicians, Gredt continues, extend their speculation to fictitious quantity, which has conceptual but not real being; for example, the fourth dimension, which by its essence positively excludes a relation to real existence. According to Gredt this is a special, transcendental mathematics essentially distinct from "real mathematics," and belonging to it only by reduction. (shrink)

After a survey of disagreements among Thomists on the nature of mathematical abstraction, the author cites Aquinas's text Scriptum super libros Sententiarum, I, d. 2, a.3 (a late text inserted in an older work). It assimilates the objects of mathematics to those of logic, thus admitting a remote foundation in reality but not the direct one of the concepts of the physical sciences.