BRADWARDINE, THOMAS.

(b. England, c. 1300; d. Lambeth, England, 26 August 1349),

mathematics, natural philosophy, theology. For the original article on Bradwardine see DSB, vol. 2.

Much has been published on Bradwardine since the original DSB article was written. New editions of several of Bradwardine’s works have appeared. In comparison to the original article, Bradwardine’s tentative birth date is here set at 1300, rather than left open within the range 1290 to 1300. This fits with his listing as a bachelor of arts in a Balliol College document of August 1321, before he moved to Merton College about 1323.

Scholarly Career Path. Although researchers still have no proof that Thomas Bradwardine’s works on mathematics and philosophy were not written early in his career, some of them at least were written after he began his theological studies. His Tractatus de proportionibus, for instance, appeared in 1328, while he was bachelor of the Sentences from 1332 to 1333. Bachelor lectures on the Sentences ordinarily came after seven years of theological study, so that he might have begun to study theology about 1325 to 1326. Like his De proportionibus, his De continuo(On the continuum), an unabashedly multidisciplinary work, running through all the main arts disciplines as well as optics and medicine to prove that no continuum is composed of indivisibles, was also written when he had already begun his studies of theology. However, Lauge Olaf Nielsen, the editor of Bradwardine’s De incipit et desinit(On beginning and ceasing), argues that William of Ockham changed his position on the exposition of propositions involving the words “to begin” and “to cease” between what he argued in Part I of his Summa logicae and what he argued in Part II, and this as a result of Bradwardine’s explicit arguments against what Ockham said in Part I. If this is the case, then Bradwardine must have composed his De incipit et desinit before 1323 when Ockham completed his Summa logicae.

In his theological works, including his question on future contingents, now recognized to have been part of his bachelor lectures on Peter Lombard’s Sentences(delivered about 1332–1333), as well as in his De causa Dei(completed about 1344), Bradwardine makes clear use of his philosophical training. He argued that God knows the future just as God knows the past. In the 876 printed folio pages of De causa Dei, Bradwardine argues against those he calls Pelagians (who argue that by their actions humans can earn salvation) that no one is saved without grace, that God is immutable and does not “change his mind,” and yet, somehow, predetermination with respect to God is not incompatible with human free will. For a full understanding of Bradwardine’s positions, it is advisable to examine his theological, as well as his philosophical and mathematical works.

Some of the logical works attributed to Bradwardine have now been published. He likely composed his De insolubilibus as a young teacher in the arts faculty at Oxford. Whereas this work on logical paradoxes, such as the statement “What I am saying is false,” belongs to a tradition of works in the same genre, Stephen Read argued in a 2002 article that Bradwardine’s ideas concerning the liar paradox form “arguably a genuine and original solution” (p. 190).

On Beginning and Ceasing. More obviously related to the history of science, although also in a logical framework, is Bradwardine’s De incipit et desinit, on beginning and ceasing. The historiography of the thirteenth and fourteenth centuries development of ideas on beginning and ceasing, or on first and last instants, is still in flux. This is, in part, because of the uncertain status of the work of Peter of Spain on the subject. Both the identity of the author and the original content of Peter of Spain’s work, before modifications in later centuries, have been opened to doubt. Nevertheless, Bradwardine’s view that neither successive entities such as motion nor permanent things such as animals have a last instant of existence was widely held.

Scholarly focus on the issue of first and last instants might have arisen both within natural philosophy and within logic (especially in view of so-called supposition theory. Aristotle had argued in the Physics that in a change from one permanent state to another, the instant of transition should always be assigned to the later state. So, for instance, if water is heated until it becomes air (a different permanent substance), there will be no last instant of being water, but only a first instant of being air (when the temperature has become so high that it is incompatible with the continued existence of water).

Within logic, there was a long tradition of analyzing the truth or falsity of propositions in terms of the supposition (Lat. suppositio) of their terms, either for things in the world, approximately corresponding to the later concept of the reference of terms, or for concepts in the mind or for the actual words (as the phrase “what I am saying” in the proposition “What I am saying is false” might be said to supposit for or refer to the verbal expression itself). Whereas some words in propositions were supposed to have supposition for things in the world, other terms, called syncategoremata, were supposed not to have supposition themselves, but to affect the supposition of other terms in the proposition. The terms incipit (begin) and desinit (cease) were considered to be syncategorematic terms. It was often said that propositions containing syncategorematic terms must be unpacked (expounded) into other propositions in order to check their truth or falsity. Thus the proposition “Air begins to be present” would be expounded into the compound proposition “Air is now present and immediately before this it did not exist.” This exposition fits with the fact that permanent things such as air have a first instant of existing.

But what about propositions containing terms referring to so-called successive entities such as motion and time? What exposition should be given for the proposition “Socrates begins to run”? According to Bradwardine and most other fourteenth-century logicians, running, like other successive entities, does not have a first instant of existence. When a person is standing still and then begins to move, there is no first instant when he has moved. Thus the proper exposition would be “Socrates is not now running and immediately after this he will be running.”

By the 1320s, scholars were well aware that William of Ockham and others had said that in the outside world there are no “successive entities” (res successivae). There are only permanent entities, such as Socrates. What is meant by saying “Socrates is running” is that Socrates is now in a given location, immediately before now he was in a different location, and immediately after now he will be in a different location. Bradwardine begins De incipit et desinit with suppositions and distinctions and then turns to conclusions, objections against each conclusion, and replies. He distinguishes between permanent things, such as “animal,” and “white,” as opposed to successives, such as “motion,” “time,” and “to traverse.” But although Bradwardine uses the term successive thing, he recognizes the ontological minimalism of thinkers such as William of Ockham, when he explains that terms are said to be of permanent things, not because they do not signify successive things, but because they can signify these things even though they are not moved. Likewise terms are said to be of successive things, not because they do not signify permanent things, but because they do not signify these permanent things unless these things are moved (p. 48). At one point Bradwardine, in a way atypical for fourteenth-century authors, even calls Ockham by name, calling him “a modern thinker, brother William of Ockham” (Bradwardine/Nielsen, 1982, p. 74).

Although Bradwardine’s De incipit et desinit had far less influence than his Tractatus de proportionibus, it was, nevertheless, a work very typical of Oxford University at this time, particularly in the objections that Bradwardine listed after every conclusion and his replies. A typical objection runs as follows: if no motion has a last instant of being, then if a body is in motion at any given instant, it necessarily continues to move after that instant (Bradwardine/Nielsen, 1982, p. 54). This makes a future contingent necessary. Bradwardine solves this problem by distinguishing between two kinds of future contingents: first, there are those that in no way can be false, and second there are future contingents that for the same instant at which they could be true could also be false. Thus, from the proposition “Socrates eats,” the future contingent “Socrates will eat” could be true or false. From the present proposition “Socrates is moving,” however, the future contingent “Socrates will move” is necessarily true in the first way and cannot be false (pp. 55–56). Besides future contingents, Bradwardine uses his discussion of beginning and ceasing to give his students practice in analyzing the relations of indivisibles and continua and in using many other logical tools favored in fourteenth-century universities.

Bradwardine’s Tractatus de proportionibus was already well-known at the time the original DSB article was written. Successive attempts to explain Bradwardine’s use of mathematics in formulating his dynamical rule may be found in the supplementary bibliography below. Developments in thinking about Bradwardine’s De continuo can also be found in the bibliography, although a printed version of the text itself continues to be available only in John Murdoch’s 1957 dissertation. George Molland’s edition of the Geometria speculativa appeared in 1989 and his expert analysis already in earlier articles is listed below.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY BRADWARDINE

De incipit et desinit. In “Thomas Bradwardine’s Treatise on ‘incipit’ and ‘desinit.’ Edition and Introduction,” edited by Lauge Olaf Nielsen. Cahiers de l’Institute du moyen âge grec et latin 42 (1982): 1–83. The same volume includes a logical work on Ockham’s doctrine of consequences and a work Opus Artis Logicae, both sometimes attributed to Bradwardine. See Jan Pinborg. “A Logical Treatise Ascribed to Bradwardine.” In Studi sul XIV secolo in memoria di Anneliese Maier, edited by Alfonso Maierù and Agostino Paravicini Bagliani, 27–55. Rome: Edizioni di Storia et Letteratura, 1981.

De insolubilibus. In “La problématique des propositions insolubles du XIII siècle et du début du XIV, suivie de l’édition des traités de William Shyreswood, Walter Burleigh et Thomas Bradwardine,” edited by M. L. Roure. Archives d’histoire doctrinale et littérarie du moyen âge27 (1970): 205–326. A new edition of the De insolubilibus is scheduled to appear in early 2008: Stephen Read, ed., Thomas Bradwardine's Insolubilia: A New Edition from the Manuscripts, with English Translation and Substantial Introduction. Dallas: Medieval Texts and Translations; and Louvain: Peeters.

De continuo. In Geometry and the Continuum in the FourteenthCentury: A Philosophical Analysis of Thomas Bradwardine’s ‘Tractatus de Continuo,’ edited by John E. Murdoch. PhD diss., University of Wisconsin, 1957. The microfilm of this dissertation from University Microfilms currently remains the best source for this text.

Arithmetica Speculativa. H. L. L. Busard attempts to identify which, if any, work by this name should be ascribed to Bradwardine in “Zwei mittelalterliche Texte zur theoretischen Mathematik: Die Arithmetica speculativa von Thomas Bradwardine und die Theorica numerorum von Wigandus Durnheimer.” Archive for History of Exact Sciences53 (1998): 97–124.

Geometria Speculativa. In Thomas Bradwardine:GeometriaSpeculativa, edited and translated by A. George Molland. Stuttgart, Germany: Franz Steiner Verlag Wiesbaden, 1989. Based on Molland’s 1967 Cambridge dissertation, this includes an introduction, English translation, and commentary. For a fuller analysis, see Molland’s 1978 article listed below.

Tractatus de futuris contingentibus. Edited by Jean-François Genest. Recherches augustiniennes 14 (1979): 249–336.

De memoria artificiali. Selections from this work are published in The Medieval Craft of Memory: An Anthology of Texts and Pictures, edited by Mary Carruthers and Jan M. Ziolkowski. Philadelphia: University of Pennsylvania Press, 2002. Latin text in Mary Carruthers, ed., “Thomas Bradwardine: ‘De memoria artificiali adquirenda.’” Journal of Medieval Latin2 (1992): 25–43. Although James Weisheipl doubted the ascription of this work to Bradwardine, Bradwardine appears as author in the two existing manuscripts. Mary Carruthers argues that the work is by Bradwardine and that it was written soon after Edward III’s victory at Berwick on 20 July 1333. See Mary Carruthers, The Book of Memory: A Study of Memory in Medieval Culture. Cambridge, U.K.: Cambridge University Press, 1990.

OTHER SOURCES

Dolnikowski, Edith Wilks. Thomas Bradwardine: A View of Time and a Vision of Eternity in Fourteenth-Century Thought. Leiden, Germany: Brill, 1995. Considers Bradwardine’s view of time as it appears in his mathematical, philosophical, and theological works.

Genest, Jean-François. Prédétermination et Liberté Crée à Oxford au XIV Siècle.Buckingham contra Bradwardine. Paris: Vrin, 1992. Includes the text of Buckingham, Determinatio de contingentia futurorum.

Genest, Jean-François and Katherine H. Tachau. “Le lecture the Thomas Bradwardine sur les Sentences.” Archives d’histoire doctrinale et littéraire du moyen âge56 (1990): 301–306. Some fragments of Bradwardine’s commentary on the Sentences, previously thought to be lost, including the question on future contingents, that had been known as a separate text.

Kaluza, Zénon. “La prétendue discussion parisienne de Thomas Bradwardine avec Thomas de Buckingham. Témoignage de Thomas de Cracovie.” Recherches de théologie ancienne et mediévale42 (1976): 219–236. Shows that, contrary to an often repeated story, the text in MS Paris BNF 16409 does not refer to a live disputatio in aula at Paris between Bradwardine and Buckingham, but instead consists of notes by a later scholar taken from written texts.

Molland, A. George. “An Examination of Bradwardine’s Geometry.” Archive for History of Exact Sciences19 (1978): 113–175. Reprinted in A. George Molland, Mathematics and the Medieval Ancestry of Physics. Aldershot, U.K.: Variorum, 1995. Also in the same volume is a reprint of Molland’s “The Geometrical Background to the ‘Merton School,’” first published in British Journal for the History of Science4 (1968): 108–125.

——— “Addressing Ancient Authority: Thomas Bradwardine and Prisca Sapientia.” Annals of Science53 (1996): 213–233.

Murdoch, John Emery. “Thomas Bradwardine: Mathematics and Continuity in the Fourteenth Century.” In Mathematics and Its Applications to Science and Natural Philosophy in the Middle Age: Essays in Honor of Marshall Clagett, 103–137. Cambridge, U.K.: Cambridge University Press, 1987. This includes the Latin text of the propositions of De continuo.

Read, Stephen. “The Liar Paradox from John Buridan Back to Thomas Bradwardine.” Vivarium40, no. 2 (2002): 189–218. Argues that Bradwardine’s solution to the liar paradox was genuine and original and that it has not been appreciated in part because the adaptation of it by John Buridan and Albert of Saxony seriously weakens it.

Sbrozi, Marco. “Metodo matematico e pensiero teologico nel ‘De causa Dei’ di Thomas Bradwardine.” Studi medievali 31 no. 3 (1990): 143–191.

Spade, Paul Vincent. “Insolubles.” Stanford Encyclopedia ofPhilosophy. Available from http://plato.stanford.edu/entries/insolubles(2005) Puts Bradwardine’s theory of insolubles into context and states that it was “enormously influential on later authors.”

Sylla, Edith Dudley. “Medieval Concepts of the Latitude of Forms: The Oxford Calculators.” Archives d’Histoire Doctrinale et Littéraire du Moyen Age 40 (1973): 223–283. Describes John Dumbleton’s use of latitudes of proportion to express Bradwardine’s rule.

———. “Compounding Ratios: Bradwardine, Oresme, and the First Edition of Newton’s Principia.” In Transformation and Tradition in the Sciences: Essays in Honor of I. Bernard Cohen, edited by Everett Mendelsohn, 11–43. Cambridge, U.K.: Cambridge University Press, 1984.

——— “Thomas Bradwardine’s De continuo and the Structure of Fourteenth-Century Learning.” In Texts and Contexts in Ancient and Medieval Science: Studies on the Occasion of JohnE. Murdoch’s Seventieth Birthday, edited by Edith Sylla and Michael McVaugh, 148–186. Leiden, Germany: Brill, 1997.

———“The Origin and Fate of Thomas Bradwardine’s Deproportionibus velocitatum in motibus in Relation to the History of Mathematics.” In Mechanics and Natural Philosophy before the Scientific Revolution, edited by Walter Roy Laird and Sophie Roux. Boston Studies in the Philosophy of Science, vol. 254. New York: Springer Verlag, 2007. Explains how Bradwardine’s rule for the proportions relating forces, resistances, and velocities takes advantage of the methods of compounding ratios found in works on music and in Latin translations of Euclid’s Elements such as that of Campanus, which was used by Bradwardine.

Edith Dudley Sylla