Several factors combined to bring about the work of the Oxford Calculators. First of all, undergraduate education at Oxford in the fourteenth century emphasized taking part in disputations as a way of training students to detect fallacies, argue logically, remember significant detail, think quickly, and so forth. The curriculum of the Faculty of Arts included grammar, logic, the quadrivium (arithmetic, geometry, astronomy, and music), natural philosophy, ethics, and metaphysics. The most important tool used in disputations was logic, especially the ‘‘new logic,’’ which made use of the theory of supposition and related tools of analysis. All students took part in disputations on sophismata, that is perplexing propositions linked to a ‘‘case’’ with reference to which the truth or falsehood of the sophismata statement was to be judged. Typical calculatory sophismata from a relatively early period are those of Richard Kilvington, starting with the sophisma sentence, ‘‘Socrates is whiter than Plato begins to be white,’’ taken together with the case that Socrates is now white in a given degree, while Plato begins to increase his whiteness from having no whiteness at all. This case and the related sophisma sentence are perplexing because ‘‘whiter’’ is understood to involve a geometric ratio between Socrates’ whiteness and Plato’s whiteness, but if Plato begins to become whiter from not being white at all, Plato will begin from zero whiteness and the initial ratios between Socrates’ finite whiteness and Plato’s very earliest whitenesses will be greater than any given ratio, or potentially infinite. To resolve the sophism, the respondent had to determine whether as Plato or any other subject begins to become white, there is a first instant of being white with some degree of whiteness or only a last instant of having no whiteness at all. If there were a first instant of being white, then there would be a finite ratio between Socrates’ whiteness and Plato’s first whiteness, but if there is no first instant of Plato’s being white, as was normally assumed, then the ratios between Socrates’ whiteness and Plato’s whiteness will be greater than any given value as one considers times closer and closer to the beginning of his whitening (see Richard Kilvington 1320s/1990).
Only a few of Richard Kilvington’s 48 sophismata do not deal with situations involving natural philosophy and related issues of continuity and infinity, and thus Norman and Barbara Kretzmann are right in claiming in their edition of his Sophismata that it ‘‘may be taken as
A paradigm of the philosophical problems and techniques that characterize the work’’ of the Oxford Calculators. Interestingly, Kilvington’s series of commentaries on the works of Aristotle in the form of a few (usually about ten) but lengthy questions track the acceptance of the second major factor in the development of the Oxford Calculatory approach, namely Thomas Bradwardine’s new theory of the relations of proportions of forces to resistances, on the one hand, and velocities produced on the other, which appeared in 1328 in his On the Proportions of Velocities in Motions. According to Bradwardine’s new theory, the proportion of force to resistance must be “duplicated” for a velocity to be doubled, “triplicated” for the velocity to be tripled, and so forth - where ‘‘duplicated’’ not infrequently was rendered ‘‘doubled,’’ and ‘‘triplicated’’ as ‘‘tripled.’’ Thus the proportion of nine to one was said to be ‘‘double’’ the proportion ofthree to one, so that nine to one would, according to Bradwardine’s theory, produce a velocity double that produced by a proportion of three to one.
Given that Richard Kilvington’s Sophismata was earlier than Bradwardine’s On the Proportions of Velocities in Motions, it follows that the context of disputations on sophismata was more important than Bradwardine’s On Proportions in leading to the Oxford Calculatory movement, and yet Bradwardine’s mathematical physics certainly contributed to the impact that the Oxford Calculators had, first at the University of Paris, and later at Italian, Spanish, and eastern European universities. For the next 200 years and more, university student textbooks would include the mathematics of proportions needed for Bradwardine’s science of motion as well as the standard topics of logic and sophismata, etc. (Not discussed here are the external factors that may have contributed to the turn to quantification, but see Kaye (1998) for an argument that economic factors were important).
In earlier surveys of the work of the Oxford Calculators, Richard Kilvington played a relatively small part, but that was because none of his work was in print until the Sophismata were edited in 1990; because his Questions on Aristotle’s Physics, ascribed in manuscript to ‘‘Magister Richard,’’ had tentatively been assigned to Richard Swineshead; and because his Questions on Aristotle's On Generation and Corruption and Nicomachean Ethics, and on Peter Lombard’s Sentences had been little noticed. More recently Elzbieta Jung has begun to edit the rest of Kilvington’s corpus beyond the Sophismata. Eventually, we will be able to see the application of the Oxford Calculatory approach throughout the curriculum, including theology and Kilvington will take a central place among the Oxford Calculators.
Jung has shown that the order of composition of Kilvington’s works is (1) the Sophismata, before 1325; (2) the Questions on Aristotle's On Generation and Corruption; (3) Questions on Aristotle’s Physics; (4) Questions Aristotle's Ethics; and finally (5) Questions on the Sentences, which she dates as before the end of 1334. In the Sophismata there is nothing to indicate that Kilvington had any idea of Bradwardine’s new theory of the proportions of velocities in motions. In his Questions on Aristotle’s On Generation and Corruption, Kilvington seems to know Bradwardine’s usage according to which the proportion of nine to one is called ‘‘double’’ the proportion of three to one, but Kilvington rejects this approach because he says that in the case of proportions of lesser inequality (i. e., where the first term is less than the second), proportions become less by being duplicated, as the proportion of one to nine is less than the proportion of one to three. In his Questions on Aristotle's Physics, however, Kilvington accepts Bradwardine’s theory of the relation of proportions of force to resistance to velocities. We may suppose then that in the years around 1330, Bradwardine’s function came gradually to be accepted, and remained the dominant view for the next 200 years, despite a minority who protested against it, mostly on mathematical grounds. Jung has argued that Kilvington’s Questions on the Physics are before Bradwardine’s De proportionibus, and hence that Kilvington influenced Bradwardine in the introduction of ‘‘Bradwardine’s function’’ rather than vice versa. There is more evidence, however, that those who rejected Bradwardine’s function often took arguments against it from Kilvington’s work. It would be safer to say, then, that Bradwardine and Kilvington were sparring partners in this matter. In later years both Bradwardine and Kilvington, along with Walter Burley and others, were employed as members ofthe circle of Richard of Bury, Bishop of Durham (see articles on these individuals).
William Heytesbury’s 1335 Rules for Solving Sophismata follows mainly in the footsteps of Kilvington’s Sophismata, with some influence from the work of Bradwardine. In addition to discussing questions of dynamics - how velocities are related to the forces and resistances causing them - Bradwardine had also discussed the measure of motion with respect to effect (tanquam penes effectum). For instance, how should a velocity of rotation be measured? Should it be measured by the motion of the fastest moved point of the body rotating, or by some average of the velocities of the various parts or points of the body? Likewise, the question might be raised as to how motions nonuniform in time, that is, motions that are accelerated or decelerated regularly or irregularly, are to be dealt with. Heytesbury, Dumbleton, and later authors accept the so-called Merton mean-speed theorem, recognizing that a uniformly accelerated body will cover the same distance as it would cover if in the same period of time it moved uniformly with the velocity it had at the middle instant of the motion.
In Heytesbury’s Rules for Solving Sophismata natural philosophical issues are involved in many, but not all of the chapters. Thus the sixth chapter concerns the measures of motion in the categories of place, quality, and quantity, but the earlier chapters concern: (1) insoluble sentences, such as the liar paradox; (2) paradoxes involving the terms to know and to doubt; (3) relative terms; (4) beginning and ceasing or first and last instants; and (5) maxima and minima. Thus it is mainly the last three chapters that deal with the natural philosophical topics with which the Oxford Calculators are primarily identified.
It is generally accepted that most of the Oxford Calculators tended to follow a nominalist, ontologically minimalist, or Ockhamist line, as opposed to a realist line. Thus they assumed that there are no Platonic forms existing outside of individual substances and they tried to understand the intension and remission of forms in terms of the actual qualitative forms inhering in a body. Although they did not all discuss the matter explicitly, it appears that most of the Oxford Calculators (but not Walter Burley) followed John Duns Scotus in holding the so-called addition theory of intension or increase of quality (see Sylla 1973). In his 1968 article ‘‘Ockham and some Mertonians’’ (Weisheipl 1968:173), James A. Weisheipl said that the Mertonians, except Walter Burley, accepted the Ockhamist teaching in logic and natural philosophy ‘‘as a matter of course.’’ In contrast to the other Mertonian Calculators, Burley was a ‘‘realist’’ in the sense that he argued against Ockham’s nominalism in logic.
If William of Ockham is not included as an ‘‘Oxford Calculator” - because he did not take a calculational and sophismatical approach to natural philosophical problems - does it make sense to include Walter Burley? Burley had already left Oxford to study theology in Paris long before the late 1320s when Bradwardine’s On Proportions appeared, so his Aristotelian commentaries are more like those of the late thirteenth century than like Kilvington’s commentaries consisting of a short list of very long questions. Actually, Burley wrote a commentary on Aristotle’s Physics with fairly long questions as well as an exposition of the text when he was still teaching at Oxford, not long after 1300. (Burley’s early Physics commentary seems to have a close relationship to that of Thomas Wylton, something that has yet to be entirely sorted out.) But Burley revised his Physics commentary while teaching in the Faculty of Arts at Paris at the same time as his work in theology, and he completed the revision of the last books after he was already a member of Richard of Bury’s circle, back in England, and associated with Bradwardine and Kilvington. It is, however, Burley’s work in logic and his separate treatises or questions that reveal his similarities to the likes of Heytesbury or Richard Swineshead. Two of his works, the so-called Tractatus primus and Tractatus secundus on the intension and remission of forms, were off-shoots of his bachelor lectures and disputations on the Sentences at Paris. Nevertheless, they pay the same attention to issues of indivisibles and continuity in a natural philosophical context as do the other works of the Oxford Calculators. Burley’s quodlibetal question On the First and Last Instant is also calculatory in nature as are some of his later works composed in the 1330s.
Of all the Oxford masters who might be included among the ‘‘Oxford Calculators,’’ it was Thomas Bradwardine, William Heytesbury, and Richard Swineshead whose works received multiple early modern editions, commentaries, and elaborations. Out of its Oxford context, Richard Swineshead’s Book of Calculations (Liber calculationum) appears to be a work of mathematical physics, covering the sub-topics that might be expected to arise from Aristotle’s natural philosophical works, but then elaborating quantitative ‘‘calculations.’’ At Oxford itself, however, the work of Bradwardine, Heytesbury, and Kilvington was closely related to works on natural philosophy, to which logical and mathematical methods might be applied. Perhaps surprisingly, early modern printers did not choose to publish the more natural philosophical works of the Oxford Calculators, except in the case of Walter Burley, whose thorough commentary on Aristotle’s Physics, as it was revised at the University of Paris in the 1320s and later, received multiple printings.
A larger gap in our knowledge comes from the fact that John Dumbleton’s Summa ofLogic and Natural Philosophy has never been printed. Dumbleton’s explanations of the theories of latitudes and degrees shared by the Oxford Calculators and of Bradwardine’s theory of the proportions of velocities in motions go a long way to reveal what is tacitly assumed about natural philosophy by William Heytesbury and Richard Swineshead. A start to fill in this gap has been made in the article on John Dumbleton elsewhere in this volume.
Of all the individuals who might be considered Oxford Calculators, this encyclopedia contains separate articles on John Dumbleton, Richard Kilvington, Richard Swineshead, Thomas Bradwardine, Walter Burley, and William Heytesbury, but it lacks an article on Roger Swineshead, probably because Roger has sometimes been lumped together with Richard Swineshead. Given this lack, something more about Roger Swineshead will be said here.
In his thorough work on medieval theories of insolubles and on the exercise of obligations, Paul Spade has made clear the significant place that Roger Swineshead held in each of these traditions. With regard to the exercise of “obligations,” Roger Swineshead directed the respondent to reply taking into account only the position, which he had been obligated to affirm, whereas Burley had earlier required that the respondent in obligations exercises take account not only of the position to which he was originally obligated, but also ofall the other propositions that he had accepted, rejected, or labeled doubtful earlier in the exercise. With regard to insolubles, Roger Swineshead took a position that might be regarded as a variant of the position held by Thomas Bradwardine. According to Bradwardine, for a proposition to be true, it must not only ‘‘signify as is the case,’’ but also not signify other than is the case. To this Roger Swineshead added that for a proposition to be true it must not ‘‘falsify itself’’ (see Spade 2005). From this position, Swinehead concluded that some insolubles do signify as is the case, despite being false - as is true for the proposition ‘‘a is false,’’ where a is the proposition itself. In some valid formal inferences, Swineshead said, falsehoods follow from truths. And, finally, in the case of insolubles, two mutually contradictory propositions can be false at the same time (Spade 2005). In his 1335 Rules for Solving Sophismata, Heytesbury added to the debate about insolubles the requirement that the circumstances in which the insoluble is asserted must be taken into account before anything can be said about its truth or falsehood.
In his On Natural Motions (c. 1337), Roger Swineshead compiled a work that included both natural philosophical theory and the particular methods of the Calculators (see Sylla 1987a). Unlike Dumbleton, who was to advocate the mainstream theories that the other Calculators simply assumed, Roger Swineshead advocated theories that, whether on purpose or not, made it easy to reach apparently paradoxical conclusions. Thus he defined ‘‘uniform degrees’’ of qualities or velocities and uniformly difform degrees, where the ‘‘uniform degrees’’ were indivisibly greater than a uniformly difform distribution that included every degree of a quality less than the given uniform degree. There is only one complete manuscript of Swineshead’s On Natural Motions, which currently is MS Erfurt Amplonian F 135, but a famous student notebook from the 1340s still existing as MS Paris, BNF 16621 includes excerpts from On Natural Motions (in a more accurate version than the Erfurt manuscript), along with excerpts from Dumbleton, Summa; Burley, On First and Last Instants; Bradwardine, On Proportions of Velocities in Motions; and other calculatory works (see Kaluza 1978).
But it is Richard Swineshead’s Book of Calculations that gives the Oxford Calculators their name (see Murdoch and Sylla 1976). That book begins by discussing the measures and intension and remission of qualities and works up to more and more complicated situations. With regard to bodies that are non-uniformly qualified with whiteness or heat, Swineshead asks what degree of quality should be assigned to the body as a whole - the maximum degree perhaps, or some sort of average degree? If there is a body that has two different qualities, say it is both hot and moist, is there an overall degree that might be assigned? And how will the substantial form of a body change as the qualities change? As water is heated, will it immediately begin to become air, or only after the hotness has reached a certain degree? And if two bodies with different degrees of various qualities like hot, cold, wet, and dry act on each other and resist each other, what will determine the force with which one body acts on another - will it involve the maximum degree? The average degree? The extent of the body? The ‘‘quantity of quality’’ if certain bodies have, so to speak a greater density of quality than others (think of specific heats)? If the force must be greater than the resistance for action to occur, how can hot and cold bodies interacting with each other each act on the other at the same time (this is the problem of ‘‘reaction’’)?
In later treatises of the work, Richard Swineshead deals with the power of light sources, as measured by the intensity of the light they cast together with the distance to which the illumination extends. He then turns to cases of local motion. After a treatise in which he shows how Bradwardine’s function works to calculate what velocity will result if it is known what velocity results from an initial situation and how the force or resistance changes, Swineshead attacks problems where the resistance to a mover results from the medium in which the mover is exerting its force. Here the Bradwardinian theory, as well as the classical Aristotelian theory, runs into problems, because both theories indicate what the ‘‘velocity’’ will be for a given force and a given resistance. If, however, the degree of resistance depends on the position of a moving body in a nonuniform resistance, there will be two, most likely contradictory ways to calculate velocity - either by the ratio of force to resistance (this is the measure of motion with respect to cause) or by the distance moved in a given time (this is the measure of motion with respect to effect).
As if this were not a sufficient problem, Swineshead then supposes that the nonuniform resistance of the medium moves, as the mobile itself moves. It is not difficult to see that if there were a medium with little resistance with a boundary layer next to a medium of high resistance, and if this boundary layer were moving in the direction the mobile is moving, then the mobile might move up very quickly to the boundary layer and get stuck there, if the boundary layer were moving faster than the mobile, given its force, was able to move in the more resistant medium. But then the velocity of the mobile within the boundary layer would not correspond to the proportion of its force to the resistance of medium, but rather simply to the velocity of the boundary layer.
In all of this, it appears that Swineshead is interested providing exercises that will give the students facility in working with such calculations as applied to physical problems. The problems are much more complex than the problems that Galileo would later deal with, assuming nonresistant mediums, and they bear some relation to hydrodynamics applied to biological systems. In later Treatises, turning from local motion to alteration, and assuming that an agent of qualitative change, such as a heat source, causes a greater effect next to itself, which decreases, perhaps uniformly, the farther away from the agent one gets, Swineshead considers how, over time, the maximum degree might be induced into an extended body. In the separate article on Richard Swineshead, reference is made to Swineshead’s Treatise 11, which discusses what would happen if a thin rod were to fall through a channel in the earth until it began to pass the center, in which case part of the rod would begin to try to move back toward the center.
How then are we to understand the context and purpose of Swineshead’s Book of Calculations? On one level, it seems to be a book of mathematics applied to various natural philosophical problems. But while the natural philosophical problems are indeed like problems that had long been raised in connection with commentaries on the natural works of Aristotle, they are often more complex and technical. Much has been made of the tendency of fourteenth-century philosophers to deal with problems secundum imaginationem, that is supposing imaginary situations, and inquiring what would happen in such a case. It has been supposed that this resulted from the condemnations at Paris in 1277, as a result of which philosophers teaching at the university of Paris were enjoined not to deny that God might do anything that is not a logical contradiction. So, it followed, God might create a vacuum by annihilating everything inside the sphere of the moon. What then? If there was a mobile without any resistance, would it not move infinitely fast, according to the theories of Aristotle or Bradwardine?
Swineshead, however, does not seem to be invoking God’s absolute power in designing the problems to which he applies his “calculations” - almost all of which, notably, involve juggling quantities in the mind without any actual numbers or arithmetic being involved. Rather, the point seems to be to exhibit what can be achieved with the available logical and mathematical tools. There really is not a special subject matter, in which the students are expected to be interested. The point is not to study the ballistics of cannons or the motion of ships through water or any other practical problem. The obvious conclusion seems to be that, just as Heytesbury’s Rules for Solving Sophismata was written as a handbook for students who would take part in disputations on sophismata, so Swineshead’s Book of Calculations was most immediately to be an aid to students in their disputations, whether in the very same disputations on sophismata as Heytesbury’s book, or in disputations connected with the curriculum in natural philosophy or ethics. In fact, we can see Richard Kilvington and then later scholastics applying calculatory techniques in their commentaries on Aristotle’s Ethics and on Peter Lombard’s Book of Sentences.
When calculatory techniques were applied in theology, was it an incredible misapplication of technical brain power to matters better left to the heart? This may very well be a reasonable judgment in the case of a theologian such as Jean de Ripa, who seems to have been influenced by the Calculators (see Coleman 1975; Kaye 1998). This issue, however, goes beyond the work of the Oxford Calculators proper, and will not be pursued here.
Very soon after the 1330s when calculatory ideas developed at Oxford, they spread to Paris, perhaps as Oxford students traveled to Paris to study theology, as was the case of Burley and Dumbleton. Nicholas Oresme and Albert of Saxony wrote treatises extending, on the one hand, and simplifying on the other, Bradwardine’s theory of the proportions of velocities in motions. Later the logical ideas of the likes of Heytesbury were widely studied in Italian universities and then in eastern European universities such as Erfurt (see Sylla 1996a). At the beginning of the sixteenth century, Alvarus Thomas of Lisbon, teaching at Paris, thoroughly mastered Swineshead’s Book of Calculations, as well as some of the extensions of his work by Nicholas Oresme, and discussed astutely some of Swineshead’s knottier problems (see Sylla 1989, 2004). Although the label “calculators” was not applied in the fourteenth century to Swineshead or the men surrounding him - at the time, ‘‘calculations’’ was more likely to bring to mind the work of astrologers - the label began to be used in the sixteenth century, when it became common to label the different philosophical approaches or ‘‘ways’’ (via modernorum, antiqui, nominales, etc. see Wallace 1969).
See also: > Insolubles > Intension and Remission of Forms > John Dumbleton > Obligations Logic > Richard Swineshead > Thomas Bradwardine > Walter Burley > William Heytesbury
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