‘‘Mathematics,’’ Bradwardine wrote, ‘‘is the revelatrix of truth, has brought to life every hidden secret, and carries the key to all subtle letters.’’ Bradwardine’s Geometria speculativa shows his fascination with the foundations of geometric and mathematical theory, containing treatises on stellated polygons, isoperimetric figures, and solid geometry, each development of medieval commentaries on ancient mathematics, and his De proportionibus, his most important contribution to medieval scientific reasoning. De proportionibus attempts to improve Aristotelian mechanics by addressing mathematical inconsistencies that had begun to trouble medieval natural philosophers. The Mertonians William Heytesbury, Richard Swineshead, and John Dumbleton would make use of Bradwardine’s study of the relations of quantities in their own speculative physics, as would Galileo in the seventeenth century. Particularly significant in the treatise is Bradwardine’s attempt to explain the relation of variation in the velocities of a moving thing to variation in the forces and resistances that affect velocity. The traditional understanding in Aristotelian physics relied on the axiom that motion occurs only when the motive force is greater than the resistance offered, so that velocity is explained as proportionate to the ratio of force to resistance. Bradwardine reasoned that if we begin with a motion in which the force is greater than the resistance, and if we continually double the resistance while holding the force constant, at some point the resistance will be greater than the force. The problem lies in the axiom of velocity being proportionate to the ratio of force to resistance; as the velocity decreases in proportion to the increase of resistance, there will still be a velocity assignable at the point that resistance is greater than force. This would mean there would be a measurable, albeit tiny, velocity assignable to a stationary object. Better, Bradwardine argued, to recognize that velocities vary arithmetically, while the ratios of force to resistance vary geometrically. The immediate significance of this realization of the need for geometric ratios in measuring velocity was the need for a more advanced mathematics than had been applied in kinematics. It had been thought that calculations based on the direct proportionality of quantities would serve, but Bradwardine’s discovery demanded a return to ancient mathematics, where he recognized the basis for logarithmic reasoning. It quickly became apparent that the use of Bradwardine’s logarithmic function in expressing the quantity defined by the relation of two other quantities was of great value not only in kinematics, but in a wide range of questions regarding quantitative and qualitative change. The formal method of logarithms would not appear until John Napier’s work in 1614 provided the mathematical foundation for their uniform and widespread application in calculation.
De causa Dei
William Ockham had argued that statements of the form ‘‘X will occur at Time N’’ have a truth value contingent upon what will happen in the future, so knowing them must be a different sort of knowledge than knowing statements about the present or the past, for which the truth value is already clear. Since this is a natural fact about such future contingent statements, God’s knowledge, too, must be that ‘‘X will occur at Time N’’ is true contingent upon X’s occurrence. The difference between our fallible and uncertain knowing and God’s infallible and perfect knowledge is that, while for us knowing X or knowing not X involves reasoned recognition of the opposition of X and not X, and the steps involved in resolving which of the two opposites are the case, for God, knowing X or knowing not X entails no reasoned recognition of the opposition, and no reasoned resolution of whether X or not X is the case. Ockham’s resolution of the puzzle was not necessarily a departure from earlier thinkers like Grosseteste or Peter Lombard, but his nuanced treatment of the relation of God’s eternal knowledge and the contingency of created action suggested that we might be able to earn salvation on our own merit, without grace. In short, it was possible to interpret Ockham’s approach as countenancing elements of Pelagianism. Bradwardine began to address the problem in the 1320s, in his Sentence commentary, and in De futuris contingentibus. Bradwardine’s interest in the subject developed, and after he had become a member of Bury’s circle, he compiled De causa Dei, a polemic refutation of ‘‘the Pelagians.’’ This massive work is not constructed along a recognizably scholastic model, but appears instead to be a Summa encompassing all that Bradwardine understood to be involved in explaining grace, merit, human salvation, and God’s knowing and willing. Each topic Bradwardine addresses finds its way back to God’s unmediated causal influence over creation. At the heart of Bradwardine’s theology is the fundamental truth that nothing occurs that is not willed by God. On the face of it, this seems so deterministic as to be fatalist. Ifall that happens is in accord with God’s will, then the revealed certainty that some will be damned and others saved amounts to double predestination. Bradwardine’s position was not so extreme. Even if God is co-agent in every created action, including the evil that men do, He is neither responsible for evil nor is His foreknowledge the cause of man’s damnation.
Bradwardine’s position that God is co-agent in all human actions rules out Pelagianism, but it also demands an account of how we, not God, are responsible for evil. This is particularly a problem if (a) God necessarily knows all that will occur in creation, (b) necessarily, all that God knows will occur, will occur, and (c) if God’s will and knowing are identical, then if God knows that a thing will occur, God necessarily wills that it occur. Bradwardine felt that necessitas consequentis, or absolute necessity, is commensurate with God’s foreknowledge without leading to a fatalistic determinism. Talk of God knowing a thing before it happens, or of a thing being necessary because God knows it ahead of time, is confused. God is eternal, and eternality is not a mode in which ‘‘before’’ or ‘‘ahead of time’’ applies. Hence, if God eternally wills that man act freely in time, the freedom of human action is not limited by the necessity of the divine willing. Bradwardine balances God’s eternal foreknowledge and human freedom by distinguishing between kinds of antecedent necessity. Bradwardine argues that some antecedent necessity is wholly absolute, but some is relative. Relative antecedent necessity can describe the secondary cause of an event, or it can describe the first cause. Peter may be free to choose to sin through an antecedent necessity relative by virtue of his being a member of the human species (a secondary cause of Peter’s existence), or by virtue of God’s willing (the primary cause of all created act). Peter’s choice may be partially determined by being a member of the human species, but his biological form does not compel his choice. God’s will is compulsory by virtue of its being the sufficient cause of every effect in creation, including human willing. Bradwardine believes he has preserved God’s eternal foreknowledge by making relative antecedent necessity commensurate with necessitas consequentis. The statement ‘‘If God knows that man would sin, then man’s sinning is necessary’’ is true, but escapes fatalism by decreeing that man’s sinning is free by virtue of God’s willing it to be so.
Relating the two sets of Bradwardine’s bodies of work has been a matter of disagreement. Until recently, the theologically oriented analysis of De causa Dei has been understood to be a significant departure from Bradwardine’s earlier work catalyzed by personal epiphany, an approach that allows the reader the freedom to disregard the treatises on mathematics and kinematics. Dolnikowski has argued that at least one commonality, namely time, unites the two bodies of work. By understanding the medium of time as the focus of his study of velocity and the mathematics by which we measure movement through it, and applying it to the relation of extratemporal divine understanding of events occurring within it, she argues, we are much better able to understand the continuity of Bradwardine’s philosophical theology.
See also: > Future Contingents > Insolubles > John Dumbleton > Modal Theories and Modal Logic > Oxford Calculators > Richard Fitzralph > Richard Kilvington
> Richard Swineshead > Robert Holcot > Walter Burley
> Walter Chatton > William Heytesbury > William of Ockham
World History
