Post-classical history

CHAPTER 11
The Merton Calculators

Richard of Wallingford and Roger Bacon were respected figures in the Middle Ages, but it was not to them that the university of Oxford owed its reputation as a leading philosophical centre. Bacon only achieved the renown he enjoys today when he was rediscovered after the Middle Ages had ended. Instead, a flock of innovative thinkers who were active from the late thirteenth century, many of whom completed their careers in mainland Europe, briefly propelled Oxford to a position of intellectual pre-eminence.

Foremost among them was John Duns Scotus (c.1265–1308), the most important developer and critic of Thomas Aquinas’s theology. He acquired the honorific ‘the Subtle Doctor’ and his work formed the foundation of an entire school of theology. Duns Scotus was educated at Oxford in the aftermath of the condemnations of 1277 and, like Roger Bacon, he joined the Franciscans. Once he qualified as a theologian, his order moved him to Paris and thence to Cologne where he died.1

In many ways, Duns Scotus drove forward the train of thought begun by Thomas Aquinas. Like Thomas, he was a great believer in the power of human rationality and insisted that we can prove the existence of God by reason alone. But Duns Scotus also thought that Aquinas had gone too far in subjecting God to the dictates of reason. In natural philosophy, Thomas thought that the workings of the universe had to reflect the character of God. Duns Scotus said that this placed unwarranted restrictions on God’s freedom of action. He could make the universe in any way that he pleased and it did not have to reflect anything else at all.2 Likewise, in ethics, to oversimplify a complicated argument, Thomas thought that God willed what was good and Duns Scotus thought that it was good because God willed it. As Duns Scotus put it, ‘the divine will is the cause of good and so by the fact that it wills something, it is good.’3

Thus, in both ethics and natural philosophy, Duns Scotus echoed the 1277 condemnations in his emphasis on the freedom of God to do as he pleases. Good is what God says it is and the universe works in the way God says it ought to. For science, this was a positive move. There was now no reason to assume that everything worked in the rational way that Aristotle said it did. Natural philosophers were free to speculate on all sorts of possibilities that they had previously ruled out.

Ockham and his Razor

The trend towards granting God absolute liberty of action and sovereignty continued with the work of another British Franciscan, William of Ockham (or Occam). He was born in Surrey, in south-eastern England, and entered Oxford University around 1300. He probably joined the Franciscans at the same time. In about 1320, he left the university before he had completed the theology course and gained his doctorate.4 He took up residence in a Franciscan convent for three years, which was when he enjoyed his most productive period as a philosophical author.5 The reasons behind his departure from the university remain obscure but may revolve around a commentary he had written on the Sentences by Peter Lombard. Putting together such a commentary was part of the standard training for theologians. If it was found to contain errors of doctrine, the candidate would have to either correct them or else drop out from the course. Simply making the required amendments would mean that the heretical statements need have no negative effect on the theologian’s future career. The procedure was no different from today’s students having to give the correct answers in order to pass their exams. However, arguing that it was the examiners who were wrong was riskier. This seems to have been what William of Ockham did, and the matter ended up before Pope John XXII whom we encountered in chapter 8. As well as banning fraudulent alchemy, he was responsible for finally crushing the ascetic wing of the Franciscans with whom Roger Bacon may have been involved.

At the time, the papacy was based in the city of Avignon, in the south of France. Pope Clement V (1260–1314), who was French, had removed himself there in 1309 to avoid the constant conflict and risk of being murdered in Rome. The king of France was delighted to have him and effectively turned the papacy into a branch of the French monarchy.

Pope John XXII, at least, had an independent spirit but he was happy to remain in Avignon rather than return to chance his luck with the unruly Romans. He summoned William of Ockham to Avignon where a committee, headed by John Luttrell (fl.1317–23), the ex-chancellor of Oxford, heard his case. Some modern scholars have suggested that Luttrell lost his job at the university as a direct result of censuring William’s work. If so, relations between the two would not have been cordial to start with. Others think it unfeasible that Luttrell would have been involved in a case if he had such a personal interest in it.6 We will never know for sure. Regardless of the political machinations, the committee declared that 51 propositions in William’s commentary on the Sentenceswere heretical. Rather than accept this and amend his work, William fled with a couple of other academics and sought protection from the prospective Holy Roman Emperor. The papacy, allied with France, was continuing its struggle with the Empire for mastery of Europe. Allegedly, William made the Emperor an offer: ‘Do you defend me with the sword and I will defend you with my pen.’7 Thus for the rest of his career, William was part of the imperial court and wrote political propaganda on behalf of his master.

Ockham wrote several important books in the period between leaving Oxford and travelling to Avignon in 1324. The gist of his most controversial work was that human reason is not a sufficiently powerful tool to discover very much about God or the world. We cannot even use reason to prove most statements about cause and effect – experience is the only way to know things.8 This makes science as well as theology very difficult. As Albert the Great had said, natural philosophy was all about trying to find out causes. William held that we can know many statements about God and the soul only through the light of faith, whereas Thomas Aquinas had thought them provable by reason. Clearly, William’s ideas were a serious challenge to the works of rational theology that were championed by Thomas and Duns Scotus. No wonder the old guard were upset.

For a younger generation, of course, William’s radical scepticism was exciting and novel. It seemed to honour God by placing him well above the deliberations of human reason. For the rest of the Middle Ages, the theology based on Duns Scotus was called the ‘Old Way’ while that which followed William of Ockham’s ideas became the ‘New Way’.9 Students in Paris were especially excited by the new innovations and their professors responded by trying to ban them in 1339.10 Theologians argued relentlessly over the relative merits of the Old and New Ways until the Reformation swept away the whole debate in the sixteenth century. At that point, the Catholic Church reverted to the work of Thomas Aquinas while Protestants disowned many of the achievements of medieval theology altogether.

Much of the argument between the supporters of Ockham and Duns Scotus was a proxy for the philosophical dispute between realism and nominalism. In chapter 3, we examined this briefly. To recap, the controversy was over the status of universals. A universal is a term used for a group of things, such as ‘dog’ for all dogs (including imaginary dogs). Realists believed that universals have a real existence, while nominalists considered universals to be merely names that humans have invented for convenience. William of Ockham, as you might expect, was a forthright supporter of the nominalist position. He insisted that we can only perceive individual things and that any connections we make between them are down to us. There is no need to postulate about the existence of real universals when we can explain the world in terms of the actual individuals it contains. Inventing new concepts, like universals, is unnecessary.

This is an application of the celebrated principle known to posterity as ‘Ockham’s Razor’. It is deservedly famous and often invoked as a reason for preferring elegant scientific theories over complicated ones. However, the term ‘Ockham’s Razor’ is another nineteenth-century coinage and was never used by William himself. His real point was rather different. He actually said: ‘Multiple entities should never be invoked unnecessarily.’11 What this means is that we should reject physical and philosophical explanations that posit the existence of things, like universals, of which we have no direct experience. From the anachronistic point of view of modern science, this is not a terribly good idea. Species, elements and electrons are all universals that have real and specific properties. The element carbon really does have a unique atomic structure and really does combine with oxygen (another universal) during combustion (yet another). Carbon is not just a collection of black lumps to which we have arbitrarily given a particular label.

On the other hand, medieval nominalists, by rejecting generalisations, tended to be more empirical than their realist rivals were. Because nominalists dealt only in particular real instances, no amount of rationalisation from first principles would convince them that something was so if they could not see it with their own eyes. As we will discover in the next chapter, this attitude meant that they were more than ready to reject Aristotle’s conclusions, however reasonable they might be. This rejection was important because very often the Philosopher’s conclusions were plain wrong.

The Errors of Aristotle

Does a heavy object naturally fall faster than a light one? Many people believe that it does. They will point out that a feather and a hammer, dropped from the same height, will not land at the same time. The hammer, of course, will hit the ground almost at once while the feather will meander gracefully to the floor. But what would happen if you were to do the same experiment with a pea and a ball bearing? Will the heavy ball bearing fall faster than the pea? If you are not sure of the answer, try releasing two objects of different weights but similar shapes, perhaps a teaspoon and a large serving spoon, from head height. You will find that they both hit the ground at almost precisely the same moment. The reason a hammer falls faster than a feather has nothing to do with their respective weights, but rather their reactions to air resistance. The American astronaut David Scott (b. 1932) carried out a famous demonstration of this on the moon, where there is no atmosphere, during the Apollo 15 mission. He dropped both of these objects and they fell at the same speed, albeit rather more slowly than they would do on earth due to the moon’s inferior gravity.

Does nature abhor a vacuum? Can a vacuum suck matter into itself? It is a staple of science fiction that the vacuum of space can suck people out of faulty airlocks. The truth is very different. A vacuum is literally nothing and so cannot do anything. All it does is provide empty space into which matter can move. It cannot suck you out of the window of a spaceship, but all of the air escaping through the hole might blow you out with it.

Aristotle was convinced that a heavier object naturally falls faster than a light one.12 He also insisted that nature abhors a vacuum. In fact, he did not think a vacuum could exist at all.13 According to him, it was completely impossible for a space to contain absolutely nothing. Although Aristotle justified his views with careful arguments, he often sided with common sense. For example, he thought the earth was stationary and located at the centre of the universe. This is a sensible position to take when we cannot perceive that the earth is moving and that the stars and planets all appear to move around it. Aristotle said the heavens were incorruptible and unvarying because all the records of his time showed that the movements of the stars and planets never changed. Neither did he believe in atoms. This was partly because a belief in atoms presupposes a belief in a vacuum for them to move around in. There was also no good reason to think that there was a minimum size of object or that matter was not endlessly divisible. Besides, no one had ever seen an atom or any direct evidence of one.

One of his most significant and long-lasting mistakes, baldly stated in Physics, was the belief that no object could continue moving without some other object moving it.14 Often, this is true. If you stop pushing a chair along the floor, it stops moving. But equally, on many occasions things keep going after you have stopped touching them. The best example is throwing a ball. Aristotle was convinced that something must be pushing it after it has left your hand. The only thing he could think of was that the air behind the ball was propelling it forward.15 This idea is easily refuted. A very powerful blast of wind would be needed to keep a ball moving through the air, and presumably we would notice this gust as we threw an object. Air, we know, actually resists motion, which would be impossible if it was also supposed to be providing the motive force to keep the object moving.

Such was Aristotle’s prestige that even his harebrained ideas had to be taken seriously. The trouble was, although critics were unconvinced by the air-pushing concept, they still accepted Aristotle’s fundamental law that a moving object must be moved by something else. This made it very difficult to come up with alternative theories. One writer who did take a different tack was John Philoponus (c.AD490–570), a Christian philosopher attached to the famous school in Alexandria. He suggested that the act of throwing a ball impressed a force onto it. This impressed force was then responsible for moving the ball forward but was gradually used up in the process.16 In this way, Philoponus could maintain the dictum that a moving object (in this case the ball) had to be moved by something (here, the force impressed onto it).

William of Ockham also realised there was a problem with the theory that a moving object had to be moved by something else. Unfortunately, his alternative was hardly more enlightening. William suggested that a thrown ball moves itself, so that it provides its own motive force. He also deconstructs the very concept of motion, claiming that it is simply an object occupying successive places. Movement as a real entity is another idea cut out by William of Ockham’s ubiquitous razor. Unfortunately, William’s whole discussion of motion is simply an aside while he is talking about a completely different subject.17 This meant that he never really developed his radical ideas.

Despite their critics, the combination of apparently sound common sense and cogent argument made Aristotle’s theories extremely attractive. They also formed a consistent whole that gave a full description of reality. Taken together, Aristotle’s philosophy makes for a deeply impressive package. This is the reason, more than any other, why it took such a long time for natural philosophers to realise that he was wrong about so many things.

The trouble is that it is impossible just to tinker with Aristotle’s natural philosophy at the edges. It goes much deeper than that. His was a complete theory of reality and rejecting any significant chunk of it would cause the whole edifice to collapse. It is hardly surprising that both ancient and medieval commentators very often gave Aristotle the benefit of the doubt rather than habitually challenging what he said. Reforming natural philosophy had to happen from the ground up. Even if someone wanted to suggest a new theory, the language of philosophy was the language of Aristotle and so that was how he had to express his ideas.

Nonetheless, in the fourteenth century medieval thinkers began to notice that there was something seriously amiss with all aspects of Aristotle’s natural philosophy, and not just those parts of it that directly contradicted the Christian faith. The time had come when medieval scholars could begin their own quest to advance knowledge; criticising and correcting their predecessors and striking out in new directions that neither the Greeks nor the Arabs had ever explored. Their first breakthrough was to combine the two subjects of mathematics and physics in a way that had not been done before. The setting for this most essential of steps towards modern science was the quadrangles of Merton College in Oxford.

The Mathematical Archbishop of Canterbury

Merton is among the oldest and grandest of the colleges of Oxford. It was founded in 1264 by Walter de Merton (d. 1277), bishop of Rochester, to provide a home for scholars studying theology for the many years that were required to complete the course and become a doctor of divinity. The college still holds a collection of hundreds of medieval manuscripts housed in a library that has been in continuous use for over six centuries. Also on display in the library is a fourteenth-century astrolabe traditionally known as Chaucer’s Astrolabe. As well as The Canterbury Tales and his other poems, William Chaucer wrote an instruction manual in English on how to use the astrolabe to tell the time of year and measure the positions of the stars.18

During the fourteenth century, Merton College was the scene of some of the most important work on natural philosophy and mathematics in the Middle Ages. A succession of its scholars was famous throughout Europe for pushing back the boundaries of physics. Collectively, these men are known as the Merton Calculators and their influence was still being felt in Italy as late as the sixteenth century. In fact, as we shall see, they almost certainly beat out the path later followed by Galileo and the other founders of modern science.

The earliest of the Merton Calculators, Thomas Bradwardine (c.1290–1349), entered Merton College in 1323 and stayed for about twelve years during which time he became a bachelor of theology. On leaving, he was appointed chaplain to Edward III who was engaged in the Hundred Years War against the French. Bradwardine followed the king on his expedition to France. He may even have been present at the Battle of Crécy in 1346, where English long-bowmen slaughtered the heavily armoured French knights. Shortly afterwards, Bradwardine was nominated as archbishop of Canterbury and, like Richard of Wallingford, he had to make the arduous journey across war-ravaged France to the papal curia in Avignon to have the appointment confirmed. He had barely arrived home when he died in 1349.19 Today his tomb lies in Canterbury Cathedral, just a few feet away from the shrine of the twelfth-century advocate of reason, Saint Anselm. The esteem in which his contemporaries held Bradwardine is illustrated by Chaucer mentioning him in the same breath as Boethius and St Augustine of Hippo in the Nun’s Priest’s Tale.20 He could hardly have bestowed any higher praise.

All of Bradwardine’s important work on natural philosophy was done while he was at Merton. His lasting achievement was to take a step that in retrospect seems blindingly obvious. But at the time it was a radical departure from the accepted norms of scholarship that had been inherited from the ancient Greeks. We saw in the last chapter how students had to spend a couple of years studying maths. Then they moved on and studied natural philosophy. The two subjects were kept separate. Although it was accepted that the stars moved according to predictable geometrical patterns, the use of formulae to produce physical theorems had been frowned upon by Aristotle. He did not believe that it was possible to make deductions in one subject, say mathematics, and use them to prove something in another subject, say physics.21 This means that there is remarkably little maths in his books of natural philosophy. His account of motion is explained with a few examples and generalisations, but with no attempt to produce a universally valid formula.

Bradwardine, in common with his colleagues, took the opposite point of view. He said that numbers were a vital ingredient of a successful natural philosophy. Mathematics, he wrote,

is the revealer of every genuine truth, for it knows every hidden secret and bears the key to every subtlety of letters. Whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start that he will never make his entry through the portals of wisdom.22

Translating this poetic language into practice, Bradwardine wanted to show that if what Aristotle said about how objects move is true, there must be some way to describe their movement as a mathematical function. Furthermore, he reasoned, for the formula to work it had to be valid for all situations, including for very large and very small numbers. All previous efforts to put Aristotle on a mathematical footing had failed because they could not be made to apply in every case. For Bradwardine, this meant that they could not be right. He decided to concentrate his efforts on finding the correct formula linking the force exerted on an object to its speed.

As it turned out, Bradwardine did finally come up with a formula that properly described Aristotle’s laws of motion in all circumstances. We should be clear, though, that the formula was completely wrong. This wasn’t Bradwardine’s fault but Aristotle’s. The basic laws of motion that Bradwardine was describing with mathematics were badly flawed to start with. He may have accurately modelled how things moved in Aristotle’s universe, but this was not how things worked in the real world. That does not stop Bradwardine’s work being an important step forward and it was recognised as such at the time.23 He had shown that it was possible to mathematically describe the laws of motion, but more importantly he had gone a long way towards demonstrating that any physical law worth its salt had to be expressible in numerical terms.

Modern historians studying Bradwardine’s work have noticed something strange about his formula of motion. Translated into the notation that we employ today, it uses a special function called a logarithm. According to the official histories of mathematics, logarithms were invented by a Scot, John Napier (1550–1617), who published his work in 1614.24 That they were being used in a limited sense for 300 years previously came as something of a surprise. In fact, Napier almost certainly made his discovery independently of his medieval antecedents, and the applications that he finds for logarithms never occurred to Bradwardine or his contemporaries.

Another area where Bradwardine made headway was in the question of falling bodies. Aristotle, of course, had thought that a heavy boulder falls faster than a pebble. The earliest record we have of someone categorically rejecting this is from the work of John Philoponus back in the sixth century. He wrote:

If you let fall from the same height two weights, one of which is many times heavier than the other, you will see that the relative times required for their drop does not depend on their relative weights, but that the difference in the time taken is very small.25

In this passage, Philoponus is clearly referring to an experiment that he has tried himself. Seven hundred years later, Bradwardine, who probably never did an experiment in his life, considered the hypothetical situation of objects falling in the absence of air resistance. Since the 1277 condemnations, natural philosophers had been considering how a vacuum might behave if God deigned to create one. Bradwardine fruitfully speculated that, in certain circumstances, a light and a heavy object in a vacuum would drop at the same speed.26 So, by the fourteenth century, it had been shown that objects of differing weights do fall at the same rate both in practice (by Philoponus) and in an idealised situation (by Bradwardine thinking about a vacuum). It would be a while yet before anyone would put these two results together and come up with a general law.

The Mean Speed Theorem

The most talented of the mathematicians at Merton was Richard Swineshead (fl.1340–55) who was probably still a fresher when Thomas Bradwardine left the college. Swineshead’s achievement led to his being granted the honorary title of ‘The Calculator’ by later authors, but we know almost nothing about him. His Book of Calculations took contemporary mathematical ideas as far as was possible and pressed them into service in a wide array of applications.

Swineshead was particularly interested in how he could analyse a situation where a quality like heat or speed was increasing and decreasing at various rates. In one chapter of his book, he uses Bradwardine’s formula to analyse what would happen if you drilled a hole through the earth and dropped a weight into it. Swineshead’s first attempt at a detailed mathematical account of the weight falling to the centre of the earth gives him the absurd result that it would slow down as it fell, so that it could never quite reach the centre.27 He rejects this and tries a simpler solution that does allow the weight to end up at rest at the centre of the earth. Actually, this is wrong too. We would expect the weight to fall faster and faster, reaching maximum velocity at the centre. Then it would slow down as it came up the other side, and finally come to rest on the far side of the earth (at least if we assume no air resistance in the hole). As this example shows, Swineshead and his fellow Merton Calculators were quite happy to apply their minds to imaginary situations. They set up a fictitious problem and then tried to work out the mathematical consequences. They were still a long way, however, from applying mathematics to a real-world situation and then trying to verify the result experimentally.

It is possible to produce mathematical equations for all kinds of situations. A general law of motion might be too difficult to solve at the first attempt, but simpler situations can be modelled in the same way. That is exactly what the last of the Merton Calculators, William Heytesbury (c.1313–73), did. Again, we know little about his life beyond his sojourn at Merton College and the fact that he wrote Rules for Solving Logical Puzzles which was published in 1335. In this book, he derived the most significant result of fourteenth-century physics. There is some doubt as to whether he actually discovered the formula himself or merely had the good fortune to be the first to write it down. For in contrast to Bradwardine’s function, Heytesbury’s has stood the test of time. If you have ever studied elementary mechanics, it will be familiar to you.

The problem he posed was: what happens when a moving object accelerates at a constant rate? Like his fellow Mertonians, Heytesbury was not very interested in empirical testing. He simply set up a problem and tried to solve it mathematically. The best way to illustrate this is to work through the question of uniform acceleration using some very simple numbers.

Suppose you drive your car at 50 miles per hour for an hour and a half. It is very easy to calculate how far you will have travelled – 50 times 1½ equals 75 miles. However, to use the jargon of the car trade, suppose you accelerate at a constant rate from nought to 60 miles an hour, in ten seconds. How far will you have travelled? First, we should convert 60 miles per hour into 88 feet per second. William showed that the correct method is to calculate how far you would go at your average speed of 44 feet per second over ten seconds, which is 440 feet. As he put it:

A moving body will travel in an equal period of time, a distance exactly equal to that which it would travel if it were moving continuously as its mean speed.28

This result, dubbed the mean speed theorem by historians, is central to physics because it describes the motion of an object, any object, falling under gravity. Note that it makes no mention of how much the object weighs. (Nor does it make allowances for air resistance, and so strictly speaking applies only to motion in a vacuum. That is why the feather and hammer fell at the same speed on the moon.)

Unfortunately, William Heytesbury and his contemporaries had no way of knowing the significance of the mean speed theorem. Nonetheless, they found it sufficiently intriguing to discuss many possible applications. Thanks to the 1277 condemnations, they could even talk about motion in a vacuum. For some reason, though, most of the natural philosophy that comes after Heytesbury took place in Paris and Italy rather than Oxford. The year 1350 marks a watershed after which no further mathematicians or philosophers of note emerged from England for almost 200 years. For this reason, we must now leave for mainland Europe and follow Roger Bacon and John Duns Scotus to the university of Paris.

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