 Hi, I am Peter Adamson and you are listening to the History of Philosophy podcast, brought to you with the support of the Philosophy Department at King's College London and the LMU in Munich, online at www.historyoffilosophy.net. Today's episode, Quadrifugal Pursuits – The Oxford Calculators. Suppose you opened a work on physics in antiquity or at a medieval university and compared it to a physics textbook from the modern day. The first big difference that was tri-key would be, well, probably the ancient or medieval text is in Greek or Latin, but the second big difference would be that the modern textbook is full of mathematics, with formulas and numbers strewn across every page. Not so with works on natural philosophy written in antiquity or the 13th century. I never had to ask you to recall ideas from high school math class when discussing Aristotle's physics, Richard Rufus, or Albert the Great. Though it might seem obvious nowadays that physics should involve doing calculations and solving equations, the earlier history of physics suggests that this approach is far from evident. Any shepherd will readily think to use numbers to keep track of the size of the flock, but it took scientists a long time to realize that it would also make sense to use numbers to keep track of how fast a sheep is working its way across the meadow as it grazes, or to compare the speed of this motion to its motion when it runs away from a wolf. To say nothing of applying mathematics to more subtle sorts of change, like the rate at which the grass in the meadow is turning brown in autumn or warming up on a summer's day. In the popular imagination, the breakthrough came suddenly, when early modern scientists first merged the study of mathematics with the study of nature. But in fact, certain natural phenomena, notably the motions of the stars and the harmonic ratios we used to produce music, had fallen under the study of mathematics since antiquity. The medievals too dealt with these phenomena in the so-called quadrivium of liberal arts, that is, the four mathematical sciences arithmetic, geometry, music and astronomy. As for the application of mathematical concepts to spatial motion and changes in quality like colour or temperature, we do not need to wait for Galileo and Newton. This step was already taken in the 14th century by a group of thinkers at the University of Oxford, especially Merton College, who historians call the calculators. We've already met most of the Oxford calculators while surveying developments in 14th century logic. One key figure, as in logic and the debato for predestination, was Thomas Bradwardine. Notice that, although even most professional philosophers have not heard of Bradwardine, he is emerging in this series as a truly pivotal figure in medieval thought, who made crucial contributions in a number of different fields. He must be one of the best illustrations we have yet come across of why it makes sense to study the history of philosophy without any gaps. In a work devoted to the proportions of motion, written in 1328, Bradwardine set forth an influential mathematical analysis of the relationship between the force applied to a moving body, the resistance the body encounters, and the velocity of the motion that will result. Also significant was that champion of realism, Walter Burley, who in addition to writing on Aristotle's physics, composed treatises analyzing qualitative change. Other calculators included Richard Kelvington and Roger Swineshead, whom we met in our discussion of 14th century logic, as well as another Swineshead whose first name was Richard and who actually wrote a treatise called Book of Calculations. I would love to be able to report that Roger and Richard were both called Swineshead as a comment on their personal appearance, disappointingly it instead indicates that they came from the same town. Still other calculators included John Dumbleton and William Hatesbury, disappointingly not named for his dislike of soft fruits. There were good reasons for medieval thinkers not to do what the calculators finally did, by pursuing a mathematical approach to physics. For starters, crucial mathematical tools were still lacking. On the bright side, algebra had been invented by the 9th century Muslim scholar al-Khwarizmi and passed into Latin in the 12th century. The name betrays the cultural origin of this branch of mathematics, it comes from the Arabic al-jabr. But calculus was still centuries away from being discovered. Then too, there were doctrinal reasons to think that mathematical techniques would be out of place in discussions of physical change. In Aristotelian terms, such items as the color of grass or the warmth of a sunny day belong to the category of quality. The category of quantity takes in an entirely different range of properties like length. So applying numbers to colors and temperatures would require contemplating a categorical monstrosity, the quantity of a quality. Worse still, Aristotle gave explicit instructions that we are not to engage in what he called metabasis, or crossing, from one scientific field to another except under certain rather strict conditions. Basically, he allows it only when one of the sciences in question is subordinated to the other. That is, it must take its principles from the other science, as we saw Aquinas thinks that natural reason takes its principles from theology. So, to take results from mathematics and apply them in natural philosophy seemed to contravene Aristotelian methodology. The calculators were aware of the problem. Radwarding pleads that his procedure is justified since we can speak in a broad sense of proportion whenever there is a question of more and less. Since grass can be more or less green and emotion can be more or less fast, this means that we apply the mathematics of proportion to these things. Possibly, the context of contemporary nominalism played a role here too. Consider that for Occam, knowledge or science operates at the level of propositions. So long as we don't engage in outright equivocation, that is, use the same term in two different senses, it should be possible to combine true propositions into valid arguments so long as they share terms whether or not these propositions were proven true in the same or in different sciences. Also, nominalism readily grants that important scientific concepts like universals may be only in the mind. Thus, the calculators lived at a time when introducing a mathematical abstraction like velocity may have seemed less problematic. It would not need to imply the existence of a real thing out there in the world, to which the word velocity applies. Given all this, we can hardly be surprised that in earlier medieval philosophy there had been no tradition of applying numbers to qualitative changes and motions. When the calculators ventured to do so, even they do it only in a rather abstract fashion. They use variables by saying suppose a motion moves A in time B, or simple numbers saying suppose a motion moves 2 in 1 hour. There is no idea here of actually going out to measure real motions or to contrive units for the measurement of things like force, acceleration, resistance and velocity. Instead, the calculator's arguments move at the level of intuition and what they take to be common sense. Furthermore, when we talk about their mathematical approach, we should not imagine that there are formulas or equations scattered through the manuscripts of their works, as in the modern-day physics textbook. Instead, everything is spelled out in Latin apart from the letters used as variables. In this respect, the calculator's discussions of motion read very much like works on scholastic logic. There's a good reason for this, these discussions often appear in works on scholastic logic. As just mentioned, these calculators were the same men who were writing about sophisms, obligations, and so on. They often took up the topic of change in motion when discussing statistical arguments. Consider the following rather strange sentence. Socrates is whiter than Plato begins to be white. It is the first item considered by Richard Kelvington in his work about sophisms. What he has in mind is that Socrates is completely white. Apparently it has been cloudy during all the time he spent in the marketplace doing philosophy. Whereas Plato is not white, but is just becoming white. Perhaps this is a young Plato turning pale with embarrassment as Socrates refutes him in the marketplace. Hence the sophistical sentence, Plato is just turning white while Socrates is completely white and so is now whiter than Plato is beginning to be. Kelvington considers an objection to this sentence, namely that there are an indefinite number of shades of white between Plato's still dusky color and Socrates's brilliant pure whiteness. Doesn't that mean that Socrates is infinitely whiter than Plato? No, says Kelvington. Just because there are an infinite number of degrees of whiteness between not white and white, doesn't mean that anything can be infinitely white. We might say that pure white is the limit of the process of whitening, and as something whitens, it goes through an indefinite number of degrees of whiteness along the way, just as something moving in a straight line can be thought of as moving across all the points that make up that line. For a nice example of a sophism concerning spatial motion, we can turn to Hatesbury, who considers the following puzzle. Suppose that Socrates moves distance b over the course of day a. Notice that as promised he uses generic variables rather than concrete units. On the same day, Plato moves exactly the same distance. But whereas Socrates is moving at a constant rate, Plato starts slow and speeds up as he goes, like a hare gradually accelerating to catch a tortoise that is plodding steadily along. Plato catches up with Socrates only at the very end of the day. So, here comes the sophistical sentence. Can we say that Socrates and Plato will begin to move equally fast? It may seem so, since they covered the same ground in the same time. But Hatesbury denies this, since Plato is not moving at the same speed as Socrates. Rather, as the calculators would put it, Socrates and Plato have the same total velocity, meaning that the complete distance and time of the motion are equal, but their velocities at each moment are different. Given the scholastics fascination with such logical puzzles, we might suppose that the calculators chose to explore motion in such depth simply to handle these and similar sophistries. Maybe they just didn't want to get caught out in a game of obligations after admitting that Plato is moving or turning white. But there was more to it. There were good theological reasons to worry about change, including one we have thought about in an earlier episode, understanding the Eucharist, which as we saw was a topic that elicited deep reflection on qualities and the way that they alter. The problem also came up when discussing the increase of God-given grace in a human being. By now I've probably said this more times than there are shades of white, but it bears repeating that the theological context of medieval thought, far from precluding scientific advance and inquiry, often provoked it. These theological worries seem to have been on the mind of Walter Burley when he wrote his treatises on change, for example. In these treatises, Burley wanted to explain in greater depth what happens when someone, or something, is turning white, or when water heats until it turns into air, or a person grows in grace, but he uses the more mundane examples in his discussion. Burley considers a theory from an unnamed opponent, perhaps Thomas Wilton, that the process of change always involves a mixture of two contrary qualities. Thus, something that starts out cold and is becoming hot would have a certain ratio of cold to hot in it. We can even assign numbers to the ratio. Starting out from zero degrees of heat, water might advance to six degrees, at which point it would transform into air. Halfway through this process, it would have three degrees of cold and three of heat, hence we could say that the temperature would result from the mixture of qualities. Now, Burley has no quarrel with the basic idea of degrees of heat and cold. This notion could have come to the scholastics through the medical tradition. The ancient Dr. Galen, and following him, Muslim thinkers whose works were translated into Latin, like Al-Kindi and Averroes, had assigned degrees of these basic qualities to drugs in their works on pharmacology. The same background helps to explain why Burley and the other calculators speak of the latitude of equality, meaning the range of intensity that a feature like white or heat may have. In medical texts by Galen and Avicenna, health is described as a state with a certain latitude, meaning that the human body can become somewhat more or less hot, dry and so on, while still remaining healthy. While Burley is happy to think of qualities in these terms, he rejects the theory of mixed qualities. Instead, he believes that qualitative change involves a succession of different qualitative forms in the changing thing. Something that is heating up has a form of heat that is slightly more intense than the one it had a moment ago, and slightly less intense than the one it will have a moment from now. On this account, we can again think of qualitative change as being very much like spatial motion, a successive passage through an indefinite number of intensities. The analogy between motion and a change in quality means that the methods applied to one should apply to the other. Just as we can introduce quantitative measures to talk about degrees of heat or white, and even assign these degrees numbers, so we should be able to assign numbers to the intensity of a motion, and that is exactly what we find the calculators doing. Here, the most important contributions are made by Bradwardine and Hatesbury. In the treatise on proportions that I mentioned earlier, Bradwardine offers a new and influential mathematical analysis of spatial motion. Aristotle had argued that the speed of such a movement would be inversely proportional to the resistance offered by a medium. This is why you would need to exert much more force to walk through water than through air at the same speed. From this, Aristotle had drawn the conclusion that motion in a void would be impossible because it would offer zero resistance, implying that an infinite speed would result. No, by the 14th century, this argument against void had been rejected by a number of thinkers, including in late antiquity John Philoponus, and in the Islamic world several thinkers including Im Baja, Abul Barakat al-Baghdadi, and Fakhra Din Arazi. They all made roughly the same complaint, which is that resistance only slows down the intrinsic speed of a motion. In a void, there would be no slowing effect at all, so the motion would simply have its basic speed, which of course would be finite. In the 14th century, a number of Latin Christian thinkers likewise accept that void is possible at least in theory, and that bodies could move in a void. As we saw a while back, the 1277 condomations may have helped popularize this view since they discouraged the notion that God would be unable to create empty space should He choose to do so. Brad Wergene would agree that void is in principle possible. He makes a telling objection to Aristotle's position, namely that if it were really true that motion in void would have to be infinite in speed, whereas a minimal resistance would make the motion finite in speed, then adding just this small amount of resistance to avoid space would somehow cause an infinite reduction in speed. Furthermore, he observes that increasing resistance more and more doesn't just make things move slower and slower. At some point, the motion will grind to a halt completely. Imagine walking through a medium whose density is increasing. First it's like walking through air, then through water, then yogurt, then molasses, and so on. Eventually, you would be unable to move at all rather than moving more and more slowly. This is unexplained by Aristotle's simple theory that there is an inverse relation between speed and resistance. Instead, Brad Wergene offers a new theory centering on what is in effect a new formula, though of course he presents it in Latin sentences and not symbols. To understand it, I'm afraid you will now need to dust off some of that high school math. Following earlier medieval ideas about proportions, he says that to double a speed, you don't have the resistance, or double the force applied to the moving body. Rather, you have to take the ratio of the force to the resistance and multiply this by itself. In other words, you have to square that ratio. To triple the speed, you would need to cube that same ratio, or multiply it by itself three times. Thus, if a ratio of force to resistance is 3 to 2, and this yields a certain speed, then to triple that speed, you would need a ratio of force to resistance of 27 to 8, that is 3 to the third to 2 to the third. Brad Wergene does not back this up with empirical proof, and in fact it has some counterintuitive results as was pointed out by other 14th century thinkers like Nicolas Rennes, on whom more is shortly. But the other calculators liked it, in part because the speed is now calculated as a function of the extent to which the force is in excess of the resistance. If the ratio of the force to resistance is 1 to 1, or even less, then the speed will be zero, which is why you cannot walk through molasses, as people learned in Boston in the year 1919, googled the phrase Great Molasses Flood to see what I mean. Unlike those Bostonians, the calculators continued to make progress with another breakthrough, first explained in writing by William Hatesbury. Think back to our earlier example of Plato and Socrates moving the same distance in the same time, but at varying speeds, because Plato starts slow and accelerates, while Socrates moves at a constant rate. The calculators would say that Socrates's steady motion is uniform, while Plato's speeding up makes his motion deform. It is pretty easy to work out how far something will move in a given time if its speed is constant, but it is not so easy to work it out if the speed is varying. What we would like to do is find a way to reduce the case of accelerated motion to the simpler case of motion with constant speed. Hatesbury accomplishes this by announcing what is now known as the Mean Speed Theorem. It states that if a body is moving while changing speed, it will cover the same distance in a given time, that it would have covered if it moved for the same time with its mean speed. Suppose a sheep moves for an hour across a meadow, slowing down and speeding up as it does so, first grazing and then lurching into the sheep version of a sprint when it sees a wolf in the next field. The different speeds can then be averaged. You might take its speed at every minute, then divide by 60 to get the mean speed. And the distance it moves will be the same as the distance it would have moved if it had been going at that rate constantly the whole hour. This is good work on Hatesbury's part, and not only because he's right. Also because to conceptualize this situation like this, he needs to introduce the idea of a speed at a time. It's a notion we find obvious, living as we do in an age where we can just look at the speedometer of the car to see how fast we are going right this moment. But as with so many apparently obvious ideas, it needed to be discovered. It has to be admitted that Hatesbury didn't actually prove the mean speed theorem, he only articulated it and left it to the reader to see that it makes intuitive sense. Before long though, something like a proof was offered by Nicole Oresme, who modeled the situation of the accelerating motion using geometrical diagrams. These diagrams anticipate the geometrical approach to these and related topics later taken by Galileo. If you're still not impressed, no less a figure than Leibniz, who of course is going to help introduce the much needed tools of calculus, was aware of the calculators and praised their pioneering work. And if even that doesn't impress you, then I guess I will need a whole further episode to do it. Join me next time, when we will look at more innovations in physics made by 14th century thinkers, including John Buridan. He proposed that, however difficult it might be to impress podcast listeners, it is at least possible to impress a force into a body. This impressed force is what we call impetus, and it will feature heavily in the next installment of The History of Philosophy Without Any Gaps.