Before we start, just a quick reminder that I’ll be taking down a bunch of old episodes from the web any day now, due to their inclusion in the upcoming Math Mutation book. So be sure to download any old ones you’re interested in ASAP. Now, on to today’s topic.
Often we think of the centuries before Copernicus and Galileo as a huge black hole of science, with little of note happening other than medieval knights spending all day jousting at dragons. But actually, the pioneers of the Scientific Revolution who became household names, like Copernicus and Galileo, did not spontaneously arise from a vacuum like some quantum particle. They owed a major debt to a number of earlier scholars who re-examined the knowledge passed down by the ancients, and explored the possibility of more carefully applying mathematical methods to understand the natural world. One of the most important such groups were the cluster of philosophers from the 1300s, led by John Dumbleton, Richard Swineshead, Thomas Bradwardine, and William Heytesbury, who became known as the Oxford Calculators.
The most critical contribution of the Oxford Calculators was probably the concept that nearly any physical attribute worth studying could be measured and quantified. Nowadays we take this for granted from the beginning of our science education in school, but this was by no means obvious for most of human history. Among the ancient Greeks, Aristotle had discussed measurements related to size and motion, but had not addressed the issue of whether other aspects of the physical world could be measured numerically. The ancients seemed to have an implicit assumption that many aspects of reality, like heat and light, could only be discussed qualitatively. The Oxford Calculators challenged this approach, and tried to quantify their discussions whenever possible, believing that it really should be possible to quantitatively specify nearly everything you could discuss. They were primarily thinking of themselves as theologians and philosophers, so did not conduct the actual experiments that would be critical to the real advances of the later scientific revolution, but even thinking in these terms was a major step.
Another important contribution was their willingness to reopen discussion of matters that had been supposedly solved by the ancients, and question teachings that had been passed down since Aristotle. One example is Artistotle’s belief that the velocity of an object would be proportional to the force exerted on it, and inversely proportional to its resistance. In modern terms, we would write this as V = kF/R, with V = velocity, K = some constant, F = force, and R = resistance. One of the Oxford Calculators, Thomas Bradwardine, pointed out that there was something fundamentally wrong with this formula: if the force and resistance precisely balanced, we should expect an object not to move at all. However, the Aristotelian formula would impart a constant velocity, since F/R would equal 1 in such a case. Bradwardine proposed an alternate formula, which we would now write as V = k log (F/R). This was also horribly wrong, but at least fixed a key flaw in Aristotle’s teachings, so we need to give him some credit. Since the log of 1 is 0, Bradwardine’s approach would at least predict objects not to move when force and resistance are balanced, a pretty important characteristic for a basic theory of motion. This wrong formula also helped advance the definition of logarithms, a major mathematical building block for further developments centuries later.
The Oxford Calculators’ most significant concrete contribution was probably the Mean Speed Theorem, an idea which is often incorrectly attributed to Galileo, but was actually first published by William Heytesbury in 1335. This came out of attempts to understand accelerated motion, quite a challenge before the development of calculus, our key tool for understanding changing quantities. You may remember the basic formula for the distance traveled under constant acceleration: S = 1/2 A T squared, where S is the distance traveled, A is acceleration, and T is time. Nowadays, we can trivially derive this using calculus. But Heytesbury was able to essentially figure out the same formula centuries before calculus was available, reasoning that the total distance traveled by an object under constant acceleration would be equal to the average speed multiplied by the time. In modern terms, the average speed is just 1/2 A T, and the time is T, so Haytesbury’s reasoning leads us to the same 1/2 A T squared formula, without the need for calculus. Heytesbury didn’t write it precisely in the modern form, but managed to achieve the correct result, centuries before Galileo.
Now, we should point out that while they provided critical stepping stones, the achievements of the Oxford Calculators do not diminish the significance of Galileo’s work or others involved in the later Scientific Revolution. If their successors sometimes failed to recognize or mention them, it wasn’t out of malice or deceit, but mostly because the modern system of scientific citation and of publication credits had not yet been established. With modern scholarship, we can give the Oxford Calculators the credit they deserve, without needing to take away credit from anyone else. They played a small but critical role in laying the foundations for the rapid mathematical and scientific advances that continue to benefit and astonish us in our own time.
And this has been your math mutation for today.