Mediaeval mathematics

It is a historical cliché that there were no intellectual achievements worth speaking of in Europe between roughly the middle of the first and the middle of the second millennium. While it is certainly true that scholarship flourished in the Islamic, Indian and Chinese worlds while it was more or less dormant in Europe (and so it is proper that historians should concentrate on these worlds), I thought that nobody believed the cliché any more.

However, a fairly recent biography of Leonardo da Vinci by Michael White (subtitled The First Scientist) is utterly dismissive of the Dark Ages. In a chapter on Leonardo’s intellectual inheritance, White refers to “Europe [cast] into a time frozen in aspic, a dark age that lasted just a little under a millennium”, “Europe lay immersed in its age of intellectual darkness”, “the mire of the Dark Ages”, and “the miasma that was the Dark Ages”; he refers, with almost no scepticism, to a “legend” that the Venerable Bede was the only literate person in England in his day.

So I was interested to see, among the Christmas fare in several bookshops, a new book, God’s Philosophers: How the Medieval World Laid the Foundations of Modern Science by James Hannam, which sets out to redress the balance. A glance at the index showed that Abelard, Bacon, Ockham, Buridan and Oresme were all discussed; so I bought it.

It is a good antidote to White’s limited view of the Middle Ages; but it is also a polemic, biased in the other direction. Over and over the myth of the Dark Ages is attacked, the Inquisition defended, and the work of the Scholastics praised.

But the book is not strong on detail. We are told that Peter Abelard was “one of the most celebrated and controversial figures in the history of philosophy”, and that “the subject at which [he] excelled was logic”. But nothing about any advances to his subject that he might have made, nor any applications of logic to philosophy he might have made (though a couple of examples of his rhetoric are given). It is all ad hominem: we learn that “Abelard was a big man with an ego to match”, and that he raped Héloïse (the only basis of this charge is a passage in one of his letters to her – and the love letters of logicians are no more factual than those of anyone else).

We learn that the “Merton calculator” William Heytesbury proved the Mean Speed Theorem (the average speed of a uniformly accelerated body over an interval of time is its instantaneous speed at the mid-point of the interval) in the 14th century, and that Nicole Oresme in the same century gave a geometric proof showing that he understood that distance travelled is the area under the graph of velocity against time. But tantalisingly, no hint of Heytesbury’s method of proof is given.

This analysis was not at first applied to the motion of falling bodies, though the problem had been considered by the Merton calculators. Galileo claimed in 1638 to have done this for the first time, although Domingo de Soto had published such a description in 1551. De Soto had learned of the work of the Merton calculators, Buridan and Oresme from his teacher Juan de Celaya. Galileo’s work was supported by detailed experiments on bodies rolling down inclined planes, timed with water clocks. It is not clear what role experiment played in the earlier work.

Thomas Bradwardine, another Merton calculator, is supposed to have invented logarithms 300 years before Napier, while solving a kinematical model derived from Aristotle’s physics. How? Did Aristotle assert that speed is inversely proportional to time, and was Bradwardine effectively integrating 1/x? Does anyone know?

On the subject of Oresme, we learn that, unusually for his time, he was opposed to astrology. However, as Karl Petersen has pointed out, his argument was based on two surprisingly modern mathematical insights: that rational numbers form a null set, and that integer multiples of an irrational number are uniformly distributed mod 1. These stunning insights are not mentioned.

Buridan and Oresme did realise, long before Copernicus, that the same phenomena can be explained by two hypotheses: the heavens revolve around the earth, or the earth rotates. They dismissed Ptolemy’s contention that if the earth rotated, it would leave the air behind and we would feel a mighty wind: the solution is simply that the air rotates with the earth. Similarly, Oresme realised that an arrow fired vertically would also share the earth’s rotation and would fall on the same spot. Since no observation or experiment can decide between the two possibilities, we have to look elsewhere; for Bishop Oresme this meant the Bible, where Psalm 93 says “The world also is stablished, that it cannot be moved.”

It is interesting that, two centuries later, Galileo stumbled at just this point. He developed a theory of the tides which involved the oceans being left behind by the earth’s rotation, even though he knew the air was not, and claimed that his theory proved that the earth rotates. (The Church permitted Galileo to teach the Copernican theory as a hypothesis; Galileo was desperate to show that it was true, and his desperation led to this fall.)

I know a little about the work of William of Ockham, since I have editions of his philosophical writings and his work on predestination and God’s foreknowledge. In the former can be found, among other things, an extremely lucid account of De Morgan’s Laws.

What brought mediaeval scholasticism to an end? According to Hannam, it was the rediscovery of classical manuscripts in the early Renaissance; the progress of the Scholastics was abandoned for uncritical admiration of the Greek philosophers.

I have another book on the subject, which has chosen now to go into hiding, so I cannot compare it in detail: David Knowles’ The Evolution of Medieval Thought. Knowles is concerned with the whole scholastic enterprise, not just natural philosophy, so he gives no more detail about the work of Abelard, Oresme and others; but he gives a far superior overview. For comparison, his view is that it was the caustic effect of nominalism which killed scholasticism, before the Renaissance. If God is all-powerful, he can manipulate an experiment to give any result he pleases; or he can manipulate the experimenter’s perceptions with the same effect. (This view has recently re-surfaced in the religion of the Flying Spaghetti Monster.)

It may seem that I am seeing the past through modern eyes; but in the absense of hard information that is all I can do. In any case, Ockham and Oresme wrote so clearly that I’m sure they had a very good idea what they were talking about.

I would love to have a book about the mathematical achievements of the scholastics; but Hannam’s book isn’t it, and so I am still looking.


About Peter Cameron

I count all the things that need to be counted.
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One Response to Mediaeval mathematics

  1. proaonuiq says:

    Interesting post.

    I do think though that the cliché is not far from the truth. Please do not expect to find a good book about the great mathematical achievements during the western middle ages: there were not more achievements than the few outliers we all know.

    Although I have not read Hannam book yet, I´m highly interested in the history of ideas in general and specially in the history of scientific ideas. To put all the authors that appears in Hannam´s book in context I suggest as a good reading Etienne Gilson´s la Philosophie au Moyen Age (I et II). That´s the book that helped me to understand christianism in deep. There we can see that christianism from its begining to today´s was nor against science, neither for science: it was about souls, God and salvation; and therefore, in general, science indifferent. Due to this, it is very difficult to proove if science predominates today thanks or despite christianism. And this is surely truth regarding any other religion.

    From any past period in history you can select some authors, and from these authors extract some sentences or facts so that the reader has the feeling that what has beeing said about the period this author lived on is not correct. Hannam is a self-declared Christian, I guess a Roman Catholic, and as such i suppose he is writing the kind of revisionist history book we are seeing frequently lately. But to have contributed to to the birth of science or on the contrary to have delayed its development is completely irrelevant from a Christian point of view: this was not in the past, it is not today, its purpose.

    Personally, as an atheist, I´m more interested in two related debates which are also revisited by scholars from time to time: was there really a western scientific revolution or just a continous evolution from antiquity to today ? was science really a exclusively western european institution until XX century ?

    Ps. I´m awaiting impatiently for the new 2011 posts on the symmetric group !

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