A History of Mathematics From Mesopotamia to Modernity

Understanding the‘ScientificRevolution’ 135

as a discipline in the eighteenth and nineteenth centuries. One historian changed the situation

completely: the reactionary catholic French physicist Pierre Duhem, writing around 1900. His key

works, based on a careful study of French medieval writers, as well as of Leonardo da Vinci, aimed

to show that there had been high-quality scientific activity from the thirteenth century on, and

that the Church had played a decisive role in promoting it.^3 Furthermore, and this was his main

contention, there had been no ‘revolution’; Galileo’s (physical) discoveries were already present in

the works of the Paris school in the fourteenth century; and, typically, the history of science is

continuous rather than catastrophic or revolutionary in nature.

[T]he mechanical and physical science of which the present day is so proud comes to us through an uninterrupted

sequence of almost imperceptible refinements from the doctrines professed in the Schools of the Middle Ages. The

so-called intellectual revolutions consisted,in most cases, of nothing but an evolution developing over long periods of

time. The so-called Renaissances were frequently nothing but unjust and sterile reactions. Finally, respect for tradition

is an essential condition for all scientific progress. (Duhem 1991, p. 9.)

The fact that many historians are still committed to some version of the Duhem thesis means

that a serious account of the scientific revolution needs to start some time before; which, in turn

tends to lead to an overload of often disparate information from a period of several centuries during

which mathematics was used in a variety of different ways. Because the material for this chapter is

rich, diffuse, and well, if unevenly, covered in various texts, we shall develop the story as a series of

snapshots, or meditations on particular themes. To try for any degree of completeness would be to

risk complete unreadability.

3. Scholastics and scholasticism

Although this is an instance of an unfounded mathematical formulation of a natural law that is not valid,
Bradwardine’s^4 argument is by no means destitute of historical significance. (Dijksterhuis 1986, p. 191)
The question naturally arises as to what the scholastics did with their interpretation of Eudoxus. What use can one
make of the useless? (Murdoch 1963, p. 257)

Medieval science is now taken seriously, if often with the kind of patronizing despair expressed

by Dijksterhuis and Murdoch. Thanks to the detail in Duhem’s research, it is not necessary

to agree with his more extreme theses (e.g. that the Church had helped scientific research by

condemning Aristotle in 1277) to see that the work of the period preceding Galileo cannot be

dismissed out of hand. However misguided his theories and specious his arguments, he con-

tributed more than anyone else to changing our ideas of the scientific revolution by enforcing

at the very least a serious consideration of Galileo’s predecessors, even if the end result was

(as with Koyré, for example, see (1978)), to conclude that therewasa decisive break rather than

a continuous evolution. The starting point of any thoughtful history is thereby pushed back; and

where, before the sixteenth century, one chooses to start is likely to be determined by something

3. There is no consensus about how one uses the terms ‘Middle Ages’ or ‘medieval’; and the problem is made worse by the fact
   that so much that is obviously ‘Renaissance’ which in some sense implies post-medieval, is also obviously ‘medieval’ in period—for
   example, the cathedral at Florence, or the work of Masaccio (both fifteenth century). Different ways of describing society may coexist
   more or less uneasily; my ‘medieval’ is roughly from 1100 to 1500, while my ‘Renaissance’, at least in Italy, is roughly from 1350 to


4. To complicate matters further, the term ‘early modern’ is now academically popular for (roughly) the period 1500–1700.


5. Thomas Bradwardine, fourteenth century Oxford physicist-mathematician, whose study of motion has often been considered
   a precursor of Galileo’s work.