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reasoning ridiculed by Galileo and Descartes, for example. Dijksterhuis makes the case against such
methods forcefully:
In fact, it had been one of the traditions of Scholasticism from the twelfth century onward to employ the so-called
sic et nonmethod, advocated especially by Abelard; its principle was that in dealing with a given subject all the opinions
that had ever been pronounced about it and all the arguments that could be advanced for or against a certain view
were enumerated and discussed as fully as possible...This method, of course, presented great advantages; it bespoke
a striving after objectivity and it helped to prevent an idea, once it had been pronounced, from falling into oblivion
again. It is, however, obvious that if the method were applied too thoroughly, the disadvantages would be bound to
preponderate. (Dijksterhuis 1986, p. 167)
This method, moreover, makes some of the most interesting medieval work on mathematics
appear peculiar in a unique way. The idea that a scientific question might be decided in this way by
logical arguments ultimately derives from Aristotle. Its great virtue is that it encourages us to think
of counter-arguments to the hypotheses to which we are committed, although the way of deciding
between alternatives tended in the Middle Ages to depend on logic rather than what would now
be called scientific method. And in mathematics, where we normally accept that there is exactly
one right answer, it may seem quite contrary to the spirit of the subject. (What arguments could one
produce against a method for solving quadratic equations? The question is worth thinking about.)
It was not the method of Islamic mathematicians, even those who most respected Aristotle, so that
the ‘mathematics’ of many of the schoolmen whatever it was worth, was genuinely a new area of
enquiry. In the course of teaching mathematics in the faculty of arts, (which led to study in one of
the advanced faculties of medicine, law, or theology), they frequently raised mathematical topics in
the form ofquaestiones, and tried to settle them by a form of debate.
The rational arguments of the schoolmen would not usually speak to today’s rational under-
standing, as they rested in general on the basis of Aristotle’s physics and logic (with a little Euclid),
rarely went far beyond, and were often quite confused. For an example, we could consider Albert of
Saxony’s discussion of whether it is possible to square the circle; for this see E. Grant’s sourcebook
(1978), a good source for the schoolmen generally. It seems clear that Albert did not know, or did
not understand the sophisticated methods of squaring by curves such as the quadratrix (for which
see Knorr 1986), since he made no reference to them. His equipment consisted basically of some
historians’ references to circle-squaring, and Archimedes’sMeasurement of the Circle; the latter he
misunderstood in the standard way to mean that the circumference was 22/7 times the diameter.
He gives four arguments for squaring and two against, and then makes—again a typical scholastic
trick—a distinction of five meanings which ‘squaring the circle’ could have. The distinctions are
important in a scholastic argument, since clearly if you have conclusive arguments for and against
a proposition, the proposition must have different meanings in the two cases. One argument for
is simply that Aristotle said that it had been done by Antiphon and Bryson (which is not what
Aristotle said in any case). The next introduces something new:
If there could not be given a square equal to a circle, it would follow that there would take place passage from ‘greater’
to ‘lesser’, or from extreme to extreme, through all the means without ever arriving at ‘equal’ or ‘middle’. But this is
false. Therefore I prove the consequence. (Grant 1978, p. 171)
What Albert is doing here is a simple version of what we would call an existence proof —it shows
by continuity that there must exist a square equal to the circle, while completely ignoring the
problem which preoccupied the Greeks, that is, how you construct it. This argument was already
known in Greek times, but Albert’s presentation has something fresh about it. One could, critically,