1 The Birth of Science - MSRI

11.10 The Removal of Ancient Science 343
model. It is not an accident that his “method of ultimate ratios” is part of
his philosophy of nature. Newton and his contemporaries were still far from
mastering the technical methodology that, two thousand years earlier, had
allowed Archimedes to compare infinitesimals of different orders within
the rigorous structure of a hypothetico-deductive model. 162
This is how Boyer, in his History of mathematics, contrasts ancient and
early modern infinitesimal methods:
Stevin, Kepler, and Galileo all had need for Archimedean methods,
being practical men, but they wished to avoid the logical niceties of
the method of exhaustion. It was largely the resulting modifications
of the ancient infinitesimal methods that ultimately led to the calculus[.]
163
How did modern infinitesimal calculus ever manage to work despite
its lack of “logical niceties” (or, to put it more bluntly, in spite of logical
contradictions)? Probably because it was created precisely by pruning
the “logical niceties” from an actual scientific theory. The “practical
men” of calculus considered raw “infinite” or “infinitesimal” quantities
because they were not in a position to obtain rigorous demonstrations
using only finite quantities, as Euclid and Archimedes did, and as contemporary
mathematical analysis would again do. It was at that point,
thanks to the “practical” mathematicians, that the idea was born that infinity
was unfathomable to the “Ancients”. 164 This misjudgement survived
in the eyes of many historians of mathematics even after rigorous infinitesimal
methods were reintroduced in the late nineteenth century. 165
11.10 The Removal of Ancient Science
Each step in the recovery of ancient knowledge was accompanied by a loss
of historical memory. The assimilation of ancient ideas, indeed, consists in
translating them into the idiom of one’s own culture, recasting them into
writings that tend to edge out the old ones, which often end up forgotten.
162 See, for example, Archimedes’ On spirals, 5 (ed. Mugler), where the comparison between infinitesimals
of different orders is an important step in determining the direction of the tangent at
an arbitrary point of the spiral. This is the first known discussion of a topic in what we now call
differential geometry, making Archimedes the founder of the subject.
163 [Boyer], p. 354 (1st ed.), p. 322 (2nd ed.).
164 Of course, when the modern recovery of the scientific method started, there reappeared mathematicians
who “didn’t understand infinity”. Boyer states, with apparent amazement, that both
“Gauss and Cauchy seem to have had a kind of horror infiniti”; see [Boyer], p. 565 (1st ed.), p. 516
(2nd ed).
165 See the quotation by Kline on page 38.
Revision: 1.11 Date: 2003/01/06 07:48:20