"The scholastic writers of the Middle Ages, especially St. Thomas
Aquinas, accepted Aristotle's _infinitum actu non datur_ [= there
is no actual(ly) infinite], but considered every continuum as
potentially divisible ad infinitum. Thus there was no smallest
line. A point, therefore, was not a part of a line, because it
was indivisible: _ex indivisibilibus non potest constare aliquod
continuum_ [= a continuum cannot consist of indivisibles]. A point
could generate a line by motion. Such speculations had their
influence on the inventors of the infinitesimal calculus in the
seventeenth century and on the philosophers of the transfinite in
the nineteenth; Cavalieri, Tacquet, Bolzano, and Cantor knew the
scholastic authors and pondered over the meaning of their ideas."
In my opinion even Aristotle believed the continuum potentially divisible
ad infinitum. The differences between Aristotle and Scholsticism are
quite subtle, and I'll say something about it later. I think instead that
he real breakthrough was between Aristotelism and Renaissance Science.
And it was centred on the idea of "real number".
I believe that real numbers were an incredible creation of Renaissance
(Descartes, but even Wallis, Stevin, etc). For the classic Greek
mathematics the continuum was not numerical. We can even translate
'diastema' as 'distance' or 'chorion' as 'area', but in Euclid they
meant only 'interval' and 'surface'. Greek mathematicians were able
to create a sort of "algebra" of those objects but the gap between them
and the numbers was insuperable.
Actually even the rationals were far from a clear analysis: in astronomy
they were reduced by the sexagesimal system, in practical computations
they were reduced to unitary fractions in an egyptian style, in geometry
they appeared as 'logoi' and sometimes as a number of 'parts'.
For us real numbers are so common that we do not realize how far they
were alien to the ancient mathematical view. They became at the
beginning of modern science the bridge between numbers and geometrical
entities, merging the integers in the continuum (as special real numbers)
and translating the 'point' as an infinite sequence (limit or decimal
expansion or section in the rationals) of integers.
The rejection of the actual infinite lasted till Renaissance. We have
plenty of authors, from St.Thomas to Nicolaus Cusanus and even Giordano
Bruno, who tried to find a way out between such rejection and the
theological need of the actual infinite. A scholastic thesis was that
infinite did get neither finite nor infinite parts: it did not get parts
at all, and for many authors (Tertullianus, Nicolaus Cusanus,etc.) the
theologically motivated acceptance of the actual infinite went parallel
with the rejection of the logical principles.
Archimedes went very near to modern ideas about the infinite: his
techniques to write very great numbers, his approximation techniques
applied to compute "pi", his study of the spiral and conics were
bridges toward modern mathematics. But his theorems about areas are
maybe the most intriguing for our theme. Let us consider the example
referred by Milo Gardner: I do not know whether it was linked to
previous Egyptian approaches, but it is revealing about Greek ideas.
Here Archimedes 'compute' the sum of the geometrical series for the
ratio 1/4, but his approach is general. In the lemma in the "Quadrature
of the parabola" the figure displays a square divided in four little
squares. Three of them (a gnomon) are called A, the fourth, whose
'area' is obviously A/3, is furtherly divided in four little squares.
Three of them are a little gnomon, of 'area' A/4, the fourth, whose
area is A/12, is furtherly divided in four squares, and so on.
>From the figure it is easy to realize that the sum of the 'infinite'
gnomons is the sum of our geometrical series and gives, at the same
time, the whole square, whose area is trivially 4A/3. But Aristotle
never went so far, he employed the lemma to compute just a general
finite approximation to be employed in the successive exhaustion-based
theorem. Nicholas Oresme (XIV century) made this step and it was the
beginning of modern science.
Was this step not rigorous? Maybe. Surely it was much more rigorous
than many substantially accepted principles in Renaissance Mathematics
(as the principle of Cavalieri). It is noteworthy that Archimedes never
tried to propose these techniques, somehow analogous to those described
in his "Method".
My opinion is that here we can trace the rigid borders of a well
established paradigm whose crisis appears during the sunset of
Scholasticism dealing with theological questions.
My best regards
Luigi Borzacchini