III. MATHEMATICS
A famous Greek encounter with infinity is the
“puzzles” (logoi) about motion by Zeno of Elea, about
the middle of the fifth century B.C. Best known is the
conundrum about “Achilles and the Turtle.” It main-
tains, against all experience, that in a race between
a quick-footed Achilles and a slow-moving Turtle, if
the Turtle has any head start at all then Achilles cannot
overtake him, ever. In fact, by the time Achilles has
reached the Turtle's starting point the latter has moved
on by a certain distance; when Achilles has covered
that distance, the Turtle has again gained a novel
distance, etc. This gives rise to an unending sequence
of distances; and the puzzle maintains that Achilles
cannot exhaust the sum of the distances and come
abreast with the Turtle (Ross, Aristotle's Physics, Intro-
duction; also A. Edel, Aristotle's Theory...).
The puzzles have an enduring appeal; but their role
in stimulating Greek rationality cannot be easily
gauged, because the Greek documentation of them is
very sparse and hesitant. The puzzles were transmitted
only by Aristotle, not in his Metaphysica, which is
Aristotle's work in basic philosophy, but only in the
Physica, and only in the second half of the latter, which
deals with problems of motion, and not with concep-
tions and principles of physics in their generality as
does the first half. Furthermore, in classical antiquity
the puzzles are never alluded to in mathematical con-
texts, and there is no kind of evidence or even allusion
that would link professional mathematicians with them.
In a broad sense, in classical antiquity the conception
of infinity belonged to physics and natural philosophy,
but not to mathematics proper; that is, to the area of
knowledge with which a department of mathematics
is entrusted today. Nobody in antiquity would have
expected Archimedes to give a lecture “On Infinity”
to an academic audience, or to his engineering staff
at the Syracuse Ministry of Defence. Also, no ancient
commentator would have said that Anaxagoras (fifth
century B.C.) had introduced a mathematical aspect of
infinity, as is sometimes asserted today (e.g., in
Revue
de Synthèse, pp. 18-19).
Furthermore, such Greek efforts by mathematics
proper as, from our retrospect, did bear on infinity,
were—again from our retrospect—greatly hampered
in their eventual outcome by a congenital limitation
of Greek mathematics at its root (Bochner, The Role
of Mathematics..., pp. 48-58). As evidenced by
developments since around A.D. 1600, mathematics, if
it is to be truly successful, has to be basically opera-
tional. Greek constructive thinking however, in math-
ematics and also in general, was basically only idea-
tional. By this we mean that, on the whole, the Greeks
only formed abstractions of the first order, that is ide-
alizations, whereas mathematics demands also abstrac-
tions of higher order, that is abstractions from abstrac-
tions, abstractions from abstractions from abstractions,
etc. We are not underestimating Greek ideations as
such. Some of them are among the choicest Greek
achievements ever. For instance, Aristotle's distinction
between potential infinity and actual infinity was a
pure ideation, yet unsurpassed in originality and im-
perishable in its importance. However, as Aristotle
conceived it, and generations of followers knew it, this
distinction was not fitted into operational syllogisms,
and was therefore unexploitable. Because of this even
front-rank philosophers, especially after the Renais-
sance, mistook this distinction for a tiresome scholas-
ticism, until, at last, late Victorian mathematics began
to assimilate it into its operational texture.
In the seventeenth and eighteenth centuries, mathe-
matics was so fascinated with its newly developing raw
operational skills, that, in its ebullience, it hid from
itself the necessity of attending to some basic con-
ceptual (ideative) subtleties, mostly involving infinity,
the discovery and pursuit of which had been a hallmark
of the mathematics of the Greeks. Only in the nine-
teenth century did mathematics sober down, and finally
turn its attention to certain conceptualizations and
delicate ideations towards which the Greeks, in their
precociousness, were oriented from the first. But even
with its vastly superior operational skills, modern
mathematics had to spend the whole nineteenth cen-
tury to really overtake the Greeks in these matters.
This raises the problem, a very difficult one, of
determining the role of the Middle Ages as an interme-
diary between Greek precociousness and modern ex-
pertise. In the realm of mathematical infinity the
thirteenth and fourteenth centuries were rather active.
But studies thus far have not determined whether, as
maintained in the voluminous work of Pierre Duhem
(ibid., p. 117), a spark from the late Middle Ages leapt
across the Renaissance to ignite the scientific revolution
which centered in the seventeenth century, or whether
this revolution was self-igniting, as implied in well-
reasoned books of Anneliese Maier. And they also have
not determined what, in this area of knowledge, the
contribution of the Arabic tributary to the Western
mainstream actually was.
