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Dictionary of the History of Ideas | ||

*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

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).

de Synthèse,

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.

Dictionary of the History of Ideas | ||