Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*Limitations of Greek Mathematics.* The reasons that

have been variously adduced for the dissolution of

ancient civilization—the overextension of the *oikou-
méne* so that “the stupendous fabric yielded to the

pressure of its own weight”; inadequacy of industriali-

zation and too much involvement with slave labor;

gradual absorption of the educated classes by the

masses”; “the pitiful poverty of Western Rome”; etc.

(Kagan, pp. xi and xii)—may all help to account for

the

*ultimate*extinction of Greek mathematics, around

500 A.D., after a gradual decline of long duration. One

may even add the view of J. B. Bury that “the gradual

collapse... was the consequence of

*a series of con-*

tigent events.No general cause can be assigned that

tigent events.

made it inevitable” (ibid.).

But, in the case of mathematics there is one peculiar

fact which no such reasons from general history can

really account for. It is the fact that in the second

century B.C., much before the ultimate extinction, the

decline of mathematics from the heights which it had

attained in the preceding century was seemingly too

large, too brusque, and too unmotivated by internal

developments to be satisfactorily explained by general

reasons of this kind. By standard criteria of advance-

ment, mathematics in the third century B.C. was in a

state of upward development, and it suggests itself that

the rather sudden break in the development after 200

B.C. may have been due, at least in part, to some

particular reasons applying to mathematics only. This

is indeed our suggestion, and we shall attempt to for-

mulate it.

In the third century B.C., Greek mathematics was

not only very good, but it also reached a climax. By

this we mean that it reached a level of development

that was maximal relative to the intellectual base,

mathematical and philosophical, on which it had been

erected and on which it rested. Therefore, mathematics

could have continued to develop in the second century

B.C. and later only if the overall intellectual base on

which it rested could also have been broadened in the

process. But of this kind of broadening of the total

intellectual setting of mathematics, Greek civilization

in the second century B.C. was no longer capable. The

general intellectual basis for Greek mathematics, which

in a sense never broadened or deepened, was laid in

the sixth and fifth centuries B.C., especially the latter,

and in this peculiarly conceived sense it can be said

that, as far as mathematics is concerned the decline

of Greek civilization reaches back even into the fifth

century B.C. (Walbank).

In order to demonstrate that the mathematics of

Archimedes and Apollonius was overripe relative to

its intellectual basis we shall compare the conceptual

setting in Archimedes and Apollonius with the corre-

sponding setting in Newton's *Principia* (1686), even if

Newton's work came nineteen centuries later. A com-

parison of the works of Archimedes and Apollonius

(and Pappus) with *La Géométrie* (1637) of Descartes,

which was published half a century before the *Prin*

*cipia,* would not serve our present purpose, because

Descartes does not retain the setting of antiquity. On

the contrary, he radically changed the technical setting

by a full recourse to the apparatus of algebraic sym-

bolism as made ready for him by Viète. Not so Newton.

He was most expert in the handling of this apparatus,

and on occasions he employed it more penetratingly

than Descartes and others; but, for reasons best known

to himself he elected to cast the *Principia* in a mold

of Archimedean technicalities, outwardly, that is. In

the *Principia* there are hardly any analytical formulas;

but there are circumlocutions and verbalized formulae

which, at times, seem to be as condensed and stero-

typed as in Archimedes. This makes for hard reading

nowadays, but it makes it easy to isolate differences

of approach and setting. The differences are enormous,

and we list the following ones.

Newton prominently introduced an underlying

overall space, his absolute space, as a background space

for both mathematics and mechanics. The Greeks

achieved nothing like it. They certainly did not intro-

duce a space for mechanics and mathematics jointly.

They did introduce a “place” for events in nature

which perhaps served as a space of mechanics, but they

most certainly did not ever introduce a space of math-

ematics, or any kind of space of perception, physical,

logical, or ontological. In mathematics, they had “loci”

for individual figures when constructed, but not a space

for such figures before being constructed. In short, the

Greeks did not have any kind of space in the sense

of Descartes, or Newton, or John Locke.

Newton expressly introduced in his mechanics a

*translational momentum* (quantity of motion), defining

it, for a mass particle moving on a straight line, as

the product *m · v* in which the factor *m* is the constant

amount of mass of the particle and *v* is its instantaneous

velocity. Archimedes, in his theory of the lever, ought

to have introduced the conception of a rotational mo-

mentum, defining it as the product *l · p* in which the

factor *l* is the length of an arm of the lever and *p* is

the weight suspended from this arm. But Archimedes

did not introduce such a concept, nor did Greek math-

ematical thought ever conceptualize a product like

*l · p;* and mechanics went on marking time for almost

2000 years.

Even more significantly, Newton had the concept

of a function constantly in his thinking, however cov-

ertly. Altogether since the seventeenth century the

concept of a function kept on occurring in many facets

and contexts, in mathematics as well as in other areas

of cognition. Greek cognition, however, never had the

notion of function, anywhere. Even the absence of

products like *l · p* from Greek thinking was part of the

absence of functions, inasmuch as in mathematics of

*l · p*for variable values of

*l*and

*p,*

is a function on the set of pairs (

*l,p*). More centrally,

in cognition today the most important component of

the concept of function is the notion of

*relation,*how-

ever elusive it may be, to define or even describe what

a relation is. Aristotle, the creator of the academic

discipline of logic, did not anticipate the importance

of

*relation*(which he terms

*pros ti*), nor did the Stoic

logicians after him. But in modern developments, the

creation of an algebra or logic by the American

philosopher-logician Charles Sanders Peirce was his

most outstanding logical achievement.

Operationally, functions occur in Newton's *Principia*

in the following way. If a mass particle moves on the

*x*-axis and *t* denotes the time variable, then Newton

covertly assumes that there is a function *x*(*t*) which is

the instantaneous distance of the particle from a fixed

origin. He forms the derivative *dx/dt* for variable *t,*

which is a new function *v = v*(*t*). It is the instantaneous

velocity of the motion. He multiplies this by the con-

stant value *m* of the mass, thus introducing the instan-

taneous quantity of motion *m · v*(*t*), which is again a

function in the variable *t.* Newton then crowns these

covert assumptions with the hypothesis, which is ap-

parently due to himself, that the force *F* which brings

about the motion is, at every instant, equal to the rate

of change of the quantity of motion, *F = d*(*mv*)/*dt.*

This hypothesis, coupled with Newton's specific law

of gravitation, created our exact science of today. The

Greeks did not conceive of any part of this entire

context of assumptions and hypothesis, not because

they were unable to form a derivative of a function,

but because they did not have in their thinking the

concept of a function that is a prerequisite to forming

the various derivatives involved. By maturity of insight,

Archimedes was better equipped than Newton to carry

out the limit process that is involved in the formation

of a derivative, if only the concept of function and

the entire prerequisite setting had been given to him.

The Greek lack of familiarity with the concept of

function does not manifest itself only in mathematical

mechanics, which, to the Greeks was a relatively eso-

teric topic, but also in the entire area of geometry,

which, by a common conception, was a stronghold of

Greek rationality. There is a purely geometrical con-

text, common to Archimedes and Newton, in which

Newton does, and Archimedes does not have functions

in his thinking. Namely, Newton views the tangent to

a curve at a point of the curve as the limiting position

of a secant through the fixed point and a variable point

of the curve, so that, in effect, he performs the opera-

tion of differentiation on “hidden” coordinate func-

tions. Greek mathematics, however, never broke

through to this all-important view, but persisted in the

view, known from Euclid, that a tangent to a curve

is a straight line which in its entire extent coincides

with the curve at one point only. Archimedes tries to

adhere to this Euclidean definition even in his essay

on (Archimedean) spirals, in spite of the complication,

of which he is apparently aware, that any straight line

in the plane of the spiral intersects it in more than

one point. Without putting it into words, Archimedes

overcomes the complication by a simple adjustment,

but he does not advance towards the modern concep-

tion of a tangent as in Newton.

Dictionary of the History of Ideas | ||