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;
declining manpower; loss of economic freedom; “the
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
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
today a product
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.
