Epilogue. In modern mathematics the Greek limita-
tions which we have adduced were overcome mainly
by conceptual innovations, namely by the creation of
abstractions, and of escalations of abstractions, which
do not conform with the cognitive texture of Greek
classical philosophy and general knowledge. There is
an all-pervading difference between modern mathe-
matical abstractions and, say, Platonic ideas; reductions
of the one to the other, as frequently attempted in
philosophy of mathematics, are forced and unconvinc-
ing. There are analogies and parallels between the two,
but not assimilations and subordinations. The Greeks
could form the (Platonic) idea of a “general” triangle,
quadrangle, pentagon, and even of a “general” poly-
gon, but the conception of a background space for
Euclid's geometry was somehow no longer such an idea
and eluded them. A Platonic idea, even in its most
“idealistic” form, was still somehow object-bound,
which a background space for mathematics no longer
is. Nor could the Greeks form the “idea” of a rotational
momentum, for it simply is no longer an “idea,” and
cannot be pressed into the mold of one. It is, quanti-
tatively, a product l · p in which l and p represent
“ideally” heterogenous objects, but are nevertheless
measured by the same kind of positive real number;
and real numbers themselves are already abstractions
pressing beyond the confines of mere “ideas.” Such
fusion of several abstractions into one was more than
the Greek could cope with; and the formation of
(Newton's) translational momentum, as presented skel-
etally above, was even farther beyond their intellectual
horizon.
If it is granted that Greek mathematics has been thus
circumscribed, it becomes a major task of the history
of ideas—and not only of the history of mathe-
matics—to determine by what stages of medieval de-
velopment, gradual or spontaneous—the inherited
mathematics was eventually made receptive to sym-
bolic and conceptual innovations during the Renais-
sance and after.
This task is inseparable from the task of determining
the originality and effectiveness of medieval Arabic
knowledge, mathematical and other, and its durable
influence on the Latinized West. Only within this kind
of setting will it be possible to comprehend the course
of mathematics in its conceptual and cultural aspects.
