Phonetic Writing. We are taking it for granted that
within the history of civilization there is a correlation,
an important one, between the rise of organized
knowledge and the emergence of adequate writing, the
degree of adequacy being measured by the degree of
phonetic articulation. With a suitable definition of
“phonetic,” we may say that Chinese writing is a fully
developed phonetic system, and has been so since its
appearance about the middle of the second millennium
B.C. (Gelb, p. 85). Now, mathematics is the oldest
organized knowledge there is, or nearly so, and our
problem is the task of assessing the role of mathematics
in this correlation between knowledge and writing.
On hard evidence, the presence of organized mathe-
matics is first attested around 1800 B.C. Now, our first
intricacy is the fact that this first evidence for organ-
ized mathematics appears in two separate areas simul-
taneously, in Mesopotamia and in Egypt (Neugebauer,
Chs. 1-3). This simultaneity of appearance cannot be
readily explained by invoking a “cultural diffusion” or
only a “stimulus diffusion” because by intent and con-
tent these two mathematics are very different from
each other (Neugebauer, loc. cit.; van der Waerden,
Chs. 1-3). Both these systems of mathematics are
documented by writing that is highly organized (see
Gelb, pp. 168ff., for Egypt, and Kramer, p. 306, for
Mesopotamia), but of seemingly different provenance.
In both geographical areas the mathematics in use
appears to be a full part of organized knowledge in
general. Also, from our retrospect, the Mesopotamian
mathematics even gives the impression of having been
in the very vanguard of organized knowledge, although
to an “average Mesopotamian” of the era, the code
of Hammurabi may have been more important, and,
above all, much more familiar.
At this point it might be expected that if writing
and knowledge reach a certain stage of organization
conjointly, then organized mathematics would do so
too. But this would be a very hasty expectation, as can
be seen from the development in an area that geo-
graphically and intellectually was very proximate to
both Mesopotamia and Egypt, namely, the land of the
Bible, where writing of various modes was organizing
itself in the latter half of the second millennium B.C.,
and in this area it was even advancing towards its
ultimate stage, namely, to the stage of becoming en-
tirely alphabetical (Gelb, pp. 134-53). At the same
time, juridical, ethical, and sacerdotal knowledge was
organizing itself too, and much of it became ultimately
knowledge for the ages. Yet, the history of mathematics
knows absolutely nothing about an indigenous mathe-
matics also springing up in this area at the same time.
This is our most serious intricacy.
A different problem arises—and it is a research task
rather than a conundrum—when we take into account
Greek achievements, from the first millennium B.C. It
is a fact that the Greeks made writing fully alphabeti-
cal. They thus created “a writing which expresses the
single sounds of a language” (Gelb, p. 197), and their
script was undoubtedly more advanced than Babylon-
ian script from around 1800 B.C. Similarly, the mathe-
matics which the Greeks began shaping within their
own thought patterns in the sixth century B.C. was,
even from the first, undoubtedly more advanced than
the Babylonian mathematics from around 1800 B.C.
Now, the research task arising is to determine whether
these two advances are commensurate in extent. Greek
mathematics drew, as heavily as it could, on all the
accumulated mathematics that preceded it. Never-
theless, it is immediately clear that the Greek intellec-
tual innovation in organized mathematics—as also in
organized knowledge of any kind—was far greater than
the parallel advance in the phonetic quality of writing.
An analysis in depth might counteract this impression.
But such an analysis could not be an easy one, because
it would also have to account for the fact that modern
mathematics is immeasurably superior to the Greek
creation, although “from the Greek period up to the
present nothing has happened in the inner structural
development of writing” (Gelb, p. 184).
Finally, we wish to observe that both for organized
mathematics and organized writing it is equally diffi-
cult to decide whether in China either came into being
independently of the West, or in direct dependence
on the West, or by a combination of the two possi-
bilities at different times. For organized mathematics
the problem of its geographic propagation in Asia was
already known, more or less, in the nineteenth century,
perhaps under the impact of the corresponding prob-
lem for language as it is spoken. In the twentieth
century the problem for mathematics has received less
attention than the corresponding problem for writing,
early or organized, probably because for mathematics
the problem has been less able to exploit achievements
in archeology than those in writing. For instance, for
writing there is a balanced account of the problem in
Gelb, Ch. VIII, but there seems to be no analogous
account for mathematics.
