Ivor G.G.'s view of the history of mathematics tends not to
extend past the Greeks to any significant extent. I prefer
the expensive Companion Encylopedia for the History and
Philosophy of Mathematics, and its insightful board of
directors, including David Pingree, Eberhard Knobloch and
several others. Within the pre-Greek views stated by C.S.
Roero, a strong acceptance of pre-600 BC number theory was
stated in an interesting manner. Have you read Roero's views?
As a comparative view to the historiography of mathematics,
as covered in Ubi's South American paper, recently posted,
it provided little sense that Incas or Mesoamericans used
any aspect of number theory. Ubi's views were important on
several levels. New World culture including its math should
be given back, if possible, as Ubi clearly stated. However,
due to Ubi's reductionist format, based in great part on
tracing Greek and European math across South America, I found
these fragments left out many easy to read Western Tradition
arithmetic, and New World mathematics, that could have shown
that ancient cultures commonly studied astronomy and developed
a rich arithmetic (well before geometry was formalized).
Could it be that ancient texts, like the Dresden Codex that
contains a clear statement of remainder arithmetic, where
fractions are not needed (to be stated as rational numbers, as
Greeks, Egyptians and Babylonians used) defines a rich form
of number theory that both Ubi and Ivor G.G. avoided for some
reason? I wonder if ancient number theory outside of the Western
Tradition is inconsistent with their views on Classical Greeks
mathematics?
That is to suggest, the historiography of mathematics should
be stated in such a manner that inadvertent reductionist readings
of ancient texts should be allowed, at some point, to be attempted
to be unfragmented when pieces obviously fit together. Given the
extent that astronomical cycles measurements and predictions were
made in the New World and the Old World, number theory should be
given a special place in history of math and science communities,
that it currently does not possess.
I await the day when number theory fragments are treated with
the same respect as Western geometric fragments have been treated.
For sure Ivor G.G. is correct in his view that a great change
has taken place in the history of math after 1970, with the
introduction of journals like HM. However, the openness and
extended de-centralization provided by the internet is a fresh
dynamic that math and science historians are still wondering
how to direct/control, as past controversial issues such as
Hultsch-Bruins patterns found in the RMP 2/nth table were pushed
aside, as if they did not exist.
Professional discussions should not be pedagogically re-directed
or controlled by gatekeepers, as seen in the Otto Neugebauer-David
Pingree debate, (1972-1989), in The Journal for the History of
Astronomy. The facts of ancient texts should be allowed to speak
for themselves, as the common Latin phrase is used by the legal
community. When there are controversial disagreements, the facts
of the discussion should be available for all to see, and ponder,
without inappropriate apologies being requested or issued.
Again, this is only a brief response.
Regards,
Milo