Paper prepared for
Summer School on the History of Algebra
Institute for the History of the Natural Sciences, Chinese Academy of Science, 1-2 September 2011◄
Abstract
In 1942, Edgar Zilsel proposed
that the sixteenth-seventeenth-century emergence
of Modern science was produced neither by the university tradition, nor by the
Humanist current of Renaissance culture, nor by craftsmen or other practitioners
but through an interaction between all three in which all were indispensable
for the outcome. He only included mathematics via its relation to the
'quantitative spirit'. The present study tried to apply Zilsel's perspective to
the emergence
of the Modern algebra of Viète and Descartes (etc.), by tracing the reception of
algebra within the Latin-Universitarian tradition, the Italian abbacus tradition
and Humanism, and the exchanges between them, from the twelfth through the
late sixteenth and early seventeenth century.
“Mesopotamian Calculation: Background and Contrast to Greek Mathematics”
Contribution to
IX Congresso della Società Italiana di Storia della Matematica
Genova, 17–19 novembre 2011
Abstract
The fourth-millennium state
formation process in Mesopotamia was intimately linked to accounting and to a
writing system created exclusively as support for accounting. This triple link
between the state, mathematics and the scribal craft lasted until the end of the
third millennium, whereas the connection between learned scribehood and
accounting mathematics lasted another four hundred years. Though practical
mathematics was certainly not unknown in the Greco-Hellenistic-Roman world, a
similar integration was never realized.
Social prestige usually goes together with utility for the
power structure (not to be confounded with that mere utility for those in power
which characterizes a working and tax/tribute-paying population), and until the
1600 BCE scribes appears to have enjoyed high social prestige.
From the moment writing and accounting was no longer one
activity among others of the ruling elite (c. 2600 BCE) but the task of a
separate profession, this profession started exploring the capacity of the two
professional tools, writing and calculation.Within the field of mathematics,
this resulted in the appearance of “supra-utilitarian mathematics”: mathematics
which to a superficial inspection appears to deal with practical situations but
which, without having theoretical pretensions, goes beyond anything which could
ever be encountered in real practice. After a setback in the late third
millennium, supra-utilitarian mathematics reached a high point – in particular
in the so-called “algebra” during the second half (1800–1600) of the “Old
Babylonian” period.
Analysis of the character and scope of this “algebraic”
discipline not only highlights the difference between theoretical and high-level
supra-utilitarian mathematics, it also makes some features of Greek theoretical
mathematics stand out more clearly. Babylonian “algebra” was believed by
Neugebauer (and by many after him on his authority) have inspired Greek
so-called “geometric algebra”. This story, though not wholly mistaken, is today
in need of reformulation; this reformulation throws light on one of the
processes that resulted in the creation of Greek theoretical mathematics.
“The World of the Abbaco. Abbacus mathematics analyzed and situated historically between Fibonacci and Stifel”. Book manuscript in progress, version 1 March 2022
“Intermediaries between Ab Kāmil’s and Fibonacci’s algebras – lost but leaving indubitable traces”
Abstract
It is an oft-repeated
claim that Leonardo Fibonacci’s algebra borrows from Abū
Kāmil.
A thorough analysis of the Liber abbaci as a whole
and, in particular of course, of its
chapter 15 part 3, the algebra, confirms
this; but it also shows that at least many, plausibly
all of the borrowings
are indirect, and almost certainly that at least one group of
borrowings go
via a Latin intermediary which is not itself a direct translation of Abū
Kāmil’s algebra.
In order to see that one has, firstly, to
know whether Fibonacci was faithful or
creative when borrowing. Elsewhere in
the Liber abbaci he demonstrates to be
deliberately faithful. If that is
taken into account in the analysis of his algebra confirms
that a number of
problems come from Abū Kāmil but not directly. Since, moreover,
a number of
the borrowings use the translation avere (a loanword from a Romance
vernacular meaning “possession”) for an initial non-algebraic unknown number
(sometimes represented afterwards by a res, “thing” in the ensuing
algebraic solution,
sometimes by a census), the ultimate intermediary can be
seen to have been Latin: there
is no reason an Arabic treatise should borrow
a Romance-vernacular term, and if against
all odds this should have happened,
then its orthography would hardly have survived
the transcription into Arabic
and Fibonacci’s ensuing retranslation undistorted.
The terminological
innovation demonstrates metamathematical acumen on part of
the internediary:
neither al-Khwārizmī nor Abū Kāmil make this distinction between
a
non-algebraic and an algebraic māl.
“Where and how did Archimedes get in? Oblique and labyrinthine
reflections