BMCR 2004.10.25, Netz, The Transformation of Mathematics

Reviel Netz, The Transformation of Mathematics in the Early
Mediterranean World: From Problems to Equations.  Cambridge:  Cambridge
University Press, 2004.  Pp. 198.  ISBN 0-521-82996-8.  $70.00.

Reviewed by Anne Mahoney, Tufts University (anne.mahoney@xxxxxxxxx)
Word count:  1987 words
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In this book, Netz argues that the history of mathematics should
consist not only of a catalog of which mathematicians worked on what
problems when, but also of an analysis of how they conceived of the
problems they were solving. He observes that many modern textbooks,
sourcebooks, and even more scholarly works on the history of
mathematics obscure the line of development when they almost
automatically translate earlier works into modern notation. The example
he develops throughout is a problem studied by Archimedes: how do you
cut a sphere, with a plane going through one of the latitude lines, so
that the volume of the bigger part has a certain given ratio to the
volume of the smaller part? For Archimedes, this is essentially a
geometric problem, to be solved by manipulating geometric objects and
their ratios -- lines, rectangles, similar triangles, and so on. But
for Omar Khayyam, the 11th-12th century mathematician perhaps better
known for "The Rubaiyat", the sphere-cutting problem is an algebraic
problem about squares and cubes.[[1]] Modern treatments of Archimedes
almost invariably say "here is where Archimedes solves cubic
equations."[[2]] This is mathematically correct, in that the problem
can in fact be expressed in terms of a cubic equation in one variable,
to be solved for x. But N's point is that this is historically false:
Khayyam was thinking (more or less) of cubic equations, but Archimedes
was not and could not. The book shows how the geometric style of posing
and solving problems gives way, in the Arab world, to an algebraic
style much more familiar to modern mathematicians.

At this point, regular readers of BMCR may be wondering what this
review is doing here. It is a mathematical book, to be sure. But this
book, like N's earlier "The Shaping of Deduction in Greek Mathematics"
(reviewed here, 2000-02-17), has much to tell us about how the Greeks
saw the world. Although Greek mathematics is an ancestor of modern
mathematics, it is also radically different, highly foreign. N is
correct that we lose much of the "Greekness" of the ancient works when
we think only in modern terms.

Where "The Shaping of Deduction" was primarily about Greek
mathematicians' use of language, however, the present book is much more
about the actual mathematics. N argues that the choice of techniques
for solving a given problem is neither arbitrary nor neutral.
Naturally, mathematicians can only use techniques they know about;
neither Archimedes nor Khayyam can solve this problem with integral
calculus as we might today.[[3]] But within the tool-kit of available
techniques, classical Greek mathematicians, in general, make a point of
choosing a different method from the last person to work on the
problem, while late antique and Arab mathematicians, on the whole, look
for the general, over-arching principles behind the problem (p.
187-190).

N suggests that among the Hellenistic Greeks (Archimedes, Dionysodorus,
Diocles, as well as their predecessors and contemporaries; 3rd-2nd
centuries BC), mathematics is a competitive sport. The goal is to
demonstrate that you have a better solution than other people: more
general, more complete, or more elegant. As N puts it, "the space of
[mathematical] communication is an arena for confrontation, rather than
for solidarity. The relation envisaged between works is that of
polemic. A Greek mathematical text is a challenge" (p. 62). While this
was not necessarily true at the very beginning, by the Hellenistic
period it does seem to be a significant part of the style of
mathematical discourse. To a modern mathematician, used to building on
theorems proven by others, even collaborating on joint papers, this
seems an unusual way of doing mathematics. One implication of this
combative style, according to N, is that mathematics comes to consist
of a series of problems to be solved, not a theory, language, or system
as we conceive of it.

On the other hand, present-day mathematicians do not behave precisely
like the next groups N studies either. These are the mathematicians of
late antiquity in the West, and then those of the Arab world. The main
figures here are Eutocius (6th c. AD), who wrote a commentary on
Archimedes which is still one of our main sources for his work, and a
series of Arab mathematicians in the ninth through the twelfth
centuries. For N, both of these groups have the same aim; their
mathematics is fundamentally deuteronomic. By this he means that these
writers crucially depend on previous writings. Eutocius, for example,
is a commentator on Archimedes in much the same way as Servius is a
commentator on Virgil. For a deuteronomic mathematician, the goal is
not to find a better solution to a known problem, but instead to
standardize and systematize the existing solutions and also their
presentations. N points out that this is one of the ways we get
"second-order terminology" for mathematical objects and problems (p.
122), terms for describing and classifying, such as characterizing
problems as linear, planar, or solid based on the kinds of objects
involved in their solutions. Mathematicians like Archimedes want to
create an "individual aura" around their solutions (p. 125, et passim),
while "Eutocius aims at contextualization, which is the removal of
aura" (p. 125). That is, while the earlier mathematicians are competing
to find the best or cleverest solution, Eutocius and his contemporaries
and successors are concerned with explaining the various solutions and
ensuring all the cases are covered.

N suggests that the Arab mathematicians' project is deuteronomic as
well, though not in quite the same way as that of Eutocius. They are
attempting to fill in the gaps in Archimedes' presentation, to complete
his work, but not to compete with him (p. 131). That is, when a Greek
mathematician, Diocles for example, reads a solution to the
sphere-cutting problem that is not fully general, his initial impulse
is to say "This is wrong and I can do better." An Arab mathematician,
according to N, might instead say "This is missing something which
should be filled in, and I can help." In N's terms, this sort of
research is just as dependent on the text of Archimedes as is Eutocius'
sort; the difference is more of attitude. It also helps, of course,
that Khayyam and the other Arab mathematicians N discusses were
considerably more creative as mathematicians than Eutocius was.

The Greek solutions to the sphere-cutting problem are all fundamentally
geometric. Archimedes begins by reducing the problem to one involving
ratios of rectangles in the plane and rectangular boxes in three
dimensions. He then uses similar triangles and conic sections to
construct a solution. The largest mathematical difference in the Arab
solutions is that they are algebraic: Al-Khwarizmi's book "On the Art
of Al-Jabr wa l-Mukabala", published in the ninth century, gave the
Arab world another way to look at calculations. The two nouns in the
title refer to the operations involved in posing and solving problems
like "a given number of quantities plus another given number of roots
equals another number," or, in more modern terms, ax2 + bx = c. The
Greeks never talked about solving an equation for a numerical answer;
for them, problems are always posed in terms of lines, squares,
rectangles, and ratios. Al-Khwarizmi starts from lines and rectangles,
but goes on to describe a method of calculating the measure of the
basic line x -- and then leaves the idea of lines and measures
behind.[[4]] Moreover, he and his successors are interested in
determining all the possible problem-patterns of a given class (for
example, involving sums of "quantities" (squares), "roots" (the things
the quantities are the squares of), and numbers), and figuring out how
to solve them.

By the time we get to Khayyam, nearly 400 years after Al-Khwarizmi, the
basics of the Art of Al-Jabr have become routine. Khayyam's "Algebra"
claims to be a complete synthesis of the field, with particular
attention to the most difficult problems (p. 146). N analyzes the
structure of Khayyam's text, which is simultaneously a scientific
autobiography, a survey of what algebra is, and a series of theorems
with their proofs and commentary. Khayyam divides, classifies, lists,
and categorizes his theorems and problems, working through each list in
a careful and systematic way. He makes references to many earlier
mathematicians, both Arab and Greek. As N points out, the method of
making exhaustive lists of kinds of problems, then treating them in
order, also operates at the level of the individual proofs (p. 153),
which are often divided into cases: for example, one length or quantity
may be larger than, smaller than, or equal to another one. As each of
these three cases may require a different diagram (with the point or
line representing the first number being to the right, to the left, or
on top of the one representing the other), it is convenient to treat
them separately, as mathematicians do to this day.

Archimedes and Khayyam, then, approach the sphere-cutting problem in
essentially different ways. "Archimedes' problem arises, as it were, in
'real-life geometry,' and its shape is determined by the demands of
this 'real-life geometry.' Khayyam's problem arises from its position
in a list of problems -- the list deriving not from an external,
geometrical investigation, but from its own independent listing
principle" (p. 183). More fundamentally, Khayyam steps back from the
particular task of cutting a particular sphere and looks at an entire
class of similar problems. Algebra itself, N argues, is a way to "step
back" from geometry and look more abstractly at problems; it is
intrinsically deuteronomic, in N's sense. In short, "Hellenistic Greek
mathematics, whose practice may be summed up by the aura, foregrounded
the local characteristics of configurations, giving rise to the
problem; medieval mathematics, whose practice may be summed up by
deuteronomy, foregrounded the global characteristics of relations,
giving rise to the equation" (p. 190; italics original). The cultural
differences between Greek and later mathematicians and their preferred
approach to mathematics produce significant differences in the kinds of
mathematics they end up doing. Simply stating, as some modern books do,
that Archimedes and Khayyam are both solving cubic equations, while
true at some level, obscures this cultural difference. Mathematical
progress involves not only the ability to solve more problems, but also
the ability (and even the desire) to pose different problems, solve
them in different ways, and compare the solutions.

N's style throughout is engaging, almost conversational, with many
rhetorical questions and other addresses to the reader. The text is
very well organized: each of the three chapters begins with an
overview, and section titles and running heads keep the structure clear
for the reader. N has translated all the Greek and Arabic texts; not a
word of Arabic, and very little Greek, appears in the book (and the few
Greek words are transliterated). As a result this book is a more
mathematical companion piece to "The Shaping of Deduction", which was
largely concerned with the language of the Greek texts. Here N is more
concerned with the mathematics itself than the words in which it is
presented, so translation is a reasonable choice and makes the book
accessible to a larger group of potential readers. He does, of course,
give references to standard editions of the various texts.

The history of mathematics is generally studied in mathematical terms,
looking at questions like who solved a given problem first, whether
someone's announced or rumored proof really existed and was sound,
whether various people worked together or knew each other's work. As N
points out, often the mathematics in question is expressed in modern
terms, however different they might be from those used by the scholars
under study.[[5]] N's program, applying the tools of philology to
historical mathematical texts, is highly original and beautifully
creative; it shows off features of Greek mathematics that many modern
mathematicians have ignored. The present book is useful reading not
only for historians of mathematics, but for anyone interested in how
the Greeks understood the world.

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Notes:


1.   I give all Arabic words, including proper names, in N's
transliteration.

2.   For example, Ivor Thomas, in the Loeb "Greek Mathematical Works"
(Cambridge: 1941, rpt. 1963), vol. 2, p. 126-163, gives the text under
the title "Solution of a Cubic Equation," and gives an interpretation
in modern notation in a footnote. Similarly, a standard introductory
textbook in history of mathematics talks of Archimedes "reducing his
cubic equation" to a quadratic (Carl B. Boyer, "A History of
Mathematics" (New York: Wiley, 1968), p. 147).

3.   Although Archimedes uses several arguments that look very much
like limit processes, infinite sums, or integrals, he never relies on
these for a proof. They are part of the analysis, not the synthesis;
part of figuring out the answer, but not part of demonstrating its
correctness. Examples and details in Thomas.

4.   As is well known, Al-Khwarizmi's name has become the standard word
for a method of working a problem or calculating something: an
algorithm. And the method of Al-Jabr, reducing a problem to canonical
form by moving a term from one side of the equation to the other, gives
its name to algebra.

5.   This is not invariably true, particularly for more recent, less
radically foreign mathematics. Studies of the development of
differential calculus, for example, generally mention the parallel
development of notation for referring to derivatives and differentials.
See for example Boyer p. 441, 533.