INTRODUCTION
Does mathematics have a history? I believe it does, and in this book I offer an example. I follow a mathematical problem from its first statement, in Archimedes’ Second Book on the Sphere and Cylinder, through many of the solutions that were offered to it in early Mediterranean mathematics. The route I have chosen starts with Archimedes himself and ends (largely speaking) with Omar Khayyam. I discuss the solutions offered by Hellenistic mathematicians working immediately after Archimedes, as well as the comments made by a late Ancient commentator; finally, I consider the solutions offered by Arab mathematicians prior to Khayyam and by Khayyam himself, with a brief glance forward to an Arabic response to Khayyam.
The entire route, I shall argue, constitutes history: the problem was not merely studied and re-studied, but transformed. From a geometrical problem, it became an equation.
For, in truth, not everyone agrees that mathematics has a history, while those who defend the historicity of mathematics have still to make the argument. I write the book to fill this gap: let us consider, then, the historiographical background.
My starting point is a celebrated debate in the historiography of mathematics. The following question was posed: are the historically determined features of a given piece of mathematics significant to it as mathematics? This debate was sparked by Unguru’s article from 1975, “On the Need to Re-write the History of Greek Mathematics”.1 (At its background, as we shall mention below, was the fundamental study by Klein, from 1934–6, Greek Mathematical Thought and the Origins of Algebra.)
At the heart of Unguru’s article was a simple claim for historicity. Theorems such as Euclid’s Elements II.5, “If a straight line is cut into equal and unequal <segments>, the rectangle contained by the unequal segments of the whole, with the square on the <line> between the cuts, is equal to the square on the half” (see fig. 1) were read, at least since Zeuthen (1886), as equivalent to the modern equation (a + b)(a − b) + b2 = a2. That Euclid had not referred to any general quantities, but to concrete geometrical figures; that he did not operate through symbols, but through diagrams; and that he reasoned through manipulations of the rectangles in the diagram, cutting and pasting them until the equality was obtained – all this was considered, by authors such as Zeuthen, as irrelevant. As a pure mathematical structure, the equivalence between Euclid’s formulation and modern algebra is straightforward. It is also indeed true that, for the modern reader, the best way to ascertain the validity of Euclid’s theorem is by correlating it with the symbolic notation. And here arrives the seduction of a-historicism: mathematics is supposed to be compelling, it overpowers its readers by the incontrovertibility of its arguments. So, the a-historicist feels, unless one is overpowered by the argument, it is not really mathematical. The real form of the mathematical argument, then, is the form through which the reader feels its validity – that is, for a modern reader, the modern form. In its geometrical cloth, the Euclidean formulation is rendered inaccessible to the modern reader, so that it is no longer, for him or her, a piece of mathematics. Zeuthen considered himself within his rights, then, in removing the dust of the ages and uncovering the real form of Euclid’s theorem, which was, according to Zeuthen, algebraic. The Greeks merely clothed their algebra geometrically, so that we may call this type of science “geometrical algebra.” According to authors such as Zeuthen, all the historian of science needed to mark, as historian of science, was that, in the field of algebra, the Greeks had obtained such equations as (a + b)(a − b) + b2 = a2. That they then clothed these equations in geometrical form belongs not so much to the history of science itself, but to the history, so to speak, of scientific dresses: the sartorics of mathematics.
It was against this a-historical view that Unguru cried aloud in his article from 1975. At the time, he became the target of attack from some of the most distinguished historians of mathematics. A little over a quarter-century later, it is already difficult to doubt the basic correctness of Unguru. The exercise of geometrical algebra appears, in retrospect, as a refutation through absurdity of the a-historicist approach to mathematics. There are many reasons for this, but the most important is perhaps the following. By transforming the geometrical relations of Elements II into an algebraic equation, they are rendered trivial: so that, instead of allowing us to see better the significance of ancient argument, we, instead, lose sight of its importance for the ancient audience. The moral seems to be that, if, indeed, the way to understand a mathematical text as mathematical is by perceiving its validity; and if indeed the perception of validity depends on historically conditioned tools (e.g., diagrams, for the Greeks, symbols, for the moderns) – then the way to understand ancient mathematics is not by transforming it into our mathematical language but, on the contrary, by becoming, ourselves, proficient in the mathematical language of the ancients. The skill of parsing arguments through diagrams is as essential to a historical understanding of Euclid as the skill of parsing Greek hexameter is essential to a historical understanding of Homer.
All of which, however, still does not get us into history. While most historians of mathematics would now agree on the need to understand mathematical texts through the language of their times, this amounts, so to speak, to a dialectology of mathematics, not to its history. Greek mathematics, granted, is different from modern mathematics: but what is the historical transformation that led from the first to the second?
The very success of Unguru, in challenging the old model of the a-historical “Geometrical Algebra,” makes the problem more acute. For an a-historical scholar such as Zeuthen, there was no significant process beyond accumulation, so that the historian merely needed to record the dates and names involved with this, essentially static process. History, for Zeuthen, did not change; it merely boarded the escalator of progress. But what if the very nature of mathematics had changed with time? In this case, there is a complicated process characterizing the history of mathematics, and the first task of the historian would be to uncover its dynamics. But no convincing account has yet been offered of this process, so that Unguru’s claim remains, at best, as a tantalizing observation, and, at worst, as a dogmatic statement of a gap between the ancient and modern “minds.” For here is the paradox: unless some specific historical account is offered of the difference between ancient and modern mathematics, Unguru’s claim can seem to be saying that the ancients are just different from us and that is it. In this way, we have come full circle to a-historicism, the single monolith “Mathematics” now broken into the two smaller a-historical monoliths, “Ancient” and “Modern.”
Why did Unguru not offer such a historical account? This perhaps may be answered by looking for his historiographical ancestry. Indeed, the very assumptions that led Unguru to criticize geometrical algebra, also led him away from studying the dynamics of the transformation from the ancient to the modern. Unguru’s premise was that of a great divide, separating ancient from modern thinking. The assumption of a great divide, in itself, is not conducive to the study of the dynamics leading from one side of the divide to the other. But more than this: Unguru’s assumption of a great divide was, in turn, adopted from Klein’s study Greek Mathematical Thought and the Origins of Algebra – which still remains the best statement of the difference between ancient and modern mathematics. It was Klein’s study, specifically, that led scholars away from studying the dynamics of the transformation from the ancient to the modern. As it were: a-historical readers required no dynamics, while historical readers were satisfied with its absence, relying on the methodology offered by Klein. We should therefore turn briefly to discuss this methodology. But I should immediately emphasize that my aim now is not to argue against Klein. On the contrary, I see my book as a continuation and corroboration of Klein’s thesis. It is, however, by seeing the shortcoming of Klein’s approach, that a way beyond him could be suggested. I shall therefore concentrate in what follows on the shortcoming of this work which, undoubtedly, remains a study of genius.
Klein’s approach went deeper than the forms of mathematics. For Klein, it was not merely that the ancients used diagrams while the moderns use symbols. To him, the very objects of mathematics were different. The ancients referred to objects, directly, so that their arithmetic (the case study Klein took) was a study of such objects as “2,” “3,” “4,” etc. The moderns, however, refer to symbols that only then, indirectly, refer to objects. Thus modern arithmetic is not about “2,” “3,” and “4,” but about “k,” “n,” and “p,” with all that follows for the forms of mathematics. Ancient mathematics (and science in general) was, according to Klein, based on a first-order ontology; modern mathematics (and science in general) is based on a second-order ontology.
To repeat, my aim in this book is not to argue against Klein’s main thesis, but rather to find a historical explanation for an observation that Klein offered mainly on a philosophical basis. However, it should be said that Klein’s study was conceived in the terms of an abstract history of ideas that left little room for persuasive historical explanations. It is typical of Klein’s methodology that he takes, as his starting point, not the mathematical texts themselves, but Plato’s statements about mathematics, and that, inside mathematical texts, he is especially interested in methodological discussions and in definitions. When studying the history of arithmetic, Klein focuses on “the concept of the number.” Klein’s assumption is that, in different epochs, different concepts are developed. From the different fundamental distinction in concepts, the entire difference in the nature of the science follows.
I am not sure how valid this very approach to intellectual history is. I doubt, myself, whether any generalizations can be offered at the level of “the Greek concept of . . .” More probably, different Greek thinkers had different views on such issues, as distinct from each other as they are from some modern views. Nor do I think that periods in the history of science are characterized by some fundamental concepts from which the rest follows. Sciences are not coherent logical systems, developed through an inert deduction from first principles: they are living structures, proceeding towards first principles, away from them, or, most often, in ignorance of them, revamping ambulando their assumptions. At any rate, regardless of what we think of Klein’s method in general, it is clear that it made it very difficult for him to approach the dynamics of historical change. The neat divide, and its grounding in sharp conceptual dichotomies, simply left no room for a historical account of the transformation leading from the ancient to the modern. The issue was primarily a matter of logic, not of history. Klein merely sketched a possible account of this divide – and it is instructive to see the impasse that Klein had faced in this brief sketch (I quote from the English translation, Klein [1968] 120–1. All italics in the original):
Now that which especially characterizes the ‘new’ science and influences its development is the conception which it has of its own activity . . . Whereas the ‘naturalness’ of Greek science is determined precisely by the fact that it arises out of ‘natural’ foundations [i.e. reference to the real world] . . . the ‘naturalness’ of modern science is an expression of its polemical attitude towards school science. In Greek science, concepts are formed in continual dependence on ‘natural’, prescientific experience . . . The ‘new’ science, on the other hand, generally obtains its concepts through a process of polemic against the traditional concepts of school science . . . No longer is the thing intended by the concept an object of immediate insight . . . In evolving its own concepts in the course of combating school science, the new science ceases to interpret the concepts of Greek epistēmē preserved in the scholastic tradition from the point of view of their ‘natural’ foundations; rather, it interprets them with reference to the function which each of these concepts has within the whole of science.
There is much in this paragraph that I find insightful, and I shall to a large extent adopt, in the following study, the basic distinction Klein offered between first-order concepts and second-order concepts. But notice how difficult it would be to sustain Klein’s thesis, historically. Klein suggests: (a) that the main original feature of modern science is that it was polemical – as if Greek science was not! (b) that Greek science was throughout tied to pre-scientific, natural objects – whereas it was often based on flights of theoretical fancy, removed from any connection with the natural world; (c) finally, that somehow, by virtue of such differences, the Greeks would deal with “2,” “3,” and “4,” while the moderns would deal with “k,” “n,” and “m” – how and why this follows, Klein cannot say.
The truth is that, aside from the shortcomings of the history of ideas as such – leading to Klein’s emphasis on concepts, and to his ignoring practices – he was also a captive of certain received ideas about the basic shape of Mediterranean intellectual history, ideas that were natural in the early twentieth century but are strange to us today. “The Greeks,” to him, were all of a piece (as were, of course, the “moderns”). History was told in terms of putting the first against the latter. What went in between was then twice misrepresented. First, it was reduced to the Latin Middle Ages (the “schools” Klein refers to), so that the most important medieval development of Greek science, in the Arab world, was ignored. Second – in part, as a consequence of the first – the Middle Ages were seen as a mere repository of ideas created in Antiquity, no more than rigidifying the past so that the modern world could rebel against the past’s rigidity. Now try to offer an account of the path leading from point A to B, when you oversimplify the nature of points A and B, and then ignore, or misrepresent, what went in between them! It would be a piece of common sense that, if we want to understand the transformation separating antiquity from modernity, we should be especially interested in what went in between: Late Antiquity and the Middle Ages. To ignore them is simply to accept uncritically the false claim of modernity to have been born directly from the Classics. And it was a mere construct of European linguistic capacities, and prejudices, that had made enlightened scholars such as Klein ignore, effectively, Arabic civilization. In reality, no balanced picture of Mediterranean history can be offered, as it were, purely on the Indo-European.
The thesis of this book is that Classical Greek mathematics went through a trajectory of transformation through Late Antiquity and the Middle Ages, so that, in certain works produced in the Arab speaking world, one can already find the algebra whose origins Klein sought in modern Europe. The changes are not abrupt, but continuous. They are driven not by abstruse ontological considerations, but by changes in mathematical practice. To anticipate, my claim, in a nutshell, is that Late Antiquity and the Middle Ages were characterized by a culture of books-referring-to-other-books (what I call a deuteronomic culture). This emphasized ordering and arranging previously given science: that is, it emphasized the systematic features of science. Early Greek mathematics, on the other hand, was more interested in the unique properties of isolated problems. The emphasis on the systematic led to an emphasis on the relations between concepts, giving rise to the features we associate with “algebra.” So that, finally, I do not move all that far from Klein’s original suggestion: it was by virtue of becoming second order (though in a way very different from that suggested by Klein!) that Classical mathematics came to be transformed.
As mentioned at the outset, the following is a study of a single case of development, illustrating the transformation of early Mediterranean mathematics. Since I believe the process was driven not by conceptual issues, but by mathematical practice, I concentrate not, so to speak, on mathematics in the laboratory – definitions and philosophical discussions – but on mathematics in the field – that is, actual mathematical propositions. The best way to do this, I believe, is by following the historical development of a single mathematical proposition.2 I take in this book a single ancient mathematical problem and study its transformation from the third century BC to the eleventh century AD – from geometrical problem to algebraic equation.
The book is informed by two concerns. First, I argue for the “geometrical” or “algebraic” nature of the problem at its various stages, refining, in the process, the sense of the terms. Second, I offer a historical account: why did the problem possess, at its different stages, the nature it possessed? The first concern makes a contribution to the debate on the historicity of mathematics, following Unguru, and my main aim there is to support and refine Unguru’s position. The second concern aims to go beyond the historiographical debate and to give an explanation for the transformation of early Mediterranean mathematics.
In Chapter 1, I describe the nature of the problem within Classical Greek mathematics. Chapter 2 discusses the degree to which the problem was transformed in Late Antiquity, while Chapter 3 discusses its transformation in Arab science. The hero of Chapter 1 is Archimedes himself. In Chapter 2 the hero is Archimedes’ commentator, Eutocius (though much mention is still made of Archimedes himself, so that the contrast between Archimedes and Eutocius can be understood). The hero of Chapter 3, finally, is Omar Khayyam, whose algebra is seen as the culmination of the trajectory followed here. Originally a problem, it now became an equation, and from geometry, algebra was created – leading, ultimately, to such authors as Zeuthen who would understand, retrospectively, Greek mathematics itself as characterized by a “geometrical algebra.”
There are advantages and drawbacks to taking a single example. Most obviously, I open myself to the charge that my case study is not typical. My main thesis, that Late Antiquity and the Middle Ages were characterized by deuteronomic culture, with definite consequences for the practice of mathematics, was argued, in general terms, in an article of mine (“Deuteronomic Texts: Late An- tiquity and the History of Mathematics,” 1998). That article went through many examples showing the role of systematic arrangement in late Ancient and medieval mathematics. In this book I attempt a study in depth of a single case, and I shall not repeat here the examples mentioned in that article. But I should say something on this issue, even if somewhat dogmatically – if only so as to prevent the reader from making hasty judgments. For the reader might be surprised now: was not early Greek mathematics itself characterized by an interest in systematic arrangements? Two examples come to mind: that Ancient Greek mathematicians had produced many solutions to the same problem, leading to catalogues of such solutions; and that Ancient Greek mathematics had produced Euclid’s Elements. As a comment to this I shall mention the following. First, the catalogues of ancient solutions are in fact the work of Late Ancient authors bringing together many early, isolated solutions.3 Second, one may easily exaggerate the systematic nature of Euclid’s Elements (I believe it is typical of early Greek mathematics that each of the books of the Elements has a character very distinct to itself: more on such deliberate distinctness in Chapter 1 below); but even so, I believe the work as we know it today may be more systematic than it originally was, due to a Late Ancient and Medieval transformation including, e.g., the addition of proposition numbering, titles such as “definitions” etc. Third, and most important, the centrality of Euclid’s Elements in Greek mathematics is certainly a product of Late Antiquity and the Middle Ages – that had fastened on the Elements just because it was the most systematic of ancient Greek mathematical works. In early Greek mathematics itself, Euclid had a minor role, while center stage was held by the authors of striking, isolated solutions to striking, isolated problems – the greatest of them being, of course, Archimedes.4
This book is dedicated to what may be the most striking problem studied by Archimedes – so striking, difficult, and rich in possibilities, that it could serve, on its own, as an engine for historical change. Time and again, it had attracted mathematicians; time and again, it had challenged the established forms of mathematics. Quite simply, this is a very beautiful problem. Let us then move to observe its original formulation in the works of Archimedes.
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