Revolutions and Continuity in Greek Mathematics
We are pleased to announce that an international conference on Ancient Greek Mathematics will be hosted by Birkbeck College, University of London.
‘Revolutions and Continuity in Greek Mathematics’
In 1962, T. Kuhn’s influential book The Structure of Scientific Revolutions challenged the dominant view of the time that scientific progress is ‘continuous’ and introduced the (rather revolutionary) term ‘revolution’ in the vocabulary of the historians of science. Considering that the scholars of ancient Greek mathematics do not (usually) work in literary isolation, it was only a matter of time before this terminology was introduced into the field; thus, when S. Unguru’s 1975 paper ‘On the Need to Rewrite the History of Greek Mathematics’ caused heated debates on the nature of Greek mathematics, some scholars rushed to support the idea that a revolution took place. An agreement, however, could not be reached, not only in regard to the current state of affairs in the discipline, but, perhaps more importantly, in regard to the usefulness of employing terms like ‘revolutions’ and ‘continuity’ in order to describe the progress of the field.
While these debates were taking place in the field of the historiography of Greek mathematics, time did not stand still in the field of its history either; in fact, the impressive number of recent publications reveals growing interest for the subject. Historians of Greek mathematics today apply methodologies, which appear as diverse as the authors themselves; i.e., in terms of language, culture, educational background and selection of topics. The aim of this two-day international conference is to bring together a number of leading scholars of Ancient Greek mathematics in order to explore the ideas of ‘revolutions’ and ‘continuity’ as they appear in/disappear from the Greek mathematics. Within this framework, we shall endeavour, through examining various case-studies, to identify and evaluate some general characteristics of the methodologies and approaches of the discipline as practiced today and, additionally, to suggest directions for future research.
Counter-Revolutions in Mathematics - Sabetai Unguru, University of Tel-Aviv
- The paper will discuss, by means of an analysis of selected historical examples, the typical attitudes of mathematicians to the historical development of their discipline. Specifically, we shall ask: Are there revolutions in mathematics? If there are, in what sense are these designated events revolutionary? An argument will be put forward to the effect that upheavals in the history and philosophy of mathematics have counter-revolutionary counterparts in mathematics.
Diagrammatizing Mathematics: Some Remarks on a Revolutionary Aspect of Ancient Greek Mathematics - Claas Lattmann, University of Kiel
One of the most remarkable features of ancient Greek mathematics is the use of diagrams: there is not one single proof in Euclid’s Elements that is not accompanied by a diagram. However, the ontological and epistemologial status of these diagrams is not yet adequately understood and is still being discussed controversially. In this paper, the riddle of the diagram is reassessed from a model theoretic perspective. It is argued that diagrams are icons as defined within the framework of Charles S. Peirce’s semiotic theory and, thus, models. Specifically, diagrams are ‘diagrammatical models’, i.e. models that lay bare specific structural relations supposed to exist in their originals. The proposed model theoretic approach allows 1) to see that the printed diagram is nothing but a model of an abstract diagram which 2) stands for a general mathematical situation which 3) consists of select general entities of mathematical theory; and, at the same time, 4) to recognize that this abstract model is being constructed and subsequently analyzed in the textual part of the proof – whose 5) purpose is accordingly to ‘show’ that the model in question has the specific sought-for mathematical properties, that 6) reveal general mathematical truth. Hence, Greek mathematical proof turns out to be an analysis of static general models rather than (as is commonly held) an inductive argumentation concerned with particular cases. In spite of the superficial similarity, the ancient Greek mathematical approach is incommensurably opposed both to the practical particularity of all the other known ancient mathematical traditions and, likewise, to the inductive generality of modern mathematics for which the concept of variable as a representation of a particular element of a set is central. In consequence, Greek mathematics is a paradigm of mathematics set apart from ancient as well as modern paradigms by fundamental scientific revolutions
Reciprocal subtraction in Classical Greek mathematics: “Revolutionary” practice or instance of continuity with Mesopotamian Mathematics? -Michael Weinman, Bard College Berlin
The “rediscovery” of the putatively unique and transformative nature of the progress in mathematical knowledge in and around Plato’s Academy during his lifetime and the century after, was integral to the self-understanding of mathematicians, historians of mathematics, classicists, and philosophers in the generations surrounding the beginning of the 20th century. Integral to much of this work—and to those who worked in its wake through the middle decades of the last century—was the conviction that while practical mathematical understanding was developed much earlier and to a much greater extent in other places (especially Mesopotamia and Egypt) than in classical Greece, it was the Greeks alone who sought to develop a proper “theoretical” understanding of these mathematical objects. Recent work in the history of mathematics and in the sociology of knowledge has made great strides in showing how much these claims owe to the prejudices of the “historical horizon” within which those researchers were working. My presentation attempts to look at one very small instance of this larger debate, and to ask: what can we learn concerning the novelty or “revolutionary character” of classical Greek mathematics from the practice of “anthyphairesis” as it developed from the earliest (scantily) recorded sources of the sixth century through to its presentation in Euclid’s Elements? Revisiting the (now no longer recent) debate between Szabo (1969) and Knorr (1975) concerning this technique and its role in the development of the theory of incommensurability—and the recent attempt to resolve (or dissolve) this debate by Fowler (1999)—I shall try to make a case for the importance of practice to mathematical understanding that helps cast a certain light on the impossible question of “exactly when did the Greeks know what about incommensurability?” Along the way, and perhaps more importantly, a new formulation of what is both new and ancient in this particular manifestation of the Greek mathematical knowledge of these centuries will be presented as a case study for these broader historiographical debates.
The anthyphairetic revolutions of the Platonic Ideas - Stelios Negrepontis, Athens University
In my lecture I will outline my interpretation of Plato’s philosophy in terms of periodic anthyphairesis. A key to this interpretation was my initial discovery, back in 1996, that the two principles ‘peras’ and ‘apeiron’ in the Philebus 23b-25e correspond closely to the concepts of finite and infinite anthyphairesis (commensurability and incommensurability, accordingly), described in Propositions 1-8 of Book X in Euclid’s Elements. The importance of the Philebean principles is that their mixture, according to the Philebus 16c, produces true Beings, namely Platonic Ideas. A Platonic Idea is the mixture of the infinite and the finite in the same way that a pair a, b of line segments commensurable in power only, is a mixture of incommensurability (a, b) and commensurability (a2, b2). It is a central theorem, presently attributed to the successive efforts of great mathematicians (including Fermat, Brouncker, Euler, Lagrange, Legendre, Galois), that such a pair possesses (palindromically) periodic anthyphairesis. In the Theaetetus 147d-148d Plato gives an account of the incommensurabity proofs, initially by Theodorus and then by Theaetetus, of a, b in case a2=Nb2 for non-square N. Some delicate arguments suggest that the method employed is anthyphairetic, and therefore that Theaetetus must somehow have proved the periodicity theorem. This is confirmed by the following discovery: Socrates, at the end of the Theaetetus passage, asks that Theaetetus imitate (‘peiro mimoumenos’, 148d4) his method in order to tackle the philosophic problem of knowledge of the Ideas (‘episteme’). The imitation, the method of Division and Collection by which ‘episteme’ is achieved, is presented in the sequel, Sophistes and Politicus, of the Platonic trilogy. The knowledge of the Angler and of the Sophist in the Sophistes are close philosophic analogues of proving periodicity of anthyphairesis by the Logos criterion (for the second one employing the fundamental analogy of the divided line of the Politeia 509d-510b). But what is nothing short of astounding is that in the Politicus the knowledge of the Statesman is achieved by a further refinement of the method, imitating not just periodicity but in fact palindromic periodicity. We are then driven to conclude that Theaetetus was in possession of the palindromic periodicity theorem. The introduction of the ‘apotome’ and ‘binomial’ lines and their conjugacy (Propositions 112-114) in Book X. provide the necessary tools for a natural reconstruction. Several Platonists, in perplexed puzzlement, rejected the notion that a Platonic Idea could be a mixture of the infinite and the finite and divisible, in fact ad infinitum, since such properties appear to preclude the sine qua non of a Platonic Idea, its Oneness. What they have failed to understand is that an entity divisible, but in endless revolution and periodicity while being divided, is an entity possessing the most satisfying and internally determined concept of Oneness, self-similarity.
From Greek numerals to number theory: a suggestion - Henry Mendel, California State University
The mark of Greek number theory, as developed in the fifth and fourth century BCE is its omission of fractions. It used to be common for historians of Greek number theory to look to a crisis that resulted from the discovery of incommensurables. Adherents of this view now number but a few; yet it is difficult to find an alternative. One might be to seek a philosophical motivation that comes out of conceiving numbers as whole numbers and an intellectual adversion to fractions. Yet, philosophical qualms usually turn current practice into ideology and not the other way around. Of course, in a deep way, alternatives are difficult because the evidence is so paltry. I would like to explore a more ordinary motivation. The dominant numeral system of the fifth and fourth century was the acrophonic system. Fractions were handled as subunits, pre-established for particular measures. As a result, the manipulation of arbitrary fractions was awkward, to say the least. The Egyptians and the Babylonians had excellent ways of treating fractions and devoted much of their pedagogy to their general manipulation. Although it is likely that the Greeks were well aware of sexigesimal numbers, they did something else. They developed ratios. I will, however, claim no more than plausibility for my suggestion.
Early conics: where did the focus come from? - A E L Davis, University College London
- The history of conic sections contains some gaps. I shall consider a particular problem concerning central conics (ellipse and hyperbola), where the sudden appearance of the parameter (orthia), as part of their definition in Apollonius: Conics Book I, has something of the effect of a conjuring trick. The placing of the parameter presupposed knowledge of the position of the focus, although that concept was not even mentioned until Book III. I shall investigate whether there is a geometrical connection that will create a linkage back to Archimedes, and to earlier work now lost. The problem over the specification of the focus may have been the source of a mathematical dichotomy between two distinct approaches to a general conic – the Archimedean, and the Apollonian. The Archimedean approach depended on orthogonal axes intersecting at the centre, each conic being defined by a pair of orthogonal length measurements; while the Apollonian definition of every conic implicitly involved the focus (a term coined in 1604) -- and the possibility of oblique axes. The persistence of that dichotomy may have been partly due to the different ways in which the sections can be cut from a scalene cone, which have not previously been distinguished. It is hoped to shed more light on the problem by looking at later developments in Europe during the Middle Ages. There is strong evidence that the two approaches were not reconciled until 1621 (but quite soon afterwards, they became so firmly melded together that more recent commentators have never analyzed them separately).
Geometer, in a landscape: Hero’s embodied mathematics - Courtney Roby, Cornell University
Hero of Alexandria never imposes a decisive boundary between “pure” and “applied” mathematics, and the remarkable work he manages to do in this liminal space has received particularly insightful attention in Tybjerg 2004. The crucial corollary to his innovative mathematization of material constructions is the way Hero focalizes these interventions in the physical world through the actors involved, whose perspective orients the acts of construction, and who may even be invoked as engaged actively in construction. In this paper, I will use recent work on embodied cognition to analyze the techniques Hero uses to embody the imposition of mathematical constructions on the physical world, with a central focus on his Dioptra. The subtlety with which Hero marshals his “personnel” can make them easy to miss: he rarely makes explicit mention of anyone like the assistants and workmen Philo refers to in his Belopoeica, and Apollodorus in his Poliorcetica. The figure of the surveyor often rather serves as a focalizer: orienting the measurement of the distance from his own side of a river to the opposite bank; precisely sighting the top of a rod to gauge the pitch of a planned tunnel; and so on. He may be equipped with hands as well as eyes, to discover a screw’s pitc h haptically, adjust a water-level, or delicately trace the path of a tunnel without disturbing the weighted cords hung along it. In all their varied activities, these embodied actors represent a striking alternative to the disembodied “ Helping Hand” described by Fowler and Taisbak as characteristic of mathematics in the Euclidean tradition. At the same time, Hero maintains a continuous connection with the mathematical tradition in which he inscribes himself: his embodied actors carry out sophisticated geometrical tasks using a panoply of mathematical techniques. Nor did embodied geometry end with Hero; comparable approaches are applied in the Roman agrimensorial texts, where the geometrical aspects of the surveyor’s work are quite often focalized through similarly embodied perspectives. I will close by discussing this legacy, which highlights the lasting impact of Hero’s deft translation of Euclidean geometry into a material world populated by embodied practitioners.
Dyadic arithmetic and digital transformation in ancient weaving - Ellen Harlizius-Klück, University of Copenhagen
- The aim of this paper is to give an outline of the culture of computation within ancient weaving and to discuss its possible impact on the logical and philosophical bias mathematics took in Greek antiquity. There is no historiography of calculation in weaving technology of the past. Moreover this calculation seems to be simple and less demanding. However, weaving needs consideration of order through numbers and measures. Everything that is to be depicted on a weave, be it a simple pattern, a geometrical pattern, or a natural motive, needs to be translated into ratios of integers handled by the dyadic principle of either lifting or not lifting warp threads. Though the issue of weaving itself is less challenging for mathematical operations, the specific character of weaving patterns (always consisting of number arrangements and relations even if the pattern will give a geometric shape) is able to raise the question of incommensurability, to give a perfect possibility of using (and maybe even inventing) the Euclidean algorithm and to show the idea of the exhaustion-method in weaving perfect circles within the grid of the weave. Only recently the topic of mathematics and weaving entered the historical discussion introduced by investigations of Ancient Andean khipus and their use in recording and administrating the Inka state (Urton, Brezine). Brezine suggested that textiles were the main medium in which Andean cultures (having no writing system) met and grappled with theoretical problems. In my paper I show that this might also be true for the situation in ancient Greece. I discuss the question of revolution versus continuity by investigating the structural concepts of ancient weaving as a cognitive culture that might have contributed to the unfolding of proof strategies in Greek mathematics.
Kuhn and Ancient Mathematics - Andrew Gregory, University College London
- Thomas Kuhn, in his Structure of Scientific Revolutions, claims that a firm paradigm for mathematics has been in place since pre-history. He never expands on what this paradigm might be though, or whether it undergoes any changes which might constitute a revolution in mathematics in the ancient world. This paper question whether there was any on such single, firm paradigm and how appropriate it is to employ Kuhnian terminology and the ‘Kuhn cycle’ of normal science, crisis, revolution, incommensurability and new normal science to ancient mathematics.
Ancient Numeracy - Serafina Cuomo, Birkbeck College
- In this paper I will discuss the questions and problems involved in the study of numeracy in ancient Greece and Rome. I will analyze the issue of what evidence is available, and highlight the significance of the topic for modern concerns. In the last part of the paper, I will look specifically at counting boards, and discuss various reconstructions of how calculations were carried out.
The axiomatization of mathematics and Plato’s conception of Knowledge in the Meno and the Republic - Vassilis Karasmanis, National Technical University of Athens
- Euclidean geometry is a system of hierarchically ordered propositions. Each proposition of this system is inferred deductively by others 'prior' or more elementary ones. But, not all-geometrical knowledge is demonstrative. In Euclidean geometry everything starts from some basic principles which are considered "plain to all" (Plato, Rep. 510d1). Therefore, there is an asymmetry in our knowledge of geometrical propositions between a) principles, which are unprovable and 'self-evident' or "plain to all", and b) derivative theorems which are justified by demonstration from the principles or other 'more elementary' theorems. We can schematically represent the Euclidean model (hereafter model-e) as a pyramid having the principles at its top. It is almost common place among platonic scholars that Plato adopts model-e in his theory of knowledge. The main evidence is found in Books VI and VII of the Republic where we find a hierarchy among the objects of knowledge (Forms) and their relations. The metaphors of 'up' and 'down' suggest the hierarchical character of his model of knowledge which may be represented as a pyramid with the non-hypothetical (Good) first principle at its apex. In the Republic we have also the first evidence that Greek mathematics were organized axiomatically. In this paper I am going to question the belief that Plato adopts model-e in the Meno. I shall argue that it is legitimate to read the Meno as presenting a different conception of knowledge that might remind us of modern coherentist theories. More specifically, I claim that there is no evidence in the Meno of an asymmetry in knowability between first principles and derivative propositions. In the Meno all knowledge seems to be inferential, requiring justification or reasoning into the causes (aitias logismon – 98a), but there is no need of fixed and unprovable first principles from which such reasoning proceeds. His system of knowledge consists of interrelated elements and several accounts can explain the interrelations among the elements of the system. Such a model can be represented as a network rather than as a pyramid (hereafter I shall call it model-n). In such a model, circular regress may not always be vicious. I am not going to maintain that Plato consciously proposes such a model of knowledge, but rather that he himself is exploring such directions. We see that Plato changes dramatically his model of knowledge in the period between the Meno and the Republic. I claim that this happens because in this period mathematics is organized in an axiomatic form for the first time. After the axiomatization of mathematics (that takes place within the Academy), Plato adopts the model-e of the mathematicians.
Plato on the Geometrical Hypothesis in the Meno - Naoya Iwata, University of Cambridge
- My paper examines the geometrical problem in Meno 86e4-87b2. This problem, despite being one of the few valuable sources on the early stage of Greek mathematics, remained an unravelled riddle fora long time due to Plato’s obscure language. At the beginning of 20th century, however, Cook Wilson presented the first and most promising interpretation, which has been followed by many scholars down to the present day. The gist of his interpretation is that Plato reduces the problem of whether a rectilinear figure can be inscribed in a given circle as an isosceles triangle to that of whether that figure can be applied to the diameter of the circle as a rectangle falling short by a rectangle similar to the applied one. The purpose of my paper is not to offer an alternative interpretation on the problem in question but to explore the implication of Cook Wilson’s interpretation from a different perspective in relation to the nature of Plato’s method of hypothesis. I argue that Plato is suggesting there that (a) a mathematician at that time used a hypothetical method for revealing a more general and essential problem which is implied in a particular problem and whose solution had not been found until then, and for putting a tentative answer to such a reduced problem to analyse the original one; and that (b) there was a rational process for reducing a particular problem to a general one and then narrowing down the plausible candidates of a hypothesis, based on the method of analysis, although the final choice of a hypothesis must have been guided by intuition. Time permitting, I hope to contain some discussion about other relevant techniques in Greek mathematics: problem reduction and a diorism, which have often been ambiguously explained in relation to the problem in the Meno.
Aristotle on the Structure of Numbers: A Metaphysical Account -Gabriele Galluzzo, University of Exeter
- Aristotle is traditionally associated with the empiricist view that numbers are just pluralities of concrete objects. More specifically, numbers are pluralities of concrete objects considered with reference to a particular mathematical property, countability. Aristotle also believes this understanding of numbers to be fundamentally in agreement with the standard Greek definition of numbers as “plurality of units”. For when they are considered as countable entities, concrete objects are just units, i.e. units for counting. My aim in this paper is to argue that the traditional picture of Aristotle’s philosophy of arithmetic is not the end of the story. In a certain number of texts from his Metaphysics Aristotle comes up with the surprising claim that numbers cannot be just sums or pluralities of units. But what could numbers be in addition to being pluralities of units? What the texts from the Metaphysics suggest is that, in order for a certain plurality of units to be a number, the units in the number must be kept together by some sort of structure, distinct in nature from the units in the number. In this paper, I wish to give an account of Aristotle’s view that numbers are not just sums or pluralities of units. In particular, I shall argue for two related claims: (i) that in the Metaphysics Aristotle wishes to provide a metaphysical account of numbers, which is mainly based on an analogy between numbers and organic wholes (substances); (ii) that this intuition leads him to apply to numbers the matter-form machinery that turns out to be so useful in the explanation of organic wholes. When applied to numbers, the matter-form machinery yields the view that the units in a certain number are the matter of the number, while the structure that keeps the units together is their form.
Et in Arcadia ego: Rethinking Hypatia and Her Mathematics - Elizabeth Kosmetatou, University of Illinois – Springfield
- Whether a pioneer mathematician or just a charismatic, revered teacher and editor, Hypatia has arguably become as much of a legend as Thales and Pythagoras, albeit for different reasons. In recent years two biographies by Maria Dzielska and Michael Deakin attempted a partial reconstruction of her life and her work. Deakin’s verdict in particular pronounced her a teacher and editor of textbook-¬‐style works, rather than an original thinker. This paper will take a fresh look at the meagre surviving evidence to address the question of how revolutionary Hypatia may have been and to offer a partial view of the state of Mathematics in the Library of Alexandria in the first four centuries of our Common Era. Recent work on perils of the Alexandrian Library by Roger Bagnall and Myrto Hatzimichali, has suggested that following its accidental destruction by fire during Julius Caesar’s Alexandrian War of 48 BCE, access to many earlier works became problematic to impossible, judging from the resources Alexandrian editors of the first century CE used in their work. While the library’s holdings may have suffered, and scholars may have concentrated on editing many texts, one could argue that knowledge in Mathematics, sciences, and technology was not entirely lost, given the fact that several centers outside Egypt had access to it. Finally, in considering Hypatia’s contribution to Mathematics and science, it is important to look at the primary sources in order to separate fact from construct. Her known work and its originality, or lack thereof, will be juxtaposed to that of other famous Greek mathematicians, while questions on priorities in scholarship, what moves science forward, what constitutes revolution versus continuity, and of knowledge regained will be addressed. Although some conclusions will be negative, due to the state of our surviving evidence, it is important for modern scholars to accept the fact that our picture will always be partial, and the puzzle pieces too random to allow any definitive interpretations.
Diophantus and pre-modern algebra: New light on an old image - Jean Christianidis, University of Athens
- As a theme of historical research, Diophantus’ work raises two main issues that have been intensely debated among researchers of the period: i) The proper understanding of his practice; ii) the recognition of the mathematical tradition to which this practice belongs. The former issue was much focused, somehow misleadingly, on the rather minor question of whether the ‘substitutions’ that are introduced in the course of the solutions have been devised as a result of a systematic methodology. However, the significant issue is not the ‘substitutions’ per se but the overall structure of a Diophantine solution to a problem, and, on account of this, the characterization of the strategy of problem-solving that Diophantus was practicing. The traditional answer to this range of questions –from Medieval Islam through the Renaissance and the Early Modern period up to the 20th century– was that Diophantus’ book is a book on algebra. This traditional approach has been recently criticized by some historians of mathematics who point out the anachronistic methodology that historians in the past were using in analyzing ancient texts. But, criticizing the methodology by which one defends a historical claim does not necessarily mean that the claim itself is wrong. This paper discusses some crucial issues involved in Diophantus’ problem-solving, thus, giving support to the traditional image about the character of Diophantus’ work, but put in a totally new framework of ideas.