Dr Tazuko van Berkel
Reading Numbers, Reading Minds: Numbers as Objects of Interpretation in Historiography
Attic oratory abounds in numbers. In oratorical contexts, numbers, quantities and calculations provided public speakers with ample opportunities to display expertise, construct a symbouleutic persona and to perform public virtues such as transparency and accountability. Yet ancient rhetorical theory is largely silent on the topic of numbers. Although the relevance and use of numbers may be implied in the questions of policy that symbouloi are expected to master (ways and means, war and peace, defense of the country, import and exports, legislation: Ar. Rhet. I.4.7-13 (1359b-60a)), the fact that rhetorical theory does not identify arithmetic or quantitative reasoning as rhetorical tools in their own right suggests an instrumentalist view on numbers according to which numbers are mainly informative, neutral and rhetorically "inert".
Literary representations of numerical rhetoric, on the other hand, reveal a more sophisticated view on the communicative effects of numbers and quantitative reasoning. This paper will explore historiographical representations of numbers as objects of interpretation. Analysis of the ways in which characters (i) understand and misunderstand numbers, (ii) dispute meanings attached to numbers and (iii) anticipate the ways in which numbers will be interpreted by others ("understanding understanding of numbers") and (iv) control the interpretation of numbers will shed light on an implicit rhetoric that recognizes numerical reasoning as a phenomenon in its own right.
Sat 3rd, 11:45-12:30
Professor Josine Blok
Ten Thousand: Numbers and Institutional Change in Fifth-Century Athens
The 'ten thousand' in my title refer to the amount in drachmae set as a fine in a group of Athenian decrees of the classical period, which marks a much-discussed shift in the institutions of Athens of the later fifth century. Most Athenians and others targeted with this fine were just ordinary citizens, to whom a fine of 10,000 drachmae must have seemed a zillion. Did the Athenians have numbers, but no math? In other words, is this amount real?
After a brief review of the main features of the decrees, I will compare them to other fines and finally sketch the political context. The 10,000 drachmae fine appears in a group of seven inscribed decrees, not including the restored or uncertain items (IG I3 34, Kleinias decree, where the amount of the 10,000 drachmae fine is restored in IG; and IG I3 117, l. 20-22 (407/6) idem). Three have certain dates, of 426/5, 425/4 and 418/7 respectively; taken with the other, now redated decrees, this groups spans ca. 429- ca. 415. This collection may be considered roughly representative of the period they were issued, which seems to be ca. 429-405. There is one outlier: IG II3 370 on the founding of a colony in the Adriatic, of ca. 325/4, so roughly a century later, to be discussed separately. I will argue that in the later decades of the fifth century the fine was real enough, but also that the Athenians revised their fining policy in the fourth century.
Fri 2nd, 14:30-15:15
Dr Kai Brodersen
Jokes on Numbers and Numeracy in the Philogelos
A graduate used to value the clothes of the people he met. When his father heard about this from others and criticized him, he said: "Father, you have been misinformed, possibly by someone who is not even human." The father replied: "So-and-so told me this." "And you listen to him whose himation is not even worth 50 drachmai?" (Philogelos, joke 36).
Jokes on numbers and numeracy abound in the only ancient collection of jokes, the Philogelos (for which, incidentally, two Leiden Codices are major textual witnesses). While the book itself dates to late antiquity, it is (pace M. Beard) not just evidence for "Roman Laughter", but collects earlier material, including some from Classical Greece. The oral tradition of jokes can cover large spans of time and place as well as many social strata: As the paper will argue, this kind of "communication of the masses" has perhaps more claim to be "mass communication" than inscriptions, oratory, drama, or speeches in historiography, and when analyzed using the questions guiding this conference, the jokes may well contribute to the topic of the conference: What communicative situation and relationship does the use of numbers presuppose in these jokes? What is the relationship between the presentation of numbers and their communicative function in them? Who is credited with a knowledge of numbers and with numeracy? Which values are being communicated, especially in the realm of numbers and numeracy in education?
After all, as Philogelos, joke 265, tells us: When a graduate was asked, how many xestai make an amphoreus, he answered: "Do you mean of wine or of water?"
Sat 3rd, 09:00-09:45
University of Coimbra
Demosthenes' Proposals in the First Philippic: Political Numeracy
This paper aims to discuss the numbers of Demosthenes' spending proposal for military aid to the Olynthians, in around 351 BC, in response to their appeals for Athenian help. In the First Philippic (IV), Demosthenes enumerates the size of the necessary force and its provisions. He also explains how much will it cost the Athenians, since the Athenian public treasury was low on funds due mostly to their participation in the Sacred and Social Wars. These conflicts, moreover, led to the loss of revenues from breakaway members of the Athenian confederacy, most importantly Chios, Rhodes and Byzantion. In this way, Demosthenes gives the assembly what looks like a detailed and realistic proposal. Scholars tend to deny the validity of individual proposals because of the overall tendentiousness of the argument (Sealey 1993, Wooten 2008), but these proposals must be put into perspective. I will therefore analyse the domestic political situation in Athens and the relations between Athens and other poleis (Sparta, Thebes, Euboea) to understand the consequences of this proposal and why it ultimately failed. I will briefly relate the proposal of the First Philippic to similar content in other Demosthenic speeches speeches, such as On Organisation(XIII), the three Olynthiacs (I, II, III), and On the False Embassy (XIX), in order to elucidate Demosthenes' overall financial strategy (war taxes versus making theoric funds stratiotic) to support not only the Olynthian campaign but an ongoing war against Philip II.
Fri 2nd, 16:30-17:15
New Europe College, Bucharest
From one to One. Plato and the Origin of Numbers
The general understanding of numbers is similar to that of any entity of a platonic kind, i.e. eternal, unchanged, and not subjected to generation. In this paper I challenge the view that Plato understood numbers fully as a Platonist, questioning possible historical reasons that may be the backdrop for the controversial hypotheses in the Parmenides that advance the idea that numbers are generated. This paper discusses Plato's argument for the generation of numbers (the Parmenides 142b-144b) in the frame of ancient Greek numerical systems. It is Plato's Greece that experiences the transition from the Acrophonic numerical system to the alphabetical numerical system, a transition, I argue, whose distinctive feature is a focus shift from a prevailing cardinal numerosity (rather of a Homeric type) to a numerosity which points more forwardly to its ordinal quality (closer to a Platonic type). An analysis of the Parmenides 142b-144b follows the argument for the generation of numbers in the light of both cardinal and ordinal features of numerosity, features instantiated, first of all, not by philosophical arguments, but by historical ways of representing numerals. I underline that the constant negotiation between cardinality and ordinality as prevailing traits of number proves Plato's various attempts to understand how multiplicity is and comes to be. The transition between the two numerical systems is mirrored in Plato's discussion of numbers not only as series in a constantly multiplying line, but rather as elements in a system whose structure is not linear, but hierarchical, heralding thus the complex ontological developments of the dichotomy One-Multiple.
Sat 3rd, 16:15-17:00
Dr Serafina Cuomo
Birkbeck, University of London
Numbers, Numeracy and Democracy (Conference Keynote)
What do numeracy and democracy in classical Greece have to do with one another? I will explore several of their possible intersections: account inscriptions, their formatting and function, and their link with accountability; the public character of calculations on the 'Greek-style' counting board; the existence and use of measuring standards. With reference to work by, among others, Osborne and Johnstone, I will discuss the political significance of numeracy in classical Greece: both the possibility that numbers and numeracy might be an exquisitely democratic form of knowledge practice, and their potential for a rhetoric of empire.
Fri 2nd, 17:45-18:45
Dr Steven Johnstone
University of Arizona
Punishing and Valuing in Ancient Greek Laws
The history of numbers and the histories of the polis and of democracy are not the same, though they articulate in various instances and configurations. Some of these articulations may surprise our expectations. This paper examines the role of numbers in the laws of various ancient Greek cities (but especially Athens) during the Classical and Hellenistic periods, particularly in regard to penalties. As ancient Greek laws used numbers, they do not seem to have promoted some of the practices usually associated with numeracy: mathematical calculation, record-keeping, etc. But they did promote numerical valuation - a practice common in other contexts, not just markets but tax assessments, marriages, and religious offerings.
I begin with a straightforward question: With regard to penalties, when did laws use numbers? A full accounting of the question, however, requires addressing a related question: When did laws not use numbers in penalties? The absence of numbers may be an important facet in the patterns of their use.
I will examine legal penalties in two areas: the penalties imposed by courts as the result of conviction and the penalties that officials could impose on their own prerogative - and the penalties that could be imposed on them. In both cases, citizens had to perform or attend to money valuations.
Athenian laws frequently avoided quantifying penalties, especially for private wrongs. Like ancient Near Eastern law codes, the laws of some other Greek cities prescribed schedules of payments for particular infractions. But Athens' laws tended not to quantify penalties that courts imposed; rather to the degree that Athenian courts imposed money fines, laws often deferred this calculation to the litigants and the jurors. In some cases, the penalty was determined by the amount of damage a man claimed to have suffered. In others, in the procedure of timesis, after conviction each of the two litigants proposed his own penalty and the jurors voted which to apply. Although these proposed penalties could take any form (from death to lifetime meals in the prytaneion), litigants often proposed competing money fines which compelled jurors to become commensurating, valuing agents.
However, at Athens and elsewhere, laws defined and limited most officials' coercive power through numbers, usually values in money: thresholds of value beyond which they could not act, precise or maximum fines they should impose for particular infractions, and enumerated fines to which they themselves were subject if they failed to act properly. At Athens, with the notable exception of the Eleven, who could arrest, imprison, and execute people, officials were constantly accountable through quantification that marked a public, intelligible limit to their power. To the degree, then, that all citizens might come under the purview of these authorities, or, in the deeply participatory system at Athens, serve as one of the hundreds of officials appointed each year, citizens repeatedly calibrated and exercised legitimate power by counting and valuing.
Fri 2nd, 11:00-11:45
Dr Lisa Kallet
University of Oxford
A Counting People: Valuing Numeracy in Democratic Athens
When Bdelykleon tells his father Philokleon, in Aristophanes' Wasps of 422 BC, 'to calculate roughly with your fingers,' all the tribute and taxes, and other revenue, we can imagine the audience wildly counting up by tens, to get to the exaggerated total of 2000 talents! Likewise, when Dikaiopolis in the same poet's earlier comedy, Acharnians, amidst his yawning, farting, and other activities while he waits for the assembly to fill up, counts, we know we are on to something, specifically, a democratic, counting people. This paper will consider a variety of evidence to suggest that the presence of counting, numbers, and the need to deal with money and finance on a regular and systematic level (e.g. in the Boule), as well as the ubiquity of inscriptions with numbers, suggest a high degree of numeracy in Athenian democracy, partly driven by the empire, and that it was socially and politically, as well as economically, valued.
Fri 2nd, 10:15-11:00
Dr Athena Kirk
Counting the Infinite in Homer and Attic Inventories
This paper explores the visual rhetoric of infinity in Greek temple inventory inscriptions in light of literary conceptions of "boundlessness". While several studies of temple inventories have them as administrative texts with a singular audience and context, this paper situates these texts within the greater cultural world and specifically within the genre of literary catalogue. While historians may characterize inventories as an ad hoc creation of 5th century Athens, this analysis suggests that their roots extend further than the Classical period and indeed perhaps beyond the epigraphic record itself. Epigraphic inventories, I argue, usurp archaic poetics surrounding the infinite to effect an impression of unending riches and ever-extendable civic wealth; in doing so, they also anticipate the Aristotelian concept of "potential infinity". This brand of intertextuality emerges in the formulaics, diction, organization, and even formatting of the inscriptions. I suggest that much of the inventories' impact on the general public emerges from these traditional poetic tactics rather than their textual contents themselves. As documents and monuments, temple inventories thus communicate notions such as 'accountability' and 'transparency' in paradoxical ways: by essentially discouraging, rather than facilitating, precise calculation and quantification on the part of the viewer. In obfuscating the minute and precise in service of the large and overwhelming, however, they ultimately reinforce their own civic impact and utility.
Fri 2nd, 13:45-14:30
Ancient Greek Geometry without Numbers: Ambivalent Value in Numbers in Classical Greece
To modern students, geometry is reduced to calculations using the arithmetic operations of addition, subtraction, multiplication, division, and exponentiation. Given the development of analytical geometry, elements of geometry have been digitized or quantified; points are positioned by coordinates, lines are measured by length, and areas and volumes are represented by numbers found with arithmetic operations. This synchronization of geometric elements with numbers has established an arithmetic framework for geometric inquiries. However, numbers are absent from ancient Greek geometry. Indeed, in the ancient Greek mathematical corpus, diagrams are presented without numbers whatsoever. In that framework, geometric inquiries are resolved only with diagrammatic elements such as lines, angles, and areas, not with numbers conferred to them. In particular, the application of area in Euclid's Elements, in which one figure is transformed into another with the same area, shows how ancient Greek mathematicians sought to compile their mathematical knowledge and shape its deductive structure without depending upon numerical calculation.
The absence of numbers in the ancient Greek geometrical discourse shows a contrast with Greek practical mathematics. By practical mathematics, I mean any mathematical practices that require counting, measuring, and weighing etc. Indeed numbers are ubiquitous in many aspects of daily life, yet strangely silenced in theoretical geometry. To investigate the functions and values that ancient Greeks assigned to their presentation of numbers, it is therefore necessary to consider the contrast between the absence of numbers in theoretical geometry and their pervasiveness in other practical fields.
Three questions should be raised: 1) How the use of numbers enables folk mathematicians to secure an objectivity of their practices? 2) Why were numbers disqualified to prove the core essence of the geometric relationship of figures? 3) How was ambivalent value conferred upon numbers in Classical Greece? My paper starts from comparing diagrams from theoretical fields and those from practical fields. The difference between geometrical diagrams and practical diagrams confirms a distinctive communicative situation in ancient Greek theoretical mathematics that allows only abstract diagrams, as per Aristotle's term aphaeresis ('abstraction'), referring to the isolation of mathematical characteristics as objects of thought distinct from perceptible, thus measurable, objects. Next, the paper investigates how knowledge from practical mathematics could be applied for theoretical mathematics or vice versa. Diagrams in the book II of Euclid's Elements and Archimedes Method are introduced as an example to prove that the passage between practical mathematics and theoretical mathematics was not impassable.
Sat 3rd, 15:30-16:15
Prof. Robin Osborne
University of Cambridge
The Appearance of Numbers
Numbers appear all over Athenian inscriptions, but they are presented in many different ways. This paper explores the different decisions that the Athenians and their stonemasons made about how numbers should appear in the inscribed records and argues that behind the decisions about the layout of inscriptions lies a greater argument about what is important in the public record. Paying proper attention to the politics of numbers at Athens depends upon taking seriously the aesthetic rhetoric of the inscribed stones themselves.
Fri 2nd, 13:00-13:45
Dr Catherine Rubincam
University of Toronto
Numeric Communication in the Greek Historians: Quantification and Qualification
What kinds of numeric communication are found in works of ancient Greek historiography? What kinds of things were regularly quantified in these works, and how was the quantification expressed?
To answer questions of this kind it is necessary to look systematically at all the numbers in the relatively complete parts of the corpus of Greek historiography. Ideally, a means must be found to compile statistics on a standard set of aspects of every number in the texts of the Greek historians so as to create a numeric profile for each author and each work. This makes it possible to quantify the numeric practice of these historians, so as to get a clear picture of what kinds of things each quantifies and how. Measurable factors include: what types of numbers are used (e.g., cardinals, ordinals, or other numeric compounds), to what kind of subject matter numbers are applied (e.g., measurements of time or distance, numbers of people engaged in military or non-military activities), and how often and in what circumstances qualifying expressions (e.g., "about", "more than", "only") are attached to a number. A database of this kind has been under construction for nearly 40 years at the University of Toronto. A substantial part of it is about to be made available for use by other scholars through the opening of public access to the online site where the data is stored.
This paper will highlight some of the major insights that emerge from this work. By using this statistical database one can view any particular number in these texts in the context of the numeric practice of the particular work in which it occurs, or that of a group of works of similar date or purpose. This enables one to simulate the situation of the original author or the original reader of that text. The use of this resource also heightens one's awareness of both the similarities and the differences between the numeric practice of the ancient Greek world and our modern western practice.
Sat 3rd, 13:30-14:15
Emeritus Prof. Richard Seaford
University of Exeter
On the metaphysics of number in the archaic and classical polis
My main focus is on the distinction between (and fusion of) the (A) ordering and (B) commercial functions of numbers in the archaic and classical polis. (A) The ordering function is especially frequent in ritual, from Homer onwards. (B) Subsequently, with the development of commerce and the genesis and spread of coined money, numbers greatly increase their function of expressing exchange value. Behind the Pythagorean idea that things are numbers is the envisaging of objects in terms of their exchange value.
However, even commercial numbers have an ordering function, to create concord between the two parties to the exchange, whose interests are qua exchangers entirely opposed (whereas what ritual number co-ordinates is an existing consensus). This numerical ordering of commerce is expressed in the Herakleitean logos that regulates the unity and transformation of opposites.
And in fact Pythagorean metaphysics also expresses the ordering function of commercial numbers: e.g. the number three not only constitutes the whole (cosmos), it also orders it - by the third term uniting the other two. This combination - in number - of substance with order is unprecedented and revolutionary.
The Pythagoreanism of Aeschylus' Oresteia takes a similar but somewhat different form: the third libation is imaginatively extended over time (the trilogy dramatises violence, counter-violence, and reconciliation) and space (heaven, earth, and underworld), thereby comprehensively absorbing the potential unlimitedness of conflict (and of commercial accumulation) into communal limit of ritual order.
Sat 3rd, 09:45-10:30
Dr. Valeria Sergueenkova
University of Cincinnati
Seeing Numbers in Herodotus' Histories
Why is Herodotus as fond as he is of counts and calculations? He works out the dimensions of the Black Sea, calculates the size of Xerxes' army together with the amount of grain needed to sustain it each day, and converts 341 generations of Egyptian priests into 11,340 years. Examples abound. To explain Herodotus' interest in producing numbers scholars have suggested that claiming to know the precise dimensions of something, let alone to have measured or calculated them oneself, is a rhetorical strategy designed to increase the one's competence and credibility. Alternatively, "Herodotus' number orgies", to borrow Catherine Rubincam's expression, can be understood in the context of what Geoffrey Lloyd called "spurious exactness", the tendency of some branches of Greek science towards gratuitous overmathematization. There is no doubt that Herodotus' arithmetical fireworks are part of his polemic against those he perceives as his intellectual rivals, but are his calculations merely for show? As I have argued elsewhere, the historian's quantification efforts are an essential part of his historical method inasmuch as they allow him to construct arguments about the past. In this paper, I explore Herodotus' interest in methods of counting and measurement as well in material representations of number. For example, a few chapter after informing his audience that Darius' army on his Scythian campaign numbered "700,000 including the horsemen", Herodotus presents us with a second census: passing through Thrace en route to Scythia, each of Darius' soldiers is ordered to place a single stone in a heap, leaving giant mounds in the army's wake. Why does Herodotus invite us to picture this new landscape? Why does he measure Darius' army twice, with a number and through its monumentalization? Or, in the case of his famous calculation of the size of Xerxes' army in 480 BCE, why does the historian record the impressive (if suspiciously exact) tally of 5,283,220 bodies, and then convert this number into its equivalent in grain per diem? Herodotus, I argue, is interested both in the abstract number and its visual measure, in reifying abstractions as well as in abstracting from the concrete.
Sat 3rd, 14:15-15:00
Dr Daniel Sicka
University of Oxford
Creative Accounting? Strategies of Enumeration in Epinician Texts
When the hour of their triumph had passed, victorious individuals turned to epinician odes and epigrams as a means of capturing that moment in time and disseminating its glory across space. A key component of the commemoration offered by those media was the victory-catalogue, which enumerated the agonistic achievements of the victor and his family at the Panhellenic and local level (Kurke (1991) 20n.14). However, in the case of Pindar at least, a degree of linguistic ambiguity is often encoded that stops short of outright falsehood, but still creates the impression of a greater tally of wins than was in fact the case (Cole (1987) 553-68).
This paper expands upon these insights in several directions, beginning with an examination of epigraphic and Bacchylidean practice, both of which are less prone to numerical exaggeration. Nonetheless, the former displays traces not only of that phenomenon (e.g. FD III.1.507 p.332), but of Pindar's most sophisticated misdirections, such as the appropriation of a family member's win in order to allow a victor to claim a full periodos (FD III.1.510 ~ Pi.I.2.12-32), while Bacchylides can boost the tally via counterfactuals (B.4.11-13 ~ Pi.N.11.22-29), set the present victory in the context of all the crowns won by the victor's polis (B.2.6-10), or interpose his personal authority to stress the qualitative uniqueness of the manner in which the tally was accumulated (B.4.14-16; B.5.42-49; B.8.19-25. The paradigms of countlessness that sometimes conclude catalogues (Pi.O.13.43-46, 112-13; Pi.N.2.23; Pi.N.10.45-46; 'Simonides' A.P.13.14.5; CEG 811.7) are not only rhetorical caps or markers of generic sublimity (cf. Porter (2016) 350-60), but metatextual breaks enacting the medium's inability to process the immensity of its message.
Explanations for the poets' idiosyncrasies are sought in their awareness that reperformance would spread their odes far beyond those able to personally verify their details (cf. Morrison (2007)), as well as in the possibility that their role owed more to the concept of partisan witnessing than to the preservation of an accurate historical record (cf. Maslov (2015) 212-45).
Sat 3rd, 11:00-11:45
University of Cambridge
Performing Numbers in the Attic Orators
Numbers were as important in democratic deliberation as they were in the functioning of democratic institutions. The evidence of the Attic orators makes it clear that Athenians valued numbers and expected orators to use them in persuasive argument. Yet numbers were not a straightforward means of quantification. They could serve a variety of functions, and presented challenges to both audience and speaker. This paper examines fourth-century rhetoric from the assembly and courts to illustrate the complexities of performing numbers before mass audiences.
Orators used numbers to impress diversity and magnitude on their audience, while at the same time advertising their own accuracy and objectivity. When calculations were meant to be followed by the audience, using numbers involved greater risk. Demosthenes' missteps in On the Symmories (Dem.14, 354/3) and his more accomplished effort in the First Philippic (Dem.4, c.351/0) reveal some of the pitfalls and possibilities of numerical performance. Numbers naturally made greater demands on the attention of listeners, so an orator needed to be mindful of their limits if his calculations were going to be intelligible. Then came the challenge of credibility. The dêmos was naturally suspicious of numbers that it could not verify. This fear exacerbated popular anxiety about the misuse of expertise and rhetoric. Orators accordingly try to win the confidence of the audience by attributing figures to trustworthy, reassuringly democratic, sources.
Fri 2nd, 15:45-16:30