ECONOMIC PATHWAYS TO NUMERACY

David A. Harper Department of Economics

New York University David.Harper@nyu.edu

First Draft: November 16, 2005

Abstract: Number systems and counting routines are important institutions that frame economic processes and regulate human relationships. They are part of the structure of knowledge that comprises the economic system. Given the ubiquity of these institutions, it is somewhat surprising that economists have not examined them. Accordingly, this paper examines how number sequences enter into the economic system and how they are adopted and diffused across structures of economic agents. It argues that the evolution of number sequences is endogenous to economic processes and therefore amenable to economic explanation. In the initial stages, number sequences arise not so much from prescientific efforts to explain the world but rather from more mundane requirements of economic production and exchange, and especially the need to track property. According to this view, the emergence of systematic number concepts is historically inseparable from the social division of labor, specialization and the expansion of markets. The expected gains from trade provide the incentive, and the division of labor generates the specialist knowledge, necessary for making improvements in numerical tools. Thus, economic factors are the main impetus for early numerical quantification in human societies. JEL Classification: D02 (Institutions: Design, Formation, Operations); Z13 (Social Norms and Social Capital; Social Networks)

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ECONOMIC PATHWAYS TO NUMERACY

[A]ll action is decided in the space of representations…. To explain human actions, both successful and unsuccessful, we often need to understand the representations on which they are based.

(Loasby 1999: 10) 1 NUMBER SYSTEMS AS INSTITUTIONS

Number systems are important institutions that frame economic processes and regulate human relationships. They perform an important function in guiding and coordinating social actions that are dispersed over space and time. They serve as ‘orientation schemes’ that figure in numerous plans of numerous actors on both sides of the market. They enable market participants to take their bearings from prices and to orient themselves to a common pattern of human interaction. To adapt Loasby’s principle stated above, we can say that many economic actions are decided in the space of numerical representations. Buyers and sellers rely on number sequences in order to communicate and bargain with one another and to fix mutually agreeable terms of exchange. Indeed, particular systems of numerical representation, such as counting by conventionalised hand gestures, can help overcome communication barriers to buying and selling that arise from mutually unintelligible languages. By imposing a consistency and a uniformity of action, number sequences also help curb opportunistic behaviour by participants. They thereby define low transaction-cost pathways for doing things (cf. Nelson and Sampat 2001). Indeed, basic rules of numerical calculation (arithmetic facts and algorithms) belong to the core connective structure of common knowledge that supports the modern market economy. Number sequences are fundamental institutions (or highly generic rules) that form a coherent and relatively stable structure with which many other types of institutions must be compatible. They thus bear complementary relationships with other bundles of rules and capabilities and are often building blocks of other institutions (cf. Dopfer, Foster and Potts 2004). For example, systems of numerical representation are often interwoven with the institutions of property, contract and money.1

1 Graphic sign-systems for representing numbers had their origin in the development of procedures for keeping track of private property:

Written numeration is probably as old as private property. There is little doubt that it originated in man’s desire to keep a record of his flocks and other goods. Notches on a stick or tree, scratches on stones and rocks, marks in clay -- these are the earliest forms of this endeavor to record numbers by written symbols.

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The evolution of money in particular presupposes the availability of rudimentary conventional systems for representing numbers. Money is ‘the numerical institution par excellence’ (Crump 1990: 12). Without sequences of cardinal numbers, ‘monetary economic calculation’ in a market economy would be impossible. In other words, there could be no numerical computation, in terms of money prices, of the consequences of actions in the market. ‘It was economic calculation that assigned to measurement, number, and reckoning the role they play in our quantitative and computing civilization’ (Mises 1966: 230). The original emergence of early number sequences also represents a decisive turning point in the evolution of human cognitive abilities, cultural transmission and the growth of knowledge. When humans first engraved sequences of notches to express numerosities on bones and ivory during the Upper Paleolithic, they were launching the use of artificial external memory systems. These primitive sequences and later symbolic forms of recording numerical information were important precursors to other institutions -- especially writing systems -- that are crucial for the transmission of social and economic knowledge and the evolution of economic systems. ‘Almost certainly homo sapiens mastered the use of numbers before mastering the use of letters…. The human race had to become numerate in order to become literate. No society which could not count beyond three ever achieved writing; at least, not by its own efforts’ (Harris 1986: 133). As well as being social technologies, number sequences are also cognitive technologies. They provide categories for parsing the world and framing novel problem situations. They become imprinted on the spectacles through which we see the world. They are highly abstract relational structures that can be used to detect patterns in the environment. ‘It is rarely, if ever, the case that the appropriate notion of pattern is extracted from the phenomenon itself using minimally-biased discovery procedures. Briefly stated, in the realm of pattern formation “patterns” are guessed and then verified’ (Crutchfield 1994: 14; emphasis added). In addition, number sequences can be employed as intellectual tools for the performance of higher-order cognitive tasks. They amplify the human capacity to quantify sets beyond the limits of our natural memory and powers of perceptual discrimination. Numbers serve as ‘a means of apprehending more easily and more sharply the difference of things’ (Dedekind 1901: 31--32). To know a number sequence is to have a virtual tally-making kit in one’s mind. Economic agents can use it to discriminate between the cardinalities (sizes) of different sets of goods, even when those sets are not simultaneously present, and they can manipulate and update their numerical

(Dantzig 2005: 21)

Similarly, Piggott (1950: 180), citing E. Speiser, says that writing in general is ‘the accidental by-product of a strong sense of private property’.

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representations to reflect changes in the numerosities of available stocks of different economic goods. In markets, entrepreneurs rely upon numerical data -- expressed as money prices -- to reduce complex situations into cognitively tractable structures that enable them to calculate the monetary profitability of alternative patterns of resource use. In spite of its abstractness, the idea of number appears to be a universal category of human thought (Butterworth 1999: 4). Given the ubiquity and centrality of number sequences in our lives, it is somewhat surprising that economists have not examined them. Economists have not sought to explain the evolution of number concepts and the systems of numerical representation and computation that have emerged and been diffused in human societies. They have not examined how novel number sequences enter into the economic system and how they are adopted, adapted and diffused across structures of economic agents. Numeracy of economic agents is simply taken as given. It is a datum of the economic system. ‘The body of economic theory does not address the ontological status of “number”’ (Zúñiga 1999: 312). There is no examination of how economic agents acquire knowledge of numbers and manage to grasp different number sequences. There is no analysis of how populations of economic agents learn to count and calculate. There is no investigation of how economic agents represent quantities and then transform these representations in a rule-based manner as situations change. The gap is all the more surprising when we consider the heroic numerical abilities that are attributed to economic agents in mainstream models. All economic agents are assumed to be highly numerate beings. The universe of economic theories and of economic agents usually contains real numbers. Accordingly, economic theories are ontologically committed to real numbers for the sake of explaining economic phenomena. They entail or presuppose that there are real numbers but never explain how economic agents come to have knowledge of these numbers. Typically, ‘representative agents’ are able to solve complex constrained maximisation problems relating to the optimal allocation of given means amongst given ends. Implicitly, agents are assumed to understand the true nature of real numbers, numbers represented by decimal expansions that possibly go on forever. In addition, agents are able to apply that knowledge in order to exactly and exhaustively describe their environment in numerical terms. They live in ‘a cosmos in which outcomes have calculable probabilities’ (Shackle 1969: 10). Agents are equipped with well-defined mathematical functions for forecasting and decision-making. ‘Indeed, the agents are these functions…. They are clockwork Bayesians, wound up with prior distributions’ (Littlechild 1977: 7--8). This paper argues that number systems and counting routines are part of the structure of knowledge that comprises the economic system. The evolution of number sequences might also be endogenous to economic processes and therefore amenable to economic explanation. The evolution of numerical knowledge is part of the

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economic process itself. The emergence and selection of numerical representation systems is part of the process by which economic systems transform themselves from within. Economic processes bring into being numerical representation systems that open up further trading opportunities (by reducing transaction costs) and that support an ever finer division of labour and specialisation. Thus, initially at least, number sequences arise not so much from prescientific efforts to explain the world but rather from more mundane requirements of economic production and exchange. Economic factors are consequently the main drivers in the creation and search for more efficient and effective methods of counting and measurement. Procedures to track and exchange property provided the impetus for early quantification. Primitive economics is thus the mother of arithmetic and mathematics. In human history, the emergence of early number sequences actually depended upon increases in the division of labour, specialisation and the expansion of markets. As a species, we could not have attained the intellectual heights of an explicit notion of abstract number, without first climbing the lower rungs of the ladder provided by economic experience. We can sum up the general argument of this paper by quoting Johnson-Laird: ‘In nature, there is no representation without evolution, and perhaps there is no evolution beyond a certain point without the capacity to represent the world’ (1983: 399). Adapting that quotation to our purposes, the central thesis becomes: In human society, there is no numerical representation without economic evolution (trade, division of labour and specialisation), and no economic evolution beyond a certain level without the capacity to represent the world numerically. The problems addressed by this paper This paper is part of a research program that focuses upon four sets of questions:

1. What types of numerical representations2 and computations are severally necessary and jointly sufficient for trade in general and ‘economic calculation’ in particular?

2. Given (1) above, how are these numerical representations distributed (in the sense of ‘spread out’) and coordinated across different elements of the extended cognitive system, broadly defined?

2 The term ‘numerical representation’ is being used here quite loosely. A better, but somewhat more technical term, is ‘cardinality representation’ -- a representation of set size. Not all cardinality representations use numbers (in the technical sense in which we will use the term ‘number’), and hence not all cardinality representations are numerical.

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3. What, if any, are the economic causes of the evolution and diffusion of number concepts in human societies and, more specifically, of the conventionalised systems of numerical representation and computation that are used in production and exchange?

4. What are the economic effects of differences between, and changes in, these conventional systems of numerical representation and computation?

This paper focuses upon questions 1 and 3 above. Section II of the paper is concerned with conceptual foundations for the entire research program. It seeks to explain the nature of representation in general and to explain the nature of cardinality (numerosity or set size), which is an important property of sets. It also examines the distinction between iconic and non-iconic systems for representing cardinality. In addition, the section provides a technical definition of ‘numbers’ and of counting. Section III is a first step towards identifying representational primitives necessary for trade. In particular, it examines the minimal set of quantitative representations and capabilities that people need to have in order to engage in economic exchange. (In general, because people can make use of calculating devices and other tools, the computations people must perform need not coincide with the set of numerical computations necessary for trade to occur.) Section IV addresses the third question listed above. It examines the causal role of economic processes in the evolution of number sequences in human societies. More specifically, it argues that the social division of labour and specialisation, which in turn depend upon the human propensity to trade, are the main causal factors in the early emergence of systematic number concepts and conventional systems of numerical representation and computation. This section argues that purposeful, calculative, economic action within the context of the division of labour was the driving force behind early numerical quantification in human societies. In other words, primitive people’s quest for rudimentary means of ‘economic calculation’ was the catalyst for the historical development of numerical tools and their application in practical problem-solving. It should be noted that this argument reverses the causal chain suggested by other scholars, who contend that purposeful action within the division of labour demands (as a logical necessity) the prior human ability to represent cardinal numbers mentally and perform computations on them. Section V lays out an agenda for future research. 2 CONCEPTUAL FOUNDATIONS

The nature of representation (and computation)

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At the most general level, a representation is something that, from the perspective of some computational system, stands for something else (Peirce 1960: §2.228). This definition implies that there are two distinct but related domains, which Palmer (1978) refers to as the ‘represented world’ and the ‘representing world’. The task of the representing world is to model some aspects of the objects in the universe being examined (i.e. the represented world). Consequently, a representation is the relation as a whole that holds between a representing world and its represented world, and it comprises five aspects:

In order to specify a representation completely, then, one must state: (1) what the represented world is; (2) what the representing world is; (3) what aspects of the represented world are being modeled; (4) what aspects of the representing world are doing the modeling; and (5) what are the correspondences between the two worlds.

(Palmer 1978: 262) The defining character of representation is that there exists a correspondence from objects in the represented world to objects in the representing world, such that at least some of the relations for objects in the represented world are structurally preserved by relations for corresponding objects in the representing world (Palmer 1978: 266).3,4

Incidentally, a broad notion of computation is also adopted that applies equally well to operations that take place inside or outside the skin of individual persons. This is especially important in the case of numerical cognition because the relevant cognitive system is an ‘extended cognitive architecture’ that includes an individual decision-maker in interaction with other human agents and material artifacts, such as an abacus, cash register, pen and paper. Following Hutchins (1995: 49), therefore, computation is defined widely as the ‘creation, transformation, and propagation of representational states’ across representational media. From this standpoint, cognition is fundamentally a type of computation because it involves rule-governed operations on representations that are physically instantiated. 3 Each of the two worlds represents a relational structure which comprises a set of objects and sets of relations defined over these objects. In terms of the Representational Theory of Measurement, a representational system is an ordered triple (A, B, f) where A and B are relational structures (and B is a numerical relational structure in the case of numerical representations) and f is a homomorphic mapping from the represented structure (A) into the representing structure (B) (Krantz et al. 1971; Roberts 1979). 4 The relations are not necessarily binary, ternary and so on. For instance, it is logically possible that they could include unary (one-place) relations, such as properties and attributes defined for each individual object. However, if we want representations to model realistically the way people construe situations and make structural analogies between them, then only relations between two or more objects, rather than attributes of objects, are preserved and mapped from the base domain into a target domain (Gentner 1983).

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What is cardinality? Cardinality is a property of sets, not of individual objects. Cardinality is a property that only sets can have. It is the property that we try to assess when we ask how many members there are in a particular set. Cardinality is the quantity or size of a set or, stated more formally, it is the scope of the extension of a set (Nesher 1988: 117, 120). It is often referred to as ‘numerosity’ in the psychological literature. If they are numerate, economic agents can apply numbers in order to identify the cardinality of a set of empirical objects (i.e. to indicate how many). For example, they may wish to indicate the cardinality of a set of goods they wish to buy (‘Two AA batteries please’). Included here are also cardinal number assignments that quantify sets of measure items (such as in ‘20 pounds of potatoes’). Numbers and counting are not always necessary, however. It is possible for an economic agent to grasp the property of cardinality without being able to apply numbers to this property. For example, the Vedda tribesman using a collection of tally sticks to determine the size of a large set of coconuts is employing a pre-numerical means of identifying cardinality. In addition, adults, human babies and many species of animals, including rhesus monkeys, domesticated dogs and rats, can distinguish the cardinality of sets of two or three objects at a glance and without counting (an innate process known as ‘subitising’ which is discussed later). This paper is concerned with how economic agents represent cardinality in their daily lives. Our focus is on ‘street mathematics’ or ‘arithmetic in the wild’ rather than on complex mathematical thought. Iconic representation of cardinality In general, a representation is said to be iconic to the extent that the representing objects bear a relation of similarity to the represented set. In the case of iconic representations of cardinality, the set of cardinal icons (e.g. a visual tally, such as notches on a bone or tokens in a hollow clay tablet) has the same cardinality as the set it represents (e.g. a herd of animals). ‘In general, there is no resemblance between the two groups beyond the analogy of a one-for-one association’ (Roche 1998: 10). Cardinal icons yield representations that are based on itemising elements (e.g. ‘an object, and another object, and another object’) rather than on an assignment of a number to the whole set (‘THREE objects’). A comic example is a TV skit in which the comedian John Belushi recites ‘Cheese burger, cheese burger, cheese burger’ in order to request three burgers in a restaurant. He could also have restricted his

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message to the iconic form ‘cheese burger, another, another’. Thus, cardinal icons are token-iterative signs because one token (whether a verbal or non-verbal tag) is added to a list for every item tallied. ‘Token-iterative lists are, in principle, lists as long as the number of individual items recorded’ (Harris 1986: 145). Representing objects must meet two conditions if they are to function as an iconic representation of cardinality: namely the ‘perspicuity condition’ and the ‘matching condition’ (Harris 1986). According to the first, cardinal icons must be well-distinguished tokens, so that the set of representing objects has the property of cardinality. Each token must be individuated from all the others. As for the second condition, there must be exactly one representing object (i.e. token) for each element in the set of represented objects (i.e. the two sets must be in ‘one-to-one reciprocal correspondence’ with each other), so that the set of representing objects has the same cardinality as the represented set (Wiese 2003a: 386). Non-iconic (symbolic) representation of cardinality In symbolic representation, the link between the representing world and the represented world is established by convention. There is no relation of similarity or of physical or temporal contiguity between the two worlds. The connection between representing objects and what they stand for is arbitrary in that their particular form is not imitative or suggestive of what is being represented. Symbolic representations of cardinality connect relationships between numbers with relationships between empirical objects. That is, when we assign numbers to identify the cardinality of a set of empirical objects, we do not establish direct links between individual numbers and individual objects; instead we link a numerical relational structure and an empirical one. ‘Number assignments are essentially links between relations’ (Wiese 2001: 1120). For example, suppose A is one set of 5 cars and B is another set of 3 cars. We could associate the cardinal relation ‘has more elements than’ between A and B with the numerical relation ‘is greater than’ between the two numbers (5, 3) identifying their respective set sizes. With cardinal number assignments, a number (e.g. the Arabic numeral 5) is assigned to a set of empirical objects (e.g. set A) such that the position of that number in the number sequence reflects the set’s cardinality. What determines the link between the number and the set of empirical objects is not some inherent feature of the number itself, but the relation that the number has to other elements in the numerical progression. Cardinal symbols are always part of a system, and they refer to objects not as individual tokens, but with respect to their positions in the number system.

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Symbolic systems represent numerical information much more efficiently than iconic systems. The costs of sign production are much lower. In a polynomial system, the number of symbols to represent a particular cardinality is proportional to the logarithm of the number of tokens needed to represent the same cardinality in an iconic system. For example, in the base-ten Hindu-Arabic system of enumeration, it takes just four digits (i.e. ‘1,000’) to represent the cardinality of a set of one thousand items whereas in a one-for-one tally system, it takes one thousand tokens to represent that numerical quantity. What are numbers? Numbers are elements in a special type of pattern. In particular, numbers are sequential positions in a progression (such as a stable counting sequence), and their identity is fully determined by their relations to other positions in the structure in which they occur (Resnik 1982).5 What is significant is the pattern or structure that the elements jointly display: ‘“Objects” do not do the job of numbers singly; the whole system performs the job or nothing does’ (Benacerraf 1965: 69).6

Indeed, numbers have no distinguishing features apart from those relating them to other elements of the same pattern, and consequently they cannot be identified outside of the structures to which they belong. The counting word ‘THREE’, for example, is completely and solely defined by its position as the third element in the conventional counting sequence (‘ONE’, ‘TWO’, ‘THREE’, ‘FOUR’, ‘FIVE’ …). Within this progression, its ‘individuality’ amounts to being that element that is preceded by just ‘ONE’ and ‘TWO’ and that comes immediately before ‘FOUR’. The counting word ‘THREE’ does not refer to the number 3; rather, it is the number 3 within the particular sequence of which it forms a part.7 For numerical purposes, we can ignore any additional properties that this element has, such as its phonological shape or number of syllables. We could all substitute the word ‘UGG’ for ‘THREE’ as the third element of the English counting sequence without having any effect on the system’s ability to function as a numerical tool. 5 A progression is a particular set N that is totally ordered by a relation R (such as the ‘less-than’ relation) and R is irreflexive, asymmetric, transitive and connected in N. For the purpose of counting objects, an important property of a progression is that each of its members has only finitely many predecessors (see Benacerraf 1965: 48; Quine 1960: §54; Wiese 2003b: 22--23, 299). 6 Of course, patterns themselves can be nodes or positions of a larger pattern, and this is as true of number sequences as of any other type of pattern. The series of natural numbers, for instance, can be represented by a position in a set-theoretic hierarchy (Resnik 1997: 240). 7 As recited in a sequence, counting words do not denote or refer to anything in the outside world. They are not names for anything. They are non-referential: ‘Questions of the identification of the referents of number words should be dismissed as misguided in just the way that a question about the referents of the parts of a ruler would be seen as misguided’ (Benacerraf 1965: 71).

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We treat number sequences as a kind of yardstick that agents can use to measure sets. ‘In a nutshell, in cardinal number assignments we map a set [of empirical objects] onto a number n such that the set of numbers less than or equal to n -- the number sequence from 1 to n -- has as many elements as that set’ (Wiese 2003b: 20). Applying numbers to identify cardinalities enables us to grasp empirical relations far beyond the perceptual limits imposed by our specifically human psycho-physical organisation. Number sequences thus serve as an apparatus for expanding what we can observe with our natural sense organs. They are like an artificial sensory radar that expands the universe of human experience. Number sequences exhibit special properties. According to Wiese’s (2003b) criteria-based approach, elements of a sequence must exhibit distinctness and sequential ordering in order for that sequence to be able to serve numerical purposes in our daily lives. More formally, there are two necessary and sufficient conditions for a sequence N to be a numerical tool for quantifying finite sets:

1. All x є N must be well distinguished. 2. N must be a progression.8

(Wiese 2003b: 304) Thus, any set that forms a progression of well-distinguished elements can be used as a number sequence. We cannot pick out a unique set of abstract objects that are the numbers:

There are not the numbers waiting to be identified by us somewhere out there, but a range of possible number sequences, sequences that we can use as numbers because they fulfil the necessary requirements for this function. According to this view, the entities we assign to empirical objects in the different kinds of number assignments -- the entities constituting the basis of our cognitive number domain -- are not particular abstract objects; rather, they represent a set of tools with particular functions and many possible instances.

(Wiese 2003b: 60; original emphasis) An important implication for our purposes is that economic agents could have many different sequences at their disposal that could potentially function as numbers. Thus,

8 More generally, Wiese specifies that N must also be infinite. However, infiniteness is not a necessary condition for numerical tools in many everyday economic contexts. ‘We can manage perfectly well with a limited number sequence as long as we do not have to cope with more extended empirical relational structures and do not want to perform mathematical operations based on higher numbers or on infinity’ (Wiese 2003b: 77). Provided economic agents are only seeking to quantify finite sets of economic goods, their number sequences do not need to be infinite. However, the finiteness of a number sequence does mean that there is an obvious upper bound to the size of empirical sets that agents can quantify.

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individual economic agents do not need to limit themselves to one progression -- they can use alternative number sequences in their economic activities, even for a single transaction. Hence, we might observe multiple number sequences being employed in the marketplace, in complementary and competitive ways. A real-world example of this is the ‘Pomor’ trade that took place during the 19th century. Every summer, the Russian peasants and merchants (the so-called ‘Pomors’) would sail to northern Norway to trade grain and wood products in exchange for fish supplied by Norwegian fishermen. Counting was an area in which there was a high degree of mutual understanding between trading partners: each trader could usually count in the other’s mother tongue even though they all spoke a mutually understood pidgin language called Russenorsk (Broch 1927: 236). In other words, all parties were familiar with the numerical expressions and the numerical relational structure inherent in both the Norwegian and Russian number sequences. ‘We assume that this was an insurance against the use of wrong numbers and prices, either for the weight of fish or the weight or price of other goods’ (Broch and Jahr 1984: 37). Although many different sequences with the proper structure could potentially perform the functions of numbers, successful communication in the market requires that buyers and sellers mutually recognise the particular sequence each of them is using and its numerical relational structure. In order to correctly interpret your request ‘SIX doughnuts, please’, I must understand the relation that the counting word ‘SIX’ bears to the rest of the progression you are using. It should be noted, however, that successful communication does not require that the buyer and seller produce numbers from the same number sequence in a particular transaction: numbers must be interpretable by the other but need not be produced by the other. While in a French market, I can point to the oranges and request ‘THREE oranges’ and the vendor can hand them to me and then demand ‘CINQ euros’ in exchange. Each of us produces a number-word construction in a different tongue but this does not impede communication as long as each hearer understands the one who speaks. The desire to produce mutually comprehensible number assignments (and to reduce transaction costs) dramatically narrows down the pool of sequences that are effectively used as numbers in economic settings. Through a process of selection, economic agents in a community will converge upon a small set of sequences that are specifically dedicated to performing numerical functions and that are conventionally recognised as such. The approach in this paper also has important implications for the ontological status of numbers. Numbers are created by us; initially, they arise from purposeful economic actions (understood in the broadest sense) and from the needs of everyday life. At the outset, number sequences are not abstract, mind-independent objects. Rather they are higher-order goods that humans have created as an indirect means to human satisfaction. They are tools for making tools. They are immaterial capital

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goods that are complementary to many economic activities and that serve to reduce the costs of economic exchange. ‘The natural numbers are the work of men, the product of human language and of human thought’ (Popper 1979: 160). They are ‘our linguistic invention; our convention; our construction’ (Popper 1992: 26). Once we have created them, however, number sequences have autonomous properties which are well-determined (e.g. properties of being ‘odd’, ‘even’, ‘divisible’ and ‘prime’), which may require great ingenuity for their discovery, but which exist independently of our awareness of them (Hersh 1986: 22). What is counting? Counting is a numerical procedure for determining the cardinality of a set. It is a process that ‘maps from the numerosities [cardinalities] of sets to symbols or states of the [numerical] representing system’ (Gallistel 1990: 338). The product of the process of counting is a cardinal number assignment (e.g. ‘This set has FIVE elements’). The transitive process of counting9 is a complex cognitive operation that involves coordination between two simultaneous actions, namely producing a string of counting tags (i.e. elements of a number sequence), and tagging a set of objects. In addition, each of these actions involves several subroutines (described by Nesher 1988: 113--115). The counting process involves following abstract rules of action that constitute important connecting principles for quantifying sets numerically:

1. Stable-order principle: Use counting tags in a stable, repeatable order; 2. One-to-one principle: Assign exactly one tag to each and every object in the

represented set; 3. Order-irrelevance principle: Tag the objects in the represented set in any

order; 4. Cardinality principle: Use the last tag employed in the count to identify the

cardinality of the represented set. Any economic agent wanting to count successfully must adhere to these principles (Gelman and Meck 1992: 175). The one-to-one principle is particularly complex because it requires that the agent tag the objects in the context of an evolving partition of the represented set. (By a partition of a set, we mean a collection of pairwise disjoint subsets of that set.) At any point in the counting process, the agent has to distinguish the subset of already tagged objects from the subset of objects still to be tagged, with the partition being updated continually after the assignment of each tag (Nesher 1988: 114).

9 ‘Learning these [number] words, and how to repeat them in the right order, is learning intransitive counting. Learning their use as measures of sets is learning transitive counting’ (Benacerraf 1965; 50; original emphasis).

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It should be noted that the tags used in counting do not have to be verbal. The only requirement is that the counting sequence form a progression of well-distinguished elements. ‘The set of count words meets these criteria, but then so do other sets of tags' (Gelman and Gallistel 1978: 76). The mere fact that a group of economic agents does not have an extended sequence of counting words does not mean that it cannot count to a large number -- that it cannot make numerical cardinality assignments. The system of counting words is not crucial to an understanding of natural numbers (Gelman and Butterworth 2005: 8). ‘Words are not necessary for real counting to occur’ (Harris 1987: 36). For instance, if a society’s language lacks a subsystem of counting words, then its members might use non-verbal, visual counting tags, such as special hand gestures, other body-parts or distinctive marks drawn in sand in order to enumerate large sets of objects (Harris 1982; Saxe 1981). In addition, a group may use a combination of verbal and non-verbal semiotic means for representing number. Taken together, these considerations imply that we should not evaluate the numeracy of a social group solely on the basis of its lexicon of number words but also in terms of its use of non-verbal counting tags. Counting is also an exemplar of human cognition as a pattern-making, connective process. Counting procedures depend on the capacity of the mind to form patterns and connections between things:

If we scrutinise closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible.

(Dedekind 1901: 31--32) Like all cognition, counting involves creating representations and connecting them to form organised networks, both internal and external to the mind. The counting process constructs, transforms and propagates representations across different representational media both inside and outside the brain. (The use of different body parts to count, as practised by the Oksapmin of Papua New Guinea, for example, involves coordinating mental representations and physical counting tags.) Counting involves mapping the relational structure of empirical objects into the relational structure of numerical signs. As in all translations of relational structures, counting involves a significant degree of abstraction from the concrete details of the situation and the particular attributes of the empirical objects:

Counting and measuring are ways of representing selected aspects of objects and situations. In order to measure, one has to choose what dimension will be quantified -- a class of objects when counting, for example, or a quality like length or weight when measuring. Like any form of representation, the initial

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choice of what to represent and what to ignore involves an abstraction: Everything will be ignored except the aspect that is being quantified.

(Nunes 1992: 558; emphasis added) Conventional number sequences and counting routines condense a great deal of wisdom from previous generations and provide us with well-tested solutions for recurrent economic problems. They contain a lot of implicit procedural knowledge developed through actual experiences. Moreover, their use economises on the scarce resources of human cognition. Ordinary economic agents do not need to know what criteria counting sequences must satisfy to fulfil numerical functions, what complex principles underpin counting routines or why they can be used to quantify sets. They just need to know that they can fulfil the relevant function. Having now laid the conceptual foundations, we are now ready to consider the economic causes of the emergence of number sequences. We shall also examine the necessary representational conditions in order for trade to occur. We do this by considering a world in which agents have a primitive grasp of cardinality but lack a fully-fledged concept of number. 3 REPRESENTATIONAL CONDITIONS NECESSARY FOR TRADE

The economic origins of our urge to quantify Quantificational activities are a human response to scarcity and an attempt to ameliorate its impacts. If people did not have multiple ends of differential importance, and if the means available to them were in unlimited supply and/or incapable of alternative application, they would have much less incentive to engage in acts of quantification, such as tallying and counting. Because quantificational activities are costly in terms of time and energy, agents usually have a larger purpose in mind when carrying them out. It is the larger purpose that makes them meaningful to us. Tallying, counting and measuring are purposeful acts intended to improve our situation. They are often driven by the ultimate goal of satisfying our needs more fully and by the intermediate goal of more efficiently allocating scarce means among competing ends. Whenever they act to satisfy their needs, economising individuals are always interested in achieving a greater possible result with a given quantity of a good or a given result with a smaller possible quantity. Economisers are also interested in finding out the quantities of means required, and available to them, for the attainment of their desired ends (Menger 1994: 94--98).

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It is the ‘goods-character’ of economic objects (i.e. their standing in a particular relation to the satisfaction of our needs) and their scarcity (i.e. the fact that available quantities of these goods are smaller than our requirements) that initially make us want to quantify them. In our daily lives, we have much less incentive to tally and count non-economic objects. Although ‘the truths of arithmetic govern all that is numerable [i.e. countable]’ (Frege 1959: 21e), and economic agents could conceivably quantify sets of non-economic objects (such as imaginary events or the thoughts they entertain), the benefit of such speculation is likely to be outweighed by the substantial opportunity cost of doing so, especially at early stages of human civilisation. Tribesmen have little interest in quantifying the superabundant trees in the forest because such a collection is of little relevance to the satisfaction of their needs, but they are interested in measuring (in the broadest sense of the term) the numbers of trees required and readily available to make a specific hut for shelter. Planning for our future needs and engaging in associated economic exchanges induces us to determine the quantities of goods at our disposal and in the hands of other persons. Human survival and flourishing depend upon our ability to gauge the difference between the quantity of goods available to us and the quantity necessary in order to successfully provide in advance for the satisfaction of our needs. Indeed, empirical evidence suggests that the need to store food for use in future periods is greater in areas of climatic instability and this induces people to develop numerical systems for representing precise and large cardinalities in order to accurately estimate their food storage requirements (Divale 1999).10 Information on such quantities is part of our personal knowledge of the particular circumstances of time and place, knowledge that is fragmented and dispersed throughout the economic system. Quantitative knowledge of economic data helps us to adjust our actions and choose the most profitable use of scarce resources. This paper highlights the economic origin of systematic number concepts and everyday methods of counting and measurement. The notion of ‘economic origin’ must not be construed too narrowly, however. By using the term ‘economic’, we are not isolating a specific sphere of human endeavour as the prime mover in numerical development. We are not trying to account for the emergence of number tools by picking out particular kinds of behaviour that qualify as economic. For example, we are not identifying barter or monetary exchange in markets as the cause of systematic number concepts. Rather, our account focuses attention on a particular aspect -- the economic aspect -- of a wide range of actions that encompasses both standard economic activities as well as behaviour not typically viewed as ‘economic’ at all (cf. Kirzner 1976). Thus, for example, in an ancient society in which animal sacrifices were routinely made to the gods, priests faced genuine economic choices in the

10 On the political economy aspects of food storage, see Smyth (1996). For a review of food storage by animals, see Vander Wall (1990).

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allocation of animals among alternative ritual uses (which animals to allocate to which gods) as well as among ends related to nutritional sustenance. There is no reason why their economising behaviour (and coordination of ritual activities) could not generate a search for more exact methods of enumeration and thereby qualify as an economic stimulus to the development of number tools. Thus, previous attempts to locate the origin of counting in ritual behaviour (e.g. Seidenberg 1962) might even be subsumed within an economic explanation of the emergence of number concepts. The upshot is that the calculative features of economising activity are relevant to all spheres of human action and can spur human creativity in the domain of number. Of course, in human history, the goal of precisely quantifying sets of economic goods may exist a long time before an effective means emerges for doing so, especially if the sets involved are large and exceed the bounds of our innate powers of perception and estimation. This is precisely the situation described in the following subsection. But the very existence of such a goal may contribute significantly to the means themselves being created (cf. Hurford 1987: 92). Economic necessity is the mother of arithmetic invention. A complete failure to grasp cardinality concepts would render futile any planning of action to improve economic wellbeing: ‘Even at the lowest levels of civilization we encounter a certain amount of knowledge of the available quantities of goods, since it is evident that a complete lack of this knowledge would make impossible any provident activity of men directed to the satisfaction of their needs’ (Menger 1994: 90--91). All the divergent human groups that we observe in the world today would not have survived to this point and withstood the test of environmental fitness so far if they were lacking the minimal quantitative capacities needed to adapt to changes in economic conditions. In the next section, we consider the state of nature (of admittedly already socialised and linguistic agents) at the dawn of numerical institutions. Starting point: A world without number sequences As a starting point, we consider a world without number sequences or counting procedures. This approach may enable us to identify the set of representational primitives upon which economic calculation is based. This exercise is also important if one agrees with Mises that ‘Economics is essentially a theory of that scope of action in which calculation is applied or can be applied if certain conditions are realized’ (Mises 1966: 199). Let us begin by supposing that as part of their neurobiological endowment, economic agents possess the concept of an object and of a set. They can individuate objects in

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terms of general concepts (i.e. classes of objects). In philosophical terms, agents possess sortal concepts (e.g. ‘jar’) that equip them with criteria of individuation and of identity for picking out particular items that fall under these concepts. The most basic concept of object that an agent can have is that of ‘any bounded, coherent, three-dimensional physical object that moves as a whole’ (Xu 1997: 365). Though some scholars (e.g. Wiggins 1980) might deny that ‘physical object’ or ‘thing’ stands for a sortal concept on the grounds that there is no principle for counting ‘things’ as such, this rough-and-ready notion will meet our purposes and we will limit ourselves to considering economic goods that are bounded, perceptible objects. ‘The immediate goal of acting is frequently the acquisition of countable and measurable supplies of tangible things’ (Mises 1966: 200; emphasis added). Any tallying or counting procedure requires agents to use individuated entities as an input to the process. ‘The disposition to recognize the thingness of the world is a very general one that is in no way restricted to the domain of counting and numerosity, although it is probably a prerequisite for the recognition of numerosity’ (Gallistel and Gelman 1990: 197--198). A set is an abstract object. It is a collection of entities that are well distinguished and which satisfy other specified conditions. Sets are special aggregates whose very identity is defined by the way they are rigidly partitioned into their elements; they have a unique cardinality. Unlike other collections, therefore, sets are ‘intrinsically numbered’ (see Yourgrau (1985) for a critique). Initially, economic agents’ knowledge of sets depends on their causal interaction with basic sets of physical objects (especially economic goods) described above (cf. Maddy 1980). As their knowledge grows, they gain knowledge of more abstract sets in their attempts to understand more concrete sets. In addition, agents are assumed to have an inborn perceptual apparatus that enables them to distinguish an object from its surrounding environment. They can separate the ‘figure’ from the ‘ground’. They can visually segment an array into cohesive, bounded and persisting objects (Spelke 1990). By means of their sensory systems, agents are thus able to distinguish between types of economic goods and to classify goods into homogeneous categories. Agents are also assumed to speak a natural language, which at this stage does not contain a numeral subsystem (i.e. there is no explicit sequence of counting words). However, the language does distinguish between singular and plural by grammatical markers on nouns and verbs and it contains quantifiers such as ‘one’, ‘another’, ‘some’, ‘a few’ and ‘many’. Agents can represent serial-order relations and can master meaningless but stable ordered lists of well-distinguished verbal and nonverbal elements (such as the ‘Eeny, meeny, miny, mo’ ritual).

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It is also assumed that agents have a reasonably advanced ‘mindreading ability’ -- the capacity to attribute beliefs and intentions to others. They can think about what other parties are thinking by constructing higher-order representations of others’ representations. (More precisely, in Dennett’s (1987) terminology, they have at least third-order intentionality.) Furthermore, let us suppose that all economic agents are equipped innately with a dual mechanism of quantification: a system for representing the exact cardinality of small sets, and a system for representing the approximate cardinality of large sets. These capacities are part of their biological endowment and correspond with what we know about the human brain (Dehaene 1997; Butterworth 1999). Thus, agents possess a ‘fractionated’ set of quantitative capacities that are biologically determined and domain-specific (Dehaene 1992: 34). These systems are also shared with other vertebrate animals and pre-verbal human infants (Hauser and Spelke 2004). The first system, often referred to as ‘subitising’, enables economic agents to recognise at a glance and without counting the exact cardinalities of very small sets. It is a perceptual-apprehension mechanism for quantifying very small collections of things. Subitising is almost instantaneous, requiring around 40 ms per object in sets of up to three elements. It is a highly accurate, automatic and pre-attentive process. With this capacity, agents can rapidly discriminate between sets of one, two and three objects and can perform simple arithmetic operations on them (such as adding or subtracting one element). In this system, agents produce a mental model of objects in an array by creating one mental token (a so-called ‘object-file’) for each element in the quantified set, so that there are as many mental tokens as there are objects. Thus, a set of two objects is represented by two object-files of the form ‘OiOj’ which are stored in short-term memory (Uller et al. 1999: 5). In this pictorial type of representation, cardinality is only implicitly represented (Carey 2001). The upper bound for this representation system is three or four elements. Agents can make one-to-one correspondences between the elements in different mental models in order to compare set sizes and establish numerical equivalence or more-than/less-than relations (Feigenson and Carey 2003). The second system is the analogue magnitude system. It enables agents to estimate set sizes with a continuous neural magnitude that is an analogue of set size (Meck and Church 1983). It is a system for representing large, approximate cardinality. The agent’s nervous system generates continuous quantities of energy in proportion to the number of elements in the quantified set. Thus, the mental analogue magnitude increases as the number of represented objects increases but the increments for each object coalesce into a continuous analogue format. This mechanism suffers from inherent variability and yields ‘fuzzy’ or ‘noisy’ magnitudes for larger sets. The fuzziness increases with increases in set size. With this system, the agent can determine which of two large sets is more numerous only if the sets are sufficiently

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different in number. In addition, agents face an upper bound on the set sizes that can be represented by mental analogue magnitudes. Both subitising and the analogue magnitude mechanism provide agents with iconic representations of cardinality. They are iconic in that each element of the quantified set is represented by a distinct mental token (in the case of subitising) or an increment of the mental analogue magnitude. ‘The size of an empirical set is represented by the cardinality of another set (a set of [mental] object tokens) or the size of an analogue magnitude (in this latter case the representation is not one of discrete cardinality, but of accumulated quantity…)’ (Wiese 2003a: 387). These two core mechanisms are also natural systems of representation that are located inside the heads of economic agents (Dretske 1988). They are natural in that (i) their elements have an intrinsic power to indicate cardinality that is objectively related to the conditions they signify and that is independent of the user’s explicit beliefs, goals and decisions, and (ii) the agent does not assign a representational function to these systems. Neither their function (what they are supposed to indicate) nor their power to perform that function depends upon any external source. These mechanisms are not the product of individual learning or cultural transmission (Feigenson, Dehaene and Spelke 2004: 307). They are independent of language. Users do not exercise rational or intentional control over these systems. Consequently, these systems differ from conventional systems for representing cardinalities, such as the series of Hindu-Arabic numerals or counting sequences in a natural language, which do depend upon us to give them their functions. Apart from natural language and these two natural systems for representing cardinality, suppose that the agents have no other means of representing quantities. They have no procedures for verbal or non-verbal counting or tallying. In particular, they have not as yet developed indirect methods for comparing sets of objects by means of an auxiliary group or tally, such as their fingers, pebbles, notches, knots or any other material artifact. Real world example: The Pirahã The above scenario that we have described might at first appear obscure. But in its broad features, it is not totally outside the realm of contemporary human experience. Recent research amongst two indigenous groups in the Amazonian region in Brazil, the Pirahã and Mundurukú, provide examples of societies without extended counting sequences (Gordon (2004); Pica et al. (2004)). The research indicates that the groups can count, but their counting abilities are very limited.

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The Piraha are predominantly hunter-gatherers who occasionally engage in economic exchange with outsiders. As they exhibit significant spatial, categorisation and linguistic skills, there is no suggestion that they are deficient intellectually (Gordon 2004: 499). The Pirahã language contains a counting sequence that comprises two words, ‘hói’ (for ‘one’), and ‘hoí’ (for ‘two’). They have no other counting words. They refer to larger cardinalities as ‘many’ (‘aibaagi’ or ‘aibai’). The Pirahã’s counting method is not recursive: they do not combine basic counting words to form higher numbers.11 In addition, the Pirahã sometimes use the word ‘hói’ to denote sets containing more than one individuated item, so that it seems to mean ‘roughly one’. Gordon also suggests that because ‘hói’ can also mean ‘small’, the Pirahã language blurs the distinction between discrete and continuous quantification of sets. Participants do use their fingers to supplement verbal counting, but Gordon notes that their use is highly inaccurate even for cardinality assessments involving less than five items (p. 497). Gordon conducted a series of experiments involving simple matching tasks. The results are broadly consistent with the dual model of quantitative capacities mentioned earlier. The Pirahã were able to perform exact enumeration for set sizes smaller than three. ‘The Pirahã inherit just the abilities to exactly enumerate small sets of less than three items if process factors are not unduly taxing’ (p. 498). Their performance with sets outside the subitising range (n > 3) is very poor. Participants had difficulty representing exact cardinalities for sets of four or five items. The evidence is also consistent with the Pirahã using an analogue magnitude mechanism to estimate larger sets. The variability of their estimates tends to increase as the target set size increases, as to be expected with an analogue procedure. It is particularly relevant to our purposes that, although the Pirahã have limited contact with outside groups, they do engage in barter by means of pidgin languages. However, they do not rely upon monetary exchanges and they do not use Portuguese counting words in their economic transactions. However, it should be noted that although the Pirahã do not use number words in their daily life, children can easily be taught to count in Portuguese (Gelman and Butterworth 2005: 9). This is consistent with other anecdotal evidence in the anthropological and linguistic literature. Several researchers have attested to the rapid adoption of systematic counting methods by other groups that formerly lacked conventionalised numeral systems (e.g. Hale 1975; Torrend 1891). Rapid uptake of systematic counting methods is also consistent with the principle that all adult humans have the requisite cognitive machinery for

11 Seidenberg (1960:222), citing Schmidt (1926: 361), suggests that systemless counting, such as that of the Pirahã, might in fact be a degenerate form of a pure base-2 system of counting. That is, a systemless method of counting is secondary and more recent than the systematic, recursive method of counting. However, Chomsky (1980) regards it as highly improbable that a biological capacity for recursive counting, which has obvious selection advantages, could remain latent or stop being used.

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acquiring combinatorial syntactic rules and for assembling them into highly recursive systems (cf. Hurford 1987: 69, 305). ‘Economic calculation’ in a world without number sequences In a world without number sequences, the calculative powers of economic agents would be very limited. They would be able to perform simple arithmetic transformations involving addition and subtraction. However, division and multiplication would place too great a computational load on their innate mechanisms of quantification (cf. Spelke and Tsivkin 2001: 89--90). Although agents can implement approximate addition by juxtaposing mental analogue magnitudes of set sizes, there is no similar algorithm for multiplication (Dehaene 2001: 27). Consequently, agents will find it difficult to grasp the multiplicative relation between quantities. In his Narrative of an Explorer in Tropical South Africa, Galton provides a possible example of this difficulty, though his account is expressed in very condescending terms. The entry for March 4th, 1851, reads:

When bartering is going on, each sheep must be paid for separately. Thus: suppose two sticks of tobacco to be the rate of exchange for one sheep, it would sorely puzzle a Damara to take two sheep and give him four sticks. I have done so, and seen a man first put two of the sticks apart and take a sight over them at one of the sheep he was about to sell. Having satisfied himself that that one was honestly paid for, and finding to his surprise that exactly two sticks remained in hand to settle the account for the other sheep, he would be afflicted with doubts; the transaction seemed to come out too ‘pat’ to be correct, and he would refer back to the first couple of sticks, and then his mind got hazy and confused, and wandered from one sheep to the other, and he broke off the transaction until two sticks were put into his hand and one sheep driven away, and then the other two sticks given him and the second sheep driven away.

(Galton 1891: 81) Without the powers of division and multiplication, agents would be unable to calculate ex post the unit price of one good in terms of another when, for instance, bulk transactions involve the direct exchange of large collections of individual goods. The representational power of their innate quantificational mechanisms is such that they cannot represent ratios or fractions. Consequently, the functional limitations of their computational apparatus would make it more difficult to detect price differences in different parts of the market. Keeping track of one’s property over time and space would be possible provided that individual objects are sufficiently differentiated and not so numerous as to be beyond the agent’s memory capacity. If a shepherd has five sheep dogs, each of which he

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knows by name, he does not have to count them in order to determine if they are all in the hut. He simply has to check that each dog on his memorised list is in the hut. The only representational resources the shepherd needs are the sortal concepts mentioned earlier (the concepts under which objects are being individuated) and the capacity to distinguish one object from another. An example of this is also provided in Galton’s Narrative. Galton writes about the ‘wonderful faculty’ of the Damara for recognising any ox that they have only seen once: ‘Yet they [the Damara] seldom lose oxen: the way in which they discover the loss of one, is not by the number of the herd being diminished, but by the absence of a face they know’ (Galton 1891: 81). Even with a multitude of items, counting may be neither necessary nor sufficient for keeping track of property. Paraphrasing Lichtenstein (1812--1815), Seidenberg writes that:

Although few [of the Koosa] could count beyond 10, they nonetheless could keep track of their herds wonderfully well. The owner of a herd of 400 or 500 cattle, when they are being driven home, knows definitely whether any are missing, how many, and even which. Obviously counting is not here in question, as that process could not tell the owner which cattle are missing.

(Seidenberg 1960: 277--278; original emphasis) However, if multiple objects are not readily distinguishable on perceptual grounds, as is the case of homogeneous products in markets approximating perfect competition, the agent will find it difficult to keep track of his property without some means of tallying or counting. The extent of trade will be much more limited in a world without conventional number sequences for representing exact cardinalities. Agents will face higher transaction costs in defining rights to large collections of goods, especially if individual items are not well distinguished. Moreover, in a pre-numerical society, all trade is by means of barter or indirect exchange. Fully-fledged money cannot exist. Money presupposes a number system -- a non-iconic system for representing cardinalities of sets and mastery of a counting sequence. ‘In order to be able to employ money, whether as a medium of exchange or a unit of account, man must first develop a certain minimum degree of arithmetic faculty’ (Einzig 1966: 333--334). The money price of a good X is the ratio of the cardinality of the agreed set of units of the money good to the cardinality of the set of X offered in exchange. A money price typically expresses exchange ratios as a single cardinal number. In order to understand what ‘$5’ means, an agent needs to know the rest of the number sequence -- that ‘5’ is the only positive integer preceded by exactly 1, 2, 3, 4 and immediately followed by 6. Trade can still take place when the quantities involved exceed the bounds of agents’ innate powers of estimation and perceptual discrimination. However, the absence of

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an exact, large-number representation system means that exchanges involving large sets of homogeneous items are likely to take place concurrently and by means of direct face-to-face contact. For example, suppose that two parties want to engage in trade of food items. The first wants to trade potatoes for carrots, the other wishes to trade carrots for potatoes. Suppose also that the sets of goods supplied and demanded are very large. Even if both parties are limited to their innate mechanisms of quantification and have no counting sequences, the two parties will be able to feel their way towards an agreement (assuming that a mutually beneficial exchange opportunity exists) by an extended and iterative process of negotiation and sequential bargaining:

1. The first trader lays out the pile of potatoes he is wishing to sell; 2. The second trader then deposits the pile of carrots he is willing to offer in

exchange; 3. The first trader evaluates this offer; if he accepts, he takes the carrots; if not,

he either removes some of the potatoes or invites the other trader to add more carrots to his original pile;

4. Further adjustment may take place before the two parties seal the deal. When the first trader asks the other, ‘If I give you this many potatoes, how many carrots will you give me?’, the second trader extends his index finger and points to his pile of carrots and says ‘That many’. The agents use the sets of the commodities themselves as nonverbal sign-vehicles to communicate their quantitative trading intentions. The downside, of course, is that this process of exchange -- involving as it does a step-by-step, turn-by-turn process of iteration -- is high in transaction costs. In addition, the procedure does not extend readily to intertemporal transactions. Because agents are unable to express precisely proposed terms of exchange by means of detached numerical representations (e.g. numeral phrases such as ‘TWENTY potatoes’), they cannot specify exact quantities of goods that are absent from the current situation. Without a conventional system for explicitly representing exact cardinalities, agents will face considerable difficulties in specifying relevant quantities in transactions that are remote in space and time. Consequently, transactions involving large sets are likely to be limited to the ‘here and now’. Trade will rely upon cued representations -- the mutual visibility of exchangeable items that occupy a shared spatio-temporal proximity. Reliance upon the dual mechanism of quantification also leads to more errors in comparing set sizes. Without counting or tallying methods, agents have to rely upon perceptual quantitative differences in order to discriminate between different set sizes. As set sizes get larger, agents find it increasingly difficult to discriminate between pairs of sets of goods whose numerical sizes are separated by the same

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absolute difference (the so-called ‘magnitude effect’). For example, an agent can instantaneously distinguish a set of two items from a set of three items. However, the agent will most likely not be able to distinguish a set of twenty-one items from a set of twenty-two, even though the absolute difference between them (one item) is equal to that of the previous comparison. When two sets of tradeable items become numerically large, agents can distinguish their size only if the difference between them is sufficiently great (the so-called ‘distance effect’). How accurately agents estimate sets will thus depend upon both magnitude and distance effects. The larger the sets of the traded items and the closer their sizes, the greater will be the error in the agent comparison of their cardinalities. The nature of these errors will affect processes of economic coordination and entrepreneurial discovery of opportunities and must be investigated in future research. To what extent are people’s ‘fuzzy’ representations of cardinality a source of communication breakdown (i.e. a failure to transmit information correctly) and a cause of discoordination in people’s plans? To what extent do people’s ‘noisy’ magnitudes for large sets contribute to ignorance of opportunities for mutually beneficial exchange and thus gaps in economic coordination? Do successful entrepreneurs have more exact representations of set sizes (and more precise methods of calculation) than do others which enables them to more quickly and accurately discover ‘waste’ and missed trading opportunities? 4 ROLE OF SPECIALIST TRADERS IN THE EMERGENCE OF NUMBER SEQUENCES

In this section, we argue that the emergence of number sequences depends upon the division of labour and specialisation, which in turn depend upon the human propensity to trade. The expected gains from trade provide the incentive, and the division of labour generates the division of knowledge, necessary for making improvements in numerical tools. The division of labour breaks down transactional activities into their component parts so that agents focus their attentions on simpler operations (including quantitative activities) in which they achieve greater dexterity and creativity. This supports the supply of new numerical technologies. Greater opportunities for mutual exchange also increase the demand for numerical tools as people seek to plan future actions in the market, enforce their contracts and keep track of a higher volume of transactions. Thus, in human experience, the emergence of systematic number concepts (and arithmetic operations based on them) is historically inseparable from the social division of labour and the propensity to exchange. But note that this reasoning sets the idea of society as a ‘rational order’ on its head, according to which ‘the very possibility of purposive action within the framework of social division of labor depends on the faculty of the human intellect to conceive cardinal numbers and manipulate them in arithmetic operations’ (Salerno

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1990: 27; emphasis added). (See too Salerno 1993: 120.) In our view, it is purposeful action (based on innate pre-numerical mechanisms of quantification) within the social division of labour that causes people to create extended number sequences. The section provides some preliminary conjectures on the economic origins of conventionalised number sequences and the causal role that economic exchange, the division of labour, and specialisation play in this evolution. It seeks to explain how number sequences might plausibly evolve from some initial iconic system of representation, such that this evolutionary process is driven mainly by the self-interested actions of economic participants (especially specialist traders). Given the preliminary nature of this undertaking, we seek only to sketch out the general features of this evolutionary process. It should also be made clear that this section does not purport to provide an accurate historical account of how number sequences actually emerged but only a story of how these patterns could have arisen. Such an explanation is instructive in its own right because it improves our understanding of how and why numerical institutions persist, are (almost) universal across human societies and show such a remarkable degree of convergence. The locus of specialisation and numerical innovation The question arises as to who are the economic actors who improve numerical-representational technologies and what distinguishes them from other participants. This paper argues that the actors who specialise in taking judgemental decisions about the allocation of scarce resources (cf. Casson 1982) have the greatest incentive to introduce innovations in numerical technologies. ‘People specialize (including specialisation in the organization of institutions) because specialisation as such produces gains’ (Buchanan and Yoon 2002: 404). More specifically, potential big players -- those who expect to be major transactors in markets -- are more likely to make deliberate investments in developing numerical institutions that reduce transaction costs. In the absence of private property rights to numerical innovations, large-scale transactors are better able to appropriate the benefits of their investment. Thus, improvements in numerical capabilities arise as specialists develop new means of securing greater control over scarce resources:

As soon as a society reaches a certain level of civilization, the growing division of labor causes the development of a special professional class which operates as an intermediary in exchanges and performs for the other members of society not only the mechanical part of trading operations (shipping, distributing, the storing of goods, etc.), but also the task of keeping records of the available quantities. Thus we observe that a specific class of people has a special

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professional interest in compiling data about the quantities of goods, so-called ‘stocks’ in the widest sense of the word, currently at the disposal of the various peoples and nations whose trade they mediate.

(Menger 1994: 91) In the story we are telling, specialists are experts in assessing the relevant dimensions of commodities. ‘Experts will tend to be dealers in the commodity in which they are experts -- and dealers will tend to be experts in the goods in which they deal’ (Alchian 1977: 136). They have a comparative advantage in determining the total amount of relevant attributes contained in a given collection of a good as well as the level of those attributes per unit of the good. In addition, they find it easier to determine legal entitlements and rights to the commodities in question (Alchian 1977, Barzel 1982). Specialist intermediaries hold inventories and keep track of goods dispatched, delivered, ordered, received, owed and paid for. By focusing their energies on market-making activities, specialist intermediaries seek to further reduce their recognition and measurement costs (compared with other participants) in identifying the characteristics of goods and sets of goods. They try to reduce their costs in identifying the quantity and quality of the goods they exchange. The locus of specialisation and numerical innovation manifests itself differently in different situations. The social position of the specialists varies greatly by historical and cultural context. Specialists can include Sumerian scribes who recorded transactions on clay tablets, medieval merchants in Europe rejecting Roman notation in favour of Hindu-Arabic numerals, Kpelle blacksmiths in Africa, Mfantse fish sellers in coastal Ghana, cloth merchants of the Dioula tribe in the Ivory Coast, and young shopkeepers in Papua New Guinea. Each of these specialist groups has or had an incentive to reduce the cost of producing and interpreting numerical signs and performing transformations on them. For an example of specialisation in numerical cognition within a society without substantial investments in material capital goods, we can look to the Aboriginal society of Groote Eylandt in Australia. In this society, women have more expertise in dealing with situations that require precision with large numbers, and for this reason, when quizzed about numbers, the men generally recommend directing difficult questions to the older women (Harris 1982: 160). Women specialise in the gathering and distribution of turtle eggs, which are found in copious quantities, usually hundreds at a time. (They also specialise in the distribution of crocodile, seagull and magpie-goose eggs.) Women use their traditional counting systems in order to allocate turtle eggs in multiples of five (the base of the system). This practice has been observed in the Anindilyakwa, Gumatj and Nunggubuyu clans of east Arnhem Land.

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In the simplest trading contexts, where the division of labour is less extensive, a single person may perform all the roles involved in the production and interpretation of numerical signs. However, in more complex contractual settings, in which there is a web of agency relationships, the production and interpretation of numerical representations may be distributed amongst networks of people rather than combined in the same physical person. Consequently, numerical innovation might occur at several different sites within the network. These sites include: the ‘principals’ (the entrepreneurs or resource-owners) whose property rights, transactions, contractual commitments and economic performance are recorded and monitored by the numerical representations produced; the agents who have de facto day-to-day control over the use of resources; the agents who interpret and analyse numerical signs on behalf of the ‘principals’; the agents who choose what numerical information is to be expressed and decide the means for expressing it; and the agents who physically produce the numerical signs by making the relevant sounds, movements or objects (e.g. knots on quipus or tokens in clay tablets). The ancient Inca state provides a good example of extensive specialisation and complex agency relationships involved in the production and interpretation of numerical signs. Within the Inca Empire, a specialist hierarchy of bookkeepers known as ‘quipu camayoc’ (knot-makers) recorded official statistics and accounting information on three-dimensional arrays of coloured knotted cords called quipus. The quipumakers reported to a suyoyoc, a bureaucratic administrator, responsible for keeping records of the communities under his charge. ‘As the area of responsibility grew, the quipu camayoc became in increasing measure full time specialists, who had passed through a long and laborious apprenticeship during the third and fourth years of study in an elite school for royalty and the bureaucracy’ (Crump 1990: 63). A quipu was a nonlinear composition (unlike the planar surfaces of papyrus sheets and clay tablets) and its construction required a great deal of tactile sensitivity and dexterity (Ascher and Ascher 1997: 61--62). The quipu embodied a place-value system of numerical representation, with units at the loose end of a cord and progressively higher powers of ten represented in vertical clusters of knots approaching the main cord.12 The quipumaker had to recognise many subtle differences in colours and arrange distinctively coloured strings in varied patterns in order to express numerical relationships of interest to the bureaucracy.

The neophyte had to learn not only the system of color, strings and knots in use in his branch of the service, but had to familiarize himself with past records. The kind of skill and feats of ‘memory’ mentioned by the early Europeans came only from long practice and full time dedication to the task at hand.

(Murra 1980: 159; emphasis added)

12 For a detailed attempt to decode numerical information represented by quipus, see Burns Glynn (2002).

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Economic specialisation also drove the evolution of the Sumerian token system, which was used to record numerical information about commodities in ancient Mesopotamia from the ninth to the second millennium BCE (Schmandt-Besserat 1979, 1992). As a result of technological innovations, industrial workshops emerged which were specialised in particular crafts, such as the production of pottery, stone tools and bitumen. This specialisation, together with more extensive local and long-distance trade, gave rise to many new settlements in the late Uruk period (circa 3100 BCE). Major innovations in the token system included an increase in the inventory of token shapes, markings on the token surface (for ‘complex tokens’), perforation of tokens to string them together for particular transactions and storage of tokens in sealed clay envelopes. Shortly after these developments, markings of the type and number of tokens were also made on the external surface of the clay envelopes, so that their contents could be determined without destroying them. Eventually, the two-dimensional graphic images of the tokens replaced the tokens themselves, and inscribed solid clay tablets supplanted the hollow envelopes. These changes in numerical representation were a response to the problems of dispersed economic knowledge and the need for specific economic data for coordinating transactions:

The new economic system based on the exchange of goods from one community to another and from one country to another, instead of the home production, increased the need for efficient recording. Production, storage and redistribution of goods to workmen and administrators needed to be computed and recorded with accuracy…. Since the date of the development of the token system through the use of graphic markings coincides with a drastic economic change, the latter is likely to be a crucial factor which led to improvements in the system.

(Schmandt-Besserat 1979: 24)13

The early token system was exclusively used for organising and storing economic data on the quantities of tangible goods. Tokens were modelled in different clay shapes, each token representing a particular commodity. ‘Plain tokens’ stood for basic primary products originating on the farm, such as grain and domesticated animals, whereas ‘complex tokens’ represented goods manufactured in workshops in urban centres, such as bread, oil, perfume, textiles, tools, weapons, furniture and storage vessels. It was a token-iterative system in that cardinality could only be expressed by one-to-one correspondence between the set of tokens and the set of units of merchandise recorded. Thus, one disc bearing a cross registers one sheep and six such discs register six sheep. The tokens were ‘extra-somatic’ (outside the human body), could refer to goods that were physically absent from the current situation,

13 Schmandt-Besserat’s work has come in for serious criticism. See, for example, Glasner (2003), Lieberman (1980), Nissen (1986) and Zimansky (1993).

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could be stored indefinitely and could be interpreted by any other specialist versed in the system. The system supported cognitive skills for scanning, manipulating and analysing economic data (Schmandt-Besserat 1999). The accountant could arrange the tokens into different visual patterns according to types of goods, producers and recipients. The token system made it easy to perform simple arithmetic operations by manually adding and removing tokens. More importantly, the system reduced the transaction costs of long -distance trade:

Because the tokens provided the physical proof of an agreement and because they were small, light and sturdy, the counters [i.e. tokens] could be transported to conduct transactions in the absence of a party involved. Communication over distance expanded the sphere and scope of human interaction.

(Schmandt-Besserat 1999: 4) Explaining differences in numerical capabilities It should be noted that specialists who develop new tallying procedures or numerical tools are not necessarily endowed by birth with superior arithmetic abilities. According to Adam Smith, increasing productivity through specialisation does not require that people be distinguished by their innate abilities and skills: ‘The very different genius which appears to distinguish men of different professions, when grown up to maturity, is not upon many occasions so much the cause, as the effect of the division of labour’ (Smith 1976: 28; emphasis added). As Smith observed, the differences in talents of a philosopher and a common street porter may be unremarkable before they commit themselves to their specialty and training. Accordingly, specialists need not differ initially from the general population in their relative capacities to perform quantitative functions. At least in the beginning, specialists exhibit the same neural organisation as everyone else and possess the same innate mechanisms for quantification described above. Specialisation takes place under generalised increasing returns even in the absence of any inherent differences among economic agents in their innate gifts for arithmetic. Thus, it is the division of labour that enables specialists to develop their numerical capacities, to increase their dexterity at quantitative tasks and to save time switching between activities. According to Smith, the division of labour also increases productivity through ‘the invention of a great number of machines which facilitate and abridge labour’ (1976: 17). Specialisation encourages people to concentrate their efforts on one line of activity and in so doing they develop artifacts (e.g. tally sticks, abacuses, counting boards) to increase their productivity in transactional activities.

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We can extend Smith’s analysis to include all sorts of innovations that economise on human cognition, including new conventions and social technologies that improve our numerical capabilities and the quantificational infrastructure of the market. ‘What is distinctive about human development is its degree of reliance on external scaffolding’ (Griffiths and Stotz 2000: 29). The use of cognitive technologies steers or equips the internal resources of the brain to playing a specific problem-solving role and to focus upon pattern recognition and pattern completion. Specialisation thus leads to a division of cognitive labour between human brains and material artifacts. It is interesting to note that prominent scholars on mathematical cognition argue something akin to the Smithian conception of personal capabilities. For example, Dehaene (1997), a leading neuroscientist in the field, argues that all humans are constrained by the same set of neurological principles:

Having reviewed the available evidence, I do not believe that much of our individual differences in mathematics are the result of differences in an initial ‘talent’. At present, at any rate, there is very little evidence that great mathematicians and calculating prodigies have been endowed with a distinct neurobiological structure for numbers.

(Dehaene 2001: 29) Both Butterworth (1999) and Dehaene (1997) explain the peculiar phenomenon of calculating prodigies by the fact that these individuals concentrate all their energies on the narrow domain of numbers and the practice of mental calculation. Through extensive training, they learn scores of number facts, develop a huge repertoire of clever shortcuts and discover efficient calculation algorithms ‘that any of us could learn if we tried, and that are carefully devised to take advantage of our brain’s assets and get around its limits’ (Dehaene 2001: 29).14 Specialisation in their topic of predilection rather than an innate biological predisposition for calculation accounts for expert numerical competence. ‘At best,’ Dehaene claims, ‘genes may minimally bias cerebral organization to aid the acquisition of numerical and spatial representations. Biological factors, however, do not weigh much when compared to the power of learning, fuelled by a passion for numbers’ (1997: 171). On a related point, in his work on expert mental calculators, S.B. Smith (1983) cites the nineteenth-century study of professional cashiers at the Bon Marché supermarket in Paris that was undertaken by the pioneering psychologist Alfred Binet. When they were originally hired as teenagers, cashiers did not exhibit exceptional skills in 14 Unconvinced by Dehaene’s explanation, Giaquinto argues that it does not account for mathematical geniuses, such as Gauss, whose abilities extend well beyond what is required for numerical calculation: ‘So the superiority of a mathematician of Gauss’s calibre cannot be explained in terms of the extra effort he put in to learning novel numerical facts and calculation shortcuts and the extra hours that he spent on calculation practice’ (2001: 64).

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arithmetic. However, their job required them to spend very long hours each week mentally multiplying a wide range of different quantities by a wide range of unit prices, such as lengths of linen by price per meter, and to add up the total cost of purchases in their heads. Binet’s study found that the cashiers achieved an extraordinary level of proficiency that matched that of two famous calculating prodigies of that era. The upshot is that the emergence of number sequences in human societies does not require exceptional individuals who are innately endowed with a different brain architecture. Because of the brain’s plasticity, extensive experience may change the extension and morphology of an economic agent’s neuronal networks. The constant use of a cerebral region that results from specialisation may deeply alter its neuronal organisation: ‘The time and effort one dedicates to a domain modulates the extent of its representations in the cortex’ (Dehaene 1997: 157). Thus, even though economic agents start life with the same neural organisation, some brain areas might undergo endogenous changes (a transformation of their internal operation) as a result of individualised specialisation in production and exchange. For example, as a result of specialising in the use of physical computational devices, such as an abacus, expert users develop a greater capacity for remembering and manipulating large numbers in mental calculations and exhibit greater activity in the neural correlates for visuospatial information processing than do non-experts (who rely upon verbal rather than visuospatial working memory) (Hanakawa et al. 2003; Tanaka et al. 2002). Specialisation involves learning by selection. In multiple phases after birth, the brain produces more connections between neurons than is necessary for the agent. Specialisation leads to different environmental task demands on the agent which affects the selection process. Those synaptic connections that the agent uses are maintained, while those that are not used are subsequently pruned out. ‘To learn is to stabilize preestablished synaptic combinations, and to eliminate the surplus’ (Changeux 1985: 249). Cortical representations are thereby selected as a function of their use for the agent. Thus, learning by selection may actually simulate neural Lamarckian instruction by the environment (cf. Popper 1979: 149). (According to a constructivist learning model, on the other hand, the cognitive and neural effects of specialisation result from the environment playing a much more instructive role in the agent’s development, so that the environment shapes the structure of the cerebral cortex and stimulates the formation of new synapses that are potentially unique to the agent (Greenough et al. 1987; Quartz and Sejnowski 1997).) Similarly, specialist traders may have more parietal-lobe neurons (a key brain area for numerical cognition) than other agents, not because they are born with more, but because they concentrate their energies on numerical tasks, with the result that their brains allocate more parietal neurons to number activities or retain more parietal neurons. ‘Being good at numbers could be the cause of more neurons rather than the consequence’ (Butterworth 1999: 248).

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Cultural transmission of numerical knowledge Specialisation often entails superimposing additional structure onto established patterns that are acquired as part of the agent’s cultural heritage. It may also require ‘representational redescription’ of existing patterns, so that they become progressively explicit ‘knowledge that’ to the specialist, whereas they remain implicit ‘knowledge how’ for others (Karmiloff-Smith 1992). In addition, specialists can benefit from making powerful analogies in problem solving. By an analogical mapping process, specialists may attempt to relate the less familiar number domain to knowledge structures that are better understood in their culture. The implication is that the development of numerical tools requires that specialists be immersed in the symbolic niche of their culture so that they do not have to start from scratch. Existing techniques of tallying, counting and measurement contain much latent knowledge on how to solve recurrent problems. Specialists need access to these cultural resources in order to develop more advanced numerical tools. The point is well expressed by Wilder, quoting an anthropologist, who quips that ‘if Einstein had been born into a primitive tribe which was unable to count beyond three, lifelong application to mathematics would probably not have carried him beyond the development of a decimal system based on fingers and toes’ (1968, p. vii). Specialists must treat many aspects of the culturally-supplied system of enumeration as given. The process of change in numerical institutions is piecemeal and incremental. In most cases, the changes they implement will amount to small behavioural adaptations at the margin as they pursue greater efficiency. Their innovations often comprise marginal adjustments to the existing system, such as modifying or adding a physical token, a counting word or a graphic symbol. It is much rarer for them to make changes in the representational medium, the particular base or sub-base of the system, or the physical properties (such as iteration, position and shape) used for representing base and power dimensions. The system of constraints imposed by existing patterns and numerical institutions channels the specialist’s discovery and creativity. ‘Structures both enable and constrain; indeed, they enable because they constrain’ (Loasby 1999: 124). For example, the inflexibility of the clay tablets as a representational medium in Mesopotamia required users to economise on the range of symbols used. This may have spurred Babylonian scribes to discover the principle of place-value as a way of overcoming this constraint. In contrast, the flexibility of papyrus as a writing material may have opened up the way for Egyptian specialists to devise new symbols to substitute for groups of iterated number signs, thereby giving rise to the principle of cipherisation (Boyer 1944: 137).

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Finally, the essential role of cultural transmission in the development of numerical sequences highlights the significance of another dimension: the organisation of specialists and how knowledge is transferred between different generations. In particular, the institutions of the family and apprenticeship emerge as important frameworks for the transfer of numerical knowledge. In Mesopotamia, for instance, experienced scribes trained young scribes in their own homes (Charpin 2002). The occupation of scribe was typically handed down from father to son. Apprenticeship in the craft involved a protracted process of practical experimentation, mastering the production and interpretation of signs, copying texts and learning multiplication tables. ‘Apprenticeship, based on the transmission of acquired knowledge, in the context in which it will later be applied, is the basis of traditional instruction in numerical skills’ (Crump 1990: 20). 5 AGENDA FOR FUTURE RESEARCH

At the conceptual level, further research is needed to investigate how the representational conditions necessary for barter differ from the prerequisites for fully-fledged monetary exchange. Attention must be directed to the number concepts that underpin the notion of a money price and ‘economic calculation’ of action in monetary terms. (Economic calculation is an individual’s numerical computation, in terms of money prices, of the outcomes of his or her economic actions.) What properties of numbers must economic agents make use of when they set money prices, interpret price differences and choose economic quantities? Do people need to possess a systematic number concept, such as an understanding of natural or rational numbers, in order to engage in monetary economic calculation? To improve our understanding of the economic causes of numerical institutions, it is also necessary to investigate how monetisation and exposure to a monetary economy affect people’s concepts of number, their counting systems and arithmetic capabilities. This comprises the second component of the research agenda. More precisely, did the spontaneous emergence of money provide a causal underpinning for the emergence of a systematic and abstract number concept in human thought? Are there reciprocal patterns of causation between the emergence of money and monetary calculation, on the one hand, and the evolution of systematic number concepts, on the other? Is the evolution of conventionalised systems of numerical representation path-dependent, and if so, how do economic factors (e.g. initial set up costs, increasing returns and network effects) affect the development and selection of alternative systems? Do idiosyncratic physical constraints of representational media (and economising responses to them) play a role in this evolution? A third cluster of issues on the research agenda relates to the extended cognitive architecture involved in numerical cognition. It focuses upon the distribution (i.e.

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propagation) and coordination of representational states and computational tasks over the resources available for representing items (‘representational media’). How are the computations that are necessary for trade-related quantificational activities in general and economic calculation in particular distributed among individuals, and between individuals and the material environment? How much of the computation is conducted ‘in the head’ of the individual agent, how much is allocated to other agents and what computational tasks do economic agents ‘offload’ to the material environment (e.g. their fingers and artifacts, such as physical tallies, an abacus or pocket calculator)? And what is the comparative advantage of human brains that explains this cognitive division of labour (in numerical computation) between the individual biological brain and other elements of the extended cognitive architecture? And how are interactions between the individual brain and other elements coordinated? The fourth and final component of the research agenda examines the economic effects of different systems of numerical representation on economic processes, including economising behaviour, patterns of entrepreneurial discovery and processes of economic coordination. Further research is required to investigate to what extent different number systems exert any influence upon transaction costs of economic exchange (including measurement and communication costs), the division of labour, specialisation, the roundaboutness of production and the extent of the market. In addition, it is necessary to examine whether people’s use of different representational systems, such as the Roman and Hindu-Arabic numeral schemes, affects the speed and quality of economic decision-making and the efficiency of economic calculation. What types of perceptual and cognitive biases and/or errors do different forms of external representation of prices and different computational techniques give rise to, and what is their likely frequency and magnitude? Does the form of representation of prices affect the types of information that are perceived, the perceptual and cognitive processes activated, the portions of the environment that are explored, and the arbitrage opportunities that are discovered? BIBLIOGRAPHY

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