Algebraic Symbolism

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    (Working paper, July 2004)

    THE CULTURAL-EPISTEMOLOGICAL CONDITIONS OF THE

    EMERGENCE OF ALGEBRAIC SYMBOLISM1

    Luis RadfordUniversit Laurentienne

    Canada

    I have often taken up a book and have talked to itand then put my ears to it, when alone, in hopes itwould answer me: and I have been very much

    concerned when I found it remained silent.The interesting narrative of the life of O. Equiana(Cited by M. Harbsmeier, 1988, p. 254)

    Abstract: The main thesis of this paper is that algebraic symbolism emerged in theRenaissance as a cultural expression of a new type of general inquisitive consciousness

    shaped by the socioeconomic activities that arose progressively in the late Middle-Ages.

    In its shortest formulation, algebraic symbolism emerged as a semiotic activity on thepart of men reflecting about some definite aspects of their world a world that, at the

    time, was substantially transformed by the use of artefacts and machines of which

    printing is, naturally, one of the paradigmatic examples. To answer the question of the

    conditions which made possible the emergence of algebraic symbolism, I will enquireabout the cultural modes of representation and look for the changes which took place in

    cognitive and social forms of signification.

    1. IntroductionThe way in which I wish to study the problem of the emergence of algebraic symbolismcan easily lead to misunderstandings.The first of these would be to understand this paper as a historical investigation of theexternal factors that made possible the rise of symbolic thinking in the Renaissance.External factors have usually been seen as economic and societal factors that somewhat

    influence the development of mathematics. They are opposed to internal factors, whichare seen as the true factors accounting for the development of mathematical ideas. Thedistinction between the internal and external dimensions of the conceptual developmentof mathematics rests on a clear cut distinction between the sociocultural on one side, andthe really mathematical on the other. Within this context, the former is seen, as Lakatos

    1 This paper is a result of a research program funded by The Social Sciences and Humanities ResearchCouncil of Canada / Le Conseil de recherches en sciences humaines du Canada (SSHRC/CRSH).

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    suggested, as a mere complementto the latter.Viewed from this perspective, it may appear that the route I am taking to investigate theemergence of algebraic symbolism belongs to the sociology of knowledge. But, as weshall see, this is not the case.The second misunderstanding may be to see my attempt as redundant. Such an attempt to

    study the conditions of the emergence of algebraic symbolism in cultural terms may seemto be redundant considering that in the last five or six years it has become a truism toconceive of mathematics as a cultural endeavour. From this perspective, it may be arguedthat there is no novelty at all in saying that, since mathematical ideas are related to theircultural settings, the emergence of algebraic symbolism is related to its own culturalcontext. Yet, I want to argue that we have thus far not been very successful in specifyinghow mathematical thinking relates to culture. I want to go further and suggest that if wedo not specify the link between culture and mathematical conceptualizations, we riskusing culture as a generic term that attempts to explain something, while in reality it doesnot explain anything.In the first part of this paper, I will outline the theoretical framework to which I will

    resort in order to attempt to answer the question of the conditions of the emergence ofalgebraic symbolism. In the second part, I will deal with the place of algebra in itshistorical setting, focusing mainly on changes in the cultural forms of signification andknowledge representation.

    2. The link between culture and knowledgeFrom the Semiotic Anthropological Perspective that I have been advocating2,mathematics is considered as human production. This claim is consonant with claimsmade by Oswald Spengler (1917/1948) almost one century ago and revitalized bycontemporary scholars such as Barbin (1996) and DAmbrosio (1996). While Barbin

    draws from Bachellards epistemology and emphasizes the role of problems in thedevelopment of mathematics, DAmbrosio draws on the work of biologists such asMaturana, Valera, Lumsden and Wilson to attempt to explain the role of culture incognition. The Semiotic Anthropological Perspective draws, in contrast, from the socio-historical school of thought developed by Vygotsky, and from Wartofskys andIlyenkovs epistemologies3.Without diminishing the importance of the role of problems in mathematicaldevelopment, the Semiotic Anthropologic Approach emphasizes the role of the humanactivities which lead mathematicians to pose and tackle mathematical problems in theway that they do. As a consequence of this theoretical orientation, in order to answer suchepistemological questions as the one that I am asking in this paper (namely, what were

    the conditions that made the emergence of algebraic symbolism possible?) the essentialelement to look at is activity.The term activity, as a unit of analysis for understanding the development ofmathematical ideas, refers here not only to whatmathematicians were doing at a certainhistorical moment but overall to how they were doing it and why. The study of thequestions (or problems) tells us why a mathematical activity occurred. The study of the

    2 Radford (1997, 1998, 1999, 2003a).3 See Vygotsky (1962, 1978, 1981), Wartofsky (1979), Ilyenkov (1977).

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    methods that were followed to answer the questions or problems tells us how thequestions were tackled.

    The concept of activity does not tell us, however, in which sense we have to understandthe link between culture and knowledge. What I have asserted about activity is good

    enough for conceiving of mathematics as a human endeavour, but it is certainlyinsufficient to bring us beyond the internal/external dichotomy of classicalhistoriography. In other words, the idea of activity expounded thus far provides room forseeing connections between mathematical knowledge and its cultural settings, but in noway tells us the nature of such connections. Without further development, theconnections cannot be explainedbut only empiricallyshown4.

    What is then exactly the relationship between culture and knowledge? This question is ofthe utmost importance here if we want to understand the emergence of algebraicsymbolism. To answer this question, I need to appeal to a key principle of the theoreticalframework of the Semiotic Anthropological Perspective here sketched. This principle can

    be stated as follows. In opposition to Platonist or Realist epistemologies, knowledge isnot considered here as the discovery of something already there, preceding humanactivity. Knowledge is not about pre-existing and unchanging objects. Knowledge relatesto culture in the precise sense that the objects of knowledge (geometric figures, numbers,equations, etc.) are the productof human thinking, and thinking is a cognitive reflectionof the world in the form of the individuals culturally framed activities.

    As we can see, activity is not merely the space where people get together to do theirthinking. The essential point is that the cultural, economic, and conceptual formationsunderpinning knowledge-generating activities impress their marks on the theoreticalconcepts produced in the course of these activities. Theoretical concepts are reflectionsthat reflect the world in accordance to the social processes of meaning production and thecultural categories available to individuals.

    What I am suggesting in this paper is that algebraic symbolism is a semiotic manner ofreflecting about the world, a manner that became thinkable in the context of a world inwhich machines and new forms of labor transformed human experience, introducing asystemic dimension that acquired the form of a metaphor of efficiency, not only in themathematical and technical domains, but also in aesthetics and other spheres of life.

    In the next section I will briefly discuss some cultural-conceptual elements of abacistalgebraic activity. In the subsequent sections, I will focus on the technological and

    4 This is the case with Eves bookAn Introduction to the History of Mathematics. In contrast to theprevious editions of the book (see e.g. Eves, 1966), in the 6th edition (see Eves 1990), a section was addedin which the cultural setting is expounded before each chapter. Connections are shown rather thanexplained. That Netz (1999) placed the cultural aspects of Greek mathematics in the lastpart of hisotherwise enlightening book, after all the mathematical aspects were explained (as if the cultural aspectswere independent of or at least not really part of mathematical thinking), is representative, I believe, of thedifficulty in tackling the theoretical problem of the connection between culture and mathematicalknowledge.

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    societal elements which underlined the changes in Renaissance modes of knowledgerepresentation.

    3. Abacist Algebraic Activity

    In his work Trattato dabaco, Piero della Francesca deals with the followingproblem:

    A gentleman hires a servant on salary; he must pay him at 25 ducati and one horseper year. After 2 months the worker says that he does not want to remain with himanymore and wants to be paid for the time he did serve. The gentleman gives himthe horse and says: give me 4 ducati and you shall be paid. I ask, what was thehorse worth? (Arrighi (ed), 1970, p. 107)

    This is a typical problem from t

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