DIARIO DE CAMPO N° 07 Área : Comunicación

the texts we analyse are English words or groups of words, notations and combinations of those, as well as figures.

The 'content of an expression' in a given text is what makes series of typographical marks meaningful in this text. The content of an expression is determined ideally by all the relationships that exist between expressions in a given text. For example, the series of marks 'the consumption set of consumer i is closed' is not meaningful if it is considered as '/the consumption set of cons’um.er i/ is closed', because in the English language a set of alphabetical marks can not be closed. In this context, the content of /consumption set of consumer i/ is that which makes it closed. Contents of expressions {e.g.: the content of /consumer/) are written in square brackets (e.g.: [consumer]). The contents of expressions are defined in connection to one another. In the sentence just mentioned, [the consumption of consumer i] comprises the grammatical identification of the expression as a subject since the sentence is grammatically correct. The contents of the expressions of a given text are inter-connected. A relationship between the expression and the content of a set of marks is called a semiotic function. The set of an expression, a content of this expression and a semiotic function define a written sign.

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Methodology of the Thesis

Both the 'level of expression’ and 'level of content’ apply to a text. The 'level of expression' of a text consists in the expressions of a text together with the relationships they have with one another, considered as expressions. For example, the numerical series (|i^) % defined by = number of graphic marks of the ith expression of the text, belongs to the 'level of expression'.

A 'level of content' of a text consists of a set of signs whose contents are interrelated, together with this relationship. A text may have different levels of content if different relationships can be identified between the contents of its expressions. We claim that because the expressions we identify in the texts are words, groups of words, notations and figures, it is impossible to know the difference between a mathematical and a literary extract of a text by simply studying their level of expression, except in a weak quantitative sense. Mathematical texts use words and sentences just as

literary texts use notations and figures. They might differ only with respect to the quantity of each kind of expression they contain, but we do not consider that this difference is significant. Consequently, the difference between mathematical and literary signs is a difference between their content and not between their expressions.

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Herreman (1996b) identifies an 'isotopy' in a text with a 'level of content'. We shall use 'isotopy' in a wider sense so as to refer to relationships at the level of expression and at the level of content altogether. Arrivé's (1973, p . 54) definition of 'isotopy' will be ours: "An isotopy consists of the redundancy of linguistic units related to the level of expression or to the level of content; these linguistic units may be tangible or not." (our translation)^® A text that has more than one isotopy is called 'poly-isotopic'.

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NOTES TO CHAPTER 2

1 "It may be that numbers do not rule the world, but at least they indicate how it is ruled". (Our translation).

2 Our views on the relationships between epistemology, history and methodology are expounded in Cheix(1996).

3 Nevertheless, there is one subfield of mathematics, namely abstract algebra, which is not widely used in economic theory. In particular, quotient sets of points or of functions are rarely used except maybe in connection with utility theory. They might be useful however since they provide a straightforward way to conduct a calculus on classes of "complex" objects. In addition, in some cases, they provide a way to consider the solutions of a system of equations as a structured space which is the result of an operator. However, Philippe Mongin, who is currently conducting research at the Delta (CNRS, Paris) and at the CORE (Université Catholique de Louvain), mentioned to us an article concerned with social choice by Chichilinsky et al (1979) that uses basic algebraic properties as a key argument in the proof of theorems they provide. Chichilinsky et al (1979) are using the decomposition of finitely generated abelian groups into a direct sum of a finite number of mathematically identifiable cyclical groups.

4 More generally we are interested in the cognitive position of numerical structures in the phenomenological division between the formal, the empirical and the physical. Aristotle in the Physics. book 4 (esp. from 219al0 onwards) analysed this position. So also does Beneze(1961), who studies this position in 'experimental sciences'. The neurosciences might also provide material for defining this place. Studies into the connection between the performance of mathematical calculus and neurophysiological functions involving language were indeed reported at the Third Annual Conference of the European Society for Philosophy and Psychology (Paris, 1-4 September 1994),

This is important for the methodology of this thesis insofar as the interpretation of mathematical texts is concerned. We believe that sometimes mathematical entities play a psychological function in reasonings. This function, which is similarly played by the use of regular notations, is that of focusing the attention of the reader. This is the case of [0, 1], in real vector space analysis and in probability theory, and it might be the case of 'O' in maximisation problems. For example, in integration analysis, the point of the proofs of general theorems is often to come down to solving a problem involving [0, IJ; then the proof of the original result is obtained thanks to structural algebraic properties. To this respect, [0, 1] has "psychological" properties. The study of Cantor ' s Set can be considered in a way to set apart these properties from the properties of [0, 1] considered as a "purely" mathematical entity.

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^ These contributions are sometimes under-estimated. Franklin(1981) deplores this lack of acknowledgement of each discipline to the other. According to Popescu (1964) and Etner(1987, pll5), the history of economic thought usually neglects the contribution of business economics.

Here are five examples which illustrate this point. One is the mathematician Giuseppe Peano (1858-1932). Peano is an important figure in the history of mathematics. He is most well known for his contribution to the axiomatization of arithmetic. Peano's interest in the foundational issues of mathematics is a feature of his work as a whole. By foundational issues it shall be understood at the same issues about mathematical logic, issues in mathematical logic and issues about their representation through symbols (Eco (1992) also consider that he is not a minor figure in the history of linguistics. As far as his contribution to economics at large is concerned, we have not noticed that it has been referred to in the literature of the history of economic thought (cf. Chapter 3). In addition, Kennedy's bibliography on Peano (Kennedy, 1980, pp.211-215) does not refer to any specialised study on this aspect of Peano's work. Our information on this subject comes from Kennedy's chronological list of Peano's publications (Kennedy, 1980, pp.195-209). From 1901 onwards until it seems, 1909, he published studies of insurance systems. For two reasons it is likely that he kept an interest in applied mathematics : the first is that he published studies on numerical approximation problems later on, and the other reason is that he published an article in a periodical specialising in financial mathematics. The titles of Peano's studies on economic subjects suggest that most of them contain mathematics. Peano is not an isolated example of a mathematician with an interest in insurance problems. According to Passemore (1966, p.129) Augustus De Morgan (1806-1871) had an interest in the extension of the theory of probability to problems of assurance.

The second example which we take from Zylberberg (1990) is Louis Bachelier. Bachelier is a historical example of the contribution of an economist to the development of a mathematical tool which revealed itself useful outside economics. According to Zylberberg (1990), who is our source of information on this matter, Louis Bachelier (1870-1946) submitted a thesis in 1900 for the degree of Doctorat Ès Sciences entitled Théorie de la Spéculation for the study in economics concerned with speculative phenomena on the stock market. In this thesis, the author analyses moves on markets in terms of stochastic processes of a

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