The rise of the concept of structure
Mathematical logic is introduced in textbooks in such a way that they induce the beginners to imagine that the formalized languages descend from heaven seeking to incarnate in some earthly interpretation.
The problems of lost beings, unavailable stories, and lost content, pointed out by Muller for the 6-View, appear in such textbooks in a dramatic form. As in the case of the 6-View, there is a (lost and long) story behind the rise of such languages. Formalized languages were created originally in order to express propositions about certain mathematical domains, mainly the domain of arithmetic. Frege’s Begriffsschrift (1879), which introduced through a graphic language the first system of elementary logic, was written with the intention of determining the logical theory out of which, and by means of which, the foundations of arithmetic had to be established. With severe logical deficiencies,1 Peano (1889) characterized arithmetic by means of axioms that have prevailed thus far. The primitive terms of his system are numerus (the set of positive integers, denoted by N), unitas (the number 1), and sequens (the successor function, S) “sive n plus 1” (ibid.: 1). These terms were introduced with their usual meaning, in order to refer to the familiar positive integers and to the operation of taking the successor of any of these numbers, starting with 1.On the other hand, the development of algebra since the middle of the nineteenth century gave rise to the concept of algebraic structure. Bartel Leendert van der Waerden’s text Moderne Algebra (1949), published originally in two volumes in 1930 and 1931, makes systematic use of the concept of algebraic structure in order to organize what is now known as universal or modern algebra. This important work, in addition to that of the logicians Leopold Low- enheim, Thoralf Skolem, and mainly Alfred Tarski, set the stage for the definition of the general concept of relational structure, or model. If it were possible to pinpoint the year in which the concept of model reached its maturity, it would be 1930, or perhaps 1931, when Tarski introduced his concept of truth for formalized languages (cf.
Tarski 1930-1931, 1956: 152-278). We shall revise this concept in the next section, giving some examples of its use in the formulation of theories relevant for economics.Even though this concept of a model is useful and important for the sciences, it is well-known that it is not sufficient. The Nicolas Bourbaki group provided, in 1957, a concept of structure that gave the pattern for the formulation of a more general concept (cf. Bourbaki 1968: chapter 4). Unfortunately, the syntactic terms in which it was defined made impossible its application in the practice of mathematics. It is probably due to it that, as Corry (1992) points out, Bourbaki made no use at all of the concept for the rest of its work. This means that, even though the concept of structure had been used successfully in an informal way in the formulation of non-first-order theories (like topology), in mathematics as well in the empirical sciences, until 1987 nobody had given a satisfactory definition of the notion. In set-theoretic, non-syntactic terms, Balzer, Moulines, and Sneed (1987) reformulated Bourbaki’s concept of structure in order to clarify Suppes’ dictum that to axiomatize a theory is to define a set-theoretical predicate. Along the same lines, one year later there appeared a paper published by Newton da Costa and Rolando Chuaqui (1988), in which they proposed a similar but more detailed definition. It was given within the framework of ZFC, but it is possible to give it in a more general setting which is even useful to define categories and functors (the collection of all structures of a given type is not a set, but a proper class). I will do this in the fifth section of this chapter.2
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