A note on Bourbaki's definition of function, in the context of Anna Sfard characterization of conceptions

The Ana Sfard influential article published in Educational Studies in Mathematics in 1991 on the dual nature of mathematical conceptions is still important to read. I recently came back to this paper while working on the conceptions of function using of the modelling framework cK¢.  Indeed there is a difference in our approaches since her approach of Ana Sfard, defining a conception as the mental counterpart of a concept, the latter being the official form of a "mathematical idea". This meaning of "concept" seems close to the usual meaning of the French word " savoir", and far from the one adopted for cK¢ -- but this is another discussion. On the other hand both approaches have in common the recognition of methodological constraints: we have no choice in order to make sense of the formation of abstract (mathematical) objects but to describe them in terms of such external characteristics as student’s behaviours, attitudes and skills (Sfard 1991 p.19).

Anna Sfard distinguishes two types of conception, operational and structural. The former is characterized in terms of processes, algorithms and actions, while the latter is "treating mathematical notions as if they referred to some abstract object" (ibid. pp.3-4). The methodological constraints gives an advantage to evidencing operational conceptions but make it delicate for structural conceptions. Following Anna Sfard, a critical indicator of the presence of a structural conception is the capacity to recognize an idea "at a glance" and "to manipulate it as a whole, without going into details" (ibid. p.4).  This emergence of a structural conception would be empirically reflected by the "attempts at translating  operational intuition into structural definition" (ibid. p.15).  Anna Sfard sees the most achieved state of development of the conception of function in "the now widely accepted, purely structural Bourbaki's definition. This simple description presented function as a set of ordered pairs and made no reference whatsoever to any kind of computational  process." (ibid. p.15)

The initial ambition of the founders of the Bourbaki group [1], was to write a treatise for the teaching of calculus (incidentally claimed to be accessible to a not so smart student obliged to work alone [2]). There is no question about the structural character of the Bourbaki’s conception of function; however its characterization by Anna Sfard (ibid. p.5 Fig.1) as "Set of pairs (Bourbaki 1934)" is a bit short. Indeed, ordered pair should have been written here instead of pair, but there is more to say. The definition of function appears in the Set theory book (ST) where it emerges, so to say, from the definition of functional relation which is a restriction of the definition of relation: 

"Let R be a relation in C  [equalitarian theory]. The relation "(ⱻx)R and there exists at most one x such that R" is denoted by "there exists exactly one x such that R". If this relation is a theorem in C, R is said to be a functional relation in x in the theory C." (ST p.48)

Then function is further defined as a set of ordered pairs under a specific condition: 

"A graph F is said to be a functional graph if for each x there is at most one object which corresponds to x under F (Chapter l, § 5, no. 3). A correspondence f= (F, A, B) is said to be a function if its graph F is a fonctional graph and if its source A is equal to its domain pr₁F. In other words, a correspondence f = (F, A, B) is a function if for every x belonging to the source A of f the relation (x, y)∈F is functional in y (Chapter l, § 5, no. 3); the unique object which corresponds to x under f is called the value of f at the element x of A, and is denoted by f(x) (or f_x, or F(x), or F_x)." (ST p.81)

By the way, the contemporary teacher may interpret the graph as a curve, and the condition as the perpendicular line criterion which is often associated to the characterization of function in Anglo-Saxon curricula. In an informal way, Bourbaki accepts here to aggregate functional relation and functional graph in one single concept: "Throughout this series we shall often use the word "function" in place of "functional graph"." (ST p.82).

Eventually, Bourbaki comes back to all definitions in the “Summary of results” of the Set theory book, with the idea of fixing terms which will be used in the remainder of the series of the treatise. He adds the following caveat as a footnote:  "The reader will not fail to observe that the "naïve" point of view taken here is in direct opposition to the "formalist" point of view taken in Chapters 1 to IV. Of course, this contrast is deliberate, and corresponds to the different purposes of this Summary and the rest of the volume." (ST p.347). The following definition of function is proposed in this context:

"Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E and a variable element y of F is called a functional relation in y if, for all x∈E, there exists a unique y∈F, which is in the given relation with x. We give the name of function to the operation which in this way associates with every element x∈E the element y∈F which is in the given relation with x; y is said to be the value of the function at the element x, and the function is said to be determined by the given functional relation." (ST p.351) 

This definition bridges the naïve (in the Bourbaki sense) understanding of function with its formal characterization. However, the word "variable", which didn't appear before in the book, is here an adjective which meaning is fixed in the first section of the Summary of results:  "variable element" means "arbitrary element" (ST p.347). By denoting "the operation", the word "function" keeps some contact with what Anna Sfard (1991 p.15) refers to as "its intuitive origin". It is close to the prototypical example of operational conception she gives in the [Fig.1] of her article, quoting Richard Skemp: "well defined method of getting from one system to another" (or computational process).

The Bourbaki construction provides an example of an explicit link between the Anna Sfard structural and operational conceptions of function. From a different perspective, it illustrates well the claim that “the terms "operational" and "structural" refer to inseparable, though dramatically different, facets of the same thing." (Sfard 1991 p.9). In this quick record of the Bourbaki enterprise to define "function", we see the coherent and explicit integration of different connotation: functional relation, functional graph, operation. In naïve words, they are facets of an object which are unified by the formal construction.  This notion of object can be easily related to that of high-level interiorization proposed by Anna Sfard.