The Wrapped Dimension of Bourbaki’s Structures Mères

Abstract

The term “structures meres” (mother structures) first appeared in a paper by Bourbaki, published in 1948, under the title “L’architecture des mathematiques”. It was introduced to name a kind of elementary building block for the “architecture” of mathematics, but in fact Bourbaki’s set-theoretic presumptions prevented the concept of structure mere from playing anything beyond a token role in Bourbaki’s Elements de mathematique and so its potential remained undeveloped. A general, principled, concept of structure and the existence of structures meres in mathematics together form a characteristic feature of Bourbaki’s species within the genus called “structuralism”, which, after its emergence early in the 20th century, flourished in many areas of research in mid-century, before undergoing a swift decline. The renaissance of structuralism in the philosophy of mathematics which has taken place over the past two decades is different from Bourbaki’s species for two main reasons: it primarily pivots on category theory to capture an even more general, but no less principled, concept of structure and does away with structures meres altogether—a notion which from the outset is inconsistent with a purely relational view of mathematics. Usually, discussions about Bourbaki’s species abstract from its genus. Here it is argued that this approach is not profitable, for the new structuralism resurrects problems which the earlier structuralism without structures meres failed to solve. So, as not to run the risk of repeating that failure, further arguments are in need. Refined arguments have been provided in support of the new structuralism, but they do not avoid that risk, since they fail to identify the meaning-roots of the notions used to understand structures, failing in particular to address the structure grounding problem. To avoid the risk, we shall employ a method of addressing the problem which calls for the existence of structures meres, defined in a suitable fashion. What follows is a continuation of a research program which, begun within a phenomenological perspective on the foundations of mathematics, is intended to take advantage of the pluralistic overview suggested by John Bell, and to meet Bill Lawvere’s philosophical demands. Indeed, Bourbaki’s claim that structures meres exist was from the start accompanied by an irreducible plurality of kinds-of-structure, which at first blush is at odds with Bourbaki’s set-theoretic assumptions. Such a plurality was open to either a gestaltist or a merely taxonomical reading but neither could suffice, as structures meres call for an inherent generativity. Thus arose a grammar of universal “schemes”, the term used in the present paper, as prime patterns of structuration, which, made precisely explicit in mathematics, play an active role in every science. Section 1 introduces the motivations for Bourbaki’s structuralism and points out the issue raised by a set-theoretic definition of structure. Section 2 deals with Bourbaki’s attitude towards foundational problems: from the lack of attention to logic to the decision to leave category theory aside, while identifying some of its basic notions and employing it in the solution of problems (e.g. in algebraic geometry, by Alexandre Grothendieck). Section 3 relates Bourbaki’s relationism to historical approaches contrasting substantialism, conceptual atomism and compositionality. Section 4 offers a quick look at the structure-centred relational approach to general linguistics since Saussure’s Cours in 1916 and then the spread of structuralism to many other fields. Section 5 reports on the “golden age” of structuralism in continental Europe and some of the general objections this view provoked. Section 6 examines the connection between structuralism and the view of axiom systems as implicit definitions of the concepts axioms express, with reference to classical debates such as those which opposed Poincare to Russell and Hilbert to Frege. Here it is argued that, when the concept of structure is intended to be implicitly defined, the view lacks explanatory power. Section 7 addresses the “structure grounding problem”, suggesting a perspective quite different both from formalism and realism, as it rests on kinesthetic schemes whose action is expressed in category-theoretic terms. Finally, Sect. 8 focuses on the semantic roots of the word “structure” to emphasise the primary role of spatial interaction patterns as generators of every variety of abstract structure.