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Structure in mathematics. (English) Zbl 0905.18001

In mathematics, the word “structure” has at least three different meanings. First, a structure means a class of objects described by axioms, such as the structure of a group, or the structure of a topological space. Second, a structure may mean an (axiomatically defined) operation, such as direct product. Third, a structure of an object may mean a class of its subobjects (with some natural operations): e.g., the class of all subspaces of a linear space, or the class of all subgroups of a group. Of these three meanings, only the first one has been formally described in the most general case [by N. Bourbaki in his famous treatise: Éléments de mathématique XXII, Livre I, Chap. 4: Structures (1957; Zbl 0100.27803)]. This formal description enables us to view structures not only as an informal term but also as a methodology (philosophy) that is in some sense alternative to the standard set-theoretic (Platonic) approach.
As a methodology, such “structuralist” approach has been very efficient: First, it enabled mathematicians to realize that linear mappings, group homomorphisms, continuous functions, and several other initially different classes are morphisms in the corresponding categories. This realization simplified and clarified many results. Second, as opposed to the set-theoretic approach, the “structuralist” approach defines many objects up to isomorphisms, without fixing one of possible representations: e.g., 3-D Euclidean space in the set-theoretical approach is the set of triples of real numbers (coordinates), while a category definition is coordinate-free, in good accordance with our intuitive geometric ideas.
However, if we try to raise this useful methodological approach to the level of philosophy, i.e., if we try to promote it as the approach of mathematics (as Bourbaki did), it often becomes counterproductive: Unlike groups and topological spaces, many useful mathematical objects (especially the objects of classical mathematics such as ordinary and partial differential equations, and algorithmic objects) are uniquely determined and thus do not belong to a general axiomatically defined class. Bourbaki’s mathematics practically excluded these objects (and the corresponding applications) from mathematics; as a result, many applied mathematicians still treat the structuralist approach as a useless game. This mutually excluding attitude (almost a split in mathematics) is really unfortunate for both sides of this split because both sides can benefit from collaboration: on one hand, structural methods can be very useful in classical problems; on the other hand, classical problems can be a great testing ground and source for new structural methods and ideas.
Philosophical conclusion: in spite of the great usefulness of structuralist methodology, the ideal philosophy of mathematics should not be extremist structuralism: it should combine its useful features with useful features of Platonism and (algorithm-oriented) constructivism.

MSC:

18-03 History of category theory
00A30 Philosophy of mathematics
00A35 Methodology of mathematics
01A60 History of mathematics in the 20th century
18C10 Theories (e.g., algebraic theories), structure, and semantics

Citations:

Zbl 0100.27803
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