The Notion of Structure

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What is to be understood, in general, by a math structure! The common character of the different concepts designated by this generic name, is that they can be applied to sets of elements whose nature has not been specified.

We take here a point of view and do not deal with the thorny questions , half-philosophical, half -math raised by the problem of the "nature " of the main "begins" or "object". Suffice it to say that the axiomatic studies of the 19th and 20th centuries have gradually replaced the initial pluralism of the mental representation of these beings - thought of at first as ideal "abstractions" of sense experiences and retaining all their heterogeneity - by an unitary concepts, gradually reducing all the math notions, first to the concept of the natural number and then, in a second stage, to the notion of set. 

This latter concept, considered for a long time as "primitive" and "undefinable", has been the object of endless polemics, as a result of its extremely general character and on account of the -very vague type of mental representation which it calls forth; the difficulties did not disappear until the notion of set itself disappeared (and with it all metaphysical pseudoproblems concerning math "beings" in the light of the recent work on logical formalism. From this new point of view, math structures become, properly speaking, the only "objects" of math.

To define a structure, one takes as  given one or several relations, into which these elements enter (in the case of groups, this was relation z = xƮy between three arbitrary elements), then one postulates that the given relation or relation, satisfy certain conditions (which are explicitly stated, and which are the  axioms of the structure under consideration).

In effect, this definition of structures is not sufficiently general for the needs of maths, it is also necessary to consider the case in which the relations which hold not between elements of the set under consideration, but also between parts of this set and even, more generally, between elements of sets of still higher "degree" in the terminology of the "hierarchy of types".

Strictly speaking, one should, in the case of groups, count among the axioms besides properties (a), (b), (c) stated above, the fact that the relation z = xƮy determines one, and only one z, when x and у are given; one usually considers this property as tacitly implied by the form in which the relation is written.

To set up the axiomatic theory of a given structure amounts to the deduction of the logical consequences of the axioms of the structure, excluding every other hypothesis on the elements under consideration (in particular, ev­ery hypotheses as to their own nature).

Each structure carries with it its own language freighted with special intuitive references derived from theories from which the axiomatic analysis has derived the structure. And, for the research worker who suddenly discovers this structure in the phenomena which he is studying, it is like a sudden modulation which orients at one stroke in an unexpected direction the intuitive course of his thought and which illumines with a new light the mathematical landscape in which he is moving about.

Let us think—to take an old example—of the progress made at the beginning of the 19th century by the geometric representation of imaginaries. From our point of view, this amounted to discovering in the set of complex numbers in well-known topological structure, that of the Euclidean plane, with all the possibilities for applications which this involved; in the hands of Gauss, Abel, Cauchy and Riemann, it gave new life to analysis in less than a century.

Such examples have occurred repeatedly during the last fifty years; Hilbert space, and more generally, functional spaces, establishing topological structures in sets whose elements are no longer points, but functions; the theory of the Hensel p-adic numbers, where, in a still more astounding way, topology invades a region which had been until then the domain par excellence of the discrete, of the discontinuous, viz., the set of whole numbers; Haar measure which enlarged enormously the field of application of the concept of integral, and made possible  a very profound analysis of the properties of continuous groups.

All of these are decisive instances of math progress, of turning points of which a stroke of genius brought about a new orientation of a theory, by revealing  existence in it of a structure which did not a priori seem to play a part in it.

What all this amounts to is that math has less than ever been reduced to  a purely mechanical game of isolated formulas; more than ever does intuition dominate in the genesis of discoveries. But henceforth, it possesses the powerful tools furnished by the theory of the great types of structures; in a single view , it sweeps over immense domains, now unified by the axiomatic method which were formerly in a completely chaotic state.