Abstract Functions

The following work intent to analyze the mode of existence of a few mathematical theories. As a starting point, we would take the word theory the naive way, mingled with some back thinking. We give ourselves the right to name theory and to use, even objects which fundamental behavioral structure hasn’t been properly observe and defined. Cauchy couldn’t catch the notion of uniform structure, but we know the expression theory of convergence in Cauchy meaning. In the same way, Euclid never defined neither group structure or the idea of body, but we give ourselves the right, for instance, to use the idiomatic Euclidean theory of proportions. Provisionally, we’ll call theory an open system of compatible propositions, naming and linking properties in a closed object domain related to some operations or relations explicitly formulated. It is to say that the standard of compatibility remains, also here, a simple idealized presupposition regulated and verified on every day base, in the merging of acts by which the mathematician chooses among useful hypothesis. Our main field of investigation would be constituted by the body of the doctrine designed under the name real variable function theory. Several reasons guided us to that choice. Le first is the « antiquity » of the theory. Besides the facts that the Greek mathematicians did not define the general concept of function, they, nevertheless, considered operations where field of validity was delimitated by objects equivalent of our body of real numbers ( negative numbers excluded, naturally). One, even, can say that real numbers usual topology, initial definition domain where are builded the real variable functions, is already present, if not truly expliclted while Eudoxe conceive un general theory of proportions including incommensurable magnitudes.( see the famous definition 5 in Euclide’s Elements, book V). Furthermore, if this theory allows the use of exhaustion méthod to solve squarring problems, we can bring back to Archimedes, the real variable function theory birth certificate. From there, the concept of definite integral is put in motion by the developpement of algebraïc geometry which found its roots, in the first place, in the domain of the reals. These premises permitted calculus operations and computing ; integration, differenciation, serial developpement and also the continuity and convergence caracteristic concepts , for instance.. The second reason of our choice is linked on more organic grounds, to the nature of the theory, to the rôle, it plays since the beginning of the XIX° century, in the analysis principles deepening et enlightment. One knows since Gauchy (to fix a starting point) that a double generalisation process has affected the functions theory. One consisted in spreading to the complex variable field, operations already defined in the reals field. Unless this extention was only used for polynoms or analytically individualised defined transcendental fonctions such as Passage from the reals to the complexes have no need for more than the substitution of x, validity of computing rules defined by the body of complex numbers and their compatibility use to pass as a guarantee of the extension operative possibilities. The well knowned Euler’s theorem testifies of the operative integration of the real field to the complex field. Troubles uprising when the problem extent to the field of the complex variables for arbitrary functions. It is to say that for expressions of the form y=f(x), where we leave f absolutely undetermined. f means that between x and y has been established some mode of correspondence. If at the variable x one substitute the variable z=x+iy, by which conditions would it be possible to extend at f(z) the classic operations of infinite calculus. Particularly, how to define the admissible conditions for each point of the function to bear a sole derivative. One know that these problems inherited from Cauchy and Weierstrass, drove to insulate à "privileged" class of functions: analytic functions undefinitely derivatives in each of the points of the domain where they are defined and representable by Taylor’s development of converging series. It’s for this reason: the analytic character (and not only for the single reason of formal calculating rules compatibility in the complex body). It just happened that the usual algebraic and transcendental functions, by a simple substitution of variables, fitted to the complex field extension. And so the endeavor to generalize came out as its converse: "generalizable" operations restricted to a limited class of functions. This bordering of possibles demanded much attention to topological and metric properties in the analytic functions definition domain. This doesn’t question analysis principles; well at the opposite it meets in the field of analysis a safety domain: a canonical area. It did not go that way for the second movement of generalization, Standards for development in mathematical physics.(partial derivative equation integration « of the vibrating strings ») and the necessity to balance operations system and object field drove the analysts of the beginning of the XIX° century( mainly Gauss, Cauchy, Abel, Dirichlet) to as ask about the problem of extension to arbitrary classes of functions of a real variable, operations usually practiced on the eulerian continuum function : integration, derivation, representation by development in trigonometric series. From Leibniz to Lagrange, indeed, (to catch the ideal) the practice of « computing » has deployed in a field whose objects were reachable and operational under a norm proper to already constituted operations : it was well understood that the « reachable functions » should and could be writable. The expression « y=f(x) » was only to name an undetermined class of analytical expressions (algébraïc or transcendental) defining the mode of correspondence between the system of « values » of « x » and the system of « values » of « y » Accessible functions were, at once, faced as « normal » functions : so to say functions whose properties must have been dominable in a normed field of algebraic operations. It was, then, a idealized common place without which the theory of functions could not have constituted itself. The areas of the operational field who were dominated the firsts ( class of the continuous derivable functions) were thematic and posed as exemplary analysis domain.(cf. Hermite’s late loath when he faced the continuous functions without derivative. « Woeful wound » did he say. But this predicate one’ s standardized, restrains the practical analysis to continuous functions or to functions only admitting a finite amount of discontinuity points. Now, the ideal of arbitrary function use to implicate that by the expression y=f(x), one understand nothing but the operation f pure concept where, to the objects system « x » correspond the objects system « f(x) ». No more restrictions ( implicit or explicit) are imposed, to « x » neither to « f ». At the instant where the needs of analysis technical (cf. The problem of computing the trigonometric series coefficients, for instance) requires that we dispose of the pure concept of such a correspondence, one can say that the idealizing presupposition formulated above, cease to behave on the permitted analytic field, as a standard function. While Dirichlet pronounce the importance to, from now on, « substitute computing for ideals »(1), he states the burst of already constituted operator norms. Generalization and extension analysis operations to more and more general classes of functions, soon calls for a overview ab ovo regarding the validity conditions of these operations themselves freed from the domain where they were defined first. To take seriously the general character of the relation y=f(x) was an all other thing than substitute the real variable « x » by the complex variable « z » and ask oneself at which conditions F(z) remains, for instance, derivable at each point of its domain. The general relation y=f(x) could as well be replaced by the assembling of signs like ? = * (x) in which « * » stands for any correspondence between the « x » and the « ? » the domain properties undetermined field defined by the correspondence. To introduce determination in such a domain, was to reverse, in analysis, the traditional follow-up gestures of mathematical invention. It was out of question to let one be guided par easily dominable properties of a privileged class of functions. But, if we begin by the pure concept of functional correspondence, the problem was to precise the hypothesis statable on this mode of correspondence and as second momentum, to generalize it. In the double movement of specification and widening of hypothesis system, was enclosed the validation domain of « classic » operations of computing. For instance, the extension of operation « integration » from Cauchy to Lebesgue. From any function one can acknowledge the possibility to integrating it. Le first specification consist to attribute a bounded amount of discontinuities to an arbitrary function; the second one is more general. The discontinuity points ensemble in infinite, in this case, one can integrate the function if the measure of this ensemble is null( that is to say that one can enclose its points in a arbitrary short total length serie of intervals. The third one is even more general : any discontinuity points ensemble is infinite. For instance nothing impeach that in the neighborhood of each defined point, the function ll’be bearing a innumerable amount of discontinuity points(2) An integrability condition demand that this function is measurable in Lebesgue’s meaning. This last condition can’t afford by itself to bound and to close the integrable functions class. Daniell defined still more general functions which mean functions, in the first place, not measured in the abstract space in which they are builded. From this moment, the object « integral » appears in all its generality : the linear function defined on the elements of an abstract space.(3). As this development is going, a new object has been put on the work bench, we had to be heedful, first, for a undetermined starting point function, to the distribution mode of its possible points of discontinuity in the domain where it is defined. A points infinite ensembles analysis mean was needed for this purpose. So could be studied the incidences of this ensembles properties on the analysis operations extension conditions of such function (for ex.Integration). This brought us to isolate specific object, That was not algebra operative field anymore, but the class, at a glance, indefinitely open of most general functions definition domains. The study of the functions, as so, subordinate to the « space » ones on which are defined corresponding modes susceptible to permit their building. After the class study of discontinued functions,the study, for an arbitrary function and its representation conditions by developing it by converging Fourier’s series, demanded a the constitution of a analysis specific instrument able to dispatch the domains structures where was defined the « variables ». The Cantorian creation before develop for itself, was conceived as a tool. The theory of a « set of points » was originally destined to elaborate rigorously the criterions permitting to treat the most general functions ( and then, in principle, discontinued). By neutralizing their set of discontinued points. One had to ask to following question : what’s the amount of the set own’s structure being a impediment to the treatment of The function by the usual operations seen above(4). In which way such a structure authorize the adequate bends to maintain, the initial operation into its whole formal generalization ? It became necessary to isolate en try to think as a mathematical object ( that is to say a system accessible by a row of compatible and regulated operations) « pure material », the undetermined cloth in what was defined the most general functions By the nature of its domain, real variables functions theory problems and methods found itself in the heart of the movement, which, since the beginning of the XIX° century, induced the re-opening of the mathematical building stand. Diving in the roots of its remotest past, in the elementary operations and in the rich structures offered by the still naive ma thematic, it was the place where gathered and merged purer and poorer structures necessary to the undertaking of a rigorous mathematic. It was also the proofing bench where these structures construction laws has been invested, actualized and verified. At a point that today, with a, from now on, classical theory, a young mathematician could not lean it without a minimal mental luggage. This toolbox do not contain anymore as in around 1850, calculation technics, but the abstract theory of sets, the general topology principles and linear transformations theory elements. Hence, We’ll say that if mathematics offers, in a pure case, ideal objects existence mode dimensions. The real variables functions theory offers, in a equally pure case, all the mathematical idealizations existing mode dimensions : related to the origine, significant opening on the theory archeology, building gestures linked regulated movement, abstract structures merging and investment. Pure material theory thematic. As much « moments » seen and asked to be thinked together in a constitutive movement of mathematical effectiveness Empiric history (Chronological of discoveries written down in memoirs and manuals) is the raw material of our reflexions. No need to say, its not our goal to step into, even partial functions theory historiography. The epidemiologist is in business, first, with his own science but his trade is not to pursue it, day by day, in the details and sinuosities of its features. The dated pieces are for him, an object of analysis. He puts them back to work for his own account aiming to bring to light the constitutional process and the linking mode of specific concepts of the science he’s busy with. The problem asked to him, at the starting point of his search is then to know which pieces to chose and the moments to attach oneself. This choice can’t be fully arbitrary neither totally regulated. It is not arbitrary because epistemology is a historical subject. He receive the science phenomenon in a given form, in a timely perspective that he do not have the power to transform to his will. He is either not rapturously determined : It would be needed, for this, at the beginning, to apprehend a thorough vision of historical becoming essence and be able to approach, it the constitutive relation of that essence, each off the becoming moments. Now, by the fact he ‘s himself a subject of history, situated in the timely skyline which envelop its object, the epidemiologist can’t, on his first steps, pretend to possess such law of essence.-, even if he believes in the ideal possibility to reach it. We’ll be wrong, anyway, to let ourselves intimidate by this apparent « impoverishment » It better serve us like a guide inasmuch as it testifies on the science historic forthcoming phenomenon. We receive science from the outside as a product. We have to learn from the product itself to decrypt and to read the follow-up of productive acts. We distinguish ourselves from of empirical historicist in this : him ranges the product in its entice positivity ; the product is there like a thing showing in time after other things. To it time is minus or at least a lineal succession empty pattern waiting to be fulled. All gap worries him : he must fill it. From the moment, he feels he’ll got or judge by himself that he possess this timely fulfillment without a crack, he find his rest in presence of concrete history itself. What’s to understand afterward is not his business anymore. It goes an all other way for us. We tackle the product by its converse mediation, that mean we take seriously our situation of historical subject, accepting, in a same move, to exercise the rights and accept the servitudes of it. Our right is of First Sight, our servitude the one of the last arrived. To hold together these two requirements is the duty and proper difficulty for someone who tackles the phenomenon of cultural becoming with the design, starting from nothing , to decrypt the meaning of his movement. « First sight », he board the product there, already though and reflected, as what’s ask to be thinked. The object, (as such Riemann’s memoir, for instance), is again processed as a learning act of a conscience who must move in it, as in a non reflected new domain, waiting, to be, to reach its cultural object status, to be seen. While he perform this glance right, the historical subject, perceiving himself as a philosopher and telling so, come along the problem where live the product as a simple dimension of his reflex ion field. He can accost it in full freedom as the field objects are reflectivity equivalents ; the same operation which permit the subject to relearn such or such Riemann’s memoir, induce him to put back on the bench such Archimedes treaty. From one to the other, the connection is only to be found in the reflexive field. The one (the philosopher), enjoys here, an ideal and rooted-up ubiquity. If he doesn’t use that right, the object remains ; for him, abolish and deaf. It would stay that way if the subject do not seek for the product by the timely mediation, where he is himself situated. Time offered « conversely », showing itself in the instant called now. And the first sight inhabiting this instant is always the « last arrived » in history and in time. The philosopher can’t do anything else but exercise his reflex ion. But while verifying, in the field where is discovered their correlation, thinkable as well, the subject seemingly appears « time free », but he isn’t ,he did not escape time. On the contrary, assigned in tense, he must face it. He starts from his side, his side of the time where he see his reflexive field opening and organize in a perspective the meaning encounters that he discover little to little. It’s impossible to discard the common place saying Archimedes couldn’t be readied before the discovery of infinitesimal computing as it was after. It testifies of the objects organization mode inside the reflexive field. To discover the meaning of the product is, for the historical subject, to let oneself guided to the characteristic product by taking his departure in the timely horizon where he is assigned. In the. While doing the job, he must undo his own horizon, see the opening of the announced reality, accept ot be guided par the demand, living each time, in the core of so dissociated moments. Rupture the present face of reason. Dismantle the mind perpetual today, is then, the initial task of the historical subject who would understand the motion which brought him there, at the point from where is given to see the non dominated open contain of already done science. That’s why it has been assert above that one could board the product by the mediation of its negative. Through the achieved form, the balance shape in which it surrender itself, it testifies on its own underachievement, on the manner he missed its own past. It’s this testimony that one have to listen and translate. The « impoverishment » signaled before, far from being an obstacle, prescribe then the approach chaining. The first, should consist to learn, of the theory, our object, to read its history. It always remains in our power to choose the moment of its grown where we can move in. But whatever is this moment, it would always assert itself as a balanced figure of this moment and that’s the figure we should submit to analysis. That is to say fragment and dissociate so it’ll be possible to epitomize the distinct timely horizons in which the so obtained elements propose themselves as the objects of a constituted mathematical business. From this path open do not depend of us anymore, but of the object itself, of the connections which made possible its learning mode and eventually exacting more wealth. The initial choice, always arbitrary, so front the ruled domain which, step by step, guide it and standardize it. The duality of both, start and end, points is define by the arbitrary functions. And our « one way ticket » the yet theory calls for return : effective history in which, if the initial analyze has been correctly driven, one should a see the mapping of dissociated structures and the paced implicated timely horizons connection. Here, we’ll on our one way travel just for a short journey. We leave(arbitrarily) from the theory in its present : offered to a maturity moment in the unity of its principles and the strict rigor of its methids. But in a present already far in off from our sight so we can retrospectively see the arising of problems which arrival ask one more effort of refinement and generalisation. Chronologicaly, this instant appeared in the years 1920-1930. Finding its expression in some treaties, classics today : for instance : H.Hahn, Theorie der reelle Funktionen(1921) ; Constantin Caratheodory, Vorlesungen über reelle Funktionen (2° ed.1927) ;E. W. Hobson, The theory of functions of a real variable and the Theory of Fourier’s series(t.I, 3° ed., 1927 ; t.II, 2° ed., 1926 ;the 3° edition of tome I includes corrections ans additions to the 2° edition of the tome II).Let us be guided, in our exam by the last of these treaties which open a wider field than the precedents ; il contains and systematize the results of the whole movement which, from Cantor to Denjoy, contributed to the theory formation ans so offers us the organic wholesomeness which must be analyzed. In this totality, we’ll choose a few elements ; objects-ideals, objects-theories. We seek which conscience modalities offers it as as much usable and chained idealizations, which mediation’s delivers them in the core of an ever open and nevertheless subsisting system. These steps constitutes the preliminaries able to assign a common starting point to idealizations sciences epistemology and to its history. To begin, we’ll remain faithful to a certain amount of naive presuppositions, we assume the mathematical works city existence, a practically non dominable ideal library, but bonded. We suppose that this order contains a classified succession, in respect of their chronology. Ourselves are aware of this library existence and we possess some keys to gain access to it. At least, in its most recent parts, the tong of the contained books is understood. We bare the freedom to move, at our will, it them and to decrypt their « messages ». These « messages » consisting in in formations that the library delivers on itself. When we begin to use that freedom , we become to be the servants of the « library ». We state that the books aren’t mute on one and another. Inside the chronological order (that we aren’t untitled to abolish) we see meaning relations instituted. In our « works city », we gaze at the constitution of « under-universe », organic unities in which cores,some works ordinates along mutual expressions links., inside their own thinking and learning domain. Following the indications, each time, presented by these back feeds, we, not only, climb back the « course » of time. We see it breaking apart : appearing and dismantling these « under-universes » and we give ourself reversely, the show of a discontinued genesis. Finally, in case while, in our free journey in the city, we’ll find a unreadable area, we’ll pose the hypothesis that, since the library isn’t mute on itself, it must contain, in some points, an access road to the meaning of the area. The use of apology, do not have, here, any other goal, than confirm our manifesto : approach the mathematic as a cultural phenomenon, as a business which essence is to be inhabited by the bonded universe of its self generated signs in a relation that one can not break the roundness. Learn to decrypt through the manifest meaning, offered by the underachieved net, with its always total and present connections which make the life a such a universe. Isn’t the main task of the epidemiologist ? And it does not matter, if, on his way, breaks the empirical and fragile picture, that while beginning, he still kept, regarding the flow of things, of history, of the subject and of time.

(1) Cité par L.Brunschvicg, Les Etapes de la Philosophie mathématique, p.339
(2) For instance in the rationals ensemble characteristic function (0,1)
(3) Of course the Daniell’s method ( the use of step-fonctions produces a measure of the considered space. Cf. The note of S.Banach ( in the appending to the Saks’work, Theory of integration, 1937, and also in Zaanen ( Introduction to the theory of integration, § 13 (1958), the initial ideal of this generalisation goes back, perhaps to Frechet, on the integration of a extended functional to an abstract ensemble, see the bulletin of France’s mathematical society, 43, 1915, p.249-267. Daniell’s memoir date from 1917-1918.
(4) about this, the famous Hankel’s memoir consacred to the discontinued functions classification : Untersuchungen über die unenlich oft oscilirenden und unstetigen functionen. (Math. Annalen, t. 30, 1882, writen in 1870

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