Abstract
The best initial approach when confronted with the task of justifying or explaining activities in any branch of science is to look back, that is, to look at the development of this branch from a historical perspective. This supposes that our predecessors were not more stupid than we are now, a reasonable assumption.
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There is a description of what happened in functional equations before 1910 in a paper by S. Pincherle in the Encyclopédie des sciences mathématiques pures et appliquées (Paris, 1912, Volume II, 1, II.). But very little is said about what Euler, Laplace or Monge did for the subject and nothing about the controversy during the eighteenth century concerning functions. In fact, it seems that a useful history of functional equations would have to study simultaneously the evolution of the concept of a function. About the history of this concept a very interesting paper was written by A.P. Youschkevitch, Arch. Hist. Exact. Sc. 16 no. 1 (1976–1977), pp. 37–85: The concept of function up to the middle of the 19th Century. In a book to appear in collaboration with J. Aczél, we include a historical chapter concerning at least the material covered in the book (Functional equations in several variables. Encyclopedia of Mathematics and its Applications, Addison-Wesley). See also the section 0.2 History, pp. 5–12 in [2].
J. Aczél, Lectures on functional equations and their applications. Academic Press, New York-London, 1966.
In a paper entitled “De usu functionum discontinuarum in analysi” published in 1767 (see Opera Omnia, Pars I, Vol. 23, p. 74), Euler made clear what he meant by a continuous curve. His definition required that all points of the continuous curve had to be determined by the same equation, as by a law: “Iam vero notissimum est, in Geometria sublimiori alias lineas curvas considerari non solere, nisi quarum natura certa quadam relatione inter coordinatas per quampiam aequationem expressa definiatur, ita ut omnia eius puncta per eandem aequationem tanquam legem determinentur. Quae lex cum principium continuitatis in se complecti censeatur, quippe qua omnes curvae partes ita vinculo arctissimo inter se cohaerent, ut nulla in illis mutatio salvo continuitatis nexu locum invenire posit…”. (“It is already well known that in advanced Geometry one usually considers only curves which are determined by some equation or law involving the coordinates. That law is supposed to conform with the principle of continuity, in as much as all consecutive parts of the curve are closely coherent so that there can be found no change breaking the bound of continuity at any point…”. Using Euler’s definition, should the functional equation (1) be viewed as the “law” governing the idea of a “continuous” function, and therefore the relation ƒ(α)=α for rational α would imply ƒ(x)=x for all real x? To say the least, Euler’s definition was not very useful for further understanding at least till the introduction of analytic extensions, which came much later.
C.F. Gauss, Theoria motus corporum coelestium, liber II, sectio III, § 175–177, 1809 (Werke, Vol. VII, pp. 240–244, Leipzig, 1906).
A second and enlarged edition, including Lacroix’s Traité des différences et des séries, was published in 1810 and moulded the mathematical culture of many generations of analysts. The role of mathematical textbooks during that period has been discussed in J. Dhombres, French mathematical textbooks from Euler to Cauchy (to appear, 86 p.).
L. Euler, Demonstratio theorematis neutoniani de evolutione potestatum binomi pro casibus quibus exponentes non sunt numeri integri. Nova Comment. Acad. Sci. Petropol. 19 (1775), pp. 103–111. (Opera Omnia, Pars I, Vol. 15, pp. 207–216). There are various other attempts in Euler’s works to prove Newton’s binomial theorem. The vicious circle of the use of fluxions or derivatives was already noticed by Colson. (See page 308 of his book: The method of fluxions by the inventor, Sir Isaac Newton. 1736, London).
For a historical record of the evolution of ideas concerning “real numbers”, see for example, J. Dhombres, Nombre, mesure et continu: épistémologie et histoire. Nathan, Coll. Cédic, Paris, 1978.
Cauchy ”proved” that the limit of a sequence of continuous functions is continuous. Cauchy needed this for balancing the architecture of his Course as it linked the two basic concepts of analysis: continuity and limit. In 1826, Abel (see [10]) tried to correct Cauchy’s error but he did not avoid all the pitfalls concerning these properties either. Gudermann was the first to sense the need for uniform convergence around 1837 and then the concept was used ten years later both by Stokes and Seidel. But their influence was weak and, surprisingly, it was Cauchy again who in 1853 gave the exact definition so that his theorem on the limit of continuous functions became true (Cauchy, Notes sur les séries convergentes…Oeuvres Complètes (1) t. 12, pp. 30–36). (See P. Dugac, R. Dedekind et les fondements des mathématiques. Vrin, Paris, 1976.)
N.H. Abel, Untersuchungen über die Reihe l+(m/1)x+((m(m-1))/2)x 2+…, Journal für reine und angew. Math. 1 (1826), pp. 311–339. See the original in French in Oeuvres Complètes, Vol. I, pp. 219–250, Christiania, 1881.
”I do not reject in this proof anything as false, but say that some points are obscure and not true by necessity.” D. Bernoulli, Examen principorium mechanicae et demonstrationes geometricae de compositione et resolutione virium, Comm. Acad. Petrop. 1 (1726) pp. 126–142.
J. d’Alembert, Mémoires sur les principes de la mécanique. Mém. Académie Royale des Sciences 1769, pp. 278–286.
J. d’Alembert, Addition au mémoire sur la courbe que forme une corde tendue mise en vibration, Hist. Acad. Berlin 1750, pp. 355–360.
H. Lebesgue, Sur les transformations ponctuelles transformant les plans en plans qu’on peut définir par des procédés analytiques. Accad. Reale delle Scienze di Torino 42 (1906/07), pp. 219–226.
This brings us to functional equations in a single variable. Such functional equations were studied from the 18th century on by mathematicians like C. Babbage. Babbage explained in 1813 (Phil. Trans. 105, pp. 389–423), “I am still inclined to think that the evolution of functional equations must be sought by methods peculiarly their own”. The fascinating domain of functional equations in a single variable has been strongly developed in the last few years leading to a reinterpretation of older results and bringing new links to many mathematical theories. We may quote here, G. Targonski, Topics in iteration theory. Studia Math., Skript 6, Göttingen-Zürich, 1981, and of course, M. Kuczma, Functional equations in a single variable. Monografie Matematyczne, Tom. 46, Warszawa, 1968.
The paper by Reynolds appeared in 1895 in the Phil. Trans. Roy. Soc. London (pp. 123–164). Contributions came from M.L. Dubreil-Jacotin, J. Kampé de Fériet, G. Birkhoff. For references, see G.C. Rota, Reynolds operators, Proc. Symp. Appl. Math. Vol. 16 (1964), pp. 70–83,
and J. Dhombres, Sur les opérateurs multiplicativement liés. Mémoire Soc. Math. France no. 27 (1971), 156p. For a nonprobabilistic approach to turbulence theory, see for example,
J. Bass, Les fonctions pseudo-aléatoires. Mémorial des Sc. Math. Fasc 153, Gauthier-Villars, Paris, 1962.
For further references, see J. Dhombres, Some aspects of functional equations. Dept. of Math., Lecture Notes, Chulalongkorn University, 1979.
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Dhombres, J.G. (1984). On the historical role of functional equations. In: Functional Equations: History, Applications and Theory. Mathematics and Its Applications, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6320-7_3
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