# Paul Painlevé's plenary lecture at ICM 1904

Paul Painlevé delivered the plenary lecture The modern problem of integrating differential equations to the 1904 International Congress of Mathematicians in Heidelberg on 11 August 1904. We give a version of the first section The problem of integration in the old analysis.

The modern problem of integrating differential equations

by Paul Painlevé, (Paris).

The problem of integration in the old analysis

The theory of differential equations was born with infinitesimal calculus. It was the scientific study of natural phenomena which had guided Newton and Leibniz, like their predecessors, to their definitive discoveries: once the two fundamental notions of integral and differential had been acquired, it was again the study of nature which was to direct its first applications. Thanks to these two notions, the experimental method, interpreted and analysed, was going to give all its fruits. This is because, through finite phenomenon - always complex, grossly simplified - differentiation reaches the elementary phenomenon; it decomposes any modification in time and space into a combination of infinitesimal modifications, I mean infinitely small modifications of interest only to infinitely small particles of matter. It is by differentiating that Galileo deduces the laws of gravity from his experiments on the inclined plane, that Newton deduces from Kepler's laws the principle of universal gravitation. The aim of integration is, on the contrary, knowing the elementary laws of a phenomenon, to reconstruct the finite phenomenon. How to superimpose, how to sum this infinity of infinitely small modifications which compose the total modification? It is this problem, reciprocal of the first, but of an otherwise profound difficulty, which constitutes the object of integral calculus: it is translated, in general, by differential equations, ordinary or partial derivatives, (depending on whether the phenomenon depends on one or more independent parameters), and all the effort consists in integrating these differential equations knowing the initial conditions or at the limits of the phenomenon. The study of the movement of a heavy solid, that of the movement of n points which attract according to the laws of Newton [$n$-body problem], here are two standard problems of integral calculus, of very different difficulty. Leaving aside the vast field of partial differential equations, so happily renewed in recent years, I will speak here only of differential equations with one variable.

The development of the new science, hardly created, is marvellous: applying to all orders of physical phenomena the principles of infinitesimal calculus, the successors of Newton and Leibniz accumulate in less than a century the most dazzling discoveries. As they are limited, in each class of facts, to simple, rudimentary examples, which appear first, the problems they have to deal with are natural and uncomplicated, reducible to known cases [quadratures, linear differential equations, etc.]. Their imagination, always supported and guided by the real problem, disentangles with admirable insight the game of elementary operations to which the integration of the encountered differential systems is reduced. By elucidating particular types, they highlight many general properties that the future will rigorously verify: degree of indeterminacy of the integrals, role of the constants, arbitrary functions, boundary conditions, etc. The theoretical and experimental sciences develop in a close connection: all progress in analysis has its immediate repercussions in physics, and vice versa. It is the most glorious and fruitful time in the history of Mathematics, the time when it really seems that it is the key to the universe. We could not compare this influx of new truths better than the movement of a wave which occupies in an instant the wide open space in front of it and which stops at the foot of a granite girdle. The wave stopped when all that was integrable, in the natural problems, was integrated.

But all the attempts made to integrate with the help of simple operations (quadratures and others) any differential equation, had failed. It was therefore more than likely that such a reduction was chimerical. The only resource left to the researchers was to directly approach the study of the integral by well-adapted successive approximation methods. This is the effort that was required of Mathematics around the time when Cauchy's work began. It is this effort that they have attempted, and which directs, explains and justifies their development throughout the past century.

Last Updated July 2020