Sigo leyendo y comentando la conferencia de Hilbert:
Fermat had asserted, as is well known, that the diophantine equation
xn + yn = zn
(x, y and z integers) is unsolvable—except in certain self evident cases. The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. For Kummer, incited by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors—a law which today, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.
Fermat, Kummer (números ideales), Dedekind (gran creador, ideales), Kronecker (siguió un camino parecido): toda una cadena para explorar. Ahora Hilbert pasa a un problema relacionado con la física:
To speak of a very different region of research, I remind you of the problem of three bodies. The fruitful methods and the far-reaching principles which Poincaré has brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.
The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason, belonging to the region of abstract number theory, the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.
But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.
Sobre el trabajo de Klein ver
Lectures on the Icosahedron (Dover Phoenix Editions)
On Klein's Icosahedral Solution of the Quintic
In order to throw light on the importance of certain problems, I may also refer to Weierstrass, who spoke of it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobi's problem of inversion on which to work.
Sobre el tema Weierstrass y el problema de inversión de Jacobi, ver:
"Algebraic truths" vs "geometric fantasies": Weierstrass' Response to Riemann http://arxiv.org/pdf/math/0305022v1.pdf
Karl Weierstrass (1815–1897) una muy interesante y corta biografía, donde se menciona el problema
Hilbert quiere decir: los problemas importan, vienen de distintas fuentes, y muestra el vigor de las ramas de las matemáticas.
Angel "Java" Lopez