Este segundo artículo lo dedicamos a David Hilbert ( nace el 23 de enero de 1862, muere el 14 de febrero de 1943). La escogencia se basa en lo siguiente:

a.)  En su libro Grundlagen der Geometrie (17 de junio de 1899) Fundamentos de Geometría. Realiza la tarea titánica de convertir - la obra de Euclides a la 

 matemática moderna axiomática – de la cual es uno de  los fundadores. Ese libro cambió radicalmente la concepción del aprendizaje e investigación en geometría euclidiana. En Estados Unidos de Norteamérica el autor Edwin Moise escribió su clásico Elementary Geometry From An Advanced Standpoint ( tiene varias traducciones al castellano) .El cual se puede ubicar comouna adaptación al inglés y en sencillo del libro de David Hilbert. Algunos profesores en la Universidad de Costa Rica usamos el texto de Moise en nuestros cursos de geometría.  Si  alguien lee los 13 libros de Euclides descubre que su lenguaje y técnicas de demostración no corresponden a los conceptos actuales. En el léxico común se han quedado algunas frases de los elementos de Euclides que muestran los problemas mencionados : la distancia más corta entre dos puntos es la línea recta (en esa frase tan corta hay al menos 5 errores graves en el uso del lenguaje y conceptos  matemáticos), el orden  de los factores no altera el producto... si la multiplicación es conmutativa. Todo estudiante en el primer curso de álgebra lineal aprende que la multiplicación de matrices no es conmutativa en general. Hace unos 42 años. Dieudonné uno de los fundadores de  la Escuela de Bourbaki al finalizar la segunda guerra mundial fue más lejos y levantó la controversial bandera:  “ Afuera Euclides”.           Un trabajo muy interesante se encuentra en la red:  http://aleph0.clarku.edu/~djoyce/java/elements/elements.html  en esta página se pueden revisar los contenidos de los 13 libros de Euclides y por medio de pequeños programas en Java y JavaScript se pueden apreciar las construcciones y como estas constituyen el concepto de demostración en Euclides..  

b) En el Congreso Mundial de Matemáticos en Paris en 1900 Hilbert presentó una serie de conjeturas y problemas que se convirtieron en los famosímos 23 problemas de Hilbert. Su influencia en el desarrollo de las ciencias matemáticas en el Siglo XX  fue total y posiblemente también en el XXI. Muchos matemáticos han ganado la medalla Fields (En honor al matemático canadiense . C. Fields)  por sus respuestas en afirmativo o en negativo a alguno de los 23 problemas.: Andrew Wiles(1998) por la solución al problema de Fermat ha sido el último . En la dirección http://elib.zib.de/IMU/medals/  se puede encontrar la lista de todos los ganadores de la medalla Fields..A la pregunta como investigan los matemáticos surge la respuesta por medio de conjeturas y problemas que se le plantean .En el caso de Hilbert  él contribuyó de manera notable al plantear problemas y conjeturas en muchos campos del quehacer matemático . A uno de los problemas el joven Goedel dió una respuesta negativa y eso va ser el tema de nuestro tercer artículo : Goedel y el Círculo de Viena

 Ahora dejemos a Hilbert  contar sus 23 problemas: 

Mathematical Problems

Lecture delivered before the International Congress of Mathematicians at Paris in 1900

By Professor David Hilbert¹

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.

The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.

It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.

Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.

The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal. They knew the value of difficult problems. I remind you only of the "problem of the line of quickest descent," proposed by John Bernoulli. Experience teaches, explains Bernoulli in the public announcement of this problem, that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems, and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne, Pascal, Fermat, Viviani and others and laying before the distinguished analysts of his time a problem by which, as a touchstone, they may test the value of their methods and measure their strength. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.

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.

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.

 1 2 3 4  5 6 7 8 9 10