Von Neumann: los tentáculos de las matemáticas

El 28 de diciembre de 1903 en el entonces Imperio Astrohúngaro, nacía en una familia judía el matemático John von Neumann. Es considerado como superdotado, habiendo ganado de niño el premio Eötvös al mejor alumno del país en matemáticas y ciencia. Fue crucial el judaísmo en su vida, pues estaría marcado por el avance nazi en 1933, tras la persecución a judíos, testigos de Jehová, homosexuales y comunistas; donde había trabajado en la Universidad de Princeton desde 1929, alternando sus estancias en Estados Unidos y Alemania.
Von Neumann es conocido en diversos campos de la ciencia y la tecnología, como la física, la computación, la química y la economía, hoy día cada experto en estas áreas puede aseverar que el nombre de John von Neumann le es más que famoso, y puede que hasta le sorprenda que sus aportes en los otros campos mencionados tienen la misma magnitud de importancia.

Pero si nos preguntamos quién es realmente John von Neumann, nos remitiremos a la certificación académica que se doctoró de matemático un 12 de marzo de 1926 en la Universidad de Budapest con la calificación summa cum laude. Además fue químico, pues en 1925 obtuvo la titulación de ingeniero químico en la Escuela Politécnica Federal de Zúrich. El purismo de las matemáticas que él manejaba estaba por delante de muchos contemporáneos, y esto sería una llave fundamental para sus aplicaciones en las áreas mencionadas precedentemente, abordando los problemas de forma excepcional.
Coincidió en la Universidad de Gotinga con Hilbert, quien sería considerado como uno de los más grandes matemáticos del mundo por aquel entonces. De hecho, en Gotinga se plantearía un elemento fundamental del quinto postulado del matemático griego Euclides (325-265 a.C.), dando una gran revolución en geometría luego de 2000 años. El planteamiento fue buscado por Gauss unos años atrás y dado con mayor formalidad más adelante por Bernard Riemann, la conocida como Geometría no euclidiana, que sería usado más adelante por Einstein para su Relatividad General.
Teoría de conjuntos
Entre finales del siglo XIX e inicios del siglo XX, la teoría de conjuntos y la lógica matemática estaban en una disputa abismal, bombardeos de teorías exquisitas y contraataques de genuinos contraejemplos. El programa de Hilbert no estaba ajeno a aquello.
El matemático alemán Ernst Zermelo (1871-1953) estableció un total de siete axiomas con los que no solo pretendía hacer consistente la teoría de conjuntos, sino también evitar conjuntos conflictivos como en la paradoja de Russell. Más tarde, esta teoría fue mejorada y ampliada por Abraham Adolf Fraenkel (1891-1955). Sin embargo, los axiomas de Zermelo-Fraenkel permitía la creación de todos los conjuntos que son necesarios en la teoría matemática, pero no excluía la posibilidad que existieran conjuntos que pertenecieran a sí mismos.
Von Neumann resolvió el problemas de dos maneras, complementándose unas a otras: el axioma de fundación y el concepto de clase. Ambos aparecieron en 1928 en su tesis doctoral por la Universidad de Budapest bajo el título de La axiomatización de la teoría de conjuntos. Mediante el axioma de fundación, von Neumann construía los conjuntos de abajo hacia arriba, siguiendo los axiomas de Zermelo, de modo que si un conjunto pertenece a otro, forzosamente tiene que ser el primero en la sucesión; de esta manera, se evitaba la posibilidad de que un conjunto pudiera pertenecer a sí mismo. Este método se sigue utilizando hoy día para muchas demostraciones fundamentales en teoría de conjuntos.
Mecánica cuántica
La física siempre ha tenido fama de ser una ciencia experimental desde sus orígenes. Con los trabajos de Galileo Galilei nacen de forma significativa esta forma de iniciar teorías más generales. En el interludio se construyen hipótesis de trabajo, se definen términos y, por supuesto, se establecen fórmulas. Tarde o temprano, la física necesita a las matemáticas para consolidarse como ciencia exacta. La aparición de nuevas teorías como la relatividad y la reciente mecánica cuántica a principios del siglo XX obligó a desarrollar unas matemáticas que se adapten a nuevos paradigmas.
Había datos experimentales que no podían ser explicados dentro del marco teórico de la física newtoniana. Especialmente dos fenómenos. El primero era la emisión de radiación por un cuerpo negro para el que no se había encontrado ninguna explicación satisfactoria. El segundo era que si un electrón giraba alrededor de un núcleo debía ir perdiendo energía paulatinamente para finalmente acabar estrellándose contra el núcleo, cosa que no sucedía. Además los experimentos comprobaban que las partículas tenían doble naturaleza, como onda y como corpúsculo.
Aparecieron entonces dos teoría capaces de explicar estos fenómenos: una mecánica matricial planteada por Werner Heisemberg (1901-1976) y una mecánica ondulatoria de parte de Erwin Schrödinger (1887-1961). Solo bastaba una unificación para no entrar en conflicto ambas teorías, no obstante esto solo se llevaría a cabo estableciendo un sistema axiomático de naturaleza abstracta que fuera capaz de englobarlas.
Axiomatización de la física. Ante la pregunta de ¿se puede axiomatizar la física? Hilbert planteó esta cuestión en su famosa lista de 23 problemas planteados en París en el Segundo Congreso Internacional de Matemáticas en 1900, donde pedía investigar los fundamentos de aquellas ciencias físicas en las que la matemática tiene un papel importante; en el primer nivel están la teoría de probabilidades y la mecánica. En 1933, el matemático ruso Andréi Kolmogórov (1903-1987) fue el primero en establecer una axiomática para las probabilidades. Von Neumann axiomatizó la mecánica cuántica de manera que los parámetros que definían el estado de una partícula pudieran ser establecidos mediante los cinco axiomas que había determinado para el espacio de Hilbert. De manera que la formulación de matemática era lo suficientemente abstracta como para permanecer separada por completo de la experimentación física. Un resultado que daría un giro, por primera vez en la historia, sobre la forma de construir las teorías físicas.
Además, John se ocupó de otro problema que incomodaba de sobremanera a los físicos, dando un avance significativo en teoría de la medida. Por muy preciso que fuese cada experimento había algún tipo de error inevitable, por lo que es importante conocer la magnitud del error, especialmente cuando se estudian comportamientos de magnitud subatómica. El sistema axiomático de Von Neumann era en sí mismo una teoría de la medida que permitía describir el proceso de observación y el objeto observado como elementos lógicos a tratar en el sistema de axiomas. Para ello consideraba la medición en un carácter atemporal.
Sin embargo, el programa de Hilbert había acabado en fracaso tras la aparición de los majestuosos teoremas de incompletitud de Kurt Gödel (1906-1978), aunque Von Neumann no compartía el pesimismo de Hilbert.
Teoría de juegos
El azar interviene en gran medida, por lo general, a todo tipo de juego y decide el grado de iniciativa de los jugadores al momento de tomar alguna decisión. El cálculo de probabilidades y la estadística son teorías que surgieron a partir del estudio sistemático de los juegos, pero más en ánimo a la predicción que la propia naturaleza del juego. Entre los grandes aportes de Von Neumann a la teoría de juegos en encuentran: el juego bipersonal de suma cero, el primer teorema de minimax, la ensilladura o punto de silla, sus estrategias puras y mixtas.

Von Neumann además encontró una estrecha relación entre el teorema de minimax y la teoría de los puntos fijos desde un punto de vista topológico. Esto no solo le ayudó a demostrar el teorema en cuestión, sino que años más tarde le permitió llevar a cabo una importante ampliación del teorema del punto fijo de Brouwer.
A pesar de que no había publicado ningún trabajo sobre economía antes de 1937, el interés de John por los temas sobre economía se despertó ya en las conversaciones que su padre, un banquero, traía con frecuencia a colación en las reuniones familiares. La teoría de juegos había sido desarrollado en un ámbito puramente matemático, pero Von Neumann quería darle una dimensión más aplicable y eligió la economía como su destino.
El libro que Von Neumann escribió conjuntamente con Oskar Morgenstern (1902-1977) apareció en 1944 bajo el título Theory of Games and Economic Behavior, y está catalogada como una de las obras más importantes de Von Neumann y en la que la teoría de juegos aparece completamente desarrollada, hasta el punto de que se la considera como punto de partida de una nueva rama de las matemáticas.
Computación
El nivel de automatismo y la complejidad de las operaciones efectuadas son dos factores determinantes en el desarrollo de la computación. La diferencia entre tener que mover las fichas de un ábaco con la mano, o que esta acción la lleven a cabo dispositivos electromecánicos, señala un gran avance tecnológico.

Von Neumann introdujo el uso de tarjetas perforadas en los primeros cálculos matemáticos con computadoras. Su hermano Nicholas afirmaba que esta idea le surgió en una conversación que tuvieron en una sobremesa, cuando todavía vivían en Budapest, en casa de sus padres.
El concepto de constructor universal fue presentado en su trabajo póstumo Teoría de los autómatas autorreproductivos. El término «máquina de von Neumann» se refiere alternativamente a las máquinas autorreplicativas. Von Neumann probó que el camino más efectivo para las operaciones mineras a gran escala, como minar una luna entera o un cinturón de asteroides, es a través del uso de máquinas autorreplicativas, para aprovechar el crecimiento exponencial de tales mecanismos.
Los trabajos realizados por Von Neumann en trayectorias balísticas y ondas de expansión le convirtieron en un experto reclamado por los estamentos militares. No obstante, esto no era más que el inicio de uno de los experimentos más trascendentales para la ciencia, pero una de las más grandes desgracias para la humanidad.
La bomba atómica
Corría el año 1944. Los alemanes se jactaban de poseer un arma secreta que podía dar un vuelco a la contienda y hacer que pudieran ganar la guerra. Los alemanes se habían propuesto construir la primera bomba atómica de la historia. Tenían a su alcance todos los medios necesarios para ello, pero habían perdido la ocasión de contar con un auténtico monopolio de mentes privilegiadas.
La primera bomba atómica se basaba en un proceso de fisión nuclear de uranio enriquecido que comportaba una disminución de masa. La teoría de la relatividad de Einstein había dejado establecido mediante su conocida ecuación que la disminución de masa comporta el aumento en la energía.
Frente a la alarma que causó el saber que los alemanes estaban trabajando en la primera bomba de fisión, Leó Szilárd (1898-1964) junto con Edward Teller (1908-2003) y Eugene Wigner (1902-1995) convencieron a Einstein para que escribiera una carta al presidente Roosevelt advirtiéndole del peligro inminente. Así nació el Proyecto Manhattan. En el proyecto Manhattan estuvo a cargo del conocido físico Robert Oppenheimer (1904-1967), donde participaron científicos como Richard Feynman, Edward Teller, Enrico Fermi, Richard Wilkins, Stanislaw Ulam, Louis Slotkin, Klaus Fuchs, y finalmente Von Neumann, quien diseñó practicamente la totalidad del mecanismo de ignición, dado que llevaba tiempo trabajando en hidrodinámica de fluidos.
En Hiroshima y Nagasaki dejaría entre unas 250.000 personas afectadas (fallecidas y heridas con gravedad) por causa de las explosiones de la bomba atómica el 6 de agosto y el 9 de agosto de 1945, respectivamente. Considerado uno de los ataques más nefastos en la historia.
El ENIAC
En julio de 1943 empezó a construirse un nuevo ordenador en la Escuela Moore de Ingeniería Eléctrica de la Universidad de Pensilvania, en Filadelfia, marcando un hito en la historia de la computación. El ENIAC (Electronic Numerical Integrator and Computer) está considerado como el primer ordenador de la historia, aunque algunos piensan que la progenitura se le debería dar a Colossus, puesta en funcionamiento a mediados de febrero de 1944 en Bletchley Park, Inglaterra, inspirada por Alan Turing (1912-1984) y diseñada por Max Newman (1897-1984), utilizada para descifrar la máquina Enigma.
El ENIAC costó cerca de 8000 dólares y fue financiado por el ejército. Medía 30 metros de largo y pesaba 32 toneladas. Únicamente podía almacenar veinte números, pero su defecto fundamental era que para cambiar un programa era necesario reconfigurar sus circuitos. Además pasaba más tiempo averiado que en funcionamiento. El ENIAC adolecía de serias limitaciones.

Después de la guerra, Von Neumann colaboró en el diseño y construcción de la nueva máquina, la EDVAC (Electronic Discrete Variable Automatic Computer), que mejoraba mucho la velocidad anterior, comparada con el ENIAC que realizaba 333 operaciones por segundo, la EDVAC hacía 20 000 operaciones por segundo. Von Neumann no se limitó al aspecto estructural de las computadoras, sino que también trabajó en el diseño de algoritmos que permitieran implementar el procesamiento de operaciones matemáticas más complejas que los puros cálculos numéricos que hasta entonces se venían haciendo. Llegó a realizar algoritmo para encontrar soluciones de sistemas de ecuaciones, hallar la inversa de una matriz, encontrar vectores propios, hasta calcular máximos y mínimos para funciones de varias variables.
En los últimos años de su vida, Von Neumann supo conjugar una matemática aplicada, relacionada básicamente con la industria armamentística, con la matemática pura, en una nueva vertiente que le llevó a estudiar la estructura lógica de la reproducción de los seres vivos -autómatas celulares- y las matemáticas que rigen el funcionamiento del cerebro, considerando a este como una red neuronal que podría llegar a ser simulada por un ordenador.
Falleció el 8 de febrero de 1957 en Washington D.C., Estados Unidos.
Referencias
Enrique Gracián Rodríguez-National Geographic, La teoría de juegos, Von Neumann. Piedra, papel, teorema. España, 2013.
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