Angel "Java" Lopez en Blog

Publicado el 14 de Mayo, 2016, 7:06

Hace unos años, mencioné en un post al libro de Hermann Weyl, muy conocido, The Theory of Groups and Quantum Mechanics. Hoy lo vuelvo a leer, y encuentro este fragmento al principio, que quiero comentar y compartir:

There exists, in my opinion, a plainly discernible parallelism between the more recent developments of mathematics and physics. Occidental mathematics has in past centuries broke away from the Greek view and followed a course which seen to have originated in India .and which has been transmitter with additions, to us by the Arabs; in it the concept of number appears as logically prior to the concepts of geometry. The result of this has been that we have applied this systematicall I developed number concept to all branches, irrespective of whether it is most appropriate for these particular applications. But the present trend in mathematics is clearly in the direction of the return to the Greek standpoint; we now look upon each branch of mathematics as determining its own characteristic domain of quantities. The algebraise of the present day considers the continuum of real or complex numbers as merely one "field ': among many; the recent axiomatic foundation of projectire geometry may be considered as the geometric counterpart of this view. This newer mathematics, including the modern theory of groups and "abstract algebra," is clearly motivated by a spirit different from that of" classical mathematics," which found its highest expression in the theory of functions of a complex variable. The continuum of real numbers has retained its ancient prerogative in physics for the expression of physicall measurements, but it can justly be maintained that the essence of the new Heisenberg-Schrodinger-Dirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense, the elements of which are the physical quantities themselves.

Es interesante notar como contrapone el desarrollo algebraico con el geométrico. Agregaría que el desarrollo algebraico, incluso de la geometría, tuvo un gran impulso con Descartes, y sus coordenadas cartesianas, que llevó el álgebra al estudio de curvas y otros elementos en el plano y en el espacio.

También es interesante destacar cómo menciona a la aplicación de los números reales a la física, pero que no necesariamente es el camino a seguir. La aparición de la no conmutatividad y las cantidades no continuas ha hecho replantear los métodos matemáticos aplicados a la física moderna. Hasta la aplicación de los números complejos es relativamente moderna (ver Números Complejos en Mecánica Cuántica, La Ecuación de Schrödinger (10) Un Comentario Sobre Números Complejos).

Podemos encontrar la revindicación de la geometría en la obra física de Penrose (leer "el Penrose"). Y las teorías de la relatividad de Einstein vuelven a poner la geometría, sin sistemas de coordenadas de base,  y las simetrías, como fundamental en la comprensión de los fenómenos físicos.

Nos leemos!

Angel "Java" Lopez
http://www.ajlopez.com
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