Angel "Java" Lopez en Blog

Enero del 2017

Publicado el 31 de Enero, 2017, 13:41

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Angel "Java" Lopez

Publicado el 30 de Enero, 2017, 15:22

Encuentro un texto (nuevo para mí) de Dirac, describiendo su experiencia como estudiante de escuela secundaria, en Merchant Venturer's Technical College (M.V.), la escuela pública donde su padre enseñaba:

The M.V. was an excellent school for science and modem languages. There was no Latin or Greek, something of which I was rather glad, because I did not appreciate the value of old cultures. I consider myself very lucky in having been able to attend the School. I was at the M.V. during the period 1914-18, just the period of the First World War. Many of the boys then left the School for National Service. As a result, the upper classes were rather empty; and to fill the gaps the younger boys were pressed ahead, as far as they were able to follow the more advanced work. This was very beneficial to me: I was rushed through the lower forms, and was introduced at an especially early age to the basis of mathematics, physics and chemistry in the higher forms. In mathematics I was studying from books which mostly were ahead of the class. This rapid advancement was a great help to me in my later career.

The rapid pushing-ahead was a disadvantage from the point of view of Games—which we had on Wednesday afternoons. I played soccer anti cricket, mostly with boys older and bigger than myself, and never had much success. But all through my schooldays, my interest in science was encouraged and stimulated.

It was a great advantage, that the School was situated in the same building as the Merchant Venturers’ Technical College. The College “took over” in the evenings, after the School was finished. The College had excellent laboratories, which were available to the School during the daytime. Furthermore, some of the staff combined teaching in the School in the daytime with teaching in the College in the evenings. (Dirac 1980, p. 9)

Se refiere Dirac, P.A.M. 1980. A little ‘prehistory.’ The Old Cothamian 1980, p. 9. Lo encuentro en el excelente "QED and The Men Who Made It", de Silvan Schweber.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 29 de Enero, 2017, 15:23

Hace un tiempo publiqué:

Dirac, Heisenberg, Paili y la religión

Hoy leo una versión más completa de la opinión de Dirac, en ese congreso Solvay de 1927:

It we are honest -and scientist have lo be—we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.... I can't for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much miscry and injustice, the exploitation of the poor by the rich and all other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but siinply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also easier to exploit... Hence the close alliance between those two great public forces, the State and the Church.

Lo encuentro en el excelente "QED and The Men Who Made It" de Silvan Schweber. Asombra un poco la postura "dura" de un Dirac circumspecto, casi tímido en otros aspectos.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 28 de Enero, 2017, 12:58

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Angel "Java" Lopez

Publicado el 27 de Enero, 2017, 11:17

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Angel "Java" Lopez

Publicado el 26 de Enero, 2017, 12:50

Dirac siempre comentó que su relación con su padre Charles nunca fue fácil. A partir de 1925, año del suicidio del hermano de Dirac, Reginald, padre e hijo se distanciaron aún más. Leo a Schwber, QED and the Men Who Made It:

Reginald, Paul"s older brother, had wanted to become a physician, but Charles forced him to study mechanical engineering at Bristol. He obtained only a third-class degree upon graduating and accepted a position as a draftsman with an engineering firm in Wolverhampton. He committed suicide when he was twentyfour years old. The death of his oldest son deeply disturbed Charles, and for a while Paul feared that his father might lose his sanity—and he resolved that he would never take his own life no matter what the circumstances. Thereafter Paul"s relationship with his father became chilled and they had very little interaction with one another. One manifestation of Paul"s feeling toward his father was that throughout his life he avoided going to Switzerland, a country he associated with his father (Mehra and Rechenberg 1982). Paul invited only his mother to attend the ceremonies in Stockholm honoring him with the Nobel Prize in 1933. Dalitz and Peierls report that when Professor Tyndall, who had headed the physics department at Bristol University for three decades, gave a set of public evening lectures on modem physics in the early 1930s, he noticed a regular listener in the front row, a man much older than the others there, who was taking careful note of all that he said. At the end of the last lecture of the series, this old man came up to him to thank him, saying: "I am glad to have heard all this. My son does physics but he never tells me anything about it." The old man was Charles Dirac. Charles Dirac died in 1935. Paul was in Russia at the time to watch an eclipse of the sun; when infomied of the seriousness of his father"s illness, Paul flew back to England, but it was too late. The first letter he wrote his wife after his father"s death was to say, "1 feel much freer now" (Margit Dirac 1987, p. 5).

No conocía la anécdota del padre tratando de conocer física. Debió ser duro para él encontrarse en la madurez, sin sus hijos. Dirac siempre tuvo una conducta lejana, sin grandes relaciones, con excepción de su esposa (recordemos que era la hermana de Wigner).

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 25 de Enero, 2017, 11:20

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Es notable todo los temas en los que se involucró Jordan, estando cerca de Born:

For his dissertation under Born, Jordan worked on a problem in the theory of light quanta (Jordan 1925) that dealt with the interaction of electrons and radiation. In it he tried to disprove Einstein"s hypothesis that in the process of absorption or emission of a photon of energy hv by an atom an amount of momentum hi"/c is transferred by or to the photon. Einstein, in a brief note to the Zeitschrift für Physik, took exception with Jordan"s work. He pointed out that Jordan"s paper was based on a hypothesis that implied "that the amounts ol radiation taken (by an atom exposed to blackbody radiation) from rays of different directions were treated as not being independent of each other" and that this would result in consequences contrary to observation (Einstein 1925). After finishing his thesis in the fall of 1924, Jordan worked with James Franck on problems connected with spectroscopy. He helped him write volume 3 of the series Struktur der Materie which Bom and Franck edited. The book was published with Franck and Jordan as coauthors, with the title Anregung von Quantensprüngen der Stosse. During that same year Jordan wrote several papers dealing with problems in atomic structure and spectroscopy. He also collaborated with Born on a paper in the quantum theory of aperiodic processes. By generalizing Kramers and Heisenberg"s dispersion-theoretic approach, they calculated the effect on an atom of an electric field whose time dependence is arbitrary. The arbitrary time dependence was to allow them to simulate the effect of a charged projectile particle during an atomic collision.

Es interesante notar que se había familiarizado con las ideas y formalismo matemático de Kramers y Heisenberg sobre la dispersión de radiación por un átomo. Parte de ese formulismo será usado por Heisenberg en su gran "paper" de 1925.

Con este post termino la serie. Luego, la historia de Jordan ya cae en un periodo más allá de sus primeros años, donde comienza a aportar nuevas ideas a la mecánica cuántica y la teoría de campos.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 24 de Enero, 2017, 11:00

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Veamos como Jordan va apareciendo en escena, a partir de 1922:

The Bohr Festspiel of June 1922 gave Jordan a taste of the drama of physics. Heisenberg, Fermi, Pauli, and Hund were also in Göttingen at the time. Fermi was evidently left out of the Göttingen intellectual community (Segre 1970, p.32) but Jordan got to know the other three well, particularly Heisenberg and Pauli, and came to appreciate the company of these brilliant young men. But he was overshadowed by their brash and confident ways. Jordan was rather short, and his presentation of self reflected his physical stature. He gave the impression of being insecure, an impression that was reinforced by his stuttering (he in fact suffered a breakdown in the early 1930s).

Although Jordan did not enjoy his courses in experimental physics—he actually stopped attending them—he found the laboratory course in zoology in which he had enrolled very satisfying; he also faithfully attended Alfred Kuhn"s lectures on heredity. In fact, he chose zoology as one of the minor subjects for his doctorate. Most of his energies, however, were spent on theoretical physics and mathematics. He helped Courant in the preparation of the book he was then writing with Hilbert, the famous Mathematische Methoden der Physik (Jordan 1963, p. 12). He also assisted Born with an article on crystal dynamics and became quite close to him.

El "festspiel" de Bohr fue el primer encuentro de Heisenberg y Bohr, y al parecer de Pauli con Bohr también. No sabía que había acudido Fermi también. Vean que Jordan era tartamudo, lo que influyó en su relación con los demás. No sabía que había ayudado en el famoso libro de Hilbert y Courant.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 23 de Enero, 2017, 12:44

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Veamos como llega Jordan a conocer a Born:

Jordan entered the Technische Hochschule of the University of Hannover in 1921 intending to study physics. He had by that time learned some special and general relativity from Moritz Schlick"s Raum und Zeit in der gegenwärtigen Physik, had mastered electromagnetic theory, and had carefully studied Sommerfeld's Atombau und Spektrallinien. Jordan found that physics was not taught well at the Technische Hochschule and he transferred to Göttingen in 1922. However, he had made good use of his year at the Hochschule taking courses in mathematics, electrical engineering, and physical chemistry. In Göttingen he attended Courant"s course on mathematical methods for physicists and became the official note taker for the course. For a while he toyed with the idea of becoming a mathematician. But he came into Bom"s orbit, and under his influence and with his guidance became more and more committed to physics. When Born died, Jordan, in a brief eulogy for him, wrote: "He was not only my teacher who in my student days introduced me to the wide world of physics—his lectures were a wonderful combination of intellectual clarity and horizon widening overview. But he was also, I want to assert, the person who next to my parents, exerted the deepest, longest lasting influence on my life"....

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Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 22 de Enero, 2017, 12:16

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Ya saben que me interesa la historia de la ciencia, y en especial, he escrito bastante sobre el "mágico" año 1925 y alrededores, con el nacimiento de la mecánica cuántica. Uno de los personajes que aparecen una y otra vez en cualquier historia sobre el periodo, pero que tal vez no es tan conocido, es Pascual Jordan. Nunca consiguió el premio Nobel, aunque colegas como Born y Heisenberg lo consiguieron, por trabajos que de alguna manera también compartieron con Jordan. Como ayudante de Born, estuvo con él cuando llego el tiempo de escribir y expandir las ideas de Heisenberg de 1925. Luego involucrado con la actividad nazi en Alemania en los treintas, se dedicó a otros temas, además de la física. Encuentro hoy un texto, relato, de sus primeros años, en el excelente "QED and the Men Who Made It", de Silvan Schweber:

Jordan was bom in Hannover. Germany, in 1902. He was the younger of the two children in the family; a sister some ten years older than Pascual was the older sibling. Both his parents were well read in the natural sciences. His father was a painter and he got the young Pascual interested in the geometrical concepts involved in the "perspective" of drawing at an early age. In his interview with T. S. Kuhn, Jordan recalled that as a young boy his father read him books from the Kosmos series that acquainted him with the writings of Darwin and Haeckel. His mother introduced him to the world of plants, animals, and stars. "From her I... learned that... light has to go eight minutes from the sun to here. She was also very interested in calculation, in numbers and so on and from her 1 learned the first steps in arithmetic and so on" (Jordan 1963, p. 1). She often took him to visit the local zoo and he remembered collecting pictures of extinct animals, particularly those of dinosaurs. In his early teens he thought of becoming a painter or an architect, but gradually his interests shifted to natural history and biology, and eventually to physics and mathematics. He was clearly quite gifted and ambitious: "At fourteen, I had a plan of writing a big book on all the fields of science linking them all together" (Jordan 1963, p. 5). He had by then read and absorbed such books as Pauly"s Darwinismus und Lamarkismus and F. A. Lange"s Geschichte des Materialismus. He had also studied by himself classical physics and a great deal of mathematics. While in Gymnasium he taught himself the differential and integral calculus from Nernst and Schoenfiiess"s Kurzgefasstes Lehrbuch der Differentialuiid Integralrechnung, and the theory of complex variables from Knoff"s Funktionentheorie. During his last year in the Gymnasium he began to study physics in depth and carefully read Mach"s Mechanik and Prinzipien der Wärmelehre. Mach"s views influenced Jordan deeply and he became an ardent positivist. He later declared that he took up physics in order to help resolve the discrepancy he felt existed between Mach"s teachings and the old quantum theory (Jordan 1936). He adopted as the central tenet of his philosophical outlook what he considered to be  the essential and decisive principle of the positivistic theory of knowledge: that scientifically sound proposilions arc limiled lo those that can be proved experimentally.

Interesante la influencia de su madre, y su inclinación a la ciencia, y varias ramas a la vez, por ejemplo, su interés en la biología evolutiva.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 21 de Enero, 2017, 13:25

Ya apareció por este blog el tema de grupos y física, en especial, cuántica:

Grupos y Física, por Dirac
Teoría de Grupos y Partículas Elementales
Hermann Weyl, Teoría de Grupos y Teoría Cuántica

Encuentro hoy unos párrafos de Abraham País, en su excelente "Inward bound, of matter and forces in the physical world", en un capítulo dedicado a grupos y la "clásica" mecánica cuántica:

Nearly a year after Heisenberg had considered the theory of one linear oscillator and so discovered quantum mechanics, he had something interesting to say about two identical oscillators symmetrically coupled to each other. The quantum states of this system, he found, separate into two sets, one symmetric, the other anti-symmetric under exchange of the oscillator coordinates. Assuming further that the oscillators carry electric charge, he noted that radiative transitions can occur between states within each set, never between one set and the other. He further conjetural that non-combining sets should likewise exist if the number n of identical particles is larger than two, but had not yet found a proof. He left this problem aside; another question was on his mind. Six weeks later he gave the theory of the hellium spectrum, that bane of the old quantum theory. To Pauli he complianed that his calculations were 'imprecise and incomplete'. It is true that others were able to refine his answer in later years. Nevertheless, his outstanding paper  contains all the basic ingredients used today. The first quantum mechanical application of the Pauli principle is given: two-electron wave functions are antisymmetric for simultaneous exchange of space and spin coordinates. That  principle is the subject of the following section...

Meanwhile in Berlin a young Hungarian chemistry engineer, Jenö Pál (better known since as Eugene Paul) Wigner, had become interested in the n > 2 identical particle problem. He rapidly mastered the case n = 3 (without spin). His method were rather laborious; for example, he had to solve a (reducible) equation of degree six. It would be pretty awful to go on this way to higher n. So, Wigner told me, he went to consult his friend the mathematician Johnny von Neumann. Johnny thought a few moments then told him that he should read certain papers by Frobenius and by Schur which he promised to bring the next day. As a result Wigner's paper on the case n (no spin), was ready soon and was submitted in November 1926. It contains an acknowledment to von Neumann, and also the following phrase: 'There is a  well-developed mathematical theory which one can use here: the theory of transformation groups which are isomorphic with the symmetric group (the group of permutations)'.

Thus did group theory enter quantum mechanics.

El propio Wigner escribió un libro "clásico" "Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra". Y ya mencioné en un post de arriba, al trabajo de Weyl.

Vean cómo el trabajo de Heisenberg fue precursor de la aplicación de grupos. Luego, él mismo se encargaría de introducir las simetrías internas, en el caso protón-neutrón, que inauguraría otra rama de aplicación, la de teoría de grupos en partículas elementales.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 20 de Enero, 2017, 14:24

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Angel "Java" Lopez

Publicado el 19 de Enero, 2017, 15:30

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Encontré datos adicionales sobre la historia de la aparición de la "segunda cuantización". Leo:

In the midst of their efforts to cobble together a new quantum mechanics to treat the physics of atoms during the mid-1920s, several theorists—Werner Heisenberg, Pascual Jordan, Wolfgang Pauli, Paul Dirac, and others—began trying to quantize Maxwell"s electromagnetic field as well. Jordan began the process in the midst of his work on Heisenberg"s matrix mechanics, composing a long and difficult section of the famed Dreim¨annerarbeit, or "three-man paper" by Heisenberg, Jordan, and Max Born of 1926, on suggestive ways to quantize vibrating strings (seen as a first step toward treating waves and fields). The following year, Paul Dirac demonstrated that the electromagnetic field"s infinite number of degrees of freedom could be decomposed as a sum over quasi-particulate oscillators, each corresponding to a specific frequency or energy—a representation, soon dubbed "second quantization," that Jordan quickly extended tomatter fields aswell.By 1927, Jordan had become convinced that all physical quantities—everything from the electrons and protons of ordinary matter to the electromagnetic fields that bound them together into atoms—arose ultimately from quantum fields. Although resisting some of Jordan"s maneuvers at first, other theorists, including Heisenberg and Pauli, soon came to share Jordan"s view.

Lo encuentro en el libro de David Kaiser, Drawing theories apart, the dispersion of Feynman Diagrams in Postwar Physics, que trata sobre cómo los diagramas de Feynman se fueron popularizando luego de la segunda guerra mundial. Menciona en una nota, estas fuentes:

Dirac, "Quantum theory of radiation" (1959 [1927]); Born, Heisenberg, and Jordan, "Quantenmechanik" (1926); Jordan, "Quantenmechanik des Gasentartung" (1927); Heisenberg and Pauli, "Quantenelektrodynamik" (1929–30); Darrigol, "Origin of quantized matter waves" (1986); Pais, Inward Bound (1986), 334–40; Schweber,QED(1994), 23–56; andMiller, "Frame-setting essay" (1994), 18–28.

El "paper" de Dirac se puede encontrar en:

El "paper" de los "tres autores" lo tengo en el libro de van der Waerden, Sources of Quantum Mechanics, pero no sé si completo. Y es muy interesante la descripción del trabajo de Dirac que hace el libro de País, Inward bound.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 18 de Enero, 2017, 16:20

Ya he escrito varios posts sobre el desarrollo de la mecánica cuántica, alrededor del año 1925, y quiero extenderlos a la época que le precedió y siguió. Ver por ejemplo:

Dirac y la teoría de Heisenberg
Dirac revisando el trabajo de Heisenberg
Erwin Schrödinger creando su ecuación
Dirac y las ecuaciones de la mecánica cuántica
Heisenberg desarrollando la mecánica cuántica
Desarrollo de la mecánica cuántica, por Max Born
La ecuación de Schrodinger

En estos días, me encuentro con el relato de Abraham País, que fue contemporáneo de los avances que vinieron más tarde, y es interesante ver cómo ya se tomaba diferente a esos "quiebres" de la física básica:

The way I was first exposed to quantum mechanics, not long after 1935 when I began my university studies, was no different from the way I learned, say, thermodynamics. There were courses on the subject and there were books, some more helpful for an understanding of the principles, some better for learning how to solve problems. I learned some experimental facts about electron behaving as particles in collision processes, as waves in diffraction effects. I was awed by the success of the Schrodinger equation for the hydrogen atom and found the introduction of quantum mechanical probabilities via the continuity equation a most plausible step. Not did I experience any difficulty in accepting Heisenberg's uncertainty relations, served up with the help of a classical picture of the dispersion of wave packets combined with E = hv and p = hk. Soon I was happily making quantum mechanical exercises. I had no sense whatever at the time of the stir and struggle which, only ten years earlier, had accompanied the introduction of the new mechanics. I knew a few dates but those seemed to belong to antiquity.

Pero cuando comenzó a tener contacto con Bohr, y otros que participaron en esa "revolución" de 1925 y cercanías, empezó a darse cuenta lo que fueron esos tiempos, lo que significó para sus protagonistas:

In 1946 I went to Copenhagen and for a brief period became Niels Bohr's close colaborator. Of that experience I have written: 'I must admit that in the early stages of collaboration I did not follow Bohr's line of thinking a good deal of the time... I failed to see the relevance of such remarks as that
Schrodinger was completely shocked in 1926 when he was told of the probability interpretation of quantum mechanics, or a reference to some objection by Einstein in 1928, which apparently had no bearing whatever on the subject at hand. But it did not take long before the fog started to lift... Bohr would relive the struggles which it took before the content of quantum mechanics was understood and accepted... Through steady exposure to Bohr's "daily struggle" and his ever repeated emphasis on "the epistemological lesson which quantum mechanics has taught us", to use a favorite phrase of his, my understanding deepened not only of the history of physics but of physics itself'.

Además de Bohr, otros le comentaron sus reacciones:

In the course of time, other physicists who were active during the years of discovery of quantum mechanics also told me occasionally of their reactions. Uhlenbeck said to me that it was as if within the span of a few years his life had changed. Wigner told me of his astonishment upon reading Born and Jordan's paper which explained that Heisenberg had unwittingly introduced matrix methods and of his sense that now there seemed to be hope after all for a rationale of quantum theory. Several members of the 1925 generation have told me that Heisenberg's paper which marks the beginning of the new era took a while, but not long, to sink it. Even now this paper, one of the most admirable contributions to physics, is hard to read without knowledge of its subsequent elaboration.

Estoy tratando de comentar el "paper" de Heisenberg en:

Entiendiendo a Heisenberg

Fueron tiempos muy interesantes. Encuentro este texto en el excelente "Inward bound, on matter and forces in the physical world" de País.

Nos leemos!

Angel "Java" Lopez

Por ajlopez, en: General

Publicado el 17 de Enero, 2017, 17:47

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Angel "Java" Lopez

Publicado el 16 de Enero, 2017, 14:16

Este tema lo conocía y hace un tiempo que quiero compartirlo por acá, pero no encontraba una referencia adecuada. En estos días de convalecencia, vuelvo a descubrir libros y lecturas, y encuentro este pasaje: 

It is worth mentioning, in passing, that in 1900, the same year in which Planck"s paper on blackbody radiation appeared, Lord Kelvin gave a lecture that drew attention to another difficulty with the classical theory of statistical mechanics. Kelvin described two "clouds" over nineteenth century physics at the dawn of the twentieth century. The first of these clouds concerned aether—a hypothetical medium through which electromagnetic radiation propagates—and the failure of Michelson and Morley to observe the motion of earth relative to the aether. Under this cloud lurked the theory of special relativity. The second of Kelvin"s clouds concerned heat capacities in gases. The equipartition theorem of classical statistical mechanics made predictions for the ratio of heat capacity at constant pressure (cp) and the heat capacity at constant volume (cv). These predictions deviated substantially from the experimentally measured ratios. Under the second cloud lurked the theory of quantum mechanics, because the resolution of this discrepancy is similar to Planck"s resolution of the blackbody problem. As in the case of blackbody radiation, quantum mechanics gives rise to a correction to the equipartition theorem, thus resulting in different predictions for the ratio of cp to cv, predictions that can be reconciled with the observed ratios.

Lo encuentro en el excelente libro de Brian C. Hall, "Quantum Theory for Mathematicians".

¡Qué puntería en señalar "nubes" que tenía Lord Kelvin! Justo apuntó a dos temas que terminarían provocando gran parte del desarrollo de la nueva física del siglo que estaba por comenzar. ¿Tenemos nubes así, ahora que estamos en el siglo XXI? Debe haber varias, pero a mí me llaman la atención dos en particular: las llamadas "materia oscura" y la "energía oscura" que aparecieron con distinto sustrato experimental, pero que no encuentran una explicación evidente en estos días. A veces pienso que la respuesta a estas "nubes" realmente será insólita, y veremos el nacer de una novísima física.

Vivimos tiempos interesantes...

Post relacionado: Lord Kelvin y Rutherford

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Angel "Java" Lopez

Por ajlopez, en: Ciencia

Publicado el 14 de Enero, 2017, 16:04

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Sigo traduciendo, y comentando, a Boyer. Sobre la aparición de las matemáticas ligadas a las explicaciones físicas, escribe:

Esta concepción de las matemáticas como la base de la arquitectura del universo fue a su tiempo modificado en los siglos dieciséis y diecisiete. En matemáticas,  la causa del cambio fue el uso menos crítico y más práctico del álgebra que había sido adoptado de los árabes en el siglo trece, y luego desarrollada en Italia. En la ciencia natural, el cambio se debió al ascenso del método experimental. La certeza en matemáticas de las que hablaban Descartes, Boyle y otros fue entonces interpretada como indicando una consistencia a ser encontrada antes en el carácter de su razonamiento que en cualquier necesidad ontológica que fuera presentada a priori.

Yo veo más la influencia de Galileo y contemporáneos como causa de la aparición de las matemáticas en las ciencias físicas. Pero entiendo que Boyer apunta a que se promovió las matemáticas sin hacer tanto énfasis en sus fundamentos sino en sus resultados.

El centro de la atención en los procedimientos antes que en las bases de las matemáticas fue enfatizada en los siglos dieciocho por el extraordinario éxito en la aplicación del cálculo a los problemas científicos y matemáticos. Un actitud más crítica fue inaugurada en el siglo diecinueve por los esfuerzos persistentes para encontrar un fundamento satisfactorio de las concepciones envueltas en este nuevo análisis del infinito. El rigor matemático fue revivido, y se descubrió que los postulados de Euclides no son juicios sintéticos, como mantuvo Kant, sino simple asunciones. Se encontró que esas premisas pueden ser elegidas de forma tan libre y arbitraria que - sujetas a la condición que sean mutuamente compatibles - pueden permitirse contradecir la aparente evidencia de los sentidos. Hacia el final del siglo, como resultado de una tendencia "aritmatizante" en el análisis matemático, se descubrió que el concepto de infinito, transcendiendo la intuición y el análisis, podría ser introducido en matemáticas sin afectar al lógica consistencia del tema.

Es notable el avance del análisis en los siglos mencionados, motivado por su aplicación a problemas físicos, desde ondas hasta movimiento planetarios y fluidos. Y notable la reacción del siglo diecinueve, volviendo en parte al rigor (digo, parte de los matemáticos, pues tanto el análisis y el álgebra siguieron su ímpetu).

Nos leemos!

Angel "Java" Lopez

Publicado el 13 de Enero, 2017, 15:29

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