Math II (Несколько текстов для зачёта), страница 6

Merchant Venturers', from its outset aimed, as such schools did across Britain, to provide a passage for students into specific trades including bricklaying, plasterwork, plumbing, metalwork, and shoemaking. Navigation had been central to its mission for decades, and continued to be of importance as did mathematics, chemistry, and physics.(n28) In every way distant from British public education, this school was not, in mission, in curriculum, or in student body, designed to prepare the upper class for their stations in empire through a study of the classics. In the school archives of 1912, for example, there survives correspondence between Merchant Venturers' and the nascent University College, about the advisability of teaching firemen and preparing students for their Mine Manager's Certificates. "The more we do for the working classes," the then headmaster wrote, "the better for the university."(n29) Like so many technical colleges around England, Merchant Venturers' held geometry front and center as a site for training in an appropriate, practical reason.

Paul Dirac entered Merchant Venturers' in 1914, at the age of 12, passing from it immediately into his study of electrical engineering at Bristol University, where the university's program was, in fact, run by Merchant Venturers' as an extension of their primary and secondary programs. Young Paul took up electrical engineering under the supervision of David Robertson; Dirac's notebooks show a diligent student, adept in the technical drawing that had accompanied geometry from France to Germany to England. Month after month, Dirac trained himself to confront the constant stream of practical problems: electrical motors, currents, shunts, circuits, generators. Graduating in June 1921, he had as his principal subjects electrical machinery, mathematics, strength of materials, and heat engines (fig. 4).(n30)

While he was in the midst of this engineering program, Dirac watched Arthur Eddington's 1919 eclipse expedition, "hit the world with tremendous impact," and Dirac, along with his fellow engineering students, desperately immersed themselves in the new theory of relativity. They picked up what physics they could from Eddington; Dirac even took a relativity course with the philosopher Charlie D. Broad. The relativity Dirac seized upon was not that presented in Einstein's 1905 paper--it was not a relativity of neo-Machian arguments and Gedankenexperimenten about trains and clocks. No, what enthralled Dirac was Hermann Minkowski's spacetime, relativity cast into the diagrams in which startling relativistic results issued from reasoning through well-defined, if not-quite Euclidean, geometry. The appeal of this geometrized relativity was no doubt doubled in virtue of the fact that Dirac himself had struggled, in vain, to formulate a consistent, physically meaningful four-dimensional space-time.(n31)

While a student, Dirac did some practical engineering work with the British Thompson Houston Works in Rugby and on graduation applied there for a job for which he was rejected. But Robertson was impressed by young Dirac and, with his engineering colleagues at Merchant Venturers', tried to lure him further into their field. They were bested by the mathematicians, who offered to include Dirac, gratis, in their courses for two years.(n32) Entranced by his Bristol mathematics instructor, Peter Fraser, Dirac seized on projective geometry as his favorite subject and immediately began applying it to relativity. More specifically, Dirac turned his attention to the geometrical version of relativity that Minkowski had developed and made so popular; with projective geometry Dirac could simplify the new space-time geometry even further.(n33)

In 1923 Dirac moved out of Bristol and up to Cambridge, where as a physics research student at St. John's, he entered the research group of Ralph H. Fowler. Fowler immediately introduced Dirac to Bohr's theory of the atom. But it took no time at all for Dirac to gravitate, on the side, back to the geometry he had come to love at Bristol. At 4:15, once a week, aspiring geometers would join the afternoon geometry tea parties held by the acknowledged Cambridge master of the subject, Henry Frederick Baker. Baker himself had just authored the first volume of his multitome text on projective geometry where he announced that whatever algebra was included, the geometry was sufficient unto itself. It was a form of mathematics that, Baker judged, would naturally appeal to engineers and physicists.(n34) Certainly this proved to be the case with Dirac; as Olivier Darrigol, Jagdish Mehra, and Helmut Rechenberg have shown, even Dirac's notation seems to follow in some detail the choices made by Baker in his 1922 text.(n35)

Sometime in 1924--the date cannot be deduced exactly from the handwritten fragment--Dirac delivered a talk to Baker's tea party. This was a tough audience to please. All of Baker's students and associates understood that silences would promptly be filled by grilling, and no quarter would be given in discussion.(n36) Dirac immediately turned to the intersection of relativity with geometry and expressed his heartfelt sense that pure mathematics had nothing over the applied. On the contrary, so Dirac contended, there was a deep mathematical beauty in the specificity of the "actual world" that was obscure to the pure mathematician.(n37) "I think," Dirac penciled onto his handwritten notes,

the general opinion among pure mathematicians is that applied mathematics consists of finding solutions of certain differential equations which are the mathematical expression of the laws of nature. To the pure mathematician these equations appear arbitrary. He can write down many other equations which are equally interesting to him, but which do not happen to be laws of nature. The modern physicist does not regard the equations he has to deal with as being arbitrarily chosen by nature. There is a reason, {which he has to find} why the equations are what they are, of such a nature that, when it is found, the study of these equations will be more interesting than that of any of the others.

Old Newtonian gravity had a force that varied as the distance squared--but from the pure mathematician's view, there was nothing special about the square--it could have been cube or the fourth power. But the new theory of gravity, built out of Riemannian geometry, was (from the physicist's perspective) anything but arbitrary?

"Again," Dirac added, "the geometrician at present is no more interested in a space of 4 dim[ensions] than space of any other number of dimensions. There must, however, be some fundamental reason why the actual universe is 4 dim[ensional], and I feel sure that when the reason is discovered 4 dimensional space will be of more interest to the geometrician than any other." Questions of applied mathematics, questions from the physical world, would, he believed, become of central concern to the mathematician. That which is arbitrary in pure terms became fixed, definite, and unique when put into the frame of a real-world geometry.(n39) To draw diagrams, to picture relationships--these were the starting points for grasping why the universe was as k was.

These words would have been music to Baker's ears, for he had little truck with the new, vastly more abstract, rigorous, and algebraic mathematics that was coming into prominence. For example, when the Indian abstract number theorist Ramanujan wrote to the leading mathematicians at Cambridge, Baker had evinced no particular interest in him or his work. G. F. Hardy and J. E. Littlewood welcomed the unknown Indian number theorist as something of a mathematical prophet.(n40) Hardy, who helped shape a generation of British mathematics, emphasized rigor, axiomatic presentations, and perfect clarity in definitions. By stark contrast, Baker began volume 5 of his famous series of works on geometry with the words, "The study of the fundamental notions of geometry is not itself geometry; this is more an Art than a Science, and requires the constant play of an agile imagination, and a delight in exploring the relations of geometrical figures; only so do the exact ideas find their value."(n41)

Dirac's fascination with the confluence of physical reasoning, geometrical pictures, and mathematical aesthetics became a theme to which he returned throughout his life. In a fragment called "The Physicist and the Engineer," Dirac contended that mathematical beauty existed in the approximate reality of the engineer, not in the realm of pure and exact proof. Mathematical beauty was the guide but it was a guide through the approximate reality of the engineer's world, the one actual world in which we live. Many times Dirac insisted that all physical laws--Isaac Newton's, Einstein's, his own, were but approximations. "I think I owe a lot to my engineering training because it did teach me to tolerate approximations," Dirac recalled. "Previously to that I thought any kind of an approximation was really intolerable.... Then I got the idea that in the actual world all our equations are only approximate.... In spite of the equations' being approximate they can be beautiful."(n42)

In a sense, Dirac's trajectory can be seen as a series of flights from world to world, flights away from home, no doubt from his dominating father specifically. Margit Dirac, his wife, recalled after Paul's death, that "The first letter he wrote to me [in 1935] after his father's death was to say, 'I feel much freer now.'"(n43) But my interest is not in reducing Dirac's views to his familial relations, but rather in following Dirac's path as it traversed a series of worlds of learning, a path that left mechanisms for circuits, circuits for geometry; projective geometry for physics, and eventually projective geometry and engineering for an algebra-inflected physics. It was a path at once ever further from trade work and from home. Schematically, one might summarize Dirac's trajectory as taking him across a surface that folded the geometrical, drawn world of pictures into a private space beneath the algebraic structures of the new quantum physics:

Merchant Venturers' (technical drawing)

Bristol Electrical Engineering (mechanical and circuit diagrams)

Bristol Mathematics (projective geometry)

Cambridge (relativity/projective geometry)

Cambridge (algebraic structures of quantum mechanics).

It was in the final transition beginning in 1925, just a few months after his tea party talk, that Dirac interiorized and privatized geometry, making public presentation purely in the mode of algebra. From this moment on, Dirac spoke the public ascetic language in which he couched all of his great contributions to quantum mechanics. But he had no, affective relation to algebra--it was, in his words, an equation language that for him "meant nothing." Reflecting back on the years since his Bristol days in projective geometry, Dirac told an interviewer: "All my work since then has been very much of a geometrical nature, rather than of an algebraic nature."(n44) These are statements characterizing Dirac as a subject in mathematical physics, carving out what is simultaneously a language, an affective structure, a form of argumentation, and a means of exploring the unknown.

The final step toward abstraction and toward the algebraic world for which he came to be considered a heroic figure in physics began in 1925 when his thesis advisor, Fowler, received the proof sheets for a new article from young Werner Heisenberg. The crux was this: he had dispensed with the Bohr orbits, he had developed a consistent calculus of the spectra emitted by various atomic transitions, and he had extended Bohr's "old" quantum theory of 1913 to cover a vastly more general domain. For Dirac there was something else that had fascinated him in Heisenberg's paper--the mathematics. In the course of his calculations Heisenberg had noted that there were certain quantities for which A times B was not equal to B times A. Heisenberg was rather concerned by this peculiarity. Dirac seized on it as the key to the departure of quantum physics from the classical world. He believed that it was precisely in the modification of this mathematical feature that Heisenberg's achievement lay. It may well be, as Darrigol, Mehra, and Rechenberg have argued, that the very idea of a multiplication that depends on order came from Dirac's prior explorations in projective geometry.(n45) Perhaps it was here that Dirac began to feel that he could recreate the public algebraic world in an interior geometrical one. In any case, from there Dirac was off and running with a new mathematics, accurate predictions, no (public) visualization at any level. On the side, geometry ruled.

Dirac's steps into the unvisualizable domain of quantum mechanics were taken with a certain ambivalence. As he generalized the basic equation of quantum mechanics to include relativity, as he accrued a sense of departing from safe land, the cost to him was movingly captured in an essay he wrote repeatedly over several years titled "Hopes and Fears in Theoretical Physics." In an early fragment Dirac scribbled:

The effect of fears are perhaps not so obvious.

The fears are of two kinds.

The first one is the fear of putting forward a new

idea which may turn out to be quite wrong.

The fear of sticking one's neck out.