Let's be clear. In 1960, the physicist Eugene Wigner wrote a classic paper entitled The unreasonable effectiveness of mathematics in the natural sciences. His philosophical concern was how it could be that a discipline pursued in isolation from outside constraint or influence, guided only by internal consistency and its own aesthetics, could find application in physical science. So far as I can tell, no one has answered that question, and yet from time to time mathematics provides tools and models for profound changes in science. Perhaps the most famous example is general relativity and differential geometry, in particular the theory as developed by Levi-Civita and his teacher, Ricci, and the great geometer Elie Cartan, on which Einstein based his theory.
But there is more to it than this. The mathematicians mentioned above had no criteria other than mathematical problems of geometry, not "spatial geometry" or the geometry of Euclid, but differential geometry, a subject with roots in the early 19th Century from Gauss and Riemann and Beltrami. The mathematicians working in differential geometry made no experiments and paid no attention to physics in their work, though Cartan did help Einstein with some deficiencies in general relativity. Their goal was to describe intrinsic properties such as the curvature of those manifolds without relying on some "surrounding" object.
In a 1972 lecture* Freeman Dyson bemoaned the fact that mathematicians of the late 19th Century had been so caught up in problems of their own that they ignored Maxwell's theory of electricity and magnetism, since if they had pursued mathematical problems stemming from this physical theory there would have been a purely mathematical derivation of special relativity well before Einstein.
The interesting story is how relativity came to be in the first place. Already it was known that Newtonian physics did not accurately predict a small disturbance in the orbit of Mercury. The Michelson-Morley experiment led to puzzling results that could not be explained, surprising results in that the experiment did not so much fail as its outcome did not make sense at all.
The great leap to relativity came from the recognition that Newton's physics was invariant with respect to a different method of changing reference frames than was Maxwell's physics. Such a discrepancy does not make physical sense, since the two theories could not coexist within the same physical universe. (In mathematical language, this is expressed by saying that Newton's equations are invariant with respect to a different transformation group than Maxwell's equations; in essence they lived within different geometries.) The great leap was to recognize that Maxwell's physics was compatible with a physics in which the speed of light was constant, resulting in consequences that made a significant number of physicists queasy. Maxwell's equations and special relativity are both invariant with respect to the Lorenz group of transformations which gives a different spatial structure than does the Galilean group under which Newton's equations are invariant. From Euclidean space under Newton where time is independent, to hyperbolic space under relativity where time and space are inseparably coupled, neither in themselves meaningful except when paired as a space-time event invariant with respect to the non-Euclidean distance.
This leap was conceptual and aesthetic; that the general theory of relativity explained the problem with Mercury's orbit added an argument for physical correctness. The special theory explained the Michelson-Morley experimental mess. It was the same sort of leap made by Newton in shattering classical philosophy as held since Aristotle by proposing a singe invisible force that made objects fall to earth and kept planets in orbit. These creative leaps are guided by an aesthetic standard for simplicity and structural integrity, the same kind of satisfying wholeness that makes a musical work like Bach's B Minor Mass or Villa-Lobos's Bachiana Brasileiara Number Five move us. The modern geometer Shoshichi Kobayashi put it well when he wrote, "All geometric structures are not created equal; some are the creations of gods while others are products of lesser minds."
This is overlooked, nay misrepresented by Powers in his description of physics and his description of David. He treats physics as if it amounted to solving some kind of equations, an outsider's viewpoint that is akin to saying poets pluck words from thesauruses and rhyming dictionaries to build poems or that composers create by formula.
In the end, David is a plot device. He is there to set up the ending of the book, which is supposed to explain the great mystery of why this marriage between a Jewish immigrant scientist and black woman who wants to be the next Marian Anderson, in racist America where such a marriage was illegal in half the states, happened at all. The answer is less than satisfying in a number of ways, all demeaning to the notion of science. For me the surprise ending was no surprise, as Powers telegraphed it early on, even before hammering on the question of why this marriage had taken place, why these star-crossed lovers continued a liaison they both knew was outside the bounds of social acceptability: his physicist did not fit, sticking out like the proverbial sore thumb.
Powers follows the same disturbing trend I note in many modern US novelists, building universes with an inner logic that they later warp in order to fool the reader. I believe this reflects a society refusing to live within a universe whose rules clash with their beliefs and expectations.
This surprise ending is bad science fiction. That it wowed the lawyer who suggested I read the book is no surprise, given he knows nothing whatsoever about physics and that as a lawyer his aesthetic sensibility is for subtle casuistry, the slight of hand argument that fools the jury.
The trick is based on what is called advanced action, the idea that future events affect the past. It was employed satirically by Thomas Pynchon in Gravity's Rainbow, skewering conditioning in psychology. It was employed with subtlety as a form of grace by José Saramago in The History of the Siege of Lisbon. In The Time of our Singing it is an obvious parlor trick.
In essence, Powers returns to the opening when David and Delia meet at the Lincoln Memorial in Washington DC and repeats the scene where they encounter a lost boy in the crowd. But this is their grandson and he is attending the Million Man March. And they know who he is and for this they marry. More or less.
This is where the notion that Powers is a genius comes into play. To me it showed not genius but a shallow trick ending, an illusion that does not seem a physical possibility at all, smoke and mirrors that leaves anyone understanding the physics laughing or disappointed. Why did he put them in the same "place" at different "times?" Does he not understand that place is without significance in relativity? That is to say, there is no same place (or same time), but only the event which requires both where and when simultaneously. It shows is an author overreaching from shallow knowledge.
Indulge me a bit more with this final bit of explication. Bertrand Russell in A History of Western Philosophy discusses Einstein in terms of a dispute between Newton and Leibniz with relation to Parmenides and Descartes and the concepts of matter and space and extension. The work of Einstein decided the debate in favor of Leibniz. Space is not about things but about relations. There is no distance between things, but only between events. There is no saying (in an absolute sense) when or where an event occurred. That is to say, time and space are inextricably entangled, so all that can be measured absolutely is the non-Euclidean separation between events. Moreover, all events already exist, are not becoming or gone, but simply are.
In relativity, there are paths through events. You, the reader, follow one, I another. These paths have causal restrictions of the sort that makes my wife laugh when she schedules grilling chicken on Saturday and making tacos with the leftovers on the following Wednesday and I want to reverse the order. All this is explained in mathematical detail in chapter six of the monograph of S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time.
This is where Powers stumbles mightily. These three world paths intersect in a way that is precluded in all the models for the Einsteinian field equations considered physically meaningful. Because all paths already exist does not mean that they can arbitrarily share an event; that is to say, it is not the case that all world paths can intersect. The causal restriction on the model precludes this particular event, that is to say, this meeting.
I say mightily stumbles because all through the book there exist minor stumbles in his attempt to give some kind of life to the plot device David, mostly with metaphors using mathematics or physics that turn out to silly or incorrect. The most telling one occurs near the end of the book, in describing a performance of Villa-Lobos's Bachiana Brasileiara Number Five. Powers writes, "Her pitch was something NASA used to guide satellites."
That made me laugh. NASA does not guide satellites. Anyone with a modicum of physics knows this. Gravity guides satellites, whether you consider it the gravity of Newton's force or a geodesic path of events in the curved space-time shaped by Einstein's field equations. There are perturbations to the orbit caused by the oblateness of the earth and by other celestial bodies, by atmospheric drag if the satellite is low enough, by solar wind on sails, by similar influences which cause the orbit to degrade and require small thrusts to adjust it. But once the body is in orbit, there is nothing for NASA or anyone to do. The position and velocity are purely a function of that orbit. There is no guidance.
It is the same misunderstanding of physics that leads Powers to put the crossing of events in the same "place" at different "times." If the paths of these two events were to cross, it would be no more compelling for them to meet in that "place" than in any other.
The mistakes of Powers remind me of the statement by the French mathematician Bruno Poizat in a text on mathematical logic, A Course in Model Theory. I recommend the Preface to the English edition of this book because it is funny and scathing. What he says about publishing and self-publishing and the irrationality of publishers in his own field will ring familiar to anyone involved with writing; his biased attitude towards truth in mathematics provides an uncommon and hilarious brush with Platonism.
Poizat writes about what he calls the amateur epistemologists who proclaim nonsense about the famous incompleteness theorem of Gödel, that it says such things as nothing can prove its own consistency or its own existence. (I have read this from people who were not amateurs.) He says, "What is certain is that those who would philosophize about this theorem would do well first to know its precise statement and, if possible its proof…"
That caveat ought to be applied as well to those who decide to use physics without understanding the theory. If one writes a fanciful bit of sword and sorcery or science fiction, it seems that such an abuse would be par for the course. But in a book like this, written with serious intent, and written carefully, with motivated characters, a survey of racist history in the latter half of the twentieth century in the US, a detailed discussion of serious music and the labor involved in creating it and attempting to earn a living from it, a literary approach that plays off the musical forms, to cheapen the end with such a gimmick is to let down the serious reader. For me, the entire edifice deflated before my eyes, the careful universe violated.
I tell people to read the book but ignore the science. David is a straw man and the ending is trite and unfair to a serious reader. A more serious challenge would have been building a bridge between the aesthetic and intellectual foundation of David's physics and that of the music of his sons. Though the muses are perhaps different, an aesthetic motivation is common to both.
Editor's Note:
* "Missed Opportunities," a Josiah Willard Gibbs Lecture, Jan. 17, 1972. Currently available in Selected Papers of Freeman Dyson, American Mathematical Society, 1996.
Jim Chaffee is an old guy who writes about what he knows: sex, violence, mathematics and dumbasses. His first science fiction pieces were proposals to the Air Force. These days he tries to be in Brazil. He has finished what seems to be a crime novel, São Paulo Blues, which pisses off a lot of people who read it. He began a publishing company The Drill Presss where you can find out about the book. It also publishes three online journals in English and one in Portuguese edited in Brazil. We seek writers and readers.