The Hollow Man

EYES

During the months of Gail’s illness, Jeremy abandons almost all mathematics except for his teaching and an investigation into the theory of chaos. The teaching keeps him sane. The research into chaos math changes forever his view of the universe.

Jeremy has heard of chaos math before Jacob Goldmann’s research data forces him to learn about it in depth, but as with most mathematicians Jeremy finds the concept of a mathematical system without formulae, predictability, or boundaries a contradiction in terms. It is messy. It is not math as he knows it. Jeremy is reassured that Henri Poincaré, the great nineteenth-century mathematician who helped stumble across chaos math while creating the study of topology, detested the idea of chaos in the realm of numbers just as much as Jeremy does today.

But Jacob Goldmann’s data leaves no choice but to follow the search for holographic wavefront analysis of the mind into the jungles of chaos math. So, after the long days of Gail’s chemotherapy and between the depressing hospital visits, Jeremy reads the few books available on the subject of chaos math and then goes on to the abstracts and papers, many of them translated from the French and German. As the winter days grow shorter and then grudgingly longer and while Gail’s illness grows relentlessly more serious, Jeremy reads the work of Abraham and Marsden, Barenblatt, Iooss, and Joseph; he studies the theories of Arun and Heinz, the biological work of Levin, the fractal work of Mandelbrot, Stewart, Peitgen, and Richter. After a long day of being with Gail, holding her hand while the medical procedures rack her body with pain and indignity, Jeremy comes home to lose himself in papers such as “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields” by Guckenheimer and Holmes.

Slowly his understanding grows. Slowly his mastery of chaos mathematics meshes with his more standard Schrödinger wave analyses of holographic perception. Slowly Jeremy’s view of the universe changes.

He discovers that one of the birthplaces of modern chaos math is in our failure to predict weather. Even with Cray X-MP supercomputers crunching numbers at a rate of eight hundred million calculations per second, weather prediction is a flop. In half an hour the Cray X-MP can accurately predict tomorrow’s weather for all of the northern hemisphere. In a day of frenzied activity the supercomputer can manage ten days’ worth of predictions for the northern hemisphere.

But the predictions begin to stray after about four days and any prediction a week into the future tends to be pure guesswork … even for the Cray X-MP crunching variables at the rate of sixty million crunches per minute. Meteorologists, artificial-intelligence experts, and mathematicians have all been irritated by this failure. It should not be there. That same computer is able to predict the motion of the stars billions of years into the future. Why then, they ask, is weather—even with its large but definitely finite set of variables—so difficult to predict?

To find out, Jeremy must do research on Edward Lorenz and chaos.

In the early 1960s a natural mathematician turned meteorologist named Edward Lorenz began using one of the primitive computers of the day, a Royal McBee LGP-300, to plot variables discovered by B. Saltzman in the equations that “control” simple convection, the rising of hot air. Lorenz discovered three variables in Saltzman’s equation that actually worked, threw away the rest, and set his tube-and-wire Royal McBee LGP-300 humming and buzzing to solve the equations at the sedate rate of about one iteration per second. The result was …

… chaos.

From the same variables, with the same equations, using the same data, the apparently simple short-term predictions degenerated into contradictory madness.

Lorenz checked his math, ran linear stability analyses, chewed his nails, and began again.

Madness. Chaos.

Lorenz had discovered the “Lorenz attractor,” wherein the trajectories of equations cycle around two lobes in apparently random fashion. Out of Lorenz’s chaos came a very precise pattern: a sort of Poincaré section that led to Lorenz’s understanding of what he called the “butterfly effect” in weather prediction. Simply put, Lorenz’s butterfly effect says that the flapping of a single butterfly’s wings in China will produce a tiny but inevitable change in the world’s atmosphere. That small variable of change accretes with other tiny variables until the weather is … different. Unpredictable.

Quiet chaos.

Jeremy instantly sees the import of Lorenz’s work and all the more recent chaos research in terms of Jacob’s data.

According to Jeremy’s analysis of that data, what the human mind perceives through the once-removed and distorted lens of its senses is little more than the incessant collapsing of probability waves. The universe, according to the Goldmann data, is best described as a standing wavefront made up of these churning ripples of probability chaos. The human mind—nothing more than another standing wavefront according to Jeremy’s own research, a sort of superhologram made up of millions of complete but lesser holograms—observes these phenomena, collapses the probability waves into an ordered series of events (“Wave or particle,” Jeremy had explained to Gail on the train back from Boston that time, “the observer seems to make the universe decide through the mere act of observation.…”), and goes on with its business.

Jeremy is at a loss for a paradigm for this ongoing structuring of the unstructurable until he stumbles upon an article talking about the complex mathematics that had been used to analyze the attitude of the orbit of Saturn’s moon Hyperion after the Voyager flyby. Hyperion’s orbit is Keplerian and Newtonian enough to be predicted accurately by linear mathematics now and for many decades to come. But its attitude, the directions in which its three axes point, is what might be politely described as a fucking mess.

Hyperion is tumbling and the tumbling simply cannot be predicted. Its attitude is controlled not by random influences of gravity and by Newtonian laws that could be plotted if the programmer were smart enough, the program clever enough, and the computer big enough, but by a dynamical chaos that follows a logic and illogic all its own. It is Lorenz’s butterfly effect played out in the silent vacuum of Saturn space with lumpy little Hyperion as its confused and tumbling victim.

But even in such uncharted oceans of chaos, Jeremy discovers, there can be small islands of linear reason.

Jeremy follows the Hyperion trail to the work of Andrei Kolmogorov, Vladimir Arnold, and Jurgen Moser. These mathematicians and dynamics experts have formulated the KAM theorem (Kolmogorov-Arnold-Moser, that is) to explain the existence of classical quasi-periodic motions within this hurricane of chaotic trajectories. The diagram resulting from the KAM theorem results in a disturbing plot that shows an almost organic structure of these classical and plottable trajectories existing like sheaths of wire or plastic, windings within windings, wherein resonance islands of order lay embedded within folds of dynamic chaos.

American mathematicians have given this model the name VAK, short for “Vague Attractor of Kolmogorov.” Jeremy remembers that Vak is also the name of the goddess of vibration in the Rig-Veda.

On the night Gail first enters the hospital not for tests but to stay … to stay until recovered sufficiently to return home, the doctors say, but both she and Jeremy know that there will be no real return … Jeremy sits alone in his second-floor study and gazes at the Vague Attractor of Kolmogorov.

Regular quasiperiodic trajectories winding above secondary sheathings of resonances, tertiary sheathings budding in the form of more delicate multiple resonances, chaotic trajectories lashing through the organism like tangled wires.

And Jeremy sees the model for his analysis of the holographic neurological interpretation of the set of collapsing probability waves that is the universe.

He sees the beginnings of the model for the human mind … and for the talent that he and Gail share … and for the universe that has hurt her so.

And above it all the butterfly effect. The sure knowledge that the entire life of a human being is like a single day in that human’s life: unplannable, unpredictable, governed by the hidden tides of chaotic factors and buffeted by butterfly wings that bring death in the form of a tumor … or, in Jacob’s case, in the form of a bullet to the brain.

Jeremy realizes his life’s ambition that evening, of discovering a profoundly new direction of mathematical reasoning and research—not for status or further academic honors, for those have been forgotten—but to advance the lighted circle of knowledge a little farther into the encroaching darkness. Islands of resonance within the chaotic sea.

But even in seeing the path of research he can take, he abandons it, tossing the abstracts and studies aside, wiping away the preliminary equations on his chalkboard. That night he stands at the window and stares out at nothing, weeping softly to himself, unable to stop, filled with neither anger nor despair, but with something infinitely more lethal as the emptiness enfolds him from within.