Even in the oldest quantum theories of gravity, the "vacuum" of empty space-time had been viewed as a seething mass of virtual worm-holes, and other more exotic topological distortions, popping in and out of existence. The smooth appearance at macroscopic lengths and human timescales was just the visible average of a hidden riot of complexity. In a way, it was like ordinary matter: a sheet of flexible plastic betrayed nothing to the naked eye of its microstructure—molecules, atoms, electrons, and quarks—but knowledge of those constituents allowed the bulk substance's physical properties to be computed: its modulus of elasticity, for example. Space-time wasn't made of atoms, but its properties could be understood by viewing it as being "built" from a hierarchy of ever more convoluted deviations from its apparent state of continuity and mild curvature. Quantum gravity had explained why observable space-time, underpinned by an infinite number of invisible knots and detours, behaved as it did in the presence of mass (or energy): curving in exactly the fashion required to produce the gravitational force.
ATM theorists were striving to generalize this result: to explain the (relatively) smooth ten-dimensional "total space" of the Standard Unified Field Theory—whose properties accounted for
Nine spatial dimensions (six rolled up tight), and one time, was only what total space appeared to be if it wasn't examined too closely. Whenever two subatomic particles interacted, there was always a chance that the total space they occupied would behave, instead, like part of a twelve-dimensional hypersphere, or a thirteen-dimensional doughnut, or a fourteen-dimensional figure eight, or just about anything else. In fact—just as a single photon could travel along two different paths at once—any number of these possibilities could take effect simultaneously, and "interfere with each other" to produce the final outcome. Nine space, one time, was nothing but an average.
There were two main questions still in dispute among ATM theorists:
What, exactly, was meant by "all" topologies? Just how bizarre could the possibilities contributing to the average total space become? Did they have to be, merely, those which could be formed with a twisted, knotted sheet of higher-dimensional plastic—or could they include states more like a (possibly infinite) handful of scattered grains of sand— where notions like "number of dimensions" and "space-time curvature" ceased to exist altogether?
And: how, exactly, should the average effect of all these different structures be computed? How should the sum over the infinite number of possibilities be
On one level, the obvious response to both questions was: "Use whatever gives the right answers"—but choices which did that were hard to find… and some of them smacked of contrivance. Infinite sums were notorious for being either intractable, or too pliable by far. I jotted down an example—remote from the actual tensor equations of ATMs, but good enough to illustrate the point:
Let S = 1-1+1-1+1-1+1- …
Then S = (1-1) + (1-1) + (1-1) + … = 0 + 0 + 0 … = 0
But S = 1 + (-1+1) + (-1+1) + (-1+1) … = 1 + 0 + 0 + 0 … = 1
It was a mathematically naive "paradox"; the correct answer was, simply, that this particular infinite sequence didn't add up to any definite sum at all. Mathematicians would always be perfectly happy with such a verdict, and would know all the rules for avoiding the pitfalls—and software could assess even the most difficult cases. When a physicist's hard-won theory starred generating similarly ambiguous equations, though, and the choice came down to strict mathematical rigor and a theory with no predictive power at all… or, a bit of pragmatic side-stepping of the rules, and a theory which churned out beautiful results in perfect agreement with every experiment… it was no surprise that people were tempted. After all, most of what Newton had done to calculate planetary orbits had left contemporary mathematicians apoplectic with rage.
Violet Mosala's approach was controversial for a very different reason. She'd been awarded the Nobel prize for rigorously proving a dozen key theorems in general topology—theorems which had rapidly come to comprise a standard mathematical toolbox for ATM physicists, obliterating stumbling blocks and resolving ambiguities. She'd done more than anyone else to provide the field with solid foundations, and the means of making careful, measured progress. Even her fiercest critics agreed that her
The trouble was, she told her equations too much about the world.
