Alex watched as the assembled workers, one by one, stepped toward him and the great swiveled bulk of the resonator. Their silent vote was unanimous. “Good,” he said, feeling a wave of warmth toward his comrades. “Let’s get to work then. I had a dream not long ago, and it gave me an idea. A possible way we just might get the good colonel’s attention.”

Goldman didn’t have much to do anymore. Others ran the scans now, reduced the data, constructed ever-subtler models of the inner Earth, even traced the involute geometries of that refulgent, renitent entity below… the thing called Beta. A midget town had sprung up around the lonely

Tangoparu dome on a rocky plain below the vast Greenland ice sheet. High-powered tech types bustled with armloads of data cubes, arguing in the arcane new language of gazerdynamics. Of the original team, only he remained now, the others having gone home to New Zealand long ago.

The NATO scientific commander had specifically asked him to stay. So Stan sat in on all the daily seminars, struggling to keep up with younger, more agile minds, even though his understanding grew more obsolete with each fast-breaking discovery. No matter. They all treated him with utter deference. Hardly a moment passed without hearing the name Alex Lustig spoken with an awe customarily lavished on the shades of Newton and Einstein and Hurt, and as the great one’s former teacher, Stan shared in that glory.

Singularities. There was a lot of talk about singularities, by which the bright young men and women meant the kind you made inside a cavitron — micro black holes and those newer innovations, tuned strings and cosmic knots. Of late, though, Stan had found himself thinking about another kind of “singularity” altogether. It was on his mind as he passed a saluting sentry and left the bustling encampment, swinging his walking stick across the moraine-strewn valley.

In mathematics, a singularity is a sudden discontinuity, where one expression suddenly ceases being valid, and a completely different one takes its place.

You got the simplest kind of singularity — a delta function — by dividing any real number by zero. The result, converging on infinity, was actually undefined, unknowable. That’s where we’re at right now… a singularity in the life history of mankind.

It wasn’t just the present crisis. Oh, certainly he was worried. Would the world’s institutions — or the planet itself — survive the next few hours or days? Stan was as concerned as the next man. Still, even if tomorrow the spectre of reborn international paranoia evaporated like a bad dream, and all the gorgeous, terrifying new technologies were tamed, nothing would ever be the same.

Earlier today, some of the youngsters had been discussing notions about gravitational circuits… equivalent, in collapsed mass and stressed space, to capacitors and resistors and transistors, for heaven’s sake! To Stan it was proof the time had come. The moment he’d secretly been waiting for all his life.

There’s another kind of singularity… having to do with society, and information.

Technological breakthroughs had happened before — when farming was invented, for example. Or metallurgy. Or writing. Each time, men and women gained new power over their lives, and thinking itself changed. With each such naissance, human beings were in effect reborn, remade… reprogrammed.

In early times, change came slowly. But each breakthrough laid a foundation for those that followed. And with the Western breakout of the sixteenth century, it became self-sustaining. Inventions bred wealth, which spread education and leisure to broader masses. Printing dispersed literacy. Transport distributed food. Food meant more people.

He paused near a sandy bank in the wind-shadow of a boulder, and used his walking stick to trace a rough figure. It was the standard doom scenario, depicting the fate forecast by Malthus for any species that outbreeds the carrying capacity of its niche.

The curve portrayed human population over time, and it rose very slowly at first. All through the late Stone Age — when Stan’s ancestors had chipped flint, scratched fleas, and thought fire the final terror weapon — there were never more than five million homo sapiens at a time. This changed with agriculture, though. Human numbers doubled, then doubled again every fifteen hundred years or so — a rapid climb — until they reached five hundred million around the time of Newton.

Impressive progress, achieved by people who had hardly a glimmer of what the laws of nature were, let alone concepts like ecology or psychology or planetary history. But then it accelerated even faster! New foods, sanitation, emigration… babies lived longer. Humans reproduced copiously. The next doubling, to a billion souls, took only two hundred years. The next, less than a century. Then, from just 1950 to 1980, two billions became four. And still the curve steepened. Stan recalled the elegant, symmetrical projections proclaimed by pessimists when he was young. No population boom can be sustained forever on a finite world. There must inevitably come crash.

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The curve never reached infinity after all. It peaked. Then, like a spent rocket, it turned over and plummeted. The great die-back, that’s where we seemed headed. After all, it happens whenever anchovies and deer breed beyond their food supply.

And we did have little die-backs. But so far we’ve escaped the big one, haven’t we?

So far.

He scratched another rude figure, identical to the first until it reached the top of the curve. At which point the population stopped growing all right, but neither did it fall! Instead of plummeting, this rocket turned sideways.

This is what they say can happen if you add intelligence and free will to the formula. After all, we aren’t deer or anchovies!

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Two graphs. Two destinies. Malthusian calamity and the so-called S-curve. On the one hand, utter collapse. And on the other, a chain of last-minute reprieves… like self-fertilizing corn, room-temperature superconductors, and gene-spliced catfish… each arriving just in time for mankind to muddle through another year, eking out a living from one brilliant innovation to the next.