Diaspora

In the Truth Mines, though, the tags weren't just references; they included complete statements of the particular definitions, axioms, or theorems the objects represented. The Mines were self-contained: every mathematical result that fleshers and their descendants had ever proven was on display in its entirety. The library's exegesis was helpful—but the truths themselves were all here.

The luminous object buried in the cavern floor broadcast the definition of a topological space: a set of points (the sparks), grouped into "open subsets" (the contents of one or more of the membranes) which specified how the points were connected to each other—without appealing to notions like "distance" or "dimension." Short of a raw set with no structure at all, this was about as basic as you could get: the common ancestor of virtually every entity worthy of the name "space," however exotic. A single tunnel led into the cavern, providing a link to the necessary prior concepts, and half a dozen tunnels led out, slanting gently "down" into the bedrock, pursuing various implications of the definition. Suppose T is a topological space… then what follows? These routes were paved with small gemstones, each one broadcasting an intermediate result on the way to a theorem. Every tunnel in the Mines was built from the steps of a watertight proof; every theorem, however deeply buried, could be traced back to every one of its assumptions. And to pin down exactly what was meant by a "proof," every field of mathematics used its own collection of formal systems: sets of axioms, definitions, and rules of deduction, along with the specialized vocabulary needed to state theorems and conjectures precisely.

When ve'd first met Radiya in the Mines, Yatima had asked ver why some non-sentient program couldn't just take each formal system used by the miners and crank out all its theorems automatically sparing citizens the effort.

Radiya had replied, "Two is prime. Three is prime. Five is prime. Seven is prime. Eleven is prime. Thirteen is prime. Seventeen is-"

"Stop!"

"If I didn't get bored, I could go on like that until the Big Crunch, and discover nothing else."

"But we could run a few billion programs at once, all mining in different directions. It wouldn't matter if some of them never found anything interesting."

"Which 'different directions' would you choose?"

"I don't know. All of them?"

"A few billion blind moles won't let you do that. Suppose you have just one axiom, taken as given, and ten valid logical steps you can use to generate new statements. After one step, you have ten truths to explore." Radiya had demonstrated, building a miniature, branching mine in the space in front of Yatima. "After ten steps, you have ten billion, ten to the tenth power." The fan of tunnels in the toy mine was already an unresolvable smear-but Radiya filled them with ten billion luminous moles, making the coal face glow strongly. "After twenty steps, you have ten to the twentieth. Too many to explore at once, by a factor of ten billion. How are you going to choose the right ones? Or would you time-share the moles between all of these paths—slowing them down to the point of uselessness?" The moles spread their light out proportionately-and the glow of activity became invisibly feeble. "Exponential growth is a curse in all its forms. You know it almost wiped out the fleshers? If we were insane enough, we could try turning the whole planet—or the whole galaxy—into some kind of machine able to exert the necessary brute computational force… but even then, I doubt we'd reach Fermat's Last Theorem before the end of the universe."

Yatima had persisted. "You could make the programs more sophisticated. More discriminating. Let them generalize from examples, form conjectures… aim for proofs."

Radiya had conceded, "Perhaps it could be done. Some fleshers tried that approach before the Introdus—and if you're short-lived, slow, and easily distracted, it almost makes sense to let unthinking software find the lodes you'd never hit before you died. For us, though… Why should we sacrifice the opportunity for pleasure?"

Now that ve'd experienced Truth Mining for verself, Yatima could only agree. There was nothing in any scape or library file, any satellite feed or drone image, more beautiful than mathematics. Ve sent the scape a query tag, and it lit the way to the Gauss-Bonnet Theorem with an azure glow for vis viewpoint only. Ve floated off slowly down one of the tunnels, reading all the tags from the jeweled path.