The library of the Institute for Advanced Study at Princeton resembled a sanctuary of knowledge—dome high and distant, walls encircled by heavy bookshelves reaching ceiling‑ward, air sedimented with the tranquility and profundity left by countless wise minds clashing across centuries. Yue'er was accustomed to this atmosphere; each breath seemed to draw lingering sparks of inspiration from departed souls. Today, however, what lay open before her wasn't some predecessor's masterpiece but her own constructed mathematical‑framework sketch attempting to connect the P‑versus‑NP problem with the Langlands program. Lines, symbols, arrows tried to weave a net reaching toward understanding's opposite shore, yet now this net seemed shrouded by an intangible mist—a mist not born from knowledge deficiency but from logic's possibly bottomless abyss.
Her research recently, like an explorer venturing deep into mysterious jungle, gradually touched a core realm that fascinated yet alarmed all mathematicians and computer scientists—the limits of computation, or rather, a territory brimming with unknowns and challenges: "undecidability."
This territory resembled shrouded in thick fog, obscuring truth and mysteries within. Yet this mist's source wasn't untraceable; it could be traced back to that genius British mathematician of last century, Alan Turing.
Turing, this legend in mathematics, with his exceptional intelligence and creativity, laid solid foundation for computer science's development. His theories and thoughts not only influenced academia then but continued sparking deep reflection on computation's essence in subsequent years.
Yue'er's fingertip gently traced the words "Halting Problem" written on scratch paper. This was a thought experiment, a most fundamental insight into computation theory. Turing asked: Does there exist a universal algorithm (or Turing‑machine program) H such that when we hand H **any** program P's code, together with that program P's input data I, H can always determine within finite steps: will program P, upon receiving input I, eventually **halt** (output a result), or will it **loop forever** (unable to stop)?
Intuitively, this should be decidable. Careful examination of program's logic, wouldn't that suffice? But Turing, with his exquisite, self‑referential proof by contradiction, demonstrated such universal program H **cannot exist**. He constructed a "paradox program": assuming H exists, one could construct another program G that uses H to determine "if I feed my own code as input to H, what will H say—halt or not halt?" If H judges G will halt, G intentionally enters infinite loop; if H judges G will not halt, G immediately stops. This creates contradiction, therefore initial assumption (H exists) is false.
This proof's depth lies not in saying some specific program is hard to analyze, but in stating: **For behaviors of all possible programs, no uniform, mechanical decision method exists.** Some programs' fate cannot be predicted beforehand by any algorithm unless you truly run it—and running it might imply endless waiting.
Yue'er attempted a more vivid metaphor to grasp this **undecidability**. Imagine an exquisitely designed labyrinth, walls engraved with complex rules and conditions. You stand at entrance, holding a "labyrinth‑behavior predictor" possibly provided by some mysterious sage. This predictor claims, for any labyrinth and any walking strategy, it can determine whether you'll ultimately emerge.
Now, replace this "labyrinth" with "a program," "emerging from labyrinth" with "program halting." Turing proved such "universal predictor" cannot exist. There always exist some labyrinths (programs) whose internal logic is so self‑referential and entwined that its final outcome (halt or not) is logically unpredictable, independent of any external decision system. You can only jump in, walk through personally, to possibly know result—but cost is, you might never emerge.
This thought like a cold stream pierced Yue'er's attempted mathematical edifice. She'd been exploring P‑versus‑NP problem's deep structure, trying to find unified mathematical framework depicting computational complexity. But what if, P versus NP question itself, bears some **undecidable** hue?
This isn't mainstream conjecture, even sounds somewhat heretical. Most researchers assume P versus NP is a proposition with definite answer (either P=NP or P≠NP), only they haven't found proof or disproof method. But Yue'er's thinking never bound by mainstream. She keenly sensed that hierarchy structures within computational complexity theory, like **polynomial hierarchy (PH)** she previously studied deeply, its infinite extensibility itself hints some "inexhaustible" nature. If PH truly infinite, each layer strictly containing next, then does this imply existence of some computational problems whose complexity so high that cannot be fully captured and decided by any fixed, finite logical system? Further, P versus NP problem, core of complexity theory, its own "truth value" (true or false), might also transcend proof capability of some specific formal system? Like Gödel's incompleteness theorems indicate, any sufficiently powerful mathematical system contains propositions neither provable nor disprovable within it.
This idea made her shudder. If holds, that would mean she and countless colleagues' diligently pursued ultimate answer might itself be an "undecidable" proposition, at least within current human‑grasped (even possibly forever‑grasped) mathematical frameworks. This doesn't imply problem lacks answer, but implies, search‑for‑answer process might resemble trying to measure infinite universe with finite ruler—forever encountering mist beyond scale.
She set down pen, felt slight dizziness. This exploration of cognitive boundaries consumed mental energy more than any specific mathematical derivation. It shook exploration's very foundation. She couldn't help recalling Mozi. He just experienced market's "black swan"—that model‑failure, order‑collapse experience, with what she now contemplated as "undecidability"—how similar philosophically!
Mozi's quantitative models, built on historical data and statistical regularities, attempted to seek order and predictability within financial markets' chaos. But "black‑swan" events precisely are those models cannot cover, even cannot probabilistically describe "undecidable" existences. They exceed models' based axiom systems and probability distributions, just like that program G that cannot be judged by universal halting‑decision program H—only when truly occur ("run"), then reveal their massive, disruptive power. Market's "chaos," its deepest roots perhaps contain such based‑on‑complex‑interactions‑and‑human‑group‑irrationality‑emerged, algorithmic‑level "undecidability."
This cross‑domain association excited her. She longed to share this discovery with Mozi, not specific mathematical details but this joint exploration of "certainty boundaries." She wondered how someone battling real‑world chaos would regard similar boundaries possibly existing in thought‑universe.
She almost eagerly connected video call with Mozi. Screen lit, showing Mozi's slightly weary yet focused face, background his office overlooking Huangpu River.
"Yue'er?" Mozi somewhat surprised—usually this time her deep‑work period.
"Disturbing you," Yue'er's voice carried a barely perceptible urgency, "I just thought of some… boundary questions, want to chat."
Mozi relaxed leaning back chair‑back, made a go‑ahead gesture, eyes revealing interested glow. He always gladly entered her thought‑world, even if merely periphery sensing that pure intellectual exploration's awe.
Yue'er didn't directly throw terms like "halting problem" or "undecidability," instead used that "labyrinth predictor" metaphor. She described that theoretically impossible universal machine judging all labyrinths' fate.
"… So, you see," she concluded, gaze burning at screen's Mozi, "within certain systems, exist some fundamental limits. Not because we're not smart enough, tools not advanced enough, but logic itself drew a line, telling us: some problems, cannot obtain answer through some uniform, mechanical way. This is 'undecidability.'"
She paused, observing Mozi's reaction. He didn't show confusion, but sank into contemplation, fingers unconsciously lightly tapping desk.
"Just like my models," Mozi slowly spoke, voice low, "they can handle 'known unknowns'—risks with historical data, probability distributions. But 'black swans'… they belong to 'unknown unknowns,' or using your words, are 'undecidable' events within model framework itself. Models cannot beforehand 'judge' they'll occur, because they essentially exceed model's 'understood' scope. Only when they truly happen, market this 'program' runs to that step, then we see result—usually catastrophic result."
His understanding so precise, so swiftly establishing connection, made Yue'er's heartbeat skip. This was the conversation she desired—a transcendence beyond professional terminology, resonance at underlying‑logic level.
"Then, Mozi," Yue'er leaned forward slightly, an unconscious gesture indicating engagement, "do you think 'certainty' itself, possesses such boundaries? Exist some domains, some layers, where absolute, predictable certainty, just like that universal labyrinth predictor, essentially doesn't exist?"
This was a grand, nearly philosophical fundamental question. Mozi silent long, outside neon casting changing lights on his face.
"In financial markets, I've long accepted fact no absolute certainty exists," he finally said, tone carrying weathered‑through‑storms' desolation and clarity, "So‑called 'certainty,' merely high‑probability expectation, and contingency plans responding to uncertainty. But…" he shifted topic, gaze turning sharp, "this doesn't mean we abandon seeking patterns and order. Quite opposite, recognizing boundary's existence enables us more clearly explore possibilities within boundary, build more resilient systems, prepare coping with shocks beyond boundary. Like you, Yue'er, even if P‑versus‑NP problem possibly 'undecidable' at some layers, would this stop you continuing exploring its structure, seeking mathematical bridges connecting it?"
"No." Yue'er's reply without hesitation, eyes gleaming firm light, "Even if final answer is 'undecidable,' exploring this process itself, revealed mathematical structures and deep connections, possesses immense value. Like climbing a mountain possibly without peak, scenery along path and understanding of mountain‑body itself, are harvest."
Two separated by screen, exchanged smile. That feeling, as if two explorers climbing respective steep mountains, amid mist‑shrouded gap, saw each other's silhouette, instantly understanding cliffs other faces and courage other holds. This collision and resonance at thought's deepest depth, more profound and moving than any superficial emotional exchange.