Night deep, most windows of the "Contemplation Building" housing the String‑Light Research Institute's Mathematics Division had fallen into darkness; only the top‑floor office belonging to Yue'er still emitted warm, persistent light. Inside, the scene was spectacular, verging on chaotic. Four huge erasable glass panels dominated the main walls, densely covered with mathematical symbols, definitions, lemmas, and connecting arrows—like overly‑lush vines intertwining, covering originally transparent surfaces. On the side desk, spread‑open books and printed‑out papers stacked mountains‑high; some page‑edges bore colorful tabs, others covered with annotations in different‑colored inks. The air held the scent of old paper, ink, and a nearly‑stagnant, intensely‑focused thought atmosphere.
Yue'er sat curled in a wide office chair, unconsciously twirling a pen that had run out of ink, her gaze unfocused on a glass panel opposite filled with mappings between algebraic varieties and complexity classes. She had maintained this posture nearly an hour. External stillness concealed a fierce, disordered storm within her skull.
She was attempting to construct a more universal mathematical framework for the "metasurface inverse‑design problem" Xiuxiu had raised. This wasn't merely solving a specific engineering‑optimization problem; she ambitiously hoped to incorporate it into the grand vision of her brewing "information‑geometric field theory," treating it as a special case and excellent validation of "encoding continuous‑field information with discrete structures."
Yet progress had bogged down. She tried multiple existing optimization theories—from classical calculus‑of‑variations, to modern non‑convex optimization, to some random‑search algorithms inspired by neural networks. Theoretically they seemed feasible, but when she attempted coupling them with Maxwell's equations for electromagnetic fields, nanostructure fabrication constraints, and stringent target‑light‑field requirements, the model either became unbearably bloated—computational‑complexity exploding exponentially—or the "optimal solutions" obtained were physically meaningless, or extremely sensitive to tiny perturbations—impossible for actual manufacturing.
The logical chain seemed to have reached a dead end. She followed strict mathematical norms, deriving step‑by‑step, checking each premise, ensuring rigorousness of every inference step, yet what finally emerged were either dead‑ends or seemingly‑correct yet useless mediocre conclusions. This brought a familiar sense of frustration—like a master craftsman possessing all the best tools, yet unable to carve that vaguely‑envisioned perfect form.
Amid this extreme mental impasse, an indescribable feeling—like deep‑water undercurrents—began swirling at her consciousness's edge. It had no shape, no sound, no logical expression. Merely a strong "sense of direction," an almost instinctive guidance. It seemed to say: *"Forget those complex constraints, temporarily set aside Maxwell's equations, look at the overall 'shape' of that high‑dimensional parameter space… notice those 'singularities'… the connection method is wrong—not linear superposition, perhaps some 'parallel transport' on a fiber‑bundle…"*
This feeling Yue'er was not unfamiliar with. In her past mathematical career, before several key breakthroughs, similar seemingly‑mystical moments had occurred. It wasn't reasoning, not calculation—more like a sudden "seeing," as if a fog of thought were briefly illuminated by a light of unknown origin, revealing contours of a hidden path. This was what mathematicians privately call, yet in formal papers never mention—"**mathematical intuition**."
She abruptly stood from the chair, walked to a relatively blank glass panel, grabbed a blue marker. Without deliberate thinking, the pen‑tip seemingly moved of its own will, swiftly sketching on the surface. She drew a schematic of an abstract high‑dimensional manifold, marked several key points, then connected them with winding curves, writing beside them key mathematical symbols and seemingly‑unrelated topological terms. The entire process flowed smoothly, with an irrational fluidity.
When she stopped, stepped back to examine, her heart involuntarily beat faster. This suddenly emerging composition—the idea of treating "meta‑atom" design‑parameter space as a fiber‑bundle with special connection, optimizing process as seeking specific "holonomy"‑action path on that bundle… it looked so bizarre, even somewhat absurd, utterly different from all rigorous paths she'd previously attempted. Logically, it was currently chaos—full of unproven assertions and huge leaps.
Yet a deep, nearly irresistible "**sense of conviction**" seized her. She "felt" that this road, though seemingly thorn‑ridden, might be directionally correct. It touched the problem's more essential geometric structure.
This strong, intuition‑born "conviction" clashed sharply with her strict mathematical training. Mathematics—this discipline hailed as humanity's rational pinnacle—its cornerstone is logic and proof. From Euclid's *Elements* to Hilbert's formalist program, every brick of mathematics' edifice must be laid through impeccable logical deduction. Intuition—that vague, subjective, inexpressible feeling—has no place in mathematics' formal hall; even considered a temptation requiring vigilance, possibly leading to error.
Yue'er plunged into deep philosophical contemplation. She began seriously examining what role "mathematical intuition" plays in mathematical discovery and proof. Is it friend or foe to logical deduction?
She walked to the bookshelf, pulled out several works on philosophy of mathematics and psychology of mathematical discovery. **Poincaré**—that great mathematical genius—vividly described in *Science and Method* his experience discovering Fuchsian functions. He mentioned that after long, conscious yet fruitless work, answers often suddenly, clearly emerged when he set work aside—riding a bus, walking by the sea. He attributed this to the "subconscious self," suggesting it works below consciousness's threshold, combining and filtering mathematical elements in ways we cannot perceive, finally delivering the most valuable combinations into consciousness domain.
**Hardy**, another mathematical giant, also acknowledged intuition's importance; he believed mathematicians first "see" truth via intuition, then seek proving paths.
Yue'er realized intuition might not be logic's opposite, but rather a precursor and catalyst for logical exploration. It's like a prospector in vast unknown territory—based on experience, wind‑direction, even a faint, elusive scent—sensing where mineral veins might lie. It itself cannot provide ore, yet indicates where to dig. Without intuition, mathematicians might blindly wander in logical labyrinths, exhausting energy without touching core; without logic, the "gold‑mine location" intuition provides could never be mined and verified—remaining mere speculation, fantasy.
Intuition provides strategic blueprint; logic handles tactical construction and quality‑check. The two complement each other; both indispensable. Especially facing extremely complex, frontier problems without ready‑made paths—like "information‑geometric field theory" or "metasurface inverse‑design"—intuition's guiding role becomes crucial. It can leap across chasms logic steps cannot yet fill, directly "perceiving" hidden harmonies and connections deep within mathematical structures.
Yet relying on intuition is dangerous. It can err; can lead astray. In history, how many genius intuitions ultimately proved beautiful fallacies? How to distinguish genuine mathematical insight from pure subjective imagination? The only touchstone remains ruthless logical proof. Intuition proposes bold conjecture; logic cautiously verifies.
Thinking here, Yue'er felt relief, also grew more cautious. Looking at the intuition‑driven sketch on the glass panel, she no longer felt anxious due to its lack of strict logical foundation, nor blindly believed it necessarily correct. She regarded it as a highly valuable "**working hypothesis**"—an exploratory starting point requiring immense effort to verify and solidify.
At that moment, the internal communicator emitted a gentle tone. Mozi. This late, he usually wouldn't disturb her, unless something urgent, or… simply sensing she might still be awake.
Yue'er answered video; Mozi's face appeared on‑screen—background his office overlooking city nightscape; he too looked somewhat weary, yet eyes still sharp.
"Still wrestling your mathematical ghosts?" Mozi asked smilingly, tone carrying familiar concern.
Uncharacteristically, Yue'er did not directly discuss mathematics; instead, with a hint of puzzlement and exploratory tone, described her recent experience—that impasse after logic's dead‑end, the sudden, inexpressible intuitive guidance, and her ensuing thoughts about intuition‑logic relationship.
"…It's like a flash—no reason, yet you feel you've seen something." Yue'er sought accurate words. "But it's so vague, so unreliable. I cannot prove it; can't even clearly express it."
Surprisingly, Mozi showed almost no lack of understanding. He listened intently, then thoughtfully nodded.
"I understand this feeling." Mozi said, leaning forward slightly. "In the trading room, sometimes it's like that. Facing massive, noisy, seemingly chaotic market data—all models may give contradictory signals, or simply fail. In that extreme uncertainty, occasionally, a very strong 'feeling,' or 'trading sense,' tells you whether to go long or short, whether a key support level might be broken. This 'feeling' often comes quickly, without complex reasoning, yet its post‑fact verified accuracy sometimes is astonishingly high."
Yue'er's eyes widened; she hadn't expected this seemingly‑different financial field to have similar phenomena.
"I've thought before why this is." Mozi continued, eyes flashing analytical light. "Later I believed perhaps this isn't mysticism. Our brains—especially subconsciousness—process information far beyond our imagination. It might, beyond our conscious awareness, have integrated thousands of tiny signals we're not even clearly aware of, hidden patterns, historical experiences. The mathematical intuition you rely on may be your brain—based on your vast mathematical‑knowledge base and long‑term deep thinking—performing a huge‑scale, subconscious 'computation' and 'pattern recognition,' finally pushing a condensed 'result'—that intuition—to your consciousness level. And my 'trading sense' might also be based on my latent understanding of market‑historical data, macroeconomic indicators, news sentiment, even counterparty‑behavior patterns—rapidly integrated and computed by subconsciousness, yielding a probabilistic judgment."
"Subconscious computation…" Yue'er murmured repeating the phrase, feeling new light illuminate her confusion. Mozi's analogy pulled that mysterious "intuition" into an understandable, almost‑scientific category. It wasn't divine revelation, not inspiration without source—but the product of brain—this extremely‑complex information‑processing system—working below threshold. It still builds upon knowledge and experience, yet its operation transcends conscious monitoring and linear expression in logical language.
"So, you needn't feel uneasy about relying on intuition." Mozi concluded, tone assured. "Just as I don't completely ignore validated 'trading sense.' What matters is how to treat it. In my domain, I test such 'feelings' with small positions, verify with real money; if effective, gradually increase investment. In your domain, perhaps it's as you're doing now—treat it as a powerful conjecture, then try proving it with the strictest logical tools, or… disproving it."
Yue'er felt the boulder in her heart about "intuition's legitimacy" seemingly gently moved by Mozi's words. She was no longer alone facing this tension between rationality and supra‑rationality. On different battlefields, her companion experienced similar processes in different forms.
"Thank you, Mozi." She said sincerely, face showing a long‑absent, relieved smile. "Your 'trading sense' analogy makes me feel much better. It makes me feel intuition perhaps isn't reason's traitor, but its silent, efficient assistant."
"Glad it helps." Mozi also smiled. "Very late now; don't over‑exhaust. Sometimes, setting aside the problem, letting your 'subconscious' work, might be more effective than continuously consuming consciousness."
After ending communication, Yue'er again turned her gaze to that glass panel filled with intuitive sketches. Her mindset had changed. She no longer fretted over its origin "orthodoxy"; she viewed it as a secret report from her deep‑level wisdom. She returned to her desk, opened a new document, began attempting to translate those vague lines and symbols into more precise mathematical language—constructing the first logical foothold for this intuition‑born conjecture.
She knew this would be a long road—requiring unimaginable mental effort to deduce, compute, verify; might even ultimately prove this intuitive path impassable. Yet now, her heart brimmed with calm and strength. She respected the intuitive spark that momentarily illuminated darkness; she trusted the logical‑paved climbing path, step‑by‑step, beneath her feet. In this eternal dance between intuition and logic, she had found her place and rhythm as an explorer. Night deepened further, yet her office's light—like the intertwined inspiration‑and‑reason thought‑fire in her mind—remained stable, brightly burning.