In antiquity, when magic was still shrouded in mysticism and arcana was indistinguishable from spellcraft, geometry's role in magic circles elevated mathematics to sacred status.
Scholars proclaimed "All is Number"—that math, derived purely from logic, was infallible.
Then came the pentagram.
A mage discovered that the ratio of a regular pentagon's side to its diagonal could not be expressed rationally. For this heresy, he was bound and hurled from a tower.
The "Irrational Number Crisis" that followed had a silver lining: it taught mages to distrust intuition, prizing deductive proof instead. This shift birthed axiomatic geometry—a foundation for arcana's eventual rise.
Second Crisis: The Infinitesimal
During the Glorious Age, Grand Arcanists Isaac and Leibniz (whose manuscripts Turing had cited) independently invented calculus.
Yet their infinitesimals—quantities vanishingly close to zero—sparked a century of debate: Were they logical? Were they even real?
The "Infinitesimal Crisis" only resolved when later arcanists like Cauchy and Weierstrass arithmetized analysis, rigorously defining limits and continuity.
The fallout? Calculus shed its mystical veneer, becoming accessible to all—even non-mages.
Third Crisis: The Paradox
The current upheaval began when Bertrand Arthur, an arcanist from the Arcane Federation, uncovered a set-theoretic paradox—epitomized by the "Barber's Paradox":
A barber shaves all who do not shave themselves. Who shaves the barber?
If he shaves himself, he violates his rule. If he doesn't, he must.
This self-referential loop erupted just two years after Grand Arcanist Poincaré's boast: "Mathematical rigor is now complete."
The crisis still rages. Turing's paper—pondering the "halting problem" (another self-referential quandary)—is one attempt to navigate it.
Lanen's Epiphany
"Literature builds cathedrals of beauty," Lanen mused, massaging his temples. "But mathematics constructs the cathedral of reason."*
To glimpse the vistas he yearned for, becoming a Grand Arcanist was essential—and that meant ascending reason's edifice.
Closing the last book, he shelved it with care. Time to switch gears.
Between study sessions, Lanen indulged in hands-on experiments—particularly with magic runes.
His current obsession?
Repeaters.