The top floor of the tower soaring into the clouds.
"The Prophet" Begner had a copy of *Mathematical Foundations* spread open before him. Well before *Nature* published its special issue, he had already gotten word and obtained a transcribed manuscript directly from Levsky.
"I have to say, although Evans's book lacks groundbreaking insights and mathematical discoveries, by introducing new concepts and redefining certain existing ones, he has ingeniously woven together the numerous fragmented research results in the current mathematical field into a cohesive whole — forming a complete system with clearly defined branches, rigorous structure, and a solid foundation..."
"...And his deeper elaboration of this system has completely dispelled the fog shrouding the mathematical field, finally allowing the entire mathematical palace to bask in brilliant sunlight, revealing its glorious golden splendor..."
Even during his first reading of this book, Begner had voiced similar sentiments, but with each rereading, he couldn't help but marvel anew. Since the era of calculus and the definition of its corresponding concepts, this was the first time someone had made such a monumental contribution to the mathematical field through a single book.
"...The era of calculus is past. This is modern mathematics..."
Begner seemed to see new branches taking shape within the mathematical palace, their origins and foundational concepts all tracing back to this *Mathematical Foundations*!
For him, two aspects were of paramount importance. First was the axiomatized system of tower geometry that Lucian had completed, which filled him with boundless passion for mathematics in similar domains — as if everything in the world could be axiomatized, could form a rigorous, self-consistent, flawless system.
The second was Lucian's deeper research into topology, groups, and set theory, which sparked a burst of inspiration in him. These didn't seem to be merely pure, independent, abstract mathematical results — they could be directly applied to solve many problems in current Archanist research, such as crystallography and the cutting-edge fields of the microscopic world.
"...This is absolutely the most groundbreaking book in mathematics of the last hundred years. It will surely be enshrined by future Archanists in the halls of mathematics, worshipped alongside *Mathematical Principles of Arcane Philosophy*. If only the ten problems in the final appendix didn't exist..." A faintly bitter smile crossed "The Prophet" Begner's face, and the copy of *Mathematical Foundations* before him lay open to "Unresolved Problems in Current Mathematical Research."
For a legendary mage like him who excelled in mathematics, these difficult problems were irresistible. After skimming through the earlier sections, he quickly sank into deep thought and calculation.
But from the day he received *Mathematical Foundations* until now, an entire week had passed, and he had made no progress. He hadn't solved a single problem — not one. This left him, who took great pride in his mathematical expertise, feeling quite defeated. Some of the problems seemed simple on the surface, as though they could be easily proven, but the actual process was fraught with unimaginable difficulties.
Of course, none of that compared to the final problem — the Barber's Paradox — which was a truly horrifying question that drove Begner into a rare hour-long stupor.
"Until determinism is completely refuted, there is nothing more painful and embarrassing for Tower Archanists than this. Just as we celebrate the final completion of the mathematical palace, its very foundation suddenly collapses — as though applying set theory and accepting the old concepts would negate mathematics itself."
Begner let out a long sigh. The moment he saw the Barber's Paradox, he had nearly succumbed to the urge to destroy *Mathematical Foundations* with magic and eliminate every last barber — as though infinite catastrophe would otherwise descend upon the world!
His gaze drifted downward, landing on the words Lucian had written after the paradox: "Paradoxes like this are deeply frustrating, but they merely indicate that our research still has many areas lacking rigor — that our previous understanding of mathematics contained certain biases. Therefore, we must not despair or lose our way. Instead, we should conduct deeper research and delve further into set theory. For mathematical problems, there is one and only one solution: continue researching mathematics itself."
"...It is Evans's Archanist spirit that forms the foundation supporting all his achievements." Begner praised with admiration, then flipped back to the section on set theory and resumed his thinking and research. The paradoxes of set theory could only be solved from within set theory itself.
......
"...You're listening to your old friend 'Nightingale,' and now it's time for 'Arcane News'..."
General school was still on holiday, so Langman had to rely on programs like "Voice of Mysteries" and "Channel of World Truth" to keep up with the latest developments in Archanist research.
"...Lord Evans has completed his magnum opus *Mathematical Foundations*, resolving one difficult problem after another in the current mathematical field and clearing the path forward. But the most eye-catching aspect isn't the earlier content — it's the ten difficult problems proposed at the end. They've sparked tremendous interest among Archanists..."
"...It must be said, the problems that have stumped Lord Evans are truly extraordinary. So far, not a single Archanist has claimed to have found an approach. Even the Chairman and Lord Brook have publicly admitted that solving these seemingly simple problems is far from easy..."
"...It appears that whoever solves all ten problems will be rewarded with generous arcane points and earn recognition as an established mathematical authority. These problems are..."
When young Langman heard about problems that had stumped Lord Evans, his eyes widened. Amid his disbelief, an eager impulse arose — if he could solve a problem that even Lord Evans couldn't...
The problems Lucian had selected were each representative in their own way, yet all belonged to the category of those that appeared simple on the surface. This meant that even with his limited foundation, Langman could follow along. He tentatively took out paper and pen and began working through calculations.