Reiliya pulled two journals from the thick stack in her hands: "The hottest topic this month is your Mathematical Foundations, Professor. The Barber's Paradox has drawn the most attention—it's been called a great crisis in the field of mathematics."
Though she had become proficient in mathematics under Lucian's tutelage, she was fundamentally still an Archanist specializing in elements and alchemy. Mathematics was something she applied rather than researched, so when it came to "a great crisis in mathematics," she didn't share the anxious, pacing bewilderment of the high-tower Archanists. Instead, there was a touch of excitement—the kind that came from watching a spectacle unfold.
Lucian took Arcane and Nature and found that the President, his mentor, Grandmother Hathaway, and other great Archanists had all written papers specifically addressing Mathematical Foundations. On one hand, they fully affirmed the vital role Mathematical Foundations played in the development of mathematics, conducting varying degrees of research on the numerous mathematical branches that had emerged because of it. On the other hand, they all pointed out, without exception, that resolving the Barber's Paradox must depend on the development of set theory itself—on its rigorous definition.
Although before Lucian wrote Mathematical Foundations, the Arcane Magic Parliament had not yet formed a comprehensive set-theoretical framework, many Archanists skilled in mathematics had already recognized that starting from set theory and combining it with other concepts, one could construct the entire edifice of mathematics. It was the bedrock of all mathematical achievements. Therefore, not only did the Great Archanists hope to preserve all valuable content within the current set theory while establishing rigorous definitions to avoid paradoxes, but the papers of other high-tower Archanists also reflected this view.
However, their papers each revealed different inclinations. Some Archanists started from the premise that mathematics was equivalent to logic, represented by Douglas and Hathaway. Others believed that mathematics could be a completely formalized symbolic system requiring no real-world content—as long as it was internally consistent, research could continue. This radical position was represented by Fernando, and Lucian's axiomatic framework concept also belonged to this camp. Beyond these, there were yet other tendencies.
Of course, these tendencies fell within the realm of one's own worldview and values. It was difficult to declare anyone definitely right or definitely wrong. All of them could promote the development of mathematics, though to varying degrees.
"It seems the Archanists haven't been frightened by the Barber's Paradox. Instead, they're motivated to research how to solve it," Lucian said with a chuckle.
Heidi, Cherry, and the others quietly shot their teacher a glance. As if the Barber's Paradox hadn't been proposed by him.
Flipping through casually, Lucian came across Prognosticator Begner's paper on "Goldbach's Conjecture." He admitted that a proof was currently impossible but believed one could approach it step by step—first proving that an even number could be expressed as the product of m primes plus the product of n primes, in other words, proving "m+n" first. Once both m and n were reduced to 1, the problem would be solved.
"Your conjecture has been jokingly called proving '1+1,' which is quite amusing," Heidi said with a grin. "But Lord Begner hasn't even proved '9+9' yet, so there's still a long way to go before solving it. Professor, you haven't actually solved the conjecture already and then put it out there just to watch everyone rack their brains, have you?"
"I really haven't solved it," Lucian said, shaking his head with an amused smile. He could already imagine Nature in the years to come, filled with "m+n" papers. Perhaps in a few decades, someone might reach "1+2"...
After discussing the latest journals, Lucian was about to enter his office when Heidi suddenly asked, "Professor, I heard the Kingdom is going to hold a Lontat Music Festival, and the opening performance will be your opera, The Valkyrie?"
"Assuming nothing unexpected happens, that should be the case," Lucian replied without denying it.
"Oh, how exciting!" Heidi, Reiliya, and the other students and assistants exclaimed, their eyes sparkling. Over these years, the professor had devoted the vast majority of his energy to research in Arcane Magic and its related arts. In the realm of music, apart from that classic and unconventional violin concerto, there had only been a few simple, short piano pieces and light compositions—truly disheartening for those among them who loved great music. This opera, which the professor had supposedly been writing for several years, was sure to be perfect, brimming with classic melodies and arias.
Sprint quietly curled his lip. He couldn't understand their enthusiasm. He listened to music when it was available and didn't care when it wasn't. Was there really any reason to get this excited?
Amid the chatter of Reiliya, Heidi, and the others, Lucian walked into his office and stared blankly at the specialized paper for academic submissions before him.
After a moment, he picked up his Feather Pen, dipped it in Arcane Magic ink, and wrote a title: "Determinism, Self-Consciousness, and the Origin of Arcane Magic."
Though the title of this paper bore a resemblance to Mr. Schrödinger's famous "Determinism and Free Will," the content shared only a few common points. After analyzing the inherent contradictions within determinism itself, Lucian introduced the concept of self-consciousness and then steered the discussion toward the relationship between the "observer effect" and "the origin of Arcane Magic." This, too, had no theoretical proof—it was a possible model in the vein of "the Negative-Energy Vacuum Sea."
"...It is precisely because consciousness exerts an influence on microscopic particles that our material foundation appears insufficiently 'stable,' allowing all manner of changes to occur under the influence of spiritual power and the corresponding Arcane Magic patterns, thereby producing the strange and wondrous Arcane Magic. For instance, some of our transformation spells can be described as the reorganization of the body at the level of microscopic particles... Therefore, Arcane Magic without a permanent attached effect always has a duration. In the past, we explained this phenomenon as being rejected by 'nature,' but now it seems more fitting to describe it as being influenced by other 'observers,' resulting in collapse..."
"The mutation of bloodlines may also stem from this kind of microscopic 'instability'—quantum leaps, for example..."
"...Of course, much Arcane Magic requires tremendous amounts of energy—far more than our spiritual power can supply. So the question is: where does this energy come from?"
"...In certain transformation spells, material decreases or increases before and after the transformation. The decrease is understandable—that material hasn't vanished but is stored in a temporary spacetime created by the Arcane Magic. Once the spell's duration expires, it naturally returns to its original state. But what about the increase? Can matter appear from nothing? Moreover, according to the conservation of mass and energy, the energy required would be staggering. So where does it come from?"
"I hypothesize that when using this type of transformation spell, what we obtain from the outside world is not energy but fundamental matter. But where does that matter come from?"
"...Perhaps the existence of the 'True World' could resolve our questions. But in what form does it exist? Does it exist in every corner of the universe, rendering absolute vacuum impossible? What is its essential nature? And why can it provide the fundamental elemental matter?"
"Can Meditation be described as a process of strengthening self-consciousness? The growth of a mage is a journey from a weak observer to a strong observer..."
After finishing this lengthy paper, Lucian picked up the sheets and blew gently on them. "It just needs to give people the impression that I drew inspiration from 'the Negative-Energy Vacuum Sea.' Hmm, and I need to maintain the paper's internal logical coherence..."
After muttering to himself, Lucian set down his pen and paper and once again stared at the paper in a daze. "How exactly should I explain transformation spells? What is their true principle..."
That afternoon, after casually submitting the paper to the Mage Administration Department, Lucian returned to the "Atomic Universe" and found Natasha sitting on the sofa in a daze.
Was today the couple's day for staring into space? Lucian chuckled to himself and waved a hand. "What's wrong?"
Natasha, dressed in a purple court gown, started awake and smiled sheepishly. "The Lontat Music Festival is getting closer and closer, so I'm feeling a bit nostalgic for Altor."
"I can tell. Sometimes I find myself reminiscing about life in Altor as well," Lucian said with a smile, taking Natasha's hand. "Since that's the case—let's turn thought into action. We'll go take a stroll through Altor right now."
"What?" Natasha was taken aback for a moment. Since when was Lucian the impulsive type—without any prior planning?
But she quickly recovered and said with a brilliant smile, "Wonderful! We'll have dinner in Altor."
A casual "stroll" like this didn't seem to require any planning!