It was precisely because he possessed advanced knowledge in mathematics that Wang Qi understood the significance of this mathematical paper more than anyone else.
In discussions of mathematics, anything that can concretely present a certain object or provide a calculation method for it is called constructible. Constructive mathematics is an important field of modern mathematics research, its fundamental characteristic being the emphasis on constructibility. Constructibility refers to the ability to concretely present a certain object or to provide a calculation method for that object.
Constructive mathematics differs from classical mathematics in that constructivism asserts "to exist is to be constructed." To be constructive, mathematicians must reinterpret the existential quantifiers and other logical connectives and quantifiers, so as to explain the meaning of proofs involving these logical expressions from a constructive point of view.
Based on constructive computational theory, there are very strong advantages. It is extremely reliable, unlike set theory and logical mathematics, whose foundations are not solid. However, on the flip side, because it is too stable, it appears very closed off. This theory rejects logical proof, rejects actual infinity, excludes countless practical, known methods. Simply put, it cuts off everything unreliable and imperfect, forming a finite "perfection."
This type of Law Gate with "too much killing power" was precisely what the Master of Calculation rejected. More importantly, it was because this way of thinking prohibited too many methods, leading to mathematicians being hampered in dealing with problems, also failing to have any practical use. As a result, this concept was widely criticized.
But the Mathematics Lord solved this problem.
The Mathematics Lord made new breakthroughs in constructive algorithms, forcefully ignoring the achievements made by Xi Baiche in this field, retaining only the constructive parts and eliminating all non-constructive parts. Such new algorithms were incredibly concise and, because of their constructive nature, they were highly actionable; their potential infinity traits were also better suited for practical applications in the field of computational science.
All along, the Li Sect had despised the continuous sect for the role of mathematical logic in promoting the development of calculators. Still, with this, the continuous sect's theory of mathematics actually surpassed the Li Sect in practicality!
"This... this isn't scientific, is it?" Wang Qi couldn't help but exclaim.
On Earth, constructive mathematics didn't emerge until the 1960s. By this stage, the worldviews of all mathematicians had been through the destructive blows of geniuses like Gödel, Turing, Church, again and again, nullifying countless wrong paths; subsequently, the Bourbaki School, Grothendieck, and many more mathematicians found many new paths. By then, recursion theory and modern mathematical logic had become foundational content, it could be said that the mathematics of this era were worlds apart from early twentieth-century mathematics. On such a fertile ground, the theory of constructive computation was able to take root and sprout.
But in this world, in Shenzhou, Gödel missed this historical gathering, the "golden diagonal" was broken, and Elder Ji Turing couldn't shine as he should have. Since no one doubted that there were discrepancies between semantics and syntax, that is, the defects of human language itself, the Master of Calculation still furiously collided with the southern wall of completeness.
Under such circumstances, such mathematics shouldn't have been possible!
However, upon further reflection, it seems not so impossible. After all, in the history of Earth, Henry Poincaré died too early, missing the major development of mathematics. He allowed Brouwer to take intuitionism into a dead end of his personal philosophical quirks, without witnessing the day when mathematics would give rise to computer science, changing the era. But Mathematics Lord Pang Jialai was always alive!
His accumulation was far beyond that of his Earth counterpart.
"Mathematics Lord's paper was proposed by Earth scientists fifty years after Henry Poincaré's death. It seems in the future, one can't use Earth's history to judge the upper limits of Shenzhou Carefree. Despite countless wrong paths, relying on his intuition for mathematics, he still forged this new path... Mathematics Lord is indeed one of the strongest geniuses in Shenzhou's history! So powerful! If I hadn't studied 21st-century mathematical theory, I wouldn't even dare to entertain the thought of comparing to him." Wang Qi inwardly marveled, while carefully observing Feng Luoyi's expression.
Now, our Genting might be in big trouble!
In the Ten Thousand Immortals Illusion Realm, no small gesture could escape Feng Luoyi's notice. Feeling Wang Qi's mood, she smiled bitterly, "How about it?"
"Very powerful. I really don't know how to describe it, but this paper can completely serve as a beacon, guiding the development of Shenzhou's mathematics, especially the development of applied mathematics."
"It is strong on both theoretical and practical levels. We originally thought with your first-order Completeness Law, we could take a step ahead in mathematical logic. We overestimated ourselves and underestimated the Mathematics Lord," Feng Luoyi shook her head, "Truly deserving of being the peerless powerhouse who once, single-handedly, subdued the entire Wanfa Sect."
She was one of the younger members of Carefree in the Wanfa Sect. When she stepped onto the Cultivation Road, it was the time when Xi Baiche rose and the transfer of power between two generations of the Wanfa Sect took place. And the time she became recognized as Carefree coincided with the formation of the Immortal Alliance. The Mathematics Lord left Shenzhou to guard other realms not long after. Therefore, Feng Luoyi never had the chance to face that Tyrant directly.
"Overestimated ourselves?" Wang Qi pretended not to understand, "Is there a mistake in the proof of completeness?"
Before Gödel, no one doubted the contradictions between semantics and syntax, some concepts are undefinable in human language, and some problems are inherently beyond the grasp of current logic. It was virtually impossible for the Master of Calculation to achieve the universally sought-after result of Completeness he dreamt of.
But science progresses just . The Master of Calculation hitting that southern wall couldn't prove universal, broad completeness. However, in this process, they would necessarily test some dead-end routes, proposing some theories that come accidentally. These outcomes would become the foundation for future developments in mathematics. If possible, Wang Qi even hoped to guide the many disciples of the Genting Sect towards the trail of the Earth's Bourbaki School, transforming Shenzhou's mathematics to be closer to the more familiar and advanced Earth mathematics he knew.
Feng Luoyi sighed, "Ximen Master is complaining. He seems to vaguely see the destination, but there's always an invisible wall blocking him, making him feel like he's going around in circles..."
Suddenly, Feng Luoyi's expression changed, staring intently at Wang Qi, "Wang Qi, Ximen Master asked me to ask you a question, and you must answer me honestly."
Wang Qi was surprised, wondering what important question this was.
Feng Luoyi asked, "When you initially proved the completeness of first-order logic perfectly within the finite framework, why did you suddenly go against the framework of Finiteism and use Transfinite methods for the next step of research?"
Wang Qi was at a loss for words, "That... was just a moment of inspiration..."
"Did you anticipate this kind of impossible situation from the start? Or are you hiding some discovery?" (To be continued. If you enjoy this work, welcome to qidian.com to vote for recommendation tickets and monthly tickets. Your support is my biggest motivation. Mobile users please go to m.qidian.com to read.)