Exploring Immortality Cultivation-Chapter 912 - 241: The Nirvana of Mathematics

With the help of Master Turing, this question-and-answer session passed without any danger.

Before Wang Qi "transmigrated," the research on modern mathematical logic in that era was far beyond what could be compared to Godel's time. There were countless mathematical systems equivalent to Turing machines and the Lambda Algorithm. Just with what Wang Qi knew, it included the cellular automata he constructed in Shenzhou, the Post Correspondence Problem, counter machines, the Markov Algorithm, and recursive functions.

On this issue, his perspective was innately superior to the people of Shenzhou.

And his own talent was not weak, coupled with Master Turing's strong assistance. In half a year, he could naturally accomplish many things.

For an ordinary Carefree Practitioner, the preparations of Wang Qi and Master Turing were flawless and impeccable.

So impeccable that just thinking about it could make them lose their courage.

In the end, it was Feng Luoyi who interrupted the exchange between Wang Qi and Master Turing. He looked around and said, "If no one has any more questions, then the question time ends here."

Master Turing was still unsatisfied: "Hold on, let us finish discussing this issue..."

"You can chat in private," Feng Luoyi glanced at the Master of Calculation and the Mathematics Lord before announcing, "Now, we wait for midnight."

This is one of the few traditions of the Modern Cultivation Method. Such a summary of a revolutionary path must be conducted at midnight.

At the dawn of the new and old, the happenings of yesterday died yesterday, and the happenings of today are born today.

Those Carefree Practitioners were still immersed in the shock brought by the incompleteness theorem, and only now did they realize that so much time had passed.

From noon to midnight, this lecture had lasted for half a day.

After discussing with Master Turing, Wang Qi again sat back on the edge of the podium, closing his eyes for a short rest. Clearly, he was still just the Golden Core Stage small cultivator from before. Yet, inexplicably, many of the Carefree felt that he now exuded an aura reminiscent of a towering mountain or abyss.

—A psychological effect?

Some people began to harbor a trace of doubt.

Even those from the Mathematics Lord's faction who did not acknowledge Wang Qi's theory had to admit that Wang Qi was indeed very talented.

No one could ignore him any longer.

"Just entering the Immortal Path, yet already intimidating the world..." someone murmured, "I seem to see another master among the common people."

"Isn't he more like the Mathematics Lord...? Both are figures who shattered the Dao Hearts of one era's Genting leaders to achieve great fame..."

"Shut up!"

That phrase seemed to stir some commotion. Most Arithmeticians looked worriedly at the Master of Calculation.

In a time of turmoil, the Master of Calculation was the one who suffered the most. Wang Qi had developed a weapon that negated his ideology, his thoughts. His life's pursuit was shattered. Now, even if this peak Carefree were to die immediately, it wouldn't be surprising.

Yet, the Master of Calculation remained motionless and silent.

The midnight bell soon rang. Wang Qi ascended the podium once more. This time, his expression showed unparalleled confidence.

"Pang Jialai of the Mathematics Lord once talked about the future of mathematics in a previous lecture. He said: 'There were once some fortune-telling demi-immortals who kept claiming that all problems had been solved, and all that remained was polishing the answers,' and then he added: 'But these melancholic types never come to a good end... I think no one today still holds this belief.'

"The words of my predecessor indeed seem apt now. Our faith in progress, our confidence in our Immortal Dao is no longer so steadfast, as these views have been profoundly shaken. To us, it doesn't seem as reasonable to unhesitatingly extrapolate, like the Mathematics Lord before us, past and present into the future—we don't even know what we will know, what we must know, or what we need to know."

Wang Qi's words were like unveiling the curtain of a dark era. He was expounding on the destructive power of incompleteness and undecidability. The Wanfa Sect had been struck to the point of almost collapsing. Aside from a few bottom-level cultivators, no one's Dao Heart remained unshaken.

Outside the venue, Wanfa Sect disciples who were still paying attention to this lecture were still sobbing. They cursed softly: "Banished Immortal! Heavenly Demon! Evil Demon Heterodox!"

Even those of the Shaoli Sect inside the venue also felt a sense of loss.

"If we were to ask a mathematician about the future view of mathematics, they would likely want to raise more fundamental questions: what kind of Immortal Dao does the Human Clan want to forge for themselves? Over the past eighty thousand years of accumulation and two thousand years of exploration in the Modern Cultivation Method, is our way of thinking merely an ephemeral moment in the cosmos? If we are worried about getting entangled in mystical discussions, preferring to stay in the not-so-reliable historical records of now, we would encounter the same issues, only expressed in a different form: are we witnessing our Immortal Dao growing old after reaching its peak, transitioning from prosperity to decline? Should we abandon pioneering efforts, at least abandon pioneering efforts for our own Dao in our hearts, and instead turn to preserving our achievements?"

The Li Sect arithmeticians led by the Genting Sect were also perplexed.

From finite to infinite, is it the barrier of our cognition?

Are we forever unable to reach the other shore of infinity?

Is this barrier insurmountable?

Or is it as the sages said, that there is no infinity in this universe?

However, the purpose of Wang Qi's speech was not to boast, but to express his views.

— Mathematics must undergo Nirvana Rebirth!

The Bourbaki School emerged as the phoenixes at the time when Godel and Turing led a new generation of geniuses who shook the foundational beliefs of the mathematical community and walked the farthest in the latter half of the twentieth century.

"Why is this happening? Undoubtedly, we have encountered a big problem, a tremendously big problem. This problem may take my entire life to solve. But at such times, I always think of what the Ximen Master once said—A branch of science is vibrant as long as it provides plenty of questions, while a lack of questions is a sign of imminent death. I think, at least we don't have to worry about witnessing the death of mathematics. What a terrifying problem we've encountered!"

Someone murmured, "You say it lightly…"

"Indeed, some elders might want to say that I—this not-yet-thirty Golden Core Cultivator—speak without consequence from a safe position. But, but! Has our mathematics, has our Wanfa Sect ever feared a crisis?"

Wang Qi's voice carried a hint of excitement.

"When ancient arithmeticians first defined the ratio of two quantities precisely enough, it raised the issue of incommensurables. They seemed to believe and insisted that all ratios were rational numbers, establishing their initial geometrical reasoning sketches based on this temporary assumption. Some of the greatest progress of ancient mathematicians was linked to their initial error at this point."

— The first mathematical crisis, the discovery of irrational numbers.

"Similarly, at the beginning of the Heavenly Transformation Formula theory and the infinitesimal calculus era, people also hoped that each analytical expression defined a Heavenly Transformation Formula, and each Heavenly Transformation Formula had a derivative; today we know these requirements are incompatible—and the person who solved this problem, Mr. Wei Er, is sitting among us now!"

— The second mathematical crisis, infinitesimal calculus' manipulation with infinities and infinitesimals, leading to contradictions.

"The most recent crisis arose from the bizarre argumentation methods provided by the emergence of Naive Set Theory, which has led us to quite good results, to such an extent that we can consider it finally established. We have learned to trace all of mathematics back to a single source, consisting of a few symbols and a few usage rules for these symbols. Undoubtedly, this is an impregnable Mountain Protection Array, though we have difficulty being confined within without risking starvation, but we can always decide to retreat into it in situations where the situation is unpredictable and there are external dangers."

— The third mathematical crisis, contradictions triggered by self-reference in set theory.

Wang Qi closed his eyes and paused here for a moment. Then, he resolutely and firmly expressed his thoughts: "Only a few backward-thinking individuals persist in the view that mathematicians must rely on their 'intuition' to derive new 'non-logical' or 'pre-logical' elements of reasoning. If some branches of mathematics have not yet been axiomatized, that is, reduced to a statement form where all terms are defined by the basic concepts of set theory, and all axioms are clearly expressed by the primitive concepts of set theory, it is only because we haven't had enough time to do so. Of course, it is very likely that one day our descendants will demand that reasoning methods unacceptable to us be introduced into set theory, or it may very well be that with our current reasoning patterns, we discover the germ of contradictions today that we haven't noticed, although the work of modern logicians indicates that the likelihood of this is very small. At that time, a comprehensive revision would be necessary, but even now, we can be sure that the most essential elements of mathematics will remain unaffected."

At the first sentence, all the Li Sect Cultivators, including the Mathematics Lord, turned pale with shock.

Wang Qi was essentially pointing at them and scolding!

However, Wang Qi had to say this.

This was the foundation for his subsequent thoughts.

"After all, he is still a Li Sect Disciple…" A complex expression appeared on Fairy Ruoche's face, unclear whether it was comfort or resentment.

"Even among the 23 questions, there are still several problems far from our reach, although they are not unachievable targets, they may continue providing topics for more than one generation of research, among which the fifth question regarding Li Quan is an example…"

"The Riemann Hypothesis, after people abandoned the idea of proving it using the Heavenly Transformation Formula method, now seems to show new light. It indicates that it is closely related to the conjecture of some Heavenly Transformation Formula, thus making these two problems become two aspects of the same arithmetic-algebra problem, and for this problem, simultaneous research on all its circular extensions for a given number field undoubtedly plays a decisive role…"

"…Calculation King Gao Si's arithmetic focuses around the law of quadratic reciprocity; now we know that the law of quadratic reciprocity is just a first example of the 'class field' laws, which are theorems determining the arithmetic expansion of algebraic number fields. We also know how to express these laws to make them appear as a compact whole…"

One problem after another was proposed. Those many arithmeticians were not foolish. They quickly realized that this was a mature theoretical system.

It included the logic and formalism of the Li Sect, as well as the low-dimensional topology and algebraic topology represented by the Lianzong. Vaguely, there was also something beyond these.

He Wai'er's eyes became fervent.

"This Wang Qi, truly seeks the Nirvana of mathematics!"