Challenging the Infinite with 'Heuristics'—What Two Great Problems Teach Us About Human Maturity in the Age of AI
In my previous article, I wrote about the importance of breaking out of the cage of perfect 'protocols' provided by AI and proactively engaging with the friction of reality using our own immature, flesh-and-blood bodies.
If we merely consume the beautiful, logical arguments (procedures) presented by AI, our intelligence and spirit will remain in a hollow state of 'eternal childhood.' It is the 'heuristics' we acquire by enduring the cold sweat and bloodshed of reality that serve as the fuel to push humanity to the next stage.
To deepen this philosophy further, I would like to draw upon the drama of two massive breakthroughs in the history of mathematics.
Herein lie hints for how we, living in this coming era, should confront the massive intelligence of AI and how we should maintain our agency to continue growing.
1. The 'Four Color Theorem' and the Co-creation with Computers
In 1976, the mathematical world was shaken to its core.
The 'Four Color Theorem', which states that any map can be colored with just four colors, was proven by Kenneth Appel and Wolfgang Haken. It was a super-difficult problem that genius mathematicians had challenged and failed to solve for over 100 years.
However, the method of proof sparked fierce criticism and intense debate in the mathematical community at the time.
They used a computer to exhaustively verify a number of configurations too vast for humans to check by hand. Mathematicians of the time lamented this method, which relied on machines for such a vast amount of verification that it appeared to be 'brute force', complaining that 'this is not beautiful' and 'this is not a proof by human hands.'
Yet, this was precisely the prototype of AI co-creation: 'the machine fills in astronomical probabilistic calculations within a framework of procedures assembled by humans.' If the patterns to be verified are 'finite,' the machine's computational power far exceeds that of the human brain. We accepted that reality and learned how to master it as a tool.
However, the history of mathematics did not allow for a total victory of machines based on that alone.
2. Computers were powerless against 'Fermat's Last Theorem'
Twenty years after the resolution of the Four Color Theorem, another legendary difficult problem fell.
It was that incredibly simple theorem left behind by the genius Fermat in the 17th century.
The equation x^n + y^n = z^n has no solutions in natural numbers (excluding 0) when n is a natural number greater than 2.
When Andrew Wiles completed this proof in 1995, the protagonist was not brute-force calculation by a computer. It was the deep mathematical intuition and structural understanding that Wiles had accumulated over years of solitary research.
Why was brute-force calculation by a computer powerless against this problem?
The reason is extremely simple. What Fermat was dealing with was not a finite set of patterns, but the 'infinite' nature of 'all numbers greater than 2'.
No matter how much technology evolves and computational speed increases, it is absolutely impossible to calculate (digest procedures for) infinitely continuing numbers one by one. Therefore, what was needed was not to eliminate numbers one by one, but to translate the problem itself into another universe.
3. An inexpressible 'premonition' that connected two universes
What Wiles used for his proof was a bridge connecting two universes of geometry and number theory—“elliptic curves” and “modular forms”—which were believed to be “completely unrelated” even by the mathematical community at the time.
The staggering “intuition (Taniyama-Shimura Conjecture)” left behind by Japanese mathematical geniuses Yutaka Taniyama and Goro Shimura, which posited that “all elliptic curves must be modular,” came from an era without proof. Into this dark space, where no one had even created a procedure (protocol) yet, Wiles applied his own brain, driven by a raw will that insisted, “There is absolutely a beautiful connection here.”
No matter how much AI evolves, the “urgency of the question” and the “aesthetic sense” that humans generate from the friction of reality cannot be easily replaced.
The answers provided by AI, no matter how high their precision, are products of a “learned information space” derived from vast amounts of past data.
On the other hand, human intuition and heuristics are “the time we have lived” itself—having navigated gritty friction, broken a sweat, and absorbed myths and art—which creates “conviction without reason (a premonition of the future)” by creating shortcuts in the brain. The aesthetic sense that notices “there is a beautiful common law” between two seemingly unrelated phenomena cannot be born from mere data processing.
Conclusion: We challenge the “infinite” using AI as a “finite whetstone.”
The Overlords, the aliens who appear in the science fiction classic *Childhood's End*, possessed science and technology and a perfect management system (procedures) that far surpassed humanity. However, no matter how wise they were, they were a completed, dead-end species that could not “evolve (grow)” to the next stage of the spirit.
Modern AI is also, fundamentally, an existence that swims within a given framework and a learned information space.
However, we humans are the “process (growth)” itself, starting from an immature state, exposed to the winds and rains of reality, and capable of changing indefinitely while getting hurt.
The realm of the Four Color Theorem: Entrusting the existing framework of procedures to AI and having it explore the information space at overwhelming speed.
The realm of Fermat's Last Theorem: In places where the form of the question has not yet been defined, humans build bridges with intuition and forcefully throw new “questions” toward the device known as AI.
Human maturity in the coming era is not just about hoarding knowledge.
AI runs fast through the finite. Humans throw questions into the infinite. It is the maturity of heuristics that gives birth to those questions.
While using the “finite mesh” of AI as the ultimate tool, we must sharpen the blade of our own heuristics, proactively continue to throw questions into the unseen infinite universe, and keep growing. It is within that process of friction that the true growth for humans to mature in the age of AI is hidden.