AI Gives Answers in a Second, But Can Never Take Away Wiles' Moment of Proving Fermat's Last Theorem

When AI Solves Everything with a Button Press, What Have We Lost?

Thirty years ago, a documentary about Andrew Wiles captured a deeply moving scene. Sitting at his desk, he described the moment he finally found the fix for the flaw in his proof of Fermat's Last Theorem—not his initial flawed announcement, but the true moment of clarity after months of struggle.

Choking up, he said: "I'll never have another moment like that at my desk for the rest of my life." He could barely finish a sentence.

This scene was repeatedly brought up by Lex Fridman in conversation to contrast with today's experience of AI generating answers with a single click. If you press a button and the machine spits out a solution, you can never have the emotional impact Wiles experienced.

Helicopter to the Summit vs. Climbing on Foot

The conversation offered a brilliant metaphor: you see a massive mountain and say, "Someday I'll reach the top," so you train day after day, inching closer. Then a helicopter descends and drops you right at the summit. You're standing at the peak, but the meaning of the climb has been hollowed out.

The answer AI provides is that helicopter. It's efficient and accurate, yet it strips away the emotional bond humans build with a problem during the long process of solving it. Wiles' tears weren't for the answer—they were for the seven years of solitary refinement.

Andrew Wiles is a British mathematician famed for proving Fermat's Last Theorem. Proposed by Pierre de Fermat in 1637, the theorem asserts that for integers n > 2, the equation x^n + y^n = z^n has no positive integer solutions. This seemingly simple proposition baffled mathematicians for 358 years. Wiles began secret research in 1986 and after seven years announced a proof in 1993, but a flaw was subsequently discovered. He spent over a year repairing it, finally completing the proof in 1994 in collaboration with Richard Taylor. Wiles' proof integrated modern mathematical tools such as modular forms, elliptic curves, and Galois representations, ushering in a new era in number theory. His work not only solved a millennium-old problem but also showcased the solitude and passion of mathematical creation. The documentary Fermat's Last Theorem captured his emotional outburst upon completing the proof, becoming one of the most moving moments in mathematical history.

Mathematics Is a Solitary Craft

The conversation highlighted a key fact: until the 20th century, almost all mathematics papers had a single author. Today, the median number of authors on a math paper is two. This stands in stark contrast to experimental sciences—mathematics is not a laboratory science; it is inherently a solitary process.

Wiles worked alone on Fermat's Last Theorem for seven years, and Grigori Perelman's work on the Poincaré conjecture is similarly known for long periods of isolation. This way of working is increasingly rare in modern academia, yet it is precisely the breeding ground for major breakthroughs.

Why Advisors Don't Assign Students Their Own Research Topics

The interlocutor shared his mentoring philosophy: the topics he assigns to graduate students deliberately do not overlap with what he himself is working on. "I want them to have their own field. I want to open doors for them." When students succeed, it should be their own success—even if the advisor initially had directional intuition, the execution and breakthrough belong to the student.

This deliberate distancing contrasts sharply with the "instant answers" culture of the AI era. True academic growth happens in the friction of independent exploration, not in the passive reception of answers.

In academic mentoring, the practice of deliberately having students research different topics from their advisor stems from an educational philosophy aimed at cultivating independent research ability. One of the core goals of graduate study is learning to pose and solve problems, rather than simply executing an advisor's instructions. If students directly inherit their advisor's research topics, they can easily fall into path dependence and lack independent judgment about research directions. Independent topic selection forces students to confront the unknown, requiring them to navigate everything from literature review to problem definition on their own—and this friction is precisely the catalyst for academic growth. Moreover, diverse topics can prevent academic inbreeding and promote interdisciplinary exchange. Many major breakthroughs in history have arisen from unexpected discoveries made by young scholars during independent exploration. Thus, an advisor's "letting go" is not indifference but respect and trust in the student's academic identity.

Problems Aren't Solved—They're Discovered

The interlocutor mentioned a paper he published a month ago that was the culmination of ten years of thought. It was a five-author paper; over a decade they made incremental progress until they could finally assemble the complete picture.

More crucially, he described the nature of the problem: at the project's outset, they couldn't even provide a precise formal statement. "We couldn't tell you at the starting point what the exact proposition to be proved was." The problem itself gradually revealed itself through repeated attacks—you chisel away bit by bit until you see enough of the outline to understand how it should be formulated.

A Fundamental Divergence from the AI Problem-Solving Paradigm

Current AI progress in mathematics is impressive, but its mode of operation differs fundamentally from the process described above. AI excels at handling already-formalized problem statements—give it a clear proposition, and it searches for a proof path. But Wiles-style mathematical creation happens precisely at the stage when the problem has not yet been clearly articulated.

This is not to say AI is useless, but that it cannot replace the human ability to discern the contours of a problem amid chaos. A decade-in-the-making paper often begins with a sentence like "There's a phenomenon here, and I don't understand what it is."

Formalized problems are those that have been precisely stated as mathematical propositions, such as "prove that there are infinitely many primes" or "solve a certain equation." Typical applications of AI in mathematics include automated theorem proving and symbolic computation, which rely on converting problems into machine-processable logical forms. However, the most critical step in mathematical creation is often translating vague intuition into a formal statement. This process requires mathematicians to abstract, analogize, and conjecture about phenomena, and AI currently lacks the ability to distill problems from chaos. Human mathematicians can perceive patterns in incomplete information, propose meaningful conjectures, and progressively refine problem statements. This capacity for "problem discovery" is the core of mathematical innovation and is difficult for AI to replace.

The Value of the Process Cannot Be Outsourced

The reason Wiles' moment is so powerful is that it carries seven years of frustration, revision, and persistence. AI can compress solution time, but it cannot compress the accumulation of meaning. As we delegate more and more cognitive labor to machines, we need to seriously consider: which processes are worth preserving for humans?

At least for pure mathematics, solitary long-term struggle is not just the way breakthroughs are produced—it is part of a mathematician's identity. Perelman retreated for years to solve the Poincaré conjecture and then refused the Fields Medal; Wiles wept on camera in the documentary—these moments define the culture of the discipline, not merely the accumulation of knowledge.

AI is a powerful tool, but it should not be a helicopter. The meaning of the mountain, in the end, lies in the climb itself.

One-line Summary

AI's one-click answers strip away the value of the process of mathematical exploration—solitary struggle is the source of both breakthroughs and meaning.

Paragraph Summary

Through the moving scene of Wiles proving Fermat's Last Theorem, the article contrasts it with the experience of AI instantly providing answers, arguing that the value of the process cannot be outsourced. Mathematics is inherently a solitary craft; long periods of isolation and repeated struggle are the breeding ground for major breakthroughs. The starting point of real research often cannot offer a precise problem statement—the problem gradually reveals itself through exploration. AI excels at handling formalized problems but cannot replace the human ability to discern the contours of a problem amid chaos. The author calls for preserving, in the AI era, those cognitive processes worth experiencing—because the meaning of the mountain ultimately lies in the climb itself.