1 Introduction

Recently, I watched a movie about the mathematician Ramanujan, *The Man Who Knew Infinity*:

This movie does not attempt to explain those abstruse mathematical formulas to the audience, but instead repeatedly emphasizes a more intuitive question:What happens to a genius born in the wrong era?

Ramanujan was an Indian mathematician of the early 20th century who, with almost no systematic training in higher mathematics, independently derived a vast number of extremely profound mathematical results. Even compared to contemporary Western mathematicians, his intuition was rare—for instance, his discoveries in infinite series, partition functions, and modular forms were on par with the classical theories established by Gauss, Riemann, and others; some results were not rigorously proven and absorbed by modern mathematicians until decades later. His name still frequently appears in research on number theory and modern cryptography today, and he is widely considered one of the rarest “purely intuitive geniuses” in the history of mathematics.

Yet his life was exceptionally short—he died of tuberculosis at the age of 33. If viewed solely from the outcome, this is a typical story of a “genius dying young”; but once we zoom out, we find that the issue goes far beyond that.

Ramanujan's life fell precisely between multiple historical fault lines: he was born in colonial India and grew up under a strict caste system; he relied on a near-divine intuition for mathematical creation, yet was thrust into a highly formalized Western academic system that emphasized proof and standards; the disease he suffered from would have effective treatments just a few decades later, but was virtually incurable at the time.

In this sense, Ramanujan did not simply “die young,” but ratherexperienced a systematic misalignment with his era。

This article does not intend to repeat the movie plot, nor does it plan to list complex mathematical details. Instead, it attempts to re-examine Ramanujan's fate from the perspective of “historical misalignment”: his background, his intuition, the cultural environment he lived in, and how these factors together shaped one of the most unique geniuses in the history of mathematics.

And all of this must begin with the social structure of his birth and how it shaped individual destiny.

Note: This article is not a biography in the strict sense, nor is it a eulogy for a genius. Ramanujan serves here more as a sample placed in an extreme position. What this article truly focuses on is what kind of misalignment occurs when a way of knowing—especially a way of thinking that relies heavily on intuition and structural perception—enters an era and system that are not yet ready to accommodate it. In other words, this is an article aboutthe relationship between cognition, tools, and the eraRamanujan is simply the person who made this issue clear enough.

2 A Brahmin Who Shouldn't Have Been So Poor

Looking only at his background, Ramanujan's life seemingly shouldn't have been so difficult: he was born into the Brahmin caste, the highest status in traditional Indian society. According to the intuition of modern readers, the “highest caste” is often equated with better social resources, higher security of livelihood, and even a kind of “starting-line advantage.” Yet the reality was quite the opposite—chronic poverty, unstable income, and continuous marginalization in the modern professional system characterized almost his entire life.

This contrast was neither accidental nor the result of poor personal choices, but rather a deeper structural misalignment. In traditional Indian society, the Brahmin was not a class aimed at wealth creation. Its core functions were religious rituals, scriptural interpretation, and the transmission of knowledge. In other words, it was a highly symbolic, rather than economic, identity system: it provided social respect and cultural legitimacy, but rarely corresponded directly to stable material rewards.

More crucially, this identity was not “freely switchable”—the status of a Brahmin was not a pass that could be redeemed for a real-world profession at any time, but a set of role constraints accompanied by taboos and boundaries: what could be done and what “should not be done” were not entirely determined by individual rationality or economic needs.

When the economic logic of modern society began to penetrate India, this identity system did not complete its transition in tandem. In colonial and post-colonial India, a modern occupational system gradually took shape: accounting, clerical, technical, and engineering positions became the foundational nodes of social operation. But for a Brahmin, entering these positions might have been an economically rational choice, yet culturally, it was a distinct “sinking.” This was not a matter of personal will, but a continuous constraint imposed by the identity system on role mobility.

Ramanujan's early life trajectory was a concrete manifestation of this misalignment. It was not that he did not try to integrate into real society, but whether in teaching, job hunting, or briefly working in accounting-related roles, he was always in a state of instability, low returns, and lack of long-term support. This was not because he was “unfit for work,” but because the identity position he occupied itself lacked an interface to connect with the modern economic system.

From this perspective, his poverty was not an “anomaly,” but precisely the natural outcome of the system's operation.

This misalignment was not an abstract social issue; it had very concrete and very cruel consequences: long-term economic instability meant persistent malnutrition, life stress, and neglect of physical health. When external systems failed to provide stable support, the cost was ultimately transferred to the individual.

It was also at this point that Ramanujan's life began to transition from a social-level misalignment to a gradual accumulation of practical impacts on his living conditions—laying an inconspicuous yet crucial premise for his subsequent learning experiences and physical challenges.

Introduction to the Indian Caste System

India's caste system is a system that divides members of society into fixed hierarchies. Traditionally, it is divided into four main categories: Brahmins (priests and scholars), Kshatriyas (warriors and rulers), Vaishyas (merchants and farmers), Shudras (servants and laborers), and the excluded “untouchables.” Each category has its established duties, codes of conduct, and social boundaries, with extremely low mobility.

This system originated from ancient Indian classical texts, such as the Vedas and the Manusmriti, and its core intent was to maintain social order and cultural inheritance, rather than being directly linked to economic capability. Because of this, even the Brahmins, the highest caste, could live in poverty due to a lack of opportunities to connect with the modern economic system.

3 What Kind of Intuition Was Shaped by a Mathematical Education of Only Formulas

If Chapter 2 describes a misalignment in social structure, then at the level of mathematical learning, what Ramanujan experienced was an equally profound but qualitatively different misalignment—he did not grow up within a complete, step-by-step modern mathematical education system.

During his youth, the mathematical resources Ramanujan could access were extremely limited. The most important, and almost only, systematic reference material was a compilation of formulas titled A Synopsis of Elementary Results in Pure and Applied Mathematics (A Synopsis of Elementary Results in Pure and Applied Mathematics). The author of this book was the British mathematician G.S. Carr. It was neither a textbook nor a monograph, but rather resembled a highly compressed “list of results”: the book listed a vast number of theorems, formulas, and conclusions, but provided almost no proofs, let alone explanations of how these conclusions were derived step-by-step.

For most people with formal mathematical training, such a book could only serve as a reference or a quick-lookup tool; but for Ramanujan, it was almost equivalent to the entirety of his mathematical world. This learning path, in itself, had already fundamentally diverged from mainstream mathematical education.

In the modern mathematical system, proof is not merely an appendage to a conclusion, but the core tool for training ways of thinking. Through proof, students are forced to understand boundaries, conditions, hypotheses, and scopes of applicability, and are constantly reminded of where things might go wrong. However, when a person is exposed from the very beginning to mathematics that has “only results and no processes,” the abilities that are reinforced in him will naturally be entirely different.

What Ramanujan faced was not “how to prove a known proposition,” but another kind of question:Is there some inherent structure among these results?

Through repeated exposure to a vast number of formulas and conclusions, he gradually developed an exceptionally strong capacity for pattern recognition: he would notice which forms recurred, which expressions exhibited similar behavior in different contexts, and which results seemed to share hidden correspondences.

This training method does not teach one how to reach an answer step by step, but rather seems to train one to directly “see” the answer. This also explains a frequently mentioned but easily misunderstood phenomenon: Ramanujan did not fail to understand the importance of proof, but rather had never been systematically trained to rely on it. In his mathematical world, conclusions were not the end point, but rather the starting point; formulas were not derived, but rather “discovered.”

But this learning method also naturally came with a price. When lacking rigorous training in proof, determining under what conditions a result holds true and at what scale it fails becomes extremely difficult. Formulas that appear perfect within a finite range only expose their hidden problems once pushed to extreme cases; this was not due to carelessness or lack of ability, but rather because the vision itself was limited by the training method.

Precisely because of this, the notebooks Ramanujan left behind not only contained a vast number of shockingly profound results, but were also inevitably mixed with some incomplete, unrigorously verified ”wrong formulas.” This “mixed state” was not an accidental mistake, but rather a direct product of his mathematical growth path.

From a broader perspective, this educational misalignment once again reflects the influence of the “era.” In early 20th-century India, what was lacking was a complete modern mathematical training system; while on Ramanujan's personal level, what was lacking was not intelligence or passion, but rather a set of tools that could provide a safe boundary for this highly intuitive exploration—tools that, at the time, only existed in a few Western mathematical centers.

The impact of this training method would not stop at personal habits, but would be amplified into a methodological conflict when he entered another mathematical system. And this was precisely one of the fundamental reasons for his subsequent encounter with Hardy, which triggered profound tension.

This is particularly evident in some of his ‘incorrect results“; those derivations later referred to as ”wrong formulas" were actually not simple calculation errors. They were more likea cognitive trace of seeing the structure first and then searching for the proof。

In modern mathematical training, the correct path is usually: definition—axiom—derivation—conclusion. Whether a formula holds true must be provable step by step; if any step is not self-consistent, it will be judged as “incorrect.” But many of Ramanujan's results were precisely the opposite:The conclusion appeared first, and the proof was only filled in later. In the context of the time, these conclusions could neither be verified nor systematically explained, so they could only be labeled as “suspected errors” or “not yet established.”

The problem is that, decades later, mathematicians discovered that a significant portion of these so-called “incorrect formulas” were not errors in direction, but rathercorrect intuitions that lacked the mathematical tools of the time to be expressedEven today, some formulas still do not conform to traditional proof paths, yet they can be explained through modular forms, analytic continuation, or higher-order theoretical frameworks. They were not calculated incorrectly, but ratherstated too early。

Therefore, the significance of these “incorrect formulas” lies not in whether they all hold true, but in how they expose a way of generating mathematics completely different from the mainstream educational system: not starting from rules, but from structure; not building step-by-step, but capturing as a whole. Ramanujan did not lack an understanding of rigor; rather, his environment, the tools he used, and the mathematical language of his timewere insufficient to bear what he had already seen。

And precisely because of this, these formulas have become a key entry point for understanding Ramanujan. They remind us that when the system of knowledge is not yet ready,ahead-of-its-time cognition can often only appear in the form of “errors”And what history truly needs is not to correct them immediately, but to wait for an era mature enough to re-understand them.

4 The Conflict Between Ramanujan and Hardy

If Ramanujan's plight in India stemmed from a mismatch between social structure and economic reality, then the conflict he encountered in Britain came from another, more hidden and modern tension—conflict within the academic system。

In 1913, Ramanujan had already been doing “informal” mathematical research in Madras for many years. He had no stable position, no academic endorsement, and struggled to even consistently obtain basic mathematics textbooks. To find a way out, he began sending letters to British mathematicians, enclosing a large number of formulas he had derived, hoping to attract attention. Most of these letters went unanswered, and some were even discarded directly as eccentric amateur hobbies.

Until one of them arrived at Cambridge. The recipient of this letter was G. H. Hardy, one of the most prestigious British mathematicians of the time.

Hardy was not an easy man to impress. He placed extreme importance on rigor and was naturally wary of “unproven results.” But precisely because of this, when he saw the formulas in the letter, he immediately realized the gravity of the matter:these things could not have been made up out of thin airSome of the results would take even professional mathematicians a considerable amount of time to derive; others, Hardy himself could not even immediately judge as right or wrong, yet he could clearly perceive the complexity and depth of their internal structure.

Hardy later recalled that he and his colleague Littlewood spent an evening checking these formulas line by line, finally concluding that the author of the letter was either a complete fraud or a mathematical genius never seen before—and the probability of the former was, in fact, lower. Thus, Hardy made a decision that was unusual at the time:he invited Ramanujan to come to England。

In 1914, Ramanujan arrived at Cambridge and entered Trinity College. This seemed like the beginning of a legendary tale: a poor Indian mathematician, discovered by the British temple of academia, stepping onto the world stage. But the real story is precisely where things began to get complicated. The root of the conflict did not lie in whether Hardy recognized Ramanujan's talent—on this point, Hardy was always a steadfast supporter. The real problem was:there was a fundamental difference in their understanding of “how mathematics should be produced”。

What Hardy represented was a highly institutionalized system of modern mathematics: proof first, clear logical chains, and results that must be embedded within existing theoretical frameworks. Mathematics is a public language; any conclusion must be capable of being reproduced, checked, and verified by others. Ramanujan's way of working, however, was almost the exact opposite. He was more like “seeing” the results rather than “deriving” them step by step; for him, formulas were not the end point, but fragments of structure that naturally manifested.

This was not a question of who was right and who was wrong, but rathera conflict between two cognitive pathsIn Hardy's view, Ramanujan had to learn how to translate his intuition into a form acceptable to modern mathematics; in Ramanujan's view, this translation process itself often seemed tedious and redundant. Thus, the partnership gradually evolved into a state of tension: one side constantly demanding supplementary proofs and rewritten derivations, while the other was forced to repeatedly “explain answers he already knew” in a language he was neither familiar with nor good at.

This tension did not exist only on paper; culture, lifestyle, and even religious customs added to its weight. Ramanujan adhered to strict vegetarianism and found it difficult to adapt to the British diet; the difference in climate, wartime shortages, and prolonged psychological stress constantly took a toll on his body. The cost of academic adaptation, compounded by the discomforts of daily life, gradually drove him toward a breakdown.

Ultimately, this conflict did not end in any kind of “reconciliation,” and Ramanujan's time in England was short. His health deteriorated rapidly, leading to several hospitalizations, and his mental state also began to suffer. In 1919, he was forced to return to India; a year later, Ramanujan passed away at the age of only 32.

In terms of the outcome, this was a failed collaboration: a genius was discovered but could not be truly “accommodated.” Yet, on a deeper level, this relationship reveals a precisely modern problem:When an extremely personalized way of knowing enters a highly standardized system of knowledge, conflict is almost inevitable.—Hardy opened the door to the world for Ramanujan; yet this world, at the same time, exposed its own boundaries.

5 A Sample That Cannot Be Replicated

After Ramanujan's death, people repeatedly asked one question: If he had been born a few decades later, would the outcome have been different? If the modern academic system were more “inclusive,” would he have lived longer and gone further? These questions seem to point toward institutional improvement, but in reality, they hide a deeper underlying assumption:Do we really want, or are we truly capable of, replicating individuals like Ramanujan?

On the surface, the modern system of knowledge seems to have absorbed the “lessons” of Ramanujan. Today's mathematical community is more international, and scholars from non-Western backgrounds are not uncommon; research pathways are more diverse, and intuition, computer experimentation, and conjecture-driven research have gained a degree of legitimacy. But if we probe further, we find that the core logic of the system has not fundamentally changed.

The modern academic system is, in essence, still ahighly standardized, verifiable, and replicable systemWhat it excels at is the large-scale production of reliable knowledge; what it requires is a talent structure that can be trained, evaluated, and replaced. In such a system, personal style can exist, but it must be compressed to a degree that does not affect the overall operation. Intuition can be used, but only on the condition that it can ultimately be translated into a public language.

And Ramanujan's problem lay precisely in the fact that his core valuewas almost untranslatableThe reason his formulas were so shocking was not just because of the results themselves, but because these results came from an extremely personalized cognitive path. This path relied on long-term internal accumulation, minimal external calibration, and a natural disregard for formalized steps—making it difficult to teach, hard to evaluate, and impossible to replicate institutionally.

In other words, Ramanujan was not a “genius yet to be absorbed by the system,” but rather atype of person the system was never designed to accommodate in the first placeThis is not a moral indictment of modern academia, but a structural fact. Any system that attempts to produce knowledge on a large scale must inevitably sacrifice its ability to adapt to “anomalous individuals.” Otherwise, it would be unable to distinguish between the reliable and the uncontrollable, ultimately losing its own stability.

From this perspective, Ramanujan's tragedy was not because the system was “not kind enough,” but because the systemhad to maintain its distance from himWhat is even more brutal is that even if a “Ramanujan-style” figure were to emerge today, we would in all probability still make similar choices. We would admire his achievements, cite his conclusions, and even write commemorative articles for him; but we would still require him to complete his proofs, integrate into the process, undergo peer review, and conform to standards. Otherwise, his work could not enter the public repository of knowledge. This is not a historical accident, but the price of modernity.

Therefore, the value Ramanujan left to posterity does not lie in providing a “more romantic image of mathematics,” nor in proving that intuition can replace rigor. On the contrary, he allows us to see a fact that is often overlooked:highly effective systems can often only function at the expense of marginal individualsAnd those we call “geniuses” are often precisely the products of the system's margins.

This is also why Ramanujan cannot become a replicable paradigm. He can only exist as a cautionary presence, reminding us of what exactly we discard when we talk about efficiency, standardization, and scalability; and whether, when we demand that all knowledge must be proven, recorded, and peer-reviewed, we are also silently closing off the possibilities of certain cognitive pathways.

Ramanujan's life did not provide a solution for the modern system of knowledge, but he left us with a far more important question:In a world that increasingly relies on standardization and processes, how much space are we truly willing to reserve for “irreplicable creativity”? And this question still has no answer to this day.

6 What If Ramanujan Had Met AI

Assuming Ramanujan lived today, an era in which AI is already involved in the process of mathematical research, what interests me is:Would his thinking become less isolated as a result?

What was truly rare about Ramanujan was never his technique or training, but his intuitive grasp of mathematical structures. He often directly “saw” a result hold true without a complete derivation. This ability is a gift in any era, but in his era, there was almost no mechanism to respond to this intuition in a timely manner. An idea was either shelved or had to wait until years later for others to provide a formalized explanation.

If placed in today's world, this situation would change significantly. With the help of AI's formalization and verification capabilities, his derivations would no longer have to remain on paper waiting to be tested, but could receive feedback in a very short time: which conclusions still hold under broader conditions, and where they have begun to deviate from the controllable range. This feedback is not a negation of intuition, but a continuous boundary prompt, so that exploration does not have to go all the way to the end every time to know if it has crossed the line.

More importantly, AI might make him realize his limitations earlier. In Ramanujan's era, many results later called “wrong formulas” were not because his intuition failed, but because there was a lack of adequate tools at the time to verify their scope of application. If, at these nodes, there was a system that could prompt: this is not an error, but requires additional conditions or a new way of expression, then the fate of many explorations might have been completely different.

At the level of proof, the intervention of AI is equally of practical significance. Ramanujan was not averse to rigor; he was simply not trained to be accustomed to that highly formalized style of writing. AI could take on a large amount of tedious and mechanical verification and deduction work, enabling him to focus his energy on what he was truly good at—the discovery of structures, the intuitive capturing of relationships, and those mathematical objects not yet fully covered by language. This is not to do the thinking for him, but to prevent his thinking from being constantly interrupted by technical details.

There is another change, more subtle but potentially more profound—with the help of AI, his thinking would no longer be a completely closed monologue. When a conjecture arises, it could immediately be placed into a larger background of knowledge, enter into a dialogue with existing theories, and even be prompted with possible directions of extension. The existence of this “instant response” would quietly change the rhythm of thinking: intuition would no longer be a one-time adventure, but an exploration that can be repeatedly revised and continuously advanced.

Of course, AI cannot replace intuition itself; true insight can still only come from an individual's cognitive structure, accumulation of experience, and sense of mathematics as a whole. What AI provides is an external stabilizer: it prevents intuition from being suspended in mid-air, ensures that exploration does not have to come at the cost of one's entire physical and mental well-being, and makes failure no longer always irreversible.

From this perspective, the change brought by AI is not to make Ramanujan another kind of mathematician, but to enable him to become himself more completely. His intuition would still be sharp, but no longer exposed; his exploration would still be bold, but with an extra layer of buffer that can be repeatedly tested.

It is also in this sense that the value of AI perhaps does not lie in “creating more Ramanujans,” but in:For the first time, it provides an external system of tools for this kind of extreme intuitive thinking that can continuously converse and calibrate.

7 Epilogue: The Boundaries of Cognition and Era

Looking back at Ramanujan's life, we may gradually realize a fact: he was not an “irreplicable myth,” but a human specimen pushed to an extreme position. His intuition, his creativity, and his limitations all truly exist within the spectrum of human cognition, only concentrated too dazzlingly.

The misalignment Ramanujan experienced was, in essence, not a personal tragedy, but the result of immature conditions of the era. His intuition walked ahead of tools, language, and institutions, and thus could only be paid for with isolation, misunderstanding, and even physical exhaustion. In that era, if a person touched structures not yet absorbed by the system too early, they were almost destined to bear all the risks—Ramanujan was not the only example; he was just an extreme specimen, a person who made the problem visible.

Entering contemporary society, we are producing more and more “semi-Ramanujans.” They may not possess such astonishing mathematical talent, but in their respective fields, they often similarly rely on intuition, cross-boundary perception, and structural insight: programmers, researchers, designers, engineers, and even ordinary knowledge workers are constantly facing complex systems and problems that have not yet been fully formalized. They can see the direction, but find it difficult to immediately give a complete expression; they can perceive the structure, but often lack the tools to verify and extend it.

The difference is that this time, the era is beginning to provide a response: modern society has not eliminated the limits of cognition, but it is gradually filling the gaps at the tool level. Formalized systems, computing power, and the maturing AI mean that individuals no longer have to bear the entire cost of exploration alone. Intuition is still the starting point, but it no longer inevitably leads to isolation; exploration is still full of uncertainty, but failure is no longer always irreversible.

In this sense, AI is not meant to replace creation, nor can it manufacture genius. What it changes is the relationship between cognitive structures and the era: allowing limited brains to continuously touch more complex and broader structures without fully understanding everything.

What Ramanujan faced was an era that was not yet ready to catch him; whereas the era we live in is learning how not to let intuitive cognition fall alone. This may not guarantee that greatness will inevitably emerge, but at least it means: when the next Ramanujan appears, he will no longer have to use his entire life to exchange for the day when the world slowly understands him.

From this perspective, this is not only a history of Ramanujan, but also a case study of “how cognition relates to the structure of the era.”