I came across it by accident on my phone screen, a reel on Instagram. A teacher was standing at a blackboard talking to his class about Leonardo da Vinci watching a river. The river was wide and slow where it began. Then the banks pressed inward, the channel narrowed, and the water quickened. Da Vinci watched and noticed that whatever volume flowed through the wide section flowed in the same unit of time through the narrow one. If the space halves, the speed doubles. The water accounts for every drop of itself.
Da Vinci’s notebooks contain what historians of science consider the earliest qualitative statement of what would later be formalised as the continuity equation: cross-sectional area multiplied by velocity is constant along a flow. Five centuries later, the formula governs every pipe, every artery, every wind tunnel in the world.
The teacher’s reel led me to think, not on the formula but on the sequence behind it: the river, the watching, the question, and then years later with notebooks dense with observations about water and the mechanics of moving bodies, the compression into symbols. That sequence reveals something specific : that knowing the journey behind a formula changes how we think, not by making us discoverers, but by making us less vulnerable to the formulas we live inside.
Let’s begin with a caveat : the observation-to-formula journey has always been made by very few people, not because the rest were poorly educated, but because the capacity to pursue a question all the way to formal abstraction is rare in any age right up to the present times : Newton, Da Vinci, Ramanujan, Al-Khwarizmi, Gödel. These are not representatives of any distribution. They are its extreme edge. Most people, across most of history, have held formulas without rivers and the world has functioned. The blacksmith understood the properties of iron without the chemistry of alloys. The navigator read the stars without the mathematics of celestial mechanics. The inherited procedure worked without the origin story.
The question this essay seeks to answer is therefore not why we are failing to produce more Da Vincis. That was never on offer. The question is narrower: does knowing that Da Vinci’s journey existed, that behind the formula there was a river, and behind the river a man troubled by something he could not yet name, does that knowledge do anything for the person who will never make such a journey themselves? I believe that the answer is yes. The evidence is in the world around us, in the failures that accumulate whenever formulas are applied by people who have no sense of the country they came from and in the rare cases where that knowledge made the difference.
A doctor looked at me across her desk and pronounced a number. My BMI, she said, indicated that I needed to lose weight. The certainty with which she said it was itself a kind of formula: a number had been computed, a threshold crossed, a conclusion followed. I was intrigued rather than persuaded. Not because I disagreed with the conclusion but because I found myself wanting to know where the number came from. What was BMI, exactly? Who had built it, and for what purpose, and on whose body? The answer, when I found it, reoriented everything. The Body Mass Index was derived by Adolphe Quetelet, a Belgian statistician, in 1832. Quetelet was not a physician. He was a social scientist building a statistical portrait of the average European man of his era as a sociological instrument, not a clinical one. It was adopted as a medical screening tool a century later by a different profession, for a different purpose, applied to populations its creator had never considered.
For South Asian bodies, the consequences of this transplantation are not minor. South Asian populations develop metabolic disease insulin resistance, Type 2 diabetes, cardiovascular risk at body compositions that Quetelet’s thresholds classify as normal or only mildly overweight. The formula, calibrated to European physiology, systematically underestimates risk in South Asian patients. The irony of my own encounter with it was this: when I understood where the formula came from, the conclusion did not become more comfortable. It became more demanding. I did not need to lose less weight because the formula had been calibrated to a different body. I needed to lose more because my risk threshold, properly understood, was lower than the number on the desk had suggested. The formula, applied without its river, had offered false reassurance rather than false alarm. This is what distinguishes knowing a formula from understanding it. Knowing it produces a number. Understanding it, knowing whose body it was derived from, what question it was built to answer, where it was never designed to travel changes what the number means and, in this case, what it demands.
This individual encounter scales into something much larger. India is currently managing what epidemiologists describe as a diabetes epidemic arriving ahead of its demographic schedule, a wave of metabolic disease in a population younger and leaner than Western disease models predicted. For decades, screening protocols built on BMI thresholds calibrated to European bodies missed the early signals in Indian patients who fell within normal ranges. By the time clinical presentation made the diagnosis unavoidable, the window for low-cost intervention had often closed. The formula had been doing its work faithfully. It simply did not know whose body it was working on.
The same failure takes a different and more catastrophic form in finance. In 2000, a mathematician at JPMorgan named David Li published a formula for pricing collateralised debt obligations, the instruments that bundled mortgage-backed securities into tranches of varying risk. It provided a way to calculate the probability that multiple borrowers would default simultaneously. It was elegant, it was tractable, and it solved a problem that had resisted quantification. Wall Street adopted it with the speed that attends any tool that makes the previously unmanageable appear manageable. The formula carried an assumption so deeply embedded that most of those using it were unaware it existed. Li had calibrated his correlation functions using data on corporate bond defaults, and applied the framework to mortgages under the implicit assumption that the two markets had similar correlation structures that defaults in Arizona and defaults in Florida were not the kind of events that happened together.
They were exactly that kind of event, under conditions that had never appeared in the historical data. When American housing prices declined simultaneously across all major markets in 2007, something that had not occurred in the post-war dataset, the formula’s correlations collapsed. The tranches rated AAA failed in sequence. Eight trillion dollars in household wealth was destroyed. Li himself had published a warning that the formula was being misused. It went largely unread by the people making the decisions, because those people did not know enough of the river to understand what the warning was about. The builders of the formula knew its derivation. The users of it largely did not. This asymmetry between the few who understood the origin and the many who only possessed the output is where the catastrophe was housed.
Not every case is a failure. Understanding where a formula came from can also explain why one works when others do not. In 1952, Virginia Apgar, an anaesthesiologist at Columbia University, watched deliveries and was troubled in exactly the productive sense that Da Vinci was troubled by his river. Decisions about newborn intervention were being made inconsistently, on the basis of clinical judgment that varied enormously between practitioners. Some infants who needed immediate resuscitation were not getting it. Others were subjected to unnecessary intervention. The variation was costing lives, and nobody had a reliable way to see it because nobody had agreed on what to measure.
Apgar went back to the river. She watched deliveries systematically until she had identified the five observable signs that best predicted a newborn’s immediate survival: heart rate, respiratory effort, muscle tone, reflex response, skin colour. She assigned each a score of zero, one, or two. A total below seven at one minute indicated a need for intervention. The formula could be applied in sixty seconds by any trained nurse in any delivery room in the world. The Apgar score was adopted almost immediately and universally. Infant mortality declined sharply wherever it was introduced. It remains in use today, essentially unchanged, more than seventy years later. What distinguishes it from the BMI and the Li formula is that Apgar built it close to its river derived from observations made on the population it would be applied to, for the specific decision it would be used to make. The formula carried the river because its builder had been standing in it. A formula built close to its river is recognisably different in use. Its failures are legible, its limits visible, its assumptions surfaced rather than buried. It behaves like a tool made for the hand that holds it.
The Green Revolution occupies a more uncomfortable position precisely because of its ambiguity. Norman Borlaug’s high-yielding wheat varieties, developed through the 1950s and 1960s, prevented famines that demographers had forecast as inevitable. The formula, semi-dwarf varieties bred for maximum yield under high-fertiliser, high-irrigation conditions, saved hundreds of millions of lives. This is not in question. But the formula carried assumptions that were not stated because its builders did not know they were assumptions. It presupposed reliable irrigation, which presupposed groundwater availability that was not infinite. It required monoculture planting at scale, which presupposed that soil could absorb continuous intensive cultivation indefinitely. The water table in Punjab has dropped by as much as a metre per year in some districts over the past four decades. Soil degradation is measurable and advancing.
The Green Revolution is not a story of error. It is a story of a formula that outran its river through necessity rather than negligence; deployed under emergency conditions, without time to examine what it was carrying. Its costs are only legible now, with enough distance to see the shape of what was silently assumed. The formula worked, and the assumptions it buried are arriving, decades later, as separate crises.
The cases above are not unified by a single villain. The doctors applying BMI were not incompetent. The traders using Li’s formula were not fraudulent. The agronomists extending Borlaug’s varieties were not reckless. In each case, the formula was applied with professional competence by people who possessed it without understanding its origin and the origin was where the limits lived.
Knowing a formula is possession of its output. Understanding it is knowing what question it was the answer to, what observations generated it, what population it was derived from, what assumptions its builders may not have known they were making. Judgment is the further capacity to sense when the landscape has changed enough that the formula, however precisely stated, is being asked to do work it was not designed for.
The physician who knows that BMI was derived from nineteenth-century European data applies it differently to a South Asian patient, not by abandoning the number, but by holding it lightly enough to ask what it might be missing. The financial risk officer who knows that Li’s correlations were calibrated on corporate bonds asks, before applying them to mortgages, whether the correlation structure actually transfers. The agricultural planner who knows that Borlaug’s varieties were developed under specific irrigation conditions asks, before recommending them for a new geography, whether those conditions obtain. What is called wisdom, in most traditions and most languages, is close to this: the ability to hold a principle lightly enough to know when it does not apply, because you know enough of the circumstances that shaped it to sense when those circumstances have changed. This is knowledge that is not decorative but load-bearing.
This is where AI enters, not as the culminating threat, but as the latest and most powerful instrument in a long sequence of tools that have delivered formulas at increasing speed and decreasing context. The teacher’s reel about Da Vinci is already a form of AI-adjacent distribution: an algorithm selected it for my feed without my asking, at a moment when I was not looking for it. A few minutes on a phone conveyed more of the journey behind the continuity equation than most classroom hours managed in a semester. The backstory of a formula, made available at the moment of encounter with the formula, this is what no previous educational technology could do at scale.
The risk is the familiar one, made sharper. AI pointed toward the formula delivers answers at a speed that makes the absence of context invisible, fluent, structured, complete, and without the scar tissue of the journey that genuine understanding carries. And AI can invent, hallucinate : a backstory fabricated with confidence is worse than none as it forecloses the question rather than opening it.
The cautionary account of AI omits something important. The observation-to-formula journey has always been gatekept not just by education, but by the material conditions of survival. For every Ramanujan who reached Cambridge, there were others who did not: too poor, too exhausted, too far from any institution that could recognise what they were. Ramanujan himself nearly did not happen. He was a postal clerk who had failed his examinations, working in Madras, unknown, until a single letter to a Cambridge mathematician changed everything. For every such letter that was sent and read, how many were never written? Da Vinci was apprenticed to Verrocchio at fourteen because his illegitimacy barred him from university. A different accident of birth and he dies a peasant, his notebooks unwritten, the river still narrowing with nobody troubled by it.
What AI removes, potentially, is the cost of the reach. The child who watches a river and feels the trouble before the words no longer needs the right teacher in the right city, the right patron, the right examiner who can see past the examination failure to what lies underneath. Whether that attention can be cultivated, or only recognised when it arrives, is the deeper question. But the removal of cost from the reach is historically new.
The question is which direction it will face. Pointed at the formula, AI produces a faster version of what industrial education already did. Pointed at the river, it might do something no classroom ever quite managed: make it possible, at scale, to know not just what the formula says but where it has been. And to find, among the billions now within reach, the ones who were always troubled by the right things and never, until now, had anyone to tell.