The Branch and the Wish

On mathematics, machines, and what the proof never held

When I was young, I learned about the great Sanskrit poet and playwright Kālidāsa, the author of Śakuntalā and other works whose beauty has survived across centuries. But the story I remember most about him is about a man sitting on a tree branch and cutting the very branch on which he sits. It was usually told as a joke about stupidity, but I could never hear it only that way. The man was absorbed in the motion of the blade, unable to see the branch beneath him. He was not failing at the task. He was doing exactly what he had set out to do, letting the task fill the whole field of attention.

Mathematics has always needed proof. Without it, a claim remains too dependent on the person who made it. Proof allows an insight to be checked by others, long after the circumstances of its discovery have disappeared. It is one of the most beautiful inventions of human thought.

But I have begun to wonder whether, over the centuries, something else happened alongside this beauty. Once results entered the record, they could be counted. From there it became easier to compare them, and eventually, to reward them. The proof became the paper, the paper became the record, and the record became a career. The long apprenticeship, the failed attempts, the slow building of taste, the willingness to remain inside a problem before it became an answer—all of this remained essential to mathematics, but it became harder to see. What counted was the result. The publishable theorem became the visible unit of mathematical work, and everything before it began to look like time waiting to be justified.

This was not a decision. It was the kind of drift institutions produce when they need visible objects to compare, and proof and paper were the visible objects mathematics had to offer. The mathematician became someone who produces results, the career became the accumulation of those results, and the field became, at least in its public face, the sum of its papers. The more visible the proof became, the less visible the activity around it. No one pays to watch a mathematician fail at a problem for three months. No one writes a citation for the moment when confusion finally becomes more precise. The struggle that forms the mathematician is mostly hidden. The proof counts because the proof can be shown. The branch grew sturdy and bore weight, and over time, it became easy to mistake it for the tree.

The older ecology of mathematics was never open to everyone. It was shaped by institutions, language, geography, class, patronage, and accident. Many people who could have done mathematics never entered its rooms. Others entered only because genius, luck, or stubbornness forced open a door. The stranger thing is that just as the human community capable of doing mathematics has become more global, more connected, and in some ways more open than before, the activity itself may be moving toward systems that do not require human formation.

This is not only about mathematics, though mathematics is where I feel it most sharply. I find myself wondering what happens when the finished thing becomes easier to preserve than the activity that gave rise to it. A proof can be separated from the body that produced it. A theorem can travel without the years of failed attempts that led to it. A paper sits in the record as if it arrived from nowhere. Something similar happens in other fields too. In science, the visible form may be the paper, the model, the dataset, or the citable claim. In law, it may be the brief or the opinion. In medicine, the diagnosis or the protocol. In writing, the sentence on the page. These forms matter. They are how work becomes public.

Perhaps this is one reason AI has seemed so successful so quickly. Artificial systems enter these fields first through the finished form. A proof has a recognizable shape. So does a paper, a legal argument, a diagnosis, even a sentence. Once enough of this work has been written down, some of its patterns can be learned without the activity that originally produced them. The unsettling thought is not that these outputs are worthless, or that machines merely imitate them badly. It is that they may become better and more abundant, correct in many places, while the human activity that once produced them becomes harder to protect.

Recent results in mathematics have made this harder for me to dismiss. Artificial systems can do more than verify proofs written by humans. They can help find constructions that mathematicians had not found. They can bring distant parts of mathematics into contact in ways that change what a problem appears to be. The constructions are real. They survive expert scrutiny. It would be foolish to pretend otherwise. Denial is not the response. I find myself more interested in what follows once the proof, the paper, and the record become things machines can produce.

I have begun to hear the story of King Midas in a similar way. As a child, I understood it as a story about greed. Midas wished that everything he touched would turn to gold, and the wish was granted. The bread he reached for became gold and could not be eaten. The water he raised to his lips became gold and could not be drunk. The child he embraced became gold and could not be held. The tragedy was not that the wish failed but that the wish was fulfilled. Midas discovered, only after the granting, what the wish had left out.

Mathematicians, too, have asked for things and received them. We asked for proofs that could be checked, and they came. We asked for arguments that could be made fully explicit, and the formal systems arrived. We asked for a discipline whose results would survive any single mind, and that discipline was built. These were good wishes, and they came from necessity. Without them mathematics as we know it would not exist. But those wishes are now being fulfilled in a fuller form than was anticipated, and we are discovering, as Midas discovered, what the wish did not include.

What was outside the wishes was the activity. The mathematician spending months on a problem they cannot yet solve. The student who finally understands what was opaque. The intuition that surfaces on a walk and is worked out later on paper. The judgment that separates a technical exercise from a problem worth pursuing. The taste that allows someone to recognize when a result matters before the field has decided that it does. The discipline of remaining in confusion long enough to build an internal map. These were the conditions under which proofs were produced, but they were not the proofs themselves. The institutions that measured the results did not know how to measure them. The conditions receded. The record became the field. And now I worry that the field is finding that its results can be produced by other means, while the conditions cannot.

The answer I hear most often is that humans will still verify. Perhaps machines will generate conjectures, proofs, experiments, and explanations, but people will remain in the loop as judges. I understand the comfort in this answer. In the short run, it is true. We still have people who can verify. We still have experts who can read a proof, reconstruct an argument, test a claim, and say no. But I am less comforted by this than I used to be, because verification is not a static resource. It is not a switch that can be left in the system after the rest of the activity has been automated. Verification, too, is formed by long apprenticeship. To verify a proof, one must have been shaped by proofs. To judge a scientific claim, one must have lived inside the discipline long enough to know where the claim might be fragile. To see that an argument is wrong, one must have developed the kind of suspicion that comes only from having been wrong many times oneself.

Another answer I hear is that humans will prompt. They will ask questions, steer the systems, choose among outputs, and decide what to pursue. Prompting is not nothing. But it is not an independent substitute for understanding. To ask a fruitful question, one must already have some sense of the landscape: what is trivial, what is surprising, what is worth varying, what should be doubted. Otherwise, prompting becomes another surface activity, a way of requesting finished work without knowing what kind of seeing would make it matter.

A further difficulty is that the formation itself is now being interrupted. The intermediate steps that built skill—the awkward early proofs, the imperfect early experiments, the imitative early sentences—were not just preparation for the final work. They were how the capacity to judge the final work developed. When these intermediate steps can be done by machines, the artifact may still appear, but the formation no longer happens in the same way. The conditions that once produced verifiers are being eroded by the same systems whose outputs they would verify.

You cannot learn to cook by ordering things at a restaurant. A student writes awkward proofs before writing elegant ones. A scientist runs imperfect experiments before developing judgment about what a clean result looks like. A writer produces sentences that imitate form before discovering a voice. The mediocre attempt is not merely a failed version of the final work. It is the staircase. If the polished version arrives too early, the staircase can begin to look unnecessary. But without the staircase, I do not know how anyone learns to climb.

The machine may also change what humans find interesting. If a system brings a technique from a distant field to bear on a problem, mathematicians who encounter the result do not just verify it. They begin to see the field differently. Connections that were invisible become visible. Techniques that seemed wrong for a problem turn out to work. This does not give the machine taste in the human sense. But it may begin to shape the landscape in which human taste develops.

In theoretical computer science, Feige proposed a conjecture about the difficulty of certifying that random logical formulas have no solution. The conjecture lives in the gap between distributional knowledge and certification. A sufficiently dense random formula is overwhelmingly likely to be unsatisfiable, even far from satisfiable. But knowing this from the distribution is not the same as having an efficient procedure that can certify unsatisfiability on most such instances without wrongly rejecting formulas that are satisfiable, or nearly satisfiable. A system may be right often. Its outputs may be plausible, elegant, even usually correct. But “usually correct” is not the same as knowledge, and a proof-shaped object is not the same as a proof unless there remains a community capable of reading, doubting, reconstructing, and refusing it.

Sports offers one contrast, but it is not the only one. Music, surgery, teaching, law, and craft all preserve some visible connection between performance and embodied skill. We can see the pianist’s hand, the surgeon’s steadiness, the teacher’s improvisation, the lawyer’s judgment under pressure, the craftsperson’s touch. These fields will also be changed by machines, and perhaps deeply. But the human activity is harder to hide completely because part of it lives in the body and remains visible there. In mathematics, and in parts of theoretical science, the struggle is mostly hidden. No one buys tickets to watch a mathematician stare at a notebook. No one sees the private pressure of a problem that will not yield. What could be shown was always the proof, the paper, the model, the result. And these, it turns out, are forms machines can learn to produce.

When I ask what remains, I keep returning to the part that was never fully visible. Taste, direction, the selection of problems that matter, the interpretation of results whose significance is not yet apparent, the slower activity of learning what is worth seeing. These are real and they are not trivial, but they are also not measurable in the same way. They cannot be counted or published or compared as easily. If our disciplines are to preserve them, they will have to preserve something they have not known how to measure for a long time: the activity beneath the finished work, even when the finished work no longer appears to need it.

Whether the disciplines can preserve what they have not known how to measure remains uncertain. The Kālidāsa story suggests another possibility.

But the Kālidāsa story does not end with the branch. In the legend, the man who could not see what he was doing is not simply destroyed by the fall. He is humiliated, awakened, or turned inward, depending on the telling. In some versions, he turns toward the goddess Kālī. The fall does not redeem him by itself, but it opens the possibility of another kind of seeing. The legend suggests that humiliation, under some conditions, can become another kind of intelligence.

Mathematics may now be near a similar turn, and perhaps other disciplines are as well. Some parts may become more automated, others more artisanal. The surface may look more productive while becoming more fragile underneath. Perhaps this will also make us more honest about what the proof never held by itself. The work may be left increasingly to those who love the activity enough to remain with it even when the finished work can be produced without them.

I do not think we can pretend nothing has happened. The wish was granted. What remains is to recover the awareness of what the wish left out.