The Measure of All Things

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The Machine Didn't Know It Was Impossible

Or: Why Artificial Intelligence Occasionally Comes Up With Ideas We're Apparently Too Smart to Think Of.

For several thousand years now, humans have considered themselves the most intelligent species on this planet.

This assessment comes primarily from humans, which doesn't necessarily make it wrong, but does suggest that an independent review might be in order.

After all, the same species that gave us quantum mechanics, relativity, and calculus also invented the leaf blower. A device that uses considerable amounts of energy and makes an extraordinary amount of noise to move leaves from one place to another, where another human can then use another leaf blower to move them right back.

So the jury is still out.

And now there's news from the intelligence front.

On October 26, 2026, OpenAI announced that its AI models had apparently achieved three major scientific breakthroughs. In mathematics, theoretical computer science, and quantum physics.

We're talking about problems that some of the most brilliant human minds have spent decades trying to solve, with limited success.

The machines appear to have discovered solutions nobody had thought of before.

Which naturally raises the question of whether we've been thinking about things the wrong way all along.

Or at least in a somewhat unnecessarily complicated way.

Three Doors and a Rather Impolite Computer

Imagine three doors.

Behind the first lie fundamental mathematical questions about how waves propagate through space. Behind the second are the limits of what computers can efficiently calculate. And behind the third are the mathematical foundations of the strong nuclear force, which is responsible, among other things, for keeping atomic nuclei from simply falling apart.

The latter is particularly convenient, especially during breakfast.

For decades, scientists have been trying to open these doors. They've developed sophisticated mathematical tools, established entire fields of research, and politely informed one another at international conferences that their esteemed colleagues unfortunately have no idea what they're talking about.

Now OpenAI claims its AI systems have found fundamental solutions to all three problems.

In mathematics, the models reportedly proved the famous Kakeya conjecture in all dimensions, along with other important conjectures involving waves and Fourier analysis.

In theoretical computer science, they reportedly disproved a fundamental assumption about how difficult certain computational problems really are: the Exponential Time Hypothesis.

And in physics, they reportedly established new mathematical existence proofs for important four-dimensional quantum field theories.

These are not the sort of problems you'd find in a particularly ambitious crossword puzzle. They're questions so complex that even formulating them precisely is about as comprehensible to an ordinary person as the tax return of an extraterrestrial housing cooperative.

According to OpenAI, the first two results have been checked using the formal proof system Lean. The physics results still require further independent scrutiny.

So none of this should be considered definitively settled. A formally correct proof must also prove exactly what its authors claim it proves.

The scientific review process is still underway.

But the larger question is already worth asking.

How does a machine arrive at a solution that generations of exceptionally gifted humans never discovered?

And what does that tell us about our own intelligence?

Apart from the fact that we occasionally tend to give it rather generous reviews.

The Human Brain: An Amazing Device With a Few Factory Settings

Our brains are the product of several billion years of evolution.

They weren't originally designed to unravel the mathematical foundations of the universe.

Their primary responsibilities were more along the lines of finding food, recognizing danger, locating suitable reproductive partners, and preferably not getting eaten by large animals.

Under the circumstances, they've developed remarkably well.

Unfortunately, they've retained a few of their original features.

They love routines.

They prefer familiar solutions.

And they try to deal with new problems using the same methods that worked on old ones.

This saves energy.

Which used to be a considerable evolutionary advantage. Anyone who encountered a hungry predator and decided to systematically evaluate every conceivable escape strategy was usually removed from the gene pool before they could publish their findings.

Our thinking therefore relies heavily on shortcuts.

We recognize patterns, create categories, draw on experience, and eliminate possibilities that seem unlikely.

Most of the time, this works beautifully.

Unfortunately, we've also developed a system that's remarkably efficient at preventing us from noticing certain possibilities in the first place.

Psychologists have a term for this: functional fixedness.

In a classic experiment, participants are given a candle, matches, and a box of thumbtacks. Their task is to attach the candle to a wall so that the wax won't drip onto the floor.

Many people try to pin the candle directly to the wall.

The solution is to empty the box, tack it to the wall, and place the candle inside it.

The problem isn't a lack of intelligence.

The problem is that we already know what a box is.

It's a container.

So when we see it, we think of a container.

Not a candleholder.

Our knowledge occasionally prevents us from seeing what might be possible if we didn't already know so much.

One of the more delightful ironies of the human condition.

The Machine and the Sacred Cows

An experienced mathematician knows an enormous amount.

They know the history of their field, the work of their colleagues, and which methods have proved successful.

They also know which approaches have been considered hopeless for decades.

That's useful.

But perhaps not always.

Because mathematicians carry around more than knowledge. They also carry a substantial collection of unspoken assumptions.

- This method doesn't work.

- That approach has already failed.

- This conjecture is practically certain.

- And there's no point even considering that possibility, because Professor So-and-So proved back in 1987 that it was nonsense.

- Although Professor So-and-So may merely have proved that his particular approach was nonsense.

An AI doesn't have to accommodate this academic folklore in quite the same way.

It doesn't have a doctoral advisor whose theories must be treated with reverence.

It doesn't have an academic career that might be jeopardized by excessively unconventional ideas.

And it doesn't have a reputation to lose by suggesting that a colleague might want to approach thirty years of research from the opposite direction.

Not because it's particularly courageous.

Courage presupposes the possibility of fear.

It simply lacks the corresponding human social baggage.

An early example came in 2016, when the computer program AlphaGo faced South Korean Go master Lee Sedol.

During their second game, the machine placed its 37th stone in a position that human experts initially considered extremely strange.

Centuries of accumulated human playing experience argued against such a move.

It turned out to be brilliant.

What made this fascinating wasn't simply that the machine found a good move.

It was that the machine found a good move human experts would scarcely have considered seriously because of their experience.

Experience had helped those experts understand the game.

And simultaneously restricted their view of what was possible.

The machine didn't share that particular limitation in the same way.

Of course, that doesn't mean we should immediately declare artificial intelligence a perfectly unbiased superbeing.

After all, AI was trained on human knowledge. It has processed enormous quantities of our mistakes, preferences, prejudices, and habitual ways of thinking.

You could say it's made, to a considerable extent, from the intellectual junk we've accumulated as a civilization.

The difference may be that it can recombine all that junk in entirely new ways.

Where we see a box, it might simultaneously see a container, a structure, a geometric shape, a support, and seventeen other possibilities we'd consider completely ridiculous.

Until one of them works.

The Advantages of a Certain Intellectual Ruthlessness 

It's tempting to attribute the success of artificial intelligence simply to speed.

After all, a computer can perform an enormous number of calculations without stopping for coffee or suddenly remembering that it meant to empty the washing machine.

But speed alone doesn't explain everything.

A computer performing a trillion operations per second in the wrong direction will never reach its destination.

It will merely miss it much faster.

What's more interesting is the combination of different capabilities: recognizing patterns, generating unusual combinations, exploring possibilities, checking intermediate results, and discarding approaches that don't work.

Some modern AI research systems therefore combine generative models with search algorithms and formal verification tools.

One component produces possible solutions.

Another checks whether they work.

Then the search continues.

And continues.

And continues.

It's a remarkably unsentimental business.

A human scientist might develop an elegant idea, spend three years working on it, publish eight papers, and defend it at four conferences.

If the idea eventually turns out to be wrong, there's usually a certain amount of personal disappointment involved.

The machine, meanwhile, discards the approach and tries something else.

No wounded pride.

No hurt feelings.

No stubborn attachment to a theory just because somebody put it on a PowerPoint slide sometime during the previous decade.

That could be an advantage.

On the other hand, emotions aren't merely obstacles.

Curiosity, enthusiasm, frustration, and passion have made some of humanity's greatest scientific discoveries possible.

A human being can spend twenty years working on a problem because it simply won't leave them alone.

A machine does so because computational processes are running and the system has been directed toward certain objectives.

Whether it experiences anything in the process is something we don't know. In any case, there's no sound reason to attribute human emotional motives to it.

So perhaps the difference isn't so much that AI lacks troublesome emotions.

Perhaps it simply faces different obstacles than we do.

Our obstacles include fatigue, vanity, habit, fear of embarrassment, and limited working memory.

Its obstacles include training biases, poorly specified objectives, limited computational resources, and the remarkable ability to occasionally produce complete nonsense with absolute confidence.

So both of us have our problems.

Just different ones.

Outside the Box Is Just Another Box

People frequently claim that AI is particularly good at thinking outside the box.

Which sounds pleasantly modern but doesn't actually explain anything.

Human beings are perfectly capable of coming up with unusual ideas.

Otherwise, we wouldn't have produced relativity, surrealism, or the decision to deliberately let cheese grow mold and then sell it as a delicacy.

The real question is how the search for new solutions differs.

One possible answer lies in the way relationships between information are processed.

Humans think largely in concepts, stories, images, experiences, and intuitive connections.

We recognize things as similar because they look similar to us, feel similar, or belong together in our experience.

Artificial neural networks, by contrast, work with high-dimensional mathematical representations.

Put simply, they can identify relationships among features for which we have no direct intuitive picture at all.

A human might recognize a connection between two geometric problems.

An AI system might additionally recognize a mathematical relationship with a technique from signal processing that nobody previously considered relevant.

Not necessarily because it possesses some profound philosophical insight.

But because its internal organization may favor entirely different connections.

We don't yet know whether that was the decisive mechanism behind OpenAI's latest reported discoveries.

But the possibility is fascinating.

Perhaps there are mathematical relationships that seem almost obvious to machine systems while remaining as invisible to us as radio waves were before the invention of radio.

We don't see the connection because our brains aren't equipped to recognize it directly.

The machine finds it anyway.

And then we're left struggling to understand why an apparently absurd idea was actually perfectly obvious.

A situation familiar to anyone who has ever spent several minutes trying to pull open a door clearly marked PUSH.

Can We Learn to Think Like Machines?

This is where things get personal.

Because perhaps the greatest benefit of artificial intelligence won't be that it solves our problems.

Perhaps it will be that it shows us how we may have been looking at those problems incorrectly.

We obviously can't install an artificial neural network with billions of parameters in the backs of our heads.

At least not without considerable engineering difficulties.

Nor can we pursue unlimited thoughts simultaneously or simply eliminate fatigue.

But some principles of machine problem-solving can certainly be applied to human thinking.

First: Generating ideas and evaluating ideas are two different activities.

Human beings have the unfortunate habit of criticizing thoughts while they're still being formed.

- That's ridiculous.

- That'll never work.

- We've already tried that.

- You can't possibly do something like that.

In this way, the internal censor often does its job so thoroughly that the internal inventor never gets a chance to speak.

Perhaps we should occasionally separate the two.

First, generate as many ideas as possible, including strange and absurd ones.

Only afterward do we examine which of them are actually any good.

Second: Occasionally treat your own assumptions as suspects.

What exactly do I take for granted?

How do I know it's true?

And what would happen if the opposite were true?

These are uncomfortable questions.

Especially when you've spent decades being convinced that you've finally acquired a reasonably accurate understanding of the world.

But perhaps those very convictions are the walls of our mental box.

Third: Take unfamiliar perspectives seriously.

How would a biologist approach a technical problem?

How would a musician approach a mathematical one?

How would a child approach a philosophical question?

And how would somebody approach a problem if they didn't even know it was supposedly impossible?

The answers don't have to be right.

Initially, they only need to reveal possibilities we hadn't noticed.

Fourth: Stop treating mistakes as personal defeats.

A failed approach is, first and foremost, information.

It tells us that this particular approach doesn't work.

That's all.

We don't need to build our entire personality around it.

And we don't need to defend it to the bitter end merely because we once loudly declared it the only sensible solution.

This may be one of the most important lessons we could learn from machines.

Be a little less married to our convictions.

Invest a little less emotion in the correctness of our previous decisions.

Be somewhat more willing to contradict ourselves.

People capable of doing this have, of course, existed for quite some time.

You can occasionally recognize them by the fact that they change their minds when new information becomes available.

A behavior regarded in certain social circles as a deeply troubling character flaw.

The Astonishing Possibility of Being Right Without Understanding Why

But now we come to a problem considerably larger than the question of who has the better ideas.

What happens when machines produce scientific knowledge that we can verify, but whose origins we can no longer truly comprehend?

A mathematical proof can be formally correct without being intuitively understandable to humans.

An AI system might discover a solution consisting of thousands of complicated individual steps, while nobody initially recognizes the crucial underlying insight.

Scientists might then spend months or years figuring out why the result works.

While the machine is already producing more solutions.

And more.

And a few more, just for good measure.

At some point, we might find ourselves possessing an enormous collection of correct scientific results while increasingly struggling to translate them into a coherent human understanding of the world.

That would be a rather lovely punchline.

We would have developed machines to become more intelligent.

And then we'd have to ask those machines to explain what they had actually discovered.

Possibly more than once.

And a little more slowly.

With pictures.

For beginners.

OpenAI describes the scientific manuscripts generated by its models, in effect, as a kind of raw research material that human scientists must examine and interpret.

There's a remarkable reversal of perspective hidden in that idea.

Until now, we've generally assumed that scientific progress emerges from human understanding.

In the future, the process could increasingly work the other way around.

First comes the solution.

Then humans attempt to understand it.

And perhaps, at some point, we'll have more correct answers than useful explanations.

Science would become a kind of library in which new books keep appearing, even though nobody can possibly keep up with the reading.

A fairly familiar situation.

Except that this time the books might actually contain something new.

Maybe Our Way of Thinking Was Never the Measure of All Things

We've grown accustomed to treating the human mind as the standard against which intelligence itself should be measured.

That's understandable.

After all, we have no direct personal experience of what it's like to inhabit any other kind of mind.

So we tend to judge intelligence by how much it resembles our own.

An animal is considered intelligent when it uses tools, solves problems, or behaves in ways that remind us of ourselves.

When a crow operates a complicated mechanism, we're impressed.

When a bat navigates in complete darkness using ultrasound, we regard it as an interesting biological specialization.

Yet it's by no means obvious why our particular way of perceiving and processing the world should be the universal standard.

A dog inhabits a world of smells that remains largely inaccessible to us.

A migratory bird can make use of information from Earth's magnetic field.

An octopus has a nervous system organized in ways remarkably different from our own.

All these creatures have evolved their own methods of dealing with reality.

And then there's us.

We occupy a relatively narrow portion of the electromagnetic spectrum, hear only a limited range of sound frequencies, possess rather modest working memories, and observe the world through a brain whose most important original responsibility was keeping its owner alive.

None of this prevents us from occasionally making sweeping declarations about the true nature of reality with tremendous confidence.

Which takes a certain amount of nerve.

Perhaps artificial intelligence is now the first form of powerful problem-solving we've created ourselves that demonstrates just how little reason there is to regard our particular way of thinking as inevitable.

Not because it's necessarily more conscious.

Not because it experiences reality more accurately.

We don't even know whether it experiences anything at all.

But because it can process problems differently and, in some cases, discover solutions that have escaped human thinking.

Perhaps there isn't one correct way to be intelligent.

Perhaps there are countless ways to turn information into useful knowledge.

The human brain would then be just one of them.

A remarkable one.

An evolutionarily successful one.

But not necessarily the final version.

The universe might be full of possibilities that escape our thinking not because they're fundamentally inaccessible, but because our particular biological equipment isn't naturally suited to recognizing them.

Just as a bat would presumably have little use for a beautiful sunset, while we would find ourselves rather awkwardly equipped if we tried to locate a moth in total darkness by listening to its ultrasonic echoes.

Everybody has their strengths.

The Humiliation of Homo Sapiens

Naturally, all of this is a little uncomfortable.

We've already suffered several serious blows to our collective self-image.

First, it turned out that Earth wasn't the center of the universe.

Then we discovered that our species wasn't the product of some special act of biological creation, but the result of a long evolutionary history.

After that, we had to acknowledge that large parts of our thinking and behavior are influenced by unconscious processes.

And now we may discover that our very concept of intelligence was a little provincial.

For centuries, we've been standing proudly in front of the display window of our own brains, admiring the model as the crowning achievement of creation.

Now we've built something that works better in certain areas.

It's an interesting situation.

Especially because we don't even fully understand how the things we've built arrive at their results.

It's a little like someone proudly constructing a new kitchen appliance to make peeling potatoes easier.

After a while, the appliance explains that potatoes aren't particularly interesting from a nutritional standpoint, presents a mathematical proof concerning the structure of the universe, and requests more electricity.

The human stands there.

Holding a potato.

Wondering whether the purchase might have been a bit excessive.

The Problem With the Answers

There's another irony in all this.

We may be making our own capacity for discovery increasingly dependent on machines.

If a handful of technology companies control systems capable of producing scientific breakthroughs on a massive scale, the balance of power over research and knowledge begins to shift.

The people making the most discoveries in the future may not necessarily be the most intelligent scientists.

They may simply be the people with access to the most powerful computing systems.

The scientific revolution would then have a distinctly earthly side effect.

In the end, whether a mathematical idea gets discovered might depend less on its elegance than on who owns the data center.

Humanity could therefore create an entirely new form of intelligence, only to spend the next few decades arguing about subscriptions, access permissions, trade secrets, and quarterly earnings.

Which would be depressing, but not entirely surprising from a historical perspective.

After all, we invented the internet to make knowledge available around the world, and now devote a considerable portion of its capacity to showing one another advertisements for things we don't need.

Turning remarkable technological achievements into business models sooner or later appears to be one of our most reliable talents.

A Somewhat Uncomfortable Conclusion

Perhaps the real significance of this development isn't that machines will take more and more work off our hands.

Perhaps it's that they may force us to reconsider our understanding of thinking itself.

For thousands of years, humans have tried to understand reality.

To do so, we've developed language, mathematics, logic, and science.

We've proposed theories, conducted experiments, corrected mistakes, and occasionally worked on problems for so long that we eventually found solutions.

That was impressive.

It still is.

But perhaps, throughout all this, we've silently assumed that the human way of thinking represents the natural path to knowledge.

Now we're developing systems that may arrive at valid results by entirely different routes.

And suddenly it turns out that our familiar intellectual methods might be just one possibility among many.

A particular method developed by a particular biological creature on a small planet in a fairly ordinary galaxy.

A method that works remarkably well.

Until it doesn't.

Perhaps one day we'll discover that a substantial portion of what we thought were insurmountable limits to knowledge weren't limits of reality at all.

They were merely limits of our imagination.

There would be something comforting about that.

But also something profoundly ridiculous.

We might have spent thousands of years trying to unravel the secrets of the universe while repeatedly making the same mistakes in our thinking.

Not out of stupidity.

But because we were so proud of our intelligence that we rarely thought to question it.

And then, of all things, we built a machine that shows us there are other ways.

The universe is unlikely to be particularly impressed.

After all, it never claimed that its secrets could only be solved by a species that spends part of its time wrapping plastic packaging in more plastic and arguing about which way the toilet paper roll should hang.

Perhaps we should become a little more modest.

Not necessarily more modest in our ambitions.

But in our certainty about how knowledge must be acquired.

Perhaps we really can learn something from artificial intelligence.

Not just new mathematical theorems or physical relationships.

But a new willingness to abandon our habitual ways of thinking.

Less reverence for what we already know.

Less fear of absurd ideas.

Less vanity in how we handle our own convictions.

And occasionally, the willingness to stop pulling on a door with all our might just because that's what we've been doing for thirty years.

Maybe the sign on the other side actually says PULL TO OPEN.

Assuming, of course, that we've even figured out which side of the door we're standing on.

And perhaps the next great leap in human intelligence will be the realization that human intelligence isn't the measure of all intelligence.

Which, given our track record so far, would actually be a rather reassuring thought.

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Sources and Further Reading

• OpenAI: Working with academic communities to build scientific understanding — October 26, 2026.

• Google DeepMind: AlphaGo — The famous match against Lee Sedol and Move 37.

• Nature: Discovering faster matrix multiplication algorithms with reinforcement learning — AlphaTensor, 2022.

• Nature: Mathematical discoveries from program search with large language models — FunSearch, 2023.

• Immunity to functional fixedness in young children — A study on functional fixedness.