One of the world's greatest mathematicians explains 6 essential concepts of math | Terence Tao

My name is Terence Tao. I'm a professor of mathematics at the University of California, Los Angeles, and I have a forthcoming book, Six Math Essentials. Today on Big Think, I'll be talking about six essential pillars of mathematics — how math interacts with science historically, how it has anticipated many of the great developments in the sciences, and how new developments in AI will impact math and science going forward.

Chapter one: the six essential elements of mathematics.

I decided to organize my book around six really fundamental concepts that have origins from thousands of years ago or centuries ago. They are very familiar to most people in the early stages, but mathematicians have developed them over time to become extremely sophisticated. Numbers is the first concept, and then algebra, geometry, probability, analysis, and dynamics.

These are very basic concepts, but they've evolved into very sophisticated mathematics. Strip away the technical complexities, though, and they are extremely intuitive ideas; the mathematics we developed is just a precise language for describing them carefully, in a way that lets you think clearly about these concepts.

Numbers are one of the oldest mathematical inventions and still the most useful. We have records of carvings on bones that predate the alphabet or other writing, and numbers were invented multiple times by multiple civilizations. Without numbers, we would always have to speak poetically when describing any situation, which would always be a little imprecise—the next person relaying what you said would get it slightly different. Numbers allow for precision. They are placeholders for concepts like quantity, size, and magnitude, turned into very portable things you can communicate to people who may never have directly interacted with the objects you're describing. Once you try to describe a very complicated scenario with many moving parts, you need numbers and everything built on top of them.

Humans are not really wired to think in numbers. If you can't think quantitatively—measuring both the benefits and the costs of an action and which is bigger—you can make life choices you'll regret later, spending a lot of resources for very little gain. The first step toward thinking more quantitatively is understanding numbers, and then more advanced topics like probability and algebra on top of that. Basic things like agriculture or trade could not have happened without numbers to measure large quantities of grain, and the ability to tax—you can't have a civilization without taxation, unfortunately—requires mathematics and numbers.

Of course, not everything is quantitative. If you want to go on a date, you shouldn't be measuring the costs and benefits of your prospective partner. Some things should remain subjective and personal. But increasingly in the modern world there are many decisions—for example in finance or medicine—where quantitative thinking is very helpful.

Numbers take on a life of their own. Once you have the concept of number, you can study numbers abstractly, divorced from their actual applications, and you find patterns—and often it's very natural to extend the number system you have, creating new numbers you wouldn't have thought applicable to your original context, but which fit very well into the system. If you're counting sheep, you want to add sheep and subtract sheep, so you soon develop the notions of addition and subtraction. But if you're only counting 1, 2, 3, 4—not even zero—you find you can always add numbers together but can't always subtract: subtracting four from three makes no sense, since you can't take away four sheep from three sheep. Yet the patterns in the numbers themselves are so regular that if you blindly apply the rules of arithmetic, it feels like you should be able to take three from four and get a new number. It took a while, but eventually people realized you could invent negative numbers—adding −1, −2, −3, and zero to the number system. Zero itself took a long time to be recognized as a good addition. And you still get all the nice laws of arithmetic: if you take a number a, subtract b, then add b back, you get a again—and that law still works even with negative numbers.

Similarly, we learned to divide numbers by one another and created fractions, which fit very well into the number system. Then came a shock: between all these rational numbers, all these fractions, there were numbers like the square root of two that could not be expressed as any ratio. This was a big shock—these numbers are literally called irrational numbers, which is Latin for "insane," not unreasonable, numbers—but they do exist and are very useful. You can never write down all their digits on a finite sheet of paper, yet it is very useful to have all these extra numbers lying around. Eventually we tried to take square roots of negative numbers, which we couldn't do in the regular number system, and we invented complex numbers—and that turned out to be…

These number systems turned out to be extremely useful for electromagnetics and quantum mechanics. It's remarkable that number systems often invented just so we could be better at solving equations and practical problems end up being the most natural language for describing very complicated real-world phenomena like quantum mechanics.

Algebra is the second layer of abstraction over numbers. With numbers, we took concrete things—a flock of sheep, a quantity of water—and replaced them with numbers that we can then apply operations to: addition, subtraction, division, and so forth. Algebra goes one step further: it stops looking at specific numbers like 7 or 17, replaces numbers with generic placeholders named x and y, and studies the operations themselves—plus, times, and the others—asking what properties the operations have, not just the numbers.

People discovered that these operations, so useful in arithmetic, have many fascinating properties. Addition has a property called commutativity: adding a to b gives the same result as adding b to a. That turns out to be an extremely useful property for solving problems involving addition. Similarly, multiplication and the other basic arithmetic operations obey these very simple laws.

Later we discovered that these laws also hold for other operations. For example, if I rotate an object by 30° and then by another 60°, I end up in the same position as if I rotated 60° first and 30° next—these two rotations commute. Even though this operation isn't addition or multiplication in the traditional sense of numbers, it has the same algebraic structure. On the other hand, some operations do not obey the commutative law: if I put on my socks and then my shoes, I get a different outcome than putting on my shoes first and then my socks. Those two operations do not commute.

So some operations obey nice laws like the commutative law, and some don't. Sometimes the laws of a new operation closely resemble laws we already understand for numbers, and then we can transfer our intuition, ideas, proofs, and theory of numbers to a different setting. Matrices, for example, are much more complicated than a single number—they're a whole square array of numbers—but they obey very similar laws of algebra. If you're good at manipulating numbers, you can manipulate matrices the same way, and many modern technologies, such as large language models, are based on manipulating matrices very efficiently.

Once you've abstracted from numbers to algebra, you work with equations involving variables like x and y. X may have had some physical meaning and a specific value, but it often clarifies your thinking to ignore the specific values and what they represent, and just manipulate the equations by pure algebra—moving symbols around. When you first learn this, it feels disconnected from actual experience, but it is a very powerful technique.

One early use of algebra is the story of Johannes Kepler, the astronomer and scientist. Walking through his hometown one day, he saw wine sellers selling wine by the barrel—some large, some small—and the market had a way of figuring out how much wine was in each barrel so sellers could be paid. You could pour a barrel out into cups to measure it, but that was very tedious. What fascinated Kepler was that the person in charge of the market had a very efficient way to measure a barrel's volume: he had a stick with markings on it, and the barrel had a bung hole in the middle. He poked the stick down through the bung hole into the corner, saw how far it went, and based on that marking could say, "This is 30 gallons of wine," and price it accordingly.

This astounded Kepler: how could a single diagonal measurement determine the volume, when some barrels are tall and skinny and others short and wide? It was a puzzle, so he went home and wrote out equations—assume the barrel has radius r and height h. Nowadays this is a question you could assign to a high school student; it's almost precisely one of the word problems we love to give students. He computed the volume and that diagonal length, and found that the length did not completely determine the volume. But reasoning that merchants want to sell as much wine as possible—maximizing volume for a given length—he added this profit incentive to the problem. Even though he couldn't quite solve the equations right away, he did some rudimentary what we would now call calculus, and the result almost exactly matched the shape of the barrels actually sold in the marketplace.

His formulas did actually match what people used with very high precision. So he explained rules that, I think, the marketplace had come up with over time just by empirical measurement. He had found a very satisfactory explanation, and I think this was part of the inspiration for the calculus developed by Newton and Leibniz a few centuries later.

Geometry is literally Greek for "measurement of the earth." Since antiquity, it was important to know how many miles it was to travel from one place to another, and how to navigate in the ocean or out in the wilderness by the stars. We needed to understand how to use things we could observe, like angles and distances, for very practical problems like transportation.

Just as numbers have patterns like a + b = b + a, once you start measuring distances between points and angles, there are lots of relations between those measurements as well. Geometry obeys laws just as numbers obey laws.

One such law is similarity. Once you know two shapes are similar—they have the same angles and so on—all their sides are proportional. If the side of the big triangle is, say, two times as big as the side of the small triangle, then all the other sides of the big triangle are also two times as big. The proportions are equal.

This is so useful because it lets you predict measurements of distances or scales you couldn't directly reach or measure. You can look at a distant mountain and estimate how far away it is, because if you know something about how tall it is and the angle of elevation, you can use a similar triangle and make a scale model. That way you can navigate and figure out how to reach a distant location even if you can't see it directly from your sailboat. Even in ancient Greek times they...

Even in ancient Greek times, people could measure distances to the moon and the sun, which they could not possibly measure directly. Using the laws of geometry they had already worked out—similar triangles and the like—they obtained reasonably good measurements without any satellites or advanced technology. Once you know geometry, you can extend your senses well beyond what you can touch and measure directly.

The fourth math essential is probability, the standard way mathematicians encapsulate one of the basic features of the real world: uncertainty. In primary or high school classes, mathematics is often presented through sanitized, predictable word problems—Annie has 30 apples and gives half to James, so how many does James have?—where everything is precise and all the information is known. But the real world contains uncertainty and unpredictability. When you flip a coin, it may be heads or tails. In principle, if you knew exactly how much force was applied and all the relevant measurements, you could simulate enough to predict exactly how the coin lands, but this is extremely difficult and often you don't have that data.

What was eventually realized is that rather than trying to compute exact answers for every single outcome, sometimes you have to accept that a measurement has a range of outcomes—and what matters is which ones are more frequent and which are less frequent. The first people to recognize this were gamblers, who bet on events like dice rolls summing to a certain number. If they calculated the odds correctly, they made money in the long run; if not, they lost it. The mathematics of probability was created through letters that gamblers wrote to their mathematician friends asking for help optimizing their gambling strategies.

Probability has blossomed well beyond its gambling roots. Anytime a system is too complicated to model from first principles, there will be some stochasticity and we want a probabilistic model—whether it's the stock market or whether a drug will be successful, we turn to probability.

Sometimes we don't know what the odds are. Probability works best for events that happen over and over again, with thousands of trials, so we can get a good measure of the odds. For an event that happens only once in a century, it may not be quite the right mathematics, and mathematicians are still working out a better replacement for probability for extremely rare events.

You might think that every different experiment—a medical trial, a die roll, the genetic traits emerging from evolution—produces entirely different distributions, some heavy-tailed and some narrow. But there are curious laws in probability called universality laws: even for very general types of random systems, common shapes emerge. The most famous of these...

The most famous universality law is the Gaussian, or bell curve: many distributions, such as the heights of men or women, form almost a perfect bell curve. We now understand probability well enough to explain quite a few of these universal laws mathematically, though some remain mysterious.

Analysis is how mathematics deals with two things. The first is inaccuracy in our measurements—sometimes it's not randomness but just imprecision. Analysis is the mathematics of error bars: we need to understand not only the numerical value of quantities but how much uncertainty surrounds them, the plus-or-minus error bars. Qualitative language of "big" and "small" only works up to a point. You might measure the length of an object as roughly 2 meters, but maybe 2 meters plus or minus 10 centimeters. Ideally the errors would be zero, but in the real world we can't always make them entirely zero. Sometimes, though, by making more and more precise measurements we can shrink the errors toward zero—it may just take an infinite amount of precision or time to get them all the way down.

Analysis is also about taking limits and dealing with infinities. The laws of algebra work fine for a finite number of operations: add five things together and you can rearrange them in any order. But once you work with an infinite number of operations, funny paradoxes appear—sometimes rearranging an infinite sum changes its value, which never happens with finite sums.

Consider betting on red and black at roulette, where you win 50% of the time and lose 50% of the time. There is a theorem that no strategy can guarantee a win in the long run as long as you have only a finite amount of money—no betting strategy can beat the house. The doubling-down strategy, where you bet bigger and bigger after each loss so that one win recovers everything, seems to beat the house constantly. The problem is that it assumes an infinite bankroll: it compresses all the risk of losing into a very small event where you keep losing, until you're betting millions and go bankrupt. Analysis helps you understand exactly what these tail risks are and how to reason with infinities in a way that avoids such paradoxes. A lot of inaccurate mathematics predated analysis, where people would just say, "If I do this infinitely often, all these problems go away." It took a while to realize that infinity is a dangerous beast if you're not trained to handle it. Yet we do need to deal with very large numbers in the real world, and infinity is a very good approximation for understanding them—as long as it's used with care.

One famous demonstration of infinity's unintuitive nature is the infinite monkeys theorem. If you have an infinite number of monkeys, or one monkey typing forever on a typewriter and hitting keys at random, the monkey usually types gibberish, occasionally a word like "the," and sometimes a sentence. The theorem says that if you wait long enough, the monkey is almost certain to eventually type whatever you like—the complete works of Shakespeare, Hamlet, Wikipedia, anything. You can prove this mathematically: as long as the probability of the monkey producing the text at least once is positive—no matter how small—then given enough time, the probability of hitting that pattern goes to one. It's like Russian roulette with one bullet: no matter how many chambers the revolver has, if you keep firing, you'll eventually get your hit.

For any given word, sentence, or paragraph, the monkey will eventually create it, but the time required grows exponentially with the size of the text. A four-letter word might take an hour of typing; a seven-letter word like "Shakespeare" may take years; a sentence might take millennia; and even a single page of Hamlet would take far longer than the age of the universe. So infinity is really a placeholder for anything potentially larger than any fixed number you could name. Even though the real world has neither infinitely many monkeys nor an infinite budget, we often reason with infinity first as an idealized case to see what is possible, and only then turn to the quantitative question of what can be done with finite resources.

When I was a kid, I played a lot of computer games, and many had cheats that gave you infinite health or infinite ammunition. It was sometimes helpful to play with those cheats first—without worrying about managing health potions or ammo—just to learn how to solve the game, and then play on a harder mode and learn to do things efficiently. Many problems in math are solved this way. One thing that distinguishes math from other disciplines is the freedom to fail, because failure is cheap: a bad business decision can bankrupt a company, a surgeon's mistake can be catastrophic, but if a math problem doesn't work out because of a wrong assumption, you just try again. In fact, a good move when solving a math problem is to first make an idealized assumption—

In math, making an idealized assumption — infinite energy, zero friction, or some other unrealistic simplification — is actually a good first move. You solve the problem in that infinite world, then work back to the finite one. This is where analysis comes in: carefully seeing which features of infinite mathematics still work in the finite world and which ones break down.

Dynamics is the mathematics of change over time. It studies how rules of incremental change cause a state to evolve from one moment to the next, and it produces all kinds of emergent and interesting behavior you might not expect from the initial rules. Even very simple rules can generate extremely complicated emergent behavior if you iterate them long enough.

Simple rules, complex behavior

Evolution is a good example from biology. You have organisms that reproduce, fitter ones survive more often than weaker ones, and organisms pass traits down to descendants. These are very simple rules, yet they produce a massive diversity of species, predator-prey relationships, and incredibly complicated dynamics.

Traffic works the same way. Each car on the freeway is doing nothing complicated — it speeds up when there are few cars ahead and slows down when there are many, simply trying to optimize its own flow. But put all the cars together and you get amazing emergent phenomena like traffic waves, compressions and expansions that behave something like waves in a slinky. If an accident causes cars to pile up, the math of the iterated dynamics shows that even after the accident is cleared and there is no obstruction, the compression wave takes a while to dissipate. Living in Los Angeles, I've often hit a slowdown with no accident or immediate cause, because hours earlier something triggered a jam and the wave simply hadn't dissipated yet.

Once you understand the dynamics well, you can model and simulate, and even make predictions — for example, will adding a lane to this freeway improve traffic? Sometimes it doesn't: paradoxically, closing off certain lanes can actually make global traffic flow faster.

Stable and unstable equilibria

Some dynamics are predictable. We have equilibria — states that stay the same for all time — and sometimes these are stable: if you move a little away from the state, you come back. A pendulum hanging straight down is a stable equilibrium; perturb it slightly and it settles back. But a pendulum balanced upside down on its tip is also technically an equilibrium — it could stay there forever — yet any slight perturbation causes it to move away over time. Knowing which equilibria are stable and which are not is important.

We now face a world of climate change where, for thousands — perhaps ten thousand — years, we have lived in a climate close to equilibrium: hotter or colder in some years, but always bouncing back. We're now in danger of leaving that equilibrium for much less stable dynamics, which is scary but needs to be modeled. We may have to adapt our agriculture and our other practices.

Why dynamics matters

Understanding which systems are stable, which are not, which are chaotic and which are predictable is extremely important. There are mundane examples like weather prediction: we take for granted accurate seven-day forecasts, but this is an amazing achievement of atmospheric scientists, who collected vast amounts of data and solved many dynamical systems problems to drive the error rate down. Forecasts a week ahead are still not 100% accurate, but far better than guessing.

Systems involving large numbers of humans, like the stock market or politics, remain well beyond current dynamical systems theory. Natural systems, and some human ones like traffic, we can model. It's a fairly advanced area of mathematics — often requiring heavy computer simulation and advanced differential equations — but it yields very valuable insights. One discovery of the field is that most systems exhibit what's called chaos, which came as a surprise. In the 17th century, Newton introduced his law of gravitation...

From the Two-Body Problem to Chaos

One of the great successes of Newton's theory was that it explained the motion of the moon around the earth and the earth around the sun. He could explain retroactively all of Kepler's curious laws, such as why planets move in ellipses. In other words, he solved what we now call the two-body problem: given two massive objects like the sun and the earth, governed by a single law of motion — Newton's inverse square law of universal gravitation — he could solve the equations using his newly derived calculus and obtain perfect formulas for the orbits, which were exact ellipses, verifying Kepler's theory. It was an amazing achievement.

Once the two-body problem was solved, it was natural that many of Newton's successors — and I think Newton himself — tried to solve the three-body problem. Newton once said this was the only problem that ever gave him a headache, because no matter what he tried, he could not get an exact solution. The leading scientific societies of the time offered major prizes for anyone who could write down a solution; it was considered one of the major open problems in mathematics. We still do not have an exact solution for these equations, and the belief now is that there isn't one that can be written as a neat formula.

When you look at the numerics, the motion is not a nice periodic pattern. It often stays periodic for a long time, but then suddenly shifts to something slightly different, and then changes yet again. We suspect that our solar system, which currently has about eight planets, once contained other planets. They mostly moved in elliptical orbits as Kepler described, but every so often the small gravitational interactions with Jupiter, Mars, and so on would jiggle these planets slightly out of their usual orbit. Occasionally they would veer off completely: sometimes two planets would collide, or one would escape the solar system. The asteroid belt, for example, is believed to be the remnant of a collision from millions of years ago. Even the most stable-looking system like the solar system, which seems not to have changed for millennia, has long-term instabilities.

Once you move beyond the simplest systems, there are many tiny, hard-to-predict deviations that occasionally pile up — just as a bunch of monkeys can occasionally write the works of Shakespeare, gravitational perturbations can occasionally set an entire planet off course. So in the most advanced forms of dynamics today, even when a system is completely deterministic with no unpredictability whatsoever, the best way to model it is often to approximate it probabilistically, assuming some random fluctuations back and forth. Eventually your predictions just get blurrier and blurrier.

This blurring of predictions seems to be a fundamental feature of chaos, which many systems have. These six essentials don't describe all of mathematics, but they do describe six of the great themes mathematics tries to encapsulate—and there is much more precision to it. This is just a taste of what goes on these days.

I view STEM as a whole ecosystem. At the bottom there's basic research, like mathematics and some other fundamental sciences, where we pursue things mostly driven by curiosity. We see a phenomenon that is crying out for an explanation or further study. It may not be something we urgently need to solve for an immediate problem, but it looks like it should have an interesting answer. Mathematics is almost entirely curiosity driven in this way: there's some pattern in numbers, some pattern in shapes, observed by people while trying to do something else, and we want to understand it better.

At some point, other scientists are able to connect that pattern to something they're studying—a mathematical or numerical pattern might show up in the behavior of an insect swarm, or in a stock market, or whatever. And sometimes, once you understand it, you can convert it into useful technology, or build a company that makes money from a service exploiting that phenomenon. We often don't see that connection; it's much further down the pipeline.

What you do need is for the people doing basic sciences to talk to the people doing applied sciences, who have to talk to the engineers, who have to talk to people in industry. If you didn't have one of these communities, you wouldn't have this pipeline from curiosity-driven questions to actual commercial results—like the ability to communicate across the planet at almost zero cost.

This is part of what Eugene Wigner calls the unreasonable effectiveness of mathematics in the physical sciences. He observed that mathematicians often discover concepts such as complex numbers or curved space simply because they seem to be a natural extension.

Mathematicians often extend the objects they study simply because it feels natural, and then 10, 20, or 50 years later scientists discover that these concepts—introduced for fun or play—were almost exactly what was needed to understand some new type of science. It's a really amazing phenomenon, and we still don't have a good explanation for why it works.

The parallel postulate and non-Euclidean geometry

One historical example of curiosity-driven mathematics leading to a deep scientific advance is the story of the parallel postulate. Around the 3rd century BC, Euclid introduced the notion of proof—explaining complicated geometric results from simpler axioms. For example, he could explain that the angles of a triangle always sum to 180° in terms of simpler axioms. He reduced all the known facts about points, angles, and lines to five statements. Four of them were very straightforward and non-controversial, like the fact that a line can always be drawn between two points.

But one axiom, the parallel postulate, gave him a lot of grief, and his original version was very complicated. Even the simplified version remained controversial: given a line and a point not on it, there is exactly one line through that point parallel to the first—meaning it never crosses the first line. Once you accept that axiom, you can derive the 180° triangle result and all the other classic results of Euclidean geometry. But compared to the four elegant axioms, it was an ugly one.

Eventually people realized there are actually multiple geometries beyond Euclidean geometry. In spherical geometry there are no parallel lines at all: instead of lines you have great circles, like the equator or a line of longitude, and these always intersect—you can never make two parallel great circles. There is also a weirder geometry that's harder to visualize, hyperbolic geometry, where lines diverge from each other: lines can start off looking parallel but move further and further apart, so from one point you can draw multiple parallel lines to a given line. Both geometries are entirely self-consistent, and eventually it was accepted that these were the first two non-Euclidean geometries.

Once we freed ourselves from the notion that there's only one geometry, the floodgates opened and people studied all kinds of curved spaces—spaces shaped like donuts or with twists in them. There are geometries where a right-handed person can travel through the universe and come back left-handed, so you can change your orientation just by traveling, which is very unintuitive, or come back smaller or larger than you started. People developed a very nice language for describing all these geometries, called Riemannian geometry after Bernhard Riemann—but it was pure curiosity, since the universe we lived in seemed completely flat.

Einstein and the curvature of spacetime

Then Einstein, trying to understand gravity, concluded that gravity bends space and time in a certain way. He needed a language to describe how space and time could bend so that light rays would become non-straight, sometimes converging or diverging. He asked his mathematician friend whether any existing mathematics could describe this, and the answer was: yes, there's this bright chap Bernhard Riemann who developed such a theory. It turned out to be almost exactly the right language for the Einstein equations. Riemannian geometry has a notion of curvature—positive or negative—and in this language the Einstein equations are extremely simple to state: mass and energy create curvature, and the curvature of space and time is proportional to how much mass and energy you have. That's basically the Einstein equations.

Solving them is a different matter—they're extremely hard to model. Even modeling two colliding black holes can barely be done with modern supercomputers. But stating the equations is extremely natural once you have this language.

Sphere packing

Another example of mathematical curiosity leading to practical developments centuries later is the story of sphere packing. A British sailor was curious about stacking round cannonballs in the hold of his ship: because the cannonballs are round, not square, stacking them wastes a certain amount of space. He wanted to know the most efficient way to pack them so the most cannonballs fit in a given space. He asked a physician friend, who happened to be Johannes Kepler. Kepler eventually proposed that the most efficient packing is the same one you see today in supermarkets when packing oranges—what's called hexagonal close packing, where you pack layer...

by layer. Each layer is a triangular grid of cannonballs or oranges, and you stack another triangular grid on top, shifted slightly, repeating in a regular pattern. This is the most natural arrangement and is about 76% efficient. Kepler believed this was the best possible—that there was no clever way to squeeze in more space—but he couldn't prove it. This became known as the Kepler conjecture, one of the most famous unsolved problems in geometry for centuries.

In two dimensions, the problem was solved by around 1900: packing discs in a plane is simpler, and the analogous triangular lattice was relatively easy to prove optimal. Three dimensions simply had too many possibilities. In fact, we still do not have a nice, simple proof of the Kepler conjecture that humans can fully understand on their own. The conjecture was eventually solved, published around 1998, but it required computers—it was one of the first computer-assisted proofs. A team of referees said they could not verify all of the computations, though they believed the strategy was correct, so lingering doubts remained. Only much more recently, in 2014, was the proof converted into a proof assistant language—a computer language specifically designed to check proofs with 100% certainty. The Kepler conjecture is now formally verified: we are now 100% certain it is true.

Mathematicians were not content with just the three-dimensional problem, so they asked what happens in four, five, or six dimensions. There are no four-dimensional oranges or cannonballs to pack, but people still ask the question. They also asked what happens in a discrete space rather than a continuous one. In particular, once computer science developed, we realized that in addition to the geometry of regular space—where coordinates are given by real numbers—we are interested in the geometry of strings of bits. This sounds very divorced from the original sphere packing problem, both because you now have thousands upon thousands of dimensions and because space is discrete rather than continuous. But it is still geometry, and many of the techniques for understanding sphere packing still work in this setting.

It turned out that packing spheres as efficiently as possible into this huge cube of bit strings is extremely practical. When cell phones became digital, every signal you send—an image, a text, whatever—is encoded as a bitstring sent over a wireless network. But other people are also sending signals, and you don't want yours corrupted by interference and mistaken for someone else's. So you want to keep each different signal as separated as possible from the others in this space of bitstrings. Mathematically, the problem of separating all these signals so that no one can be confused for another is almost exactly the sphere packing problem—except in high dimensions and discrete. A lot of the mathematics used to understand sphere packings could be used to design really efficient sphere packing codes, and not just to design codes, but also to tell engineers the theoretical limit of communication: the maximum number of bits per second you could possibly hope to send in a given wireless spectrum. That gave excellent benchmarks for measuring how efficient your protocol was. You could price how many billions of dollars a wireless spectrum was worth, because you knew exactly how much data you could push through it. The entire wireless telecommunication industry is based on being able to pack oranges in really high dimensions.

One of the applications of mathematics I was involved in that I'm most proud of is the story of compressed sensing. I was once at an interdisciplinary program at a math institute here in Los Angeles, where I met a friend of mine, a statistician, who was working with an electrical engineer to improve medical imaging—specifically MRI scans. At the time, MRI scans were quite slow: you had to sit in the scanning machine for about three minutes so enough data could be collected from all different angles for the scan to reconstruct a good image of your body and pick up tumors, cysts, or anything else medically important. If you sat in the machine for only a short period, say half a minute, you would not get enough data. If you tried the standard reconstruction algorithm of the time, least squares approximation, you would get an image so blurry and low-resolution that nothing useful could be read from it for diagnosis. So people had to sit in these machines for minutes on end—and children sometimes had to be sedated, because after about the second minute they would wriggle around and not follow instructions.

The engineer and statistician were trying a new technique, not least squares but total variation minimization. They had a hunch that this other method might perform a little better, so they tried it on some test data. They were expecting a slightly sharper image than least squares approximation, but they got perfect resolution—almost exactly the correct image—even though they had taken only a few

It would be like giving someone a crossword puzzle where you had only filled in 10% of the letters, and suddenly they could fill in all the other letters without having to look up the clues. They couldn't explain this, and when they showed it to me, my first instinct was: you made a mistake. You could not possibly have done what you said you did. In fact, I was going to prove to you that there was not enough information in the data you took to make this measurement possible.

So I went home that night and tried to write down a proof that there was no way to correctly guess the right image from the small amount of data they were measuring. And while writing it down, I found that one of my steps did not work. In fact, it showed the opposite: if a certain measurement matrix had a certain property, then what they were doing was actually going to work. I then checked that their measurement matrix did seem to obey this property. So I understood how the method worked, went back to them the next day and explained it. They got very excited, we wrote a couple of papers, and that got everyone else excited.

This method they had stumbled upon was not completely new. Seismologists had discovered a similar method: they had a different problem, trying to locate the fault lines of the Earth's crust based on a small amount of seismic data. Astronomers had a similar problem, trying to measure the location of stars using a very small amount of observed data. There were a couple of other disciplines where this problem of extracting a high-quality image from a very small amount of signal had occurred. In each case, they had found some ad hoc fix that could squeeze better resolution out of the data they had, but they could not explain mathematically why it worked. The seismologist thought, "Here's a trick, but it only works for seismographs," and the astronomers had a trick, but it only worked for astronomy.

But once we found the mathematical explanation, we found this was a general technique, which we now call compressed sensing. It's useful for MRI, but also for wireless broadband and certain types of sensor networks.

Once we figured out the underlying mathematics of compressed sensing, we could see all the other applications it was useful for. Now compressed sensing is taught in textbooks right next to least squares: sometimes least squares is the right tool, sometimes compressed sensing, sometimes neither. It has become a very well-developed theory, and I was pleased to be involved at the very beginning. It's a fascinating interplay between mathematics and science.

Why mathematics is so effective

We have this "unreasonable effectiveness of mathematics," where mathematical discoveries often turn out to be the best way to explain physical phenomena. Philosophers and historians have debated why. One theory of mine is that whenever we learn anything—math, science, or any other subject—the first explanations we make are often not the best. Before you understand something completely, you may have an overly elaborate explanation for why something is true, while the true explanation is more elegant, shorter, and simpler than your initial attempts. But finding the short explanation takes time, because you have to unlearn assumptions that turn out to be incorrect.

For example, with Einstein's theory of relativity, a key assumption people held before Einstein was that time was universal—everyone had the same notion of time, an hour for me is the same as an hour for you. That mindset really blocks you from finding the right way to explain gravity in particular. Once you accept that everyone has their own relative notion of time, you can find the right language to explain things properly.

Mathematicians likewise take phenomena they first understand in very inefficient language and try to condense them, finding the most concise, elegant explanation. Since there are only so many ways to say things concisely, it often happens that the concise way to describe a mathematical phenomenon is also a concise way to describe a physical one.

That is my theory. Unfortunately, we only have one timeline of science: with maybe a hundred turning points, that's only a little data to build theories on. I would love it if, in the far future, we meet other civilizations and see their histories of science—whether they had their own versions of Kepler, Einstein, and Newton, and whether they followed a similar track or a completely different one. I don't know.

How mathematics actually gets done

There's a stereotype that mathematicians are all geniuses who get stuck on a problem and then have a eureka moment, a light bulb going on with a genius idea out of nowhere. I would love for that to happen to me, but it doesn't happen often. What actually happens is I try something and it doesn't work. I try something else, and it kind of works but gets stuck at a certain point—at least now I know there's an obstacle, and I need to find a tool to deal with it. Maybe I identify a sub-problem with the same type of difficulty but simpler, solve that first, and then try to scale back up to the original problem, going back and forth.

Often much of the work is exploring the negative space of the problem—all the techniques that don't work. Eventually, with enough of the negative space mapped, the path forward becomes clear almost by elimination: there's only one thing left that could work. Sometimes nothing can work, and you give up on the problem. This repeated "failure" is really just understanding the limitations of different approaches. After weeks or months of this, the answer becomes clear—but by then the difficulties are so internalized that it doesn't feel amazing anymore. It feels natural: of course you had to go around this pothole, of course you must do this step first, because in five lines you'll need this hypothesis. You become so attuned to the problem that everything becomes natural. Or sometimes you never solve it, because you never attune and you give up.

The feeling I get is never so much eureka as "how did I miss this early? I was so stupid." It's the constant experimentation and failure that primes you to find and accept the right solution.

High standards for outcomes, messy process

There's a dramatic contrast between the standards we assign to outcomes and those we assign to process. For outcomes, mathematics famously has a very high standard of correctness: for a given problem there's a correct answer and lots of incorrect ones, and when we grade students' homework, a wrong sign in the final answer earns negative marks and criticism. One consequence is that many students who go through high-school-level mathematics become very averse to making any mistakes at all when approaching a math problem.

Paradoxically, the process of arriving at the answer is almost the complete opposite: you have to make mistakes over and over, and try the stupid things first to appreciate why the clever things work. There's a quote by the physicist Niels Bohr:

An expert is someone who has made all the mistakes that can be made in a very narrow field.

You don't publish these mistakes; you execute them in your process in order to locate the correct answer—but only after exploring a lot of incorrect answers first. It's very important.

I think we need to normalize failure in the process, and disclose that behind every successful solution to a problem there are dozens of incorrect attempts — not because the people trying these problems were stupid, but because this is often just part of the learning process.

Science and mathematics have changed a lot over the centuries. Traditionally, the two major paradigms in science were theory and experiment. You would create a theory — Kepler might create a theory of how planets move, or Newton a theory of gravity — and then there was experimental data: you would run an experiment and see whether the theory and the experiment fit. Math was a little different in that it was almost entirely theory; there are very few experiments done purely in mathematics. There were a few — Gauss famously computed the first 100,000 prime numbers, and that was a data set he used to make predictions, leading to what we now call the prime number theorem.

Later, simulation came along: instead of running a big, expensive experiment, you could simulate, say, a hurricane in a supercomputer rather than in real life. Then big data arrived — rather than doing a small number of experiments to confirm or deny a specific theory, you could take megabytes or petabytes of data and try to discern patterns, to extract laws from massive data sets. That is a more emerging type of science.

Now all these modes of science are being transformed because we also have AI to help us. In the past, every one of these ways of doing science had to be done by human scientists: someone had to perform the experiments, do the theoretical calculations, run the simulations, or go through the data. You could use computers for some of that, but even then you had to program the data analysis tools, and you still needed a lot of expertise. Today we have automated labs that can perform experiments automatically, coding agents that can run simulations for you, and automated data analysis. Increasingly, you can also do automated theory: you can take a mathematical problem and ask what the consequences of these hypotheses and axioms should be. These tools can now operate at scale and much faster — potentially running far more theoretical analyses than any one human scientist could.

On the other hand, this is not the only thing we want. There is value in doing things the slow way. A scientist who spends hours working things out on pen and paper, doing experiments in the field with their bare hands, and actually debugging the simulations often learns a lot of extra insight beyond just getting the answer they were seeking. They can discover new phenomena, see connections, notice similarities to something previously studied elsewhere in the literature, and communicate what they are finding to other people.

So there is a paradox: AI is becoming more powerful and capable, making fewer mistakes, and ostensibly achieving many of the goals we think scientists are trying to accomplish — running experiments, analyzing data, writing papers. But it may come at a cost: the AI picks up some skill, yet no human scientist gets any better at doing the science. No human can communicate exactly what just happened, why this scientific discovery is interesting, why this proof is new, what features it has, and how it connects. We may have to redesign our conception of what science is and what we actually want out of it. What exactly is science for? What are we trying to do? Is there a danger that we are optimizing the wrong thing when we point our AI tools at science?

One analogy I have given in the past is that science is a little like going on a hike. You have heard there is some interesting, beautiful waterfall out there, so you decide to hike with some friends to find it, and you need to make a map. You get lost a little bit — but maybe while getting lost, you discover something else interesting and make a note of it. On the way to the waterfall, you spot an even more spectacular one in the distance. You cannot get there yet, but maybe some future hiker will figure out a way. There is a whole process involved in reaching your goal, and that process is also very valuable.

But these AI tools can be like helicopters that fly you directly to the waterfall: you see it, fly back, and learn nothing about how to get there. You may not see any other interesting phenomena beyond the specific thing you asked for. Even though you technically achieve your goal much more efficiently, something may be lost.

Modern AIs are powered by a type of algorithm known as machine learning, which tries to predict patterns in data. A very simple example is regression. Say you observe that if you feed an animal more food, it gets bigger. You plot how much food you give various animals against their weight, get dots on a graph, and if you're lucky they fit a line—and that line becomes your prediction: feed a dog this much food, it gains this much weight. In the real world you don't always get nice linear relationships; often there are many inputs and outputs and the relationship can be really complicated. But sometimes the data has a shape, and we now have all kinds of clever ways to detect that shape and fit curves to these input-output pairs.

Large language models, which power chatbots and similar tools, are just playing the game of naming the next word in a sentence. Roughly speaking: if I say "roses are red, violets are blank," what word fills the blank? You can probably guess the answer is blue. Imagine a giant plot where the inputs are all these incomplete sentences and the outputs are the completing words—dots in a high-dimensional space—and you fit a curve that predicts the most likely next word. Sometimes there's more than one answer: "Hello, my name is..." could be followed by many names, so you don't always get a single answer, but you can try for the most plausible one. People have tried this; the autocomplete feature on your phone does exactly it—sometimes it's right, sometimes it's silly. Once you have an operation like this, it creates dynamics: many people have pressed autocomplete over and over and gotten gibberish sentences—monkeys typing on typewriters.

The magic of LLMs is that if you train them on enough data—trillions and trillions of data points, fitting as good a curve as possible, which takes millions of dollars of computing power and months of time—then suddenly, even when you iterate, it stays coherent. It begins to sound not like monkeys but like a human speaking. We don't fully understand why. What seems to be true is that natural languages like English contain a lot of hidden patterns we're not consciously aware of. We know some laws of English, like grammar, but there are unwritten rules of language that humans pick up: a child, never taught what a noun or verb is, learns what order English words go in just through continual exposure. It seems you can teach these models to pick up those patterns too, to the point where you can give them math questions—"the answer to 2 plus 3 is"—and they will say five. They've been trained to get correct answers to at least simple math.

Once a model has a little ability to speak English, it can go in loops, check its work, and make fewer mistakes. You can prompt it to proceed step by step and not say something unless it's been double-checked, so it becomes a little smarter, quote unquote—able to solve many complicated tasks. But it's still just guessing the next word to say. It isn't grounded in any deep understanding of the real world; it has simply absorbed the patterns of English or another language so well that it can mimic people speaking intelligently—presenting as intelligent long enough to fool us, but also long enough to actually do useful things. We can now solve certain math problems by asking an LLM to provide a proof; sometimes the proof is complete rubbish, but if you loop it enough with enough checks, you can start getting a positive success rate.

It's a very strange way of solving problems, completely orthogonal to how we normally think of intelligence as grounded, methodical, first-principles thinking. It's like having someone who knows a lot but is slightly drunk and throwing out ideas — with enough guidance, you can actually extract useful output. It's not the most advanced mathematics out there, but you give it a lot of data, a lot of time, and a lot of band-aids, and it works pretty well.

Some of the debate about AI's role in science defaults to a one-dimensional view: easy tasks, hard tasks, very hard tasks, and humans can do tasks up to a certain level while AIs can do tasks up to a certain level — so which is better? But working with AIs and comparing their problem-solving to humans', I've found they are really quite complementary.

Depth versus breadth

Human experts focus on depth. A human mathematician could work on thousands of problems, but they pick one or two that are difficult but not impossible — difficult enough that trying to make progress reveals all kinds of insights they can share, which students or collaborators can build upon.

When we point AIs at really difficult problems where none of the standard techniques apply, they are still very bad — they're just randomly guessing. But they excel at breadth. Point them at a thousand problems of various difficulties: some may be too hard, but some are within reach of existing methods — there's some method in the literature that will solve the problem, or maybe two separate methods need combining. There just aren't enough human experts to look at all these problems, and the experts who do may not realize that an obscure paper from a journal in 1970 holds the key idea. They don't have the patience or time to go through all the combinations of which technique might work on which problem.

AIs, by contrast, take educated guesses at what techniques might work. Some guesses are stupid, but some work, and through all these combinations we're finding they can sometimes catch a solution all the humans have missed. Occasionally the conventional wisdom of experts is wrong — everyone thinks a problem has a positive answer, but it actually has a negative one, and nobody looked closely at the negative case because everyone assumed the answer was true. An AI may not have that preconception, so sometimes it just serves as an independent pair of eyes. Some problems we thought were very difficult turned out to have surprisingly simple solutions which, in retrospect, we should have found ourselves.

Raw numbers of solved problems

AIs are beginning to succeed when pointed at a very broad range of problems, solving some percentage of them. Point them at a thousand problems and they might solve 5% — that's still 50 problems solved. You can already have tools that, by raw number of problems solved, outperform human mathematicians. The 50 problems solved may not be the 50 you most want solved — they could be 50 random ones — but it's still very impressive.

As a profession, we will have to find ways to incorporate this new capability—solving some problems at broad scales—and figure out how to make it mesh with our existing capability to solve a few deep problems very slowly.

Kepler: theory versus data

Kepler's story of how he found his famous laws of motion shows how important the process is. Kepler learned of Copernicus' theory of the motion of the planets, in which Copernicus had roughly worked out how far the Earth was from the Sun, how far Mars was, and so forth. Kepler noticed that the ratios of these orbits looked a little like certain ratios in geometry, and eventually proposed that if you take one sphere for every planet—six planets were known at the time—you could inscribe the five Platonic solids, like a dodecahedron, a cube, and a tetrahedron, between these six spheres. He thought it would be a perfect fit, and this explained the shape of the solar system in terms of the five Platonic solids. It was a beautiful geometric idea.

It was only after he got his hands on some really high-quality observational data of Tycho Brahe—which he had to fight for, and possibly even steal—and tried to fit his theory to it, that he found it didn't quite fit. With the precision Tycho's data offered, he could not get these spheres to fit, and in fact he discovered from that process that the orbits of Mars and Earth could not be circles at all; there had to be some other shape. He spent many years figuring out what to do—I don't know how long he held on to the Platonic solids theory—and you can see in his writings that he tried many other things. He tried making the circles off-center, and at some point he landed on the ellipse, and then suddenly everything fit.

This shows there is an interplay between theory and experiment: you can pose a theory, but if it doesn't fit the data, it may not be a good theory. But it's more complicated than that. Before Kepler, one of the criticisms of Copernicus's theory was that Copernicus himself acknowledged his measurements were worse than the best predictions available at the time. The best models were the geocentric ones, developed by the Greeks and then by the Arabs and Indians, with many adjustments and fine-tuning—a very precise model that could predict, in a very complicated way, where all the planets would be. Copernicus's model was worse. Just agreement with data is not necessarily the only metric. Only after Kepler found his revised model, with orbits that were ellipses rather than circles, did the heliocentric model become more accurate than the geocentric one.

Science doesn't give instant feedback

What this tells you is that you can't always get instant feedback on whether you've solved a scientific problem. If Kepler and Copernicus had had AIs and asked them to predict a model for the universe, it could be that the AIs that generated the correct heliocentric model would be discarded, because initially their predictions were not as good as the geocentric ones. It takes time to really digest these theories and see how they fit with everything else we know about planets, motion, gravity, and everything else.

One concern is that AIs are too fast. There's a danger they will do what's called overfitting—creating a very complicated model that has nothing to do with what's actually going on but fits your data extremely well, and doesn't extrapolate beyond that data set.

Accelerating components isn't accelerating science

How we incorporate AI into the scientific discovery process will be a challenge. It can certainly accelerate individual steps: you can make experimentation faster, write code faster, write your papers faster. But science as a whole may not necessarily accelerate just because every single component gets faster. There's a danger that we will optimize the wrong thing when we point AI at science—we will get all these amazing successes on paper, but find out that science is not actually advancing the way it used to. Still, it's better to have these tools than not have them; we're just learning how to use them most efficiently.

The life cycle of a proof

Part of what we do is solve problems and prove things, and proofs go through a certain life cycle. First you have to generate a proof or solution, which used to be quite hard—and some of the proofs you generate are incorrect, so then you have to verify them, which also used to be quite tedious. Both of these tasks are becoming more and more automated, so we're beginning to see more and more proposed solutions to various problems, many of which are actually correct.

But proofs are also getting longer, and when they're written by AIs they are often not very pleasant to read. An AI-generated proof might spend a lot of time on something very trivial and very little on the most interesting portion of the paper—probably because the AI can't distinguish what is hard, since by brute force everything takes the same amount of time for it. A human who has naturally had to struggle at the most difficult step of a paper would naturally spend a lot of time on that step. So you need to write a proof in a way that reads well and can be explained to other people.

Then other people have to get excited by it and accept it—that it's really interesting, that it will help them solve their own problems, or that it really clarifies why a phenomenon is true. This is where we traditionally have the peer review process: we send papers to referees, and if the referees are excited by the result, the paper gets accepted.

A paper can be technically correct, readable, and fine, yet answer a question that no one cares about. Finally, a result needs to be completely polished, put into textbooks, and taught to students. Often the first version of a proof is not suitable for a textbook: it may be done inefficiently, and the ordering of the steps may not be logical. There is a digestion process in which someone spends a lot of time thinking hard about the completely right way to organize and edit the paper so that it flows well — a bit like editing a documentary or a movie.

What we're finding is that AI tools are accelerating the early stages of this process, but not the late stages. We are now generating many proofs and verifying a bunch of them, but understanding them and putting them into final textbook form is still done by humans. In fact, we're now experiencing what you might call proof indigestion: suddenly there are lots and lots of pending solutions to problems that should be understood and should go into textbooks, but we're flooded with too many of them. We have to pick and triage, which has never had to happen before. Solutions used to come out so rarely that when a major problem was solved, all the experts would drop everything and read it, trying to digest it as quickly as possible, because it was so rare and valuable that it was worth doing. Now we're just flooded with possible solutions.

I myself have had to stop trying to stay current with all the latest developments in my field. Sometimes there's just so much going on that I can no longer promise to read every single one.

This was already becoming a problem before AI, but AI has really accelerated the sheer volume of content being generated, so we're going to need much better curation and filtering. It's a good problem to have — it's better to have more food than you can eat than not enough — but it is still a problem.

AI has become increasingly capable in mathematics. For those of us following developments over the last few years, there's been a steady progression: four years ago they could solve middle school math problems, then high school problems, then high school Olympiad-level problems, then graduate-student qualifying exam problems, and then they started solving some minor unsolved problems — the kind someone like Paul Erdős might have proposed but no one really looked at. That's a lot of low-hanging fruit. Then, just recently, there have been one or two occasions where they solved problems people had genuinely tried very hard to solve. Somehow, collectively, humans had taken the wrong turn, and the AIs — with a different set of biases — managed to cobble together a solution that was quite clever and has already had some impact. For instance, nearby variants of the unit distance problem have since been solved by humans who adapted the AI's technique. I found that quite exciting.

For some of my colleagues, though, it was very concerning, especially if they hadn't been following the earlier developments. If a colleague had only seen what ChatGPT could do in 2023 and asked it a difficult math question, it would give you complete rubbish — and they are quite different now. It's still fundamentally the same technology, but they have found ways to reduce the error rate and become genuinely useful.

It's still unclear how replicable this is. Many of these achievements are done by private companies that don't disclose how much compute they used — was it $100,000? A million dollars? We don't know. We also don't know their success rate: was the problem they solved the only one they looked at, or did they try ten, or a hundred? The results are impressive, but we don't have enough data to gauge whether this will become a regular occurrence, or whether you only get results like this by spending $100,000 over several months with a team of ten people. Maybe only 1% of the problems we care about are amenable to this method. We don't know, but there are efforts to benchmark this more scientifically. The most recent is the FrontierMath challenge: they tested the latest models against a set of ten research-level questions, and the best models could solve about five or six out of ten — medium-difficulty problems that already had a solution, kept secret.

I think a lot of the routine tasks we do every day in research can now, in some percentage of cases, be done by AIs. It can be expensive, though: many of these tools require a couple hundred dollars of compute before they produce a solution, and sometimes they fail — they spend all that compute and end up with nothing useful.

In programming, many expert programmers report that their ability to write code has increased by a factor of five, ten, or a hundred with these tools. But they can also feel themselves losing the ability to code by hand, and sometimes they cannot review the code that comes out of these agents. There's a trade-off — speed is not everything.

I very much like the collaborative aspect of mathematics. I didn't realize how important it was until relatively late in my career: a lot of the mathematics I learned after grad school came from working with mathematicians and scientists in different fields. I teach them what I know, they teach me what they know, and I become much broader as a result. When you've worked with a collaborator for a long time, you become almost mentally attuned — like a close friend or family member you've talked to for years, you can sometimes complete each other's sentences. You can throw out an idea and, before you even finish the sentence, the other person gets it and runs with it. People have tried to use AIs this way, but you can't really converse with them. They make mistakes, they can be sycophantic and only tell you what you want to hear, and the interactions right now are kind of impersonal. I've tried collaborating with people in person while also having an AI present, but it breaks up the rhythm.

These tools aren't really conversational yet—not as fluid as working with human collaborators, though maybe they will get there. Until recently, they don't learn from your conversations. With a collaborator, you can resume the next day and pick up very quickly, and sometimes even after years without a call, you can pick up some very old threads. AIs have a certain amount of context, can remember some things, and can make notes to simulate this memory, but you can't attune to an AI the same way you can to a really close collaboration.

I actually don't use these tools so much for the actual problem-solving process. To date, I've found them much better at secondary tasks like literature searches, checking a proof, writing some code, or proofreading something I wrote to see if there's an opportunity to make things a little bit tighter. The rhythm of working with an AI is not quite the rhythm I prefer with a human collaborator, but that could just be the current state of technology. Maybe future AIs will be much more conversational and more human to interact with.

We're at a somewhat risky point in the structure of funding the scientific enterprise. These tools are allowing us to create the outputs of science—or what seems to be the outputs of science—at a much accelerated rate, but it could come at the cost of nurturing our seed corn for the next generation of scientists. There's a real concern that the training problems we give graduate students as their first projects, to gain recognition, career training, and experience, are exactly the types of problems AIs can now replicate, producing many grad-student-level papers. But if we replace grad students with these AIs, we won't get the next generation of students. And if we don't continue digesting all this AI output to build the next base of knowledge for the next generation of humans and AIs to build upon, we may end up stagnating as a scientific society—able to optimize everything we can do with current technology, but not actually developing really original new ideas anymore.

We need a much more open discussion about what basic science is, what it's useful for, and why it's important to still have curiosity-driven research—why we still need a community of humans to explore things, sometimes slowly, sometimes in ways that aren't as efficient as the latest model AIs. I also think we should share the insights we gain more and do more outreach to the general public. Today, people can see the visible outputs of science—they have a cell phone, the internet, GPS—but many don't see the whole process, or how a basic understanding of math and science makes the world around them a lot less scary and a lot clearer. A lot of people now live in a state of anxiety because the world is so complicated. We haven't emphasized these softer values of science as much as the hard technological outputs, but science does add a certain amount of clarity to one's thinking. These are valuable things, and they need to be supported.