Transcription
Have you ever thought about pi? And I'm not talking about the delicious bakery item. I'm actually talking about the ratio of a circle's circumference to its diameter. It's famous for the fact that its decimal actually never comes to an end.
But it got me thinking, what if it did stop? What if tomorrow some computer or some mathematician found the last digits of pi? What if after millions and trillions of numbers, it just comes to an end or it ends in a string of zeros? Would math fundamentally break? Would the stars fall out of the sky? Would it prove that we're living in a simulation?
Now, honestly, nothing physical would happen. Your GPS would still work and bridges would still stay up. Your mortgage payment or anything that you owe would still be there, but the entire foundation of modern mathematics would actually collapse. But that's actually the interesting part. So, let's talk about that.
Has it actually been proven that it doesn't come to an end? And how do we actually know for sure? And does any of this stuff actually matter to you or affect you in your daily life? Or is it kind of just a cool number kind of like the Fibonacci sequence that we put on t-shirts? So, let's get into that.
Now, before we actually break pi, we kind of need to understand what it actually is because most of us learned it in middle school and memorized that 3.1459 depending on how far you got. We used it on a test and then we honestly never thought about it again. But pi is just a ratio. You take any circle, any circle in the entire universe, and you measure the distance around the circle, and then measure the distance straight across the middle and divide that number by the second, and you get pi, 3.14, every single time. And here's the kicker. It doesn't matter if the circle is the size of a coin or the circle is the size of an entire orbit of a planet, the ratio is always the same number. 3.14159 and so on and so on.
And here's the thing that's easy to actually miss. Pi is not big. People hear infinite digits and picture this enormous cosmic number that's kind of stretching across the universe. But pi is actually smaller than four. It lives permanently between the lines of three and four. It really is a precise address on a number line. It's trapped in this tiny neighborhood. And what's infinite about pi isn't its actual size. It's its precision. It's the number of decimal places you need to write it down exactly. That's the part that never ends. The decimal places go on and on.
Think of it like a zoom lens. You're looking at the space between 3.14 and 3.15. Now, if you zoom in, there's more. And if you zoom in again, there's even more. And every time you zoom in, pi just keeps going. It keeps going and becoming more specific. It never leaves that tiny sliver of the number line. And it's infinitely deep, but not infinitely wide.
But here's where it gets weird. Because pi doesn't just describe circles. It actually shows up everywhere. It's in the equations for things like how gravity works. It's in the math behind electrical signals, sound waves, and GPS satellites. And the bell curve, you know that one? The curve that staticians use to describe everything from the test scores to stock prices. Pi is woven into every operating system of reality in ways that go beyond the measure of a circle. Which is exactly why the question, what if it ended, isn't actually a silly question at all. It's kind of a terrifying one.
All right, so let's run this thought experiment. Tomorrow a computer spits out a digit, then another, then a zero, and then nothing. Pi just stops. What happens? The first thing that actually breaks is the concept of a perfect circle. If pi ends, it means it can be written as a clean fraction. And that means the ratio between a circle circumference and its diameter is actually exact and finite. And that implies something just a little bit unsettling. A perfect smooth continuous curve doesn't actually exist. If you zoomed in far enough on any circle, you'd actually eventually see the edges. You would see flat segments, tiny straight lines stitched together like a polygon, but with billions of sides. The circle would be really convincingly fake.
Now, think of it like the screen you're watching this on. Now, from a distance, the image looks smooth. But if you zoom in enough, and you'll actually see the individual pixels, a square grid making up the picture itself. If pi ended, it would mean the universe works the same way. That space itself would have a maximum resolution and geometry would be pixelated.
The second thing that would actually break is calculus. And I mean all of it. Calculus is built on the idea that you can divide things infinitely, that curves are smooth, that you can always zoom in one more level. Pi's infinite precision is not a quirky side effect. It's actually loadbearing. It holds up the math that we use to describe things like gravity and light, orbits, and even quantum mechanics. If you pull that out of the equations that describe the physical universe, and then it would contain a fundamental contradiction.
Now, here's the part where I have to be honest with you. After all that drama, your daily life actually wouldn't change a whole lot or really even at all because NASA actually uses just 15 digits of pi to navigate spacecraft across the solar system. If you use just 39 digits, you can measure the circumference of the entire observable universe. And you could do that with an air smaller than a single hydrogen atom. I mean, engineers already round pi off. And the practical world runs on approximations and it works just fine. So the collapse would be theoretical, not structural. Bridges stay up, planes keep flying, and the chalkboards, the equations, the foundations of logic. Well, that's kind of the stuff that would be in ruins. Go figure.
Which raises kind of a fair question. Is this something that we just assume? Or has someone actually proven that pi never ends? So, how do we know for sure? Because here's something that actually surprised me when I started researching this. We didn't discover pi goes on forever by calculating a bunch of digits and never finding an end. That's actually not how this whole thing works. You could calculate digits until the heat death of the actual universe, and that wouldn't prove anything because you can't honestly calculate your way to infinity. The crazy part is we actually proved it with logic. on paper in 1761.
So in 1761, a Swiss mathematician named Johan Heinrich Lambert sat down with a quill and paper and through a series of logical steps, he proved that pi cannot be written as a fraction. And I mean not a complicated fraction, not a long fraction, no fraction period. And if a number can't be written as a fraction, its decimal never actually ends. It just continues to repeat on forever. And that's what mathematics means when they say the number is actually irrational.
But the version of this proof that really is important here came actually in 1947 from a mathematician named Ivan Naven. Naven published a paper called a simple proof that pi is irrational. Makes sense. The entire thing fits on a single piece of paper on a single page. One page. And I want to talk to you about the ideas related to that. Not the math in general, just the ideas because it's actually really elegant.
So here we go. It's a technique called proof by contradiction. So you start by assuming the opposite of what you want to prove. So let's say you assume pi does end. Let's say it's a clean fraction, something over something. Then Naveen feeds that assumption into a basic calculus equation and the equation spits out a result that is mathematically impossible. It basically claims that there is a number an integer between 0 and one. Now there is no whole number last time I checked between 0 and 1. I mean that's almost like saying there's a whole floor between the ground floor and the first floor. It just doesn't exist. And since the assumption produced an impossibility, the assumption itself must be wrong. Pi can't actually be a fraction. It's irrational. It doesn't end.
And this isn't some like ancient claim we're talking about here. We're talking about it on faith. Calculus students all over the world can actually reproduce this proof as a classroom exercise. And it's been checked by millions and millions of people. And more recently, computer scientists have actually run it through automated theorem proving software. programs that actually check every illogical step against the fundamental rules of math without any human bias. The machines verify it. So, the proof is short. It's transparent. It's widely tested and machineverified. That's honestly about as solid as I think you can get when it comes to this.
But the thing is, does that mean we should stop questioning it because there's a pretty uncomfortable historical precedence for what happens when we actually stop checking and testing things in science? Let me explain. It's called the fourcolor theorem. And here's the claim. The claim is take any flat map, any arrangement of regions, and you will never need more than four colors to fill it in so that no two neighboring regions share the same color. So even though the fourcolor problem actually sounds easy, the problem is proving that no map out of every possible map that could ever be imagined, including more specifically designed by mathematicians trying to force a fifth color would ever need more than four. And that right there fundamentally changes the challenge. It is resisted proof for over 120 years. And in 1879, a British lawyer and part-time mathematician named Alfred Kemp published what he claimed was the proof. It was clever, his proof. It was published in all the top journals at the time, leading mathematicians of the era. They reviewed it and everyone accepted it. And the problem at that time was declared solved. Everyone checked the boxes and moved on. And for 11 years, nobody went back and checked, at least not carefully, not the hard parts. They just assume the credentialed folks actually got it right. But then in 1890, a mathematician named Percy Hewood actually sat down, worked through the logic step by step, and found a specific type of the mapper arrangements where Kemp's method actually broke. The proof had a hole in it, and it had been broken the entire time. The theorem itself actually proved to be true, like you only need four colors, but it took until 1976 and the help of a supercomput to actually prove it correctly. The answer was right for almost a century. Just the proof was wrong.
Now, if you're thinking, why does this matter for pi? It's because it shows us something uncomfortable about how science and math actually work. We are biased towards accepting things that once credible authority says the problem is solved. And that bias just doesn't live in Victorian mathematics. It actually shows up again and again. And I have proof because just last year in a physics lab with the help of AI, they proved something that everybody thought was settled for a very long time. So in mid 2025, there was a team at Emory University and they published a study in PNAS. It's one of the most prestigious scientific journals in the world and they had been studying something called dusty plasma, the fourth state of matter, an ionized gas filled with tiny charged particles. And this is the stuff that actually exists in Saturn's rings and comet tails, even wildfire smoke. So for decades, physicists had been using a set of simplified equations to describe how the particles in dusty plasma actually interact. Their approximations were considered good enough and they were in textbooks and nobody actually questioned them. The Emory University team built a neural network, but they did something a little bit clever. They didn't just throw data at a black box. They designed the AI with the fundamental laws of physics hardwired into the actual structure. So, think of things like gravity, drag, basic symmetry. They fed all of that into real tracking data of particles moving in this plasma chamber. and they said, "Figure out the forces. Don't use the textbook equations. Just look at the data. Just observe." And then the AI came back with force descriptions that were over 99% accurate. And when the researchers compared those results to the standard textbook approximations, well, the textbooks were wrong. The way particle charge scales with size was off. The relationship between force and distance was off. assumptions that had been in the literature for decades were all inaccurate. And the AI didn't do anything crazy like break the laws of physics. I mean, Newton's laws still hold. What it broke were the human shortcuts, the approximations that physicists had been copying and pasting from one paper to the next because nobody went back and actually checked.
Now, tell me that doesn't sound familiar. So, it's the same pattern. Kemp's proof in 1879 and the dusty plasma experiment in 2025. They have credible sources. They have accepted answers and everyone just moved on. Nobody rechecks until something forces them to do it. So when we say pi, the proof has been verified by students and machines. Stuff like that matters. That's the rechecking that didn't happen in those other cases.
But there's also a group of mathematicians who have an even deeper objection. They don't think the proof has an error. They think the entire method is the actual problem. Okay, this is kind of one of the parts of the research that genuinely uh surprised me and I think it's the part that most people have never actually even heard. There is a real ongoing philosophical war, mind you, philosophical war in mathematics and it's been going on for over a hundred years. And on one side you have classical mathematicians. They use a rule from Aristotle that basically states every statement is either true or false. There is no middle ground and because of that they are allowed to use proof by contradiction. The same technique Naveen used. If assuming something is false leads to nonsense then it must be true. But on the other side of that there's a group called the constructivists. And in the early 1900s, a Dutch mathematician named Brower looked at classical math and said, "Hold on, you can't just prove something exists by showing it's impossible for it not to exist. You have to actually build it. You have to show me the thing. Hand me the blueprint. If you can't construct it step by step, then you actually haven't proven anything."
And here's an example of what that difference looks like. Because the question is, can you raise one irrational number to the power of another and get a rational number? Classical math answers this with kind of a slick trick using the square root of two. Without going through the whole thing, the argument actually boils down to either this case works or that case works. But either way, the answer is yes. Done. Now constructivist looks at that and says which case is it? You don't know. You prove the answer. exists without actually finding the answer. That's like proving there's money in a vault because it's logically impossible for that vault to be empty. But you never actually open the door to check.
So what does this mean for pi? Does the proof actually survive? And here's the twist. It does because both sides actually agree on pi because the word rational is a negative claim. It just means not rational, not a fraction. When Naveen's proof assumes pi is a fraction and the math breaks, he's not actually claiming to have found some infinite object. He's proving the vault is empty. Pi is not in the set of fractions. And constructivists accept that. Proving an absence through contradiction is fair game, even by their strict rules. Now, where they disagree on is what pi actually is. A classical mathematician says pi is a complicated infinite object that exists somewhere in the abstract or eternal realm. Now a constructivist says that's a fantasy to them. Pi is not a number. It's an algorithm. It's a recipe. It's a set of instructions that generates the next digit whenever you ask for it. The code is the construction and the code is the only pie that's actually real.
Which leads me to my next question that I have not been able to stop thinking about since I started this research. Now, we've spent a lot of time talking in the abstract. So, let's bring this down to earth just a little bit because does pi actually matter to you? Like, does it matter to you in your everyday life? And the short answer is kind of, but you'll never actually notice it because pi is baked into the math behind so many technologies that you interact with it constantly without even realizing it. And every time your phone calculates a GPS route, pi is actually in the equation. And every time an MRI machine generates an image of someone's brain, pi is in the signal processing, the audio compression, image rendering, wireless communications, structural engineering. Pi is in the plumbing of all of them.
But here's the most important part. None of those applications uses Pi's infinite precision. They use a handful of decimal places. Your phone's GPS is not running 300 trillion digits of pi. It's running maybe 15. And that's actually more than enough because I mentioned earlier NASA navigates spacecraft across billions of miles of space just using 15 digits. And also mentioned if you use 39 digits, you could measure the circumference of the entire observable universe with an margin of error smaller than a hydrogen atom. So in a sense, pi's infinite digits are completely unnecessary for any type of practicality. The universe just doesn't need them. Engineering doesn't need them and your phone doesn't need them.
But in another sense, the fact that pi is infinite is actually what makes the math underneath all of that and those technologies actually work. Calculus, the math that makes GPS and MRIs and signal processing possible, is built on the assumption that you can always zoom in one more level. The precision itself is limitless. That's the loadbearing wall. But if pi ended, the approximations would still work for a while. The same way a house with a cracked foundation still stands for a while, but the structure underneath it would actually be compromised. So pi affects your daily life the way the foundation of your house affects your daily. You never see it. You never think about it, but everything depends on it being solid.
Which brings us to the actual biggest version of this question. Does math die with us? This is a philosophical one because everything in the universe, everything in the physical universe, it ends. Stars burn out, galaxies scatter, and according to thermodynamics, the universe itself will eventually hit a heat death. A state where all energy has spread out evenly. Nothing moves and time effectively stops. So if everything dies, does pi does that die? If pi is a real thing, an eternal truth that exists independently of us, then technically no. The ratio of a circle's circumference to its diameter was pi before humans even existed and will still be pi long after we're gone. The logic doesn't need oxygen or atoms to be true. That's the position of what philosophers call mathematical platonism. Math is discovered, not invented. It lives in some abstract realm that doesn't depend on physical reality.
But remember constructivist, if pi is not a number but a recipe, and recipes only exist inside brains and computers, then when the last brain shuts off and the last machine loses power, the recipe itself is gone forever. Pi actually dies with us. Math was a game, a spectacularly useful game, but a game that let us split the atom and land on the moon. And a game that has rules that only existed because someone was around to actually play. Then here's the part that gets me. Neither side can prove the other wrong. The Platonist can't show you where the abstract realm of math physically exists. You can't point a telescope at it. The constructivist can't explain what philosophers call the access problem. How does a biological brain, a lump of carbon and water and electrical signals, reach into some supposedly non-physical realm and pull out the truth? Nobody has a good answer for that. And I'll be honest, I myself don't know which side to come down on. But I find something comforting in the uncertainty because it means the question is still open. We're not done. The deepest questions about the most basic number in mathematics, a number we teach 12-year-olds, is still really unresolved. And that's kind of amazing.
Now, before we close, there's one more piece of this story that I think will bring all this together. Because while mathematicians are debating whether pi is eternal, there's a team of engineers that use pi to do something that has actually nothing to do with math at all. In late 2025, a tech lab called storage review calculated pi to 314 trillion digits. And they did this on a single server. One machine about the size of a pizza box running non-stop for 110 days. There were no crashes. There were no errors. There were no restarts. And here's the thing, because they didn't actually do this to find the end of pi. They already knew it doesn't end. They didn't do it for the math. 15 digits handles the solar system. 39 handles the observable universe. They did it to break the machine. And believe it or not, the machine didn't break. Calculating pi at that scale turns the algorithm into the most brutal stress test you can run on hardware. The system has to move massive amounts of data between the processor and the storage drives continuously for months. If one drive fails, if one bit flips, if the cooling can't hold, the entire calculation itself is worthless and you have to start over. A server that survives that is a server you can actually trust with things that actually matter. Think of things like AI training runs that take months of genome sequencing or climate simulations. The pi calculation was never about the number. It was about proving the machine could take the worst case scenario and not flinch. And that I think is the whole story of pi in one image. A number that could never be finished being chased by machines that are getting better specifically because they tried.
Now we will never find the end of pi. And we've known that since 1761. But the question it forces us to ask is what can we trust? What counts as proof? And whether the math is discovered or invented. Those questions don't have an end either. And honestly, I think that's the whole point of it all. In any sense, thank you for watching another episode of the archive. Happy Pi Day.