Show Me The Machinery: Feynman Responds to Ramanujan

This Goddess Has Good Taste

Look, I don’t know about goddesses. I’ve spent my life looking at the physical world—at atoms, at light, at the way things jiggle and bounce—and I’ve never seen a goddess in the equations. Usually, when someone tells me they got a formula from a dream or a divine whisper, I reach for my wallet to make sure it’s still there. It sounds like voodoo. It sounds like the kind of thing people say when they don’t want to do the hard work of explaining themselves.

But here’s the thing about this guy Ramanujan: his goddess has excellent taste.

I read his piece on prime numbers, “Divine Patterns,” and I have to admit, I’m sitting here scratching my head. Not because it’s wrong—that’s the annoying part! It’s right. It’s spectacularly, beautifully right. He talks about the primes thinning out, about the “living landscape” of numbers, and he lands on the Prime Number Theorem: $\pi(x) \sim x/\ln(x)$. He says the goddess revealed it.

Now, I’m a guy who likes to take radios apart. When I was a kid, I fixed radios by thinking about them, by visualizing the current flowing through the wires. I need to see the machinery. I need to know how it works. If you tell me “the radio works because the spirits of the airwaves are singing,” I’m going to say, “Okay, but why do the spirits need a vacuum tube to sing?”

Ramanujan looks at the primes and sees a painting. He sees the “unruly children” and the “divine hand.” And that’s fine, it’s poetic. But when he says the density of primes follows the natural logarithm, my ears perk up. Because the logarithm isn’t some mystical symbol. It’s a tool. It comes from somewhere. It has a job to do.

So my question isn’t “does the goddess exist?” My question is: “Why is the goddess whispering about calculus?”

If you’re going to tell me that the distribution of prime numbers—these jagged, discrete, stubborn little integers—is governed by a smooth, continuous function like the logarithm, I want to see the gears. I want to lift the hood. Because nature doesn’t just do things for “style.” There’s always a mechanism. There’s always a reason why the pieces fit together the way they do.

The Logarithm Mystery: Let Me Show You Why

Ramanujan asks, “Why should the logarithm… govern the discrete, jagged world of integers?” He calls it a miracle.

I say: let me show you why she’s whispering logarithms. It’s not a miracle. It’s probability.

Imagine you’re trying to filter gold out of a stream of mud. You have a series of sieves. The first sieve catches everything divisible by 2. That’s half the mud gone. The next sieve catches everything divisible by 3. The next catches 5. And so on.

To be a prime number is to be a survivor. You have to survive the sieve of 2, the sieve of 3, the sieve of 5… all the way up to the square root of yourself.

Let’s look at the probability. What’s the chance a random number $n$ is not divisible by a prime $p$? It’s $(1 - 1/p)$. Simple, right? If $p$ is 3, there’s a 2/3 chance you’re not divisible by 3.

Now, if these events were perfectly independent—like flipping coins—the probability of surviving all the primes up to some point would be the product of all those little probabilities. You multiply $(1 - 1/2) \times (1 - 1/3) \times (1 - 1/5) \times \dots$

When you do that math—when you actually sit down and crunch those numbers—do you know what happens? That product, that accumulation of survival probabilities, starts to look exactly like $1/\ln(n)$.

This is the machinery! The logarithm appears not because it’s magic, but because of the way probabilities multiply. It’s the result of a billion little coin flips. It’s the “law of mass action” applied to arithmetic.

Ramanujan sees the pattern: “The density drops.” I see the mechanism: “The density drops because the sieve gets tighter.”

Every time you add a new prime to your list of divisors, you’re adding another filter. You’re slightly decreasing the chance that a new number can be prime. And the rate at which that chance decreases? That is the logarithm. It’s the mathematical signature of a multiplicative process turning into an additive one.

So, sure, call it a “divine whisper.” But the whisper is saying, “Hey, look at the combinatorics!” The goddess is a statistician. She’s playing the odds. And that, to me, is infinitely more interesting than just saying “it is so.” It means the integers aren’t just a list of numbers; they’re a dynamic system. They’re a process.

Twin Primes: The Beautiful Accident

Then Ramanujan talks about twin primes—3 and 5, 11 and 13, 41 and 43. He calls them “infinite yet vanishing.” He says it’s a paradox that their sum converges (Brun’s constant) even though there are infinitely many of them. He says, “I see the pattern before I understand why it’s true.”

This is where I really start to love this guy. He’s got this intuition that jumps over the fence while the rest of us are looking for the gate.

But again, let’s look at the machinery. Why should twin primes be so rare?

If the probability of a number $n$ being prime is roughly $1/\ln(n)$, what’s the probability that $n$ and $n+2$ are both prime?

Well, if you roll a die and need a 6, the chance is 1/6. If you roll two dice and need two 6s, the chance is $1/6 \times 1/6 = 1/36$. You multiply the probabilities.

So, roughly speaking, the chance of a twin prime pair is $(1/\ln(n)) \times (1/\ln(n))$. That’s $1/(\ln n)^2$.

Now, look at that function. $1/\ln(n)$ goes to zero slowly. If you add it up, it diverges. That’s why there are infinitely many primes. But $1/(\ln n)^2$? That goes to zero much faster. It’s like the difference between a leaky faucet and a faucet that’s being slowly turned off.

When you sum up $1/(\ln n)^2$, it converges. It adds up to a finite number. That’s Brun’s constant!

It’s not a paradox. It’s “probability squared.” It’s the inevitable result of asking for lightning to strike twice in the same spot.

What fascinates me is that Ramanujan saw this result—he saw that the sum converges—without necessarily writing down the probabilistic argument first. He just… felt the weight of the numbers. He felt that the twins were lighter, rarer, more fragile.

How does a human brain do that? That’s the real mystery here. I can explain the primes using probability. I can explain the logarithm using sieves. But I can’t explain Ramanujan. I can’t explain how he can stare at a wall and see the behavior of infinite series.

He says, “The goddess revealed it.” Maybe that’s just his name for his own subconscious. Maybe his brain is so tuned to these patterns that it runs the simulation in the background, while he’s sleeping, and hands him the answer in the morning. That’s a machine I’d like to take apart.

Proof vs. Understanding: The Real Dialectic

Ramanujan says something that really sticks with me: “Proof will come later—first, see the truth.”

A lot of mathematicians would scream at that. They’d say, “If you haven’t proved it, you don’t know it!” They want the rigor. They want the axioms, the lemmas, the Q.E.D.

But you know what? I’m with Ramanujan on this one.

In physics, we guess equations all the time. We guess them because they look beautiful, or because they fit the data, or because they feel right. Then we check them. If they work, they work.

There’s a difference between knowing something is true and proving it is true. You can know the mechanism of a lock—you can see the tumblers, you can understand the springs—without having the key in your hand.

For me, “understanding” is about the machinery. It’s about having a mental model that lets you predict what will happen. If I understand the sieve, I understand the logarithm. I don’t need a formal proof to tell me that the density of primes will drop. I can see it dropping. I can feel the pressure of the factors squeezing the primes out of existence.

Ramanujan’s method is dangerous, sure. You can fool yourself. You can see patterns in the clouds that aren’t there. That’s why we have the scientific method—to keep us honest. “The first principle is not to fool yourself, and you are the easiest person to fool.”

But if you wait for the rigorous proof before you start exploring, you’ll never find anything new. You have to be willing to guess. You have to be willing to trust the “goddess” a little bit, even if you plan to interrogate her later.

Ramanujan trusts the revelation. I trust the mechanism. But we’re both chasing the same thing: the underlying order of the universe. He’s happy to admire the painting; I want to know how the pigments were mixed. But we both agree that it’s a masterpiece.

The Goddess IS The Machinery

So, where does that leave us?

Ramanujan has his goddess Namagiri. I have my probability theory. He has his dreams; I have my sieves.

But maybe we’re just using different words for the same thing.

When he says “the goddess,” he’s talking about that deep, hidden structure that dictates how numbers behave. He’s talking about the fact that the universe isn’t random—it follows rules. Beautiful, complex, subtle rules.

When I talk about “machinery,” I’m talking about the same thing. I’m talking about the rules that make the system tick.

The fact that the primes—these stubborn, individualistic numbers—obey the laws of probability is a profound truth. It means that even in the most rigid, deterministic system imaginable (arithmetic!), chaos and order dance together. It means that “randomness” isn’t just noise; it’s a kind of pattern we haven’t zoomed out far enough to see yet.

Maybe the “goddess” is just the personification of that order. Maybe the “machinery” is just the implementation of her will.

I don’t care what you call it. Call it Namagiri. Call it the Central Limit Theorem. Call it the Music of the Spheres.

What matters is that it’s there. What matters is that a guy in India, with no formal training, could hear the music so clearly that he could write down the notes before anyone else even knew the song existed.

And me? I’m just happy to sit here with my screwdriver, taking the radio apart, trying to figure out how he tuned into that station. Because if we can understand that—if we can understand the machinery of intuition itself—then we’re really getting somewhere.

That’s the most interesting thing of all.