Note: This is an AI-generated translation from my original Italian article: L'equazione che nessuno ha scritto

Why does mathematics describe the universe? And who wrote the rules that the rules themselves can't explain?

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The rules of the game

Mathematics, for me, has always been a matter of trust.

I'm not talking about school math, the kind with expressions and surprise quizzes. I'm talking about that feeling I've had since childhood, similar to what I felt sitting in front of the CPC 464: the feeling of a world where the rules are clear. Where if something doesn't add up, there's a reason. And the reason can be found.

It's the same thing I like about chess. The same thing I like about programming. In a world full of "it depends" and "maybe," mathematics looks you in the eye and says: this is true, and I can prove it. No ambiguity. No interpretation. You can trust it.

Or at least, that's what I thought.

Incompleteness

In high school I had wandered into strange territory. The thesis I presented at my final exam was called "From Gödel's Incompleteness Theorems to Turing's Real Computers." Stuff that didn't have much to do with the curriculum, probably nothing to do with what was expected of a graduating student. (My systems professor took it well, at least he did.)

But the point wasn't the thesis. The point was what I'd found inside it.

Kurt Gödel, in 1931, had proven something that never stops surprising me. In any formal system powerful enough to express arithmetic, there exist true statements that the system cannot prove from within. Not because of a flaw. Not because of an oversight. By structure. Mathematics, that perfect world where the rules are clear and everything can be proven, at a certain point looks at itself in the mirror and discovers it has a limit it cannot overcome.

The thing fascinated me so much that I wrote about it in my first exam essay too, the one for literature. For me there was no separation between mathematics and philosophy. They were the same question seen from two different angles: what can we know? And where does certainty end?

Turing took that thread and carried it even further. He built a perfectly deterministic model of computation, a machine where every step follows precise rules, no ambiguity, no margin. And he proved that even so, with perfect rules, there are problems the machine cannot solve. Not because it's broken. Because the problem itself doesn't admit an algorithmic solution. Determinism isn't enough to guarantee completeness.

At eighteen I didn't have the words to say it like that. But I remember the feeling well. That perfect system I had blindly trusted, mathematics, had just admitted it couldn't prove everything. And instead of disappointing me, it had fascinated me even more. Because a system that knows its own limits is more honest than one that pretends it doesn't have any.

The equation that shouldn't work

Then years passed. The curiosity didn't change, but the questions did.

After high school I enrolled at university. In the mathematics department there was a professor who struck me immediately: Neculai Sinel Teleman. He was the department chair and an internationally renowned mathematician. His lectures had something different: he didn't just explain mathematics, he made it breathe. At some point I managed to get the chance to talk to him about my doubts, about that tension between Gödel's limits and the power of mathematics in describing the world. He gave me an essay in English, printed on stapled sheets. It had been written by Eugene Wigner, physicist, Nobel laureate, and the title said it all: "The Unreasonable Effectiveness of Mathematics in the Natural Sciences."

Reading it, I realized that the question I'd asked myself at eighteen was only half the story.

Gödel had shown me the limit: mathematics cannot fully ground itself. But Wigner posed the opposite question, and perhaps the more unsettling one: if mathematics has all these limits, how does it describe the universe with such absurd precision?

Think about it for a moment.

A mathematician sits at a desk. No telescope. No laboratory. Just paper and pen. Playing with symbols, manipulating abstract structures, following the thread of formal beauty. No intention of describing nature. Pure intellectual pleasure, what Wigner called "showing one's ingenuity and sense of formal beauty."

And then a physicist takes those structures, built for fun, and discovers they describe exactly how stars behave. Not approximately. Exactly.

How is that possible?

There it is: this is the question I've been carrying with me for years. In school I'd discovered that mathematics has limits. As an adult I discovered that, despite those limits, it describes reality with a precision that, in Wigner's words, "borders on the mysterious" and for which "there is no rational explanation."

Gödel showed me the boundary. Wigner showed me what lies beyond.

The book of nature

I'm not the first to be thrown by this question. In 1623, Galileo Galilei wrote a sentence in The Assayer that after four centuries hasn't lost an ounce of weight:

"Natural philosophy is written in this grand book that stands continually open before our eyes, I mean the universe, but it cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics."

The universe is a book. And the language it's written in is mathematics. Not poetry, not philosophy, not rhetoric. Mathematics. That abstract stuff, made of symbols and structures, that kids in school think is useless.

But Galileo also said something else, less quoted and perhaps more important. He believed that God had written two books: the Bible and the Book of Nature. The Bible was to be interpreted by theologians. The Book of Nature, written in mathematical language, was to be interpreted by mathematicians. Two books, one author.

Keep this idea in mind. I'll come back to it.

When mathematics predicted the future

The most impressive thing about Wigner's observation isn't philosophical. It's concrete. There are documented, repeated cases in which mathematics described something that nobody had yet observed. It predicted the future. Here are some examples.

Mercury and the 43 arc-seconds

In November 1915, Einstein published the field equations of general relativity. On paper, these equations describe how mass curves spacetime. Nobody had tested them yet. They were pure mathematics.

But there was a problem that physicists had been carrying around for decades. Mercury's orbit didn't behave as predicted by Newtonian mechanics. The perihelion — the point closest to the Sun — advanced by 574 arc-seconds per century. Newton predicted 531. Forty-three arc-seconds were missing, and nobody knew where they came from. For decades, people had even hypothesized the existence of a hidden planet, Vulcan, disturbing the orbit. It was never found.

Einstein's equations predicted exactly those 43 arc-seconds. Not approximately. Exactly. Einstein recounted that when he saw the result his heart was pounding. He knew the equations were right. Not because someone had confirmed them in a laboratory, but because the mathematics worked out.

Light that bends

The same equations predicted something else: that light, passing near a mass, should curve. It's not intuitive. Light seems like the straightest thing there is. But if space itself curves near a mass, then the light passing through it must follow that curvature.

In 1919, during a total solar eclipse, astronomer Arthur Eddington led an expedition to verify this. He photographed the stars near the edge of the Sun and compared their positions with those they had when the Sun wasn't in the way. The stars were displaced. Exactly by the amount predicted by the equations.

Einstein, overnight, became the most famous man in the world. Modern measurements with radio waves confirm the prediction with less than 1% deviation.

Waves in spacetime

In 1916, Einstein noticed that his field equations, in the weak-gravity limit, reduced to a wave equation. This meant that spacetime could oscillate, like the surface of a pond when you throw in a stone. He predicted the existence of gravitational waves: ripples in the very fabric of reality.

An idea on a piece of paper, in 1916.

On September 14, 2015, at 9:50:45 UTC, the LIGO detector recorded a signal. Ripples in spacetime caused by the merger of two black holes 1.3 billion light-years away from us. The signal lasted less than a second. The deformation of space that LIGO measured was smaller than the diameter of a proton.

One hundred years. An equation on a sheet of paper and a signal in a laboratory, separated by a century. And the mathematics was right.

Dirac's impossible particle

In 1928, Paul Dirac was trying to combine quantum mechanics with special relativity. He wrote an equation that worked, but it also produced solutions with negative energy. Mathematically coherent, physically absurd. Any reasonable person would have discarded them as calculation artifacts.

Dirac didn't. In 1931, he proposed that those solutions described a never-observed particle: identical to the electron, but with opposite charge. It was called the positron.

In 1932, Carl Anderson found it in cosmic ray tracks, in a cloud chamber. A particle that shouldn't have existed, existed. Because mathematics had seen it before eyes did.

Dirac said something worth quoting: "God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe." It should be said, in fairness: Dirac was not a believer. It was a conventional phrase, not a statement of faith. But the fact that a non-believer spontaneously uses this image — that he arrives at this formulation following only mathematics — says something about the force of the intuition.

The boson that had to exist

In 1964, Peter Higgs, along with other physicists including Brout and Englert, predicted the existence of a field and its associated particle. The prediction was purely mathematical: that field was needed to explain why particles have mass. Without it, the Standard Model equations didn't work. With it, everything added up.

But the particle had never been seen. For 48 years.

On July 4, 2012, CERN announced they had found it at the Large Hadron Collider. Officially confirmed on March 14, 2013. A particle that for nearly half a century had existed only in equations was now real.

Think about it. A physicist writes an equation in 1964. He says: "There must be something here, otherwise the numbers don't add up." Forty-eight years pass. The most complex machine humanity has ever built is constructed, a 27-kilometer tunnel beneath the French-Swiss border. Matter is smashed into matter at speeds approaching light. And there, among the fragments, they find exactly what the equation said.

Coincidence?

The numbers that allow existence

There's another aspect of the universe's mathematics that goes even deeper. It's not about the equations that describe nature. It's about the constants that nature itself seems to have chosen.

The universe works thanks to a set of fundamental numbers: the speed of light, the gravitational constant, the electron's charge. Numbers that don't derive from any theory. They are what they are. We measure them, but we don't know why they have those values.

The unsettling thing is this: if those numbers were even slightly different, we wouldn't be here.

The ratio between gravitational force and electromagnetic force, for example, must fall within an extremely narrow window. If gravity were just a little stronger, stars would burn too fast to allow life to develop. If it were weaker, they wouldn't form at all. No stars, no heavy elements, no planets, no us. The same goes for the electron's mass, for the strong nuclear force, for a series of constants that all seem calibrated with a precision that makes your head spin.

Science calls this phenomenon "fine-tuning." And it has various proposals to explain it, none definitive.

The first is the multiverse: infinite universes exist with different constants, and we find ourselves in the one where they work. It's a legitimate hypothesis, suggested by some interpretations of string theory and inflationary cosmology. But it's not experimentally verifiable. For now it's a conjecture, not a confirmed theory.

The second is the anthropic principle: we shouldn't be surprised to find ourselves in a universe compatible with life, because in an incompatible one we wouldn't be here to wonder. It's logically impeccable. But it doesn't explain why the constants have those values. It describes a tautology, not a cause.

The third is physical necessity: there might exist a deeper theory, not yet discovered, that explains why the constants must have exactly those values. It would be the most elegant answer. But we haven't found it yet.

Nobody has the answer. Science is honest about this. And this honesty, for me, is one of the most beautiful things about science.

Discovery or invention?

Before I get where I want to get, there's one last question worth asking. A question philosophers have been asking for centuries and that physicists have lately stopped ignoring.

Did we invent mathematics, or discover it?

Roger Penrose, recipient of the 2020 Nobel Prize in Physics for demonstrating that black hole formation is a robust prediction of general relativity, is one of the most important living mathematicians. And he's a declared Platonist. He believes that mathematical structures exist independently of the human mind.

Speaking of the Mandelbrot set, that fractal structure of infinite complexity that emerges from the simplest of equations, Penrose wrote: "It is not an invention of the human mind: it is a discovery. Like Mount Everest, the Mandelbrot set is simply there!"

And in his book The Emperor's New Mind he went even further: "The notion of mathematical truth goes beyond the whole concept of formalism. There is something absolute and 'God-given' about mathematical truth." In The Road to Reality, Penrose describes three worlds: the physical world, the mental world, and the Platonic world of mathematics. When a mathematician "sees" a truth, according to Penrose, their consciousness breaks into this world of ideas and experiences it directly.

Here too, in fairness: Penrose is not a traditional theist. He uses "God-given" in a philosophical, not religious, sense. But the question he poses is the same one I ask myself, and the one Galileo asked four centuries earlier: if we didn't invent mathematics but discovered it, if it was already there before us, who put it there?

The foundation of evidence

At this point the conversation changes register. Up to here I've talked about science, numbers, experimental verification. Now I'm talking about myself.

Those who read this blog know I'm a believer. And I know that for some, the phrase "I love science and I believe in God" sounds like a contradiction. As if at some point the reasoning breaks off and something else kicks in.

For me it's the opposite. It's the same reasoning that continues.

My journey started from observation. I look at the world: the mathematical order, the calibration of constants, the equations that describe reality before reality has even been observed. And I ask myself a question: is there an intelligent cause behind all this? Accepting that possibility doesn't close the conversation. It opens it. Because it leads to other questions, and those questions require other answers, and those answers need to be verified.

It's a process. It's a method.

There's a verse I often use when people ask me what faith means to me. It's in Hebrews 11:1, and it's worth reading carefully:

"Faith is the assured expectation of what is hoped for, the evident demonstration of realities that are not seen." (NWT)

In the original Greek, the two key words are ὑπόστασις (hypostasis) and ἔλεγχος (elegchos). Hypostasis, rendered in English as "assured expectation," literally means "that which stands under" — a concrete foundation. In the commercial documents of the time it was a legal term: the title deed, the concrete guarantee of a future possession. Elegchos, rendered as "evident demonstration," indicates a convincing proof, the evidence through which something is verified — particularly something contrary to what it seems.

The apostle Paul is not describing a feeling. He's not talking about a leap of faith. He's describing a process based on evidence. Biblical faith is not believing despite the evidence. It's believing because of the evidence, even when what the evidence points to is not directly visible.

Now, doesn't that remind you of something?

The scientific method works like this: you observe a phenomenon, formulate a hypothesis, make predictions, verify the predictions, accept the most plausible explanation until proven otherwise. No serious scientist claims to have absolute truth. Science accepts the best available theory, and remains open to the possibility that tomorrow something better will come along. Einstein didn't "replace" Newton: he showed that Newtonian gravitation was a special case of something deeper.

My journey of faith works the same way. I observe the world: creation, the mathematical order, the impossible calibration of constants. I formulate a hypothesis: there exists an intelligent cause. I look for confirmation. The Bible: I've studied it. I find it a reliable book. A book that has defied time, that has survived millennia of attempts to destroy and discredit it. A book that speaks of God, but in a human way. A book that answers those questions that arise naturally when the creation around you makes you realize there's an intelligent cause. The way we are made reveals a great deal about the one who made us.

And then more questions arise. And those questions require more experiments, more verification, more comparisons. And so on. Nobody can claim to have absolute truth. But you can choose the most plausible explanation, with honesty, and keep it open to verification.

I don't claim this journey works for everyone. I don't claim it's the only valid one. I'm just saying that for me there's no contradiction between science and faith. On the contrary. The more I study science, the more I observe the mathematics behind the universe, the more the evidence, for me, points in one direction.

The boundary and what lies beyond

At eighteen I thought mathematics was the perfect system. Then Gödel showed me it wasn't. Then Wigner showed me it worked anyway, and nobody could explain why.

Galileo called it the language in which the universe is written. Dirac, a non-believer, found himself saying that God must be a mathematician of the highest order. Penrose, another non-believer, wrote that in mathematical truth there is something "God-given." Neither of them meant literally what they said. But neither of them could find better words.

Einstein's equations predicted Mercury's behavior, the bending of light, gravitational waves, a century before anyone managed to measure them. Dirac found the positron by following mathematics, not a microscope. Higgs predicted his particle 48 years before anyone found it in a tunnel beneath Switzerland.

Mathematics precedes reality. It doesn't just describe it: it anticipates it. As if the rules were written before the game.

I'm that same curious kid who sat on the floor at the Feltrinelli bookstore leafing through books he half understood. The one who dove into incompleteness theorems in high school, who saw no separation between mathematics and philosophy, who always looked for the rules beneath the rules. The question has always been the same. I just phrase it differently now.

Science keeps answering the "how," and each answer is more precise than the last. But the question I truly carry with me, the one Wigner called mysterious and for which he found no rational explanation, isn't "how."

It's "who."

Science accepts the most plausible explanation until proven otherwise. I do the same.

And the most plausible explanation, for me, has a name.