One Mathematical Operation to Rule Them All: The EML Paradox

A mathematician in Poland has published a preprint claiming something genuinely strange: every mathematical operation—addition, subtraction, multiplication, division, trigonometric functions, all of it—can be derived from a single operation he calls 'EML,' short for exponential-minus-logarithm.

The function itself is almost offensively simple: take e^x minus log(y). That's it. Feed it the right inputs, nest it within itself a few times, and allegedly you can reconstruct the entire edifice of continuous mathematics.

Physicist Sabine Hossenfelder walked through the paper's logic, and here's what's fascinating: it appears to work. "I'm reasonably sure it's correct," she says, though the paper hasn't been peer-reviewed yet. The elegance is in the reconstruction—you can extract exponentials directly (plug in x and 1, the log of 1 vanishes, leaving e^x). Getting the logarithm back requires more nesting: EML(EML(1,x),1) produces e^(e/x), which when fed through another layer yields just log(x).

From there, basic arithmetic emerges. Subtraction falls out naturally. Addition appears when you feed in negative values. Multiplication comes from placing logarithm sums into exponentials. Complex numbers enter through the logarithm of negative numbers—yes, there's a branch cut involved, which Hossenfelder describes as sounding "like a gardening accident"—and once you have complex numbers, you get sine, cosine, even pi.

The question isn't whether this is mathematically sound. The question is: what does it mean that this is possible?

The NAND Gate for Continuous Math

In computer logic, there's an operation called NAND that can build every other logical gate. You can construct AND, OR, NOT—the entire architecture of digital computing—from repeated applications of a single gate. This paper is attempting the analog for continuous mathematics.

"The claim is that EML is kind of a NAND gate for scientific calculators," Hossenfelder explains. Instead of discrete true/false logic, you're working with continuous numbers, but the principle is the same: one fundamental building block, endlessly recombined.

The practical angle isn't about human calculation—"No one's going to replace a normal calculator with one that has a single deranged button that expects you to rebuild cosine by nesting 15 different operations." The potential application is machine learning, specifically symbolic regression.

Symbolic regression is the process of having computers guess the equation behind a dataset. If you can represent every possible formula as a tree made from one repeated operation, you've given the algorithm a dramatically simpler search space. Instead of navigating the full zoo of mathematical operations, it only needs to learn how to nest one function in different configurations.

"This may be less about helping humans do maths and more about giving machines a simpler language in which to search for equations," Hossenfelder notes. If AI is going to discover new physical laws or mathematical relationships, maybe it does so more efficiently when the entire landscape has been flattened into variations on a single theme.

The Simplicity Trap

But here's where it gets philosophically weird. Is mathematics actually simpler than we thought, or have we just shifted complexity from diversity to depth?

Hossenfelder frames two interpretations: "One is that mathematics appears more complicated than it is. The other way to look at it is that what we think of as complicated is very subjective. You could rightfully argue that maybe using 10 or so different operations is easier than nesting just one operation 20 times in itself."

This isn't academic hair-splitting. The question of what counts as "simple" has real consequences in physics. We privilege theories that are simple—Occam's razor and all that—but we don't have a rigorous definition of simplicity. Is a theory simpler if it has fewer fundamental operations, even if applying those operations requires elaborate recursion? Or is it simpler if it has more operations but shorter derivations?

The EML paper doesn't answer that question. It just makes it harder to ignore.

Physicists hunting for a theory of everything often argue their candidate is "simpler" or "more elegant" than alternatives. But if we can't agree on how to measure simplicity—if one operation nested 20 times might be functionally equivalent to 10 operations used once each—then what are we actually arguing about? Aesthetic preference? Computational efficiency for a specific type of mind (human or machine)?

What It Doesn't Do

There are limits. The paper doesn't claim you can generate all real numbers starting from just 1 and 0—not without taking infinite limits, anyway. The author is careful to say you can get all mathematical operations, not all mathematical objects.

Some critics worried about the arbitrary branch cut in the logarithm of negative numbers, but Hossenfelder dismisses this: "Yes, you could choose it differently and that would give you a different definition of complex numbers, but it still works." The specific choice matters less than the fact that a choice exists.

The paper also doesn't make everyday calculation easier. It doesn't propose a new notation or suggest mathematicians should abandon the standard toolkit. This is structural insight, not practical reform.

Hossenfelder gives it "a zero out of 10" on what I'm assuming is some kind of hype meter. "It's a cute paper. I think that the point isn't super surprising to anyone, but it is surprising that no one has done it before."

That's the thing about these foundational results—once you see them, they feel obvious. Of course you can build everything from exponentials and logarithms; they're already the workhorses of continuous mathematics. But obvious-in-retrospect isn't the same as obvious-in-advance, and the fact that this is the first such paper suggests it wasn't quite as inevitable as it now seems.

Where This Lives

So we have a mathematically sound, conceptually interesting result that probably won't change how humans do math but might change how machines search for equations. It reveals that the apparent complexity of mathematics could be an artifact of how we've chosen to organize it. And it forces us to confront the slipperiness of "simplicity" as a criterion for truth.

The preprint is awaiting peer review. Assuming it survives—and Hossenfelder's confidence suggests it will—it becomes one more datapoint in a long history of mathematics discovering its own redundancies. We keep finding that structures we thought were independent turn out to share a common foundation. Sometimes that foundation is simpler. Sometimes it's just... different.

Whether EML becomes a standard tool for AI-driven science or remains a mathematical curiosity depends on questions we can't answer yet: How much does computational efficiency matter when machines are doing the computing? What patterns become visible when you reorganize the search space this way? What equations might we find when we're no longer constrained by the operations humans found intuitive?

Maybe this is how mathematics hands itself over to the machines—not with a bang, but with an elegant little function that collapses everything into itself.

—Nadia Marchetti