Could All Of Math Be Reduced To A Single Operation? This Theoretical Physicist Says Yes, And He's Found It

“Everyone learns many mathematical operations in school: fractions, roots, logarithms, and trigonometric functions […] each with its own rules and a dedicated button on a scientific calculator,” notes Andrzej Odrzywołek’s new paper.

But “higher mathematics reveals that many of these are redundant,” he writes: “for example, trigonometric ones reduce to the complex exponential.”

“How far can this reduction go?” he asks. “We show that it goes all the way: a single operation, eml(x, y), replaces every one of them.”

Very simple; very complicated

Imagine a calculator with only two buttons: 1, and something that says “EML”. It may not be easy or time-efficient, but from those two, you could feasibly reproduce any computation possible on a normal scientific calculator.

So what is this mysterious function? Well, “EML” is actually a very descriptive name: the letters stand for “exponent minus log,” and the operator looks like this:

 eml(x, y) := exp(x) – ln(y).

Now, we’re really not kidding when we say this baby isn’t easy to use. If you want to produce the number 0, for example – pretty much one of the foundational bits of any mathematical system – you’d have to type this into your two-button calculator:

eml(1, eml(eml(1, 1), 1))

… which is a mite longer than just pressing a simple “0” button, let’s face it.

But to be fair, efficiency and ease for humans isn’t really the point. “The discovery was a side result of a larger project on exhaustive search, using methods known as symbolic regression,” Odrzywołek told IFLScience.

That’s an area of math – or, really, computer science – that is more like a treasure hunt than a brainteaser: it “seek[s] exact laws and formulas in data,” he explained.

You can think of symbolic regression methods as a kind of “backwards” math – you start with the end results and use symbolic regression to work towards the equation that produces them.

Except, by “work towards,” we mean “trawl wildly through a vast space of potential expressions until you find one that fits.” It’s not something you’d want to try as a human, basically.

But if you’re already deep in the theoretical computing weeds, why not keep going? “Along the way, I became curious about how small the basis for such a search could be,” Odrzywołek said. “EML was the answer.”

The first step on an unknown path

On the face of it, this is little more than a mathematical trick – a particularly neat one, to be sure, but nothing super helpful for day-to-day life.

But that’s really selling it short: Odrzywołek “is essentially proposing a ‘NAND gate for continuous mathematics’,” Martin Benning, Professor of Inverse Problems in the Department of Computer Science at University College London, explained to IFLScience. A NAND gate is a type of logic gate in computing that is "functionally complete," meaning any function can be performed with only this type of gate.

(Benning was not involved in Odrzywołek's project; his work focuses on inverse problems and continuous optimization, an area that is fairly close to the concepts in this paper, but that he stresses does not make him an authority on them.)

“By doing this, [he] turns what is usually a discrete and combinatorial search problem into a uniform continuous optimization problem,” Benning told IFLScience. “From the perspective of machine learning, that is an interesting and appealing shift because it allows us to theoretically use standard neural network training techniques to discover exact mathematical formulas from data.”

In that sense, the paper is more of a proof-of-concept than an end result – a demonstration that something unexpected is indeed possible, even if not necessarily practical right now. After all, if the EML operator is ill-suited to human use, it’s not all that much better for computers either: in experiments to see how complex he could go with EML, Odrzywołek found that past six nested sets of brackets, his network failed completely to provide a solution to his input.

That’s not really surprising: there’s a minimum amount of information needed to convey any idea, and if you reduce the breadth of the grammar available, then the scales will insist you increase the depth. It means that “even simple functions require many, deeply nested constructions,” Benning explained – and past a certain point, “algorithms struggle finding the right path to the solution.”

Still, it’s a notable first step – one that has proven a path exists at all. “It demonstrates that such a reductionist approach is feasible,” says Odrzywołek. “I expect follow-up work will find further operators with even better properties.”

“Time will show”

While the headline may be attention-grabbing, the truth is it’s hard to say where this preprint will lead. That’s not unusual for a result like this: “It's hard to predict which direction will pay off,” Odrzywołek told IFLScience.

Perhaps it can be used in analogue computing, or genetic programming, he suggested; maybe it will just pave the way for similar discoveries with symbolic regression methods. “Time will show,” he said.

But future applications will need more work behind them – ways to make the operator work with more realistic problems, for one thing, past six sets of nested brackets. For now, the result, while “intriguing,” is “currently an interesting theoretical work rather than a practical path to solving applied problems yet,” said Benning.

For Odrzywołek, however, more familiar pastures await. He plans to submit the preprint to a specialist journal, and then “I'll likely return to astrophysics projects for a while,” he said.

The moral of the paper, then? Perhaps it’s just to keep an open mind. “Exhaustive search methods are underused in mathematics, and they can occasionally reveal something that was hiding in plain sight,” said Odrzywołek. “Something that looks obvious in retrospect, like the wheel.”

The preprint, which has not yet been peer-reviewed, is available on ArXiv.