The 30-year fight over how many numbers we need to describe reality

It all started one summer lunchtime in 1992 on the terrace outside the CERN cafeteria. If you had happened to be there at the physics research lab near Geneva at the time, you might have overheard conversations about the enormous new particle accelerator being planned – a machine that would become known as the Large Hadron Collider – or about a fledgling information-sharing project, the World Wide Web, which had launched a few months earlier. But on this particular day, there was also an argument going on among three physicists.

There was Gabriele Veneziano, the Italian who helped invent string theory; Lev Okun, the Soviet scientist who coined the term “hadron” to refer to particles made of quarks; and Michael Duff, a British theorist who had been instrumental in developing string theory’s even more ambitious cousin, M-theory.

They were arguing over a deceptively simple question: how many numbers do you really need to describe reality? Veneziano had recently floated the idea that, if string theory were true, nature would contain only two fundamental constants. Okun disagreed. Three, he insisted, was the bare minimum any respectable theory needed. Duff scoffed at both of them. For him, the answer was obviously zero.

This lunchroom banter ballooned into a decades-long trialogue that took the trio of physicists into deep intellectual waters. After all, to ask how many numbers we need to properly define the universe is to ask what its true nature consists of. The debate still prompts a good deal of head-scratching today. Even recently, a new set of researchers stuck their oar in and gave their own unexpected answer to this enduring question.

Open any physics textbook and there will be no shortage of numbers floating around. Many will be what physicists call “constants”, specific numbers that get plugged into equations to make them spit out useful answers. The mass of a proton. The charge of an electron. The radius of a hydrogen atom. The Committee on Data of the International Science Council, often seen as the keeper of fundamental constants, maintains an exhaustive list of hundreds of values. Yet how many are truly indispensable is a slippery question.

Fundamental constants

Around the time of the contretemps in the cafeteria, textbooks tended to put special emphasis on three constants because of their centrality to physics. One crops up as the last term in E = mc², Albert Einstein’s famous equation that shows how the speed of light, a constant called c, connects energy and mass. It is a cornerstone of Einstein’s special theory of relativity, which explains the workings of causality. Special relativity states that the speed of light is the same for all observers, regardless of their relative motion. This is only possible if space and time aren’t independent of one another, so c also binds space and time into a single fabric: space-time.

The second number is Planck’s constant, denoted with the letter h, which performs a similar kind of alchemy, this time between the energy and frequency of a wave. Physics paints waves and particles as interchangeable descriptions of the same phenomenon, and h can be used to toggle between the two, setting the foundations for quantum mechanics. Physicists also often use a related constant called h-bar, which can be used to define the scale at which quantum effects come into play.

And then there’s Isaac Newton’s gravitational constant, G – often fondly called “big G” ­– which quantifies the attractive force between masses and anchors our understanding of gravity. It shows how things with mass are affected by the curvature of space-time.

There is a pattern here. These constants don’t just define relationships; they merge concepts together. Space becomes time, matter becomes energy, waves become particles. Physics, at its best, is minimalist and leaves us with the most essential features of nature.

That was partly the spirit behind Veneziano’s 1986 paper, the one that lit the fuse on the squabble at CERN. He was inspired by string theory, which had seen huge advances in the previous few years and paints particles as just vibrations of one-dimensional strings. “There were big hopes that this was a theory of everything, and it could explain everything, the standard model and beyond,” he says. Working from the logic of that theory, he argued that you don’t need all three constants – c, h and G – to describe nature. Concepts like mass and energy could be reduced to the action of strings. As a result, he argued, there are just two essential constants: the length of those strings and the speed of light.

Okun wasn’t having it. He saw all three original constants as the irreducible core of physics. Together, they tied together relativity, quantum mechanics and gravity. Any theory of everything worth its salt would have to accommodate all three. Okun saw great value in keeping them well away from the abstractions of string theory. He proposed a conceptual map of physical theories, with the constants acting like toggle switches. Classical mechanics sits at one extreme, with all three set to zero: no relativity, no quantum mechanics and no gravity. Switch on c, and you step into special relativity. Turn on h, and you’re in the quantum realm. Combine both, and you get quantum field theory. Gravity enters the picture when G is added, first giving you general relativity, and finally, a hypothetical theory of quantum gravity where all three constants are in play. For Okun, these weren’t just numerical conveniences – they were the scaffolding on which all known theories hang.

The banter over constants went on and on. The three physicists would often see each other at conferences and other events, and it became a habit to revisit the question. Okun passed away in 2015, but Duff and Veneziano both remember it as a playful disagreement, at once inconsequential – it wasn’t going to change the outcome of any calculations – and strangely deep. Veneziano remembers one time when the three of them ran into each other on a skiing trip. He met Okun just as he was about to step off a chairlift. “And even before saying hello, Okun would point at me,” he says, “and ask: ‘two or three?’”

In 2001, with the disagreement still not settled, the trio wrote a paper summarising their positions. But what was behind Duff’s view that there were no constants at all? He actually had a distinctly different take on the whole problem. For him, the issue wasn’t how many constants are needed to describe the universe, but which ones represented something intrinsically real, rather than human convention. Imagine we encountered an alien civilisation with its own language, history, culture and modes of cognition – but an accurate grasp of physics. What numbers would they unavoidably have to use in their equations? That is one way of understanding how Duff approached the question.

To get a better grasp of his answer, we need to know that there is a dividing line between two different types of constants. Some are just ratios of numbers. For example, the ratio of the mass of a proton to the mass of an electron is a constant, but because you are dividing one mass by another, the units drop away, leaving it dimensionless. But c, h and G aren’t like that. They come with units attached, and so are called dimensional constants. Take c, defined as 299,792,458 metres per second. The trouble with this, says Duff, is that this number gets inked in only because we have already defined what a metre and a second are. If we used some other way to measure distance, it would change. “A committee in Paris decides what we call a metre, but nature doesn’t care what that committee is doing,” he says. (That committee, incidentally, is the International Bureau of Weights and Measures, which turns 150 this year.)

The trouble with units

In fact, it goes further than that. You can deliberately choose your units so the constant becomes 1. This is actually a common practice in some areas of high-energy physics, known as using “natural units”. Because anything multiplied or divided by 1 is itself, the result is that the constants effectively vanish from equations. It isn’t that physicists think the speed of light or any other constant has literally vanished; it is just that they have redefined their rulers so the constant becomes the baseline.

Duff’s point is that if a constant can be rescaled out of existence, it was never fundamental to begin with. Much better, he reckons, to stick with dimensionless measures, which remain unchanged. These, he concedes, we may need a few of. But how many depends on your chosen theory, and pinning down the exact number isn’t too important. The standard model has up to 25 of these dimensionless parameters, depending on the exact formulation.

The back-and-forth between Duff, Okun and Veneziano and their 2001 paper became physics folklore. But George Matsas at São Paulo State University in Brazil thinks it is time to put this question to bed. “It is a kind of scandal that we know so much about basic physics and are still discussing this controversy,” he says.

So, in 2024, he and his colleagues endeavoured to settle the matter, returning to first principles and reframing the question. If a physicist were stranded on a desert island and had to measure everything in the universe, what would be the minimum number of independent yardsticks they would need? That, he says, is what “fundamental” means. To measure the volume of a box, for example, you don’t need to invent a new device. A ruler, used in three directions, gets you there. Length is more basic than volume.

The original big three, when taken together, would allow a scientist to define independent measures of length, time and mass – effectively acting as a ruler, a clock and a set of scales. But in their paper, Matsas and his colleagues see redundancies.

Take mass, says Matsas. Because gravity pulls masses together in a predictable way, you would be able to infer the mass of an object just by timing how it falls. Similarly, relativity links time and space so tightly that measuring one gives you the other. A clock can give you a measure of length. At that point, you don’t need to ask whether mass or length is “real” in a metaphysical sense. What counts as fundamental is what you can’t get rid of. And by the logic of the most brutal minimalism, all you need is a clock. This means you don’t need to bother with c, h or G – you can do everything you need just by using a clock and a constant that helps define time, such as the frequency of an atomic clock. Matsas’s answer to the question of constants is not three, not two and not zero – just one.

For Matsas, this solution cuts through the decades-old argument. Duff’s view that no dimensional constants are fundamental may be logically coherent, but it leaves physicists with little guidance on measuring anything. “By saying that you don’t have any fundamental standard, you basically say that space-time doesn’t provide you any procedure to measure its structure,” says Matsas.

Still, the dust may not have fully settled. Even Matsas admits his team’s argument breaks down at the quantum scale. In theory, a single clock might be enough to measure the universe but, in practice, it isn’t that simple. You can’t build a clock with arbitrarily fine resolution. The Heisenberg uncertainty principle makes sure of that: the more precisely you try to measure time, the more energy your clock must expend. Push too far, and gravity intervenes. Your ultra-precise timepiece might pack so much energy into so small a space that it collapses into a black hole. For that reason, Matsas sees his backing of just one constant as contingent on future developments. “It may be that when we discover quantum gravity, the answer to this question might change, and then it could be zero,” he says.

That is precisely the problem with philosophical discussions like these, according to João Magueijo at Imperial College London, who has found himself debating with Duff in the past. “It’s just basically your prejudices about theory,” he says. “It’s so arrogant to say that what we know is going to be the last word forever.” He points out that in Galileo’s time, Earth’s gravity was considered a universal constant, whereas we now know it varies depending on how high up you go.

And perhaps that is what these decades of discussion, which kicked off in the CERN cafeteria all those years ago, have taught us. How many numbers do we need to describe the universe? Well, that depends on what you believe about the foundations of reality. It was so long ago now that Duff and Veneziano don’t remember the finer details of what was said during that leisurely summertime lunch – but it was clearly a lot to chew on.