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

The discussion had its genesis one summer day in 1992 at CERN’s cafeteria terrace. If anyone was present in the vicinity during lunch, they might have heard talks about the unprecedented particle accelerator set to be built—later known as the Large Hadron Collider—or about the budding project, the World Wide Web, which had just launched. That day was marked not only by scientific excitement but also by a debate among three renowned physicists.

Among them were Gabriele Veneziano, the Italian contributor to string theory; Lev Okun, the Soviet scientist who coined the term “hadron” for quark-based particles; and Michael Duff, the British theorist pivotal in developing the ambitious M-theory stemming from string theory.

What sparked their discussion was a seemingly simple yet profound question: how many numbers are essential to capture the essence of reality? Veneziano proposed that if string theory holds true, nature may only need two basic constants. Okun opposed that idea, insisting that a minimum of three constants was vital for any serious theory. Duff dismissively claimed it could be zero.

This lighthearted cafeteria chatter spiraled into a decades-long philosophical inquiry, guiding these physicists through deep intellectual challenges. The crux of the matter—how many numbers one needs to effectively define the universe—complicates the understanding of its fundamental nature. This debate continues to evoke contemplation, with new scholars recently chiming in with their surprises to this ongoing inquiry.

Open nearly any physics textbook, and numbers abound. Many are termed “constants”, which are specific figures used in formulas to derive useful results. Examples include the mass of a proton, the charge of an electron, or the radius of a hydrogen atom. The Committee on Data of the International Science Council is regarded as the custodian of fundamental constants, boasting an extensive catalog of hundreds of values. However, determining which numbers are truly necessary poses a complex challenge.

Fundamental constants

During the time of the debates in the cafeteria, textbooks typically highlighted three specific constants due to their significance in physics. One of these appears in the iconic equation E=mc², Albert Einstein’s representation of how the speed of light (denoted as c) relates energy and mass. This concept is foundational to Einstein’s special theory of relativity, which delineates the principles of causality. According to special relativity, the speed of light is uniform for all observers, no matter their relative motion, necessitating the intertwining of space and time into what we call space-time.

The second constant is Planck’s constant, represented as h, which serves a similar transformative role, relating the energy and frequency of waves. Physics regards waves and particles as two interchangeable manifestations of the same phenomenon, with h allowing transitions between the two, laying the groundwork for quantum mechanics. Physicists additionally refer to a related metric known as h-bar, which helps define when quantum impacts manifest.

Lastly, there’s Newton’s gravitational constant, G—endearingly termed “big G“—which measures the gravitational attraction between masses, forming the basis for our comprehension of gravity. It illustrates how masses respond to the geometrical curvature of space-time.

A discernible pattern emerges: these constants not only define relationships but also unify concepts. Space translates to time, matter to energy, and waves to particles. Physics, at its zenith, embodies minimalism that distills nature’s core attributes.

This notion partly inspired Veneziano’s 1986 paper, igniting the CERN squabble. He was motivated by string theory, which saw significant advancements in prior years and conceptualizes particles as mere vibrations of one-dimensional strings. “There were high hopes for this theory being a comprehensive one that could elucidate everything, including the standard model and beyond,” he asserts. Operating on the logic of string theory, he maintained that all three constants—c, h, and G—were unnecessary for describing nature. Instead, Veneziano posited only two essential constants: the length of these strings and the speed of light.

Okun, however, rejected this idea. For him, all three original constants represented the essential foundations of physics, collectively linking relativity, quantum mechanics, and gravity. A credible theory of everything needed to incorporate all three. Okun valued the preservation of these constants away from the abstractions of string theory. He envisioned a conceptual framework of physical theories, with the constants acting as toggles. Classical mechanics stood on one end, with all three constants set to zero—no relativity, no quantum mechanics, and no gravity. Activating c transitioned one to special relativity. Introducing h brought one into the quantum domain. The addition of G completed the picture with general relativity, culminating in a hypothetical quantum gravity theory where all three constants come into play. For Okun, these constants represented not mere conveniences but the scaffolding upon which all known theories rest.

The discourse regarding these constants persisted, often resurfacing whenever the three physicists met at conferences and events. Although Okun passed away in 2015, both Duff and Veneziano recall their exchanges fondly—an amusing disagreement that held both trivial and profound implications. Veneziano recounts a moment during a skiing trip when he encountered Okun, who, before exchanging pleasantries, pointed at him and asked, “two or three?”

In 2001, with their disagreement unresolved, they crafted a paper outlining their positions. Yet, what underpinned Duff’s assertion that no constants exist? He held a distinct perspective on the problem. For him, the core issue wasn’t the quantity of constants necessary to describe the universe but identifying which constants signified something fundamentally real, rather than conventionally defined. Consider a scenario where an alien civilization with its own language and culture also possesses a precise understanding of physics: which numbers would they inherently employ in their equations? That question is pivotal to Duff’s reasoning.

To grasp his analysis better, one must recognize the distinction between two varieties of constants. Some are mere ratios. For example, the mass ratio of a proton to that of an electron is a constant, but due to division, the units cancel out, rendering it dimensionless. However, c, h, and G are not dimensionless; they come with units. Take c, defined as 299,792,458 meters per second. Duff argues that this definition hinges on previously established definitions of a meter and a second. If different units were employed, the speed of light’s numeric representation would shift. “While a committee in Paris determines what constitutes a meter, nature remains indifferent to their decisions,” he remarks. (This committee, coincidentally, is the International Bureau of Weights and Measures, which celebrates 150 years this year.)

The trouble with units

Moreover, one can intentionally select units to make a constant equal to one. This method is frequently adopted in high-energy physics, referred to as using “natural units.” Since any value multiplied or divided by one remains unchanged, the constants effectively disappear from equations. Physicists don’t believe the speed of light or any other constant has truly vanished; instead, they have adjusted their measurement systems so that the constant becomes the baseline value.

Duff’s argument asserts that if a constant can be scaled out of existence, it was never fundamental. Instead, he argues for a reliance on dimensionless quantities, which remain invariant. While he admits that a few may be necessary, their quantity is theory-dependent, and establishing an exact count is less critical. The standard model features up to 25 dimensionless parameters, varying with precise formulations.

The ongoing exchanges among Duff, Okun, and Veneziano, alongside their 2001 document, have taken on a life of their own within physics circles. Nevertheless, George Matsas from São Paulo State University in Brazil believes it’s time to resolve this question definitively. “It’s somewhat scandalous that, despite our substantial understanding of fundamental physics, we are still engaging in this debate,” he states.

Thus, in 2024, Matsas and his team sought to bring clarity by revisiting foundational principles and reframing the question. If a physicist found themselves marooned on a deserted island, which minimal number of independent measures would they need to define everything? That, he contends, outlines what is “fundamental.” To ascertain the volume of a box, one wouldn’t need a novel device; a ruler applied in three directions would suffice. Length remains more elementary than volume.

The original trio of constants collectively enables a scientist to define independent measures for length, time, and mass—functioning as a ruler, clock, and balance. Yet, Matsas and his team identified redundancies in this arrangement.

Matsas argues concerning mass: because gravity creates a predictable pull between masses, one could determine an object’s mass simply by timing its fall. In a similar vein, the relationship between time and space in relativity means that measuring one inherently gives insight into the other. A clock can function as a measure of length. Here, philosophical considerations about the nature of mass versus length become unnecessary. The fundamental nature is tied to what can’t be eliminated. Through a lens of extreme minimalism, one concludes that merely a clock suffices. This implies that constants like c, h, or G are not required—everything can be resolved using just a clock paired with a time-defining constant, including the frequency from an atomic clock. Matsas’s conclusion regarding constants isn’t three, two, or zero—but simply one.

For Matsas, this perspective resolves the longstanding dispute. While Duff’s assertion that no dimensional constants are fundamental may hold logical merit, it offers little practical guidance for measurements. “By stating that there is no fundamental basis, you essentially claim that the structure of space-time offers no means to measure itself,” Matsas observes.

Nevertheless, there remains lingering uncertainty. Matsas himself acknowledges that his team’s argument falters at the quantum level. Theoretically, a single clock might suffice for universal measurements, but practicalities complicate matters. It isn’t straightforward to craft a clock with infinite precision. The Heisenberg uncertainty principle ensures that striving for time measurement precision often requires greater energy consumption from the clock. Push this limit too far, and gravitational effects intervene. An overly refined clock could concentrate so much energy in a confined space that it collapses into a black hole. For these reasons, Matsas views his support for just one constant as contingent upon future advancements. “When we achieve breakthroughs in quantum gravity, our understanding of this question could shift, potentially leading to a consensus of zero constants,” he states.

This encapsulates the challenge presented by such philosophical deliberations, according to João Magueijo of Imperial College London, who has engaged in dialogues with Duff before. “These discussions are often rooted in theoretical biases,” he remarks. “It’s quite hubristic to assume that our current understanding will be the end-all narrative.” He reminds us that in Galileo’s era, Earth’s gravity was perceived as a universal constant, whereas we now recognize it varies with altitude.

What these decades of discourse, originating from that lunch at CERN, reveal is that the answer to how many constants are needed to describe the universe largely hinges on one’s conceptualization of reality’s foundations. Duff and Veneziano may not recall every minute detail of their lengthy summer debate, but it’s evident that the dialogue was rich with insights.