The universal constants and mathematics They form an odd pair: on the one hand, we have physical numbers that seem written into the very fabric of the cosmos; on the other, an abstract edifice of theorems and proofs that doesn’t need an atom of matter to exist. When we ask ourselves why the speed of light, Planck’s constant, or gravity are exactly what they are, we inevitably end up stumbling upon questions about the role of mathematics in reality. At its core lies a profound doubt: if we were to change the fundamental constants of the universeWould the same mathematics still be valid? Or could mathematical logics and structures emerge that are as different as the physics of that other universe? Understanding what physical constants are, how they relate to our units of measurement, which parts of them are arbitrary and which are truly “natural” takes us on a journey through the history of science, philosophy, and modern cosmology.
What is a universal constant, really?
When physicists talk about a physical constantThese refer to the value of a quantity that, within the physical processes we know, does not change with time or place. The speed of light in a vacuum, the gravitational constant, or the elementary charge of the electron are typical examples: wherever you measure, on Earth or in a distant galaxy, you obtain the same number (within experimental precision). That value, however, is expressed in arbitrary human unitsMeters, seconds, kilograms, coulombs… Today we use the International System of Units (SI), formalized in 1960 and refined since then, but throughout history we have described the same quantities with very different systems. Hence the usefulness of distinguishing between two concepts: on the one hand, numbers that depend on our units (such as 299,792,458 m/s); on the other, pure, dimensionless numbers, which do not change even if we change the system of units. This distinction connects with a famous idea attributed to Einstein in a letter to Ilse Rosenthal-Schneider: there are apparent constants and real constantsThe apparent ones arise from the choice of units and can be “eliminated” with a suitable redefinition; the real ones would be the numbers that “God had to choose” when creating the universe, authentic fundamental parameters that determine what everything that exists is like. The so-called fundamental constants They are linked to essential physical phenomena and, as far as we know, cannot be calculated from other constants. We can measure them with ever-increasing precision, but we cannot deduce their value from deeper principles. This enigmatic nature fuels a debate that goes beyond physics: are they the product of deeper laws that we still don’t understand, or are they simply the result of “rolling the dice” in the Big Bang?
Units, consensus and the role of constants
Before seriously discussing universal constants, it is necessary to understand that our units of measurement They are, to a large extent, a convention. The case of the Mars Climate Orbiter probe, lost due to a confusion between metric and imperial units, is a dramatic example of how costly it can be to be unclear on this point. Through a long historical process, the International System of Units with seven base units: meter, kilogram, second, ampere, kelvin, mole, and candela. From these, dozens of derived units (newton, joule, pascal, etc.) that we use daily are defined. Interestingly, many of these units have been redefined precisely by fixing the value of certain universal constants. Today, the speed of light in a vacuum, c, is taken as exact value299,792,458 m/s. This doesn’t mean that light “has” to travel at that speed, but rather that we have fixed its value and defined the meter in a way that is consistent with it. Something similar occurs with Planck’s constant, Avogadro’s number, or the frequency of the hyperfine transition of cesium-133, which is used to define the second. To organize the “zoo” of constants, since 1966 there has been the CODATA (Commission on Data for Science and Technology)It compiles and recommends numerical values for hundreds of physical constants. One of its recent compilations includes around 230 constants, but only a minority have truly profound conceptual weight: c, G, h, the fine-structure constant, the masses of elementary particles, etc.
Some key examples of fundamental constants
Among all the constants, there is a small group that acts as backbones of the physical buildingTheir values are incorporated into the most basic theories and influence everything from the structure of atoms to the evolution of the universe. speed of light in a vacuum, c, is approximately 3·108 m/s. The Michelson-Morley experiment proved that this speed is independent of the motion of the emitting source, discarding the hypothesis of the “ether” as the propagation medium. From then on, the Lorentz transformations and Einstein’s special relativity took c as the insurmountable limit for information transmission. universal gravitational constant, GIt appears in Newton’s law of gravitation and in Einstein’s equations. It is perhaps the most poorly measured constant of all: we only have a few significant figures firmly established. Even so, we know that gravity is an extremely weak force compared to the other interactions; if it weren’t always attractive and acted on large scales, we would barely notice it. Planck constant, hIt marks the quantum scale: it sets the minimum size of the “quanta of action” and enters into the famous relationship E = hν between energy and frequency. Its value has become so fundamental that today it forms part of the very definition of the kilogram. Other basic constants are the elementary charge e, the Boltzmann constant k, Avogadro’s number NA and luminous efficacy Kcd for a monochromatic radiation of 540·1012 Hz. In addition to their relevance in fundamental physics, they all have very concrete manifestations in chemistry, biology, ecology and technology: they determine how molecules are organized, how living systems exchange energy or how we calibrate sensors and devices.
Apparent constants, real constants, and pure numbers
Not all constants are equally profound. Some, like the Boltzmann constantThese can be interpreted essentially as conversion factors between units of energy and temperature. Their numerical value depends on our system of units; if we change it, the number changes. These are the “apparent constants” to which Einstein referred. true universal constants In the strictest sense, they should be dimensionless, pure numbers that are unaffected by changing units. A canonical example is the fine structure constant αwhich measures the intensity of the electromagnetic interaction. Its approximate value is 1/137, and more precisely, CODATA recommends 137.035999084. This number is independent of meters or seconds: it is the same for any civilization using any reasonable system of units. α combines three dimensional constants: Planck’s constant, the elementary charge, and the speed of light. In a sense, condensed into a single number quantum mechanics (h), electromagnetism (e), and special relativity (c). Hence, physicists like Feynman called it a “magic number that comes to us without being understood,” or Dirac described it as “the deepest unsolved problem in physics.” Another way to “naturalize” the constants is to use the Planck unitsThese constants are constructed from c, G, and h (along with the Boltzmann constant if we introduce temperature). The Planck length, Planck time, or Planck mass define the scales at which quantum gravity is expected to become relevant. At that scale, our current theories cease to be valid, and we suspect that a more complete description of spacetime must come into play. From this perspective, we can see many constants that we use daily, such as… derived parameters of a handful of truly basic values. The permittivity of free space, the Bohr radius, or the Faraday constant would be different manifestations of the same underlying physics, recoded in combinations of essential constants.
Are constants truly immutable?
A fascinating part of the history of 20th-century physics revolves around the possible variation of the constants in space or cosmic time. It’s not just a numbers game: if any of them were to change even slightly, chemistry and life as we know them could become impossible. Arthur Eddington, famous among other things for experimentally confirming general relativity during the 1919 eclipse, was obsessed with the idea of deducing the values of the constants from purely mathematical principles. He attempted to construct elaborate numerical proofs that, by playing with relationships between numbers, “explained” why certain constants took on the observed values. For a time, colleagues like Einstein himself regarded these attempts with curiosity, but it soon became clear that Eddington He forced the mathematics to obtain the desired results. His constructions were more reminiscent of numerology than physics. Even so, he sowed a seed of unease that others picked up on: the suspicion that behind the values of the constants there might be still unknown mathematical structures. Paul Dirac was one of those who took up the mantle, albeit with a different approach. He noticed that several combinations of fundamental constants These coincidences resulted in enormous, seemingly related numbers, the so-called “large coincidence numbers.” This led him to conjecture that perhaps not everything was pure chance and that a simple mathematical relationship might exist behind these coincidences. In 1937, Dirac published an article in Nature in which he proposed that the gravitational constant G It could have varied over cosmic time. If that were true, some constants we treat as universal would actually be slowly changing parameters. This line of thinking connects with modern ideas that consider the possibility that certain constants do indeed depend on the history of the universe or on as-yet-unidentified background fields. Observational, evidence of variations in constants such as the fine structure constant Analyzing the spectra of distant galaxies, some studies have suggested the existence of a “preferred direction” in the universe where α could take slightly different values, which would directly contradict the equivalence principle and general relativity. However, most of the scientific community considers the current evidence inconclusive and believes that these results could be due to systematic errors or hasty interpretations.
Constants, life, and the anthropic principle
There is one particularly delicate aspect: the sensitivity of life to the values of the constants. A small change in the mass of the proton, the charge of the electron, or the intensity of the electromagnetic force could prevent the formation of stable atoms, complex molecules, or nuclear processes in stars such as those that produce carbon. It has been calculated, for example, that if the fine-structure constant α were to vary by only a few parts per ten million, chemistry as we know it would change drastically. With an increase of just 4%, it is estimated that certain nuclear reactions in stars would cease to produce carbon efficiently, causing our biochemistry based on that element to collapse. This apparent “fine-tuning” leads some to appeal to the anthropic principleThe fact that we can observe the universe already implies that its constants must be within the appropriate range to allow observers. In more robust versions, there is talk of a possible “multiverse” where different universes would have different values for these constants, and we simply inhabit the one (or those) where life is viable. Other scientists are more cautious and prefer to view fine-tuning as a open problemPerhaps a deeper theory, yet to be discovered, will necessarily fix the values of these constants, without resorting to multiple universes or anthropic arguments. For now, we don’t have such a theory, so the question remains entirely open. In any case, our current experience indicates that the fundamental constants they don’t seem to vary Within the measurement uncertainties achieved, neither in our environment nor in the observable cosmos. If variations exist, they must be extremely small and difficult to detect with current technology.
The cube of theories and the boundary of Planck units
A very illustrative approach to understanding the role of the constants c, G and h is the so-called cube of theoriesIntroduced by Bronstein, Gamow, Ivanenko, and Landau. Imagine a three-dimensional cube where each axis corresponds to “turning on” or “turning off” one of these fundamental constants in the sense of considering its effects relevant or negligible. If we simultaneously ignore gravity (G), relativity (c), and quantum mechanics (h), we remain in the corner of the classical Newtonian mechanicsIf we only turn on c, we enter special relativity; if we turn on G without c or h, we have Newtonian gravitation; if we turn on h without G or c, we move into non-relativistic quantum mechanics; and so on until we reach the most “complete” corner, where all three constants play an essential role and we should have a fully relativistic theory of quantum gravity. Planck units They precisely mark the corner where these three axes intersect: lengths on the order of 10-33 cm, times of 10-43 and temperatures close to 1032 K. It is thought that the universe, in its first infinitesimal fractions of a second after the Big Bang, reached these extreme conditions. Beyond that point, our current theories—general relativity and quantum field theory—cease to be reliable separately. Hence, so much current effort is directed toward formulating a quantum gravity theoryWhether through string theory, loop theory, or approaches to “double” or warped special relativity, all these proposals attempt to extend general relativity within a quantized or discrete spacetime structure, in which the constants c, G, and h would be unified in a new conceptual framework. Meanwhile, we live in a universe where general relativity describes large-scale behavior with great precision—galaxies, clusters, cosmic expansion—and quantum field theory does the same in the microscopic realm of subatomic particles. universal constants act as a bridge between both extremes, establishing scales of mass, length and energy that give coherence to the whole.
Mathematics and constants: are mathematics universal?
All of the above brings us back to the initial question: if we were to modify the values of the fundamental constantsWould mathematics still be the same? To answer this, it’s helpful to distinguish between two levels: that of abstract mathematical structures and that of the concrete physical models we build with them. Basic truths such as 1 + 1 = 2 Or, elementary properties of arithmetic (commutativity, associativity, etc.) are derived from very general logical axioms. These axioms make no reference to protons, electrons, or physical constants. From this point of view, many parts of mathematics seem independent of physical realityWe could imagine a universe with other constants, or even without matter, and the proofs would still be valid within their axiomatic system. The question is what Mathematical structures would be relevant or “natural” in a universe with different physics. In our universe, Euclidean geometry is a useful approximation at small scales, while curved Riemannian geometry is fundamental for describing gravity. Symmetry group theory is crucial in the Standard Model of particle physics. If the physical laws were different, perhaps the dominant geometry would be different, or other types of logic would be used to describe exotic phenomena. In the philosophy of mathematics, this debate is often summarized in contrasts such as Platonism vs. FormalismPlatonism holds that mathematical entities exist independently of us and any physical universe; we simply discover them. From this perspective, mathematics would indeed be “universal” in a strong sense: any intelligence in any cosmos that followed consistent reasoning would arrive at equivalent theorems. Formalism and related positions see mathematics more as systems of rules that we construct to organize symbols. In that case, the parts of mathematics we use most intensively would be strongly conditioned by the structure of the universe in which we live. Other intelligences in other universes could develop very different mathematics because their physical reality would “demand” other conceptual tools. Whatever position we adopt, the experience of modern physics suggests something unsettling: mathematics fit together surprisingly well with the structure of the world. A handful of constants, embedded in relatively compact equations, allow us to describe an enormous range of phenomena, from the orbit of a star to the emission of an atom. This “unreasonable effectiveness” of mathematics, as Wigner called it, is one of the greatest philosophical mysteries of science.
Constants, human mind and knowledge
Authors such as Max Planck emphasized that universal constants These are numbers that “were not invented by men,” but discovered in nature, and any intelligence in any corner of the cosmos should find them to be the same, regardless of the methods or instruments employed. For Planck, this invariance was proof that an objective physical reality exists, separate from our minds. At the same time, the fact that we depend on constants like c, G, or o for define our measurement standards It shows the extent to which our knowledge is intertwined with what we don’t know. As Jesús Navarro summarized in his book on universal constants, these numbers reflect both what we understand about the universe and what we still cannot explain: we know their values with great precision, but we don’t know where they “come from.” In practice, these constants mark the limits of our current theories. We know that general relativity works very well over a huge range of scales, but it breaks down conceptually when we approach times shorter than the Planck time or lengths shorter than the Planck length. We know that quantum mechanics successfully describes the microscopic world, but it is unclear how to elegantly couple it with gravity at extreme scales. Meanwhile, the visible universe continues to evolve under the silent guidance of these constants. apparently immutable magnitudesThey give rhythm to the cosmos, allow for the stability of atoms and molecules, make information, chemistry, and ultimately, beings capable of asking questions about all of this possible. If we ever achieve a theory that naturally derives the values of the constants, perhaps we will also discover why certain mathematical structures—and not others—describe our world with such precision. Until then, the universal constants will remain a meeting point between our best-established physics and our deepest doubts about the nature of reality and mathematics itself.