Math is the Compelled Speech of Physics: How This Limits Progress
Science — the provision of natural explanation for phenomena — is one of the most reliable and benevolent institutions that humans have erected. Science is, afterall, a monument to objective understanding. If we imagine it as an actual edifice, the foundation of that building is physics. Biologists explain interactions in terms of chemicals, referring details to chemists who explain matters with atoms. But atoms reside in the foundation of the shrine. While explaining their behavior is a duty that falls squarely, and terminally, on the shoulders of physicists, it is by no means their concern alone. Therefore, crises in foundational physics potentially affect, however indirectly, the precision and potency of all other hard sciences. A wider scientific attention to insurmountable problems in physics might be just the medicine for everyone involved.
Recently, physics has been obsessed with unification — a simplified, elegant conception of reality that behaves consistently at all scales and speeds. The hunger for a unified theory arises from the fact that General Relativity (GR) and Quantum Mechanics (QM), the most promising physical works of the 20th century, remain at a century-long detente. GR, Einstein’s conception of gravity, improved the accuracy of Newtonian descriptions while QM offered a detailed description of atomic behavior that has allowed engineers to harness the inner workings of atoms in the service of extraordinary technical ends. But despite their myriad successes, the two modes of describing reality have remained stubbornly irreconcilable.
For over a hundred years, physicists have worked to construct a robust theory of quantum gravity that could unify the very small with the very large — but all attempts at this yield paradoxical results. These dead-ends include singularities, the breakdown of a theory under extreme conditions, as well as infinitely expanding calculations. Seth Lloyd, Professor of Quantum Mechanical Engineering at MIT, once wrote that “long-time practitioners of quantum gravity have advised me that if one wishes to publish in the field, any advance… must be offset by making other problems worse, so that net effect is negative. If economics is the dismal science, then quantum gravity is the dismal physics.”
In a recent Quanta Magazine piece, Sara Cremonini, a theoretical physicist at Lehigh University echoed Lloyd’s sentiment about the quantum treatment of relativity in suggesting that the math was rock solid — but that it was missing some of the “key features and ingredients” of gravity. Some have gone so far as to suggest that many of these theoretical approaches to unification, which rely on fundamentally novel mathematics, have departed from science and are approaching philosophy.
Reached by phone, Dr. David Lindley, author of The Dream Universe and former editor at Science and Nature, wonders if progress won’t come from somewhere beyond the physics department completely. He recounts the familiar tale of theoreticians endlessly re-working the foundational conceptions of relativity, thinking “Oh, we just need to tweak this theory a little bit, understand this other detail. But this is fundamentally based on hopes for finding some kind of mathematical model for reality. And I don’t think we’re any closer to that than we were 40 years ago. We need an Einstein, we need a Galileo, we need a Faraday to come along and say, ‘No, you’re looking at this completely wrong.’”
Could mathematics be the wrong approach for breaking up the stagnation? It may seem absurd, since physics is by most contemporary accounts an inherently quantitative discipline. Yet, Michael Faraday, one of the greatest physicists of all time, brought almost no mathematics to the table when he developed the idea of the electromagnetic field. Although the mathematician James Clerk Maxwell eventually ascribed quantitative dynamics to Faraday’s fields, the revolution in our notions of electricity and magnetism emerged from a qualitative understanding, not the equations themselves.
Faraday astutely identified the perpendicular relationship between magnetism and electricity — but rather than simply describing the interaction between them, he imagined physical materials that he called “lines of force” that were responsible for stirring the “electric fluid.” In 1852, in a correspondence with the Royal Institution, he reflected on the “action in curved lines” that iron filings take in the vicinity of a magnet or current carrying wire. He noted, it “seems to me to imply at once that the lines have a physical existence... There must, I think, be a previous state, a state of tension or a static state, as regards the wire, which, when motion is superadded, produces the dynamic state or current of electricity.”
His dedication to uncovering the physical nature of electromagnetism, though, would remain just out of reach. In his letters, he wrote that “by the use of iron filings, numerous pictorial representations of the same general result may be obtained.” On the basis of these experimental studies, he imagined what the invisible relationship could look like. He simplified the fields into closed rings, called sphondyloids, that intersect each other at 90 degrees. In his model, shown below, the motion of the electric sphondyloid incites the magnetic ring into action. The drawing itself is mildly anatomical, and brings to mind a biological dissection of a specimen that’s used to illuminate the mechanics of a body.
Michael Faraday’s sphondyloids
Faraday’s dedication to the underlying physical mechanism of observed phenomena was clear in his writing, but was insufficient to halt the tide of progress - no matter how many lectures he gave at the Royal Institute. Forty years later, when Maxwell published his first versions of the equations that described the behavior of electromagnetic fields, the question of what was causing the action still had not been addressed - and was slowly buried by the motion of countless pens, hard at work describing.
One hundred and ninety years after Faraday’s discovery of electromagnetic induction, the takeover of physics by the describers rather than explainers has led some to question the validity of the premise in the first place. Faraday observed something, and Maxwell mathematically described it, but neither one of them explained what exactly was causing the phenomenon. The gradual creep of mathematics into the heart of physics, which can be traced back to Newton himself, has led to an arms-race of elegance and increasingly abstruse descriptive detail. But has it helped us come to a mechanistic understanding?
Author Dr. Lindley reflects on the fixation of modern physics with simplicity by recalling how an old friend who studied the immune system pointed out to him that reality is not always elegant. In biology, “nobody would say, ‘Oh, this is a beautiful, elegant piece of biochemical machinery.’ It's just this weird contraption that's come about through many years of evolution, and it does its job...But the question of looking for elegance, I don't know if that really comes up in molecular biology. They're saying, Well, here's this molecule, what does it do? How does it communicate? What is it attached to? You try to tease the whole thing out. So this question of elegance is something that really began to afflict physics, especially as it grew further and further away from basic empirical testing, which is what's happened, particularly in the last several decades.” Dr. Lindley’s concern that physics has moved away from mechanisms in favor of elegant calculus, and ends in a question - are alternative methodologies to mathematics?
While Faraday did not succeed in identifying the actors comprising his beloved sphondyloids, his visual approach allowed him to break through the previously stagnant understanding of the relationship between magnetism and electricity. All at once, he came to consider that they were two sides of the same action. And yet the realization gave him no peace, for the detailed nature of the physical actor behind the phenomenon was still mysterious. At best, he was left to speculate on “what that surrounding magnetic medium, deprived of all material substance, may be.” He concluded, “I cannot tell, perhaps the aether. I incline to consider this outer medium as essential to the magnet.” Concerning the ringed structures composing his fields, Faraday wrote, “It may be a vibration of the hypothetical aether, or a state or tension of that aether...”
Faraday’s mechanical conception of fields has been replaced with progressively more precise mathematical descriptions, but still no physical, surface-bound material actors. To this day, mathematical fields are central to all frontiers of physical theory, including General Relativity and Quantum Mechanics. Dr. David Tong, Cambridge Professor of Theoretical Physics, explained at his 2017 Royal Institution lecture that “the fundamental building blocks of nature are fluid-like substances which are spread throughout the entire universe and ripple in strange and interesting ways...These fluid-like substances we have a name for. We call them fields.”
He continues, “the physicist's definition of a field is the following. It's something that is spread everywhere throughout the universe. It's something that takes a particular value at every point in space. And what's more, that value can change in time.” At this point he shrugs and squeezes his palms together for a moment before moving on to Faraday’s contribution to our understanding of the field - without returning to the inherent contradiction. What is the architectural mediation that’s responsible for the changing values of the fields? What is the something that is spread throughout the universe?
There have been many theories that have sought to settle that question. One old favorite was that action was transmitted by an invisible fluid substance known as the “luminiferous aether.” Famously, though, the theory was dealt a death blow in 1887 when Michaelson and Morely failed to demonstrate the existence of the static aether. In a final analysis of their results, they wrote that “it appears, from all that precedes, reasonably certain that if there can be any relative motion between the earth and the luminiferous aether, it must be small.”
But a few decades later, Albert Einstein drew on history to propose a new version of the aether: one with dynamic qualities. Unlike the static luminiferous aether, Einstein’s was responsive to material and in return, instructed its motion. Einstein addressed the physical implications of his then five-year-old General Theory of Relativity at the University of Leyden in 1920, saying that “to deny the aether is ultimately to assume that empty space has no physical qualities whatsoever. The fundamental facts of mechanics do not harmonize with this view.” Simply put, the static aether of old, presumed by thinkers from Aristotle to Faraday and through Michaelson Morley, was a tapestry upon which material existence unfolded. But Einstein imagined the situation somewhat differently.
For him, there was no static universal reference point as prescribed by the luminiferous aether. During the same 1920 address, Einstein continued: “What is fundamentally new in the ether of the general theory of relativity as opposed to the aether of [the past] consists in this, that the state of the former is at every place determined by connections with the matter and the state of the ether in neighbouring places…” All points in Einstein’s aether were in motion relative to others.
Much like Faraday, Einstein arrived at his revelation of General Relativity by imagining a new kind of mediator, space. And he, too, could not answer the question of what is space. In his speech at Leyden he conceded that “the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν), has, I think, finally disposed of the view that space is physically empty.” But the difficulty of conceptualizing that physical material, and the apparent impossibility of its empirical detection, led to a repeat of Maxwell. Einstein provided a mathematical description, and it has served as the basis of the last century of physics. It works. But the question remains - if we can’t touch what’s out there, does that mean there’s nothing there at all?
Widespread acceptance of relativistic space that can be neither directly measured nor observed, and a sense that the aether is a long-discredited theory in the first place, may have some responsibility for the elevation of descriptive mathematics over mechanistic explanations. Beyond the Leyden address, Einstein seems to have devoted less concern for the physical nature of invisible mediation than Faraday. And yet, in utilization of a hypothetical physical substance, space, in contemplation of invisible phenomena, gravity, Einstein managed to replace the paradigm of static ether with a mental image of something that was responding to the motion of bodies. However, he never approached a physical conception of that material, he simply described its effects upon observable matter.
Dr. Lindley places a premium on empirical thought specifically because “empirical testing is basically shorthand for the fact that there are objects that are doing something that you can interrogate. They're behaving in a certain way.” Faraday’s lines of force, empirically derived, imply the action of some material or materials with definite locations. While an interpretation of Einstein’s GR as immaterial space has served physics for over a century, our inability to unify it with QM poses a serious problem for modern physics. The issue may be further compounded by the fact that both disciplines are mathematics-first in their interpretation of apparent phenomena, and so all hope for their reconciliation is based on various quantitative approaches.
Perhaps whoever manages to rectify this conflict will need to return to an earlier age, like Einstein did in his own conception. Perhaps our next Einstein will be the person who is able to provide a physical interpretation of the “imponderable” materials responsible for invisible actions within space. Given how visual interpretations of reality have preceded mathematical ones in both the case of Einstein and Faraday, perhaps we can expect the next generation of physical inquiry to be launched from well beyond the mathematically fortified halls of a physics department. Dr. Lindley puts it best when he says that “some people like to think in terms of just mathematics and say, well, that represents reality, and that's all we need to know. But then if you're Faraday, then you want to say, but what is it really?”
