Is Ultimate Truth an Equation? Nah.

If I believed ultimate truth is mathematical, I would convey that idea with this groovy image, the Ulam spiral, which plots prime numbers.

January 19, 2023. I have a friend, Richard Gaylord, a curmudgeonly chemist/physicist, whom I've never met. Our relationship consists of him emailing me science-y essays, videos and screen shots that he finds online, to which I react with cheers, hoots or growls. Richard loves making the point that if you don't understand something mathematically, you don't understand it. This claim, perhaps because I studied literature in college and teach humanities courses now, bugs me. My rebuttal follows.

Richard is in fancy company when he contends that the deepest truths are mathematical. Pythagoras and Plato both implied as much, and Galileo famously wrote that you can only read "the grand book of the universe" if you understand the language in which the book is written, that is, mathematics. In 1931 James Jeans, a British physicist, expressed the math-is-truth idea in especially adamant terms. "The final truth about a phenomenon resides in the mathematical description of it," Jeans wrote. He speculated that "the Great Architect," that is, God, "seems to be a mathematician."

Richard passed the Jeans quote along to me, as well as similar comments from Richard Feynman: "To those who do not know mathematics," Feynman wrote in The Character of Physical Law, "it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature." But here's an irony: Feynman's comments on quantum physics contradict the claim that mathematics illuminates nature.

In a book on quantum electrodynamics, which he helped formulate, Feynman reiterates that you can't comprehend quantum theory without the math. But he adds that you can't understand it with the math either. "I don't understand" quantum physics, Feynman confesses. "Nobody does." He suggests that physicists' advanced mathematical "tricks," although they make calculations easier, can obscure what is actually happening in nature.

Also, if God is a mathematician, in what dialect does She/He/They/It speak? Quantum phenomena are described with differential equations, matrices and path integrals, a method invented by Feynman. Each of these dialects employs imaginary numbers, which are constructed from the square roots of negative numbers.

You can also represent quantum events without differential equations, matrices, integrals and imaginary numbers, as physicist Terry Rudolph demonstrates in his wonderful little book Q Is for Quantum. Rudolph models superposition, entanglement and other quantum puzzles with arithmetic and a little algebra.

Another problem: quantum theory accounts for electromagnetism and the nuclear forces, and general relativity describes gravity. Quantum theory and general relativity are conveyed in radically different lingos, which are hard to translate into each other. Some physicists still dream of a unified theory, possibly embodied in a single formula, that describes reality. That is the theme of Michio Kaku's recent bestseller The God Equation: The Quest for a Theory of Everything. 

Kaku's vision of a mathematical theory of everything seems increasingly quaint, given all we've learned about the limits of mathematics. In the 1930s, Kurt Gödel proved that all but the simplest mathematical systems are inconsistent, posing problems that cannot be solved within the axioms of that system. Extending the work of Gödel, mathematician Gregory Chaitin points out that mathematics, rather than being a unified, logically consistent whole, is riddled with randomness, contradictions and paradoxes.

Chaitin even questions whether real numbers are, well, real. Real numbers, which correspond to points on a line running from negative infinity to positive infinity, are mathematical beasts of burden; they help us model things like neurons, rainbow trout and missiles. But whereas most scientific measurements are approximations, which come with error bars, real numbers are impossibly precise. Between any two integers lies an infinite number of real numbers, most of which must be specified with an infinite number of digits. 

Physicist Nicolas Gisin makes similar points in a recent essay in Nature. If you model time with real numbers, the present moment becomes infinitesimal, and time ceases to exist. Models based on real numbers, Gisin argues, commit us to a rigid determinism that rules out the possibility of creativity. Like Chaitin, Gisin emphasizes that real numbers are "marvelous tools." But just because real numbers help us solve problems, he proposes, does not mean they reflect reality.

The philosopher Bertrand Russell, early in his career, revered mathematics. "Too often it is said that there is no absolute truth, but only opinion and private judgement," Russell wrote over a century ago. "Of such skepticism mathematics is a perpetual reproof." Toward the end of his life, perhaps because of the influence of Gödel and the ultra-skeptic Ludwig Wittgenstein, Russell arrived at a darker view of mathematics. "I fear that, to a mind of sufficient intellectual power," he wrote, "the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal."

That's far too bleak a view. If mathematics reduces to a tautology, 1 = 1, it is a fantastically fecund tautology. Mathematics has led to countless intellectual, aesthetic and material advances, on which our civilization depends. But mathematics, like ordinary language, is a human invention, a powerful but limited tool, not a divine gift. Many mysteries resist mathematical analysis, especially those related to the human mind. As physicist Sabine Hossenfelder says in her provocative book Existential Physics, it is "presumptuous" to assume that "humans have already discovered the language in which nature speaks, basically on the first try." 

And let's not forget that some of science's greatest advances have been non-mathematical. Take the theory of evolution by natural selection, which philosopher Daniel Dennett has called "the single best idea anyone has ever had." Darwin, who never liked math, spelled out the theory in On the Origin of Species. That monumental work does not include a single equation.

For all these reasons, we should doubt physicists who say that truth must be expressed in equations. Physicists would say that, wouldn't they? That's like a poet saying that truth can only be expressed in meter and rhyme, or an economist saying that everything comes down to money.

Alec Wilkinson, author of A Divine Language, a lovely memoir about his attempt to learn calculus in his 60s, says mathematics, rather than giving him answers, has deepened his sense of the mystery of things. "None of what I studied," he writes in The New Yorker, "illuminated anything for me so much as the idea that I don't know, that there is more to life than I think." Studying mathematics related to quantum mechanics makes me feel the same way.

Back for a moment to my pal Richard Gaylord. Although a math-o-phile, Richard does not share the belief of many physicists that there is—must be!--a single, true mathematical description of the world. He adheres to a position called theoretical pluralism. There can be many ways to model nature and to solve a scientific problem, Richard says, and insisting that there must be one correct way can impede scientific progress. On this point, Richard and I agree.

Further Reading:

My book Mind-Body Problems makes the case for theoretical pluralism, as does my column "Pluralism: Beyond the One and Only Truth," which was inspired by Richard Gaylord. See also my followup post: "Is the Schrodinger Equation True?," which was previously behind a paywall.

Comment from physicist Peter Woit of "Not Even Wrong": About the math/reality business, my point of view is very different. Pretty conventionally, I'd argue that you need an appropriate language for talking about physics. Natural language just isn't up to it, for instance the fact that "string theory" "multiverse" "wormhole", now don't now refer to anything specific has made much discussion of fundamental physics just incoherent streams of bullshit. There are many different mathematical idioms, you can describe quantum mechanics in very different mathematical languages. Less conventionally, it seems to me that the fact that some mathematical idioms work better in describing fundamental physics means they capture something fundamental about the real world. And the fact that these specific idioms turn out to be the ones that best express the deepest ideas about number theory is more evidence that there's some fundamental congruence between the physical world and some specific mathematics. But I should acknowledge that even my mathematical colleagues aren't convinced by this argument. And it does sound too much like Michio Kaku's God Equation, and if Kaku believes something, that's a good reason to be skeptical. 

Comment from physicist Nigel Goldenfeld: Richard Gaylord sent my column to Nigel Goldenfeld, who replied with the following message: "I believe John Horgan is wrong when he says (paraphrasing) that we should doubt that truth must be expressed in equations.  The reason is this.  He thinks we physicists use mathematics because it is an aesthetic choice. That's not the reason at all.  We use mathematics because that is the only way you can make predictions that are precise and can be compared to experiment. Only by doing that can we figure out what is a practically good description of nature, based on minimal models.  We only know something is true empirically when we can calculate and compare precisely with experiment, with full understanding of the uncertainties in both the experiments and the calculations.  I don't take an exalted view of physics. We are at heart engineers of the natural world.  

"It is true that you can convince yourself that you understand quantum mechanics without calculus (or using simpler mathematics than that which professional physicists use) but that's totally delusional.  Good luck calculating the Lamb shift!  Because popular science books focus on quasi-religious stuff like multiverse and other pathologies of physics, they completely overlook how we know it all works. But I guess you don't sell as many books if you tell people that if you want to understand the natural world you need the appropriate level of mathematics, which may require some hard work.  I feel like I understand something when I can calculate it.  I don't want to fly in a plane built by someone with a different definition of understanding!"

Synchronicity: I'm reading The Idiot, the marvelous novel by Elif Batuman, and I just came across this passage: "math is a language that started out so abstract, more abstract than words, and then suddenly it turned out to be the most real, the most physical thing there was. With math they built the atomic bomb. Suddenly this abstract language is leaving third-degree burns on your skin. Now there's this special language that can control everything, and manipulate everything, and if you're the elite who speaks it—you can control everything."