I take inspiration where I can get it. My girlfriend recently alerted me to a viral video in which a teenage girl complains about mathematics. “I was just doing my makeup for work,” Gracie Cunningham says while dabbing makeup on her face, “and I just wanted to tell you guys how I don’t think math is real.”
Some of the math she’s learning in school, Cunningham suggests, has little to do with the world in which she lives. “I get addition, like, if I take two apples and add three it’s five. But how would you come up with the concept of algebra?” While some geeks mocked Cunningham, others came to her defense, pointing out that she is raising questions that have troubled scientific heavyweights.
Gracie’s complaints struck a chord in me. Since last May, as part of my ongoing effort to learn quantum mechanics, I’ve been struggling to grasp eigenvectors, complex conjugates and other esoterica. Wolfgang Pauli dismissed some ideas as so off base that they’re “not even wrong.” I’m so confused that I’m not even confused. I keep wondering, as Cunningham put it, “Who came up with this concept?”
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Take Hilbert space, a realm of infinite dimensions swarming with arrow-shaped abstractions called vectors. Pondering Hilbert space makes me feel like a lump of dumb, decrepit flesh trapped in a squalid, 3-D prison. Far from exploring Hilbert space, I can’t even find a window through which to peer into it. I envision it as an immaterial paradise where luminescent cognoscenti glide to and fro, telepathically swapping witticisms about adjoint operators.
Reality, great sages have assured us, is essentially mathematical. Plato held that we and other things of this world are mere shadows of the sublime geometric forms that constitute reality. Galileo declared that “the great book of nature is written in mathematics.” We’re part of nature, aren’t we? So why does mathematics, once we get past natural numbers and basic arithmetic, feel so alien to most of us?
More to Gracie’s point, how real are the equations with which we represent nature? As real as or even more real than nature itself, as Plato insisted? Were quantum mechanics and general relativity waiting for us to discover them in the same way that gold, gravity and galaxies were waiting?
Physicists’ theories work. They predict the arc of planets and the flutter of electrons, and they have spawned smartphones, H-bombs and—well, what more do we need? But scientists, and especially physicists, aren’t just seeking practical advances. They’re after Truth. They want to believe that their theories are correct—exclusively correct—representations of nature. Physicists share this craving with religious folk, who need to believe that their path to salvation is the One True Path.
But can you call a theory true if no one understands it? A century after inventing quantum mechanics, physicists still squabble over what, exactly, it tells us about reality. Consider the Schrödinger equation, which allows you to compute the “wave function” of an electron. The wave function, in turn, yields a “probability amplitude,” which, when squared, yields the likelihood that you’ll find the electron in a certain spot.
The wave function has embedded within it an imaginary number. That’s an appropriate label, because an imaginary number consists of the square root of a negative number, which by definition does not exist. Although it gives you the answer you want, the wave function doesn’t correspond to anything in the real world. It works, but no one knows why. The same can be said of the Schrödinger equation.
Maybe we should look at the Schrödinger equation not as a discovery but as an invention, an arbitrary, contingent, historical accident, as much so as the Greek and Arabic symbols with which we represent functions and numbers. After all, physicists arrived at the Schrödinger equation and other canonical quantum formulas only haltingly, after many false steps.
Imagine you are the Great Geek God, looking down on the sprawling landscape of all possible mathematical ways of representing the microrealm. Would you say, “Yup, those clever humans found it, the best possible set of solutions.” Or would you exclaim, “Oh, if only they had taken a different path at this moment, they might have found these equations over here, which would work much better!”
Moreover, the Schrödinger equation is far from all-powerful. Although it does a great job modeling a hydrogen atom, the Schrödinger equation can’t yield an exact description of a helium atom! Helium, which consists of a positively charged nucleus and two electrons, is an example of a three-body problem, which can be solved, if at all, only through extra mathematical sleights of hand.
And three-body problems are just a subset of the vastly larger set of N-body problems, which riddle classical as well as quantum physics. Physicists exalt the beauty and elegance of Newton’s law of gravitational attraction and of the Schrödinger equation. But the formulas match experimental data only with the help of hideously complex patches and approximations.
When I contemplate quantum mechanics, with all its hedges and qualifications, I keep thinking of poor old Ptolemy. We look back at his geocentric model of the solar system, with its baroque circles within circles within circles, as hopelessly kludgy and ad hoc. But Ptolemy’s geocentric model worked. It accurately predicted the motions of planets and solar and lunar eclipses.
Quantum mechanics also works, better, arguably, than any other scientific theory. But perhaps its relationship to reality—to what’s really out there—is as tenuous as Ptolemy’s geocentric model. Perhaps our descendants will look back on quantum mechanics a century from now and think, “Those old physicists didn’t have a clue.”
Some authorities have suggested as much. Last fall I took a course at my school, Stevens Institute of Technology, called “PEP553: Quantum Mechanics for Engineering Applications.” In the last line of our textbook, Introduction to Quantum Mechanics, David Griffiths and a co-author speculate that future physicists will look back on our era and “wonder how we could have been so gullible.”
The implication is that one day we will find the correct mathematical theory of reality, one that actually makes sense, like the heliocentric model of the solar system. But maybe the best we can say of any mathematical theory is that it works in a particular context. That is the subversive takeaway of Eugene Wigner’s famous 1960 essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”
Wigner, a prominent quantum theorist, notes that the equations embedded in Newton’s laws of motion, quantum mechanics and general relativity are extraordinarily, even unreasonably effective. Why do they work so well? No one knows, Wigner admits. But just because these models work, he emphasizes, does not mean they are “uniquely” true.
Wigner points out several problems with this assumption. First, theories of physics are limited in their scope. They apply only to specific, highly circumscribed aspects of nature, and they leave lots of stuff out. Second, quantum mechanics and general relativity, the foundational theories of modern physics, are mathematically incompatible.
“All physicists believe that a union of the two theories is inherently possible and that we shall find it,” Wigner writes. “Nevertheless, it is possible also to imagine that no union of the two theories can be found.” Sixty years after Wigner wrote his essay, quantum mechanics and relativity remain unreconciled. Doesn’t that imply that one or both are in some sense incorrect?
The “laws” of physics, Wigner adds, have little or nothing to say about biology, and especially about consciousness, the most baffling of all biological phenomena. When we understand life and consciousness better, inconsistencies might arise between biology and physics. These conflicts, like the incompatibility of quantum mechanics and general relativity, might imply that physics is incomplete or wrong.
Here again Wigner has proven prescient. Prominent scientists and philosophers are questioning whether physics and indeed the basic paradigm of materialism can account for life and consciousness. Some claim that mind is at least as fundamental as matter.
Wigner is questioning the Gospel of Physics, which decrees, “In the beginning was the Number….” He is urging his colleagues not to confuse their mathematical models with reality. That’s also the position of Scott Beaver, one of the commenters on Gracie Cunningham’s math video. “Here’s my simple answer about whether math is real: No,” said Beaver, a chemical engineer. “Math is just a way to describe patterns. Patterns are real, but not math. Nonetheless, math is really, really useful stuff!”
I like the pragmatism and modesty of Beaver’s view, which reflects, I’m guessing, his background in engineering. Compared to physicists, engineers are humble. When trying to solve a problem—such as building a new car or drone—engineers don’t ask whether a given solution is true; they would see that terminology as a category error. They ask whether the solution works, whether it solves the problem at hand.
Mathematical models such as quantum mechanics and general relativity work, extraordinarily well. But they aren’t real in the same sense that neutrons and neurons are real, and we shouldn’t confer upon them the status of “truth” or “laws of nature.”
If physicists adopt this humble mindset, and resist their craving for certitude, they are more likely to seek and hence to find more even more effective theories, perhaps ones that work even better than quantum mechanics. The catch is that they must abandon hope of finding a final formula, one that demystifies, once and for all, our weird, weird world.
Further Reading:
See also “Tragedy and Telepathy,” a chapter in my free online book Mind-Body Problems.
And for more ruminations on quantum mechanics and other puzzles, see my new book Pay Attention: Sex, Death, and Science.