I look for inspiration everywhere I can find it. My girlfriend recently informed me about a viral video in which a young girl expresses her dissatisfaction with the mathematics. “I was just doing makeup for work,” Gracie Cunningham says as she applies foundation to her face, “and I just wanted to tell you guys how I don’t think math is real.”
Cunningham claims that some of the algebra she’s learning in school has nothing to do with the world she works in. “I understand addition; for example, if I take two apples and add three, the result is five. However, how did you come up with idea of algebra?” Although some geeks ridiculed Cunningham, others defended her, pointing out that she addresses issues that science heavyweights have debated.
I may relate to Gracie’s grievances. I’ve been trying to understand eigenvectors, complex conjugates, and other esoterica since last May as part of my continuing attempt to study quantum mechanics. According to Wolfgang Pauli, some thoughts are so far off the mark that they’re “not even wrong.” I’m so perplexed that I’m not even perplexed. “Who came up with this concept?” I keep asking, as Cunningham put it.
Consider Hilbert space, which has infinite dimensions and is densely packed with arrow-shaped abstractions known as vectors. I feel like a lump of weak flesh stuck in a filthy, 3-D dungeon when I think about Hilbert’s space. I can’t even find a window from which to peer into Hilbert space, let alone explore it. I see it as luminous heaven where luminant cognoscenti glide back and forth, telepathically exchanging witticisms about adjacent operators.
Great sages have told us that reality is fundamentally mathematical. Plato believed that we and other entities in this universe are merely shadows of reality’s sublime geometric shapes. The “best book of nature is written in mathematics,” according to Galileo. Isn’t it true that we’re all a part of the natural world? Then why does algebra sound so foreign to most of us until we get beyond natural numbers and simple arithmetic?
More specifically, how true are the equations we use to describe existence, as Gracie points out? As accurate as, if not more valid than, nature itself, as Plato claimed? Were quantum mechanics and general relativity awaiting our discovery in the same way as gold, gravity, and galaxies awaited our discovery?
The hypotheses of physicists are correct. They predict planet arcs and electron flutters, and they’ve shown us smartphones, H-bombs, and—well, what else do we need? But scientists, mostly physicists, aren’t only looking for realistic solutions. They’re looking for the truth. They like to think that their ideas are accurate—and only accurate—representations of reality. This yearning is shared by physicists and religious people, who all need to feel that their road to heaven is the Only True Path.
Can a statement be said to be valid if no one understands it? Physicists are also arguing over what quantum mechanics teaches us about reality a century after it was invented. Consider the Schrödinger equation, which can be used to determine an electron’s “wave function.” The wave function, in essence, produces a “probability amplitude,” which, when squared, gives the probability of finding the electron in a specific location.
An imaginary number is embedded in the wave function. Since an imaginary number is the square root of a negative number, which by definition does not exist, the name is fitting. The wave function does not correlate to something in the physical world, even though it gives you the response you seek. It works for some reason, but no one knows why. The Schrödinger equation is in the same boat.
Perhaps we should regard the Schrödinger equation, like the Greek and Arabic symbols in which we describe functions and numbers, as an innovation, an arbitrary, dependent, historical mistake. After all, scientists took several false moves to arrive at the Schrödinger equation and other canonical quantum formulas.
Assume you’re the Great Geek God, surveying the vast geography with all imaginable mathematical representations of the micro realm. “Yup, those brilliant humans made it the finest possible collection of ideas,” you might think. Or would you exclaim, “Oh, if only they had followed a different direction right now, they would have discovered these calculations over here that would work even easier!”
In addition, the Schrödinger equation isn’t all-powerful. The Schrödinger equation, although excellent at modeling hydrogen atoms, is incapable of accurately describing helium atoms! Helium, which is made up of a positively charged nucleus and two electrons, is an example of a three-body puzzle that can only be solved with additional mathematical tricks.
And three-body problems are only one kind of N-body problem that plagues both classical and quantum physics. Physicists praise Newton’s law of gravitational attraction and the Schrödinger equation for their beauty and elegance. However, the algorithms can only balance experimental data using excruciatingly complicated fixes and approximations.
I keep thinking of dear old Ptolemy while I hear of quantum mechanics, with all its caveats and credentials. We now see his geocentric solar system concept, with its elaborate circles within circles within circles, as hopelessly kludgy and ad hoc. The geocentric model of Ptolemy, on the other hand, was successful. Planetary movements and solar and lunar eclipses were correctly forecasted.
Quantum mechanics functions as well as any other physical theory, if not stronger. However, it’s possible that its relation to reality—to what’s out there—is almost as shaky as Ptolemy’s geocentric paradigm. Perhaps in a century, our descendants would look back on quantum mechanics and say, “Those ancient scientists didn’t have an idea.”
Some officials have proposed this. Last fall, at Stevens Institute of Technology, I took a class called “PEP553: Quantum Mechanics for Engineering Applications.” Future scientists will “wonder how we could’ve been so gullible,” David Griffiths and a co-author ponder in the last paragraph of our textbook, Introduction to Quantum Mechanics.
The implication is that, as the heliocentric model of the solar system, we will one day discover the correct scientific explanation of truth, one that makes sense. But maybe the best thing we can claim of any mathematical theory is that it functions in a specific situation. Eugene Wigner’s controversial 1960 essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” has a subversive takeaway.”
According to Wigner, a famous quantum theorist, the equations contained in Newton’s laws of motion, quantum mechanics, and general relativity are incredibly, even unreasonably, accurate. What makes them so effective? Wigner acknowledges that no one knows. But, he stresses, just because these models perform doesn’t mean they’re “uniquely” right.
This assumption has some flaws, according to Wigner. For starters, physics theories have a finite spectrum. They refer to primary, narrowly defined facets of nature, and they leave a lot out. Second, the foundational ideas of scientific physics, quantum mechanics, and general relativity are mathematically incompatible.
According to Wigner, “all physicists conclude that a union of the two theories is inherently feasible and that we can discover it.” “However, it is easy to imagine that no compromise between the two hypotheses can be found.” Quantum mechanics and relativity have not been reconciled sixty years since Wigner published his thesis. Doesn’t this mean that one or both of them was mistaken in some way?
Wigner adds that physics’ “rules” have nothing to say about biology, especially consciousness, the most perplexing of all biological phenomena. Inconsistencies between biology and physics can emerge as we gain a better understanding of life and consciousness. These inconsistencies, such as the incompatibility between quantum mechanics and general relativity, could indicate that physics is wrong or incomplete.
Wigner has once again proved to be foresighted. Leading scientists and thinkers are debating whether mechanics or even materialism as a whole will account for existence and cognition. Some argue that the intellect is as fundamental as matter.
Wigner is challenging the Gospel of Physics, which states, “In the beginning, there was the Number….” He warns his colleagues not to mistake their mathematical models for truth. Scott Beaver, one of the commentators on Gracie Cunningham’s math video, holds the same opinion. Beaver, a chemical engineer, said, “Here’s my easy answer to whether math is real: No.” “Math is merely a tool for describing trends. Patterns exist, but the math does not. Math, on the other hand, is extremely valuable!
Beaver’s perspective is pragmatism and modesty, which I assume represents his engineering experience. Engineers are modest as compared to physicists. Engineers don’t question if a given solution is correct when attempting to solve a problem, such as constructing a new car or drone; they would consider the language a category mistake. They want to see how the answer works and if it fixes the issue at hand.
Quantum mechanics and general relativity are examples of mathematical theories that perform exceptionally well. They aren’t true in the same way as neutrons and neurons are, and we shouldn’t give them the status of “fact” or “natural rules.”
If physicists follow this modest approach and resist their need for certainty, they would be more able to search out and thereby discover more and more valuable hypotheses, perhaps much better than quantum mechanics. The catch is that they must give up hope of ever finding a final formula that demystifies our strange, strange universe once and for all.
