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!” Imagine you are the Great Geek God, looking down on the sprawling landscape of all possible mathematical ways of representing the microrealm. After all, physicists arrived at the Schrödinger equation and other canonical quantum formulas only haltingly, after many false steps. 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. The same can be said of the Schrödinger equation. Although it gives you the answer you want, the wave function doesn’t correspond to anything in the real world. That’s an appropriate label, because an imaginary number consists of the square root of a negative number, which by definition does not exist. The wave function has embedded within it an imaginary number. 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. Consider the Schrödinger equation, which allows you to compute the “wave function” of an electron. Physicists share this craving with religious folk, who need to believe that their path to salvation is the One True Path.īut 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. They want to believe that their theories are correct- exclusively correct-representations of nature. 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. 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? 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? Plato held that we and other things of this world are mere shadows of the sublime geometric forms that constitute reality. Reality, great sages have assured us, is essentially mathematical. I envision it as an immaterial paradise where luminescent cognoscenti glide to and fro, telepathically swapping witticisms about adjoint operators. Far from exploring Hilbert space, I can’t even find a window through which to peer into it. Pondering Hilbert space makes me feel like a lump of dumb, decrepit flesh trapped in a squalid, 3-D prison. Take Hilbert space, a realm of infinite dimensions swarming with arrow-shaped abstractions called vectors. I keep wondering, as Cunningham put it, “Who came up with this concept?” 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.
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. Gracie’s complaints struck a chord in me.
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. “I get addition, like, if I take two apples and add three it’s five. Some of the math she’s learning in school, Cunningham suggests, has little to do with the world in which she lives. “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.” My girlfriend recently alerted me to a viral video in which a teenage girl complains about mathematics.