CHAPTER FOUR

I Understand That I Can’t Understand

Last month, a policeman in Minneapolis choked to death a black man suspected of buying cigarettes with a counterfeit $20 bill. His name was George Floyd. Since then, protests have erupted across the country. My daughter Skye joined a march in Brooklyn that got ugly. Protesters scuffled with cops and set a police car on fire. Skye got jostled and knocked down.

Black Lives Matter protest in Brooklyn, June 2020. I’m not sure if this is the one in which my daughter got knocked down.

I feel admiration for my daughter and relief that she’s all right, plus fresh moral qualms. Should I pursue my quantum experiment when the world is falling apart? We’re facing racism and other forms of injustice, Covid-19, climate change, the corrosive effects of capitalism and militarism. I don’t want to read/think/write about these problems. I want to read/think/write about quantum mechanics to escape the fucked-up world. But I feel obliged to say something about the protests, so I write a column for Scientific American that urges protesters of police violence to remain nonviolent.

Proud of myself, I email my column to Skye. She doesn’t like it. I shouldn’t tell black protesters what to do, she says, because I can’t comprehend what they endure. Brooding over her reaction, I remember when Leonard Susskind bashed me for bashing string and multiverse theories. Susskind told me, in effect, Shut the fuck up, Horgan, you’re too ignorant to have an opinion. My daughter is essentially telling me the same thing.

Now I’m wondering which is greater: my ignorance of physics or my ignorance of what black men experience in America. Or what women of all colors experience at the hands of men. Emily, triggered by Trump, has been denouncing the patriarchy a lot lately. She says all straight men, even “nice guys” like me, are “rapey,” in that we see women as objects of sexual gratification. She and I recently fought over the finale of Ulysses, in which Joyce dresses in drag, imagining what it is like to be a woman. To me, Molly Bloom’s soliloquy is a masterpiece within a masterpiece, a sublime feat of the literary imagination. To Emily, Molly’s soliloquy is just another rapey male fantasy, as I would know if I weren’t a typical rapey guy.

Should I accept these rebukes from my girlfriend and daughter? I can in principle learn enough physics to know what it’s like to be a physicist. Sort of. A really shitty physicist. But I can never know what it’s like to be a black person or a woman. Another reason, perhaps, for me to stick to quantum mechanics. Shut up and learn linear algebra.

The Manga Guide to Linear Algebra

Reiji (foreground) and Misa. The guy in the background is Misa’s older brother and Reiji’s karate teacher. The algebra is linear, the human relations, not so much.

Sabine Hossenfelder calls linear algebra “the key to understanding quantum mechanics” in one of her videos. Wait, didn’t she say elsewhere that differential equations, like the Schrodinger equation, are the key? I guess you need two keys to enter the quantum sanctum. My sense from Susskind is that linear algebra is preferable to differential equations for certain applications of quantum mechanics, like quantum computing. I’ve picked up a little linear algebra from Susskind, Wikipedia and the website “Math Is Fun,” but I need a better grounding; I need a primer, something like Quick Calculus.

After diligent research, i.e., scanning customer reviews on Amazon, I purchase The Manga Guide to Linear Algebra. It’s a comic strip in which Reiji, a math whiz, teaches linear algebra to Misa, a girl on whom he has a crush. Misa and Reiji are charmingly unpretentious. Misa says, “Matrices aren’t as exciting as they seem in the movies.” Reiji replies, “Yeah, just numbers, no Keanu.”

I relate to Misa, the teenage girl, even though she usually gets things faster than I do. She frowns anxiously when Reiji first imparts a chunk of knowledge, like how to transpose, or flip, a matrix; beads of sweat fly off her face. But she always gets it eventually, beaming her beautiful smile. She never accuses Reiji of mansplaining, talking down to her. They have a great relationship; I can sense the chemistry between them.

Reiji explains how to multiply matrices. What seems magical is reduced to mechanical, step-by-step procedures. As Reiji says, linear algebra is “just numbers, no Keanu.”

Reiji warns Misa that linear algebra is “pretty abstract,” but I think I’m getting it. Algebra is a method for representing things with numbers, and linear algebra lets you carry out certain algebraic calculations more efficiently than ordinary algebra. What makes linear algebra linear? Here’s how I explain it to myself:

I’ll start with matrices, the main component of linear algebra. A matrix is an array of numbers organized in horizontal rows and vertical columns. You can have as many rows and columns as you like. The numbers in the rows and columns correspond to coordinates in a space with a certain number of dimensions. These are mathematical dimensions, which are not necessarily the spatial dimensions in which we move. A matrix represents something in that space.

Let’s say you have a matrix, Q, representing an object on a two-dimensional grid with a horizontal x axis and vertical y axis. The magic happens when you manipulate Q with other matrices by adding, subtracting or multiplying them. You end up transforming Q by transforming its space. Picture one square in Q’s grid. You can expand or shrink the area of the square. You can slant the square, so it turns into a parallelogram. You can rotate the square 90 degrees to the right or to the left. You can flip the square around the x axis or the y axis, or around a diagonal cutting across the grid.

Now here’s where things get freaky. You can turn a square in the grid into a cube by adding an extra axis, or dimension, to the space; and you can turn a 2D parallelogram into a 3D parallelopiped. You can turn a cube into a tesseract, a 4D cube. Conversely, you can take dimensions away, turning a tesseract into a cube, or a parallelopiped into a parallelogram.

Linear algebra lets you have as many dimensions as you like; you don’t have to restrict yourself to the two and three dimensions across which we skate or through which we fly. To turn a 2D grid into a 3D grid, you add another axis, z, that is perpendicular, or orthogonal, to the x and y axes. You can keep adding more dimensions to a mathematical space by adding more orthogonal axes. All this just by manipulating your original matrix, Q, with other matrices. You can also transform your grid by shifting the point where axes cross and changing the length of units.

So why is linear algebra “linear”? Because the transformations must conform to certain strict rules. These rules have abstract mathematical definitions, but the result is easy to visualize. No matter how you transform the grid of the original matrix, Q, the lines of its underlying grid remain straight and parallel to each other. They do not cross or get squiggly. A square unit of the grid does not become an ellipse; a cube does not become a sphere. That is what it means to be linear.

I’m so proud of my realization that I try to convey it to Emily, but she’s not interested. She’s perfectly content to know nothing about linear algebra, just as I was a few months ago.

In linear algebra, you can transform a grid so that square units become parallelograms, as shown in these images from 3Blue1Brown.com, but your grid lines must remain straight and parallel.

Criminal Justice in Cleveland

During lunch breaks, I’ve been listening to a Serial podcast on criminal justice in Cleveland. I find parallels between The Manga Guide to Linear Algebra and the Serial podcast. Each tries to pull you into a complicated topic by getting you to empathize with characters. Manga features the imaginary characters Reiji and Misa, whereas Serial teems with real defendants, cops, public defenders, prosecutors, judges. Each narrative warns when you are approaching a deeper level of complexity, pausing to explain jargon, such as “eigenvectors” (a key concept in linear algebra) and “stacking charges” (overcharging suspects to intimidate them into a bad plea bargain).

In the case of linear algebra, the complexities and confusion yield, slowly, to clarity. When I multiply two 3 x 3 matrices (each consisting of 3 rows and 3 columns of numbers), I know there is a solution, a correct answer. Not so with criminal justice in Cleveland. As the narrator, Sarah Koenig, walks you through all the dimensions of different criminal cases, which range from a bar fight to armed robbery and murder, “truth” and “justice” become more and more elusive. Did this person actually commit this crime? And if so, what is a fair punishment?

Cleveland’s criminal-justice system is a gigantic, kludgy machine for grinding up cases and generating just outcomes. Ideally. The machine’s efficiency is degraded by bureaucratic inertia, incompetence and corruption, as well as the racism of police, attorneys and judges. Most operators of the machine are white, most defendants black. The machine’s efficiency is improved by the compassion and idealism of some operators. Yes, there are compassionate idealists in Cleveland’s criminal justice system.

Race complicates efforts to achieve justice in Cleveland. Judges and lawyers tend to be white and defendants black. I found this photo of a Cleveland court on the Ideastream website.

I imagine Cleveland’s criminal justice system as a vast, multi-dimensional matrix. With such a model, scientists might identify ways to make the machine more efficient, by identifying police prone to excessive violence, for example, or by speeding up the processing of defendants. But can scientists tell us whether justice is actually served in Cleveland? Not according to Steven Weinberg, the great seeker of a theory of everything. He once told me that science can only tell us what is and not what should be. “We’ve learned to absolutely disentangle value judgments from truth judgements,” Weinberg said. Science “can certainly help you find what the consequences of your actions are, but it can’t tell you what consequences to hope for. And that seems to me to be an absolute distinction.”

Here's one problem with that distinction: physics is about matter in motion, and a theory that doesn’t include human values can’t tell us much about our motions. Socrates made this point while awaiting execution in his prison cell. He complains to his buddies about philosophers who explain everything in terms of physical stuff, such as “air and ether and water.” Yes, 2,400 years ago, proto-physicists were claiming that everything is reducible to physical stuff pushed and pulled by physical forces.

How, Socrates asks, would such a philosopher account for what he is doing in this cell? The philosopher might point out that he, Socrates, “is made up of bones and muscles… and as the bones are lifted at their joints by the contraction or relaxation of the muscles, I am able to bend my limbs, and this is why I am sitting here in a curved posture.” But that would be a lousy explanation, Socrates points out, because the “true cause” of his situation “is that the Athenians have thought fit to condemn me, and accordingly I have thought it better and more right to remain here and undergo my sentence.”

If your theory of everything doesn’t include moral values, in other words, it’s not a theory of everything. Socrates’s assertion is as true today as it was back then. But how do we define morality? How do we decide what is right or wrong to do? It is one thing to model the effects of our values on our actions; it is quite another to determine what our values should be.

Socrates claimed that we can reason our way to moral truth. So do some modern philosophers, like utilitarians. For a hard-core utilitarian, morality is no big deal; it is reducible to what maximizes pleasure and minimizes pain across a population. Estimating the pleasure and pain resulting from a given action might be tricky, but you can substitute objective proxies for these subjective states: wealth and health for pleasure, poverty and illness for pain.

But utilitarianism doesn’t do justice to the immense, ever-shifting complexity and diversity of human values. And neither utilitarianism, Unitarianism, Marxism, capitalism nor any other ism has been compelling enough to eliminate our disagreements about what is right and wrong. Moral proclamations don’t have the status of mathematical or scientific truth; they are ineradicably subjective. I have strong ideas about what is right and wrong—for example, I think Donald Trump is a bad person and a bad President—but many people disagree with me, and that gives me pause. Coming up with clear-cut moral rules has proved difficult. As soon as one smart person, like Kant or Jeremy Bentham, erects a system of ethics, another smart person, like Nietzsche or Wittgenstein, smashes it.

What Is It Like to Be a Trump Voter?

In 2016, I attended a symposium at New York University, “Ethics of Artificial Intelligence.” Scientists and philosophers talked about how to instill ethical rules into machines, including autonomous cars, robotic soldiers and sexbots. Sitting in the audience with a skeptical frown was Thomas Nagel, an elderly philosopher. He is famous for a paper about the mind-body problem, published in 1974, with one of the best titles ever: “What Is It Like to Be Bat?” Near the end of the meeting, Nagel stood up and reminded everyone that after millennia of effort, philosophers have failed to agree on any system of ethical rules. Everyone looked at Nagel as though he had just farted.

Our inability to reach agreement on moral rules, it occurs to me, is related to the issues that Nagel explores in “What Is It Like to Be a Bat?” Consciousness is unlike other problems tackled by science, Nagel says, because it involves subjective, first-person experience. No matter how much we learn about the neurophysiology of a bat, we can’t know what it is like to be a bat. We can imagine, but we can’t know.

Nagel’s analysis applies to humans trying to understand other humans. I can’t really know what it is like to be a black man hassled by a cop, or a woman bullied by her boss into having sex, or a devout Christian who adores Trump. This fundamental human limitation has moral/social/political implications. We can never really understand each other, and that’s why we can’t agree on what is right and wrong and on how we should live together. The hard problem of consciousness is related to the hard problem of morality.

Vectors and Minigolf

Back to The Manga Guide’s tutorial on vectors. I assumed matrices and vectors are two different things, but a vector, Reiji tells Misa, is just an “interpretation” of a matrix, a way to visualize it. A vector is commonly described as an arrow of a certain length pointing in a certain direction. The vectors used in quantum mechanics—for example, to model the spin of an electron or the reflection of light off glass--are expressed in complex numbers, which, you will recall, consist of imaginary numbers combined with real numbers. Reiji mentions complex numbers but says he’s going to “avoid” them because they might “make it harder to understand the important parts” of linear algebra. Thanks, Reiji!

Reiji explains vectors with an analogy involving minigolf. You map the golf course as a two-dimensional grid, marked in meters, say, with a horizontal x axis and vertical y axis. You want to get your ball from a starting point, where x and y equal 0, to the cup, which corresponds to the coordinates 7 and 4. That’s the spot you’d reach if you walked 7 meters along the x axis and then turned left and walked another 4 meters. An arrow extending from the hole to the cup forms one vector, V, which you represent as two numbers in brackets: [7, 4]. This vector is the hypotenuse of a right-angled triangle whose two arms are 7 and 4 meters long. The Pythagorean theorem tells you that the vector’s length equals the square root of 65 (which you get by adding 49 + 16). 

Reiji uses minigolf to explain vectors. Each shot represents a new vector, with a specific length and direction.

Now you start hitting your ball toward the cup. Each new shot represents a new vector, expressed as a distance along the x and y axes from the ball’s previous position. Let’s say you have three shots, represented by three vectors: [6, 3], [2, 2] and [-1, -1]. You overshot the hole on your second shot, so you come back to it on your third shot, hence the negative numbers. Add the three xs, then add the three ys, and you get 7 and 4, the coordinates of the cup. If you plotted your shots on a grid, you’d have three end-to-end arrows beginning at 0, 0 and culminating at 7, 4.

Congratulations, you just learned how to add and subtract vectors.

Note how easy this accounting method is. You don’t have to keep track of the direction of each shot, that is, its angle relative to the x axis. Nor do you have to calculate the length of each shot based on the Pythagorean theorem. You just keep track of how far each shot advances along the x and y axes. Vectors can model movement in three dimensions, too, along x, y and z axes. You represent a three-dimensional vector with three numbers in brackets, such as [4, 3, 1]. Alternatively, and usually, you represent vectors with numbers stacked on top of each other, forming vertical columns.

Vectors can do much more than help you keep track of positions. You can use them to model the forces acting upon the ball. Your putter strikes the ball with a certain force in the direction of the cup. Meanwhile, gravity exerts a downward force. So you have two different vectors acting on the ball. If you know the strength, or length, of these two vectors, and their direction relative to each other, you can calculate the length and direction of a new vector representing the total force acting on the ball. If you wanted to be more precise, you’d include vectors representing friction and air resistance.

Minigolf involves a ball moving in real space, the space through which we move. Vectors can also represent more abstract things, which require many dimensions defined with real or complex numbers. In the case of criminal justice in Cleveland, vectors could quantify the poverty of defendants, the mendacity of police, the racism of prosecutors, the exhaustion of defense attorneys, the impatience of judges. These vectors might help you predict the outcome of cases. If you want to predict who will win the Presidential election in November, you can model voters’ positions on crime, race, sexuality, abortion, immigration, inequality and so on. But according to Steven Weinberg, science can’t tell us what our positions should be, and how we should vote.

On January 20, 2017, a variety of vectors propelled me and Robert Hutchinson—the friend who wanted to go to Namibia with me—to Washington, D.C. At one point during that crazy day, Robert and I found ourselves briefly, inadvertently, running with a gang of black-masked protesters who smashed windows with hammers and threw firecrackers at cops. Some carried signs identifying themselves as anarchists. As I ran, I felt exhilaration, admiration, fear, revulsion and an overwhelming sense of unreality. I was in a very complex space. I was also just trying to keep up with Robert, who kept darting ahead of me. Robert, fortunately, is very tall, and he was wearing a pink pussy hat (as was I).

My commitment to nonviolence trumps my loathing of Trump. And so, safely back in Hoboken, I wrote a column deploring the violence of the anarchists. Trump-haters responded to my column by saying, in effect, Shut the fuck up, Horgan, sometimes you need to resist violence with violence. Trump-lovers said I should be fired from my teaching job and arrested for sedition.

Sexism and Physics

On January 21, 2017, the day after Robert and I ran with anarchists protesting Trump’s inauguration, we joined the Women’s March on Washington, which was part of a global protest against sexism. Physics has always been a bit rapey. Erwin Schrodinger preyed on girls, including one he started tutoring when she was 14; not cool, even by the lax standards of his time. An essay celebrating the 100th birthday of Richard Feynman, in 2018, bemoans his “predatory behavior toward women.” In Genius, his biography of Feynman, James Gleick deplores the physicist’s tendency to describe nature as a woman “waiting to be penetrated.”

Feminist philosophers have knocked physicists’ sexualization of nature. In her 1986 essay “The Science Question in Feminism,” Sandra Harding notes that “rape and torture metaphors” are common “in the writings of Francis Bacon and others (e.g., Machiavelli) enthusiastic about the new scientific method.” Nature is depicted as “a woman indifferent to or welcoming rape.” Harding asks, “Why is it not as illuminating and honest to refer to Newton’s laws as ‘Newton’s rape manual’ as it is to call them ‘Newton’s mechanics’?”

Harding took flack for this line, especially in the science wars of the 1990s, when scientists lashed out against critiques from feminist, anti-colonial and queer scholars. How dare these uppity humanities professors impugn the noble men of science! Science isn’t male or European or white; it is universal! Objective! True! And so on. That used to be my attitude, too, to be honest. In 2020, the era of #MeToo and Donald “grab ‘em by the pussy” Trump, Harding’s complaints have more weight.

Amrou Al-Kadhi, a.k.a. Glamrou, says quantum mechanics subverts the whole notion of fixed identities. Right on.

As the #MeToo movement unfolded in 2017, countless women in science blurted out their stories, and it became clear that science is sexist in two related ways: First, male scientists contend—more in private now than previously, but still--that females are innately inferior to males when it comes to mathematics and other hard-science skills. Second, male scientists harass female scientists, especially young ones. If these women falter, sexists say: See? Women just aren’t cut out for science.

“Yes, women in science still have a disadvantage,” Sabine Hossenfelder wrote in 2018. “Women today still face obstacles men don’t encounter and often don’t notice. I see this every day at my front door, in physics, where women are still underrepresented.” One of my male quantum advisors, Chris Search, a physics professor at Stevens, thinks physics has been harmed by its lack of diversity. In a Q&A for Scientific American, I asked Chris if he thought physics would “look different if more non-western, non-male, non-white physicists were involved.” His response:

My belief is that different cultural traditions and less homogeneity of thought (i.e. group think) would have led to more diverse avenues of research within physics and would have enriched the philosophical interpretations by drawing on more non-Western philosophies and systems of belief… Perhaps we would have by now a working theory of quantum gravity?

A theory of quantum gravity is another name for a unified theory. Here’s a thought: Maybe if physics were more diverse, it would be less obsessed with unification. Maybe unification is a product of white, patriarchal thinking, and maybe quantum mechanics is too. Uh oh. Maybe when I study quantum mechanics, I am immersing myself in a paradigm that reinforces my white, male, rapey mindset.

Or maybe not. Emily, unexpectedly, gets me to see quantum mechanics in a more positive light. After I tell her I’ve been mulling over the intersection of quantum mechanics and gender, she sends me a video of Amrou Al-Kadhi, who goes by “Glamrou” and they/them. Born in London to an Iraqi Muslim family, Amrou Al-Kadhi became a writer, drag performer, filmmaker and queer activist.

In the video Glamrou tells an interviewer that learning quantum mechanics helped them embrace their contradictions. If Newtonian physics is “heteronormative,” then quantum mechanics is “non-binary.” “If subatomic particles defy constructs all the time,” Glamrou asks, “why should we believe in fixed constructs of gender?” Good question. Maybe quantum mechanics can subvert oppressive social paradigms instead of reinforcing them.

3Blue1Brown

To get a more intuitive sense of vectors and matrices, I check out 3Blue1Brown, which posts video tutorials on linear algebra, calculus and other mathematical tools. The guy behind 3Blue1Brown is Grant Sanderson, who graduated from Stanford with a math degree in 2015. 3Blue1Brown refers to the unusual pigmentation of Sanderson’s right eye, which is three quarters blue and one quarter brown.

Sanderson is a superb mansplainer. He has the perfect temperament and voice for a teacher; he sounds calm and excited at the same time. He loves math, its reasonableness, interconnectedness, beauty; he revels in revealing the logic underpinning familiar mathematical tools. He wraps up riffs by saying, Isn’t that cool? or words to that effect. YouTube commenters, who are notoriously snarky, love 3Blue1Brown; their comments are effusive, gushy. They thank Sanderson and ask how they can donate to him.

In 3Blue’s videos, a cute, animated 𝝅 with googly eyes guides you through undulating cartesian landscapes as Sanderson explains the math corresponding to what you’re seeing. I watch the first in a series of videos on linear algebra, and within minutes I see why linear algebra is called linear algebra (as I explain above). 3Blue shows how the grid lines for a matrix remain straight and parallel as you transform them. As I watch the video, I’m nodding along, thinking, Yeah, I get it, this makes sense. A tougher, more sensible part of me knows I’m not really getting it. I need to solve thousands of problems under the critical eye of a teacher before I “get” linear algebra. I nonetheless find 3Blue’s videos entrancing. They are works of art, ends in themselves.

Now I’m watching a 3Blue video on eigenvectors and eigenvalues. At the beginning of the video, 3Blue says many students find eigenvectors and eigenvalues un-intuitive, hard to understand. “Questions like, ‘Why are we doing this?’ and ‘What does this actually mean?’ are too often left just floating away in an unanswered sea of computations.” Yes, that’s how I feel. That Germanic prefix “eigen” intimidates me; it epitomizes technical hardness. But eigen just means “own,” “self” or “same.”

If you multiply a vector, V, by a special kind of matrix, you get a new vector, call it Vn, that is n times as long as the original and points in the same direction. Vn is the eigenvector of V, and n is its eigenvalue. That’s what I glean from 3Blue and from Manga. Eigenvectors, I learn elsewhere, are analogous to the different ways in which a string of a given length, under a given tension, vibrates when plucked with a given force.

3Blue and Manga don’t mention quantum mechanics, but my guess is that quantum states can be represented by a vector consisting of many eigenvectors in superposition, like the notes that form a chord. Eigenvectors have something to do with the lumpiness highlighted in Quantum Physics for Babies. The different energy levels of an electron are represented by different eigenvectors.

I can’t understand quantum mechanics, and I probably never will. And yet I understand a little more every day, in my half-assed, metaphorical way.

Superdeterminism

If you’re talking about justice and morality, about the difference between what is and what should be, you must talk about free will. I define free will as our capacity to perceive and choose different trajectories. Example: I could quit studying linear algebra right now and volunteer to work for Joe Biden, the likely Democratic candidate for President.

Free will is always constrained; no one has absolute freedom or infinite choices. Free will is a variable that can shrink or expand in response to a range of factors, which can be physiological, psychological, political, moral, spiritual. I have more choices now, more free will, than I had when I was an infant and fewer than I will have when I sink into senility. I have more freedom, more options, than the average queer, poor woman of color in our unjust society.

None of this, to my mind, should be controversial. But some physicists reject free will because they are committed to physical determinism, the idea that physical causes underpin everything that happens. They say quantum mechanics, although superficially probabilistic, is deterministic, if you look at it the right way. Determinism is an implication of what Susskind calls the minus first law, which says that information is always conserved, and of the principle of least action. Then there is the quantum interpretation called superdeterminism, which is determinism espoused with first-thumping adamance. Forget free will, choices, values and all that nonsense; reality consists of bits of matter pushed and pulled this way and that by physical forces.

Superdeterminism was proposed by John Bell. He is the physicist renowned for a 1964 theorem, now named after him, that dramatically exposes quantum nonlocality, the apparent ability of a measurement of one particle to affect the measurement of a distant, “entangled” particle described by the same wave function. Bell said in 1985 that the puzzle of nonlocality vanishes if you assume that “the world is superdeterministic, with not just inanimate nature running on behind-the-scenes clockwork, but with our behavior, including our belief that we are free to choose to do one experiment rather than another, absolutely predetermined.” 

Bell, I’m guessing, proposed superdeterminism to highlight the wackiness of quantum mechanics, as a reductio ad absurdum. And yet prominent physicists, including my friend Sabine Hossenfelder, embrace superdeterminism. She says the results of quantum experiments only seem random; “hidden variables” determine exactly how the physicist will carry out her experiment and exactly what she will observe. All randomness, and all choices, are illusory.

Conversely, in their 2009 paper “The Strong Free Will Theorem,” mathematicians John Conway and Simon Kochen “prove” that we have free will because particles have free will; particles choose which way to go in experiments. I suspect that Conway and Kochen are being ironic, just as Feynman was when he said that photons “decide” whether to bounce off glass. That is, Conway and Kochen don’t actually believe in free will; they are trying to cast doubt on it with their clever reductio ad absurdum. Can physics and mathematics be ironic? Of course they can, although the irony is often unintended.

Defenders of free will often equate it with unpredictability. To my mind, unpredictability is irrelevant, and so are all the physics-based arguments for and against free will. I believe in free will because I constantly make what I consider to be meaningful choices, meaningful to me if no one else. My choices are based not on physical forces but on my fears, desires and values, like my intimations of death, my love of my children, my antipathy to violence. My choices might be quite predictable to those who know me, like Emily or my kids.

My own life provides abundant evidence for free will. I also believe in free will for pragmatic, political reasons. To my mind, determinism--denial of our capacity to make choices, and especially moral choices, which promote the wellbeing of others--is harmful. The last thing we need at this dark time in our history is an ideology that feeds our fatalism and despair.

What I love about the Serial show on criminal justice in Cleveland is that it resists fatalism and despair. The narrator, Sarah Koenig, is a compassionate idealist; she clearly believes that there is such a thing as truth, as hard as it is to discern; and there is such a thing as justice, as hard as it is to achieve. And, I would add, there is such a thing as free will, as hard as it is to find in the equations of physics. Fuck superdeterminism, and fuck fate.

The Traveling Salesman’s Problems

When I was pre-pubescent lad, I was a rockhound. I loved identifying minerals with the Mohs hardness test, named after the mineralogist who invented it. You take a known specimen, like quartz, and scratch an unknown specimen with it. If the quartz scratches the mystery specimen, you know it’s softer than quartz; it could be calcite or pyrite. If the specimen scratches the quartz, it might be beryl or corundum, which are harder than quartz.

Could there be a Mohs-like method for objectively ranking the hardness of all the problems we face? Subjective assessments of hardness are no help, because they vary with each person’s experience and aptitude. You’re a whiz at multiplying matrices; I’m better at riffing on Plato’s parable of the cave. What I have in mind is an objective method for quantifying the complexity of tasks. How does multiplying two 4 x 4 matrices compare, say, to talking to my daughter about Black Lives Matter or to my girlfriend about #MeToo without irritating them?

Researchers in the field of computational complexity rank problems by estimating how long it would take a hypothetical computer to find a solution. One famously hard problem involves a traveling salesman seeking the shortest route between cities. The problem’s hardness balloons dramatically with the number of cities. If the salesman has to visit 15 cities, he has 87 billion possible routes to consider. Yeesh. When the number of cities rises into the thousands, the world’s fastest computer would take virtually forever to find the shortest route.

Coming up with a time-saving itinerary is easy compared to other problems that the traveling salesman might face. Examples: If he is lonely, should he approach a woman in the hotel bar? If he feels bad about cheating on his wife, what should he tell himself to relieve his guilt? And how does he justify selling shoddy, overpriced kitchen appliances to people who can ill afford them?

What makes these problems especially hard is their moral dimension. Like most of us, the traveling salesman wants to believe he is a good person, but what does that even mean? The play Death of a Salesman explores the moral dilemmas of a traveling salesman. But like most works of art, the play does not solve moral riddles; it rubs our faces in them.

Black Lives Matter in Hoboken

I’m in Hoboken, fussing over a problem in The Manga Guide to Linear Algebra, when the world barges into my private Hilbert space. I hear a clamor, people shouting and chanting. When I look out my window, I discover to my astonishment that Sinatra Drive, which separates my building from the Hudson River, is thronged with people. They are marching slowly south, wearing shirts and holding signs that say, “Black Lives Matter.” I feel like calling my daughter Skye to tell her: There’s a Black Lives Matter protest right outside my window, and I’m going to join it!

I take the elevator down and enter the stream of people coursing down Sinatra Drive. All colors, ages, genders are here. I feel exhilarated. Some people hold signs that say, “I understand that I can’t understand. But I stand.” I’m puzzled, then I get it. The people holding these signs are white, like me, and they’re acknowledging that they can’t know what it’s like to be black in America, just as Skye told me. But we can still stand with those struggling for justice. We can offer our sympathy and support. The same is true of straight men who care about the oppression of women and non-binary people.

When I return to my apartment, I keep dwelling on “I understand that I can’t understand. But I stand.” This dictum applies, I realize, to the mind-body problem, the riddle of what we are, can be and should be. The mind-body encompasses the hard problems of consciousness, free will, morality, social justice. The mind-body problem is the hardest problem, the problem that scratches all other problems.

Quantum mechanics, the hard problem at the core of physics, makes the mind-body problem even harder. But I see a way in which quantum mechanics can further the cause of social justice. Quantum mechanics is democratizing, in the sense that we’re all equally ignorant of its meaning. Sure, some of us are more ignorant than others. But Glamrou, the Muslim Iraqi cross-dresser, is as entitled to see their reflection in quantum mechanics, and to find solace in it, as Schrodinger or Feynman. Quantum mechanics subverts hierarchy. A quantum government has no president, a quantum religion no pope. The theory fosters doubt, humility, open-mindedness. It encourages us to care for each other, and especially for victims of oppression. Yes, this is wishful thinking, but all progress begins with wishful thinking.

In other words, I understand that I can’t understand. But I stand.

Black Lives Matter protest in Hoboken, New Jersey, just down the street from where I live. Photo Hoboken Girl.

Notes

  1. Nothing to see here, folks, move on.

 
 

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