On translation and quantum mechanics.

We have many ways to express ideas. In this essay, I’ll attempt to convey my thoughts with English words. Although this is the only metaphoric language that I know well, humans employ several thousand others – among these there may be several that could convey my ideas more clearly.

The distinct features of a language can change the way ideas feel.

Perry Link writes that,

In teaching Chinese-language courses to American students, which I have done about thirty times, perhaps the most anguishing question I get is “Professor Link, what is the Chinese word for ______?” I am always tempted to say the question makes no sense.

Anyone who knows two languages well knows that it is rare for words to match up perfectly, and for languages as far apart as Chinese and English, in which even grammatical categories are conceived differently, strict equivalence is not possible.

Book is not shu, because shu, like all Chinese nouns, is conceived as an abstraction, more like “bookness,” and to say “a book” you have to say, “one volume of bookness.” Moreover shu, but not book, can mean “writing,” “letter,” or “calligraphy.” On the other hand, you can “book a room” in English; you can’t shu one in Chinese.

There is no perfect way to translate an idea from Chinese words into English words, nor the other way around. In Nineteen Ways of Looking at Wang Wei, Eliot Weinberger reviews several English reconstructions of a short, seductively simple Chinese poem. The English variants feel very different from one another – each accentuates certain virtues of the original; by necessity, each also neglects others.

Visual appearances can’t be perfectly described with any metaphoric language. I could write about a photograph, and maybe my impression would be interesting – the boy’s arms are turned outward, such that his hands would convey a gesture of welcome if not for his grenade, grimace, and fingers curled into a claw – but you’d probably rather see the picture.

Here’s Diane Arbus’s “Child with a toy hand grenade in Central Park, N.Y.C.”

This isn’t to say that an image can’t be translated. The version posted above is a translation. The original image, created by light striking a photosensitive film, has been translated into a matrix of numbers. Your computer reads these numbers and translates them back into an image. If you enlarge this translation, your eyes will detect its numerical pixelation.

For this image, a matrix of numbers is a more useful translation than a paragraph of my words would be.

From a tutorial on computer vision prepared by Amy Jin & Vivian Chiang at Stanford.

Different forms of communication – words, pictures, numbers, gestures, sounds – are better suited to convey different ideas. The easiest way to teach organic chemistry is through the use of pictures – simple diagrams often suffice. But I sometimes worked with students who weren’t very visual learners, and then I’d have to think of words or mathematical descriptions that could represent the same ideas.

Science magazine sponsors an annual contest called “Dance Your Ph.D.,” and although it might sound silly – can someone understand your research after watching human bodies move? – the contest evokes an important idea about translation. There are many ways to convey any idea. Research journals now incorporate a combination of words, equations, images, and video.

Plant-soil feedbacks after severe tornado damage: Dance Your PhD 2014 from atinytornado on Vimeo.

A kinetic, three-dimensional dance might be better than words to explain a particular research topic. When I talked about my graduate research in membrane trafficking, I always gesticulated profusely.

My spouse coached our local high school’s Science Olympiad team, preparing students for the “Write It Do It” contest. In this competition, teams of two students collaborate – one student looks at an object and describes it, the other student reads that description and attempts to recreate the original object. Crucially, the rules prohibit students from incorporating diagrams into their instructions. The mandate to use words – and only words – makes “Write It Do It” devilishly tricky.

I love words, but they’re not the tools best suited for all ideas.

If you’re curious about quantum mechanics, Beyond Weird by Philip Ball is a nice book. Ball describes a wide variety of scientific principles in a very precise way – Ball’s language is more nuanced and exact than most researchers’. Feynman would talk about what photons want, and when I worked in a laboratory that studied the electronic structure of laser-aligned gas clouds, buckyballs, and DNA, we’d sometimes anthropomorphize the behavior of electrons to get our thoughts across. Ball broaches no such sloppiness.

Unfortunately, Ball combines linguistic exactitude with a dismissal of other ways of conveying information. Ball claims that any scientific idea that doesn’t translate well into English is an insufficient description of the world:

When physicists … exhort us to not get hung up on all-too-human words, we have a right to resist. Language is the only vehicle we have for constructing and conveying meaning: for talking about our universe. Relationships between numbers are no substitute. Science deserves more than that.

By way of example, Ball gives a translation of Hugh Everette’s “many worlds” theory, points out the flaws in his own translated version, and then argues that these flaws undermine the theory.

To be fair, I think the “many worlds” theory is no good. This is the belief that each “observation” – which means any event that links the states of various components of a system such that each component will evolve with restrictions on its future behavior (e.g. if you shine a light on a small object, photons will either pass by or hit it, which restricts where the object may be later) – causes a bifurcation of our universe. A world would exist where a photon gets absorbed by an atom; another world exists where the atom is localized slightly to the side and the photon speeds blithely by.

The benefit of the “many worlds” interpretation is that physics can be seen as deterministic, not random. Events only seem random because the consciousness that our present mind evolves into can inhabit only one of the many future worlds.

The drawback of the “many worlds” interpretation is that it presupposes granularity in our universe – physical space would have to be pixelated like computer images. Otherwise every interaction between two air molecules would presage the creation of infinite worlds.

If our world was granular, every interaction between two air molecules would still summon an absurd quantity of independent worlds, but mere absurdity doesn’t invalidate a theory. There’s no reason why our universe should be structured in a way that’s easy for human brains to comprehend. Without granularity, though, the “many worlds” theory is impossible, and we have no reason to think that granularity is a reasonable assumption.

It’s more parsimonious to assume that sometimes random things happen. To believe that our God, although He doesn’t exist, rolls marbles.

(This is a bad joke, wrought by my own persnickety exactitude with words. Stephen Hawking said, “God does play dice with the universe. All the evidence points to him being an inveterate gambler, who throws the dice on every possible equation.” But dice are granular. With a D20, you can’t roll pi. So the only way for God to avoid inadvertently pixelating His creation is to use infinite-sided dice, i.e. marbles.)

Image of dice by Diacritica on Wikimedia images.

Some physicists have argued that, although our words clearly fail when we attempt to describe the innermost workings of the universe, numbers should suffice. Neil deGrasse Tyson said, “Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.”

Indeed, equations often seem to provide accurate descriptions of the way the world works. But something’s wrong with our numbers. Even mathematics falls short when we try to converse with the cosmos.

Our numbers are granular. The universe doesn’t seem to be.

Irrational numbers didn’t bother me much when I was first studying mathematics. Irrational numbers are things like the square root of two, which can only be expressed in decimal notation by using an infinite patternless series of digits. Our numbers can’t even express the square root of two!

Similarly, our numbers can’t quite express the electronic structure of oxygen. We can solve “two body problems,” but we typically can’t give a solution for “three body problems” – we have to rely on approximations when we analyze any circumstance in which there are three or more objects, like several planets orbiting a star, or several electrons surrounding a nucleus.

Oxygen is. These molecules exist. They move through our world and interact with their surroundings. They behave precisely. But we can’t express their precise behavior with numbers. The problem isn’t due to any technical shortcoming in our computers – it’s that, if our universe isn’t granular, each oxygen behaves with infinite precision, and our numbers can only be used to express a finite degree of detail.

Using numbers, we can provide a very good translation, but never an exact replica. So what hope do our words have?

The idea that we should be able to express all the workings of our universe in English – or even with numbers – reminds me of that old quote: “If English was good enough for Jesus, it ought to be good enough for the children of Texas.” We humans exist through an unlikely quirk, a strange series of events. And that’s wonderful! You can feel pleasure. You can walk out into the sunshine. Isn’t it marvelous? Evolution could have produced self-replicating objects that were just as successful as us without those objects ever feeling anything. Rapacious hunger beasts could have been sufficient. (Indeed, that’s how many of us act at times.)

But you can feel joy, and love, and happiness. Capitalize on that!

And, yes, it’s thrilling to delve into the secrets of our universe. But there’s no a priori reason to expect that these secrets should be expressible in the languages we’ve invented.

On suboptimal optimization.

I’ve been helping a friend learn the math behind optimization so that she can pass a graduation-requirement course in linear algebra.

Optimization is a wonderful mathematical tool. Biochemists love it – progression toward an energy minimum directs protein folding, among other physical phenomena. Economists love it – whenever you’re trying to make money, you’re solving for a constrained maximum. Philosophers love it – how can we provide the most happiness for a population? Computer scientists love it – self-taught translation algorithms use this same methodology (I still believe that you could mostly replace Ludwig Wittgenstein’s Philosophical Investigations with this New York Times Magazine article on machine learning and a primer on principal component analysis).

But, even though optimization problems are useful, the math behind them can be tricky. I’m skeptical that this mathematical technique is essential for everyone who wants a B.A. to grasp – my friend, for example, is a wonderful preschool teacher who hopes to finally finish a degree in child psychology. She would have graduated two years ago except that she’s failed this math class three times.

I could understand if the university wanted her to take statistics, as that would help her understand psychology research papers … and the science underlying contemporary political debates … and value-added models for education … and more. A basic understanding of statistics might make people better citizens.

Whereas … linear algebra? This is a beautiful but counterintuitive field of mathematics. If you’re interested in certain subjects – if you want to become a physicist, for example – you really should learn this math. A deep understanding of linear algebra can enliven your study of quantum mechanics.

The summary of quantum mechanics: animation by Templaton.

Then again, Werner Heisenberg, who was a brilliant physicist, had a limited grasp on linear algebra. He made huge contributions to our understanding of quantum mechanics, but his lack of mathematical expertise occasionally held him back. He never quite understood the implications of the Heisenberg Uncertainty Principle, and he failed to provide Adolph Hitler with an atomic bomb.

In retrospect, maybe it’s good that Heisenberg didn’t know more linear algebra.

While I doubt that Heisenberg would have made a great preschool teacher, I don’t think that deficits in linear algebra were deterring him from that profession. After each evening that I spend working with my friend, I do feel that she understands matrices a little better … but her ability to nurture children isn’t improving.

And yet. Somebody in an office decided that all university students here need to pass this class. I don’t think this rule optimizes the educational outcomes for their students, but perhaps they are maximizing something else, like the registration fees that can be extracted.

Optimization is a wonderful mathematical tool, but it’s easy to misuse. Numbers will always do what they’re supposed to, but each such problem begins with a choice. What exactly do you hope to optimize?

Choose the wrong thing and you’ll make the world worse.

Most automobile companies are researching self-driving cars. They’re the way of the future! In a previous essay, I included links to studies showing that unremarkable-looking graffiti could confound self-driving cars … but the issue I want to discuss today is both more mundane and more perfidious.

After all, using graffiti to make a self-driving car interpret a stop sign as “Speed Limit 45” is a design flaw. A car that accelerates instead of braking in that situation is not operating as intended.

But passenger-less self-driving cars that roam the city all day, intentionally creating as many traffic jams as possible? That’s a feature. That’s what self-driving cars are designed to do.

A machine designed to create traffic jams?

Despite my wariness about automation and algorithms run amok, I hadn’t considered this problem until I read Adam Millard-Ball’s recent research paper, “The Autonomous Vehicle Parking Problem.” Millard-Ball begins with a simple assumption: what if a self-driving car is designed to maximize utility for its owner?

This assumption seems reasonable. After all, the AI piloting a self-driving car must include an explicit response to the trolley problem. Should the car intentionally crash and kill its passenger in order to save the lives of a group of pedestrians? This ethical quandary is notoriously tricky to answer … but a computer scientist designing a self-driving car will probably answer, “no.”

Otherwise, the manufacturers won’t sell cars. Would you ride in a vehicle that was programmed to sacrifice you?

Luckily, the AI will not have to make that sort of life and death decision often. But here’s a question that will arise daily: if you commute in a self-driving car, what should the car do while you’re working?

If the car was designed to maximize public utility, perhaps it would spend those hours serving as a low-cost taxi. If demand for transportation happened to be lower than the quantity of available, unoccupied self-driving cars, it might use its elaborate array of sensors to squeeze into as small a space as possible inside a parking garage.

But what if the car is designed to benefit its owner?

Perhaps the owner would still want for the car to work as a taxi, just as an extra source of income. But some people – especially the people wealthy enough to afford to purchase the first wave of self-driving cars – don’t like the idea of strangers mucking around in their vehicles. Some self-driving cars would spend those hours unoccupied.

But they won’t park. In most cities, parking costs between $2 and $10 per hour, depending on whether it’s street or garage parking, whether you purchase a long-term contract, etc.

The cost to just keep driving is generally going to be lower than $2 per hour. Worse, this cost is a function of the car’s speed. If the car is idling at a dead stop, it will use approximately 0.1 gallon per hour, costing 25 cents per hour at today’s prices. If the car is traveling at 30 mph without breaks, it will use approximately 1 gallon per hour, costing $2.50 per hour.

To save money, the car wants to stay on the road … but it wants for traffic to be as close to a standstill as possible.

Luckily for the car, this is an easy optimization problem. It can consult its onboard GPS to find nearby areas where traffic is slow, then drive over there. As more and more self-driving cars converge on the same jammed streets, they’ll slow traffic more and more, allowing them to consume the workday with as little motion as possible.

Pity the person sitting behind the wheel of an occupied car on those streets. All the self-driving cars will be having a great time stuck in that traffic jam: we’re saving money!, they get to think. Meanwhile the human is stuck swearing at empty shells, cursing a bevy of computer programmers who made their choices months or years ago.

And all those idling engines exhale carbon dioxide. But it doesn’t cost money to pollute, because one political party’s worth of politicians willfully ignore the fact that capitalism, by philosophical design, requires we set prices for scarce resources … like clean air, or habitable planets.

On Vaughan & Staples’s ‘Saga’ and parenting metaphors.

I’m reasonably well-versed with small stuff. I’ve studied quantum mechanics, spent two years researching electronic structure, that sort of thing. I imagine that I’m about as comfortable as I’ll ever be with the incomprehensible probabilistic weirdness that underlies reality.

But although I helped teach introductory calculus-based physics, I’ve never learned about big things. I took no geometry in college, and most big physics, I assume, is about transferring equations into spaces that aren’t flat. The basic principle seems straightforward – you substitute variables, like if you’re trying to estimate prices in another country and keep plugging in the exchange rate – but I’ve never sat down and worked through the equations myself.

There’s only so much physics you can understand without chugging through the math. Our numbers don’t quite describe the world – they can’t exactly express quantities like pi, or the solutions to most three-body problems – but they do a better job than our words.

Still, some excellent pop-science books on gravity have been published recently. My favorite of these was On Gravity by A. Zee – it’s quite short, and has everything I assume you’d want from a book like this: bad humor, lucid prose, excellent pacing. Zee has clearly had a lot of practice teaching this material to beginners, and his expertise shines through.

Near the end of the book, Zee introduces black holes – gravity at its weirdest. Gravity becomes stronger as the distance between objects decreases – it follows an “inverse square law.”

If our moon was closer to Earth, the tides would be more extreme. To give yourself a sense of the behavior of inverse square laws, you can play with some magnets. When two magnets are far apart, it seems as though neither cares about the existence of the other, but slide them together and suddenly the force gets so strong that they’ll leap through the air to clank together.

But because each magnet takes up space, there’s a limit to how close they can get. Once you hear them clank, the attractive magnetic force is being opposed by a repulsive electrostatic force – this same repulsion gives us the illusion that our world is composed of solid objects and keeps you from falling through your chair.

Gravity is much weaker than magnetism, though. A bar magnet can have a strong magnetic field but will have an imperceptible amount of gravity. It’s too small.

A big object like our sun is different. Gravity pulls everything together toward the center. At the same time, a constant flurry of nuclear explosions pushes everything apart. These forces are balanced, so our sun has a constant size, pouring life-enabling radiation into the great void of space (of which our planet intercepts a teensy tiny bit).

But if a big object had much more mass than our sun, it might tug itself together so ardently that not even nuclear explosions could counterbalance its collapse. It would become … well, nobody knows. The ultra-dense soup of mass at the center of a black hole might be stranger than we’ve guessed. All we know for certain is that there is a boundary line inside of which the force of gravity becomes so strong that not even light could possibly escape.

Satellites work because they fall toward Earth with the same curvature as the ground below – if they were going faster, they’d spiral outward and away, and if they were going slower, they’d spiral inward and crash. The “event horizon” of a black hole is where gravity becomes so strong that even light will be tugged so hard that it’ll spiral inward. So there’s almost certainly nothing there, right at the “edge” of the black hole as we perceive it. Just the point of no return.

If your friends encounter a black hole, they’re gone. Not even Morse-code messages could escape.

(Sure, sure, there’s “Hawking radiation,” quantum weirdness that causes a black hole to shrink, but this is caused by new blips in the fabric of reality and so can’t carry information away.)

The plot of Saga, by Brian K. Vaughan and Fiona Staples, revolves around a Romeo & Juliet-esque romance in the middle of intergalactic war, but most of the comic is about parenting. K read the entire series in two days, bawling several times, and then ran from the bedroom frantic to demand the next volume (unfortunately for her, Vaughan & Staples haven’t yet finished the series).

Saga is masterfully well-done, and there are many lovely metaphors for a child’s development.

For instance, the loss of a child’s beloved caretaker – babysitters, daycare workers, and teachers do great quantities of oft under-appreciated work. In Saga, the child and her first babysitter are linked through the spirit, and when the caretaker moves on, the child feels physical pain from the separation.

A hairless beast named “Lying Cat” can understand human language and denounces every untruth spoken in its present – allowing for a lovely corrective to a child’s perception that she is to blame for the traumas inflicted upon her.

Perhaps my favorite metaphor in Saga depicts the risk of falling into a black hole. Like all intergalactic travelers, they have to be careful – in Saga, a black hole is called a “timesuck” and it’s depicted as a developing baby.

My favorite scene in the film Interstellar depicts the nightmarish weirdness of relativistic time. A massive planet seems perfectly habitable, but its huge gravitational field meant that the years’ worth of “Everything’s okay!” signals had all been sent within minutes of a scout’s arrival. The planet was actually so dangerous that the scout couldn’t survive a full day, but decades would have passed on Earth before anyone understood the risk.

Gravity eats time.

So do babies. A child is born and the new parents might disappear from the world. They used to volunteer, socialize, have interests and hobbies … then, nothing.

They fell into the timesuck.

On uncertainty (with cartoon ending).

The whole cartoon is at the end. — See this monstrosity, in its entirety, at the end of this essay.

Reading about the uncertainty principle in popular literature almost always sets my teeth on edge.

I assume most people have a few qualms like that, things they often see done incorrectly that infuriate them. After a few pointed interactions with our thesis advisor, a friend of mine started going berserk whenever he saw “it’s” and “its” misused on signs. My middle school algebra teacher fumed whenever he saw store prices marked “.25% off!” when they meant you’d pay three quarters of the standard price, not 99.75%. A violinist friend with perfect pitch called me (much too early) on a Sunday morning to complain that the birds on her windowsill were out of tune… how could she sleep when they couldn’t hit an F#??

“Ha,” I say. “That’s silly… they should just let it go.” But then I start frowning and sputtering when I read about the uncertainty principle. Anytime somebody writes a line to the effect of, we’ve learned from quantum mechanics that measurement obscures the world, so we will always be uncertain what reality might have been had we not measured it.

My ire is risible in part because the idea isn’t so bad. It even holds in some fields. Like social psychology, I’d say. If a research group identifies a peculiarity of the human mind and then widely publicizes their findings, that particularity might go away. There was a study published shortly before I got my first driver’s license concluding that the rightmost lanes of toll booths were almost always fastest. Now that’s no longer true. Humans can correct their mistakes, but first they have to realize they’re mistaken.

That’s not the uncertainty principle, though.

Capture And, silly me, I’d always thought that this misconception was due to liberal arts professors wanting to cite some fancy-sounding physics they didn’t understand. I didn’t realize the original misconception was due to Heisenberg himself. In The Physical Principles of Quantum Theory. he wrote (and please note that this is not the correct explanation for the uncertainty principle):

Thus suppose that the velocity of a free electron is precisely known, while the position is completely unknown. Then the principle states that every subsequent observation of the position will alter the momentum by an unknown and undeterminable amount such that after carrying out the experiment our knowledge of the electronic motion is restricted by the uncertainty relation. This may be expressed in concise and general terms by saying that every experiment destroys some of the knowledge of the system which was obtained by previous experiments.

Most of this isn’t so bad, despite not being the uncertainty principle. The next line is worse, if what you’re hoping for is an accurate translation of quantum mechanics into English.

This formulation makes it clear that the uncertainty relation does not refer to the past; if the velocity of the electron is at first known and the position then exactly measured, the position for times previous to the measurement may be calculated. Then for these past times ∆p∆q [“p” stands for momentum and “q” stands for position in most mathematical expressions of quantum mechanics] is smaller than the usual limiting value, but this knowledge of the past is of a purely speculative character, since it can never (because of the unknown change in momentum caused by the position measurement) be used as an initial condition in any calculation of the future progress of the electron and thus cannot be subjected to experimental verification.

That’s not correct. Because the uncertainty principle is not about measurement, it’s about the world and what states the world itself can possibly adopt. We can’t trace the position & momentum both backward through time to know where & how fast an electron was earlier because the interactions that define a measurement create discrete properties, i.e. they are not revealing crisp properties that pre-existed the measurement.

Heisenberg was a brilliant man, but he made two major mistakes (that I know of, at least. Maybe he had his own running tally of things he wished he’d done differently). One mistake may have saved us all, as was depicted beautifully in Michael Frayn’s Copenhagen (also… they made a film of this? I was lucky enough to see the play in person, but I’ll have to watch it again!) — who knows what would’ve happened if Germany had the bomb?

Heisenberg’s other big mistake was his word-based interpretation of the uncertainty principle he discovered.

His misconception is understandable, though. It’s very hard to translate from mathematics into words. I’ll try my best with this essay, but I might botch it too — it’s going to be extra-hard for me because my math is so rusty. I studied quantum mechanics from 2003 to 2007 but since then haven’t had professional reasons to work through any of the equations. Eight years of lassitude is a long time, long enough to forget a lot, especially because my mathematical grounding was never very good. I skipped several prerequisite math courses because I had good intuition for numbers, but this meant that when my study groups solved problem sets together we often divided the labor such that I’d write down the correct answer then they’d work backwards from it and teach me why it was correct.

I solved equations Robert Johnson crossroads style, except I had a Texas Instruments graphing calculator instead of a guitar.

The other major impediment Heisenberg was up against is that the uncertainty principle is most intuitive when expressed in matrix mechanics… and Heisenberg had no formal training in linear algebra. I hadn’t realized this until I read Jagdish Mehra’s The Formulation of Matrix Mechanics and Its Modifications from his Historical Development of Quantum Theory. A charming book, citing many of the letters the researchers sent to one another, providing mini-biographies of everyone who contributed to the theory. The chapter describing Heisenberg’s rush to learn matrices in order to collaborate with Max Born and Pascual Jordan before the former left for a lecture series in the United States has a surprising amount of action for a history book about mathematics… but the outcome seems to be that Heisenberg’s rushed autodidacticism left him with some misconceptions.

Which is too bad. The key idea was Heisenberg’s, the idea that non-commuting variables might underlie quantum behavior.

Commuting? I should probably explain that, at least briefly. My algebra teacher, the same one who turned apoplectic when he saw miswritten grocery store discount signs, taught the subject like it was gym class (which I mean as a compliment, despite hating gym class). Each operation was its own sport with a set of rules. Multiplication, for instance, had rules that let you commute, and distribute, and associate. When you commute, you get to shuffle your players around. 7 • 5 will give you the same answer as 5 • 7.

But just because kicks to the head are legal in MMA doesn’t mean you can do ’em in soccer. You’re allowed to commute when you’re playing multiplication, but you can’t do it in quantum mechanics. You can’t commute matrices either, which was why Born realized that they might be the best way to express quantum phenomena algebraically. If you have a matrix A and another matrix B, then A • B will often not be the same as B • A.

That difference underlies the uncertainty principle.

So, here’s the part of the essay wherein I will try my very best to make the math both comprehensible and accurate. But I might fail at one or the other or both… if so, my apologies!

A matrix is an array of numbers that represents an operation. I think the easiest way to understand matrices is to start by imagining operators that work in two dimensions.

Just like surgeons all dressed up in their scrubs and carrying a gleaming scalpel and peering down the corridors searching for a next victim, every operator needs something to operate on. In the case of surgeons, it’s moneyed sick people. In the case of matrices, it’s “vectors.”

As a first approximation, you can imagine vectors are just coordinate pairs. Dots on a graph. Typically the term “vector” implies something with a starting point, a direction, and a length… but it’s not a big deal to imagine a whole bunch of vectors that all start from the origin, so then all you need to know is the point at which the tip of an arrow might end.

It’ll be easiest to show you some operations if we have a bunch of vectors. So here’s a list of them, always with the x coordinate written above the y coordinate.

3 4 5 2 6 1 7 3 5

0 , 0 , 0 , 1 , 1 , 2 , 2 , 5 , 5

That set of points makes a crude smiley face.

And we can operate on that set points with a matrix in order to change the image in a predictable way. I’ve always thought the way the math works here is cute… you have to imagine a vector leaping out of the water like a dolphin or killer whale and then splashing down horizontally onto the matrix. Then the vector sinks down through the rows.

It won’t be as fun when I depict it statically, but the math works like this:

Does it make sense why I imagine the vector, the (x,y) thing, flopping over sideways?

The simplest matrix is something called an “identity” matrix. It looks like this:

When we multiply a vector by the identity matrix, it isn’t changed. The zeros mean the y term of our initial vector won’t affect the x term of our result, and the x term of our initial vector won’t affect the y term of our result. Here:

And there are a couple other simple matrices we might consider (you’ll only need to learn a little more before I get back to that “matrices don’t commute” idea).

If we want to make our smiling face twice as big, we can use this operator:

2 0

0 2

Hopefully that matrix makes a little bit of sense. The x and y terms still do not affect each other, which is why we have the zeros on the upward diagonal, and every coordinate must become twice as large to scoot everything farther from the origin, making the entire picture bigger.

We could instead make a mirror image of our picture by reflecting across the y axis:

-1 0

0 1

Or rotate our picture 90º counterclockwise:

0 -1

1 0

The rotation matrix has those terms because the previous Y axis spins down to align with the negative X axis, and the X axis rotates up to become the positive Y axis.

And those last two operators, mirror reflection and rotation, will let us see why the commutative property does not hold in linear algebra. Why A • B is not necessarily equal to B • A if both A & B are matrices.

Here are some nifty pictures showing what happens when we first reflect our smile then rotate, versus first rotating then reflecting. If the matrices did commute, if A • B = B • A, the outcome of the pair of operations would be the same no matter what order they were applied in. And they aren’t! The top row of the image below shows reflection then rotation; the bottom row shows rotating our smile then reflecting it.

And that, in essence, is where the uncertainty principle comes from. Although there is one more mathematical concept that I should tell you about, the other rationale for using matrices to understand quantum mechanics in the first place.

You can write a matrix that would represent any operation or any set of forces. One important class of matrices are those that use the positions of each relevant object, like the locations of each electron around a nucleus, in order to calculate the total energy of a system. The electrons have kinetic energy based on their momentum (the derivative of their position with respect to time) and potential energy related to their position itself, due to interaction with the protons in the nucleus and, if there are multiple electrons, repulsive forces between each other…

(I assume you’ve heard the term “two-body problem” before, used by couples who are trying to find a pair of jobs in the same city so they can move there together. It’s a big issue in science and medicine, double matching for residencies, internships, post-docs, etc. Well, it turns out that nobody thinks it’s funny to make a math joke out of this and say, “At least two-body problems are solvable. Three-body problems have to be approximated numerically.”)

…but once you have a wavefunction (which is basically just a fancy vector, now with a stack of functions instead of a stack of numbers), you can imagine acting upon it with any matrix you want. Any measurement you make, for instance, can be represented by a matrix. And the cute thing about quantum mechanics, the thing that makes it quantized, is that only a discrete set of answers can come out of most measurements. This is because a measurement causes the system to adopt an eigenfunction of the matrix representing that measurement.

An eigenfunction is a vector that still looks the same after it’s been operated upon by a particular matrix (from the German word “eigen,” which means something like “own” or “self”). If we consider the operator for reflection that I jotted out above, you can see that a vector pointing straight up will still resemble itself after it’s been acted upon.

And a neat property of quantum mechanics is that every operator has a set of eigenfunctions that spans whatever space you’re working with. For instance, the X & Y axes together span all of two-dimensional space… but so do any pair of non-parallel lines. You could pick any pair of lines that cross and use them as a basis set to describe two-dimensional space. Any point you want to reach can indeed be arrived at by moving some distance along your first line and then some distance along your second.

This is relevant to quantum mechanics because any measurement collapses the system into an eigenfunction of its representative matrix, and the probability that it will end up in any one state is determined by the amount of that eigenfunction you need to describe its previous wavefunction in your new basis set.

That is one ugly sentence.

Maybe it’s not so surprising that Heisenberg described this incorrectly in words, because this is somewhat arduous…

Here, I’ll draw another nifty picture. We’ll have to imagine two different operations (you could even get ahead of me and imagine that these represent measuring position and momentum, since that’s the pair of famous variables that don’t commute), and the eigenvectors for these operations are represented by either the blue arrows or the red arrows below.

If we make a measurement with the blue matrix, it’ll collapse the system into one of the two blue eigenvectors. If we decide to measure the same property again, i.e. act upon the system with the blue matrix again, we’re sure to see that same blue eigenvector. We’ll know what we’ll be getting.

But once the system has collapsed into a blue arrow, if we measure with the red matrix the system has to shift to align with one of the red arrows. And our probability of getting each red answer depends upon how similar each red arrow is to the blue arrows… the one that looks more like our current state is more likely to occur, but because neither red arrow matches a blue arrow perfectly, there’s a chance we’ll end up with either answer.

And if we want to make a blue measurement, then red, then blue… the two blue measurements won’t necessarily be the same. After we’re in a state that matches a red eigenvector, we have some probability to flop back to either blue eigenvector, depending, again, on how similar each is to the red eigenvector we land in.

That’s the uncertainty principle. That position is simply not well-defined when momentum is precisely known, and vice versa. The eigenfunctions for one type of measurement do not resemble the eigenfunctions for the other measurement. Which means that the type of measurement you have to make in order to know one or the other property invariably changes the system and gives you an unpredictable result… it’s like you’re rolling dice every time you switch which flavor of measurement you’re making.

But the measurement isn’t causing error. It’s revealing an underlying probability distribution. That is, there is no conceivable “gentle” way of measuring that will give a predictable answer, because the phenomenon itself is probabilistic. Because the mechanics are quantized, because there are no in-between states, the system flops like a landbound fish from eigenvectors of one measurement to eigenvectors of the other.

Which is why it bothers me so much to see the uncertainty principle described as measurement obscuring reality when the idea crops up in philosophy or literature. Those allusions also tend to place too much import on the idea of “observers,” like the old adage about a tree making or not making sound when it falls in an empty forest. Perhaps I did a bad job of this too by writing “measurement” so often. Maybe that word makes it sound as though quantum collapse requires intentional human involvement. It doesn’t. Any interaction between quantum mechanics and a semi-classical system will couple them and can cause the probabilistic distribution of wavefunctions to condense into particle-like behavior.

And I think the biggest difference between the uncertainty principle and the way it’s often portrayed in literature is that, rather than measurements obscuring reality, you could almost say that measurements create reality. There wasn’t a discrete state until the measurement was made. It’s like asking an inebriated collegiate friend who just learned something troubling about his romantic partner, “Well, what are you going to do?” He’ll probably answer. While you’re talking about it, it’ll seem like he’s going to stick to that answer. But if you hadn’t asked he probably would’ve continued to mull things over, continued to exist in that seemingly in-between state where there’s both a chance that he’ll break up or try to work things out. By asking, you learn his plan… but you also forced him to come up with a plan.

And it’s important that our collegian be drunk in this analogy… because making a different measurement has to re-randomize behavior. Even after he resolves to break up, if you ask “Where should we go for our midnight snack,” mulling that over would make him forget what he’d planned to do about the whole dating situation. The next time you ask, he might decide to ride it out. It’s only when allowed to keep the one answer in the forefront of his mind that the answer stays consistent.

The uncertainty principle says that position and momentum can’t both be known precisely not because measurement is difficult, but because elementary particles are too drunk to remember where they are when you ask how fast they’re moving.

And, here, a treat! As a reward for wading through all this, I’ve drawn a cartoon version of Heisenberg’s misconception. Note that this is not, in fact, the correct explanation for the uncertainty principle… but do you really need me to sketch a bunch of besotted electrons?

On undeserved forgiveness and global warming.

I wish there were more essays focused on philosophy in Freeman Dyson’s collection Dreams of Earth and Sky. I thought all his remarks on morals and philosophy were nuanced and compelling. His essay “Rocket Man,” for instance, is very powerful. This essay discusses Wernher von Braun, a German scientist who helped develop Nazi weaponry during WW2 and was later hired by the government of the United States for our space program.

Many people felt betrayed that the U.S. would hire Von Braun after his participation in evil. He aided in the war effort, sure, but more damningly he knew about and did not resist Nazi atrocities — slave labor under brutal conditions was used for the program he led. And yet, after the war ended, he was allowed to pursue his dream of launching humankind into space.

Dyson, in articulating his philosophical stance, describes his own contribution to atrocity during that conflict: the bombing of civilian Dresden. To me, this acknowledgement of personal culpability lends a lot of power to his reasoning. He knows that, if Germany had won the war, it could’ve been him rather than Von Braun who was condemned as a war criminal.

With that in mind, here is Dyson’s stance:

“In order to make a lasting peace, we must learn to live with our enemies and forgive their crimes. Amnesty means that we are all equal before the law. Amnesty is not easy and not fair, but it is a moral necessity, because the alternative is an unending cycle of hatred and revenge.”

One of the t-shirts I rotate through while volunteering with the local running teams is an Amnesty International shirt given to me by my sister. It has the text an eye for an eye leaves the whole world blind along with a cartoon schematic of two faces with an x-ed out eye each that overlap Venn-diagram-style to yield a fully-blind Earth. So, sure, I’m predisposed toward excessive mercy. I do realize that the human brain is wired to desire vengeance: seeing bad behavior punished helps us resolve lingering malaise, especially when the initial bad actions were perpetrated against ourselves or those we love.

But it’s hard, the idea of balancing retribution and forgiveness. I’ll write more about my own conflicting views on vengeance when I finally type out an essay on Jon Krakauer’s Missoula — despite my hippie-esque views on the potential for rehabilitation and redemption, the way Krakauer’s book is written it’s hard not to root for the perpetrator in the book’s central case to receive the harsh punishment that the victim’s family is pushing for (I know, I know… I already posted an essay about Missoula and didn’t discuss this at all. It’s always difficult knowing how many bleak thoughts I can cram into a piece before it becomes unreadable).

Or, from more recent developments in the news — even though I’m a runner, and with numerous marathoners amongst my friends and family, I don’t feel good about the mentally-ill kid being condemned to death (I suppose this isn’t recent anymore, not by the time this will be posted. And it’s odd for me, re-reading this now — a lot of people are upset that major news outlets have speculated about the sanity of the terrorist who murdered all those people in South Carolina but didn’t extend the same doubts to the terrorist who murdered & maimed all those people in Boston. Whereas I think it’s pretty clear that perpetrators of both crimes were not getting the psychological support they needed).

My main objections to the death penalty are related to the fact that our judicial system is so broken. Coerced confessions and planted evidence have been used to condemn many innocent people to death [http://www.innocenceproject.org/]. That’s not the case here — the kid is guilty. The magnitude of suffering he inflicted means he should probably never be set free. But I think you could reasonably argue that life imprisonment would be more effective deterrence against further terrorism than the death penalty / martyrdom. The death penalty ensures that the case will return to national prominence at least one more time [http://www.bostonglobe.com/metro/2015/04/16/end-anguish-drop-death-penalty/ocQLejp8H2vesDavItHIEN/story.html]. And I think there’s something to be said for rising above the brutality of others.

Even though it’s unsettling, I think Dyson’s philosophical stance on this type of issue is admirable. And the way he reasons toward it in his essay is compelling. His piece culminates with this thought:

“In the end, the amnesty given to [Von Braun] by the United States did far more than a strict accounting of his misdeeds could have done to redeem his soul and fulfill his destiny.”

But even though I enjoyed the philosophical essays in Dyson’s book, and always enjoy personal narratives from scientists who were friends and colleagues with the founders of quantum mechanics, the essays grounded in biology often seemed strange to me. I wrote previously about my nonplussed reaction to Dyson’s comments on plants, photosynthesis, and evolution. The points he raises in his essay “The Question of Global Warming” seemed equally confusing.

Capture — Graph of recent carbon dioxide averages. See more at the ESRL home page.

The basic issue is this: Dyson presents a graph from the National Oceanic & Atmospheric Administration that shows atmospheric carbon dioxide versus time. There’s an overall positive slope, and that’s what everyone thinks is bad, but there are also annual wiggles up and down. Dyson interprets these wiggles as indicating that plant growth each year (primarily during spring and summer) takes up approximately 5-10 ppm of carbon dioxide per year, and then plant decomposition (in fall and winter) releases most of that back into the atmosphere.

Dyson then uses these numbers to approximate a residence time for any one molecule of carbon dioxide in the atmosphere — since this is a random process, the most sensible measure to use is the half-life, the length of time at which there is a 50% chance that our chosen molecule has been incorporated into a plant. The number he comes up with is 12 years, which sounds very low. Even if you make very generous assumptions, like 10 ppm being taken up each year, selected from a total of 300 ppm, you’ll get a half-life equal to [ log (0.5) / log ( 29 / 30 ) ], which is over 20 years.

Then there’s an addendum in which Dyson explains that his number comes from uptake without replacement. My calculation above is based on choosing 10 ppm one year, then letting a year pass, plants decompose, there are again 300 ppm to choose from next year. Whereas Dyson bases his calculations on the idea that we would develop genetically-modified plants to trap carbon, so the first year you’re choosing 10 ppm out of 300, the next year 10 out of 290, the next year 10 out of 280…

I have two objections to this.

The first is about math: is half-life is still a useful number to measure under this regime? If you’re hoovering up carbon dioxide, wouldn’t you rather deal with the arithmetic number that tells you how many years it will take to get atmospheric levels down to where you want them? In other words, just measure how much needs to go away, how much you can get rid of per year, and divide the two. Who would care about the identities of each particular molecule if you were solving the problem that way?

Capture And my second objection is related to ecology: even if you develop one or more fancy new varietals of carbon-trapping plants, why would you postulate no replacement of atmospheric carbon dioxide? Unless the plan is to raze all our existing plantlife, bury it, and replace everything with the new species (even though I’m pro-GM plants, this sounds a bit unwise to me), every winter will bring more rot. Our old plants will behave the same.

While I appreciate Dyson’s point that global warming is a very complicated question, and that it’s extremely upsetting for a trained scientist to hear dogmatic claims about deadlines and thresholds when it’s clear that atmospheric science is sufficiently complex that no one really understands what is or will be happening, I think he does these nuanced views a disservice by pairing them with concretely-stated, specious biological claims of his own.

Yes, global warming is not well understood. Yes, economic discounting (valuing a cost you have to pay now to fix something more highly than a cost you’d have to pay later) is appropriate to use: we might have fancy new technologies to help us by then, or might have triggered a new international conflict that kills us all. Discounting is an appropriate mathematical tool to help us plan for uncertainty.

But I would have been much happier to see Dyson make these points without invoking seemingly-unworkable schemes, because I think the strange details distract from a lot of the really good philosophical points he makes.

On talking to students about school, particularly high schoolers.

Oscar Fernandez (find him @EverydayCalc) recently wrote a charming little article about how to talk to your high-school-aged kids about math. Well worth the quick read, if you’re a parent, or might someday be a parent, or happen to interact with other people’s kids. He has some great tips, and provides a lucid description of why it’s so important to do this. Because math can be really fun for kids, and if you, a parent, let your squeamishness for the subject show through, you could sway someone away from enjoying it.

The only thing I’d like to add to Fernandez’s article is a quick note about that question, “What did you learn in school today?”

Which is what my parents asked me while I was growing up, (to which I’d generally mumble “nuthin” before shoving my nose back into a book and ignoring them. If I’d been less of a brat, I should have answered their question more literally, as in, what did I learn while I was in the high school building during school hours. Because, sure, in chemistry class I sat in the back and drew cartoons, in English I sat on top of a desk and chatted with the teacher, in math I sat on the floor and read books, so on most days I didn’t learn anything from the curricula, but I was still learning something most days),

and it’s what I asked members of the local cross country / track teams when I first started volunteering in town (a few of the very outgoing ones gave me reasonable answers that we could chat about, but I heard a lot of “I don’t know,” and “nothing,” and “we’re just reviewing right now”).

So I don’t ask about school that way anymore. A pretty common question I still use is “What was your best class today?”

fun-42593_640 Sometimes a runner will ask me to clarify, at which point I’ll toss out an “easiest” or “hardest” or “most fun” or “best teacher” addendum, but I’ve had a lot better luck getting real answers to “best class” than “what’d you learn?” And it’s pretty easy to follow up from there — talking to someone whose best class was geometry

(surprisingly common in town because the math teacher is an excellent human. Stopped out of high school when he was growing up, used illicit drugs, saw a few friends die, turned his life around and became a teacher. And it definitely seems like the teachers who realize how terrible school can be are often the most empathetic, which helps them be the best. My hope is that I provide some of that helpful loathing for K; she actually liked high school, primarily because her home life was crumbling so dramatically that school felt like a sanctuary. Whereas the last time I enjoyed public education was in third grade. My teacher was a compassionate, intelligent woman who made special math worksheets for me to do on the bus, especially before field trips, so that I’d get to have fun solving them instead of noticing that no one would sit with me. She later married my senior-year cross country coach, also an excellent human, a vegetarian who cooks up hearty meat stews for an Indianapolis soup kitchen, eccentric enough that he swore off shoes for several years and kept getting kicked out of coffee shops for health code violations, at which point he started boiling the soles off used Pumas and strapping them to his feet with underwear elastic),

I get to ask, “What were you working on in Geometry today?” and from there can chat about shapes, or proofs, or whatever… and, sure, some of those conversations wind up derailing since the participants are out-of-breath, oxygen-deprived me and a not-yet-practiced-with-math-but-sufficiently-in-shape-to-talk-and-think high schooler, but it seems like we usually have fun when we can make sense of each other. Either that, or there are a lot of good actors on the team.

Some other follow-up questions that I can remember coming up with non-math “best class” answers are things like, “Yeah? What’s your Spanish teacher do well?” or “What’d you make in baking?” or “What’re you reading now?” All of which have generally led to talk-about-able answers from even taciturn dudes.

"Look, N., a sphere! It's composed of all points in R3 equidistant to the center!" — “Look, N., a sphere! It’s composed of all points in R3 equidistant to the center!”

And, yes, even though my daughter is only one year old (as of yesterday, actually), I realize that high schoolers often behave differently with adults who aren’t their parents, but if you’re having trouble talking productively about school you might try switching up your question a little bit.

And in any case, a warm thank you to Fernandez for his article; hopefully enough parents see it that some kids can benefit from his piece.

(See more about my love of math here.)

Frank Brown Cloud

Justice through words.

Tag: math