On translation and quantum mechanics.

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 Ann Leckie’s ‘The Raven Tower.’

On Ann Leckie’s ‘The Raven Tower.’

At the beginning of Genesis, God said, Let there be light: and there was light.

“Creation” by Suus Wansink on Flickr.

In her magisterial new novel The Raven Tower, Ann Leckie continues with this simple premise: a god is an entity whose words are true.

A god might say, “The sky is green.”  Well, personally I remember it being blue, but I am not a god.  Within the world of The Raven Tower, after the god announces that the sky is green, the sky will become green.  If the god is sufficiently powerful, that is.  If the god is too weak, then the sky will stay blue, which means the statement is not true, which means that the thing who said “The sky is green” is not a god.  It was a god, sure, but now it’s dead.

Poof!

And so the deities learn to be very cautious with their language, enumerating cases and provisions with the precision of a contemporary lawyer drafting contractual agreements (like the many “individual arbitration” agreements that you’ve no doubt assented to, which allow corporations to strip away your legal rights as a citizen of this country.  But, hey, I’m not trying to judge – I have signed those lousy documents, too.  It’s difficult to navigate the modern world without stumbling across them).

A careless sentence could doom a god.

But if a god were sufficiently powerful, it could say anything, trusting that its words would reshape the fabric of the universe.  And so the gods yearn to become stronger — for their own safety in addition to all the other reasons that people seek power.

In The Raven Tower, the only way for gods to gain strength is through human faith.  When a human prays or conducts a ritual sacrifice, a deity grows stronger.  But human attention is finite (which is true in our own world, too, as demonstrated so painfully by our attention-sapping telephones and our attention-monopolizing president).

Image from svgsilh.com.

And so, like pre-monopoly corporations vying for market share, the gods battle.  By conquering vast kingdoms, a dominant god could receive the prayers of more people, allowing it to grow even stronger … and so be able to speak more freely, inured from the risk that it will not have enough power to make its statements true.

If you haven’t yet read The Raven Tower, you should.  The theological underpinnings are brilliant, the characters compelling, and the plot so craftily constructed that both my spouse and I stayed awake much, much too late while reading it.

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In The Raven Tower, only human faith feeds gods.  The rest of the natural world is both treated with reverence – after all, that bird, or rock, or snake might be a god – and yet also objectified.  There is little difference between a bird and a rock, either of which might provide a fitting receptacle for a god but neither of which can consciously pray to empower a god.

Image by Stephencdickson on Wikimedia Commons.

Although our own world hosts several species that communicate in ways that resemble human language, in The Raven Tower the boundary between human and non-human is absolute.  Within The Raven Tower, this distinction feels totally sensible – after all, that entire world was conjured through Ann Leckie’s assiduous use of human language.

But many people mistakenly believe that they are living in that fantasy world.

In the recent philosophical treatise Thinking and Being, for example, Irad Kimhi attempts to describe what is special about thought, particularly thoughts expressed in a metaphorical language like English, German, or Greek.  (Kimhi neglects mathematical languages, which is at times unfortunate.  I’ve written previously about how hard it is to translate certain concepts from mathematics into metaphorical languages like we speak with, and Kimhi fills many pages attempting to precisely the concept of “compliments” from set theory, which you could probably understand within moments by glancing at a Wikipedia page.)

Kimhi does use English assiduously, but I’m dubious that a metaphorical language was the optimal tool for the task he set himself.  And his approach was further undermined by flawed assumptions.  Kimhi begins with a “Law of Contradiction,” in which he asserts, following Aristotle, that it is impossible for a thing simultaneously to be and not to be, and that no one can simultaneously believe a thing to be and not to be.

Maybe these assumptions seemed reasonable during the time of Aristotle, but we now know that they are false.

Many research findings in quantum mechanics have shown that it is possible for a thing simultaneously to be and not to be.  An electron can have both up spin and down spin at the same moment, even though these two spin states are mutually exclusive (the states are “absolute compliments” in the terminology of set theory).  This seemingly contradictory state of both being and not being is what allows quantum computing to solve certain types of problems much faster than standard computers.

And, as a rebuttal for the psychological formulation, we have the case of free will.  Our brains, which generate consciousness, are composed of ordinary matter.  Ordinary matter evolves through time according to a set of known, predictable rules.  If the matter composing your brain was non-destructively scanned at sufficient resolution, your future behavior could be predicted.  Accurate prediction would demonstrate that you do not have free will.

And yet it feels impossible not to believe in the existence of free will.  After all, we make decisions.  I perceive myself to be choosing the words that I type.

I sincerely, simultaneously believe that humans both do and do not have free will.  And I assume that most other scientists who have pondered this question hold the same pair of seemingly contradictory beliefs.

The “Law of Contradiction” is not a great assumption to begin with.  Kimhi also objectifies nearly all conscious life upon our planet:

The consciousness of one’s thinking must involve the identification of its syncategorematic difference, and hence is essentially tied up with the use of language.

A human thinker is also a determinable being.  This book presents us with the task of trying to understand our being, the being of human beings, as that of determinable thinkers.

The Raven Tower is a fantasy novel.  Within that world, it was reasonable that there would be a sharp border separating humans from all other animals.  There are also warring gods, magical spells, and sacred objects like a spear that never misses or an amulet that makes people invisible.

But Kimhi purports to be writing about our world.

In Mama’s Last Hug, biologist Frans de Waal discusses many more instances of human thinkers brazenly touting their uniqueness.  If I jabbed a sharp piece of metal through your cheek, it would hurt.  But many humans claimed that this wouldn’t hurt a fish. 

The fish will bleed.  And writhe.  Its body will produce stress hormones.  But humans claimed that the fish was not actually in pain.

They were wrong.

Image by Catherine Matassa.

de Waal writes that:

The consensus view is now that fish do feel pain.

Readers may well ask why it has taken so long to reach this conclusion, but a parallel case is even more baffling.  For the longest time, science felt the same about human babies.  Infants were considered sub-human organisms that produced “random sounds,” smiles simply as a result of “gas,” and couldn’t feel pain. 

Serious scientists conducted torturous experiments on human infants with needle pricks, hot and cold water, and head restraints, to make the point that they feel nothing.  The babies’ reactions were considered emotion-free reflexes.  As a result, doctors routinely hurt infants (such as during circumcision or invasive surgery) without the benefit of pain-killing anesthesia.  They only gave them curare, a muscle relaxant, which conveniently kept the infants from resisting what was being done to them. 

Only in the 1980s did medical procedures change, when it was revealed that babies have a full-blown pain response with grimacing and crying.  Today we read about these experiments with disbelief.  One wonders if their pain response couldn’t have been noticed earlier!

Scientific skepticism about pain applies not just to animals, therefore, but to any organism that fails to talk.  It is as if science pays attention to feelings only if they come with an explicit verbal statement, such as “I felt a sharp pain when you did that!”  The importance we attach to language is just ridiculous.  It has given us more than a century of agnosticism with regard to wordless pain and consciousness.

As a parent, I found it extremely difficult to read the lecture de Waal cites, David Chamberlain’s “Babies Don’t Feel Pain: A Century of Denial in Medicine.”

From this lecture, I also learned that I was probably circumcised without anesthesia as a newborn.  Luckily, I don’t remember this procedure, but some people do.  Chamberlain describes several such patients, and, with my own kids, I too have been surprised by how commonly they’ve remembered and asked about things that happened before they had learned to talk.

Vaccination is painful, too, but there’s a difference – vaccination has a clear medical benefit, both for the individual and a community.  Our children have been fully vaccinated for their ages.  They cried for a moment, but we comforted them right away.

But we didn’t subject them to any elective surgical procedures, anesthesia or no.

In our world, even creatures that don’t speak with metaphorical language have feelings.

But Leckie does include a bridge between the world of The Raven Tower and our own.  Although language does not re-shape reality, words can create empathy.  We validate other lives as meaningful when we listen to their stories. 

The narrator of The Raven Tower chooses to speak in the second person to a character in the book, a man who was born with a body that did not match his mind.  Although human thinkers have not always recognized this truth, he too has a story worth sharing.

On the question of whom to blame for the paucity of women in science.

On the question of whom to blame for the paucity of women in science.

978-080704657-9I was super excited to read Eileen Pollack’s The Only Woman in the Room.  There are a lot of problems with academic science, and these have been getting better much more haltingly than one might expect.  And the problem isn’t just individuals with retrograde attitudes — although that’s clearly an issue — but also structural and cultural arrangements that bias against neurotypical females.

I’d hoped that the bulk of Pollack’s book would be devoted to documenting these problems and offering suggestions for corrective measures.  If we as a society value science enough that we want for the best and brightest of all genders, upbringings, personality types, etc., to participate in the field’s advancement, I think there’s a dire need for investigative journalism that’d produce that sort of book.

Pollack’s book is primarily a memoir, however.  This is useful, too.  There’s a reason why medical journals still publish narrative-driven case studies in addition to the charts detailing aggregate patient response and recovery rates.  Details can be presented in stories that might be overlooked or ignored when many people’s experiences are moshed together to make a statistic.  After all, if we want the statistics to change, it’s women’s experience, actual lived experience, that we need to fix.

si-sexisminscienceBut I felt displeased while reading Pollack’s book.  My major complaint is that most of the book castigates scientists for the paucity of women in STEM fields… but the narrative suggests clearly that, in this case, the biggest problem is the behavior of non-scientists.

I’ll get back to that point in a moment, but first I should make clear that I’m not writing from the standpoint of an apologist who thinks the current state of things is fine.

Where I studied, first-year Ph.D. students had weekly tea with the founder of the department.  These were advising / advice sessions.  Students could talk about their interests, ask questions about the history of the field, get input on their courses, their research, their search for an advisor whose interests and outlook matched their own.  All told, a valuable experience for budding scientists.  But the advisor, an elderly male, invariably asked a female student to serve tea to everyone else in the room.  Even if he believed that the advice he dispensed next was gender neutral, that initial request (reasonable enough at the first meeting, because someone has to pour tea, and even at the second, but disheartening by the nth time the same young woman is asked to serve her classmates) discolored everything he said next.

Or there were the monthly lunchtime research talks.  A modestly-dressed fourth-year student gave a presentation on her research, fecal analysis of mothers and infants to learn when and with what species a newborn human’s intestinal track is colonized, and after the talk a female faculty member said to her, “That was a nice talk, but your breasts were very distracting.”

Individuals with that sort of retrograde attitude make science worse.  And it’s not just elderly professors who’re like that.  The individual from the tea incident, for instance, has since been retired by the reaper (the prevailing mood in the department was very somber after he passed.  For most, but not all.  When we rode in the elevator together, a UPS deliveryman told me, “You know, I’d feel bad too, except the old guy yelled at me just last week.”).  But it’s not as though there’ve been no young misogynists to replace the retiring ones.

tumblr_npqoiwAaVi1r83d7lo1_500.jpg

And there are structural problems.  There’s a particular way that advisors expect scientists to talk about their research — brash, confident, competitive, as though it is magnitudes more important than anything else — that seems to come easier to the average male than the average female.  People who don’t have that sort of competitive attitude, whether male or female, can be marginalized… but for a host of both biological and cultural reasons, men in this country are more likely to have that sort of attitude than women.

Maybe this would be fine if brash, stereotypically masculine behavior resulted in better science.  It doesn’t.  Good science is intensely collaborative.  Competitive attitudes, like the race aspect of modern academic science to publish findings first before someone else “scoops” your work, diminish the quality and quantity of data that everyone has to work with.  And contributes to the irreproducibility of modern science, because researchers are pressured to specialize in niche techniques that are used on a particular problem in only one laboratory.

Of course, individual scientists don’t have the freedom to rebel from this system … if only because granting agencies are set up to fund only researchers who conform.  If one researcher decided to behave more collaboratively, the lab would probably run out of money and die.

Academic science could be changed in ways that would make it more inviting to women and would result in better science.  And those are changes that I think scientists will need to make.

Whereas Pollack’s book, despite castigating scientists, felt quite short on recommendations for changes that scientists should make to their behavior.  (I.e., changes to the behavior of a scientist who isn’t explicitly prejudiced against women, but has simply absorbed the cultural norms of modern academia.)

The most important corrective that Pollack offers is that scientists should be more emotive in complimenting students on work they’ve done well.  This is probably true.  In K.’s science class, for instance, she makes a conscious effort to praise students for their successes.  Praise them with words, not just a high score marked at the top of an exam.

Reading Pollack’s narrative, for instance, we learn that after a successful physics internship, the professor said only, “We’d like to have you back next year.”  After a successful research project in mathematics, her advisor didn’t praise her — a stark contrast to the lavish praise articulated by her writing professor.

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But I think it’s worth considering a possible reason why Pollack’s physics professors may have been less effusive than her humanities professors.  While working in physics, the primary language is mathematics.  Quite a bit of physics doesn’t make much sense when expressed in a metaphorical language like English — the language most of us use to express our feelings, or to praise people, is simply maladapted to conveying a clear understanding of the universe.  So the practice of physics enriches for people much more adept with numbers than words.

Whereas humanities professors work with words full-time.  They really ought to be able to praise people with words more effectively than scientists can.

But the problem isn’t just that evaluating their competence for verbal praise is like judging both a carpenter and a welder on their skill with a blowtorch — is it fair to blame someone for relative inexperience compared to a full-time user? — it’s that many scientists have narrative experiences of their own that train them not to be effusive.

In part because the language of science is mathematics, science enriches for people who’re vaguely on the autism spectrum (I’d much rather use the term “Asperger’s” here, but that’s a topic for another post).  And many of those people experienced bewildering derision in response to their attempts to compliment people while growing up.  There are numerous examples of this in Steve Silberman’s Neurotribes, and I certainly have stories of my own.  I learned that it was safe to state facts (akin to the physics professor’s “We’d like to have you back next year”) but that emotional content often led to mockery.

2981Indeed, much of Pollack’s book is devoted to frustration that so few people wanted to date or have sex with her.  The book is sprinkled with lines like, “The only reason I could see that I wasn’t datable was that I was majoring in a subject they saw as threatening,” or a description of a woman who “hated when her sister introduced her as an astrophysics major, because the boys would turn away.

A big reason why women and minorities need to be praised to keep them excited about STEM fields is that stigma from the outside world.  But that’s not scientists’ fault!  I felt sad, reading the book, because so much of it seemed to blame scientists and praise humanities people, yet those same humanities people create the problems that weigh most heavily on Pollack’s mind.  Yes, it’s crummy that most boys at parties considered her not date-able.  But those boys were by and large humanities majors.  Because non-scientists were mean to her, Pollack needed for scientists to give her more praise.

Sure, it’s a big problem that scientists didn’t work hard enough to retain her in the field.  But it’s a bigger problem that non-scientists were so mean that, by the time she arrived at college, those science professors needed to work to retain the two (!) female students who enrolled in the introductory physics lecture instead of trusting that a reasonable fraction of 60 female enrollees (her lecture had 120 students) would stay in the field.

I was sad that this wasn’t stated explicitly until page 254 of a 257-page book, and even then in only two sentences in the middle of a paragraph:

It’s the larger society that needs to change.  No American of either gender will want to become a scientist if studying science or math makes a middle schooler so nerdy he or she becomes undatable, or if science and math are taught in a way as to seem boring or irrelevant.

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.

CaptureI 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.

CaptureAnd, 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.

CaptureHis 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.

CaptureBut 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.

graph-1

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:

Picture 2

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:

Picture 4

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:

Picture 5

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.

graph-2

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…

Elliptic_orbit(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.

graph-3

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?

cartoon-title

cartoon-1003

cartoon-summary

On talking to students about school, particularly high schoolers.

Sketch by davespineapple on deviantart
Sketch by davespineapple on deviantart

Oscar Fernandez (find him @EverydayCalcrecently 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_640Sometimes 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.)

On Cosmos and working through the math.

CosmosK and I have been watching that new Cosmos television show.  The library had the whole set of DVDs, and she and I have both been tired enough that it’s felt nice to zonk out with some television in the evening while N is having her fifth dinner.

K really likes the show.  Things were perhaps stacked in her favor: she is a scientist, she likes Neil deGrasse Tyson, and she teaches Earth & Space Science, which covers a lot of the same topics as the Cosmos show (and is, incidentally, not the field of science either she or I was trained in – E&SS just happens to be what you need to teach if you want to work with people who get signed up for the most introductory level science class in this state.  Elsewhere I think the equivalent course is Integrated Chemistry & Physics or something?).

So, K likes Cosmos.

Whereas I have mostly failed to enjoy the show as much, even though it’s clear that a  lot of people worked very hard on it and it has some very redeeming qualities, too.

So, last night we watched the episode about Faraday’s experiments with electricity and magnetism.  And then this morning I went berserk and had to set N up for some self-directed play while I watched a video deriving the Maxwell equations.  Which is something I used to know, and probably still should, but almost a decade has passed since I last used them.  And when they were flashed up on the screen at the end of the Cosmos episode, I realized that I couldn’t think through the derivation.  That felt sad.  I’ve already been feeling sad about being very unpracticed with my math these days, to the point that shortly before N was born I picked up a book on differential equations and another covering specifically Fourier transforms.  I did not work through very much of either before she was born and have not picked them up since – as it happens, taking care of a baby can be a fair bit of work.

But my reason for feeling uncomfortable about Cosmos isn’t just that it reminds me of my own current ineptitude.

For me, the problem is that the show often seems to be contrasting stories from mythology, often Christian mythology, against stories as elucidated by science.  But all of it is presented without math, and much of it is presented without data of any kind, or even descriptions of the necessary experiments.  I have to admit, the particular episode that made me sad did a better job of that than most – many of Faraday’s experiments were described.  But, still, the Maxwell equations were tossed up at the end with no attempt at an explanation.

And I know that mathematical and scientific literacy is often described as, um, not good in this country.  And television is not a very good medium for conveying math.  I’ve watched some math videos online (Dr. Arthur Mattuck at MIT made some charming ones), and, yes, sometimes you might space out, or maybe want to pause them, or glance at the screen and read an equation wrong and feel confused.  But not everybody can pause broadcast television.  And a lot of mathematics presupposes familiarity with basic concepts that many people haven’t had a chance to feel comfortable with.

But the problem is, in my opinion, that science without math is very akin to mythology.

It becomes an expert telling you what you ought to think because he happens to be blessed with great truths, either because he can do the math or because he has studied his bible or communed with god or read the portents from an eagle overhead.  I am teaching you, and you should believe.  Whereas I think that science should be presented more humbly.

Yes, to my mind, having (once upon a time) worked through the math, science is more real than any religious claims.  But for someone who has not worked through the math, and currently can not work through it, I can see how scientific or religious claims would seem like equal mysteries.  You have to trust an expert.  Maybe you choose an expert in a white coat, maybe you chose one in a black cassock.

So I would much rather see science outreach like Cosmos attempt to guide people through the details – this is how the math works, and these are resources you could consult if this is hard, and these are experiments that you could do to reproduce these findings.  Which is hard, obviously.  I am (was?) pretty good at math, but I don’t have the training to follow serious astrophysics.  I did well with the physics of small things (quantum mechanics) during college and graduate school, but I’ve never studied the physics of big things.  I don’t really know gravity, for instance.  That’s something I’d like to take a few courses in, after I finish the novel I’m working on.  So, sure, there is a lot of complicated math involved.

But I’d like to think there are better ways to convey a respect for people’s lack of exposure to math than trying to convey science without it.  Obviously, it’s tricky.  I’m not trying to argue that the team that made Cosmos didn’t try very hard, or aren’t clever, or anything like that.  I’m just not super happy with the final product, even if I couldn’t necessarily do it better.  Like, okay, the introduction to Christian Rudder’s Dataclysm seem to do a good job of walking cheerily through some basic statistics.  And, sure, there are differences – statistics is an easier branch of mathematics for people to understand visually than some of the math you need for astrophysics, and, more importantly, Dataclysm is a book, which means there’s less worry that someone will get bored and change the channel during a brief math interlude – but I felt as though Rudder conveyed a more respectful attitude about wanting his audience to know where his claims were coming from.

And, right, it’s not as though this problem – trying to convey science sans numbers to non-scientists – is unique to Cosmos.  I feel a lot of similar frustration in reading editorials by scientists or medical doctors regarding the recent upsurge in Americans opting out of vaccines.  Which, again, I should make clear: I am a scientist.  I have been vaccinated against many diseases.  My daughter has received all her recommended vaccines on time.  But my impression is that scientists and medical doctors promoting vaccines do not always show sufficient sympathy.

Like, okay, there was a research paper claiming that vaccines cause autism, which I am not linking here.  There are research papers claiming that vaccines do not cause autism.  Sure, there are more of the latter, but for someone without enough scientific literacy to evaluate the data for themselves, you’re stuck picking someone to trust.  And, honestly, scientists and medical doctors have done some pretty terrible things, often in the guise of vaccines – you could have a pretty unpleasant weekend reading about some of these in Against Their Will: The Secret History of Medical Experimentation on Children in Cold War America.  So there are reasons why someone who was just picking whom to trust would not choose the doctors.

So, it’s not that I think the Earth is only 6000 years old.  Or that I think anyone should believe that.  I just think that, by trying to present science without math, the product is a competing narrative that still relies on faith.  As with vaccines, as with climate change.  And, who knows, maybe I am watching their show with the wrong attitude – maybe they were not hoping that people would watch and say, “oh, the Earth was actually formed 4.5 billion years ago,” maybe they were hoping that people would watch and think, “gee, that’s cool, I should learn how to figure this stuff out myself.”  But that’s not the impression that I got.