The fork in the road

One of the most bizarre (and misunderstood) features of quantum physics is indeterminacy.

This is because we live in a macroscopic universe that -- most of the time, at least -- behaves in a determinate fashion.  Now, that doesn't mean we necessarily know everything about it.  For example, if we drop balls into a Galton board -- a device with a grid of pegs to deflect the ball's path -- eventually we'll get a normal distribution:

[Image licensed under the Creative Commons Matemateca (IME USP), Galton box, CC BY-SA 4.0]

With a device like a Galton board, we can accurately predict the probability of any given ball landing in a particular slot, but the actual path of the ball can't be predicted ahead of time.

Here's where the difficulty starts, though.  When people talk about quantum phenomena and describe them as probabilities, there's a way in which the analogy to macroscopic probability breaks down.  With a Galton board, the problem with predicting a ball's path doesn't mean it's not completely deterministic; it has to do with our (very) incomplete knowledge about the ball's initial state.  If you knew every last detail about the game -- each ball's mass, spin, air resistance, elasticity, the angle and speed of release, the angle at which it strikes the first peg, as well as the position, shape, and composition of every peg -- at least in theory, you could predict with one hundred percent accuracy which slot it would land in.  The ball's path is completely controlled by deterministic Newtonian physics; it's only the complexity of the system and our lack of knowledge that makes it impossible to parse.

This is not the situation with quantum systems.

When a particle travels from its source to a detector -- such as in the famous double-slit experiment -- it's not that the particle really and truly went through either slit A or slit B, and we simply don't happen to know which.  The particle, or more accurately, the wave function of the particle, took both paths at the same time, and how the detector is set up determines what we end up seeing.  Prior to being observed at the detector, the particle literally existed in all possible paths simultaneously, including ones passing through Bolivia and the Andromeda Galaxy.

To summarize the difference -- in a determinate system, we may not be able to predict an outcome, but that's only because we have incomplete information about it.  In an indeterminate system, the probability field itself is the reality.  However tempting it is to say that a particle, prior to being observed, took a specific fork in the road, and we just don't know which, completely misses the truth -- and misses how utterly bizarre the quantum world actually is.

People who object to this admittedly weird model of the world usually fall back on a single question, which is surprisingly hard to answer.  Okay, so on the one hand we have deterministic but complex systems, whose outcome is sensitively dependent on initial conditions (like the Galton board).  On the other, we have quantum systems which are probabilistic by nature.  How could we tell the difference?  Maybe in a quantum system there are hidden variables -- information about the system we don't have access to -- that make it appear indeterminate.  (This was Einstein's opinion, which he summed up in his famous statement that "God does not play dice with the universe.")

Unfortunately for Einstein, and for anyone else who is uncomfortable with the fact that the microscopic basis of reality is fundamentally at odds with our desire for a mechanistic, predictable universe, research at the Vienna University of Technology, which was described in a paper this week in Physical Review Letters, has shown conclusively that there are no hidden variables.  Our reality is indeterminate.  The idea of particles having definite positions and velocities, independent of observation and measurement, is simply wrong.

The experiment hinges on something called the Leggett-Garg Inequality -- described in a 1985 paper by physicists Anthony James Leggett and Anupam Garg -- which clearly distinguishes between how classical (determinate) and quantum (indeterminate) systems evolve over time.  Correlations between three different time measurements of the same system would show a different magnitude depending on whether it was behaving in a classical or quantum fashion.

The problem is, no one was able to figure out how to create a real-world test of it -- until now.  The team developed a neutron interferometer, which splits a neutron beam into two parts and then recombines it at a detector.  And the results of the experiment showed conclusively that contrary to our mental image of neutrons as hard little b-bs, that of course have to take either the left or the right hand path, every single neutron took both paths at the same time.  This violates the Leggett-Garg Inequality and is a crystal-clear hallmark of an inherently indeterminate system.

"Our experiment shows that nature really is as strange as quantum theory claims," said study co-author Stephan Sponar.  "No matter which classical, macroscopically realistic theory you come up with, it will never be able to explain reality.  It doesn't work without quantum physics."

Now, mind you, I'm not saying I completely understand this.  As Richard Feynman himself put it, "I think we can safely say that no one understands quantum physics."  (And if the great Feynman could say this, it doesn't leave much room for a rank amateur like me to pontificate about it.)  But perhaps the most fitting way to end is with a quote by the brilliant biologist J. B. S. Haldane: "The world is not only queerer than we suppose, it is queerer than we can suppose."

****************************************

Coincidence and meaning

A friend and loyal reader of Skeptophilia sent me a link to an interview with author Sharon Hewitt Rawlette about her recent book, The Source and Significance of Coincidences, along with a note saying, "Would love to hear your thoughts about this."

I'm usually loath to give my opinion about a claim after reading a summary, book review, or interview without reading the book itself, but considering that I had issues with just about everything in the interview I can say with some confidence that it's unlikely the book would make me any less doubtful.  Rawlette's idea is that coincidences -- at least some of them -- "mean something."  Other than two events coinciding, which is the definition of coincidence.  Here's how she defines it:

For me, a coincidence is something that is not blatantly supernatural. It could be just chance. But there’s part of you that says, "This seems more meaningful than that."  And maybe just seems a little too improbable to be explained as chance.  It seems too meaningful to you, personally, given where you are in your life.  It’s something that makes you wonder, "Is there something more?"

Coincidences can certainly be startling, I'll admit that.  I was on my way to an appointment a while back and was listening to Sirius XM Radio's classical station "Symphony Hall," and one of my favorite pieces came on -- Beethoven's Moonlight Sonata.  I was maybe two-thirds of the way through the first movement when I arrived, and I was short on time so regretfully had to turn the music off and get out of the car.

When I opened the door to the waiting room, there was music coming over the speakers.  Beethoven's Moonlight Sonata -- at almost precisely the same spot where I'd turned off the radio.

Immediately, I wondered if they were also listening to Sirius XM, but they weren't.  It was the usual selection of calming music you hear in doctors' offices everywhere.  It really had been... "just a coincidence."

[Image licensed under the Creative Commons Karry manessa", Coincidence with Smile, CC BY-SA 4.0]

But did it mean anything?  How would I know?  And if it did mean something... what?

Rawlette tells us what her criteria are:

I don’t think there’s a really cut and dry answer.  There are a variety of factors that I look at in my own life when I’m trying to figure out whether something is just a coincidence or something more.  One of those is how improbable it really is...  But I also think an important element is how you feel about it.  What is your intuition telling you?  How strongly do you feel about it?  And is it telling you something that really seems to help you emotionally?  Spiritually?  Is it providing you with guidance?

Here, we're moving onto some seriously shaky ground.

First of all, there's improbability.  How do you judge that?  I'd say that the probability of a random selection on a classical music station being the same as the selection playing in a doctor's office at the same time is pretty damn low, but that's just a hand-waving "seems that way to me" assessment.  Amongst the difficulties is that humans are kind of terrible at statistical reckoning.  For example, let's say you throw two coins twenty times each.  With the first coin, you get twenty heads in a row.  With the second coin, you get the following:

HTTHHHTHTHTTHHHTHTTH

Which one of those two occurrences is likelier?

It turns out that they have exactly the same probability: (1/2)^20.  A very, very small number.  The reason most people pick the second as likelier is that it looks random, and comes close to the 50/50 distribution of heads and tails that we all learned was what came out of random coin-flips back in the seventh grade.  The first, on the other hand, looks like a pattern, and it seems weird and improbable.

The second problem is that here -- as with Rawlette's coincidences -- we're only assessing their probability after the fact.  In our coin flip patterns above, after they happen the probability that they happened is 100%.  I'll agree with her insofar as to say that in the first case (twenty heads in a row), I'd want to keep flipping the coin to see what would come up next, and if I keep getting heads, to see if I could figure out what was going on.  The second, corresponding much more to what I expected, wouldn't impel me to investigate further.

But the fact remains that as bizarre as it sounds, if you throw a (fair) coin a huge number of times -- say, a billion times -- the chance of there being twenty heads in a row somewhere in the array of throws is nearly 100%.  (Any statisticians in the studio audience could calculate for us what the actual probability is; suffice it to say it's pretty good.)

Third, of course, is that we run smack into our old friend dart-thrower's bias -- our hard-wired tendency to notice what seem to us to be outliers.  We don't pay any attention to all the times we walk into the doctor's office (or anywhere else) and the music playing isn't what we were just listening to, because it's so damn common.  The times the music is the same stand out -- and thus, we tend both to overcount them and weigh them more heavily in our attention and our memories.

Rawlette also doesn't seem to have any sort of criteria for telling the difference between random coincidence, meaningful coincidence, and something that is a deliberately targeted "sign" or "message" directed at you personally, other than how you feel about it:

I think the most impactful coincidences in people’s lives tend to be most improbable.  It’s very hard to explain them away.  But, the counterpart to that is that those coincidences also seem to have a very strong emotional impact on us.  They’re not only very improbable—very strange—but they carry a very strong emotional weight.  And we can’t escape that they’re significant somehow, even if we’re not exactly sure what the message is.  And, often, they do turn out to be life-changing.

So you are estimating how likely something is, assessing whether it was likely after the fact, deciding what the event's significance is, and deciding what the message (if any) consisted of.  It's putting a lot of confidence in our own abilities to perceive and understand the world correctly.  And if there's one thing I've learned from years of teaching neuroscience, it's that our sensory/perceptive and cognitive systems are (as Neil deGrasse Tyson put it) "poor data-taking devices... full of ways of getting it wrong."  I don't trust my own brain most of the time.  It's got a poor, highly-distractible attention span, an unreliable memory, and gets clogged up with emotions all too easily.  It's why I went into science; I learned really early that my personal interpretations of the world were all too often wrong, and I needed a more rigorous, reliable algorithm for determining what I believed to be true.

Now, I won't say I'm never prone to giving emotional weight to events after the fact.  As an example, I was quite close to my Aunt Pauline, my grandfather's youngest sister (youngest of twelve children!).  Pauline was a sweet person, childless and ten years a widow, when I was going to college at the University of Louisiana.  Every once in a while -- maybe every two or three months or so -- I'd stop by her house on the way home from school.  It wasn't far out of the way, and she was always thrilled to see me, and would bring out the coffee and a tray of cookies to share as we chatted.  One day, it occurred to me that it'd been a while since I'd seen her.  I don't know why she came to my mind; nothing I can think of reminded me.  I just suddenly thought, "I should stop by Aunt Pauline's and see how she's doing."

So I did.  She was cheerful as ever, and we had a lovely visit.

Two days later, she died of a heart attack at age 73.

I don't think I'd be human if the thought "how strange I was impelled to visit her!" didn't go through my mind.  But even back then, when I was twenty years old and much more prone to believe in unscientific explanations for things, it didn't quite sit right with me.  I visited with Aunt Pauline regularly anyhow; it certainly wasn't the first time I'd gotten in my car at the university and thought, "Hey, I should drop by."  I had lots of other older relatives who had died without my being at all inclined to visit immediately beforehand.  The "this is weird" reaction I had was understandable enough, but that by itself didn't mean there was anything supernatural going on.

I was really glad I'd gotten to see her, but I just didn't --and don't -- think I was urged to visit her by God, the Holy Spirit, the collective unconscious, or whatnot.  It was simply a fortuitous, but circumstantial, coincidence.

Rawlette then encourages us not to passively wait around for meaningful coincidences to occur to us, but to seek them out actively:

I think one of the most important things, when you experience a coincidence, is to keep an open mind about where it’s coming from and what it might mean.  Because it’s very easy to try to fit a coincidence into the way of thinking about the world that we already have—whatever our worldview is.  And coincidences generally come into our lives to expand that worldview.  They generally won’t fit neatly into the boxes that we have.  We might try to shove them in there, so we can stop thinking about it and make them less mysterious, but they generally are going to make us question some things that we thought we knew about the world.

What this puts me in mind of is the odd pastime of being a "Randonaut" -- using a random number generator to produce a set of geographical coordinates near you, going there, and looking for something strange -- about which I wrote a couple of years ago.  People report finding all sorts of bizarre things, some of them quite disturbing, while doing this.  I won't deny that it's kind of a fun concept, and no intrinsically weirder than my wife's near-obsession with geocaching, but it suffers from the same problems we considered earlier when you try to ascribe too much meaning to what you find.  If you're told to go to a random location and look around until you find something odd, with no criteria and no limitations, you're putting an awful lot of confidence in your own definition of "odd."  And, as I point out in the post, in my experience Weird Shit is Everywhere.  Wherever you are, if you look hard enough, you can find something mysterious, something that seems like a coincidence or a message or (at least) a surprise, but all that means is you had no real restrictions on what you were looking for, and that the world is an interesting place.

As an aside, this reminds me of my college friend's proof that all numbers are interesting:

• Assume that there are some numbers that are uninteresting.
• Let "x" be the first such number.
• Since being the first uninteresting number is itself interesting, this contradicts our initial assumption, and there are no uninteresting numbers.

Anyhow, all this rambling is not meant to destroy your sense that the universe we live in is mysterious and beautiful.  It is both, and much more.  I am just exceedingly cautious about ascribing meaning to events without a hell of a lot more to go on than my faulty intuition.  I'd much rather rely on the tried-and-true methods of science to determine what's out there, which for me uncovers plenty enough stunningly bizarre stuff to occupy my mind indefinitely.

But like I began with: I haven't read Rawlette's book, and if you have and I'm missing the point, please enlighten me in the comments section.  I don't want to commit the Straw Man fallacy, mischaracterizing her claim and then arguing against that mischaracterization.  But from her interview, all I can say is that I'm not really buying it.

On the other hand, if the next few times I go from my car to an office, exactly the same music is playing again and again, I'll happily reconsider my stance -- all arguments about the statistics of flipping twenty heads in a row notwithstanding.

**************************************

This week's Skeptophilia book-of-the-week is a bit of a departure from the usual science fare: podcaster and author Rose Eveleth's amazing Flash Forward: An Illustrated Guide to the Possibly (and Not-So-Possible) Tomorrows.

Eveleth looks at what might happen if twelve things that are currently in the realm of science fiction became real -- a pill becoming available that obviates the need for sleep, for example, or the development of a robot that can make art.  She then extrapolates from those, to look at how they might change our world, to consider ramifications (good and bad) from our suddenly having access to science or technology we currently only dream about.

Eveleth's book is highly entertaining not only from its content, but because it's in graphic novel format -- a number of extremely talented artists, including Matt Lubchansky, Sophie Goldstein, Ben Passmore, and Julia Gförer, illustrate her twelve new worlds, literally drawing what we might be facing in the future.  Her conclusions, and their illustrations of them, are brilliant, funny, shocking, and most of all, memorable.

I love her visions even if I'm not sure I'd want to live in some of them.  The book certainly brings home the old adage of "Be careful what you wish for, you may get it."  But as long as they're in the realm of speculative fiction, they're great fun... especially in the hands of Eveleth and her wonderful illustrators.

[Note: if you purchase this book from the image/link below, part of the proceeds goes to support Skeptophilia!]

Miraculous mathematics

I've blogged before about "miraculous thinking" -- the idea that an unlikely occurrence somehow has to be a miracle simply based on its improbability.  But yesterday I ran into a post on the wonderful site RationalWiki that showed, mathematically, why this is a silly stance.

Called "Littlewood's Law of Miracles," after John Edensor Littlewood, the man who first codified it in this way, it goes something like this:

• Let's say that a "miracle" is defined as something that has a likelihood of occurring of one in a million.
• We are awake, aware, and engaged on the average about eight hours a day.
• An event of some kind occurs about once a second.  During the eight hours we are awake, aware, and engaged, this works out to 28,800 events per day, or just shy of a million events in an average month.  (864,000, to be precise.)
• The likelihood of observing a one-in-a-million event in a given month is therefore 1-(999,999/1,000,000)^1,000,000 , or about 0.63.  In other words, we have better than 50/50 odds of observing a miracle next month!

Of course, this is some fairly goofy math, and makes some rather silly assumptions (one discrete event every second, for example, seems like a lot).  But Littlewood does make a wonderful point; given that we're only defining post hoc the unlikeliness of an event that has already occurred, we can declare anything we want to be a miracle just based on how surprised we were that it happened.  And, after all, if you want to throw statistics around, the likelihood of any event happening that has already happened is 100%.

So, like the Hallmark cards say, Miracles Do Happen.  In fact, they're pretty much unavoidable.

You hear this sort of thing all the time, though, don't you?  A quick perusal of sites like Miracle Stories will give you dozens of examples of people who survived automobile accidents without a scratch, made recoveries from life-threatening conditions, were just "in the right place at the right time," and so on.  And it's natural to sit up and take notice when these things happen; this is a built-in perceptual error called dart-thrower's bias.  This fallacy is named after a thought experiment of being in a pub while there's a darts game going on across the room, and simply asking the question: when do you notice the game?  When there's a bullseye, of course.  The rest is just background noise.  And when you think about it, it's very reasonable that we have this bias.  After all, what has the greater evolutionary cost -- noticing the outliers when they're irrelevant, or not noticing the outliers when they are relevant?  It's relatively obvious that if the unusual occurrence is a rustle in the grass, it's far better to pay attention to it when it's the wind than not to pay attention to it when it's a lion.

And of course, on the Miracle Stories webpage, no mention is made of all of the thousands of people who didn't seem to merit a miracle, and who died in the car crash, didn't recover from the illness, or were in the wrong place at the wrong time.  That sort of thing just forms the unfortunate and tragic background noise to our existence -- and it is inevitable that it doesn't register with us in the same way.

So, we should expect miracles, and we are hardwired to pay more attention to them than we do to the 999,999 other run-of-the-mill occurrences that happen in a month.  How do we escape from this perceptual error, then?

Well, the simple answer is that in some senses, we can't.  It's understandable to be surprised by an anomalous event or an unusual pattern.  (Think, for example, how astonished you'd be if you flipped a coin and got ten heads in a row.  You'd probably think, "Wow, what's the likelihood?" -- but any other pattern of heads and tails, say, H-T-T-H-H-H-T-H-T-T -- has exactly the same probability of occurring.  It's just that the first looks like a meaningful pattern, and the second one doesn't.)  The solution, of course, is the same as the solution for just about everything; don't turn off your brain.  It's okay to think, at first, "That was absolutely amazing!  How can that be?", as long as afterwards we think, "Well, there are thousands of events going on around me right now that are of equally low probability, so honestly, it's not so weird after all."

All of this, by the way, is not meant to diminish your wonder at the complexity of the universe, just to direct that wonder at the right thing.  The universe is beautiful, mysterious, and awe-inspiring.  It is also, fortunately, understandable when viewed through the lens of science.  And I think that's pretty cool -- even if no miracles occur today.