Skeptophilia (skep-to-fil-i-a) (n.) - the love of logical thought, skepticism, and thinking critically. Being an exploration of the applications of skeptical thinking to the world at large, with periodic excursions into linguistics, music, politics, cryptozoology, and why people keep seeing the face of Jesus on grilled cheese sandwiches.
Showing posts with label scientific method. Show all posts
Showing posts with label scientific method. Show all posts

Saturday, July 12, 2025

Mental models and lying stones

Richard Feynman famously said, "The first principle is that you must not fool yourself -- and you are the easiest person to fool."

This insightful statement isn't meant to impugn anyone's honesty or intelligence, but to highlight that everyone -- and I'm sure Feynman was very much including himself in this assessment -- has biases that prevent them from seeing clearly.  We've already got a model, an internal framework by which we interpret what we experience, and that inevitably constrains our understanding.

As science historian James Burke points out, in his brilliant analysis of the scientific endeavor The Day the Universe Changed, it's a trap that's impossible to get out of.  You have to have some mental model for how you think the world works, or all the sensory input you receive would simply be chaos.  "Without a structure, a theory for what's there," Burke says, "you don't see anything."

And once you've settled on a model, it's nearly impossible to compromise with.  You're automatically going to take some things as givens and ignore others as irrelevant, dismiss some pieces of evidence out of hand and accept others without question.  We're always taking what we experience and comparing it to our own mental frameworks, deciding what is important and what isn't.  When my wife finished her most recent art piece -- a stunning image of a raven's face, set against a crimson background -- and I was on social media later that day and saw another piece of art someone had posted with a raven against red -- I shrugged and laughed and said, "Weird coincidence."

Quoth the Raven, pen/ink/watercolor by Carol Bloomgarden (2025) [Image used with permission]

But that's only because I had already decided that odd synchronicities don't mean anything.  If I had a mental model that considered such chance occurrences as spiritually significant omens, I would have interpreted that very, very differently.

Our mental frameworks are essential, but they can lead us astray as often as they land us on the right answer.  Consider, for example, the strange, sad case of Johann Beringer and the "lying stones."

Johann Bartholomeus Adam Beringer was a professor of medicine at the University of Würzburg in the early eighteenth century.  His training was in anatomy and physiology, but he had a deep interest in paleontology, and had a large collection of fossils he'd found during hikes in his native Germany.  He was also a devout Lutheran and a biblical literalist, so he interpreted all the fossil evidence as consistent with biblical events like the six-day creation, the Noachian flood, and so on.

Unfortunately, he also had a reputation for being arrogant, humorless, and difficult to get along with.  This made him several enemies, including two of his coworkers -- Ignace Roderique, a professor of geography and algebra, and Johann Georg von Eckhart, the university librarian.  So Roderique and von Eckhart hatched a plan to knock Beringer down a peg or two.

They found out where he was planning on doing his next fossil hunt, and planted some fake fossils along the way.

These "lying stones" are crudely carved from limestone.  On some of them, you can still see the chisel marks.


More outlandish still, Roderique and von Eckhart carved the word "God" in Hebrew on the backs of some of them.  Making it look like the artisan had signed His name, so to speak.

One colleague -- who was not in on the prank -- looked at the stones, and said to Beringer, "Um... are you sure?  Those look like chisel marks."  Beringer dismissed his objections, and in fact, turned them into evidence for his explanation.  Beringer wrote, "...the figures... are so exactly fitted to the dimensions of the stones, that one would swear that they are the work of a very meticulous sculptor...[and they] seem to bear unmistakable indications of the sculptor's knife."

They were so perfect, Beringer said, that they could only be the work of God.

So as astonishing as it may seem, Beringer fell for the ruse hook, line, and sinker.  Roderique and von Eckhart, buoyed up by their success, repeated their prank multiple times.  Finally Beringer had enough "fossils" that in 1726, he published a scholarly work called Lithographiae Wirceburgensis (The Writing-Stones of Würzburg).  But shortly after the book's publication -- it's unclear how -- Beringer realized he'd been taken for a ride.

He sued Roderique and von Eckhart for defamation -- and won.  Roderique and von Eckhart were both summarily fired, but it was too late; Beringer was a laughingstock in the scientific community.  He tried to recover all of the copies of his book and destroy them, but finally gave up.  His reputation was reduced to rubble, and he died twelve years later in total obscurity.

It's easy to laugh at Beringer's credulity, but the only reason you're laughing is because if you found such a "fossil," your mental model would immediately make you doubt its veracity.  In his framework -- which included a six-thousand-year-old Earth, a biblical flood, and a God who was perfectly capable of signing his own handiwork -- he didn't even stop to consider it.

The history of science is laden with missteps caused by biased mental models.  In 1790, a report of a fireball over France that strewed meteorites over a large region prompted a scientific paper -- that laughingly dismissed the claim as "impossible."  Pierre Bertholon, editor of the Journal des Sciences Utiles, wrote, "How sad, is it not, to see a whole municipality attempt to certify the truth of folk tales… the philosophical reader will draw his own conclusions regarding this document, which attests to an apparently false fact, a physically impossible phenomenon."  DNA was dismissed as the genetic code for decades, because of the argument that DNA's alphabet only contains four "letters," so the much richer twenty-letter alphabet of proteins (the amino acids) must be the language of the genes.  Even in the twentieth century, geologists didn't bother looking for evidence for continental drift until the 1950s, long after there'd been significant clues that the continents had, in fact, moved, largely because they couldn't imagine a mechanism that could be responsible.

Our mental models work on every level -- all the way down to telling us what questions are worth investigating.

So poor Johann Beringer.  Not to excuse him for being an arrogant prick, but he didn't deserve to be the target of a mean-spirited practical joke, nor does he deserve our derision now.  He was merely operating within his own framework of understanding, same as you and I do.

I wonder what we're missing, simply because we've decided it's irrelevant -- and what we've accepted as axiomatic, and therefore beyond questioning?

Maybe we're not so very far ahead of Beringer ourselves.

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Tuesday, February 4, 2025

The riddle of the sun stones

When you think about it, it's unsurprising that our ancestors invented "the gods" as an explanation for anything they didn't understand.

They were constantly bombarded by stuff that was outside of the science of their time.  Diseases caused by the unseen action of either genes or microorganisms.  Weather patterns, driven by forces that even in the twenty-first century we are only beginning to understand deeply, and which controlled the all-important supply of food and water.  Earthquakes and volcanoes, whose root cause only began to come clear sixty years ago.

Back then, everything must have seemed as mysterious as it was precarious.  For most of our history, we've been at the mercy of forces we didn't understand and couldn't control, where they were one bad harvest or failed rainy season or sudden plague from dying en masse.

No wonder they attributed it all to gods and sub-gods -- and devils and demons and witches and evil spirits.

As much as we raise an eyebrow at the superstition and seeming credulity of the ancients, it's important to recognize that they were no less intelligent, on average, than we are.  They were trying to make sense of their world with the information they had at the time, just like we do.  That we have a greater knowledge base to draw upon -- and most importantly, the scientific method as a protocol -- is why we've been more successful.  But honestly, it's no wonder that they landed on supernatural, unscientific explanations; the natural and scientific ones were out of their reach.

The reason this comes up is a recent discovery that lies at the intersection of archaeology and geology, which (as regular readers of Skeptophilia know) are two enduring fascinations for me.  Researchers excavating sites at Vasagård and Rispebjerg, on the island of Bornholm, Denmark, have uncovered hundreds of flat stone disks with intricate patterns of engraving, dating from something on the order of five thousand years ago.  Because many of the disks have designs of circles with branching radial rays extending outward, they've been nicknamed "sun stones."  Why, in around 2,900 B.C.E., people were suddenly motivated to create, and then bury, hundreds of these stones, has been a mystery.

Until now.

[Image credit: John Lee, Nationalmuseet, Copenhagen, Denmark]

Data from Greenland ice cores has shown a sudden spike in sulfates and in dust and ash from right around the time the sun stones were buried -- both hallmarks of a massive volcanic eruption.  The location of the volcano has yet to be determined, but what is clear is that it would have had an enormous effect on the climate.  "It was a major eruption of a great magnitude, comparable to the well-documented eruption of Alaska’s Okmok volcano in 43 B.C.E. that cooled the climate by about seven degrees Celsius," said study lead author Rune Iversen, of the Saxo Institute at the University of Copenhagen.  "The climate event must have been devastating for them."

The idea that the volcanic eruption in 2,900 B.C.E. altered the climate worldwide got a substantial boost with the analysis of tree rings from wood in Europe and North America.  Right around the time of the sulfate spike in the Greenland ice cores, there's a series of narrow tree rings -- indicative of short growing seasons and cool temperatures.  Wherever this eruption took place, it wrought havoc with the weather, with all of the results that has on human survival.

While the connection between the eruption and the sun stones is an inference, it certainly has some sense to it.  How else would you expect a pre-technological culture to respond to a sudden, seemingly inexplicable dimming of the sun, cooler summers and bitter winters with resultant probable crop failures, and even the onset of wildly fiery sunrises and sunsets?  It bears keeping in mind that our own usual fallback of "there must be a scientific explanation even if I don't know what it is" is a relatively recent development. 

So while burying engraved rocks might seem like a strange response to a climatic change, it is understandable that the ancients looked to a supernatural solution for what must have been a mystifying natural disaster.  And we're perhaps not so very much further along, ourselves, given the way a substantial fraction of people in the United States are responding to climate change even though the models have been predicting this for decades, and the evidence is right in front of our faces.  We still have plenty of areas we don't understand, and are saddled with unavoidable cognitive biases even if we do our best to fight them.  As the eminent science historian James Burke put it, in his brilliant and provocative essay "Worlds Without End":

Science produces a cosmogony as a general structure to explain the major questions of existence.  So do the Edda and Gilgamesh epics, and the belief in Creation and the garden of Eden.  Myths provide structures which give cause-and effect reasons for the existence of phenomena.  So does science.  Rituals use secret languages known only to the initiates who have passed ritual tests and who follow the strictest rules of procedure which are essential if the magic is to work.  Science operates in the same way.  Myths confer stability and certainty because they explain why things happen or fail to happen, as does science.  The aim of the myth is to explain existence, to provide a means of control over nature, and to give to us all comfort and a sense of place in the apparent chaos of the universe.  This is precisely the aim of science.

Science, therefore for all the reasons above, is not what it appears to be.  It is not objectively impartial, since every observation it makes of nature is impregnated with theory.  Nature is so complex, and sometimes so seemingly random, that it can only be approached with a systematic tool that presupposes certain facts about it.  Without such a pattern it would be impossible to find an answer to questions even as simple as "What am I looking at?"
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Tuesday, October 8, 2024

Borley Rectory, and the problem with anecdote

There's a reason skeptics have a problem with anecdotal evidence and eyewitness testimony.

It's not that that it's impossible that you saw a ghost, or Bigfoot, or an extraterrestrial spacecraft.  What we're saying is that we need more than your assurance that you did.  Not only do we have the potential for outright lies and hoaxes -- some of them very subtle and clever -- we have the fact that the human sensory apparatus more or less sucks.

To put not too fine a point on it.

I mean, it works well enough.  It keeps us sufficiently aware of our surroundings to stay alive.  But we're easily tricked, we miss things, we misinterpret what we see and hear.  As astronomer Neil deGrasse Tyson put it, "The human perceptual system is rife with all sorts of ways of getting it wrong."

As an illustration, let's consider one of the most famous "haunted house" stories in the world -- the infamous Borley Rectory, of Borley, Essex, England.

Borley Rectory always shows up on those websites with names like, "Ten Most Terrifying Real Ghost Stories!", usually somewhere near the top of the list.  So here are the bare bones of the story, just in case you don't know it.

Borley Rectory was built in 1862 by Reverend Henry Dawson Ellis Bull, Rector of Borley Parish.  He designed the building to replace an earlier rectory that had burned down in 1841, and also to accommodate his wife and family of fourteen children, which indicates that Reverend Bull put a lot of stock in the "be fruitful and multiply" thing from the Book of Genesis.

Be that as it may, the parish was certainly steeped in history.  The parish church is thought to date to the twelfth century, and the town was the site of Borley Hall, the ancestral seat of the Waldegrave family.  But here's where truth starts twisting in with fabrication; because the additional claim that the rectory had been built on the site of an old Benedictine monastery appears to have no basis in reality.

Which means that the tale that is the basis of the haunting also is of dubious provenance.  Because the story goes that a monk in the (almost certainly non-existent) monastery was having an affair with a nun from a nearby convent.  They made plans to elope, and had in fact arranged a coach driven by a friend of the monk's in order to get away, but the plan was discovered.

Sexual indiscretion by the clergy was a major no-no back then.  The coachman was beheaded, the monk hanged, and the nun bricked up in a wall inside the convent.

Except... none of them existed, remember?  Because there's no evidence there ever was a monastery on the rectory grounds.

But that didn't stop the tale from growing. Here's one account of what Reverend Bull et al. saw:
On July 28th, 1900, three Bull daughters reportedly saw a figure on a path, which later became known as the "Nuns Walk", to the rear of the rectory.  They were joined by a fourth sister to help greet the stranger, but the apparition disappeared.  Harry also told of seeing the nun, together with the phantom coach in which she had eloped.
 
She was also seen wandering the grounds around the Rectory, in and out of the bushes, dressed in grey.  There are reports of the Monk and Nun passing across the grounds.  Several people said they observed "A lady in grey cloak" and "A gentleman with a sort of bald head, dressed in a long black gown."
Once the story of the haunting began to spread, others reported seeing spectral nuns and monks.  But then events accelerated.  A later rector of the parish, one Lionel Foyster, moved in in 1930 with his wife Marianne, and they began to experience poltergeist activity in addition to the continuing presence of ghostly figures loping about.  Marianne began to receive messages written on walls and scraps of paper, such as the following:


Both of the Foysters reported having peculiar experiences:
During the first year of their tenancy, Lionel described many unexplained happenings including; bell ringing, the appearance of Harry Bull [son of the first rector of Borley], glass objects appearing out of nowhere and being dashed to the floor, books appearing, and many items being thrown, including pebbles and an iron.  After an attempt at exorcism, Marianne was thrown out of bed several times.
The Foysters eventually moved out, apparently because of Lionel Foyster's declining health, and afterwards no one could be found who was willing to live in the rectory, almost certainly because of its reputation.

And then Harry Price got involved.

Price was a psychic investigator of significant fame, who had founded the National Laboratory of Psychic Research as a rival to the far more reputable Society for Psychical Research.  Price himself was a strange mixture of skeptic and sketchy.  He was instrumental in unmasking outright hoaxers such as Helen Duncan, who used cheesecloth and paper soaked in egg white to simulate "ectoplasm."  But his investigation of Borley Rectory, leading to the publication of a book in 1940, was unequivocally in support of its having been haunted -- despite a stinging critique by researchers for the SPR who said that Price himself was a trained conjuror (which was true), and had "salted the mine" by faking some of the evidence from Borley, in collusion with Marianne Foyster, who "was actively engaged in fraudulently creating [haunted] phenomena."

Price, of course, denied any such thing, but further inquiries by the SPR left his role in the alleged haunting in serious question.  And the matter came to an unexpected close when the rectory burned in 1939 because of an accident with an oil lamp.

The remnants of the building were demolished in 1944.  But people still visit the site and the adjacent cemetery, and still report ghostly appearances, lo unto this very day.

See what I mean about anecdote?  We have a story that started out with a most-likely-false claim of three executions on the rectory grounds, followed by what many believe was an outright hoax perpetrated by Harry Price and Marianne Foyster.  Blend that together with overactive imaginations, and the rather dubious quality of the human perceptual systems, and you have a mishmash out of which any kernel of truth -- if there is one there -- becomes impossible to discern.

So is Borley haunted?  The most honest answer is "there's no way to know for sure," with a strong corollary of "... but probably not."  There's nothing here that any unbiased individual would consider hard evidence, just tall tale piled upon unsubstantiated claim, mixed with "I heard that people saw ghosts there."

If this is "one of the best-authenticated haunted sites in Britain," as one website claimed, we've got some serious problems.

To return to my initial point, it's not that I'm saying that any of the claims of the paranormal are impossible.  What I'm saying is that thus far, no evidence I've seen has been convincing, at least not to someone who wasn't already convinced.  But despite all that, I'm hoping to visit Borley next time I'm in the UK, and if I do, I'll definitely report back with anything I happen to see.

Not that it should make a difference.  Because eyewitness testimony is still subject to all of the caveats I've mentioned -- even if it comes from yours truly.

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Tuesday, July 2, 2024

Measure for measure

In yesterday's post we looked at one bizarre human obsession, which is drawing lines all over the place and pretending they represent something real.  Today we're going to look at another, which is our penchant for quantifying everything.

Certainly, accurate measurement is critical in science; data, for the most part, is numerical, and most models these days are mathematical representations of reality.  But still, there's a strange aspect to it, which British science historian James Burke got at in his brilliant series The Day the Universe Changed:

[T]he structural view of things at the time controls what science does at every level.  From the cosmic questions about the whole universe, to what bits of that universe are worth investigating, to how far you let the questions take you, what experiments to do, what evidence you can and can't accept.  And down at that detailed level, the control still operates, because it even tells you what instruments you should use.  And of course, at this stage, you're looking for data to prove your theory, so you design the kind of instruments to find the kind of data you reckon you're going to find.  The whole argument comes full circle when you get the raw data itself.  Because it isn't raw data.  It's what you planned to find from the start.

He goes on to make the important point that true leaps in understanding occur when the unexpected occurs, and some piece of the data doesn't fit with the existing model; then (assuming the data are verified and found to be correct), there's no choice but to revise the model -- or trash it entirely and start over.

[Image is in the Public Domain]

But what this has done is created a morass of different units of measurement, and I'm not referring solely to my own country's pig-headed insistence on avoiding the use of the metric system.  Imperial units -- feet, miles, pounds, quarts, and so on -- are certainly cumbersome (check out this hilarious video if you want to find out just how awkward they are), but they're not the weirdest ways that humans have chosen to subdivide the natural world.  So for your edification, here are a few of the stranger units of measurement I've run into:

  • the micromort -- defined as a one-in-a-million chance of death.  For example, smoking a cigarette and a half increases your chance of dying by about one micromort.
  • a jiffy is 1/60 of a second, from the vertical refresh period on NTSC analog video hardware running on American (60 Hertz) alternating current.  So next time someone tells you, "I'll be back in a jiffy," you can confidently respond, "I seriously doubt that."
  • so many people in Britain publicly compared the areas of geographical regions to the size of Wales that it led to a unit of area, the nanowales -- one billionth the area of Wales, or about 20.78 square meters.
  • the Sverdrup, named after Norwegian oceanographer Harald Sverdrup, at least has its basis in metric units.  It's a unit of flow rate, equal to one million cubic meters per second.  Being as huge as it is, you might imagine it has limited utility -- in fact, it's pretty much only used in oceanography and meteorology.  (For reference, the flow rate of the Gulf Stream varies between 30 and 150 Sverdrup, depending on where you measure it and what you consider its boundaries to be.)
  • the dolor is a unit of pain.  One dolor is equal to the difference between two levels of pain that is just noticeable.  The subjective nature of pain has resulted in it not being widely accepted in the medical community.
  • a millihelen is a unit of beauty, named after Helen of Troy -- the amount of beauty required to launch one ship.
  • when I taught dimensional analysis in physics, I had students practice converting from one set of units to another -- a useful skill when doing science.  I always made a point of having them convert velocities from meters per second to furlongs per fortnight, which firmly cemented in their brains that I have a screw loose.  (For what it's worth, a furlong is 660 feet, or about 201.17 meters; a fortnight is fourteen days, so 1,209,600 seconds.  Thus, the speed of light is about 1.8 terafurlongs per fortnight, a factoid you can bring out at the next cocktail party you attend, especially if you want people to find ways to avoid you for the rest of the evening.)
  • one mickey is the smallest resolvable movement possible with a computer mouse.  Most of them have a sensitivity of about five hundred mickeys per inch.
  • a Smoot is a unit of length, named after Harvard student Oliver R. Smoot.  The story is that one day in 1958, Smoot got falling-down drunk, and his buddies (who were also snookered but not as badly as Smoot was) were basically dragging him home, and decided to measure the length of the Harvard Bridge in Smoot-lengths (about 170 centimeters).  The bridge, they found, was 364.4 Smoots in length plus a little bit, so there's now a plaque saying "364.4 Smoots and an ear" on the bridge.  (Smoot went on, I shit you not, to be the chairperson of the American National Standards Institute and president of the International Organization for Standardization.  Talk about being destined for a particular career.)
  • the weirdest unit of volume I've ever heard of is the Hubble-barn.  This combines the Hubble length -- the radius of the known universe -- with a unit of area called the barn, which is used to measure the scattering cross-section of atomic nuclei and is equal to 10^-28 square meters.  One Hubble-barn is the volume of a rectangular solid that has a square face with an area of one barn stretching across the entire known universe.  If you do the calculation, it's way less volume than you'd think -- on the order of 13.1 liters.
  • last, we have the ohnosecond, which is the time elapsed between making a mistake and recognizing it, such as pressing "send" on an email describing details of some illicit but highly pleasurable activities you want to experience with a coworker with whom you're having a clandestine dalliance, and realizing too late that you forgot to change the "to" line from "Reply All."

So there you have it -- some ways to measure the world, some serious, some not so much.  In any case, I'd better wrap this up.  So far I've had only about 0.02 Hubble-barns of coffee, so I'm moving at a velocity of around a furlong per fortnight.  I should post this, and hope that there are at least a few ohnoseconds between hitting "Publish" and seeing what I've wrought.

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Thursday, September 8, 2022

Continental bombardment

One of the reasons science is so useful is that our intuition about how things work is so often wrong.

A good example is the classic physics thought-experiment about taking two bullets -- one loaded in a gun that has the barrel perfectly horizontal, the other one held in your hand at the same height.  You fire the gun over level ground, and simultaneously let go of the bullet.  Which hits the ground first?

It seems like they should take different amounts of time; the one shot from the gun is traveling much farther, for one thing.  Most people think because of that, the dropped bullet would hit the ground first.  In fact, you undoubtedly know that (omitting the effects of air resistance or uneven terrain), the two bullets hit the ground at precisely the same time; Isaac Newton showed that the horizontal and vertical components of velocity are completely independent of one another.  It doesn't matter that the shot bullet is traveling rapidly in the horizontal direction; it and the dropped bullet have exactly the same vertical acceleration, namely 9.8 meters per second per second downward, starting from rest.  Thus they take exactly the same amount of time to hit the ground.

I was reminded of another example of this by some cool new research (which I will get to presently) I ran across yesterday.  It has to do with geology, namely, what the crust and mantle of the Earth are like.  It seems common-sensical that the surface of the Earth is uniformly cool and rocky, and the interior (judging by volcanoes) is molten; and while that isn't wrong in a broad-brush sort of way, what it misses is that there's a big difference between the rocks currently under your feet and the rocks at the bottom of the deep ocean.  Continental crust is thick, and extends both upwards into the air and downward into the mantle, a little like an iceberg; the rocks that make up the continents are, on the whole, lighter than oceanic crust, which is thin, brittle, and dense.  So the continents are literally floating in the liquid rock of the upper mantle.

This, of course, is what gives rise to plate tectonics; those iceberg-like blobs of floating rock we call continents, and the thin, heavy slabs of deep oceanic crust, jostle around on the magma of the upper mantle, colliding, pulling apart, shifting, and subducting (one piece going underneath another), and that gives rise to most of the geologic processes you've heard about.

But here's where we run into a fascinating question; why is the chemistry of continental rock (and thus its density) so different than oceanic rock?

[Image licensed under the Creative Commons Eric Gaba (Sting - fr:Sting), Tectonic plates boundaries detailed-en, CC BY-SA 2.5]

A piece of research out of Curtin University (Australia), published this week in Geology, suggests a surprising answer: the material that makes up the cratons -- the large, stable blocks of rock that form the nuclei of continents -- is extraterrestrial in origin.

Chris Kirkland, lead author of the study, was looking at the age of rocks in cratons around the world, and found something curious; their production seemed to occur at (roughly) two hundred million year intervals.  The formation of these blocks of rock coincide with the points at which the Solar System was passing through an area of dense stars in the spiral arm of the Milky Way as it orbited the Galactic Center.

"From looking at the age and isotopic signature of minerals from both the Pilbara Craton in Western Australia and North Atlantic Craton in Greenland, we see a similar rhythm of crust production, which coincides with periods during which the Solar System journeyed through areas of the galaxy most heavily populated by stars," Kirkland said.  "When passing through regions of higher star density, comets would have been dislodged from the most distant reaches of the Solar System, some of which impacted Earth.  Increased comet impact on Earth would have led to greater melting of the Earth’s surface to produce the buoyant nuclei of the early continents... Linking the formation of continents, the landmasses on which we all live and where we find the majority of our mineral resources, to the passage of the Solar System through the Milky Way casts a whole new light on the formative history of our planet and its place in the cosmos."

Of course, we've known for a while that all of the rock on Earth ultimately came from the coalescence of asteroids as the Solar System formed; but it's weird to think that the rock we're currently sitting atop may have been thrown at us by the near passage of other stars to our Sun as the entire Solar System hurtled its way around its host galaxy.  Whether Kirkland's claim will bear out under scrutiny, I don't know; but what's certain is that the methods of science has opened our eyes to a myriad processes that would have been entirely opaque to our so-called common sense.  Yes, scientists do get it wrong sometimes; they're fallible, and can misinterpret data or get hung up on their biases just like anyone.  But only science provides a protocol for catching and fixing those mistakes.

So it may not be perfect -- but for getting near to the truth, science really is the only game in town.

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Thursday, June 23, 2022

The domino effect

I find it fascinating how many important discoveries were made more or less by accident -- either because a researcher was looking for something and stumbled upon something else, or because (s)he was just playing around in the lab and noticed something cool.

Here are a few of my favorite examples:

  1. Two researchers, George Beadle and Edward Tatum, were researching nutrition in a mold called Neurospora, and were particularly interested in why some strains of Neurospora starved to death even when given adequate amounts of food.  Their research generated the concept of "one gene-one protein" -- the basis of our understanding of how genes control traits.
  2. Charles Richet was studying how the toxin of a rare species of jellyfish affects the body.  His research led to the discovery of how anaphylactic shock works -- and the development of the epi pen, saving countless lives from death because of bee sting allergies, nut allergies, and so on.
  3. Wilhelm Röntgen was researching the newly-invented cathode-ray tube, which at that point had no practical applications whatsoever.  That is, he was playing around.  He noticed that when he activated the tube, even though it was completely covered, some fluorescent papers at the other end of the room began to glow in the dark.  He had just discovered x-rays.
  4. In 1945 an engineer named Percy Spencer was working with a device called a magnetron that looked like it might have applications in ground-based radar systems.   While messing about with it, he noticed that a chocolate bar in his pocket had melted.   He patented a design that year that we now call the "microwave oven."
  5. Alexander Fleming was something of a ne'er-do-well in the scientific world.  He did a lot of raising of bacteria on plates, and his favorite hobby was to take brightly-colored species of bacteria and paint them on agar media to make pictures.  One day, a mold spore blew in and landed on one of his picture-cultures and spoiled it.  His further investigation of how the mold spoiled the culture led to the discovery of the first antibiotic, penicillin.
  6. Roy Plunkett was working with gases that could be used to quickly cool vessels in scientific experiments, and after one failure he found that the vessel was left coated with a slick substance.  He eventually named it "Teflon."
Stuff like this is one good reason to support pure research.  The criticism "I don't see what possible application this can have" is best answered, "you don't see what possible application this can have yet."  We never lose by finding out more about the universe we live in.  My own opinion is that there's a benefit to knowing stuff even if it never does have a practical use; but even if you're a pure utilitarian, there's no question that putting money into pure research pays back far more than it costs.

This came to mind because of a cool study that appeared two weeks ago in the journal Physical Review Applied about what controls the speed of collapse of a row of dominoes.

[Image licensed under the Creative Commons Nara Cute, Wallpaper kartu domino, CC BY-SA 4.0]

It's hard not to be fascinated by the phenomenon.  For example, take five minutes and watch this amazing record-setting collapse -- of thirty-two thousand dominoes:



Watching this is kind of mesmerizing, but it did bring up two things: (1) I would never have the patience to do this, and (2) if I did, knowing my luck, I'd be somewhere around domino #31,500 and my dog would come galumphing in, knock the whole thing down, and I'd have to start over.

The researchers into the domino collapse phenomenon found that the speed with which the row collapses is dependent upon two things -- the friction between the dominoes and the surface they're standing on, and the friction between one domino and the next one in line.  If you want the collapse to propagate quickly, you want a high coefficient of friction between the dominoes and the surface, but a low coefficient between the dominoes and each other.  The former means that there won't be much slip-back as the domino falls -- as it tips over, the bottom corner of the domino stays put, and the piece pivots around that fulcrum so that most of the energy of the fall is transmitted into the next domino in line.  The latter means that once a domino hits the next one, the slipping of the two surfaces across each other doesn't get "hung up" and lose energy from friction as the top corner of one domino slides across the surface of the next one.

The result lines up pretty well with common sense, but it's cool to have it confirmed experimentally, and that the researchers actually came up with a mathematical model that predicts the velocity of the collapse.

So, what use is it?  We don't know.  Maybe it could be used as a model of systems you don't want to collapse -- like buildings.  But one of the fantastic things about science is when a model in one area of study turns out to illuminate something completely different.  Maybe the mathematical principles of domino collapse rate could be applied to other phenomena that are caused by a phase transition -- a sudden change between one state and another, often caused by a tiny input of energy (like a finger pushing over a single domino).  There are a number of familiar phenomena that involve rapid phase transitions, including earthquakes, explosive volcanic eruptions, breakup of the ice sheet over a river during the spring thaw, and the firing of a neuron once the stimulus threshold is crossed.

Like I said earlier: the beauty of pure research is that we don't know what kind of applications it might have.  With the brilliant minds in the scientific community, you never know where it might lead.  One little paper on a curiosity of physics might trigger new research and go places you'd never expect.

You might even call it...

... the domino effect.

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Saturday, March 12, 2022

Re-examining the ganzfeld

Today's post asks a question not because I'm trying to lead you toward a particular answer, but because I honestly don't know the answer myself.

In Wednesday's post, I discussed some alleged claims by psychics (which a team in Australia evaluated for accuracy and found seriously wanting) and made the statement, "I'm all for keeping an open mind about things, but at some point you have to conclude that a complete absence of hard evidence means there's nothing there to see."

A friend of mine responded, "What if there's hard evidence out there that you're choosing not to accept because you've already made your mind up?"  He wasn't being combative; like me, he was just asking a question, and it's actually a reasonable thing to ask.  And he sent me a link to a post over at Paranormal Daily News that looks at one of the most famous experimental setups for detecting psychic powers: the ganzfeld experiment.

"Ganzfeld" is German for "complete field," and refers to the fact that the test subjects are placed in near-complete sensory deprivation, in order to keep them from receiving any information accept (allegedly) from telepathy.  Padded goggles are placed over the eyes; earpieces play recordings of white noise.  The subjects lie flat in a place with no drafts or other air movement.  (Some have even had subjects floating in a sensory deprivation tank.)  Then the "sender" -- usually the researcher conducting the experiment -- looks at some kind of pattern, often the famous "Zener cards" (cards with five different geometric patterns in five different colors), and the "receiver" (the test subject) reports what (s)he sees/experiences.

A participant in a ganzfeld experiment [Image is in the Public Domain]

The Wikipedia page for the ganzfeld experiment (linked above) is unequivocal; it says, "[It] is a pseudoscientific technique to detect parapsychological phenomena...  Consistent, independent replication of ganzfeld experiments has not been achieved."  However, the article my friend sent is equally unequivocal in the opposite direction -- that it has generated results that are far outside of what would come out of a random-choice statistical model, and has been done over and over with the same outcome.  The author, Craig Weiler, writes:

This works.  Not perfectly, but certainly well enough for an experiment.  It’s been done more than enough times by more than enough people to rule out any statistical anomalies.  The success rate is typically between 32 and 35%.  That’s pretty normal for a successful, statistics based experiment.  There have been six different meta analyses from skeptics and researchers alike, all showing positive results.  From an objective scientific perspective, this is an ordinary successful scientific experiment.

While I can't say I warm to the sneery tone of the article -- Weiler really needs to learn the difference between a "skeptic" and a "scoffer" -- it does bring up the question of who's right, here.  The critics of the ganzfeld experiment and other such attempts to prove the existence of paranormal abilities claim that no sufficiently-controlled experiment has ever generated positive results; the supporters claim that there are plenty of positive results that all the scientists are ignoring because they can't explain them (or, worse, because those results contradict their own biases).

Weiler is right that there have been meta-analyses done of the ganzfeld results, and that they have changed the minds of neither the pro- nor the anti- factions.  Finding a truly unbiased analysis has turned out to be not to be easy.  A September 2020 article in Frontiers in Psychology by Thomas Rayberon comes the closest of anything I've seen, but unfortunately tries to steer a middle course of "maybe, maybe not" by agreeing with both sides at once even though they're saying opposite things.  Rayberon writes (citations have been removed in the interest of space; go to the original article if you're interested in seeing them):

Psi research can be considered as a subfield of consciousness studies concerned with interactions between individuals and their environment that transcends the ordinary constraints of space-time.  Different lines of research have been developed for more than a century to tackle psi using experimental research, spontaneous cases, clinical cases, selected participants, and applications.  Several meta-analyses of studies conducted under controlled conditions examine precognitive dreams, telepathy, and presentiment and have demonstrated statistically significant effects...

While these results support the existence of consistent anomalous experience/behavior that has been labeled “psi,” there is currently no consensus in the scientific community concerning their interpretation and two main positions have emerged so far.  The “skeptics” suppose that they are the consequences of errors, bias, and different forms of QRPs [questionable research practices].  The “proponents” argue that these results prove the existence of psi beyond reasonable doubt and that new research should move on to the analysis of psi processes rather than yet more attempts to prove its existence.  This absence of consensus is related to the difficulty of drawing firm conclusions from the results of psi research.  Indeed, they represent an anomaly because there is currently no scientific model – based on physical or biology principles – to explain such interactions even if they exist.
Which reminds me from the quote from Lord of the Rings, "Go not to the Elves for advice, for they will say both yes and no."  The last bit -- that there is no current scientific model that could account for psychic phenomena -- is certainly true; but if there are statistically significant effects (which Rayberon says explicitly in the preceding paragraph), then surely there must be some protocol for devising an experiment that meets the minimum criteria of the true skeptics (people who base their understanding on the evidence, regardless of what their preconceived notions might have said).  The fact that there is no current scientific model to explain telepathy is, while correct, entirely irrelevant.  The first thing to do is to determine if the phenomenon itself is real.  There was no scientific model to explain radioactivity when it was discovered by Henri Becquerel, nor the apparent constancy of the speed of light when it was demonstrated by Michelson and Morley, nor the patterns of inheritance uncovered by Gregor Mendel.  The first thing was to determine if what they were seeing was accurate.  Once that happened, the scientists moved on to trying to figure out a model that accounted for it.

Rayberon then goes on to make quite a puzzling statement that implies it might be impossible even to tell if the phenomenon is real.  Science, he says (again, correct most of the time) uses experimental protocols that eliminate any possibility of interference by the experimenter.  That's impossible in psi research (italics are the author's):
Thus, if psi exists, the problem is the following: an advertent or inadvertent “direct” interaction between the researcher and the object of study could be possible.  This destroys the conditions necessary for the convincing scientific demonstration of psi itself.

Rayberon says this "paradox" makes psi research impossible to confirm or disconfirm.  But isn't an interaction between the researcher and the test subject what the psi researchers themselves are trying to demonstrate?  What an honest psi researcher -- well, any honest researcher, really -- needs to do is to isolate the variable (s)he's studying so that, as far as is possible, whatever results come out of the experiment can only be attributable to that variable.  So in a properly-conducted ganzfeld experiment, the researcher has eliminated any possibility of the test subject getting information about the pattern from anywhere except the mind of the "sender."

And from my admittedly layperson's viewpoint, that can't be all that hard to do.  If there have been multiple instances of positive, statistically-significant results from ganzfeld trials -- and Weiler and Rayberon agree that there have been -- then they deserve some explanation other than shrugging and saying, "I don't see how it could work."  If there are "errors, biases, and questionable research practices" generating the results, the "skeptics" (using the word in the sense both Weiler and Rayberon use) need to determine what those are.  If, on the other hand, the results aren't from poor experimental design or outright cheating, then let's have the "skeptics" and "proponents" team up to find a protocol they can both agree to.

Figuring out a model for what's going on can wait until we see if there is anything going on.

So after accusing Rayberon of playing both sides, I'm honestly not doing much better.  My inclination is to doubt the existence of psi abilities because the evidence seems sketchy for such a wild claim.  But that inclination is a bias I'm well aware of, and all it would take is one sufficiently well-designed experiment to convince me I was wrong.  Right now, all that seems to be happening is both sides becoming more entrenched and yelling at each other across no-man's land, which doesn't accomplish much but pissing everyone off.

So come on, folks.  Either psi exists or it doesn't.  If it doesn't, we can go on to studying actual real phenomena.  If it does, it will overturn pretty much everything we know about psychology, and would be one of the most colossal discoveries in the past hundred years.  How about teaming up and settling this question once and for all?

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Saturday, January 1, 2022

The perimeter of ignorance

In the past week, I watched two things that were interesting in juxtaposition.

One of them came my way because for my holiday gift my wife got me a subscription to Master Class, which has hundreds of online video classes on everything from political science to cooking.  I signed up for and watched a series of lectures by the eminent astrophysicist Neil deGrasse Tyson, director of New York City's Hayden Planetarium, on the scientific endeavor and how to think effectively about science and how to talk about it to others.

In his class, Dr. Tyson says the following:
The frontier of discovery is a messy place.  You don't know what the next step is, sometimes you don't even know what question to ask.  As the area of our knowledge grows, so too does the perimeter of our ignorance.  It's thrilling and scary at the same time...  The scientific method is whatever it takes to not fool yourself into thinking something is true that is not, or into thinking something is not true that is.  That pathway, it's not straight; it's curved, it has off-ramps to nowhere, and you don't know which of the paths in front of you are going to lead to the right place...  The cool thing about it is that nature is the ultimate judge, jury, and executioner.  You can argue all you want, but if nature disagrees with you, you're wrong.  If you care about critical thinking and science literacy, the degree to which you believe something is true should be proportional to the evidence that supports it.  If after all the experiments are done, there is convergence in a result, you have successfully winnowed out the effects of bias on that result.  No one is without bias -- just be ready to get your stuff checked.  And be ready to abandon your cherished thoughts and ideas in the face of conflicting evidence.
The other is the trailer to a video called Gods Among Us that was sent to me by a friend who is evidently trying to cause my brain to explode.  Here are a sampling of quotes (you can watch the trailer yourself if you dare, and want to know more about the context and sources, but I can promise you these were not cherry-picked to make the video sound ridiculous -- they all sound like this):
  • There are quite a few extraterrestrials walking around, humanoid ones, so we've got them walking amongst us.  You may just think they look like nice people, or they may feel a bit different to you, but they're there and you see them every day. 
  • How is it that these higher-dimensional energies can be brought down, can be downloaded, into our ordinary four-dimensional space-time experience?
  • There are thousands, possibility millions, out there leading these double lives.  They will lead us into telepathic abilities, they will lead us into being able to heal ourselves, even to being able to change our bodies.
  • The DNA in us can exist as a toroid.  It can be used as a tool to bring higher-dimensional energy into our physical bodies, convert it into electromagnetic fields that can then be used to convert the physiological and biochemical processes in us.
  • I was contacted by a being who said he was from the constellation of Orion.
  • You want to know what your DNA is?  It's 34% human, 28% tall white Zeta, and 38% Annunaki.  
The people interviewed seemed to fall into three categories: (1) researchers, all of whom seem somehow to have earned Ph.D.s; (2) people who claim to have been contacted by aliens; (3) people who claim to be human/alien hybrids themselves.  The whole thing was accompanied by music that sounds like it was rejected from Music From the Hearts of Space on the basis of being too ethereal.

Okay, I'm scoffing, but there's a serious point to be made here.  A number of claims in Gods Among Us are empirically testable.  I'm not referring to the eyewitness testimony of things like alien contact; as Dr. Tyson also points out, eyewitness testimony may be the highest standard of evidence in the court of law, but it's the lowest form of evidence in science.  "I saw it with my own eyes" is simply not enough in science.  We have far too many ways of getting it wrong to trust one person's word for something.

But there are many other kinds of statement in this video that could be tested.  DNA can become a toroid that funnels energy from outside of us into our bodies and changes our biochemistry?  Fine, demonstrate it in the lab.  There are beings who can communicate with you telepathically?  Set up a situation where they tell you something you couldn't have otherwise known, and have it verified by an independent researcher.  Over half of our DNA is extraterrestrial?  Sequence it and show me that it doesn't overlap, gene for gene, with 99% of the DNA from our nearest primate relatives (and in the 70-80% range with all other mammal species).

Oh, and you can't "be from a constellation."  A constellation is a random assemblage of stars sitting at wildly varying distances from the Earth that only appear to be near each other from our perspective.  Saying you're "from a constellation" makes about as much sense as someone asking you how to find your house, and your answering them, "You'll find it on the horizon."

One example of how a constellation would look from an altered perspective; the Big Dipper as seen after a ninety-degree rotation around the entire group

However, the most insidious problem with the people who make claims like these is their belief that mainstream science rejects their conclusions out of hand not because there's insufficient evidence, but because the claims contradict scientific orthodoxy.  They seem to think that scientists are sitting in this never-changing edifice they've built, and they'll fight you tooth and nail if you try to change one thing about it.  Contrast this to Dr. Tyson's statement about the scientific frontier; in science, you are always on the boundary between what is known and what is unknown.  Scientific orthodoxy changes every time we get a new body of evidence, which is all the time.  In fact, that's how scientific careers are made.  If there really was evidence of all the stuff Gods Among Us claims, the scientists would be trampling each other to death to be the first to publish it in a peer-reviewed journal.

Consider, as only one of many illustrative examples, how the theory of plate tectonics arose.  The belief -- the "scientific orthodoxy," if you will -- was that the Earth was static.  The continents stayed put.  Even if there were periodic events like earthquakes and volcanoes to shake things up, everything was more or less in the same place as it always had been.

Why, in fact, would you think the opposite?  A static Earth seemed common sense.  How could continents move in solid rock?

But in the 50s and 60s, the evidence from a variety of sources -- where exactly volcanoes and earthquakes took place, the position and age of hotspot island chains like Hawaii, the contours of Africa and South America, the fossil record, and (most importantly) the evidence from magnetometer readings near the Mid-Atlantic Ridge -- had piled up to the extent that there was no choice but to overturn our understanding of how geology worked.  In other words, faced with hard, verifiable, repeatable scientific research, the "scientific orthodoxy" had to change drastically.  And far from being suppressed by the scientific establishment, this put rocket fuel into the careers of the first geologists who wrote a paper about it -- Frederick Vine and Drummond Matthews -- and today, they're in every introduction-to-geology textbook written.

So if there was demonstrable evidence that over 50% of our DNA came from a non-terrestrial source?  That's Nobel-Prize-material, right there.

Could the people in Gods Among Us be right?  I suppose so.  But thus far they have not met the minimum threshold of evidence that it would take to convince anyone who wasn't already convinced.  I'll end with a quote from another physicist, Richard Feynman, which seems particularly apposite here: "The first principle of science is that you must not fool yourself.  And you are the easiest person to fool."

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Neil deGrasse Tyson has become deservedly famous for his efforts to bring the latest findings of astronomers and astrophysicists to laypeople.  Not only has he given hundreds of public talks on everything from the Big Bang to UFOs, a couple of years ago he launched (and hosted) an updated reboot of Carl Sagan's wildly successful 1980 series Cosmos.

He has also communicated his vision through his writing, and this week's Skeptophilia book-of-the-week is his 2019 Letters From an Astrophysicist.  A public figure like Tyson gets inundated with correspondence, and Tyson's drive to teach and inspire has impelled him to answer many of them personally (however arduous it may seem to those of us who struggle to keep up with a dozen emails!).  In Letters, he has selected 101 of his most intriguing pieces of correspondence, along with his answers to each -- in the process creating a book that is a testimony to his intelligence, his sense of humor, his passion as a scientist, and his commitment to inquiry.

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



Wednesday, September 29, 2021

Illuminating Hessdalen

In his wonderful poem/performance piece Storm, Tim Minchin said: "Throughout history, every mystery ever solved has turned out to be 'not magic'."

As I've pointed out many times here before, it's not that I'm saying any of the thus-far-classified-as "out there beliefs" are impossible; it's that if they actually do exist, they should be accessible to scientific inquiry.  Auras, qi, chakras?  Demonstrate they're detectable by something other than a subjective viewer.  Hauntings?  Ditto.  Cryptids of various shapes and descriptions?  Give me something analyzable other than blurry photos and anecdotal eyewitness accounts.  Psychic abilities?  Show they work under controlled conditions.  

Interestingly, there was just an article in The Skeptic asserting that parapsychology has grown to the point that it deserves the title of science rather than pseudoscience.  I'm sure that the author, Chris French, professor of psychology at the University of London, will receive some blowback from this essay, as will The Skeptic in general for publishing it; but I agree with his central thesis, which is that parapsychological claims stand and fall on exactly the same basis as scientific claims do -- evidence.

And, as Minchin says, if a supernatural explanation turns out to be scientifically demonstrable, then it's no longer supernatural, is it?  It's just natural.  After that, it can be studied by the methods of science, just like every other feature of our weird, wonderful, amazingly complex universe.

What brings this up is a recent paper in Meteorology and Atmospheric Physics that considered the odd phenomenon of the "Hessdalen Lights" which occurs in a valley in central Norway, wherein people report seeing free-floating balls of light.  I'd written about the Hessdalen Lights (and various other accounts of lights in the sky) back in 2017, and described it as follows:

The Hessdalen Lights have been seen since the 1940s in the valley of Hessdalen in Norway.  They're stationary, bright white or yellow lights, floating above the ground, sometimes remaining visible for over an hour.  With such a cooperative phenomenon, you would think it would be easily explained; but despite the efforts of scientists, who have been studying the Hessdalen Lights for decades, there is yet to be a convincing explanation.  Hypotheses abound: that it is the combustion of dust from the valley floor; that it is a stable plasma, ionized by the decay of radon from minerals in the valley; or even that it is an electrical discharge from piezoelectric compression of quartz crystals in the underlying rock.  None of these is completely convincing, and the Hessdalen Lights remain one of the most puzzling natural phenomena I know of.

The lack of a convincing explanation opens the door to all sorts of wild speculation, and those abound -- ghosts, aliens, portals in time and space, you name it.  

Photograph of the Hessdalen Lights

As usual, my fallback position was, "I may not know what the scientific explanation is, but I'm certain that one exists."  Given how many times this phenomenon has been reported and photographed, it seemed pretty likely that it wasn't a hoax, or even misattributing it to something purely prosaic (like Neil deGrasse Tyson's story of a cop who was driving down a winding country road, chasing a "weird light in the sky" -- which turned out to be the planet Venus).  So accepting that the Hessdalen Lights actually occur as advertised, what the hell are they?

Much was my delight when I ran across the recent paper, by atmospheric chemist Gerson Paiva of Federal University Pernambuco (Brazil), which seems to have solved the mystery, using...

... wait for it...

... science.

Here's what Paiva writes:

Hessdalen lights are unusual, free-floating light balls presenting different shapes and light colors, observed in the Hessdalen valley in rural central Norway.  In this work, it is shown that these ghostly light balls are produced by an electrically active inversion layer above Hessdalen valley during geomagnetic storms.  Puzzling geometric shapes and energy content observed in the HL phenomenon may be explained through a little-known solution of Maxwell’s equations to electric (and magnetic) field lines: they can form loops in a finite space...  “Natural battery”, aerosols and global atmospheric electric circuits may play a crucial role for the electrification of the temperature inversion layers.
Now, I hasten to add that I don't know if Paiva's explanation is right.  But that's the other great thing about science; it's falsifiable.  When a researcher publishes something like this, it's immediately analyzed and taken to pieces by other experts in the field.  Unlike us fiction writers, who basically want everyone to read our writing and tell us how awesome it is, scientists are looking for rigorous criticism; they want their colleagues to try to tear it down, to see if their analysis is robust enough to withstand attempts to refute it.  So time will tell if Paiva has found the answer to this enduring mystery of atmospheric science.

But even if he hasn't, I'd bet cold hard cash that like Tim Minchin said, the answer will still turn out to be "not magic."

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Mathematics tends to sort people into two categories -- those who revel in it and those who detest it.  I lucked out in college to have a phenomenal calculus teacher who instilled in me a love for math that I still have today, and even though I'm far from an expert mathematician, I truly enjoy considering some of the abstruse corners of the theory of numbers.

One of the weirdest of all of the mathematical discoveries is Euler's Equation, which links five of the most important and well-known numbers -- π (the ratio between a circle's circumference and its diameter), e (the root of the natural logarithms), i (the square root of -1, and the foundation of the theory of imaginary and complex numbers), 1, and 0.  

They're related as follows:

Figuring this out took a genius like Leonhard Euler to figure out, and its implications are profound.  Nobel-Prize-winning physicist Richard Feynman called it "the most remarkable formula in mathematics;" nineteenth-century Harvard University professor of mathematics Benjamin Peirce said about Euler's Equation, "it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

Since Peirce's time mathematicians have gone a long way into probing the depths of this bizarre equation, and that voyage is the subject of David Stipp's wonderful book A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics.  It's fascinating reading for anyone who, like me, is intrigued by the odd properties of numbers, and Stipp has made the intricacies of Euler's Equation accessible to the layperson.  When I first learned about this strange relationship between five well-known numbers when I was in calculus class, my first reaction was, "How the hell can that be true?"  If you'd like the answer to that question -- and a lot of others along the way -- you'll love Stipp's book.

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