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The Flake Equation

Estimating the number of people who have
experienced the paranormal or supernatural

The Drake Equation is the famous formula developed by the astronomer Frank Drake for estimating the number of extraterrestrial civilizations:

N = R × fp × ne × fl × fi × fc × L where…

  • N = the number of communicative civilizations,
  • R = the rate of formation of suitable stars,
  • fp = the fraction of those stars with planets,
  • ne = the number of earth-like planets per solar system,
  • fl = the fraction of planets with life,
  • fi = the fraction of planets with intelligent life,
  • fc = the fraction of planets with communicating technology, and
  • L = the lifetime of communicating civilizations.

The equation is so ubiquitous that it has even been employed in the popular television series The Big Bang Theory for computing the number of available sex partners within a 40-mile radius of Los Angeles (5,812). My favorite parody of it is by the cartoonist Randall Munroe as one in a series of his clever science send-ups, entitled “The Flake Equation” (on for calculating the number of people who will mistakenly think they had an ET encounter.

Such multiplicative equations for calculating the product of an increasingly restrictive series of fractional values are effective tools for making back-of-the-envelope calculations to solve problems for which we do not have precise data. To that end I thought it a useful addition to the Skeptic toolbox to create a Flake Equation for all paranormal and supernatural experiences (and in the Flake Equation I’m interested not in beliefs but in actual experiences that people report and that we hear about, because this becomes the foundation of paranormal and supernatural beliefs):

N = Pw × fp × fm × ft × nt × no × fm where…

  • N = Number of people we hear about who report having experienced a paranormal or supernatural phenomena,
  • Pw = Population of the United States (January 1, 2012: 312,938,813),
  • fp = Fraction of people who report having had an anomalous psychological experience or witnessed an unusual physical phenomena (1/5),
  • fm = Fraction of people who interpret such experiences and phenomena as paranormal or supernatural (1/5),
  • ft = Fraction of people who tell someone about their experience (1/10),
  • nt = Number of people they tell (15),
  • no = Number of other people told the story by original hearers (15), and
  • fm = Fraction of such stories reported in the media or on Internet blogs, tweets, and forums (1/10).

N = 28,164,493, or about 9 percent of the U.S. population.

To compute this figure I used the 2005/2007 Baylor Religion Survey, which reports that

  • 23.2% say that they have “witnessed a miraculous, physical healing,”
  • 16.3% “received a miraculous, physical healing,”
  • 27.5% “witnessed people speaking in tongues at a place of worship,”
  • 7.7% “spoke or prayed in tongues,”
  • 54.5% experienced being “protected from harm by a guardian angel,”
  • 5.9% “personally had a vision of a religious figure while awake,”
  • 19.1% “heard the voice of God speaking to me,”
  • 26.1% “had a dream of religious significance,”
  • 52% “had an experience where you felt that you were filled with the spirit,”
  • 22.1% “felt at one with the universe,”
  • 25.7% “had a religious conversion experience,”
  • 13.8% “had an experience where you felt that you were in a state of religious ecstasy,”
  • 14.2% “had an experience where you felt that you left your body for a period of time,”
  • 40.4% “had a dream that later came true,” and
  • 16.7% “witnessed an object in the sky that you could not identify (UFO).”

This works out to an average of 24.4 percent, thereby justifying my conservative 20 percent figure for fp and fm. The other numbers I gleaned from research on gossip and social networks, conservatively estimating that 10 percent of people will tell someone about their unusual experience, and that within their average social network of 150 people they will tell at least 10 percent of them (15) who in turn will pass on the story to 10 percent of their social network of 150 (15). Finally, I estimate that 10 percent of such stories will be reported in the media or recounted in blogs, tweets, forums, and the like.

Of course the final figure for N will vary considerably depending on what numbers are plugged into the equation, but the result will almost always be a number in the tens of millions, which goes a long way toward explaining why belief in the paranormal and supernatural is so ubiquitous. Experiencing is believing!

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What is Pseudoscience?

Distinguishing between science and pseudoscience
is problematic
magazine cover

CLIMATE DENIERS ARE ACCUSED OF PRACTICING PSEUDOSCIENCE, as are intelligent design creationists, astrologers, UFOlogists, parapsychologists, practitioners of alternative medicine, and often anyone who strays far from the scientific mainstream. The boundary problem between science and pseudoscience, in fact, is notoriously fraught with definitional disagreements because the categories are too broad and fuzzy on the edges, and the term “pseudoscience” is subject to adjectival abuse against any claim one happens to dislike for any reason. In his 2010 book Nonsense on Stilts (University of Chicago Press), philosopher of science Massimo Pigliucci concedes that there is “no litmus test,” because “the boundaries separating science, nonscience, and pseudoscience are much fuzzier and more permeable than Popper (or, for that matter, most scientists) would have us believe.”

It was Karl Popper who first identified what he called “the demarcation problem” of finding a criterion to distinguish between empirical science, such as the successful 1919 test of Einstein’s general theory of relativity, and pseudoscience, such as Freud’s theories, whose adherents sought only confirming evidence while ignoring disconfirming cases. Einstein’s theory might have been falsified had solar-eclipse data not shown the requisite deflection of starlight bent by the sun’s gravitational field. Freud’s theories, however, could never be disproved, because there was no testable hypothesis open to refutability. Thus, Popper famously declared “falsifiability” as the ultimate criterion of demarcation. (continue reading…)

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The Woo of Creation: My evening with Deepak Chopra

Shermer and Deepak

On Thursday, March 31, Deepak Chopra and I squared off for a second time in person in a public venue, this time accompanied by the physicist Leonard Mlodinow on my side and Stuart Hameroff on his side (along with other panelists). The question on the table was this:

“Is there an Ultimate Reality?” and if yes, “Can it be accounted for by science such as mathematics, biology and physics?”

My answers: YES and YES

I explained that I am a Materialist and a Monist. I do not believe that there is a body and a soul, there is just a body. There is no brain and mind, just brain. The mind is just a word we use to describe what the brain does. I said, “you know I’m right” (which got a surprising laugh from the audience) because of the evidence from strokes, tumors, brain damage, senility, dementia, and Alzheimer’s, all of which kill brain cells, and along with the loss of brain comes the loss of mind. I asked Deepak and Stuart where Aunt Millie’s mind goes when her brain slowly disappears from the effects of Alzheimer’s disease.

I noted that consciousness is just a word we use to describe our inner thoughts about the workings of the brain, and that our “soul” is just a pattern of information stored in our genes and our brains. Consciousness is just an emergent property of integrated brain modules and patterned firing of neural networks.

By contrast, I believe that Deepak’s use of the word “consciousness” is very anthropocentric, once again returning humans to a central place in the cosmos as the “observers” who, in quantum mechanics, brings things into existence. If Deepak is right then the moon doesn’t exist unless it is observed, and yet, quoting that great scientist Bill O’Reilly, “times come in, tides go out—never a missed communication—and they would do so whether or not humans, or any other conscious (or unconscious) being existed.

In fact, I said, Deepak’s quantum consciousness is not holistic but reductionistic in the extreme. We don’t need to go down that far. Quantum mechanics is not needed to explain brain functions: the neuron is the individual unit of thought, the “atom” of mind. I then worked in a little joke I wrote earlier in the day:

Quantum mechanics is spooky and weird.
Consciousness is spooky and weird.
So what? Charlie Sheen is spooky and weird, but we don’t need quantum mechanics to explain his behavior. His “tiger blood” theory works just fine.


In Deepak’s worldview, everything is conscious, which means that there is no way to distinguish between consciousness and unconsciousness, which is how I often feel when I listen to Deepak.

Thought Experiment:

  • If humans went extinct instead of Neanderthals, how does that effect the universe?
  • What if the Earth were suddenly demolished by a rogue planet (as in 2012)? Would that mean the end of the universe because observers would disappear?
  • Are whales, dolphins, gorillas and chimps conscious and therefore integral to the universe?
  • What can it possibly mean to say that the universe is conscious? If you will pardon the nerd science pun, that is such a vacuous concept!

Before the debate Deepak asked me to read a paper by himself and Menas Kafatos and Rudolph Tanzi published in the Journal of Cosmology, entitled: “How Consciousness Becomes the Physical Universe.” Deepak asked me to comment on it, which I did in the second half of the debate. I noted that given the prominence of “consciousness” to the central theme of the paper that one might expect it to be defined with semantic precision. Nope. Here is what the authors write:

“We will sidestep any precise definition of consciousness, limiting ourselves for now to willful actions on the part of the observer.”

What can it possibly mean for the universe to be conscious in the sense of having willful actions? The universe behaves with willful action? The universe is an observer? As well, quantum mechanics only requires an observation of any kind: an electron microscope will do. Is an electron microscope willful? Does an electron microscope take action? The authors of this paper write:

Werner Heisenberg concluded that the atom “has no immediate and direct physical properties at all.” If the universe’s basic building block isn’t physical, then the same must hold true in some way for the whole. The universe was doing a vanishing act in Heisenberg’s day, and it certainly hasn’t become more solid since. And Heisenberg again: “The atoms or elementary particles themselves … form a world of potentialities or possibilities rather than one of things or facts.”

No, sorry, these are different levels of analysis. To prove it I challenge Deepak to climb to the top of this building and jump off and see if the ground is a potentiality or a thing! They also write:

Heisenberg: “What we observe is not nature itself, but nature exposed to our method of questioning.” Reality, it seems, shifts according to the observer’s conscious intent.

Once again, NO! This would imply that anyone’s method of questioning is just as valid as anyone else’s, which would mean that the way astrologers question the universe is just as valid as that of astronomers. I concluded by saying that if you want to get a spacecraft to Mars the questions that astronomers ask are absolutely objectively really better than those of astrologers. Q.E.D.!

In Deepak’s rebuttal, in discussing quantum mechanics, he actually used the phrase “the womb of creation.” Nice. It’s that sort of precise language that makes people all gushy and mushy about science. I pressed him for a definition of consciousness, which he gave me as “consciousness is the ground of existence.” I replied that this sounded tautological to me: since reality needs consciousness to come into existence, this means that reality = consciousness = existence; or existence = existence. A is A. Very Aristotelian. But what does that really tell us?

In the end I pressed both Deepak and Stuart Hameroff for an answer as to where Aunt Millie’s mind goes during the ravages of Alzheimer’s disease. Stuart’s answer was so rapid fire and jargon laden (something about the collapse of the wave function inside the microtubules in the neurons inside Aunt Millie’s brain) that I couldn’t quite get an answer, so Deepak clarified it for me later: Aunt Millie’s mind is in the matrix. Okay, I asked, how does poor Aunt Millie access the matrix. “We’re working on that,” was the reply. Okay, fine, and if our memories really are stored somewhere outside of our brains, then that would indeed be one of the greatest discoveries ever made in the history of science: Nobel worthy. But, until that is proven, I remain … skeptical.

Post Script

I am often asked if I believe that Deepak believes what he says, with an underlying assumption behind the question that Deepak is knowingly selling snake oil and doesn’t really believe his public patter. Having gotten to know Deepak over the years I can assure you that he absolutely positively believes what he says, and that while he may make a lot of money in the process of writing books, giving lectures, hosting radio and television shows, and running his various business enterprises (but, hey, that’s not exactly something anathema in America), this fact is quite orthogonal to his deeper mission in life: to shift the Western worldview Eastward.

I had never met Stuart Hameroff before, but I liked him as well, sharing a beer after the debate while watching a Laker game and schmoozing about science. Although I do not accept his theory of consciousness (most neuroscientists are skeptical as well), it would be fun to engage him again in a spirited debate over the brain and the mind.

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The Skeptic’s Skeptic

In the battle for ideas, scientists could learn from Christopher Hitchens
magazine cover

SCIENCE VALUES DATA and statistics and champions the virtues of evidence and experimentation. Those of us “viewing the world with a rational eye” (as the new descriptor for this column in Scientific American reads) also have another, underutilized tool at our disposal: rapier logic like that of Christopher Hitchens, a practiced logician trained in rhetoric. Hitchens—who is “leaving the party a bit earlier than I’d like” because of esophageal cancer, as he lamented to Charlie Rose in a recent PBS interview—has something deeply important to offer on how to think about unscientific claims. Although he has no formal training in science, I would pit Hitchens against any of the purveyors of pseudoscientific clap trap because of his unique and enviable skill at peeling back the layers of an argument and cutting to its core. (continue reading…)

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The Eternally Boring Hereafter

A review of Clint Eastwood’s film Hereafter

/// ATTENTION! Spoiler Alert! ///

After a string of highly successful and critically acclaimed films by Clint Eastwood (Million Dollar Baby, Gran Torino, Invictus, Flags of Our Fathers, Letters from Iwo Jima, etc.), I fully expected his latest, Hereafter, to be so well written (screenplay by Peter Morgan—Frost/Nixon, The Queen) and so compelling that stories about near-death experiences would skyrocket and that I would be preoccupied for months dealing with media inquiries about “true stories” of the hereafter. Alas, and with some relief, this will not happen as Hereafter is possibly the worst film Eastwood has ever directed.

If the hereafter is anything like its filmic namesake, then it will turn out to be glacially slow, eternally boring, and pointless, with seemingly random plot lines aimlessly wandering about the ethereal landscape. I wanted to like this film, despite my skepticism on its subject, because I like Clint Eastwood productions and I’m a sucker for a well-produced story, able and willing to suspend disbelief long enough to get emotionally involved. I tried but failed to do so with this film. It’s a bomb. Don’t bother to see it in the theaters, and don’t even waste a couple of bucks on a Netflix rental.

The only redeeming part of the film was the striking opening scene of the tsunami in Southeast Asia that sets the background for the first plot line. An attractive French reporter leaves her lover in their hotel room to go shopping for his kids among the street vendors below. When he hears a disturbing sound and looks out the window he sees the ocean receding, followed by a massive body of water rushing back in to the shore and slamming into buildings and leveling everything in its path. From the woman’s street level view tucked in among buildings she can only see trees felling and chaos approaching with only enough time to realize that there is no time to do anything about it. She is swept up in the tsunami’s leading edge and slammed about cars, building debris, trees, and the like, until she is whacked on the head unconscious. Cut to minutes later when she is being given mouth-to-mouth resuscitation by rescuers, to no avail. They give up and move on to the next victim, whereupon she comes to life, after a brief encounter with the hereafter, which Eastwood portrays as a fuzzy, nebulous place with people walking about aimlessly. It’s a portent of things to come.

The second plot line is Matt Damon’s psychic character George, a former psychic who gave up fame and riches because his “gift” is also a curse. A cross between James Van Praagh and John Edward, George concedes to a reading for a client of his sleazy brother (Jay Mohr) and scores several hits. The brother encourages George to quit his job at a San Francisco dock and return to the psychic world, but he will have none of it as it’s just too emotionally traumatic to read people’s inner thoughts (that much I suspect is true, if any of it were true, which it isn’t). Matt Damon’s love interest is the beautiful Bryce Dallas Howard, whom he meets at a cooking class, but after nearly an hour’s worth of romantic buildup to some sort of coming together, she departs the film for good after George reads her and conveys the message that her deceased father is sorry for the naughty things he did to her as a young girl.

The third plot line develops around 12-year old twins named Marcus and Jason, who live with their drug-addicted mother in London, England. Jason is hit by a car and killed, leaving Marcus to wander about the city in search of a psychic who can connect him to his brother. Here at least Eastwood had the good sense to depict what most psychics are like—scammers and flimflam artists conning their marks out of a few bucks by talking twaddle with the dead through standard cold-reading techniques. Marcus is dismayed by the idiocy of these pretenders and finally returns to the foster home where he struggles to keep his sanity.

For an hour and forty-five minutes all three of these plot lines run parallel, leaving audience members to wonder when—oh please when?!—will they finally be brought together. Finally, after what feels like an interminable marathon of tedium, George quits his job and takes a vacation in London to visit the home of his favorite author, Charles Dickens. While there he notices a flyer for a lecture about Dickens at a book fair in London, where, per chance, the French reporter is doing a signing for her new book on life after death, which she was inspired to write after an hour and a half of futzing around with her mundane reporter’s job distracted by her experience with the hereafter in the tsunami. By chance, little Marcus finds himself drawn to the book fair where he recognizes George from his web page photos, and begs him for a reading, which he finally gets. Naturally, George is better than those phony psychics, and Marcus encourages George to seek out the French woman so that they may all connect to the dead. George and Marie find a love connection as well and the story ends happily ever after.

Never have I been so relieved for a movie to end. There was one memorable moment, however, and that was the opening line of the opening trailer before Hereafter even started. The trailer was for a January 2011 release called The Rite, staring Anthony Hopkins as an American priest who travels to Italy to study at an exorcism school. (You can watch the trailer here). The line that rather caught my attention as I was settling into my seat, was, “You know the interesting thing about skeptics?” To which I blurted out “No, what?” The answer: “It’s that we’re always looking for proof. The question is, What on earth would we do with it if we found it?” I know what I do with proof when I find it. I publish it! Another character in the trailer then says “I believe people prefer to lie to themselves than face the truth.”

Here, then, in this trailer is the message for belief in the hereafter. If there were proof of it, we would publish it to the high heavens. But, since there isn’t, most people prefer to lie to themselves about it rather than face the truth that it is what we do in this life that counts.

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