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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 xkcd.com) 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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The Natural and the Supernatural

Cartoonist Sidney Harris once illustrated two scientists at a chalkboard. One has written, among mathematical equations, “Then a miracle occurs,” to which his colleague replies, “I think you need to be more specific here in step two.” This nicely sums up the relationship between science and religion: one deals in the natural while the other deals in the supernatural. And never the twain shall meet.

Were only it were so. Unfortunately, religions routinely make claims about the natural world that are in direct conflict with the scientific evidence. Young-Earth Creationists, for example, believe that the world was created around 6,000 years ago, about the same time that the Babylonians invented beer. These claims cannot both be correct, and anyone who thinks the former is right has relegated all of science (along with brains) to the dumpster of life. Many people of faith believe that prayer can cajole the deity into taking action in our world to do everything from healing cancers to winning wars. Yet a comprehensive controlled scientific study on the efficacy of prayer on healing, funded by the religiously-based Templeton Foundation and conducted at the prestigious Harvard Medical School, found no relationship between the two: subjects in the non-prayed for group did just as well (or poor) as those in the prayed for group. And why is it, scientists want to know, that prayer only seems effective for things that might have happened anyway, such as tumors going into remission. A more dramatic and unmistakably religious miracle that would shock even the most skeptical of scientists would be if prayers for amputees (especially our brave wounded Christian soldiers returning from Iraq and Afghanistan) resulted in renewed whole limbs; i.e., a true miracle.

How, then, can we reconcile the natural and the supernatural? Most people keep them separated in logic-tight compartments, even scientists. Surveys conducted in 1916 and again in 1997 found that 40 percent of American scientists said they believe in God. As well, hundreds of millions of practicing Protestants, Catholics, Jews, and members of other faiths both believe in God and fully embrace science, even evolution: a 2005 Pew Research Center poll found that 68 percent of Protestants and 69 percent of Catholics accept the theory. So, demographically speaking, most people find no conflict between science and religion.

However, the natural world does not bend to the demographics of belief. Millions of people also believe in astrology, ghosts, angels, ESP, and all manner of paranormal piffle, but that does not make them real. The veracity of a proposition is independent of the number of people who believe it.

In conclusion, I go so far as to conclude: There is no such thing as the supernatural or the paranormal. There is only the natural, the normal, and mysteries we have yet to explain. God is a mystery, and the God of Abraham may very well be an eternal mystery for the simple reason that any God explicable through science and the laws of nature would, by definition, lose the status of supernatural and enter the realm of the natural. A God definable by science is not a God at all.


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Weirdonomics & Quirkology

How the curious science of the oddities
of everyday life yields new insights
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Using an index finger, trace the capital letter Q on your forehead. Which way did the tail of the Q slant?

What an odd thing to ask someone to do. Exploring weird things and why people believe them, however, is what I do for a living. Coming at science from the margins allows us to make an illuminating contrast between the normal and the paranormal, the natural and the supernatural, and the anomalous and the usual. The master at putting uncanny things to the experimental test — the man I call the Mythbuster of Magical Thinking — is University of Hertfordshire psychologist Richard Wiseman. His new book, Quirkology: How We Discover the Big Truths in Small Things (Basic, 2007), presents the results of his numerous (and often hilarious) experiments on all matters peculiar. (continue reading…)

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Michael Shermer & Michio Kaku
Discuss Science & Pseudoscience

The famous theoretical physicist and science popularizer Dr. Michio Kaku interviews Skeptic publisher and science writer Dr. Michael Shermer, in which they explore a variety of topics between science and pseudoscience, the normal and the paranormal, the natural and the supernatural.

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Mr. Skeptic Goes to Esalen

Science and spirituality on the California coast
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The Esalen Institute is a cluster of meeting rooms, lodging facilities and hot tubs all nestled into a stunning craggy coastal outcrop of the Pacific Ocean in Big Sur, Calif. In his 1985 book, “Surely You’re Joking, Mr. Feynman”, the Nobel laureate physicist Richard Feynman recounts his experience in the natural hot spring baths there, in which a woman is being massaged by a man she just met. “He starts to rub her big toe. ‘I think I feel it,’ he says. ‘I feel a kind of dent — is that the pituitary?’ I blurt out, ‘You’re a helluva long way from the pituitary, man!’ They looked at me, horrified … and said, ‘It’s reflexology!’ I quickly closed my eyes and appeared to be meditating.”

With that as my introduction to the Mecca of the New Age movement, I accepted an invitation to host a weekend workshop there on science and spirituality. Given my propensity for skepticism when it comes to most of the paranormal piffle proffered by the prajna peddlers meditating and soaking their way to nirvana here, I was surprised the hall was full. Perhaps skeptical consciousness is rising! (continue reading…)

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