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

N = R × f_p × n_e × f_l × f_i × f_c × L where…

• N = the number of communicative civilizations,
• R = the rate of formation of suitable stars,
• f_p = the fraction of those stars with planets,
• n_e = the number of earth-like planets per solar system,
• f_l = the fraction of planets with life,
• f_i = the fraction of planets with intelligent life,
• f_c = 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 = P_w × f_p × f_m × f_t × n_t × n_o × f_m where…

• N = Number of people we hear about who report having experienced a paranormal or supernatural phenomena,
• P_w = Population of the United States (January 1, 2012: 312,938,813),
• f_p = Fraction of people who report having had an anomalous psychological experience or witnessed an unusual physical phenomena (1/5),
• f_m = Fraction of people who interpret such experiences and phenomena as paranormal or supernatural (1/5),
• f_t = Fraction of people who tell someone about their experience (1/10),
• n_t = Number of people they tell (15),
• n_o = Number of other people told the story by original hearers (15), and
• f_m = 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 f_p and f_m. 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!

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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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.

For instance, Wiseman explains that the Q test is a quick measure of “self-monitoring.” High self-monitors tend to draw the letter Q with the tail slanting to their left, so that someone facing them can read it. By temperament, they tend to focus outwardly: they are concerned with how other people see them, enjoy being the center of attention and adapt their actions to suit the situation. They are also skilled at manipulating others, Wiseman says, which makes them good at deception. And self-deception, apparently, which he discovered when he told these subjects what the experiment is supposed to measure — given that high self-monitors tended to claim (and apparently believe) that they traced the Q the opposite direction to how they actually drew it.

If that is not quirky enough, Wiseman once spent a day in Londons King’s Cross railway station asking the following question of individuals and of couples reuniting in a passionate embrace: “Excuse me, do you mind taking part in a psychology experiment? How many seconds have passed since I just said the words ‘Excuse me?’ ” Wiseman discovered that people in love significantly underestimated the passing of time. In other words, as the poets already know, times passes quickly when you’re in love.

Paranormal anomalies have long been a target of Wiseman’s experimental bow. To test the psychology of ghostly experiences, for example, Wiseman spent 10 days at Hampton Court Palace, having individuals walk through specific locations and describe any unusual experiences. He discovered that people who have a vivid imagination and are easily hypnotized reported a sensed presence and an uneasy feeling in the exact same locations where those with dry imaginations reported nothing. In a related study, Wiseman’s psychologist colleague James Houran of Southern Illinois University had subjects walk though an abandoned cinema and describe how it made them feel. One group of subjects was told that the building was haunted, and the other group was told that it was being renovated. The “haunted” group reported significantly more unusual experiences than the other group.

In search of a normal explanation for such apparently paranormal enigmas, Wiseman conducted an experiment in a London concert hall in which he had participants listen to and rate the emotional experience of a performance by acclaimed Russian pianist GéNIA. At two different times during the performance, Wiseman piped in extremely low frequency infrasound waves that are inaudible to the human ear but are known to cause an internal vibratory feeling in the head and chest that can be experienced in a deeply emotional way. (NASA once tested infrasound waves on astronauts to measure the effect of rocket engines during launch.) He found that 22 percent of the 400 subjects noted unusual experiences during the infrasound conditions, reporting such feelings as “shivering on my wrist, odd feeling in stomach”; “increased heart rate, ears fluttering, anxious”; “felt like being in a jet before it takes off”; and “preorgasmic tension in body and arms, but not in legs.”

Other quirkiness reported by Wiseman includes why there are a disproportionate number of marine biologists called Dr. Fish (names do matter, it turns out); the best wording of a donation solicitation (adding “even a penny helps” doubles the giving rate); superior pickup lines (not boring, such as “Do you come here often?” but silly, such as “If you were a pizza topping, what would you be?”); the most effective personal ads (a 70 to 30 ratio between “this is me” and “this is what I’m looking for”); and the world’s funniest joke: “Two hunters are out in the woods when one of them collapses… ,” which I’ll finish next month, when I explain what weirdonomics and quirkology reveal about how science actually works.

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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!

It was in the extracurricular conversations, however, during healthy homegrown meals and while soaking in the hot tubs, that I gleaned a sense of what people believe and why. Once it became known that Mr. Skeptic was there, for example, I heard one after another “How do you explain this?” story, mostly involving angels, aliens and the usual paranormal fare. But this being Esalen — ground zero for all that is weird and wonderful in the human potential movement — there were some singularly unique accounts.

One woman explained the theory behind “energy work,” a combination of massage and adjusting the body’s seven energy centers called chakras. I signed up for a massage, which was remarkably relaxing, but when another practitioner told me about how she cured a woman’s migraine headache by directing a light beam through her head, I decided that practice and theory do not always match. Another woman warned about the epidemic of satanic cults. “But there’s no evidence of such cults,” I countered. “Of course not,” she explained. “They erase all memories and evidence of their nefarious activities.”

One gentleman recounted a lengthy tantric sexual encounter with his lover that lasted for many hours, at the culmination of which a lightning bolt shot through her left eye followed by a blue-light-being child entering her womb, ensuring conception. Nine months later friends and gurus joined the couple in a hothouse, sweating their way through their own “rebirthing” process before the mother gave birth to a baby boy. The father then told him that he would need to become an athlete in order to get into college; two decades later this young man became a professional baseball player. “How do you explain that?” I was asked. I quickly closed my eyes and appeared to be meditating.

People have and share such experiences and impart larger significance to them, because we have a cortex big enough to conceive of such transcendent notions and an imagination creative enough to concoct fantastic narratives. If we define the spirit (or soul) as the pattern of information of which we are made — our genes, proteins, memories and personalities — then spirituality is the quest to know the place of our essence within the deep time of evolution and the deep space of the cosmos.

There are many ways to be spiritual, and science is one in its awe-inspiring account about who we are and where we came from. “The cosmos is within us. We are made of star stuff. We are a way for the cosmos to know itself,” began the late astronomer Carl Sagan in the opening scene of Cosmos, filmed just down coast from Esalen, in referring to the stellar origins of the chemical elements of life. “We’ve begun at last to wonder about our origins, star stuff contemplating the stars, organized collections of ten billion billion billion atoms contemplating the evolution of matter, tracing that long path by which it arrived at consciousness … Our obligation to survive and flourish is owed not just to ourselves but also to that cosmos, ancient and vast, from which we spring.”

That is spiritual gold.

Have you ever noticed how much easier it is to do a newspaper crossword puzzle later in the day? Me neither. But according to Rupert Sheldrake, it is because the collective successes of the morning resonate through the cultural morphic field.

In Sheldrake’s theory of morphic resonance, similar forms (morphs, or “fields of information”) reverberate and exchange information within a universal life force. “As time goes on, each type of organism forms a special kind of cumulative collective memory,” Sheldrake writes in his 1981 book A New Science of Life (JP Tarcher). “The regularities of nature are therefore habitual. Things are as they are because they were as they were.” In this book and subsequent ones, Sheldrake, a botanist trained at the University of Cambridge, details the theory, which is again hotly debated in the recent June Journal of Consciousness Studies.

Morphic resonance, Sheldrake says, is “the idea of mysterious telepathy-type interconnections between organisms and of collective memories within species” and accounts for phantom limbs, how dogs know when their owners are coming home, and how people know when someone is staring at them. “Vision may involve a two-way process, an inward movement of light and an outward projection of mental images,” Sheldrake explains. Thousands of trials conducted by anyone who downloaded the experimental protocol from Sheldrake’s Web page “have given positive, repeatable, and highly significant results, implying that there is indeed a widespread sensitivity to being stared at from behind.”

Let us examine this claim more closely. First, science is not normally conducted by strangers who happen on a Web page protocol, so we have no way of knowing if these amateurs controlled for intervening variables and experimenter biases.

Second, psychologists dismiss anecdotal accounts of this sense to a reverse self-fulfilling effect: a person suspects being stared at and turns to check; such head movement catches the eyes of would-be starers, who then turn to look at the staree, who thereby confirms the feeling of being stared at.

Third, in 2000 John Colwell of Middlesex University in London conducted a formal test using Sheldrake’s experimental protocol. Twelve volunteers participated in 12 sequences of 20 stare or no-stare trials each and received accuracy feedback for the final nine sessions. Results: subjects could detect being stared at only when accuracy feedback was provided, which Colwell attributed to the subjects learning what was, in fact, a nonrandom presentation of the trials. When University of Hertfordshire psychologist Richard Wiseman also attempted to replicate Sheldrake’s research, he found that subjects detected stares at rates no better than chance.

Fourth, confirmation bias (where we look for and find confirmatory evidence for what we already believe) may be at work here. In a special issue of the Journal of Consciousness Studies devoted to “Sheldrake and His Critics,” I rated the 14 open peer commentaries on Sheldrake’s target article (on the sense of being stared at) on a scale of 1 to 5 (critical, mildly critical, neutral, mildly supportive, supportive). Without exception, the 1s, 2s and 3s were all traditional scientists with mainstream affiliations, whereas the 4s and 5s were all affiliated with fringe and pro-paranormal institutions. (For complete results, see Table 1 in the online version of this column at www.sciam.com)

Fifth, there is an experimenter bias problem. Institute of Noetic Sciences researcher Marilyn Schlitz—a believer in psychic phenomena — collaborated with Wiseman (a skeptic of psi) in replicating Sheldrake’s research and discovered that when they did the staring Schlitz found statistically significant results, whereas Wiseman found chance results.

Sheldrake responds that skeptics dampen the morphic field’s, whereas believers enhance it. Of Wiseman, he remarked: “Perhaps his negative expectations consciously or unconsciously influenced the way he looked at the subjects.”

Perhaps, but wouldn’t that mean that this claim is ultimately nonfalsifiable? If both positive and negative results are interpreted as supporting a theory, how can we test its validity? Skepticism is the default position because the burden of proof is on the believer, not the skeptic.