THE MATHEMATICS OF THE PARANORMAL

James D. Stein. The Paranormal Equation: A New Scientific Perspective on Remote Viewing, Clairvoyance, and Other Inexplicable Phenomena. New Page Books, 2013.

Dr Stein is a professor of mathematics and he believes that there are truths about the natural world that nominally fall within the domain of science that might be impossible for science to discover. He thinks that some as yet undiscovered natural laws might account for some of what appear to be paranormal phenomena.
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Unlike some scientists and mathematicians who have studied such things, though, he is not fooled by the tricks of magicians, or the psychic researchers who resort to cheating and fiddle their results. He gives as one of his examples a BBC television show in which the physicist John Taylor appeared as a "scientific hatchet man" during a performance by the Israeli "psychic" and entertainer Uri Geller. Geller did his fork-bending act, followed by telepathy, followed by apparently mending broken watches just by holding them in his hands. Taylor was baffled by these tricks, and thought Geller had psychic powers, even though, like other skilled and experienced magicians, he could make his tricks seem inexplicable.

One of the first problems he considers is how to decide what is meant by paranormal or supernatural phenomena. This is rather difficult. In the case of ghosts, for example, Stein points out that there is no scientific framework in which one can prove that there is no such thing as a ghost. If one could provide scientific proof of the reality of ghosts, they would then be regarded as natural rather than supernatural. It seems to me, though, that this introduces the idea that they could have some sort of subtle physical existence, independent of the people who see them.

Stein, though, says that a law which is a law that cannot be discovered would legitimately be supernatural. "It would be a principle governing behaviour in the Universe which would be beyond our ability, or the ability of any sentient creature, to discover."

As Dr Stein is a mathematician, he gives us a discussion on arithmetic, a subject with which most of us have some acquaintance, and he shows us that some problems with numbers are not as simple as they look. He includes two interesting examples.

The Goldbach Conjecture states that every even number is the sum of two prime numbers. For example, 84 is the sum of prime numbers 41 and 43. Mathematicians have been trying to prove or disprove this for about 300 years, without success. It could, of course, be proved false by finding an even number which is not the sum of two primes, but as the natural numbers are an infinite series, this is not practicable.

To test the Collatz Conjecture, pick any number. If it is even, divide by 2. If it is odd, triple it and add 1. Apply these rules to the result and keep going, and you will always eventually end up with 1. Mathematicians have so far failed to establish whether this conjecture is decidable.

Dr Stein describes some of the counterintuitive results of experiments in quantum physics, and his belief that the universe is infinite. In his concluding remarks, he writes: "The time may come when we have made enough measurements to conjecture a relationship involving information transfer that has all the earmarks of paranormality". This book provides interesting ideas on attempts to discover how some kinds of apparently paranormal phenomena might be compatible with scientific principles. 

• John Harney