Between Mathematics, the hardest science, and the Paranormal, a liminal region between Reality and Illusion, there seems to be an abyss. The solidity of results in Mathematics have no counterparts in the ambiguities and controversies tipically linked with parapsychological topics. The "structural" distance between the two may have had a role in keeping separated the careers of mathematicians and parapsychologists. There are a few exceptions to this rule, and they had a key role in throwing some light on their common areas.

Martin Gardner, science writer and columnist for Scientific American ("Mathematical Games"), has always been fond of conjuring, and in particular of the strong relation between trickery and the paranormal. His duplex interest for recreational mathematics and magic tricks has strongly influenced his literary works, filled with mathematical principles which can be used to create the illusion of the paranormal. Here, Mathematics stays behind the curtains, being the trick thanks to which it is possible to create a (pseudo) paranormal feat. From the point of view of conjuring, the trick should be kept secret: if someone discovers the role of numbers in producing the feat, amazement and surprise end.

Switch on a calculator, press the button 9, then + and then 0. On the display you'll see the number "0", as if the calculator had just been switched on. You are ready to astonish your friends. Tell someone that with your psychocynetic powers you'll be able to move an object with the power of your mind, and add that you cannot move a heavy object like a stone, but they work properly on lighter objects, like liquid crystals. In order to prove it, press the button 1 and then 2: on the display you'll see the number 12. Now pretend to concentrate, and explain that you'll try to swap the two digits with the power of your mind. Keep the calculator with your thumb and second finger, and put the thumb near the button =. Shake a bit the calculator, and take advantage of this moment to press that button: the display will show the number 21, as if you were able to swap the two digits with your mind! Obviously your work was more trivial: you performed a simple sum (9 + 12 = 21), the first half of which has been kept secret behind the curtains, during the preparation of the calculator.(1)

American mathematician Persi Diaconis has also dedicated many articles to this topic, using Mathematics and Statistics with a different role: the one of "controller". Joseph Rhine (1895-1980) was the first who introduced statistical analyses in parapsychological tests on telepathy, clairvoyance, precognition and psychokinesis; many of his results had been firmly criticised by other reaserchers who found heavy statistical flaws in the procedures followed by Rhine. Persi Diaconi's article "Statistical problems in ESP research"(2) offers an interesting and detailed analysis of systematical errors in ESP experiments.

Here is its abstract:

In search of repeatable ESP experiments, modern investigators are using more complex targets, richer and freer responses, feedback, and more naturalistic conditions. This makes tractable statistical models less applicable. Moreover, controls often are so loose that no valid statistical analysis is possible. Some common problems are multiple end points, subject cheating, and unconscious sensory cueing. Unfortunately, such problems are hard to recognize from published records of the experiments in which they occur; rather, these problems are often uncovered by reports of independent skilled observers who were present during the experiment. This suggests that magicians and psychologists be regularly used as observers. New statistical ideas have been developed for some of the new experiments. For example, many modern ESP studies provide subjects with feedback - partial information about previous guesses - to reward the subjects for correct guesses in hope of inducing ESP learning. Some feedback experiments can be analyzed with the use of skill-scoring, a statistical procedure that depends on the information available and the way the guessing subject uses this information.

Science writer Raymond Smullyan, mathematician but also philosopher and conjurer, has offered in his books some of the best "logical trick" which can be used to create the illusion of paranormal feats.

Bibliography

One of the richest book on the matter is Martin Gardner, Mathematics, Magic and Mystery, Dover, 1956.

Joseph Rhine's ESP experiments are analysed from the mathematical point of view in Martin Gardner, Fads and Fallacies in the Name of Science, Dover, 1957.

Persi Diaconis' article "Statistical Problems in ESP Research" is available in Science (7/14/1978), Vol 201, Issue 4351, 131-136.

Raymond Smullyan, What is the name of this book?, Prentice Hall, 1978 is full of good logical puzzles and tricks which can be used in a (pseudo) paranormal context.

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⁽¹⁾ This feat is a simplified version of a trick firstly appeared on Genii (September 1997) and written by The Napoleons. Many thanks to P.G. Varola.

⁽²⁾ Science, Vol 201, Issue 4351, 131-136.