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INNUMERACY: Mathematical Illiteracy and Its Consequences

INNUMERACY: Mathematical Illiteracy and Its Consequences

When asked why he doesn't believe in astrology, the logician Raymond Smullyan responds that he's a Gemini, and Geminis never believe in astrology.

Sample of supermarket tabloid headlines: Miracle Pickup Truck Can Heal the Sick. Giant Bigfoot Attacks Village. Seven-Year-Old Gives Birth to Twins in Toy Store. Scientists on Verge of Creating Plant People. Incredible Swami Has Stood on One Leg since 1969.

Inspect every piece of pseudoscience and you will find a security blanket, a thumb to suck, a skirt to hold. What have we to offer in exchange? Uncertainty! Insecurity!

– Isaac Asimov in the tenth-anniversary issue of The Skeptical Inquirer

To follow foolish precedents, and wink with both our eyes, is easier than to think.

– William Cowper


Innumeracy and pseudoscience are often associated, in part because of the ease with which mathematical certainty can be invoked to bludgeon the innumerate into a dumb acquiescence. Pure mathematics does indeed deal with certainties, but its applications are only as good as the underlying empirical assumptions, simplifications, and estimations that go into them.

Even such fundamental mathematical verities as "equals can be substituted for equals," or "1 and 1 are 2," can be misapplied: one cup of water plus one cup of popcorn are not equal to two cups of soggy popcorn, and "Infant Physician Duvalier" just doesn't have the same impact as "Baby Doc." Similarly, President Reagan may believe that Copenhagen is in Norway, but even though Copenhagen equals the capital of Denmark, it can't be concluded that Reagan believes the capital of Denmark is in Norway. In so-called intentional contexts like the above, the substitution doesn't always work.

If these basic principles can be misinterpreted, it shouldn't be surprising that more esoteric mathematics can be, too. If one's model or one's data are no good, the conclusions that follow won't be either. Applying old mathematics, in fact, is often more difficult than discovering new mathematics. Any bit of nonsense can be computerized-astrology, bio-rhythms, theI Ching-but that doesn't make the nonsense any more valid. Linear statistical projections, to cite a frequently abused model, are often invoked so thoughtlessly that it wouldn't be surprising to see someday that the projected waiting period for an abortion is one year.

This sort of careless reasoning is hardly limited to the uneducated. One of Freud's closest friends, a surgeon named Wilhelm Fliess, invented bio-rhythmic analysis, a practice based on the notion that various aspects of one's life follow rigid periodic cycles which begin at birth. Fliess pointed out to Freud that 23 and 28, the periods for some metaphysical male and female principles respectively, had the special property that if you add and subtract appropriate multiples of them, you can attain any number. Stated a little differently: any number at all can be expressed as 23X + 28Y for suitable choices of X and Y. For example, 6 = (23 x 10) + (28 x -8). Freud was so impressed with this that for years he was an ardent believer in biorhythms and thought that he would die at the age of fifty-one, the sum of 28 and 23. As it turns out, not only 23 and 28 but any two numbers that are relatively prime-that is, have no factors in common-have the property that any number can be expressed in terms of them. So even Freud suffered from innumeracy.

Freudian theory suffers from a more serious problem as well. Consider the statement: "Whatever God wills, happens." People may be able to take solace from it, but it's clear that the statement is not falsifiable and hence, as the English philosopher Karl Popper has insisted, not part of science. "Plane crashes always come in threes." You always hear that, too, and if you wait long enough, of course, everything comes in threes.

Popper has criticized Freudianism for claims and predictions which, though perhaps comforting or suggestive in one way or another, are, like the above statements, largely unfalsifiable. For example, an orthodox psychoanalyst might predict a certain kind of neurotic behavior. When the patient doesn't react in the predicted way, but in a very different manner, the analyst may attribute the opposite behavior to "reaction-formation." Likewise, when a Marxist predicts that the "ruling class" will act in an exploitive manner and instead something quite contrary takes place, he may attribute the outcome to an attempt by the ruling class to co-opt the "working class." There always seem to be escape clauses which can account for anything.

This is certainly not the place to argue whether or not Freudianism and Marxism should be deemed pseudosciences, but a tendency to confuse factual statements with empty logical formulations leads to sloppy thought. For example, the statements "UFOs contain extraterrestrial visitors" and "UFOs are unidentified flying objects" are two entirely different assertions. I once gave a lecture in which a listener thought that I subscribed to a belief in extraterrestrial visitors, when all I had said was that there undoubtedly were many cases of UFOs. A similar confusion is satirized by Moliere when he has his pompous doctor announce that his sleeping potion works because of its dormitive virtue. Since mathematics is the quintessential way to make impressive-sounding claims which are devoid of factual content ("Scientists reveal that 36 inches equal 1 yard on the planet Pluto"), it's perhaps not surprising that it is an ingredient in a number of pseudo-sciences. Abstruse calculations, geometric forms and algebraic terms, unusual correlations-all have been used to adorn the silliest drivel.


Interest in parapsychology is very old, yet the simple fact is that there have been no repeatable studies which have demonstrated its existence, Uri Geller and other charlatans notwithstanding. ESP (extrasensory perception) in particular has never been shown in any controlled experiment, and the few "successful" demonstrations have occurred in studies that were fatally flawed. Rather than rehash them, I'd like to make some general observations.

The first one, which is embarrassingly obvious, is that ESP runs afoul of the fundamental common-sense principle that the normal senses must somehow be involved for communication to take place. When confidential information leaks out of an organization, people suspect a spy, not a psychic. Hence, it is the presumption of common sense and science that these ESP phenomena don't exist, and the burden of proof is on those who maintain that there are such phenomena.

This raises probabilistic considerations. Because of the way ESP is defined-communication without any normal sensory mechanisms-there is no way to tell the difference between a single incidence of ESP and a chance guess. They look exactly the same, just as a particular correct answer on a true-false test looks the same whether the test taker is a straight-A student or someone who's guessing at every question. Since we can't ask the ESP subjects to justify their responses, as we can the true-false test takers, and since by definition there's no sensory mechanism into whose functioning we can inquire, the only way we can demonstrate the existence of ESP is by statistical test: run enough trials and see if the number of correct responses is sufficiently large so as to rule out chance as the explanation. If chance is ruled out and there are no other explanations, ESP will be demonstrated.

There is, of course, a tremendous will to believe which accounts for many of the faulty experiments (such as J. B. Rhine's) and much of the outright chicanery (such as S. G. Soal's) that seem to characterize the paranormal field. Another factor is what is sometimes referred to as the Jeane Dixon effect (after the self-described psychic Jeane Dixon), whereby the relatively few correct predictions are heralded and therefore widely remembered, while the much more numerous incorrect predictions are conveniently forgotten or deemphasized. Supermarket tabloids never provide an end-of-year list of false predictions by psychics, nor do the more upscale New Age periodicals, which, despite a veneer of sophistication, are just as fatuous.


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