14 1. THE ORIGIN AND PREHISTORY OF MATHEMATICShead; another tossed its head back; yet another made pecking motions toward the
floor of the cage.
The difficulties that people, even mathematicians, have in understanding and
applying probability can be seen in this example. For example, the human body
has some capacity to heal itself. Like the automatic timer that eventually provided
food to the pigeons, the human immune system often overcomes the disease. Yet
sick people, like hungry pigeons, try various methods of alleviating their misery.
The consequence is a wide variety of nostrums said to cure a cold or arthritis. One
of the triumphs of modern mathematical statistics is the establishment of reliable
systems of inference to replace the inferences that Skinner called "superstitious."
Modern logic has purged the concept of implication of all connection with the
notion of cause. The statement "If Abraham Lincoln was the first President of
the United States, then 2 + 2 = 4" is considered a true implication, even though
Lincoln was not the first President and in any case his being such would have no
causal connection with the truth of the statement "2 + 2 = 4." In standard logic
the statement "If A is true, then Β is true" is equivalent to the statement "Either
Β is true, or A is false, or both." Absolute truth or falsehood is not available
in relation to the observed world, however. As a result, science must deal with
propositions of the form "If A is true, then Β is highly probable." One cannot infer
from this statement that "If Β is false, then A is highly improbable." For example,
an American citizen, taken at random, is probably not a U. S. Senator. It does not
follow that if a person is a U. S. Senator, that person is probably not an American
citizen.Questions and problems
1.1. At what point do you find it necessary to count in order to say how large a
collection is? Can you look at a word such as tendentious and see immediately how
many letters it has? The American writer Henry Thoreau (1817-1863) was said
to have the ability to pick up exactly one dozen pencils out of a pile. Try as an
experiment to determine the largest number of pencils you can pick up out of a pile
without counting. The point of this exercise is to see where direct perception needs
to be replaced by counting.
1.2. In what practical contexts of everyday life are the fundamental operations
of arithmetic—addition, subtraction, multiplication, and division—needed? Give
at least two examples of the use of each. How do these operations apply to the
problems for which the theory of proportion was invented?
1.3. What significance might there be in the fact that there are three columns of
notches on the Ishango Bone? What might be the significance of the numbers of
notches in the three series?1.4. Is it possible that the Ishango Bone was used for divination? Can you think
of a way in which it could be used for this purpose?
1 .5. Is it significant that one of the yarrow sticks is isolated at the beginning of
each step in the Chinese divination procedure described above? What difference
does this step make in the outcome?
1.6. Measuring a continuous object involves finding its ratio to some standard unit.
For example, when you measure out one-third of a cup of flour in a recipe, you are