- MATHEMATICAL INFERENCE 13
FIGURE 2. (a) The diagonal AC divides the rectangle ABCD into
congruent triangles ABC and CD A. (b) When the congruent pairs
(ΑΕΙ, IGA) and (IHC, CFI) are subtracted from the congruent
pair (ABC, CDA), the remainders (rectangles EBHI and GIFD)
must be equal.always induction or experiment. Perminov (1997) points out that solutions of com-
plicated geometric problems which can be shown to be correct are stated without
proof—but apparently with absolute confidence—by the writers of the very earliest
mathematical documents, such as the Rhind Papyrus from Egypt and cuneiform
tablets from Mesopotamia. The fact that an author presents not merely a solution
but a sequence of steps leading to that solution, and the fact that this solution can
now be reconstructed and verified by mathematical reasoning, justify the conclu-
sion that the result was arrived at through mathematical deduction, even though
the author does not write out the details.
4.2. Chance and probability. Logic is concerned with getting conclusions that
are as reliable as the premises. From a behavioral point of view, the human tendency
to make inferences based on logic is probably hardwired and expressed as the same
mechanism by which habits are formed. This same mechanism probably accounts
for the metaphysical notion of cause. If A implies B, one feels that in some sense
A causes Β to be true. The dogs in Pavlov's experiments, described above, were
given total reinforcement as they learned geometry and came to make associations
based on the constant conjunction of a given shape and a given reward or lack
of reward. In the real world, however, we frequently encounter a weaker type of
cause, where A is usually, but not always, followed by B. For example, lightning
is always followed by thunder, but if the lightning is very distant, the thunder will
not be heard. The analog of this weaker kind of cause in conditioning is partial
reinforcement. A classical example is a famous experiment of Skinner (1948), who
put hungry pigeons in a cage and attached a food hopper to the cage with an
automatic timer to permit access to the food at regular intervals. The pigeons at
first engaged in aimless activity when not being fed, but tended to repeat whatever
activity they happened to be doing when the food arrived, as if they made an
association between the activity and the arrival of food. Naturally, the more they
repeated a given activity, the more likely that activity was to be reinforced by the
arrival of food. Since they were always hungry, it was not long before they were
engaged full-time in an activity that they apparently considered an infallible food
producer. This activity varied from one bird to another. One pigeon thrust its head
into an upper corner of the cage; another made long sweeping movements with its