The Foundations of Psychiatry

By Silvano Arieti | Go to book overview

CHAPTER 50

MATHEMATICS
AND CYBERNETICS

Anatol Rapoport

THE FUNDAMENTAL CONTRIBUTION Of mathematics to science has been to provide a precise and contentless language in which to describe events, to formulate generalizations, and to deduce consequences of assumptions. Precision and independence from content are interdependent. The vocabulary of everyday language depends on the way perceptions and concepts are organized; for instance, on the particular way objects are classified or relations among them are interpreted. In attaching names to objects, properties, or actions, we fix the categories in which we think. These categories are of necessity too crude to capture the infinite variety of events that constitute "objective reality." Thus a content-bound language may impose a structure on our perceptions of the world and on the abstract concepts we form, and this structure may or may not correspond to the structure of reality.

Because mathematical language is contentless, that is, totally abstracted from perceptions, its structure is entirely transparent. In the exact (mathematicized) sciences the structure of a mathematical theory is constantly compared with the structure of a portion of the world under study. Mathematics itself, however, is concerned with the structure of relations independent of empirical content.

As an example consider the equation (a mathematical statement) relating the area of a circle to its radius, A = π R2. It says that whatever be the radius of a circle, the ratio of the area to the square of the radius is always constant, equal approximately to 3.1415926. The statement is actually a composite of a potentially infinite number of statements, since it specifies the magnitude of the area of all possible circles. Assuming that the radius can be specified with infinite precision, the area can also be specified with infinite precision, because the number π can be calculated with infinite precision. However, the statement cannot refer to anything in the empirically observable world, because there are no perfect circles and because physical measurements cannot be made with infinite precision.

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