Thursday, October 9, 2008

feller on mathematics

Axiomatically, mathematics is concerned solely with relations among undefined things. This property is well illustrated by the game of chess. It is impossible to "define" chess otherwise than by stating a set of rules. the conventional shape of the pieces may be described to some extent, but it will not always be obvious which piece is intended for "king." The chessboard and the pieces are helpful, but they can be dispensed with. the essential thing is to know how the pieces move and act. It is meaningless to talk about the "definition" or the "true nature" of a pawn or a king. Similarly, geometry does not care what a point and a straight line "really are." They remain undefined notions, and the axioms of geometry specify the relations among them: two points determine a line, etc. These are the rules, and there is nothing sacred about them. We change the axioms to study different forms of geometry, and the logical structure of the several non-Euclidean geometries is independent of their relation to reality. Physicists have studied the motion of bodies under laws of attraction different from Newton's, and such studies are meaningful even if Newton's law of attraction is accepted as true in nature.

Newton's notions of a field of force and of action at a distance and Maxwell's concept of electromagnetic "waves" were at first decried as "unthinkable" and "contrary to intuition." Modern technology and radio in the homes have popularized these notions to such an extent that they form part of the ordinary vocabulary. Similarly, the modern student has no appreciation of the modes of thinking, the prejudices, and other difficulties against which the theory of probability had to struggle when it was new. Nowadays newspapers report on samples of public opinion, and the magic of statistics embraces all phrases of life to the extent that young girls watch the statistics of their chances to get married. Thus everyone has acquired a feeling for the meaning of statements such as "the chances are three in five." Vague as it is, this intuition serves as background and guide for the first step. It will be developed as the theory progresses and acquaintance is made with more sophisticated applications.

The concepts of geometry and mechanics are in practice identified with certain physical objects, but the process is so flexible and variable that no general rules can be given. The notion of a rigid body is fundamental and useful, even though no physical object is rigid. Whether a given body can be treated as if it were rigid depends on the circumstances and the desired degree of approximation. Rubber is certainly not rigid, but in discussing the motion of automobiles text books treat the rubber tires as rigid bodies. Depending on the purpose of the theory, we disregard the atomic structure of matter and treat the sun now as a ball of continuous matter, now as a single mass point.

. . . The manner in which mathematical theories are applied does not depend on preconceived ideas; it is a purposeful technique depending on, and changing with, experience.

Feller, William (1950) An Introduction to Probability Theory and Its Applications, 2nd ed.

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