G11. Math As A Tool And A Toy

Pure mathematics as an independent structure of thought can best be looked at as a game, for it is precisely that. Its axioms define the game, which is played according to rules of logic. The mathematician’s symbols are the pieces to be moved about. The wealth of relationships that can be derived from the basic axioms correspond to the rich variety of situations that can arise in a game with comparatively simple rules. In some particularly simple area of mathematics, such as the study of a group of only three objects, all possible consequences of the axioms can be derived, just as every possibility in tic-tac-toe can be spelled out. In more complex fields of mathematics, the seemingly limitless wealth of consequences that flow from the axioms continue to challenge the minds of generations of mathematicians, just as the inexhaustible possibilities in bridge and chess fascinate generations of players. To pursue this deep-going analogy a bit further, consider the game of chess. To make it correspond to a branch of mathematics, we must regard the board as the writing tablet of the mathematician, and the chessmen as the symbols representing the abstract objects of study. One set of chessmen may be of ivory, another of wood, another of plastic; but every knight, regardless of his size or shape or material, represents the same essence. There is but one single abstract object called “knight.” The piece on the board is no more that basic concept than is a pencil stroke on a piece of paper the abstract object of mathematical study. The statements of the way in which each chessman moves are the axioms of the game. A minor change of a single axiom produces a drastic change in the game, just as a single alteration of the axioms of arithmetic produces a whole new kind of mathematics. Finally, there is the question of what constitutes complete knowledge of a concept or of a chess piece. When the way in which a knight moves, the way he takes other pieces, and the way he can be taken are specified, that constitutes a complete definition of the abstract concept of a knight. There is nothing else to know. Questions such as how would a knight move on a triangular board are meaningless, for they are outside the game of chess. There is still much that can be learned about the usefulness of a knight and his relation to other chessmen, but nothing more that can be said to define what he is. In the same way, a mathematical concept is completely defined in terms of a few axioms. Its very abstraction rests on the fact that so few of its properties are specified. The beginning student of mathematics might say, “Yes, I understand how vectors behave, but what is a vector?” The only answer is that a vector is an idea defined by a few rules of its behavior. A scientist, more interested in making the idea visualizable and useful than in sticking to logical rigor, might give the answer, “Well, force, for example, is a vector.” What the scientist means by this is that there exists a physical concept called force that behaves like the mathematical concept called vector. The mathematical concept has been found to correspond to a physical concept and thereby to have acquired external truth as well as inner truth. Calling force a vector is exactly the same as saying about a particular carved piece of ivory, “This is a knight.” More precisely, the piece is a physical object that behaves like the abstract concept of knight. The essence of knight is an idea quite independent of whether the piece is black or white, large or small, ivory or plastic. In the same way, the mathematical ideas of number, vector, length, or angle are abstractions quite independent of any correspondence to things in the “real” world.

Mathematics, then, can be regarded as a creation of the human mind independent of science and independent of nature. As such, it is not merely like a game; it is a game. Its rules are arbitrary, and its only criterion of truth is the inner truth of self-consistency. In terms of the nature of truth, mathematics and science can be clearly separated. Mathematics, no matter how numerous its applications to the real world, finds its ultimate criterion of validity only in the harmony of inner truth. Science, no matter how abstract its concepts or how theoretical its reasoning, is ultimately justified only by the external truth of experimental confirmation. In human terms, the dichotomy can be expressed this way. Science rests basically upon our awareness of an orderly world outside ourselves. Mathematics rests basically upon our awareness of an orderly world within ourselves.

To the scientist, of course, mathematics is more than a game. It is half a tool, and half a toy. Science without mathematics is unthinkable, for it is mathematics that gives to science its quantitative character and its predictive power. But taking the modern point of view about the nature of mathematics, it must be regarded as a miraculous chance that mathematics has found useful application in the description of the physical world.

Or is it a chance? Of course, it is no surprise that arithmetic and algebra and Euclidean geometry have something to do with the real world. They were invented and developed for practical ends, in order to describe reality. Application came first, abstraction later. But the question goes deeper than that. Basically it is: Are we humans capable of conceiving the nonphysical? Since we are a part of the physical world, does it make any sense to distinguish our inner world and our external world? Must any mathematics the human mind is capable of inventing have some connection with the physical world, whether or not that connection has yet been discovered? In other words, is the popular “misconception” about the external truth of mathematics not a misconception after all? Whatever the future holds, the fact is that there now exist branches of mathematics with no known external truth; that part of mathematics remains, so far, an intellectual exercise.

At the same time, much of the “nonphysical” mathematics of the nineteenth century has become the physical mathematics that we now use. Our three-dimensional space has been generalized to a four-dimensional spacetime. The general theory of relativity has shown that space is not really Euclidean, although it seems so in any small region. The mathematical formulation of the quantum theory has required the use of imaginary numbers, of quantities that do not commute with each other, and of vectors in a space of infinitely many dimensions. Impressed by this great expansion in the mathematical basis of physics, Paul Dirac¹ wrote in 1931: “Non-Euclidean geometry and non-commutative algebra, which were at one time considered to be purely fictions of the mind and pastimes for logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and that advance in physics is to be associated with a continual modification and generalization of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.” Dirac does not say that all mathematics will eventually turn out to be useful in the scientific description of nature, but he expresses his belief that ever wider ranges of mathematics will prove to be physical mathematics, that is, to have an external truth as well as an inner truth. Even if it were true that any self-consistent mathematical scheme that scientists can invent necessarily bears some correspondence to the physical world (a proposition that seems unlikely to this author), it still seems impossible that this could be proved to be true. In the race for truth, the mathematicians would remain always a lap ahead of the scientists. At all times there would exist some branches of mathematics that had as yet found no physical application and would remain, at least for a time, “purely fictions of the mind and pastimes for logical thinkers.”

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