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.”
1 P. A. M. Dirac, an outstanding theoretical physicist in England, was himself responsible for the introduction into physics of some previously nonphysical mathematics. In 1928 he discovered that the electron must be described not only by its position in ordinary space and time, but by its position in a peculiar abstract “spinor space.” This new mathematics led to an improved theory of the electron and to the prediction of the positron. For this work, Dirac received the Nobel Prize in physics in 1933. (The quotation above appeared in the Proceedings of the Royal Society, A133 (1931), 60.)