In a great book, “The Truth Of Science”, the author, Roger G. Newton, writes on the nature of mathematics.
“Mathematics is certainly a potent calculational tool and a convenient language for abstract concepts, but it is vastly more than that. Einstein went so far as to declare that “the creative principle (of science) resides in mathematics. If mathematics cannot tell us any facts about the world, and it surely cannot, then why is it so invaluable for science? Poincare's insight is a first step toward an answer: the greatest objective value of science lies in the discovery, not of things or facts, but of relations between them. “Sensations are intransmissible . . . But it is not the same with relations between these sensations. . . Science . . . is a system of relations . . . it is in the relations alone that objectivity must be sought. Mathematics” is the most appropriate vehicle for the description of relations and for their logical exploitation. Physicists do not use it (other than numbers) for the description of experimental facts, but for the manipulation of the interrelations between these facts-the theories. Mathematics is only the means by which we use one set of facts to explain another,” writes Steven Weinberg, “and the language in which we express our explanations.” The power of mathematics resides in its versatility in dealing with an enormous variety of connections between things, concepts, and ideas. I therefore cannot agree with the view of John Ziman that “physics defines itself as the science devoted to discovering, developing, and refining those aspects of reality that are amenable to mathematical analysis.” Physics deals with relations between aspects of reality, and relations are always amenable to mathematical analysis. Whenever new kinds of relations do not seem to lend themselves to such manipulation, invariably a fresh branch of mathematics is discovered (or invented) that is suitable.
We are still left, however, with the question of why most branches of mathematics, often developed with no thought of application, have turned out to be so valuable in science, and especially in physics. “The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and. . . there is no rational explanation for it,” Eugene Wigner declared in awe. To try to solve that mystery we have to look, first of all, at the nature of mathematics.
Many practitioners regard proving a new theorem as discovering something that has an independent existence-mathematicians do not invent their structures but discover them. This school of thought, called Platonism, has a long history in the discipline, and many of the greatest mathematicians were, and are, its adherents. Some of them have found Platonism a great inspiration in their work-it is exhilarating to know that when you arrive at a long-sought insight, you have discovered a new piece of knowledge about the universe-a universe, of course, not of the external world of the senses but of Plato's realm of concepts and enduring truths. In this view, the work of mathematicians is quite analogous to that of experimental physicists, except that its scene of action is not Nature, as it is for scientists, but the eternal world of ideas. “I believe,” wrote the English mathematician G. H. Hardy, “that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’ are simply our notes of our observations.”
The branch of mathematics in which this Platonic view is most convincing is the theory of numbers, which deals with nothing but the most basic entities of mathematics, the natural numbers-the integers. Since it is hard to believe that the integers are human inventions-they must surely be used even by an alien civilization on a planet of Alpha-Centauri-theorems about the prime numbers, for example, must have an independent existence, which mathematicians discover. “The integers were made by God: all else is the work of man,” declared the German mathematician Leopold Kronecker, and he followed the precept of the second part of his assertion by arguments that led to the school of intuitionism.
For the intuitionists, mathematicians are architects and engineers rather than explorers; their theorems are of their own making, and the tools they can use are correspondingly limited to those appropriate for construction. They must not make use of indirect inferences, such as proving a proposition by demonstrating that as denial would lead to a contradiction. During the early part of this century, the intuitionist school of mathematical thinking became fairly influential and had a number of prominent adherents. It also had passionate enemies, who opposed the implication that a number of important mathematical theorems, proved by methods whose validity the intuitionists denied, would have to be sacrificed. Intuitionism “seeks to break up and to disfigure mathematics,” bitterly complained David Hilbert. For example, the classical proof that there are infinitely many prime numbers is indirect: the assumption that their number is finite leads to a contradiction. Intuitionists deny the validity of this proof and will accept only a direct, constructive demonstration.”