The Beauty of Mathematics

by Harold Morowitz

in "Mayonnaise and the Origin of Life", pages 232-235

In an age dominated by scientific technology there is a continuing problem of communicating the content and meaning of science to a broader public. Researchers still find it difficult to translate the latest technical perspective into language that is understandable to nonspecialists. It often takes 50 years or more for novel theoretical ideas to become incorporated into the thinking of the greater intellectual community. The viewpoint of quantum mechanics, for example, put forward in the late 1920s, has still not had its full impact on humanistic thought. The material is that hard to digest. Yet there are few philosophers of science who doubt that this theory has and will continue to exert a deep effect on our views of man and nature.

The one aspect of science that is most enigmatic and difficult for outsiders to comprehend is the use of mathematical abstractions to describe events that are very concrete. A physicist sits at his desk writing a series of symbols and numbers, while out in space a planet orbits the sun. Exact instruments are used to take very careful measurements of the planetary positions, and voila, the numbers obtained agree with the pencil-and-paper calculations. The whole space program, for example, depends on our ability to calculate exact orbits. Scientists' familiarity with this correlation between theory and experiment dulls us to the strangeness of the relation between the observed world and the little marks we put on paper.

There is a beautiful experience, available to physics students in their sophmore year of college, that illustrates the emotional impact of being able to deal with predictive mathematical theory. The first step in this exercise is set down on a piece of paper in mathematical form Newton's second law of motion, which states that force equals mass times acceleration. Next, one writes the equation that means there is a universal force of gravity, an attraction between all bodies that is proportional to their masses and inversely proportional to the square of the distance between them. The final step is to carry out a few lines of mathematical manipulations that quickly yield equations for Johannes Kepler's laws of planetary motion.

The act of personally deriving the motion of the bodies of the solar system from a few simple assumptions is a profound experience. When the result emerges, there is in the minds of many students a sense of deja vu, the feeling of a return to some primordial knowledge.

Some scientists greeted with awe the act of predicting the future of celestial systems. Others responded with arrogance. After Pierre-Simon Laplace's Celestial Mechanics was published, for instance, the scientist met with Napoleon, who noted that he saw no mention of God in the book. Laplace is reported to have replied, "I have no need of that hypothesis".

The most noteworthy achievement of mathematics in celestial mechanics did not occur until more than 150 years after Newton's laws of motion and universal gravitation were published. Astronomers studying the planet Uranus were unable to explain its motion by the laws of mechanics. They postulated that the orbit was being influenced by an unknown planet farther out from the sun. Astrophysicists used the mathematical methods of Laplace to predict the path of the unknown object. This information was communicated to observers who in 1846 aimed their telescopes in the predicted direction and discovered the planet Neptune. This discovery, one of the great triumphs of the human intellect, was totally dependent on the curious relation of the abstractions of mathematics to the concrete world of physics.

One one understands and feels the power of the derivation of Kepler's laws and subsequent developments, he may become more mystical about mathematics or more committed to hardnosed materialism. But one way or another the mathematical analysis is a moving experience.

Throughout physics, chemistry, and parts of biology, this strange relation between equations and real world events keeps appearing. The scientist uses mathematics because it gives him both an abbreviated way of representing experience and a feeling of deep understanding. Nevertheless there remains a whimsical wonder about why mathematics works. Theoretical physicist and Novel laureate Eugene Wigner once captured this feeling in an essay entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

Wigner's first argument begins with a statement of Einstein's that "the only physical theories which we are willing to accept are the beautiful ones". Mathematics seems to be the only study that generates sufficient beauty for the physicist. This tells us more about physicists than about physics, but in Eugene Wigner's view of science the subject material is, in any case, inseparable form the mind of the scientist.

After giving several examples of the applications of very abstract mathematics to specific problems of atomic structure and spectra, Wigner comes to the conclusion that "the unreasonable effectiveness of mathematics" in physics is an empirical law of epistemology. Somehow the structure of mathematics and the structure of the universe are close enough that one may represent aspects of the other with great precision.

Wigner's essay concludes with the following:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Thus the scientist uses this wonderful gift of mathematics but, like the layman, remains puzzled as to why it works. Knowing that even the experts wonder will perhaps lessen the fear and loathing of mathematics on the part of so many people outside science. To some of us, equations are objects of beauty, but to all of us they are tools of the trade.

A difficulty in coping with the mathematical side of science is worth overcoming if one wishes to be fully sympathetic with contemporary scientific thought. Even a rudimentary knowledge of mathematics will go a long way toward giving one a feel for the language of natural philosophy.