The late S. Chandrasekhar was best known for his discovery of the upper
limit to the mass of a white dwarf star, for which he received the Nobel
Prize in Physics in 1983. He was the author of many books, including
The Mathematical Theory of Black Holes and, most recently,
Newton's Principia for the Common Reader.
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Truth and Beauty
Aesthetics and Motivations in Science
By S. Chandrasekhar
The University of Chicago PressCopyright © 1987 The University of Chicago
All rights reserved.
I must confess at the outset to a feeling of apprehension at being included in this series on "Works of the Mind," as I am deeply aware of my shortcomings to speak either with assurance or with authority on a subject as wide and comprehensive in its scope as a discussion of the creative works of the scientist must be. But while I have these misgivings about the appropriateness of my representing the scientist in this series, I have no such misgivings in the choice of astronomy and astrophysics to represent the exact sciences. For astronomy among the exact sciences is the most comprehensive of all, and perfection in its practice requires scholarship in all its many phases. In another respect also, astronomy holds a unique position among the sciences,
as it is the only branch of the ancient sciences which has come to us intact after the collapse of the Roman Empire. Of course, the level of astronomical studies dropped within the boundaries of the remnants of the Roman Empire, but the traditions of astronomical theory and practice were never lost. On the contrary, the clumsy methods of Greek trigonometry were improved by Hindu and Arabic astronomers, and new observations were constantly compared with those of Ptolemy and so on. This must be paralleled with the total loss of understanding of the higher branches of Greek mathematics before one realizes that astronomy is the most direct link connecting the modern sciences with the ancient. Indeed, the works of Copernicus, Tycho Brahe, and Kepler can be understood only by constant reference to the ancient methods and concepts, while the Greek theory of irrational magnitudes and the Archimedes method of integration were understood only after being independently discovered by the moderns. (O. Neugebauer)
The sponsors of this series have indicated that each contributor will demonstrate the value of his art or profession by elucidating its nature, formulating its purpose, and explaining its techniques. But before I discuss these questions, I want to draw your attention to one broad division of the physical sciences which has to be kept in mind: the division into a basic science and a derived science. You will notice that my distinction is not between a "pure science" and an "applied science." I shall not be concerned with the latter, as I do not believe that the true values of science are to be found in the conscious calculated pursuit of the applications of science. I shall, therefore, be concerned only with what is generally called "pure science"; it is the division of this into a basic science and a derived science that I wish to draw your attention. While an exact or a sharp division between the two domains cannot be made or maintained, that it exists all the same will become apparent from the examples I shall presently give. But, broadly speaking, we may say that basic science seeks to analyze the ultimate constitution of matter and the basic concepts of space and time. Derived science, on the other hand, is concerned with the rational ordering of the multifarious aspects of natural phenomena in terms of the basic concepts. Stated in this manner, it is evident, first, that the division is dependent on the state of science at a particular time and, second, that there may be, and indeed are, different levels in which natural phenomena can be analyzed. For example, there is the domain of the Newtonian laws in which an enormous range and variety of phenomena find their direct and natural explanation. And then there is the domain of the quantum theory in which other types of problems receive their solution. When there are such different levels of analysis, there exist criteria which will enable us to decide when one set of laws is appropriate and the other set inappropriate or clumsy, as the case may be.
But to return to the division itself, I do not believe that there exists a better example of a basic discovery than Rutherford's discovery of the large angle scattering of α-particles. The experiment he performed was quite simple. Using a source of high-energy α-particles emitted by a radioactive substance, Rutherford allowed them to fall on a thin foil and found that sometimes the α-particles were actually scattered backward—rarely but certainly. Recalling this later in his life (1936), Rutherford said: "This was quite the most incredible event that has ever happened to me in my life." He has described his immediate reactions in the following words: "It was almost as incredible as if you fired a 15-inch shell at a tissue paper and it bounced back and hit you." He further records:
On consideration I soon realized that this scattering backwards must be the result of a single collision, and when I made calculations I saw that it was impossible to get anything of that order of magnitude unless you took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus. It was then that I had the idea of an atom with a minute massive center carrying a charge. I worked out mathematically what laws scattering should obey, and I found that the number of particles scattered through a given angle should be proportional to the thickness of the foil, the square of the nuclear charge, and inversely proportional to the fourth power of the velocity. These deductions were later verified by Geiger and Marsden in a series of beautiful experiments.
And so it was that the nuclear model of the atom, which is at the basis of all science, was born. A single observation and its correct interpretation led to a revolution in scientific thought unparalleled in the annals of science.
I would suppose that the discovery of the neutron by Chadwick is in the same category, as it is now believed that these together with protons are the basic constituents of all nuclei. But you should not suppose from these two examples that all basic scientific facts are necessarily to be gathered only in the field of atomic physics. Indeed, the first example of a law which may be called "basic" is astronomical in origin. I refer to the discovery by Kepler of the laws of planetary motion after long and patient analysis of the extensive observations of Tycho Brahe. These laws of Kepler led Newton to his celebrated laws of gravitation, which occupied the central scientific arena for over two hundred years. I shall return to this matter presently in a somewhat different context, but the example suffices to indicate that only in the gravitational domain can astronomy lead directly to results of a basic character. A further illustration of this is the fact that the minute departures exhibited by the motion of the planet Mercury from the predictions based on Newtonian laws indicated and later confirmed the far-reaching changes in our concept of space and time implied by the general theory of relativity. In the same way, it is not impossible that the discovery by Hubble of the recession of extragalactic nebulae with velocities proportional to their distances may lead to further modifications in our basic concepts.
The examples which I have just given might suggest that the true values of science are to be found in those pursuits which lead directly to advances which I have termed "basic." Indeed, there are many physicists who seriously take this view. For example, a very distinguished physicist, apparently feeling sorry for my preoccupation with things astronomical and with an intent to cheering me, said that I should really have been a physicist. To my mind this attitude represents a misunderstanding of the real values of science. And the history of science contradicts it. From the time of Newton to the beginning of this century, the whole science of dynamics and its derivative celestial mechanics have consisted entirely in amplification, in elaboration, and in working out the consequences of the laws of Newton. Halley, Laplace, Lagrange, Hamilton, Jacobi, Poincaré—all of them were content to spend a large part of their scientific efforts in doing exactly this, that is, in the furtherance of a derived science. A derision of the derived aspects of science, implying as it would a denial of the values which these men have so earnestly sought, is to my mind sufficiently absurd to merit further consideration. Indeed, it must be apparent to an impartial observer that there is a complementary relationship between the basic and the derived aspects of science. The basic concepts gain their validity in proportion to the extent of the domain of natural phenomena which can be analyzed in terms of them. And, in limiting the domain of validity of these concepts, we recognize the operation of other laws more general than those we have operated with. Looked at in this way, science is a perpetual becoming, and it is in sharing its progress in common effort that the values of science are achieved. With these remarks, I think that I may state in a more formal manner what I regard as the true values of science which a scientist in the practice of his profession seeks to attain.
Scientific values consist in the continual and increasing recognition of the uniformity of nature. In practice this only means that the values are attained in a larger or smaller measure in extending, or equivalently limiting, the domain of applicability of our concepts relating to matter, space, and time. In other words, a scientist seeks continually to extend the domain of validity of certain basic concepts. In so doing, he attempts to discover the limitations, if any, of these same concepts, and in this way he tries to formulate concepts of wider scope and generality. These values which are the quest of a scientist take in practice one of three distinct forms which I shall discuss under the headings: "Universality of the Basic Laws," "Predictions Based on the Basic Laws," and "Identifications Resulting from the Basic Laws."
Let me illustrate each of these by some examples.
UNIVERSALITY OF THE BASIC LAWS
In some ways the universality of the laws of nature is best illustrated by showing how this was achieved with regard to the law of gravitation.
It is found that, all over the earth, objects are attracted toward the center of the earth. How far does this tendency go? Can it reach as far as the moon? These were the questions which Newton asked himself and answered. Galileo had already shown that uniform motion is as natural as rest and that deviation from such motion must imply a force. If, then, the moon were relieved of all forces, it would leave its orbit and go off along the instantaneous tangent to the orbit. Consequently, if the motion of the moon is due to the attraction of the earth, then what the attraction really does is to draw the moon out of the tangent into the orbit. As the period and the distances of the moon are known, it is easy to compute how much the moon falls away from the tangent in one second. Comparing this with the speed of falling bodies on the earth, Newton found the ratio of the two speeds to be about as 1 to 3600. As the moon is sixty times as far from the center of the earth as we are, this implies a force that decreases with the square of the distance.
The next question Newton asked himself was how universal this property is. In particular, does a similar force reside in the sun, keeping the planets in their orbits as the earth does the moon? The answer is to be found in Kepler's laws. Newton showed that Kepler's second law—that planets describe equal areas in equal times—implies a central force, that is, a force directed toward the sun; that the first law—that the planetary orbits are ellipses with the sun at one focus—is a consequence of the inverse square law of attraction; and, finally, if the same law holds from planet to planet, then the periods and distances should be related as in Kepler's third law. It was in this manner that Newton was able to announce his law of gravitation that every particle in the universe attracts every other particle with a force that varies inversely as the square of their distance apart and directly as the masses of the two particles. You will notice the use of the word "universe" in this formulation and the clear indication that its importance arises from its universality.
One further related observation. In 1803 William Herschel was able to announce from his study of close pairs of stars that in some instances the pairs represented real physical binaries, revolving in orbits about each other. Herschel was further able to show that the apparent orbits were ellipses and that Kepler's law of areas was also valid. In other words, this observation extended the validity of the laws of gravitation from the solar system to the distant stars. It is difficult for us to imagine how tremendous the impression was which this discovery of Herschel made on his contemporaries.
A good deal of the progress of astronomy since Newton's laws were announced had been concerned with their application to the motions in the solar system. Newton himself pointed out many of their chief consequences. To mention only two of these, he found the correct explanation for the phenomenon of ocean tides, and also for the precession of the equinoxes, a phenomenon which had been discovered twenty centuries earlier by Hipparchus.
The application of Newton's laws throughout the solar system is a task of incredible difficulty and one that has taxed the powers of such giants as Lagrange, Laplace, Euler, Adams, Delaunay, Hill, Newcomb, and Poincaré.
I have already referred to the fact that the motion of the planet Mercury cannot be fully accounted for on Newtonian theory. Departures are found in the sense that there appears to be a slow revolution of the orbit as a whole at a rate which is in excess of what can be accounted for on Newtonian theory by 42 seconds of arc per century. It would seem that this is now satisfactorily accounted for in terms of Einstein's theory of general relativity.
There are still many fields of astronomy to which Newton's laws could be profitably applied. The newest of these relates to the motions in the galaxy as a whole and the new branch of dynamics called "stellar dynamics," which is rapidly growing in range and scope. I shall have a few things to say about this later.
Let me turn from this classic example of the universality of nature's laws to a more recent advance which in some ways is as striking. I hardly need point out to an audience in 1946 that the phenomenon of nuclear transformation (more commonly called "atom-smashing") has been studied extensively in recent years. Using the information obtained from such studies, Bethe was able to announce a few years ago that certain nuclear transformations involving carbon and nitrogen can lead indirectly to the synthesis of a helium nucleus from four protons. He was further able to show that, under the conditions for the interior of the sun which had been derived earlier by astrophysicists and with the cross sections for the reactions found in the laboratory, we can account in a most satisfactory way for the source of energy of the sun—a striking example of the synthesis of many types of investigations.
Let me consider one further example. In 1926 Fermi and Dirac were led to a reformulation of the laws of statistical mechanics as they applied to an electron gas and showed that departures from classical laws should be expected at high densities and/or low temperatures. The nature of the departures predicted was this: According to classical laws, the pressure is proportional to the concentration and temperature. If at a given temperature we increase the concentration, then departures set in, in the sense that the pressure begins to increase more rapidly with the concentration and eventually becomes a function of the concentration only. When such a state is reached, one says that the electron gas has become degenerate. These new laws have found extensive applications in the theory of metals and are of the greatest practical importance. But the first application of the new laws was found in an astrophysical context by R. H. Fowler, who used the laws of a Fermi-Dirac gas to elucidate the structure of very dense stars such as the companion of Sirius. These dense stars, commonly known as white dwarfs, have densities of the order of several tons per cubic inch. The most extreme example is a star discovered some years ago by G. P. Kuiper which is estimated to have a density of 620 tons per cubic inch. Fowler immediately recognized that under these conditions the electrons must be degenerate in the sense of the Fermi-Dirac statistics. And with this discovery of Fowler it became possible to work out the constitution of the white dwarf stars.
This subject of the structure of the white dwarf stars has interested me personally, and I may be pardoned if I dwell on it a little longer. An extension of Fowler's discussion soon made it apparent that the laws of Fermi and Dirac required further modifications to take account of the fact that, at the high densities prevailing in the white dwarf stars, there will be a considerable number of electrons moving with velocities comparable to that of light. When modifications resulting from such high velocities are included, it was found that there is an upper limit to the mass of dense stars. This upper limit is in the neighborhood of 1.4 solar masses. The reason for the appearance of this upper limit is that for larger masses no stable equilibrium configuration exists. The recognition of this upper limit raises many questions of interest concerning stellar evolution. And it is not impossible that the occurrence of the supernova phenomenon is in some ways related to this. I shall not go into these matters now, but I mention these only to draw your attention to the manner in which the domain of validity of certain basic laws is continually being extended.
In the three illustrations I have given, I have discussed the applicability of the same laws. But sometimes we have the application of the same set of ideas to problems which may appear entirely unrelated at first sight. For example, it is surprising to realize that the same basic ideas which account for the motions of microscopic colloidal particles in solution also account for the motions of stars in clusters. This basic identity of the two problems, which is far-reaching, is one of the most striking phenomena I have personally encountered, and I would like to say a few words about it.
Excerpted from Truth and Beauty by S. Chandrasekhar. Copyright © 1987 The University of Chicago. Excerpted by permission of The University of Chicago Press.
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Table of ContentsPreface
1. The Scientist 
2. The Pursuit of Science: Its Motivations 
3. THE NORA AND EDWARD RYERSON LECTURE
Shakespeare, Newton, and Beethoven, or Patterns of Creativity 
4. Beauty and the Quest for Beauty in Science 
5. MILNE LECTURE
Edward Arthur Milne: His Part in the Development of Modern Astrophysics 
6. ARTHUR STANLEY EDDINGTON CENTENARY LECTURES 
Eddington: The Most Distinguished Astrophysicist of His Time
Eddington: The Expositor and the Exponent of General Relativity
7. KARL SCHWARZSCHILD LECTURE
The Aesthetic Base of the General Theory of Relativity