Albert Einstein, although Jewish, received his earliest education in a Catholic grammar school in Munich, Bavaria, to which city his family moved while he was still quite young. Like Newton, with whom he is often compared (and certainly he is the only scientist since Newton's time who can bear the comparison), he showed no particular intellectual promise as a youngster. As a matter of fact, he was so slow in learning to speak that there was some feeling that he might prove retarded.
In 1894 his father (who had failed in business) left for Milan, Italy, while
Albert stayed behind to finish his high school studies. However, he did very
badly in Latin and Greek and was interested only in mathematics, so he left
school by invitation of the teacher who said, "You will never amount to
anything, Einstein." The young man thus became the most unusual "dropout" in
the history of science.
After an Italian vacation he began his college work in Switzerland (not without difficulty, for only in mathematics was he really qualified for entrance). Nor did he enjoy the experience. He cut most of the lectures, preferring to concentrate on independent reading in theoretical physics. That he could pass his courses at all was due to the excellent lecture notes of a friend.
Once graduated, he tried to find a teaching post but that wasn't easy, for he was not a Swiss citizen and he was Jewish besides. In 1901, thanks to the influence of the father of the same friend whose lecture notes Einstein had used Einstein accepted a position as a junior official at the Patent Office at Berne Switzerland, and in that year became a Swiss citizen.
Therefore, without any academic connections, he began his work and for it fortunately, he required no laboratory but only a pencil, some paper, and his mind. The year 1905 was his annus mirabilis, for it saw the publication of five of his papers in the German Yearbook of Physics, involving three developments of major importance (and in that same year, he earned his Ph.D.).
One paper dealt with the photoelectric effect, whereby light falling upon certain metals was found to stimulate the emission of electrons. Lenard had in 1902 found that the energy of the emitted electrons did not depend upon the intensity of the light. A bright light might bring about the emission of a greater number of electrons, but not of more energetic ones. There was no satisfactory explanation for this in terms of classical physics.
Einstein, however, applied to the problem the quantum theory worked out five years earlier by Planck and disregarded since. Einstein maintained that a particular wavelength of light, being made up of quanta of fixed energy content, would be absorbed by a metallic atom and would force out an electron of fixed energy content and no other. Brighter light (more quanta) would then bring about the emission of more numerous electrons, but all still of the same energy content. Light of shorter wave length, however, would have more energetic quanta and would bring about the emission of more energetic electrons. Light that had wavelengths longer than a certain critical value would be made up of quanta so weak as to bring about no electron emission at all. The energy content of such long wavelength photons would be insufficient to break electrons away from the atoms of which they formed a part. This "threshold wave length" would be different for different metals, of course. Planck's theory was thus, for the first time, applied to a physical phenomenon (other than the black-body problem which had occasioned its development in the first place) that it could explain and classical physics could not. This went a long way, perhaps even all the way, toward establishing the new quantum mechanics. For this feat Einstein was eventually awarded the 1921 Nobel Prize in physics, and yet it was not his greatest work of that year.
In his second paper of 1905, published two months after the first, Einstein worked out a mathematical analysis of Brownian motion, first observed by Brown three quarters of a century earlier. Einstein showed that if the water in which the particles were suspended was composed of molecules in random motion, according to the requirements of the kinetic theory of Maxwell and Boltzmann, then the suspended particles would indeed jiggle as they were observed to do. Svedberg had suggested this molecular explanation of Brownian motion three years earlier, but it was Einstein who worked matters out in mathematical detail.
All objects in water (or in any liquid or gas) are continually bombarded from all sides by molecules. Through the workings of chance, the number of molecules striking any object of ordinary size from one angle is about the same as the number from another angle, the differences in number that do exist being insignificant in comparison with the truly vast total numbers involved. For that reason there is no overall effect (or at least no detectable one) upon objects of ordinary size.
As an object grows smaller, the total number of molecules bombarding it decreases and small differences in bombardment from this direction or that grow appreciable. Grains of pollen or particles of dye are small enough to be pushed first this way by a slight excess of molecules striking in that direction, then in another, then in still another. The motion is quite random, attesting to the random motion of the molecules themselves.
The larger the average size of the molecules, the larger the body for which this difference in bombardment can produce detectable effects. Therefore, the equation deduced by Einstein to describe Brownian motion could be used to work out the size of molecules and of the atoms that compose them. Three years later Perrin conducted experiments on Brownian motion which confirmed Einstein's theoretical work and which gave the first good values of atomic size. The atomic theory of Dalton was a hundred years old by then and had been accepted by all but a few diehards such as Ostwald, and yet this was the first time the effect of individual molecules could be directly observed. Even Ostwald gave in.
Einstein's greatest accomplishment of the year involved a new outlook on the universe, replacing the old Newtonian view, which had reigned supreme for two and a quarter centuries.
Einstein's work climaxed the famous experiment of Michelson and Morley, who had been unable to detect any difference in the velocity of light with changes in its direction through the ether. So Einstein began with the assumption that the measured velocity of light in a vacuum is always constant despite any motion of its source or of the individual measuring the light. Furthermore, he canceled out the ether as unnecessary by assuming that light traveled in quanta and therefore had particle-like properties and was not merely a wave that required some material to do the waving. This particle-like form of light was named a photon a decade later by Compton. It represented a retreat from the extreme wave theory of light, moving back toward Newton's old particle theory and taking up an intermediate position that was more sophisticated, and more useful, than either of the older theories.
Einstein also pointed out that without the ether there was certainly nothing in the universe that could be viewed as at "absolute rest," nor could any motion be considered an "absolute motion." All motion was relative to some frame of reference chosen, usually, for its convenience, and the laws of nature held unchanged for all such frames of reference. His theory, because of the "all motion is relative" idea, is therefore called relativity. In this particular paper he dealt only with the special case of systems in uniform nonaccelerated motion, so it is called the special theory of relativity.
He showed that from this simple assumption of the constancy of the velocity of light and the relativity of motion, the Michelson-Morley experiment could be explained and Maxwell's electromagnetic equations could be kept. He showed also that the length-contraction effect of FitzGerald and the mass-enlargement effect of Lorentz could be deduced, and that the velocity of light in a vacuum was therefore the maximum speed at which information could be transferred.
All sorts of peculiar (in appearance) results followed. The rate at which time passed varied with velocity of motion; one had to give up notions of simultaneity, for one could no longer say, under certain conditions, whether A happened before B, after B, or simultaneously with B. Space and time vanished as single entities and were replaced by a fused "space-time." All this was against "common sense" but common sense is based on a limited experience with objects of ordinary size moving at ordinary velocity. Under such conditions the difference between Einstein's theory and the ordinary Newtonian view (which is "common sense") becomes indetectably small. In the vast world of the universe as a whole and the tiny world within the atom, however, common sense is no guide; there is a detectable difference between the two views; and it is Einstein's view and not Newton's that is the more useful.
In the special theory of relativity, Einstein worked out an interrelationship of mass and energy in a famous equation. Since the velocity of light is a huge quantity, a small amount of mass (multiplied by the square of the velocity) is equivalent to a large amount of energy.
With mass and energy thus interpreted as different aspects of the same phenomenon, it was no longer sufficient to speak of Lavoisier's conservation of mass or of Helmholtz's conservation of energy. Instead there was the greater generalization of the conservation of mass-energy. Or, if one still speaks simply of the conservation of energy, it must be understood that mass is but one more aspect of energy.
This new view at once explained the energies given off by radioactive elements as a consequence of the slight loss of mass involved, a loss so slight as to be undetectable by ordinary chemical procedures. The interrelationship of mass and energy was quickly confirmed by a variety of nuclear measurements and has, ever since, proved fundamental in atomic studies. Once only did its usefulness seem to flag and then Pauli postulated the existence of the neutrino to save it.
The value of the new generalization in everyday affairs, and not merely in the highly esoteric work of the atomic physicists, was overwhelmingly shown when the conversion of mass to energy on a large scale made possible the devastation by atomic bombs a generation later, a denouement to which Einstein was to contribute directly, and which he was to find horrifying.
Despite this triple thunderbolt of papers, it was four more years before Einstein could finally obtain a professorship (and a poorly paying one) at the University of Zürich. His reputation continued to grow, however, and in 1913 a position was created for him at the Kaiser Wilhelm Physical Institute in Berlin, thanks to Planck, who was greatly impressed by the young Einstein. For the first time Einstein was to be paid generously enough to make it possible for him to devote his life to science.
World War I broke out but Einstein was little affected, since he was at the time a Swiss citizen. However, when many German scientists signed a nationalistic pro-war proclamation, Einstein was one of the few to sign a counterproclamation calling for peace.
Einstein was then working on the application of his theory of relativity to the more general case of accelerated systems and in so doing worked out a new theory of gravitation of which Newton's classic theory was but a special case. He published it in 1915 in another tremendous paper usually referred to as the "General Theory of Relativity." The equations set up in this theory allowed grand conclusions to be drawn about the universe as a whole and Sitter was to use those equations to better effect than Einstein himself.
In the general theory, Einstein pointed out three places where his theory predicted effects that were not like those predicted by Newton's theory. The phenomena concerned could be measured and in that way a decision between the two theories could be reached.
First, Einstein's theory allowed for a shift of the position of the perihelion of a planet, a shift that Newton's theory did not allow. Only in the case of Mercury (closest to the sun and its gravitational influence) was the difference large enough to be noticeable. And, as a matter of fact, the motion that Leverrier had detected and tried to explain by supposing the existence of an infra-Mercurian planet, was explained on the spot by Einstein's theory. This, however, was not so impressive as it might be since Einstein knew about the discrepancy of Mercury's motion to begin with and could have "aimed" his theory at it.
Secondly, however, Einstein pointed out that light in an intense gravitational field should show a red shift. This had never been looked for or observed so the coast was clear for a fair test. Only extreme gravitational fields could show a shift large enough to measure at the time and, at Eddington's suggestion, W. S. Adams demonstrated the existence of this Einstein shift in the case of the white-dwarf companion of Sirius, which had the intensest gravitational field then known.
(In the 1960s, with improvement in measuring devices, the much smaller Einstein shift of the light of our own sun was measured and found to match Einstein's prediction. In addition, the shift in gamma-ray wavelength, worked out by Mössbauer in the late 1950s, was essentially an Einstein shift and it too has been measured and found to be in accord with the prediction.)
Thirdly and most dramatically, Einstein showed that light would be deflected by a gravitational field much more than Newton predicted. There was no way of testing this in the midst of World War I. However, with the war over (and Germany, but not Einstein, defeated ) the opportunity arose on March 29, 1919, when a solar eclipse was scheduled to take place at just the time when more bright stars were in the vicinity of the eclipsed sun than would be there at any other time of year.
The Royal Astronomical Society of London made ready for two expeditions, one to northern Brazil and one to Principe Island in the Gulf of Guinea off the coast of west Africa. The positions of the bright stars near the sun were measured. If light was bent in its passage near the sun, those stars would be in positions that differed slightly from those they occupied six months before, when their light passed nowhere near the sun as they rode high in the midnight sky. Again the comparison of positions backed Einstein.
Einstein was now world-famous. Ordinary people might not understand his theories and might only grasp dimly what it was all about but there was no question that they understood him to be the scientist. No scientist was so revered in his own time since Newton. This, however, was not to save Einstein from the malevolent forces that were beginning to sweep Germany.
In 1930 Einstein visited California to lecture at the California Institute of Technology and was still there when Hitler came to power. There was no point in returning to Germany, and he took up permanent residence in Princeton, New Jersey, at the Institute for Advanced Studies where, a year before, he had already been offered a post. In 1940 he became an American citizen.
The final decades of his life were spent in a vain hunt for a theory that would embrace both gravitation and electromagnetic phenomena (the unified field theory) but this, to his increasing distress, eluded him and, so far, it has eluded everyone else. Nor did Einstein succeed in accepting all the changes that were sweeping the world of physics, despite his own role as intellectual revolutionary. He would not accept Heisenberg's principle of indeterminacy, for instance, for he could not believe that the universe would be so entirely in the grip of chance. "God may be subtle," he once said, "but He is not malicious."
In 1930 he had argued that the uncertainty principle implied that time and energy could not be simultaneously determined with complete accuracy. He presented a "thought experiment" to show that this was not so and that time and energy could be determined simultaneously to any degree of accuracy. The next day, however, Bohr, having spent a sleepless night, pointed out an error in Einstein's argument. Now the time-energy indeterminacy is accepted.
With the beginning of World War II, Einstein was instrumental in achieving something he did not want. Uranium fission had been discovered in 1939 by Hahn and Meitner, and Szilard could see quite well what that implied. Szilard did not want the horrors of nuclear bombs to be released on mankind but, on the other hand, the possibility that Hitler might come into possession of such bombs had to be reckoned with.
Einstein, as the most influential scientist in the world, was persuaded by Szilard to write a letter to President Franklin D. Roosevelt, urging him to put into effect a gigantic research program designed to develop a nuclear bomb. The result was the Manhattan Engineering District, which, in six years, did develop such a bomb, the first being exploded at White Sands near Alamogordo, New Mexico, on July 16, 1945. By that time Hitler had been defeated, so the second and third bombs were exploded over Japan the next month.
The nuclear bombs remained to threaten postwar mankind, and five countries -- the United States, the CIS (former countries of the Soviet Union), Great Britain, France, and China -- now have such weapons. (India has exploded a "peaceful nuclear device. Many believe that Israel is only "the turn of a screw" from having workable nuclear weapons, although she has never exploded any such device.) To the end of his life Einstein fought stubbornly for some world agreement to end the threat of nuclear warfare. He also expressed his strong opposition to the temporary aberration of McCarthyism that swept the United States in the early 1950s. His ability to revolutionize physics was greater, however, than his ability to change man's heart, and at the time of his death the peril was greater than ever before.
The element, atomic number 99, was named einsteinium in his honor, shortly after his death, and in 1966 the United States Post Office placed his face on a stamp.
General Relativity, which deals so successfully with the large scale aspects of the universe, "is essentially a geometrisation of physics," to quote Whittaker.1 In a more restricted way so is the Special Theory of Relativity. Discussing before the German Association in 1908 the space-time manifold of relativity, Minkowski could think of no better way to comfort those who found the abandonment of the traditional views on space and time too painful than to remind them "of the idea of a pre-established harmony between pure mathematics and physics"2 so forcefully displayed by the new theory.
It was again precisely this aspect in Einstein's famous paper of 1905 on the electrodynamics of moving bodies that appeared so decisive to Jeans when he remarked that "the study of the inner workings of nature passed from the engineer scientist to the mathematician."3 For as time went on, it became increasingly evident that both relativity and wave mechanics can only draw a purely mathematical picture about nature that, let it be known, is no picture at all in the obvious visual sense of the word.
Such a deep split between the visualizing inclination of man and the belief in the fundamentally mathematical structure of the universe, however, is far from being disastrous as regards the rational understanding of nature. For, as Jeans aptly put it, "nature seems very conversant with the rules of pure mathematics,"4 with rules that are constructed without any reference to the visual features of the outside world. In fact, this harmony between mathematics and nature is so deep-going that in a sense the understanding of the universe seems to be open only to the mathematician. This is what prompted Jeans to fancy that "the Great Architect of the Universe now begins to appear as a pure mathematician."5
The word pure in this context was particularly well chosen. For much of the mathematics that so well suited the needs of modern physics was worked out long before physics became cognizant of its need of various mathematical theories without which modern physics would be simply unthinkable. The case was distinctly different from classical physics, which developed by and large its own mathematics according to the needs presented by problems of physics waiting for solution.
Classical calculus, as is well known, sprang from an age-old desire to handle the problem of infinitesimal changes in physical processes. Logarithms were similarly developed to facilitate trigonometrical calculations of already extant problems in astronomy. The development of the great variety of differential equations in the eighteenth century betrays at every point the concern with actual problems of mechanics. To mention only one of the numerous cases, it was the study of a bar clamped at one end that revealed to Daniel Bernoulli the existence of differential equations of the fourth order.
The development of the Bessel functions, so indispensable in almost every department of mathematical physics, tells the same story. Its form with coefflcients of order zero was formulated in 1732 by Bernoulli in the course of his investigations of the oscillations of heavy chains. The analysis of the vibrations of stretched membranes by Euler in 1764 led to its type with more general coefficients, and problems of perturbations in celestial motions prompted Bessel to investigate further the properties of the function that bear his name.
Again, it was the isoperimetric problems of physics that gave rise to the calculus of variations. In general the development of analysis throughout the eighteenth century betrayed the same motivation, as evidenced by the works of Euler, Clairaut, and Lagrange. Their essential aim in mathematics was to find more refined tools to handle problems ranging from fluid motion to the motion of planets. It was especially these latter problems that spurred the extensive development of perturbation methods. In all this an attitude asserted itself that, as J. Bertrand put it over a hundred years ago, was misled by the successes it had in coping with the phenomena of the physical world into the false belief that "the mathematician, without being longer occupied in the elaboration of pure mathematics, could turn his thoughts exclusively to the study of natural laws."6
This relation between mathematics and physics remained essentially unchanged well into the opening decades of the nineteenth century, as witnessed by Gauss' theory of the method of least squares Laplace's work on the theory of probabilities, Hamilton's work on the quaternions, and Fourier's analysis of heat convection. Mention should also be made of the development of potential theory, the theory of functions of a complex variable, functional analysis and differential geometry, the theory of line, surface, and volume integrals as some of the most notable instances of the creative role of physical problems in mathematical research. Again, it was the speculation on the ether that inspired the extensive work of Poisson, Navier, and Cauchy on the mathematical properties of wave motion. The general theory of coordinates as developed by G. LamŽ was an outgrowth of concern with problems of physics. He viewed the introduction of various coordinates -- rectangular, spherical, elliptical, general curvilinear -- as indicators of the advancing phases of physical science.7
Yet, in all this it was clearly mathematics that seemed to respond to the promptings of physics. Even Riemann's essay "On the hypotheses which lie at the bases of geometry" (1854), which brought to a culmination Gauss' work on the geometry of curved surfaces, had an eye on physics. As a matter of fact, Riemann insisted that purely theoretical analysis of the notion of space not only freed the study of the geometry of space "from becoming hampered by too narrow views" but also made necessary "the successive changes required by facts which it cannot explain."8
The assumed connection between Riemannian geometry and physics, however, could hardly have convinced Riemann's audience with perhaps the exception of Gauss. And if there were some physicists who might have worried about the lack of such a connection, the possible application in physics of a mathematical theorem was no longer the principal aim of mathematicians. Their new attitude was best exemplified by Weierstrass who lent his full authority as the leading mathematician of the age to the view that the basic aim of a science cannot be placed outside that science.
Not that he stopped referring in his courses to the problems of mechanics. He even held that it was the glory of mathematics to be indispensable in physics. Yet, as he warned in his famous address of 1857, one should think of "the relation between mathematics and physics in a deeper manner than is the case when a physicist sees in mathematics only an indispensable auxiliary discipline, or when a mathematician is willing to see only a rich source of illustrations for his method in the questions posed to him by the physicist."9 Just as these words had a prophetic ring, so did the reference of Weierstrass to the fact that the studies of conic sections by the Greeks had to wait until Kepler found their "useful application" in physical science.
Clearly, no one realized in 1857 to what extent this pattern would dominate the future relation of mathematics and physics. At that time one merely noted that mathematics was resolved to strike out on a path of its own and choose its problems regardless of whether they had a physical meaning or not. Little could one surmise that the major achievements of this "independent" mathematics would have to wait for over half a century before physics could find them "useful."
Yet, such was the case with Gauss' hypergeometric functions, Hermite's polynomials, and Cayley's theory of invariance. Another major contribution of Cayley to mathematics, the theory of matrices, found its way into physics only in 1925 when Heisenberg realized that the array of spectral terms he had set up was nothing but a matrix. The group concept, which F. Klein once considered as most characteristic of nineteenth-century mathematics, proved itself decades later to be a tool of fundamental importance for quantum physics, since quantum numbers were indices characterizing representations of groups. It was in fact through the group theory that quantum mechanics could reveal, as H. Weyl put it, "its essential features which are not contingent on a special form of the dynamical laws nor on special assumptions concerning the forces involved."10
If one recalls that for several decades the only use made of the theory of groups consisted of the description of the symmetry of crystals, one will perhaps have an inkling of how difficult it is to assess the potentialities of a modern mathematical theory in physics. A case in point is the publication in 1924 of the first volume of the Methoden der Mathematischen Physik by R. Courant and D. Hilbert, which aimed at showing as completely as possible the applicability in physics of the modern methods of linear transformation, expansion of arbitrary functions, bilinear and quadratic forms, and the like.
Yet the richness of the content of the book revealed itself only two years later when Schršdinger derived his now famous equation, of which neither of the authors could have had the slightest premonition in 1924. More recently an algebraic theory developed almost a hundred years ago by Sophus Lie turned up as the mathematical formalism best suited to the "eightfold way," one of the most promising systematizations of the set of fundamental particles. Today there is hardly a branch of the various "esoteric" sections of modern mathematics that has not yet found some use in modern physics.
Mathematical theories, like set theory and topology, appeared when first formulated to be highly arbitrary mental constructs based on postulates that had little to do with experience and common sense. Yet, today, they are beginning to play an indispensable role in physics. In view of all this, how could one fail to sense the prophetic truth in the remark of Lobachevsky who held that there was no branch of mathematics, however abstract, that might not some day be applied to the phenomena of the real world.11
The way in which the classical physicist viewed the role of mathematics in physics was distinctly different. Searching everywhere for "commonsense machines" in the physical world, his mathematics also had to reflect a common sense, however refined. Exaggerated though it was, Kelvin's dictum of classical mathematical analysis was in keeping with the basic assumptions of the mechanistic concept of physics." Do not imagine," he declared, "that mathematics is hard and crabbed and repulsive to common sense.
It is merely the etherealization of common sense."12 To most classical physicists mathematics was a very useful and commonsense tool, but hardly a device that could do on its own magic tricks for physics. Compared to the role it was to play in modern physics, the role of mathematics in classical physics was relatively modest. As Rowland summed it up very characteristically: "A mathematical investigation always obeys the law of the conservation of knowledge: we never get out more from it than we put in. The knowledge may be changed in form, it may be clearer and more exactly stated, but the total amount of the knowledge of nature given out by the investigation is the same as we started with."13
It must be admitted, however, that even within the framework of late nineteenth-century physics, this was not always the case. Of course the mathematics Rowland spoke of was the ordinary analysis used in nineteenth-century physics and not the "extravagant" creations mathematicians started to produce from the 1840s at an ever-increasing rate. About the "physical merit" of those extravagant theorems such a mathematically minded physicist as Gibbs felt prompted to offer this little aside: "A mathematician may say anything he pleases, but a physicist must be partially sane."14
At the same time, however, Gibbs in his work on the thermodynamics of chemical equilibrium resorted to steps that might have induced some of his fellow physicists to class him with the "mathematicians." For it was precisely the mathematical boldness of Gibbs that welded thermodynamics and chemistry into one. The partial differential coefflcients Gibbs introduced had no physically realizable notion. In the situation that he investigated, the entropy and volume of a system were supposed to remain constant while the mass of the system was changing. Such a procedure, however, is purely mathematical, for there is no experimental way of adding or subtracting mass from a system without changing its entropy. As E. A. Milne noted, Gibbs' step was the first instance of the presence in physics "of an 'unobservable' (namely a partial potential) suggested by mathematics. Once this idea has been worked into the subject, the whole of classical analytical thermodynamics follows."15
The work of another giant of classical physics, Maxwell, similarly contained indications about the novel role mathematics was to play in the physics of the future. Maxwell for one was deeply puzzled by the thought that while the molecules in an ordinary body had a mean velocity equal to that of a cannonball and moved through very short distances at high speed in every direction, the body as a whole still remained, as far as observation showed, in its fixed position. Reflecting on this problem, Maxwell was led to an extensive study of stable and unstable phenomena that showed him that unstable configurations in the physical world, such as a rock on a mountain top, or a match starting a forest fire, were actually flaws in the deterministic picture of physics. The conclusion he drew was nothing short of prophetic:
If, therefore, those cultivators of physical science... are led in the pursuit of the arcana of science to study the singularities and instabilities, rather than the continuities and stabilities of things, the promotion of natural knowledge may tend to remove that prejudice in favor of determinism which seems to arise from assuming that the physical science of the future is a mere magnified image of the past.16
Maxwell's equations of the electromagnetic field were particularly effective to convey the impression to his more perceptive colleagues that certain mathematical equations indeed contained more than what was put into them. No one diagnosed this better than Hertz to whom Maxwell's equations appeared as if they "had an independent life and an intelligence of their own, as if they were wiser than ourselves, indeed wiser than their discoverer, as if they gave forth more than he had put into them."17
Extraordinary as this might have been, it was not without some perplexing aspects. PoincarŽ for one could point out in Maxwell's papers on electromagnetism not only algebraic errors but also illogical steps and conceptual inconsistencies. As PoincarŽ put it, all this made the French reader fond of strict rigor feel rather uneasy.18 Yet, as Duhem noted, PoincarŽ too was willing to admit that in view of the incontestable brilliance of Maxwell's work mathematical physics had the right to shake off the yoke of too rigorous logic.19
The most outstanding mathematical physicists of the day began, in fact, to realize that the mathematical code of the physical world transcended the framework of customary mathematical logic and method. Gibbs himself hinted at this in words worth quoting: "If I have had any success in mathematical physics, it is, I think, because I have been able to dodge mathematical difficulties."20
Yet, for all the perplexities, confidence remained on the whole unshaken about the possibility of finding one day that self-consistent, all-embracing mathematical synthesis that would impose itself of necessity on the physicist. The physical universe, however, which modern physics stumbled upon, turned out to be almost despairingly distant from the fulfillment of such hopes.
Not that modern physics had not done its utmost to make its world picture thoroughly mathematical. It adopted as its principal tools operators and functions which all too often were very far removed from the rather commonsense framework of the mathematical physics of yesteryear. Modern physics indeed went all the way in embracing the abstract world of mathematics in renouncing unequivocally the desirability of visualization in the physical method. In fact, the basic incompatibility between visualization and modern physics was recognized as soon as quantum mechanics was developed.
As early as 1925 Bohr noted that quantum mechanics implied far more than the mere modification of mechanical and electrodynamical theories. In quantum theory, he warned, one is faced "with an essential failure of the pictures in space and time on which the description of the natural phenomena has hitherto been based."21 The unanimity on this point could not be more complete among those who were the chief architects of quantum theory. As Heisenberg put it, "the limits of visualization" have been reached by modern physics.22
Dirac was no less struck by the novelty of the situation: "There is an entirely new idea involved, to which one must get accustomed and in terms of which one must proceed to build up an exact mathematical theory without having any detailed classical picture."23 One had to recognize that pictures like that of the electron as a rotating ball could not be taken literally. Nor could one assign in the framework of quantum mechanics any mechanical meaning to expressions like the "structure of the electron."
Modern physics merely wanted to assert when speaking of the spin of the electron that the electron has an inner degree of freedom, but as W. Heitler noted, "no further conclusions should be derived from this picture and questions of what the 'radius' of such ball would be, etc., are void of any physical meaning."24 It was then only natural that Dirac placed the main objective of physical science not in the provision of pictures, or models. but simply in the formulation of mathematical laws predicting new phenomena. Finding "pictures" is no longer of much concern." If a picture exists," noted Dirac, "so much the better; but whether a picture exists or not is a matter of only secondary importance. In the case of atomic phenomena no picture can be expected to exist in the usual sense of the word 'picture' by which is meant a model functioning essentially on classical lines."25
This unconditional willingness to abandon attempts aimed at visualizing basic physical processes, underlined all the more the seemingly unlimited effectiveness of mathematics in providing an adequate description of physical phenomena. As Heisenberg put it confidently in 1930: "It has been possible to invent a mathematical scheme -- the quantum theory -- which seems entirely adequate for the treatment of atomic processes."26 Confidence in mathematics could hardly be more robust, and it was hoped that theoretical physics would soon come into possession of a definitive, all-inclusive mathematical formalism. With the ultimate mathematics in hand, one hoped to trace out the bedrock pattern of the physical world as well.
To attribute such power to mathematics was not a wholly new persuasion. The praises of mathematics in effect kept growing ever more fervid as physical science progressed. As God calculates, so the world is made, was a favorite saying with Leibniz as the progress of physics began to gain momentum. In the late 1870s, when physics seemed to approach rapidly its final stage, W. Spottiswoode found no better way to eulogize mathematics than to stress its one-to-one correspondence with the physical world. As he put it at the Dublin meeting of the British Association in 1878:
Coterminous with space and coeval with time is the kingdom of mathematics; within this range her dominion is supreme; otherwise than according to her order nothing can exist, in contradiction of her laws nothing takes place. On her mysterious scroll is to be found written for those who can read it that which has been, that which is, and that which is to come.27
The mathematics of which Spottiswoode spoke was the mathematics defined about the same time by B. Peirce as "the science which draws necessary conclusions."28 This unsuspecting confidence in and respect for mathematics was reflected of course even more strikingly in the statements of non-mathematicians and humanists in general." Mathematical truths are immutable and absolute," wrote Claude Bernard, who was firmly convinced that "the science of mathematics grows by simple successive juxtaposition of all acquired truths."29
Poor anticipation of the future indeed. But for the time being the cracks and perplexing complications in the edifice of mathematics were nowhere in sight. Thus Macaulay, in his famous essay on Ranke's History of the Popes, felt perfectly justified in making a reference to mathematics in which, as he put it, "once a proposition has been demonstrated it is never afterwards contested."30 How little did Macaulay suspect that a few decades later Weierstrass would initiate what he thought simply impossible: "a reaction against Taylor's theorem."
E. Everett, the historian, president of Harvard and the first American to earn a doctor's degree at Göttigen, also described the statements of pure mathematics as absolute truths that existed "in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven."31 The passage had only one lasting merit: it rendered splendidly the state of mind prevailing at that time among both scientists and non-scientists about the absolutely superior qualities attributed to mathematics. Although the soaring style of Everett and Macaulay could not find its way into the twentieth century, the belief in the absolute perfection of mathematics survived at least for a while as witnessed, for instance, by Whitehead's statement calling mathematics "the most secure and authoritative of sciences."32
If prior to the 1930s some prominent modern mathematicians poked fun at mathematics in an unguarded moment, it was never taken as a serious danger sign indicating hidden cracks in the safe structure of mathematical thought. No one was alarmed when D. Hilbert characterized mathematics as a game played according to certain simple rules with meaningless marks on paper.33 Hardly a mathematician saw a sign of crisis in Russell's characterization of mathematics "as the subject in which we never know what we are talking about, nor whether what we are saying is true."34
Mathematics at that time (1901) was still enjoying its pre-Gödel era when the idea of a self-consistent, universally valid mathematical theory was firmly adhered to. Such a theory existed, however, only in faith. and the "stubborn facts" of nature, the data of observations, continued to have their supreme say over the mathematical constructions of theoretical physics. Speaking of the physicists who found it inconceivable that the principles of mechanics could ever be corrected or changed, Hertz aptly said "that which is derived from experience can again be annulled by experience."35
Long indeed would be the list of theories, for instance, proposed to explain radioactivity that were discarded by the facts, and such and similar cases did in fact force on the modern physicist a considerable detachment about pet theories. Rutherford once remarked, according to Soddy, that the theory of nuclear disintegration should be abandoned as soon as a single experimental fact emerged contrary to it.36 Actually there was no physical theory that could claim an absolutely definitive confirmation by experimental proofs, numerous as these proofs were. As J. von Neumann noted about quantum mechanics: "One can never say that it has been proved by experience but only that it is the best known summarization of experience."37
It should also be noted that as regards the strictly mathematical features of a physical theory, neither simplicity nor symmetry are in themselves enough to secure the unconditional validity of a given theory. Not that such features were not immensely fruitful in finding the right path, but it was always the facts and the often completely unforeseen facts that had the last say. After the experimental evidence forced on physics the concept of the electron most physicists naturally adopted the simplest assumption in which the electron appeared as a single point charge surrounded by a structureless medium.
Undoubtedly this conception allowed a mathematics much simpler than was the case with other models of the electron. Yet, as J. J. Thomson noted in this connection, the simplicity of mathematics could not be taken as a peremptory proof in favor of a theory, as "there is no evidence that the convenience of the mathematicians has been a dominant factor in the scheme of the universe."38
The chief convenience or preference of the mathematical physicist are simplicity and symmetry, but as their modes of formulations are almost unlimited in number, no particular mathematical formalism based on any of them can claim in advance an exclusive and ultimate validity." The mathematical physicist," as Milne forcefully expressed it, "does not dictate to the world what it must be like. But he is guided by mathematical form to make suggestions to the experimenter. His peculiar role then ends. The experimenter decides."39
In 1929 when Milne made this remark it must have already been abundantly clear to what a large extent the development of quantum theory depended on unexpected experimental data and on the refinement of experimental technique. In fact, H. Weyl, one of the pioneers who shaped quantum mechanics in terms of the group theory, felt compelled to voice his admiration "for the work of the experimenter and for his fight to wring significant facts from an inflexible Nature, who says so distinctly 'No' and so indistinctly 'Yes' to our theories."40
It is in this connection of experimental data and mathematical formalism that an all-important feature of the mathematical concept of physics comes to the fore. For just as the notions of organism and mechanism failed to prove themselves as the final word in the conceptualization of physics, no different is the case with its mathematical formalization. Successful and wide-ranging as this may be, no particular formulation of it can claim to itself on purely intrinsical grounds the glory of being the true representation of the structure of physical reality.
Einstein's pregnant words refer to this state of affairs: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."41 This lack of certainty is not of epistemological nature. It rather indicates that out of the large number of mathematical systems mathematics itself does not have the criterion to choose the simplest one that at the same time would translate perfectly the assumed basic simplicity of the laws of nature. In other words, the confidence that mathematics might find such a criterion can be supported only by a sort of faith in mathematics not by strict arguments.
It is well to remember, as Einstein put it, that "our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas."42 Yet, to use his words again, in the actual carrying out of this program experiments and the observed facts remain "the sole criterion"43 and "the supreme arbiter."44 Furthermore as there is no telling what unsuspected experimental data will turn up in the course of physical research, there can be no strict scientific basis to claim that the "greatest of all aims," as Einstein called the physical system of the greatest conceivable unity will one day be achieved by science.
The conviction that nature takes basically the character of a well-formulated puzzle therefore was characterized by Einstein as an "outcome of faith."45 All the successes achieved by physics up to now could give, he insisted, but "a certain encouragement for this faith."46 Clearly then, the confidence in the correctness of the mathematical world picture of physics rests ultimately, just as was the case with the organismic and mechanistic conceptions of physics as well, on a state of mind rooted in an intellectual faith.
To all those who at one time or another spoke elatedly of the imminent formulation of an all-embracing mathematical synthesis, the persistent impotence of finding it no doubt must have been frustrating. The final answer was not forthcoming either in the mathematical formalism of quantum mechanics or in the Theory of General Relativity. What is more, it became evident that there are mutually exclusive groups of phenomena at the bottom of these two theories.
On the macroscopic level, to which relativity theory mainly applies, the coincidence or collision is a primitive event, in the sense that it defines a point in space-time with the provision that the colliding particles are infinitely small. On the microscopic level, however, where quantum mechanics is valid, the event of coincidence is not defined sharply in space-time. This difference is manifested in the fact that quantum mechanics utilizes the infinite-dimensional Hilbert space whereas the theory of relativity uses the four-dimensional Riemann space. Attempts to find a mathematical theory from which these two theories can be derived as approximations have failed so far.
Will such a theory be found in the future? Expressing the prevailing sentiment, E. P. Wigner said: "All physicists believe that a union of the two theories is inherently possible and that we shall find it." Yet he was quick to add with commendable objectivity that "it is possible also to imagine that no union of the two theories can be found."47
Which of these two possibilities will materialize no one can tell today. At any rate, for the present, a perplexity of such depth should provide a clear warning against sanguine optimism that assumes that the world taken as an embodiment of a unique mathematical function will contain what the organismic and mechanistic syntheses of physics failed to provide: a definitive, all-embracing explanation of the physical universe. Furthermore, it should not be forgotten that the intimate union of modern physics with mathematics has aspects that are almost embarrassing from the point of view of strict logic.
As is well known, quantum electrodynamics has to fall back on the technique of renormalization, which, to use the succinct characterization of Dirac, "has defied all the attempts of the mathematicians to make it sound."48 Renormalization, to be sure, is a highly successful technique. It helped to make the Dirac theory account not only for the Lamb shift but for all known phenomena falling within its range. Still, it is highly unsatisfactory. It almost amounts to cheating, as it replaces infinite quantities (mass and charge of photon cloud surrounding the electron) arrived at by the theory, with the very small quantity established by observation.
The fact that such an arbitrary procedure gives results that in their agreement with observational data surpass, as Dirac put it, the precision characteristic of astronomy, does not constitute a peremptory argument in its favor. It is of little comfort to recall that classical physics too had to rely, until Cauchy's time, on a calculus, the basic theorem of which, the theory of limits, was lacking an unassailable foundation. The difficulty implicit in the method of renormalization could be something basically different. Indications are that it might defy any attempt to resolve it in a perfectly consistent way. As Dirac commented: "It seems to be quite impossible to put this theory on a mathematically sound basis...the remarkable agreement between its results and experiment should be looked on as a fluke."49
Harsh as this stricture may appear, the unfinished business of renormalization was recalled by R. Feynman, one of the physicists who received the Nobel Prize for making the infinities of quantum electrodynamics more manageable. "It may be a funny thing to say after receiving the prize," he noted, "but as far as I am concerned, the math isn't solved completely yet."50
Consequently, the "understanding" of nature, claimed by a purely mathematical expediency, is to be taken by proper reservations. It still remains true as I. I. Rabi warned that "the theory does not converge but is made to agree with experiment through systematic mathematical manipulation."51 Renormalization in quantum electrodynamics is therefore a basically ad hoe procedure, and as such it can offer little in the way of understanding the physical reality.
It is not surprising that such "unorthodox" aspects of mathematical techniques are encountered precisely by that type of physics for which the physical universe is primarily a pattern in mathematics. After all, the baffling nature of a tool comes more readily to light when subjected to the most exacting demands, and modern physics through its problems poses a far greater challenge to mathematics than its classical counterpart.
But irrespective of physics mathematics stumbled upon a purely mathematical consideration that doomed the hopes of ever arriving at a fundamental and completely self-consistent mathematical system that could also serve as the bedrock layer both in mathematics and in physics. Interestingly enough this setback came when the infinite-dimensional Hilbert space was being taken more and more as the realization of Hilbert's ambitious program, which "aimed at nothing less than to banish once and for all from the world," as he put it in 1922, "the widespread doubts besetting the certainty of mathematical conclusions."52
The mathematicians present at the 1922 meeting of the German Association were well aware of these doubts. The rude shocks by which Weierstrass and his followers had awakened the community of mathematicians "from their dogmatic slumber"53 were still fresh in their memories. Hilbert felt that it was important to remind his audience of the overthrow of several principles of mathematics that had seemed securely established a few decades earlier.
Yet, for Hilbert such reminiscences were by no means a reason to despair of the ultimate success of mathematical research. For, as he insisted in an address given in 1921, mathematics owed its greatest advances to the discussions of its fundamental principles, and he viewed all these advances as steps toward the definitive formulation of the ultimate principles of mathematics. About these his was a conviction that in "mathematical matters there can be basically no room for doubts, half-truths, or truths of essentially different kinds." The foundations of mathematics, as he put it, "are capable of perfect clarity, understanding, and definitive solution," difficult as they may be to obtain.54
Whatever the immensity of such a task, how could one doubt its basic soundness in the opening decades of the twentieth century, which was inaugurated for mathematicians with the confident words of PoincarŽ speaking of the arithmetization of mathematics at the Second International Congress of Mathematics in 1900: "We may say today that absolute rigor has been attained."55 Such a self-assured appraisal of the status of mathematical rigor created a most favorable atmosphere for a program like Hilbert's.
A no less competent observer of such aspirations than H. Weyl recalled the "optimistic expectations"56 that prevailed in the late 1920s about the feasibility of such an undertaking. Hilbert in particular tried to fuse Einstein's Theory of General Relativity with G. Mie's work on pure field physics. To many in Hilbert's circle the prospect of formulating a universal law valid both for the structure of the cosmos as a whole and for the structure of the basic units of matter seemed near at hand. Yet the arbitrariness implicit in Hilbert's Hamiltonian function could not be ignored, although no less competent men than Weyl, Eddington, and Einstein tried to eliminate it.
Hilbert's program was no doubt as ambitious as any seen before in mathematics, and the talents engaged in it were no less extraordinary. Hilbert's work was carried on by younger collaborators, two of whom, W. Ackermann and J. von Neumann, succeeded in proving the consistency of that part of arithmetic in which the axiom about the conversion of predicates into sets is not yet introduced. This was a notable achievement, yet the high hopes generated by it were soon dashed. The deluge came without warning.
In December, 1930, Hilbert was still confidently telling the Philosophical Society of Hamburg how the ultimate certainty can be achieved in number theory, that fundamental branch of mathematics, by formalizing it in a superior form of mathematics, or metamathematics.57 Yet, a month earlier Gödel's historic paper had already been submitted for publication.58 In that paper Gödel proved that the formalism of the Principia Mathematica, or any other formal system that is not too narrow, cannot have in itself its proof of consistency.
What was so shocking in Gödel's procedure was the fact that his proof concentrated on the domain of whole number arithmetic. To declare, as Gödel did, that even in such a basic and elementary domain of mathematics, a proposition can be undecidable unless extraneous assumptions are made had to be upsetting, to say the least. Attentive readers of his paper could not fail to perceive that such a state of affairs would cast a shadow of inconsistency on any formal logical system that encompassed such basic mental operations as the addition and multiplication of integers and zero. This meant, however, that formal mathematics as a whole had to be regarded from then on as incomplete in substance, a far cry from its exalted position as the most consistent and self-sufficient of all intellectual disciplines.
Gödel for one did his best to soothe the impact of his conclusions. Actually he went on to declare in Proposition XI of his paper that his findings represent "no contradiction of the formalistic standpoint of Hilbert." Little did he suspect how ineffective his disclaimer was. For an essential part of his procedure consisted in showing that the symbols and sequences of formulas in Hilbert's formalism can be enumerated in a way in which the assertion of consistency appears in the form of an arithmetic proposition.
But as Gödel found, such propositions can be neither proved nor disproved within the formalism. Invariably, some of the arguments needed to prove the consistency of a given system will have no formal counterpart in that system. One is therefore compelled to conclude that either the formalization of the procedure of mathematical induction is not yet wholly known, or that a strictly definitive proof of consistency is out of man's reach. In short, if the game of mathematics is actually consistent, then the formula of consistency cannot be proved within this game.
To prove this consistency one needs another class of games, or a metamathematics. Yet, to prove the consistency of metamathematics one again needs a supertheory or a metamathematics, and there is no end to such steps. One is, in fact, caught in a process of endless regression when trying to formalize a metamathematical theory of proof as a set of symbols manipulated according to specified rules. Each set of rules points beyond itself for its proof of consistency. This is why one has to consider dim the prospect of mathematics ever becoming established as the system of "absolute truths."
Such is, in brief, the gist of Gödel's incompleteness theorem that cast serious doubts on any attempt aiming at an a priori mathematical theory that might claim to be the ultimate mathematical pattern of the physical world on the grounds of its built-in consistency. Such considerations could not fail to dampen some sanguine hopes in the circles of mathematicians and theoretical physicists. H. Weyl once described Gödel's theorem as a "constant drain on the enthusiasm" with which he pursued his work, and expressed the belief that his experience was shared "by other mathematicians who are not indifferent to what their scientific endeavours mean in the context of man's whole caring and knowing, suffering and creative existence in the world."59
In that creative existence of man the mathematical and scientific reflections play a prominent part. This is especially true of the mathematics that Einstein viewed as the creative principle of physical science. His was a conviction shared by most of his colleagues: since nature is the realization of the simplest conceivable mathematical ideas, we can therefore discover "by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena."60
And in a veiled reference to the Pythagoreans, he stated his belief that in a certain sense "pure thought can grasp reality as the ancients dreamed."61 In a certain sense to be sure, for Einstein was not blinded by the successes of the antiphenomenological constructive method to its limitations. His belief that we can grasp the physical reality for the purposes of science only indirectly, that is, by speculative means or mathematical formalism, impelled him to state that for this very reason the notions of science about physical reality could never be final." We must always be ready," he warned, "to change these notions -- that is to say, the axiomatic substructure of physics -- in order to do justice to perceived facts in the most logically perfect way ."62
It is on the ultimate success of such a quest that Gödel's theorem casts the shadow of judicious doubt. It seems on the strength of Gödel's theorem that the ultimate foundations of the bold symbolic constructions of mathematical physics will remain embedded forever in that deeper level of thinking characterized both by the wisdom and by the haziness of analogies and intuitions. For the speculative physicist this implies that there are limits to the precision of certainty, that even in the pure thinking of theoretical physics there is a boundary present, as in all other fields of speculations.
An integral part of this boundary is the scientist himself, as a thinker, with the ever-changing patterns of his various states of mind. For just as the shifting moods of the physicist's state of mind bespeak the imperfect, human character of science, so do the basic modes of his search for certainty demonstrate the same about his achievements. Wizardry, however great, with the techniques and principles of mathematics, is not a magic means "to get away from ourselves," that is, from the not always clear-cut operations of the human intellect, as Bridgman, a Nobel laureate physicist, acknowledged in connection with Gödel's theorem.63 Yet, only a mistaken rationalism can see in this a cause of despair.
For one thing, Gödel's theorem casts light on the immense superiority of the human brain over such of its products as the most advanced forms of computers. Clearly, none of these machines can ever yield an answer comparable in its breadth and depth to Gödel's theorem. For another, despair can grow only in a soil where a rigid rationalism has already killed off the seeds of intellectual humility. Such a soil cannot nurture the recognition that there is no escape from admitting that in mathematics and a fortiori in physics certainty is not the fruit of a "pure rationalistic" procedure alone.
Gödel's proof clearly indicates in which direction one is to face a blind alley when investigating the ultimate source of what there is of validity and certainty in the concepts of mathematics. The setback suffered by the thorough-going formalists in the hands of Gödel's theorem should help prevent our forgetting that the mind thrives on sensory experience and that postulates, however abstract or mathematically esoteric, are rooted somewhere, no matter how remotely, in experience.
In a sense, this lesson, underscored by Gödel's investigation, completes what has been clear to all those who, following Einstein, concentrated on the geometrization of physics. For by the time geometry proved itself an indispensable tool for the development of modern physics, it had already become recognized that geometry was far from being that paradigm of unchangeable, a priori principles as claimed by Descartes. The analysis of the foundations of geometry in the hands of Riemann and Helmholtz made short shrift of the Kantian belief that the a priori synthetical cognitions of pure geometry were to be accepted as absolutely certain and fundamental.
Riemann warned in effect in his paper of 1854 that "the properties by which space is distinguished from other thinkable three-dimensional continua can only be proved by experience."64 Before long his conclusion received powerful support in Helmholtz's work on the genesis of geometrical notions, and it became unavoidable to recognize that insofar as the axioms of geometry are the results of observation, they command no greater certainty than the statements of physics.
Mindful of this, Minkowski was careful to emphasize that the new views of relativity that made both space and time as independent entities "to fade away into shadows" have sprung "from the soil of experimental physics and therein lies their strength."65 In the great strides made since then in the geometrization of physical theories this awareness of the not-at-all absolute character of a particular form of geometry has been clearly noticeable.
Einstein for one carefully distinguished between a mathematics or a geometry, "the laws of which," as he put it in 1921 (still the pre-Gödel era), "are absolutely certain and indisputable," and a mathematics or a geometry that is a branch of natural science. The affirmations of the latter type of geometry rest, in his words, "on induction from experience not on logical inferences only."66 This is why he insisted so emphatically that a decision about the Euclidean or non-Euclidean geometry of the universe could be made ultimately on an experimental basis alone. Again, it was this "sensory" substratum of the geometry physics has to use, that kept suggesting to him that the scientific explanation of physical reality can never be final.
The awareness of this should be no less vivid today when one can witness the development of geometrodynamics, which by definition is "the study of curved empty space" and is based on the assumption that mass and electricity can in a sense be fashioned out of curved space."67 Clearly, a physical theory claiming that the starting point for physics at the very small distances is the "vacuum, complex in geometry and rich in dynamics,"68 must be peculiarly aware of its ultimate dependence on the data of experimental evidence.
Only on such a basis will it be able to stay within what J. A. Wheeler described as the "traditional modest spirit of theoretical physics with all openness to recognizing and formulating the new concepts which are hidden in it."69 It is in this spirit of openness that one should also remember that the type of geometry that reaches back to the curvature of abstract vacuum to fashion the description of a concrete world out of it is and must be based ultimately on the common human experience imbedded in the concrete plenum.
The concreteness of nature, however, is rich beyond comprehension in aspects and features. This is why even the most bizarre sets of mathematical postulates and geometrical axioms can prove themselves isomorphic with some portion of the observational evidence and useful in systematizing it. This is why the physicist is apt to find himself time and again, as Wigner noted, "in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial."70 This is why the physicist might even be overcome by a mood of skepticism concerning the uniqueness of coordination between his mathematical tools and the actual features of the universe.
Again, in view of the extreme richness of the features of the physical world, one should not marvel inordinately how space can be non-Euclidean or how complex numbers can be so successful in dealing with alternating currents and an almost endless array of physical processes. The novelty present in those processes is far from being exhausted. No one would dare assume today that there is nothing new for man to observe in the physical world. Consequently, the formulation of new mathematical theories useful for physics will very likely go on indefinitely. For it is not only himself that the mathematician cannot get away from; he cannot get away from the physical world either. It is there, in an immensely variegated nature and not in his finite intellect where ultimately lies the never-ending challenge for the mathematician.
At any rate, the data being collected through experimental physics today are so variegated and increase in number so rapidly that it is doubtful whether an axiomatic approach in the Hilbertian fashion can ever hope to gain a firm foothold in physics except perhaps in its most consolidated parts. As H. Weyl once noted, "Men like Einstein or Niels Bohr grope their way in the dark toward their conceptions of general relativity or atomic structure by another type of experience and imagination than those of the mathematician, although no doubt mathematics is an essential ingredient."71
This process of groping in the dark should indicate in itself that the modern, emphatically mathematical conception of physics is, in spite of all its successes, far from intimating that the long road of physical discoveries is near its end. It was mathematics that unveiled such startling aspects of the physical world as the mass-energy equivalence and the uncertainty principle. Yet, when commenting on the grave puzzles raised by these cornerstones of modern physics, Max von Laue felt compelled to remark: "Here we feel with particular intensity that physics is never completed, but that it approaches truth step by step, changing forever."72
In spite of the immense expansion of the applicability of the laws of physics that took place when the mechanistic concept of physics yielded to the mathematical operationalism of quantum mechanics, "the state of theoretical physics," as Max Born put it in 1943, "is just as problematical as it was at any time."73 Two decades later the British physicist, Sir Henry Massey, could only dismiss the naive complacency according to which "all that is necessary [in physics today] is further application of established laws which nature follows exactly."74
Only the superficial observer could fall easy prey to spurts of sanguine expectations such as the one that manifested itself in the early 1930s when to some everything in physics suddenly seemed to find an explanation. Those who would not be swayed by specious appearances realized not only that as physics progresses it requires for its theoretical formulation an ever more advanced type of mathematics but also that this mathematics grows more abstract and continually shifts its foundations.
As Dirac noted in 1931, this increasing abstraction will make it very unlikely that the future advance of physics will be associated "with a logical development of any one mathematical scheme on a fixed foundation." As for the solution of the then outstanding problems of physics, he forecast a "more drastic revision of our fundamental concepts than any that have gone before."75 Thirty years later his awareness was just as keen about "the drastic changes," as he put it, to be undertaken in theoretical physics if its new but no less vexing riddles were to be solved. Dirac felt that not only would the principle of indeterminacy fail to survive in its present form under the impact of such changes but that physics would also have to part with the four-dimensional manifold as the fundamental framework of its laws.76
The persistently recurring need of drastic changes in the mathematical formalization of modern physics clearly indicates that modern physics, by being shaped along mathematical operationalism, is not spared thereby ever fresh crises. Theories, however successful, can alleviate only temporarily that feeling that Pauli disclosed to a friend five months before Heisenberg's first paper on quantum mechanics appeared." At the moment," wrote Pauli, "physics is again terribly confused. In any case it is too difficult for me, and I wish I had been a movie comedian or something of the sort and had never heard of physics." Yet, five months later he realized that Heisenberg's theory left many questions unanswered. It was, no doubt, a powerful shot in the arm but not the final panacea. As Pauli put it, "Heisenberg's type of mechanics has again given me hope and joy in life. To be sure it does not supply the solution to the riddle but I believe it is again possible to march forward."77
In this forward march of physics the final account of the physical phenomena has not yet been secured by basing physics on mathematical symbolism. So far nothing warrants that a mathematical conception of physics will be more successful than its mechanistic and organismic antecedents were in providing man with a final scientific explanation of the material universe, or at least not in the foreseeable future. In view of the crises in which mathematics finds itself, one cannot help feeling with particular force the incongruity present in unsuspecting statements, such as the one uttered by Jeans: "The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomena is complete."78 Apart from the philosophical poverty of such a flat declaration about what constitutes the full knowledge of a phenomenon, did mathematics succeed in coming up with its final and definitive form?
Men of science who are only too ready to attribute definitiveness to this or that type of mathematics would do well to ponder a little on how badly Diderot fared in prognosticating an impending completion of the science of geometry: "I almost dare to assert," he stated in 1754, "that in less than a century we shall not have three great geometers left in Europe. This science will very soon come to a great standstill where Bernoullis, Eulers, Maupertuis, Clairauts, Fontaines, d'Alemberts, and La Granges will have left it. They will have erected the columns of Hercules. We shall not go beyond that point."79
Yet, a century later, geometry had already been steered on an entirely new course by three of the many great geometers of the nineteenth century, Gauss, Bolyai, and Lobachevsky. There are other similar instances worth remembering. Sir William Hamilton, who was a professional mathematician, scored no better than Diderot when he declared his quaternions to be both the last word on the generalization of the concept of number and the master key to geometry and mathematical physics. While he was devoting the rest of his life to the study of this "ultimate" concept in mathematics, a contemporary of his, H. G. Grassmann, was developing the concept of even more generalized numbers of which the quaternions were only a minor subdivision.
Again, mathematicians had already been busy with five-, six-, and n-dimensional space when many physicists and non-physicists flattered themselves that in the four-dimensional space-time manifold they possessed the definitive, exclusively true structure of the universe. Today the pillars of Hercules, marking the ultimate frontiers of geometry and the universe are located only in wishful thinking, not in facts. Yet, the waves washing the feet of those chimerical pillars are very much in evidence, and they resemble anything but a mighty stream rushing in one direction. Of mathematics as it stood two hundred years after Diderot, H. Weyl said that it is "more like the Nile delta, its waves fanning out in all direction."80
What is more, mathematics, for all its success in correlating and predicting recondite phenomena of nature, is a stream that by becoming ever more abstract pulls steadily away from what shall forever remain a basic human goal in the quest of understanding: the description and explanation of nature. It is an unavoidable consequence of the modern mathematical concept of physics that "its only object," to use Dirac's words, is "to calculate results that can be compared with experiment." Undoubtedly this is a statement that squares with the way in which physics is being cultivated at present. Yet, the rest of Dirac's statement deserves further comment. There one is told that "it is quite unnecessary that any satisfying description of the whole course of the phenomena should be given."81 Clearly in such a statement more is involved than the fact that the basic concepts of modern mathematical physics defy visualization. What is also implied there is that as mathematics grows more effective in coping with the problems of physics, it also becomes more evident how limited is that aspect of the world of phenomena that can be grasped, ordered, and correlated by mathematics.
More than two millennia ago the science of physics was born under the symbol of organism in the hope that man would thereby secure the full intelligibility of the physical world. Not only was this goal not achieved by the organismic approach to the inanimate world, but the approach did not even yield a minimum of control over the external world. Through the thorough mathematization of physics control over nature reached proportions that in at least some respects are not far from the measure of fullness.
Yet, to expect the full intelligibility of nature from the mathematical concept of physics would be just as illusory as were the hopes and beliefs entertained about its mechanistic and organismic antecedents. Only a limited range of the full reality of things can ever be accommodated in the molds of mathematics, advanced and esoteric as these might be. Unwittingly, however, even Russell recognized this when he stated that "physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover."82
That little that mathematics can say about the phenomena of nature was illustrated by Maxwell long ago in a memorable passage. He was struck by the fact that the equations for the uniform motion of heat in homogenous media are identical in form with those of attractions varying inversely as the square of the distance." We have only to substitute," he noted, "source of heat for centre of attraction, flow of heat for accelerating effect of attraction at any point, and temperature for potential, and the solution of a problem of attractions is transformed into that of a problem of heat."
Obviously the two phenomena are not identical although they call for exactly the same mathematical formalism. Yet, as Maxwell noted, "If we know nothing more than is expressed in the mathematical formulae, there would be nothing to distinguish between the one set of phenomena and the other."83 Of the many examples of this sort only one more will be mentioned: the case of Mathieu's equations handling equally well the vibrations of an elliptical stretched surface and the dynamics of an acrobat balancing himself on the top of a sphere. Is it possible to deny that what these two phenomena have in common is enormously less than in what they differ?
Unfortunately, there is no lack of scientists who take a stance that comes very close to such a denial. They are the ones who resolutely keep trying to supplement all physicochemical categories with mathematical equations. Their efforts were duly characterized by F. S. C. Northrop as being as ridiculous as the procedure of a pure mathematician "who would expect to derail the Twentieth Century Limited, by attempting to think the equation for a disembodied switch across its pathway."84
How little mathematics can replace reality is perhaps best illustrated by the way in which the concept of the ether has stolen back into modern physics. Discredited two generations ago because of the persistent failure of attempts to detect it, the ether was replaced by the vacuum, and its demise became the much talked about symbol of the advent of an entirely new era in physics. Before long however, modern physics began to embellish space, which the departure of ether allegedly left totally empty, with properties disguised in mathematical nomenclature. In quantum electrodynamics today, the vacuous space, or the mere emptiness, is the seat of zero-point oscillations of the electromagnetic field, of the zero-point fluctuations of the electric charge, and of several even more recondite "properties." It is in fact this reinstatement of the ether in the modern mathematical framework of physics that best shows how little the physicist can dispense with what are commonly called physicochemical categories.
Truly, "if we know nothing more than is expressed in the mathematical formulae," to recall once more Maxwell's phrase, one would be lost not only in the real world but in physics as well. Mathematics can translate only one aspect of things. It can count the atoms in an apple, it can tell whether this number is odd or even, but mathematics cannot provide either the apple or its atoms. Much less can it say anything about what are known as the pleasing features of an apple.
But apart from such "anthropomorphic" although indispensable aspects of reality, it is still true that "There is no valid inference," as Whitehead aptly remarked, "from mere possibility to matter of fact, or in other words, from mere mathematics to concrete nature."85 For contrary to the dreams and hopes of ancient and latter-day Pythagoreans, numbers depend on the concreteness of things instead of generating those things. And for this elementary reason alone, regardless of the problems and uncertainties besetting modern mathematics, the mathematical concept of physics will remain as incapable of providing a final, exhaustive inteligibility of nature, as were the organismic and mechanistic concepts of physics.
Those who seem to forget that there is immensely more in a thing than its quantitative or numerical aspect will of course speak and write as if numbers could tell the whole story about nature. In such a state of mind it is only natural to project the notion of number into an absolute hallowed image. The Pythagoreans composed sacred incantations to honor and worship the number, as the full intelligibility and source of all things: "Bless us divine number, thou who generatest gods and men," went their misguided admiration for numbers. Today one encounters a similar philosophical poverty in the statements of prominent physicists who define God as a mathematician.86
Of the mathematical expertise of God no mortal can say much. Much more, however, can be said about a specific lesson provided by the history of physics. Outstanding spokesmen of the three main types of physics time and again claimed that the "perfect intelligibility of nature" represented by the physics of the day had some divine haIlmark on it. Aristotle, the supreme architect of organismic physics, referred to that ascending staircase of "affections" that supposedly connected both the strivings of ordinary matter and the affections of the ether to the fullness of life in the Prime Mover.
Yet, the perfection of divine life did not guarantee perennial validity for organismic physics. The most persuasive salesman of mechanistic physics, Voltaire, called God "the eternal machinist."87 Little did he suspect that the infinite mechanical skill of God belonged to an incomparably higher level than all the power of Newtonian physics let alone its irremediable shortcomings. It will not be otherwise with the mathematical concept of physics, which may very well prove itself the ultimate form of physics as a science, but certainly not the ultimate intelligibility of the physical world. Such would still be the case were modern physics to find one day its final mathematical form and synthesis. At present the prospects for this are extremely meager.
The persistent failure of a priori syntheses of physics, the evidence of the fundamentally experimental roots of geometry, the basic subordination of the heuristic values of mathematics to the experimental observation, and the relative uncertainty in which mathematics is ultimately enveloped all seem to indicate that the replacement of theories in physics will continue as before. This means, however, that only the kernel of scientific truth will become better defined as time goes on. The great aim of physical science, the overall synthesis of the scientific understanding of the universe will remain for all practical purposes what it has always been, the ever-remote objective of an intellectual faith.
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Fred L. Wilson (Shangha@physics.org)