Physicists From Germany

"Can you, or anyone else, reach the central order of things or events, whose existence seems beyond doubt, as directly as you can reach the soul of another human being? I am using the term 'soul' quite deliberately so as not to be misunderstood. If you put your question like that, I would say yes. ... the word 'soul' refers to the central order, to the inner core of a being whose outer manifestations may be highly diverse and pass our understanding."

"The problem of values is nothing but the problem of our acts, goals and morals. It concerns the compass by which we must steer our ship if we are to set a true course through life. The compass itself has been given different names by various religions and philosophies: happiness, the will of God, the meaning of life-to mention just a few. The differences in the names reflect profound differences in the awareness of different human groups. I have no wish to belittle these differences, but I have the clear impression that all such formulations try to express man's relatedness to a central order."

"In general, scientific progress calls for no more than the absorption and elaboration of new ideas — and this is a call most scientists are happy to heed."

"One evening I reached the point where I was ready to determine the individual terms in the energy table, or, as we put it today, in the energy matrix, by what would now be considered an extremely clumsy series of calculations. When the first terms seemed to accord with the energy principle, I became rather excited, and I began to make countless arithmetical errors. As a result, it was almost three o'clock in the morning before the final result of my computations lay before me. The energy principle had held for all terms, and I could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and I felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me. I was far too excited to sleep, and so, as a new day dawned, I made for the southern tip of the island, where I had been longing to climb a rock jutting out into the sea. I now did so without too much trouble, and waited for the sun to rise."

"An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them."

"Quantum theory provides us with a striking illustration of the fact that we can fully understand a connection though we can only speak of it in images and parables."

"Every experiment destroys some of the knowledge of the system which was obtained by previous experiments."

"Light and matter are both single entities, and the apparent duality arises in the limitations of our language. It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for, as has been remarked, it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. Furthermore, it is very difficult to modify our language so that it will be able to describe these atomic processes, for words can only describe things of which we can form mental pictures, and this ability, too, is a result of daily experience. Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme — the quantum theory — which seems entirely adequate for the treatment of atomic processes; for visualisation, however, we must content ourselves with two incomplete analogies — the wave picture and the corpuscular picture."

"The more precise the measurement of position, the more imprecise the measurement of momentum, and vice versa."

"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated."

"Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built."

"The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'"

"Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist."

"Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians."

"According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically."

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds."

"It is to Gauss, to the Magnetic Union, and to magnetic observers in general, that we owe our deliverance from that absurd method of estimating forces by a variable standard which prevailed so long even among men of science. It was Gauss who first based the practical measurement of magnetic force (and therefore of every other force) on those long established principles, which, though they are embodied in every dynamical equation, have been so generally set aside, that these very equations, though correctly given... are usually explained... by assuming, in addition to the variable standard of force, a variable, and therefore illegal, standard of mass."

"On demandait à Laplace quel était selon lui le plus grand mathématicien de l'Allemagne. C'est Pfaff, répondit-il. - Je croyais, reprit l'interlocuteur, que Gauss lui était supérieur. - Mais, s'écria Laplace, vous me demandez quel est le plus grand mathématicien de l'Allemagne, et Gauss est le plus grand mathématicien de l'Europe."

"Toward the ends of their lives, Euler, D'Alembert, and Lagrange agreed that the realm of mathematical ideas had been practically exhausted and that no new great minds were appearing on the mathematical horizons. Of course, these men had grown old and their vision was already dimmed, for Laplace, Legendre, and Fourier were in young manhood. In one respect, however, these elder statesmen were correct... their immediate successors continued to explore and polish the very same ideas which the mid-eighteenth century had pursued. But history shows that the human mind is fertile, ingenious, and creative beyond all possible anticipations. ...even the richest vein of thought is ultimately exhausted, and then, indeed, a period of stagnation may ensue. Inevitably, however, there arise new conceptions and new periods of feverish and rewarding research. Euler and his contemporaries failed to reckon with history. ... The man who was to change the course of mathematics was but six years old when Euler and D'Alembert died in 1783... Gauss is commonly ranked with Archimedes and Newton. ...all three of these men were as much devoted to physical research as to mathematics."

"There is no doubt... that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgement has caused mathematicians to equate elegance and conciseness at the cost of intelligibility. Gauss himself wrote elegant, but highly compact, carefully polished papers with no hint of the motivation, meaning, or details of the steps. When criticized, he said that no architect leaves the scaffolding after completing the building. But the fact is that even excellent mathematicians found the reading of Gauss's papers very difficult, and the same is true of many other mathematicians."

"Not only could nobody but Gauss have produced it, but it would never have occurred to anyone but Gauss that such a formula was possible."

"Everything Gauss writes is abomination, as it is so obscure that it is almost impossible to understand it."

"He is like the fox, who effaces his tracks in the sand with his tail."

"Some of the discoveries of Abel and Jacobi were anticipated by Gauss. In the Disquisitiones Arithmeticæ he observed that the principles which he used in the division of the circle were applicable to many other functions, besides the circular, and particularly to the transcendents dependent on the integral \displaystyle \int \frac{\,dx}{\sqrt{1-x^4}}. From this Jacobi concluded that Gauss had thirty years earlier considered the nature and properties of elliptic functions and had discovered their double periodicity. The papers in the collected works of Gauss confirm this conclusion."

"Bei Gegenstdnden mit denen ich mich noch nicht lange beschaftigt habe, bin ich gegen meine eigenen Ansichten, zumal wenn sie einem Laplace widersprechen, misstrauisch und nehme gern die von anderen entgegen. ["I am suspicious of my own views on subjects with which I have not long occupied myself, and gladly accept those of others, especially when my views contradict one of Laplace."]"

"Procreare jucundum, sed parturire molestum ["to beget is pleasant, but to give birth is painful"]"

"Pauca sed matura ["few, but ripe"]"

"Abraham Gotthelf Kästner is] the best mathematician among poets and the best poet among mathematicians"

"Gauss's aim was always to give his investigations the form of perfect works of art. He would not rest sooner and never gave a piece of work to the public until he had given it the perfection of form he desired for it. A good building should not show its scaffolding when completed, he used to say. In his demonstrations he used almost entirely the synthetic method, which he had come to prize through his studies of Archimedes and Newton. It is distinguished from the analytic method by its brevity and comprehensiveness. But the road leading to the discovery remains veiled; and indeed it often seems that Gauss frequently and intentionally turned aside from the road that led to mere instruction."

"That this subject [of ] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1, -1, √-1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible) such an obscurity would have been out of question."

"The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it."

"If the object of all human investigation were but to produce in cognition a reflection of the world as it exists, of what value would be all its labor and pains, which could result only in vain repetition, in an imitation within the soul of that which exists without it?"

"In such apparent accidents which finally produce such a decisive influence on one s whole life, one is inclined to recognize the tools of a higher hand. The great enigma of life never becomes clear to us here below."

"Dark are the paths which a higher hand allows us to traverse here... let us hold fast to the faith that a finer, more sublime solution of the enigmas of earthly life will be present, will become part of us."

"One cannot reduce to concepts the distinction between two systems of three straight lines each (directed lines, of which the one system points forward, upward to the right, the other forward, upward to the left) but one can only demonstrate by holding to actually present spatial things. Two minds cannot reach agreement about it unless their views connect up with one and the same system present in the real world"

"I am almost amazed that you consider a professional philosopher capable of no confusion in concepts and definitions. Such things are nowhere more at home than among philosophers who are not mathematicians, and Wolff was no mathematician, even though he made cheap compen- diums. Look around among the philosophers of today, among Schelling, Hegel, Nees von Esenbeck, and their like; doesn t your hair stand on end at their definitions? Read in the history of ancient philosophy what kinds of definitions the men of that day, Plato and others, gave (I except Aristotle). But even in Kant it is often not much better; in my opinion his distinction between analytic and synthetic theorems is such a one that either peters out in a triviality or is false."

"The austere sides of life, at least of mine, which move through it like a red thread, and which one faces more and more defenselessly in old age, are not balanced to the hundredth part by the pleasurable. I will gladly admit that the same fates which have been so hard for me to bear, and still are, would have been much easier for many another person, but the mental constitution belongs to our ego, which the Creator of our existence has given us, and we can change little in it."

"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length."

"I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect. . . Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics."

"A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed."

"One day he said: For the soul there is a satisfaction of a higher type; the material is not at all necessary. Whether I apply mathematics to a couple of clods of dirt, which we call planets, or to purely arithmetical problems, it's just the same; the latter have only a higher charm for me."

"One is forced to the view, for which there is so much evidence even though without rigorous scientific basis, that besides this material world another, second, purely spiritual world order exists, with just as many diversities as that in which we live-—we are to participate in it."

"Even though much error and hypocrisy may often be mixed in such pietistic tendencies, nevertheless I recognize with all my heart the business of a missionary as a highly honorable one in so far as it leads to civilization the still semisavage part of earth s inhabitants. May my son try it for several years."

"All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced."

"Yes! The world would be nonsense, the whole creation an absurdity without immortality."

"You say that faith is a gift; this is perhaps the most correct thing that can be said about it."

"I scarcely believe that in psychology data are present which can be mathematically evaluated. But one cannot know this with certainty, without having made the experiment. God alone is in possession of the mathematical bases of psychic phenomena."

"I believe you are more believing in the Bible than I. I am not, and, you are much happier than I. I must say that so often in earlier times when I saw people of the lower classes, simple manual laborers who could believe so rightly with their hearts, I always envied them, and now tell me how does one begin this?"

"Finally, a few days ago, it has been achieved - but not by my cumbersome search, rather through God’s good grace, I am tempted to say. As the lightning strikes the riddle was solved; I myself would be unable to point to a guiding thread between what I knew before, what I had used in my last attempts, and what made it work."