Academics From Germany

"In the limiting case, entropy] stays the same. If it increases.., the process is irreversible. If it remains the same.., the process is reversible... [i.e.,] you can let it run backwards."

"Energy remains constant and entropy always grows and can never be reduced... [T]his is the essence of the second law of thermodynamics...the entropy of a system of bodies can... only increase."

"I was interested in the concept introduced by Clausius, entropy.., (in addition to energy,) one of the most important variables of nature."

"[T]he writings of Rudolf Clausius... through their simplicity, clarity, and... preciseness made me... dedicate myself... to the... laws of theory."

"I... turned to the main topic of my... work... the study of... thermodynamics."

"Religion and natural science are fighting a joint battle in an incessant, never relaxing crusade against scepticism and against dogmatism, against disbelief and against superstition, and the rallying cry in this crusade has always been, and always will be: "On to God!""

"No matter where and how far we look, nowhere do we find a contradiction between religion and natural science. On the contrary, we find a complete concordance in the very points of decisive importance. Religion and natural science do not exclude each other, as many contemporaries of ours would believe or fear. They mutually supplement and condition each other. The most immediate proof of the compatibility of religion and natural science, even under the most thorough critical scrutiny, is the historical fact that the very greatest natural scientists of all times—men such as Kepler, Newton, Leibniz—were permeated by a most profound religious attitude."

"Long and tedious reflection cannot enable us to shape our decisions and attitudes properly; only that definite and clear instruction which we gain can form a direct inner link to God. This instruction alone is able to give us the inner firmness and lasting peace of mind which must be regarded as the highest boon in life. And if we ascribe to God, in addition to His omnipotence and omniscience, also the attributes of goodness and love, recourse to Him produces an increased feeling of safety and happiness in the human being thirsting for solace. Against this conception not even the slightest objection can be raised from the point of natural science, for as we pointed it out before, questions of ethics are entirely outside of its realm."

"Natural science wants man to learn, religion wants him to act."

"Both religion and science require a belief in God. For believers, God is in the beginning, and for physicists He is at the end of all considerations… To the former He is the foundation, to the latter, the crown of the edifice of every generalized world view."

"Science cannot solve the ultimate mystery of nature. And that is because, in the last analysis, we ourselves are part of nature and therefore part of the mystery that we are trying to solve."

"Again and again the imaginary plan on which one attempts to build up that order breaks down and then we must try another. This imaginative vision and faith in the ultimate success are indispensable. The pure rationalist has no place here."

"Anybody who has been seriously engaged in scientific work of any kind realizes that over the entrance to the gates of the temple of science are written the words: Ye must have faith. It is a quality which the scientist cannot dispense with."

"The first and most important quality of all scientific ways of thinking must be the clear distinction between the outer object of observation and the subjective nature of the observer."

"But science sets out confidently on the endeavor finally to know the thing in itself, and even though we realize that this ideal goal can never be completely reached, still we struggle on towards it untiringly. And we know that at every step of the way each effort will be richly rewarded."

"In other words, the fundamental principles and indispensable postulates of every genuinely productive science are not based on pure logic but rather on the metaphysical hypothesis — which no rules of logic can refute — that there exists an outer world which is entirely independent of ourselves. It is only through the immediate dictate of our consciousness that we know that this world exists. And that consciousness may to a certain degree be called a special sense."

"It is not the possession of truth, but the success which attends the seeking after it, that enriches the seeker and brings happiness to him."

"To believe in science, one must believe in the rationality of the world."

"That God existed before there were human beings on Earth, that He holds the entire world, believers and non-believers, in His omnipotent hand for eternity, and that He will remain enthroned on a level inaccessible to human comprehension long after the Earth and everything that is on it has gone to ruins; those who profess this faith and who, inspired by it, in veneration and complete confidence, feel secure from the dangers of life under protection of the Almighty, only those may number themselves among the truly religious."

"Experiments are the only means of knowledge at our disposal. The rest is poetry, imagination."

"It [science] has as its highest principle and most coveted aim the solution of the problem to condense all natural phenomena which have been observed and are still to be observed into one simple principle, that allows the computation of past and more especially of future processes from present ones. ...Amid the more or less general laws which mark the achievements of physical science during the course of the last centuries, the principle of least action is perhaps that which, as regards form and content, may claim to come nearest to that ideal final aim of theoretical research."

"Under these conditions it is no wonder, that the movement of atheists, which declares religion to be just a deliberate illusion, invented by power-seeking priests, and which has for the pious belief in a higher Power nothing but words of mockery, eagerly makes use of progressive scientific knowledge and in a presumed unity with it, expands in an ever faster pace its disintegrating action on all nations of the earth and on all social levels. I do not need to explain in any more detail that after its victory not only all the most precious treasures of our culture would vanish, but — which is even worse — also any prospects at a better future."

"Experimenters are the shocktroops of science… An experiment is a question which science poses to Nature, and a measurement is the recording of Nature’s answer. But before an experiment can be performed, it must be planned – the question to nature must be formulated before being posed. Before the result of a measurement can be used, it must be interpreted – Nature’s answer must be understood properly. These two tasks are those of theorists, who find himself always more and more dependent on the tools of abstract mathematics."

"The energy and entropy of the world have no meaning, because such quantities admit of no accurate definition."

"Eine neue wissenschaftliche Wahrheit pflegt sich nicht in der Weise durchzusetzen, daß ihre Gegner überzeugt werden und sich als belehrt erklären, sondern vielmehr dadurch, daß ihre Gegner allmählich aussterben und daß die heranwachsende Generation von vornherein mit der Wahrheit vertraut gemacht ist. … Eine neue große wissenschaftliche Idee pflegt sich nicht in der Weise durchzusetzen, daß ihre Gegner allmählich überzeugt und bekehrt werden — daß aus einem Saulus ein Paulus wird, ist eine große Seltenheit —, sondern vielmehr in der Weise, dass die Gegner allmählich aussterben und daß die heranwachsende Generation von vornherein mit der Idee vertraut gemacht wird. Auch hier heißt es wieder: Wer die Jugend hat, der hat die Zukunft."

"Als Physiker, der sein ganzes Leben der nüchternen Wissenschaft, der Erforschung der Materie widmete, bin ich sicher von dem Verdacht frei, für einen Schwarmgeist gehalten zu werden. Und so sage ich nach meinen Erforschungen des Atoms dieses: Es gibt keine Materie an sich. Alle Materie entsteht und besteht nur durch eine Kraft, welche die Atomteilchen in Schwingung bringt und sie zum winzigsten Sonnensystem des Alls zusammenhält. Da es im ganzen Weltall aber weder eine intelligente Kraft noch eine ewige Kraft gibt—es ist der Menschheit nicht gelungen, das heißersehnte Perpetuum mobile zu erfinden—so müssen wir hinter dieser Kraft einen bewußten intelligenten Geist annehmen. Dieser Geist ist der Urgrund aller Materie."

"New scientific ideas never spring from a communal body, however organized, but rather from the head of an individually inspired researcher who struggles with his problems in lonely thought and unites all his thought on one single point which is his whole world for the moment."

"I also knew the formula that expresses the energy distribution in the normal spectrum. A theoretical interpretation therefore had to be found at any cost, no matter how high. It was clear to me that classical physics could offer no solution to this problem, and would have meant that all energy would eventually transfer from matter to radiation. ...This approach was opened to me by maintaining the two . The two laws, it seems to me, must be upheld under all circumstances. For the rest, I was ready to sacrifice every one of my previous convictions about physical laws. ...[One] finds that the continuous loss of energy into radiation can be prevented by assuming that energy is forced at the outset to remain together in certain quanta. This was purely a formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result."

"I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness."

"We have no right to assume that any physical laws exist, or if they have existed up to now, that they will continue to exist in a similar manner in the future."

"Farsighted theologians are now working to mine the eternal metal from the teachings of Jesus and to forge it for all time."

"Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics."

"Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen. Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heiße im Gegenteil unsere Losung:"

"Hilbert was on the verge of retirement. He was the dignified chairman of the mathematical society's meetings, though he no longer came up with those caustic quips that people would repeat afterwards, imitating as best they could his Baltic accent. It is a pity these were not recorded before it was too late. The samples cited in English translation, in Constance Reid's biography of Hilbert, give only the palest idea of his biting wit."

"Hilbert's problems have the characteristics of any good founding document. Each one is a short essay on its subject, not overly specific, and yet Hilbert makes his intent remarkably clear. He leaves room for change and adjustment. Hilbert's goal was to foster the pursuit of mathematics."

"A more thorough study of Euclid's axioms and postulates proved them to be inadequate for the deduction of Euclid's geometry. ...Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid's list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid's parallel postulate, succeeded in constructing a new geometry known as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions."

"David Hilbert—the undisputed, foremost living mathematician in the world and lifelong close friend and collaborator of the by then deceased Minkowski—had already presented to the Göttingen Academy his own version of the same equations a few days earlier [than Einstein]. Although Minkowski and Hilbert accomplished their most important achievements in pure mathematical fields, their respective contributions to relativity should in no sense be seen as merely occasional excursions into the field of theoretical physics. Minkowski and Hilbert were motivated by much more than a desire to apply their exceptional mathematical abilities opportunistically... On the contrary, Minkowski's and Hilbert's contributions to relativity are best understood as an organic part of their overall scientific careers."

"More decisive than any other influence for the young Hilbert at Königsberg was his friendship with Adolf Hurwitz and Minkowski. He got his thorough mathematical training less from lectures, teachers or books, than from conversation."

"Physics is too difficult for physicists!"

"The various branches of geometry are all interrelated closely and quite often unexpectedly. This shows up in many places in the book. Even so... it was necessary to make each chapter...self-contained... We hope that... we have rendered each chapter taken by itself... understandable and interesting. We want to take the reader on a leisurely walk... in the big garden that is geometry, so that each may pick for himself a bouquet to his liking."

"[M]athematics is not a popular subject... The reason for this is to be found in the common superstition that [it] is but a continuation... of the fine art of arithmetic, of juggling with numbers. [We] combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book... bring[s] about a greater enjoyment of mathematics, by making it easier... to penetrate the essence of mathematics without... a laborious course of studies."

"[O]ur purpose is to give a presentation of geometry... in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems... beyond this, it is possible... to depict the geometric outline of the methods of investigation and proof, without... entering into the details... In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes... and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists."

"In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry."

"An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us."

"The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!"

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly."

"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus."

"If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems."

"To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts."

"It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning."