Mathematicians From Austria

"The essential point in the definition of an algebra is that it is a vector space of finite dimension over a field. This fact allows us to conclude that of subalgebras will terminate. After the great success that Emmy Noether had in her ideal theory in rings with ascending chain condition, it seemed reasonable to expect that in rings where the ascending and the descending chain condition holds for left ideals one should obtain results similar to those of . As one of the papers written from this point of view we mention E. Artin, Zur Theorie der hyperkomplexen Zahlen (Abh. Math. Sem. Hamburgischen Univ. vol. 5 (1926)). In 1939 showed (Rings with minimal condition f or left ideals, Ann. of Math. vol. 40) that the descending chain condition suffices."

"I think he would have made a great actor. His lectures were polished: He would finish at the right moment and march off the scene. A very lively individual with many interests: music, astronomy, chemistry, history.... He loved to teach. I had a feeling that he loved to teach anybody anything. Being his student was a wonderful experience; I couldn’t have had a better start to my mathematical career. It was a remarkable accident. My favorite theorem, which I had first learned from Bell’s book, was Gauss’s law of , and there, entirely by chance, I found myself at the same university as the man who had discovered the . It was just amazing."

"Olga Taussky is remembered by many for her lectures. One was 's in 1981; this had a special resonance, for she had known Emmy Noether both at Göttingen and at Bryn Mawr. Others remember Olga as author of some beautiful research papers, as teacher, as collaborator, and someone whose zest for mathematics was deeply felt and contagious. The field she is most identified with—which might be called "linear algebra and applications," though "real and complex matrix theory" would be preferred by some—did not have autonomous existence in the 1930s, despite the textbook by . Her service in that field is the very highest, as was palpable in the standing ovation after her survey talk at the second Raleigh conference."

"The work at school was really not that difficult if one applied oneself to it, but it was so uninteresting that you could not wish to apply yourself. I felt there was another mathematics. I later found that the yearning for and the satisfaction gained from mathematical inslight brings the subject near to art. While talent is undoubtedly needed by itself, it does not always make a person a mathematician. Yet most people who go into mathematics do it because they are know they are good at it. When their talent slowly declines they find themselves occasionally quite lost. This happens to some people at an early age. But what are they to do then?"

"In spite of my admiration for , I myself left his seminar and even his private circle, to which I had been admitted. I was the youngest in age in the , but I was disappointed that these gatherings could not give me guidance for my work in number theory. Had I realized what Gödel would achieve later, I would not have run away. For Gödel's results show that logic is not a subject that stands alone and is a basis for mathematical thinking; it is in fact part of mathematics."

"The duality theorems for a manifold will be derived from the corresponding ones for nets and co-nets of group systems ... Of the two types of duality theorems — the and s — the first could be obtained more easily from the corresponding one for a group system and its character system ... But when applied to the Alexander type this method fails. Also in using the net and co-net theory for both types, the less complicated Poincaré theorem serves as an introduction for the Alexander theorem."

"... I think Walther Mayer got a designation that has never been used for anyone else. He was the only associate professor the Institute has ever had. He was called "associate professor." Yes, and the reason was: Einstein wanted him very much. Clearly the mathematicians did not feel that he really should be a full-fledged faculty member. This was at an early stage, so they took a designation that was used in American universities. Only later did it occur to them to create a new thing called "permanent member.""

"Time is that which is measured by a clock. This is a sound way of looking at things. A quantity like time, or any other physical measurement, does not exist in a completely abstract way. We find no sense in talking about something unless we specify how we measure it. It is the definition by the method of measuring a quantity that is the one sure way of avoiding talking nonsense about this kind of thing."

"On the most usual assumption, the universe is homogeneous on the large scale, i. e. down to regions containing each an appreciable number of nebulae. The homogeneity assumption may then be put in the form: An observer situated in a nebula and moving with the nebula will observe the same properties of the universe as any other similarly situated observer at any time."

"The landscape has been so totally changed, the ways of thinking have been so deeply affected, that it is very hard to get hold of what it was like before... It is very hard to realize how total a change in outlook Isaac Newton has produced."

"The kind of lecture which I have been so kindly invited to give, and which now appears in book form, gives one a rare opportunity to allow the bees in one's bonnet to buzz even more noisily than usual."

"All science is full of statements where you put the best face on your ignorance, where you say: true enough, we know awfully little about this, but more or less irrespective of the stuff we don't know about, we can make certain useful deductions."

"It seems to me that if somebody intends to marry and wants to find out 'scientifically' if his choice will probably be successful, then he can be helped, perhaps, by psychology, physiology, eugenics, or sociology, but surely by a science which centres around the word 'probable'."

"The main interest of physical statistics lies in fact not so much in the distribution of the phenomena in space, but rather in their succession in time."

"Equally possible cases do not always exist, e.g, they are not present in the game with a biased die, or in life insurance. Strictly speaking, the propositions of the classical theory are therefore no applicable to these cases."

"The mean and variance are unambiguously determined by the distribution, but a distribution is, of course, not determined by its mean and variance: A number of different distributions have the same mean and the same variance."

"I do not want to defend the occult sciences; I am, however, convinced that further unbiased investigation of these phenomena by collection and evaluation of old and new evidence, in the usual scientific manner, will lead us sooner or later to the discovery of new and important relations of which we have as yet no knowledge, but which are natural phenomena in the usual sense."

"Apart from this older generation, there is scarcely a modern mathematician who still adheres without reservation to the classical theory of probability. The majority have more or less accepted the frequency definition."

"We can only hope that statisticians will return to the use of the simple, lucid reasoning of Bayes's conceptions, and accord to the likelihood theory its proper role."

"No contradiction exists, if the events are correctly interpreted."

"If the concept of probability and the formulae of the theory of probability are used without a clear understanding of the collectives involved, one may arrive at entirely misleading results."

"Starting from a logically clear concept of probability, based on experience, using arguments which are usually called statistical, we can discover truth in wide domains of human interest."

"The theory of probability can never lead to a definite statement concerning a single event."

"The whole financial basis of insurance would be questionable if it were possible to change the relative frequency of the occurrence of the insurance cases (deaths, etc.) by excluding, for example, every tenth one of the insured persons, or by some selection principal."

"It is useful to have a short expression for denoting the whole of the probabilities attached to the different attributes in a collective. We shall use for this purpose the word distribution."

"Insurance companies nowadays apply the principle of so-called 'selection by insurance'; this means that they take into consideration the fact that persons who enter early into insurance contracts are on the average of a different type and have a different distribution of death ages from persons from persons admitted to the insurance at a more advanced age."

"I am prepared to concede without further argument that all the theoretical constructions, including geometry, which are used in the various branches of physics are only imperfect instruments to enable the world of empirical fact to be reconstructed in our minds."

"In games of chance, in the problems of insurance, and in the molecular processes we find events repeating themselves again and again. They are mass phenomena or repetitive events."

"It has been asserted - and this is no overstatement - that whereas other sciences draw their conclusions from what we know, the science of probability derives its most important results from what we do not know."

"Mass phenomena to which the theory of probability does not apply are, of course, of common occurrence."

"Because certain elements of geometry have for a long time been included in the general course of education, every educated man is able to distinguish between the practical task of the land surveyor and the theoretical investigation of the geometer. The corresponding distinction between the theory of probability and statistics has yet to be recognized."

"Remember that algebra, with all its deep and intricate problems, is nothing but a development of the four fundamental operations of arithmetic. Everyone who understands the meaning of addition, subtraction, multiplication, and division holds the key to all algebraic problems."

"The way he taught statistical mechanics and electromagnetic theory, you got the feeling of a growing science that emerged out of conflict and debate. It was alive, like his lectures, which were full of personal references to men like Boltzmann, Klein, Ritz, Abraham, and Einstein. He told us at the beginning that we should teach ourselves vector analysis in a fortnight—no babying. Ehrenfest's students all acknowledge how much his method of exposition has influenced their own teaching."

"In passing I have to mention a typical Ehrenfest anecdote, not such a nice one, perhaps. Lorentz lived in Haarlem and all these celebrities, Rutherford, Madame Curie, Bohr, Einstein and very many others travelled by train, a special train, from Leiden to Haarlem. And the week before one of those rare fatal train accidents had occurred and I said to Ehrenfest, "Wouldn't it be dreadful if that train had an accident?" And Ehrenfest replied: "Yes, that would be dreadful, but think of all the young physicists who then could get jobs .... "."

"Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on his work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously."

"Einstein, my upset stomach hates your theory — it almost hates you yourself ! How am I to provide for my students ? What am I to answer to the philosophers ?"

"You will get your difficulties with the point electron."

"There have been applied sciences throughout the ages. … However this so-called practice was not much more than paper in nearly all of these cases, and the various applied sciences were only lacking a bagatelle, namely proper scientific practice. The applied sciences show the application of theoretic doctrines in existing events; but that is precisely what it does, it merely shows. Whereas the scientific practice autonomously puts to use these theories."

"\frac{\partial \psi}{\partial t} = -\frac{2 \pi i}{h} E \psi"

"For thousands of years men have striven and suffered and begotten and women have brought forth in pain. A hundred years ago, perhaps, another man sat on this spot; like you he gazed with awe and yearning in his heart at the dying light on the glaciers. Like you he was begotten of man and born of woman. He felt pain and brief joy as you do. Was he someone else? Was it not you yourself? What is this Self of yours? What was the necessary condition for making the thing conceived this time into you, just you and not someone else?"

"I am very astonished that the scientific picture of the real world around me is deficient. It gives a lot of factual information, puts all our experience in a magnificently consistent order, but it is ghastly silent about all and sundry that is really near to our heart, that really matters to us. It cannot tell us a word about red and blue, bitter and sweet, physical pain and physical delight; it knows nothing of beautiful and ugly, good or bad, God and eternity. Science sometimes pretends to answer questions in these domains, but the answers are very often so silly that we are not inclined to take them seriously."

"The stages of human development are to strive for: (1) Besitz [Possession] (2) Wissen [Knowledge] (3) Können [Ability] (4) Sein [Being]"

"No self is of itself alone. It has a long chain of intellectual ancestors. The "I" is chained to ancestry by many factors … This is not mere allegory, but an eternal memory."

"You may ask — you are bound to ask me now: What, then, is in your opinion the value of natural science? I answer: Its scope, aim and value is the same as that of any other branch of human knowledge. Nay, none of them alone, only the union of all of them, has any scope or value at all, and that is simply enough described: it is to obey the command of the Delphic deity: gnothi seauton... get to know yourself!"

"We do not belong to this material world that science constructs for us. We are not in it; we are outside. We are only spectators. The reason why we believe that we are in it, that we belong to the picture, is that our bodies are in the picture. Our bodies belong to it. Not only my own body, but those of my friends, also of my dog and cat and horse, and of all the other people and animals. And this is my only means of communicating with them."

"Nirvana is a state of pure blissful knowledge... It has nothing to do with the individual. The ego or its separation is an illusion. Indeed in a certain sense two "I"'s are identical namely when one disregards all special contents — their Karma. The goal of man is to preserve his Karma and to develop it further... when man dies his Karma lives and creates for itself another carrier."

"'If finally we look back at that idea of [Ernst] Mach [1838-1916], we shall realize that it comes as near to the orthodox dogma of the Upanishads as it could possibly do without stating it expressis verbis . The external world and consciousness are one and the same thing.'"

"It seems plain and self-evident, yet it needs to be said: the isolated knowledge obtained by a group of specialists in a narrow field has in itself no value whatsoever, but only in its synthesis with all the rest of knowledge and only inasmuch as it really contributes in this synthesis toward answering the demand, "Who are we?""

"What we call thought (1) is itself an orderly thing, and (2) can only be applied to material, i.e. to perceptions or experiences, which have a certain degree of orderliness. This has two consequences. First, a physical organization, to be in close correspondence with thought (as my brain is with my thought) must be a very well-ordered organization, and that means that the events that happen within it must obey strict physical laws, at least to a very high degree of accuracy. Secondly, the physical impressions made upon that physically well-organized system by other bodies from outside, obviously correspond to the perception and experience of the corresponding thought, forming its material, as I have called it. Therefore, the physical interactions between our system and others must, as a rule, themselves possess a certain degree of physical orderliness, that is to say, they too must obey strict physical laws to a certain degree of accuracy. PHYSICAL LAWS REST ON ATOMIC STATISTICS AND ARE THEREFORE ONLY APPROXIMATE"

"But it is quite easy to express the solution in words, thus: the plurality that we perceive is only an appearance; it is not real. Vedantic philosophy, in which this is a fundamental dogma, has sought to clarify it by a number of analogies, one of the most attractive being the many-faceted crystal which, while showing hundreds of little pictures of what is in reality a single existent object, does not really multiply the object... 'You may suddenly come to see, in a flash, the profound rightness of the basic conviction of Vedanta: … knowledge, feeling and choice are essentially eternal and unchangeable and numerically one in all men, nay in all sentient beings.'"