Logicians From Austria

"In the end we search out the beginnings. Established, beyond comparison, as the most important logician of our times by his remarkable results of the 1930s, Kurt Gödel was also most unusual in the ways of his life and mind. Deeply private and reserved, he had a superb all embracing rationality, which could descend into a maddening attention to detail in matters of everyday life."

"Secondly, even disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely, inductively by studying its "success." Success here means fruitfulness in consequences, in particular in "verifiable" consequences, i.e. consequences verifiable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs. The axioms for the system of real numbers, rejected by the intuitionists, have in this sense been verified to some extent, owing to the fact that analytic number theory frequently allows one to prove number-theoretical theorems which, in a more cumbersome way, can subsequently be verified by elementary methods. A much higher degree of verification than that, however, is conceivable. There might exists axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems, (and even solving them constructively, as far as that is possible) that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same sense as any well-established physical theory."

"Gödel published comparatively little, but almost always to maximum effect; his papers are models of precision and incisive presentation."

"The progenitor of information theory, and perhaps the pivotal figure in the recent history of human thought, was Kurt Gödel, the eccentric Austriac genius and intimate of Einstein who drove determinism from its strongest and most indispensable redoubt; the coherence, consistency, and self-sufficiency of mathematics. Gödel demonstrated that every logical scheme, including mathematics, is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself. In an elegant mathematical proof, introduced to the world by the great mathematician and computer scientist John von Neumann in September 1930, Gödel demonstrated that mathematics was intrinsically incomplete. Gödel was reportedly concerned that he might have inadvertently proved the existence of God, a faux pas in his Viennese and Princeton circle. It was one of the famously paranoid Gödel's more reasonable fears."

"The completeness theorem, mathematically, is indeed an almost trivial consequence of Skolem 1923a. However, the fact is that, at that time, nobody (including Skolem himself) drew this conclusion (neither from Skolem 1923a nor, as I did, from similar considerations of his own)."

"Religions are, for the most part, bad-but religion is not."

"There are other worlds and rational beings of a different and higher kind. The world in which we live is not the only one in which we shall live or have lived."

"Fifty years ago Kurt Gödel... proved that the world of pure mathematics is inexhaustible. No finite set of axioms and rules of inference can ever encompass the whole of mathematics. Given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. This discovery... came at first as an unwelcome shock to many mathematicians. It destroyed... the hope that they could solve the problem of deciding by a systematic procedure the truth or falsehood of any mathematical statement. ...Gödel's theorem, in denying ...the possibility of a universal algorithm to settle all questions, gave... instead, a guarantee that mathematics can never die. ...there will always be, thanks to Gödel, fresh questions to ask and fresh ideas to discover."

"Toward the end of his life, Gödel feared that he was being poisoned, and he starved himself to death. His theorem is one of the most extraordinary results in mathematics, or in any intellectual field in this century. If ever potential mental instability is detectable by genetic analysis, an embryo of someone with Kurt Gödel's gifts might be aborted."

"Not even mathematics can be considered as a closed and complete system of axioms and theorems. The mathematical world is inexhaustible, no finite set of postulates and deductions will ever be able to give us the answer to all questions. Gödel's theorem, whose statement dates back to about half a century ago, brutally put an end to all attempts to condense mathematics into a list of axioms from which the truth or falsity of each of its assertions should follow. If the same mathematical language that physics uses to describe the world remains intrinsically incomplete, it is not reasonable to expect that the universe can be describable starting from a finite set of natural laws. The incompleteness of mathematics and consequently that of physics is repugnant to many, but it must be said that for the exact sciences, Gödel's theorem is by no means a defeat: on the contrary, it provides us with an intellectual push towards ever broader and more fruitful developments."

"Ninety percent of [contemporary philosophers] see their principal task as that of beating religion out of men's heads. … We are far from being able to provide scientific basis for the theological world view."

"Either mathematics is too big for the human mind, or the human mind is more than a machine."

"To every ω-consistent recursive class κ of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg (κ) (where v is the free variable of r)."

"The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation]."

"But every error is due to extraneous factors (such as emotion and education); reason itself does not err."

"I like Islam, it is a consistent idea of religion and open-minded."

"The meaning of the world is the separation of wish and fact. Wish is a force as applied to thinking beings, to realize something. A fulfilled wish is a union of wish and fact. The meaning of the whole world is the separation and the union of fact and wish."

"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."

"After Einstein's death, Gödel's sense of exile must have deepened enormously. When Einstein had been ordered by his doctor to take a rest cure, there had been nobody, as Gödel complained to his mother, for him to speak to. Now there would permanently be nobody."

"... according to what Veblen told me, the association between Einstein and Gödel arose in the following way. Veblen felt that he had to look out for Gödel, and spent quite a lot of time talking with him. And then, he thought that he might perhaps get Einstein to take over part of this responsibility. And that seemed to go so extremely well that Veblen removed himself, essentially, from the picture. Einstein and Gödel remained very close. They tended to come to the Institute together, and leave the Institute together, very often. Of course, Gödel's interest in the theory of relativity theory undoubtedly goes back to this association with Einstein. ... I don't think he had any interest in physics before that. I know he had some philosophical interests, but I think the specific interest in the theory of relativity, in which he did write some papers and create some results of significance, that goes back to that association."

"The one man who was, during the last years, certainly by far Einstein's best friend, and in some ways strangely resembled him most, was Kurt Gödel, The great logician. They were very different in almost every personal way — Einstein gregarious, happy, full of laughter and common sense, and Gödel extremely solemn, very serious, quite solitary, and distrustful of common sense as a means of arriving at the truth. But they shared a fundamental quality: both went directly and wholeheartedly to the questions at the very center of things."

"In the 1970s I even got to meet Kurt Gödel a few times. The king of the logicians. Gödel once told me, “The a priori is very powerful.” By this he meant that pure logic can take you farther than you might believe possible."

"The more I think about language, the more it amazes me that people ever understand each other at all."

"I appeal to the philosophers of all countries to unite and never again mention Heidegger or talk to another philosopher who defends Heidegger. This man was a devil. I mean, he behaved like a devil to his beloved teacher, and he has a devilish influence on Germany. ... One has to read Heidegger in the original to see what a swindler he was."

"Bold ideas, unjustified anticipations, and speculative thought, are our only means for interpreting nature: our only organon, our only instrument, for grasping her. And we must hazard them to win our prize. Those among us who are unwilling to expose their ideas to the hazard of refutation do not take part in the scientific game."

"I think so badly of philosophy that I don't like to talk about it. ... I do not want to say anything bad about my dear colleagues, but the profession of teacher of philosophy is a ridiculous one. We don't need a thousand of trained, and badly trained, philosophers — it is very silly. Actually most of them have nothing to say."

"For it was my master who taught me not only how very little I knew but also that any wisdom to which I might ever aspire could consist only in realizing more fully the infinity of my ignorance."

"If in this book harsh words are spoken about some of the greatest among the intellectual leaders of mankind, my motive is not, I hope, the wish to belittle them. It springs rather from my conviction that, if our civilization is to survive, we must break with the habit of deference to great men. Great men may make great mistakes; and as the book tries to show, some of the greatest leaders of the past supported the perennial attack on freedom and reason. Their influence, too rarely challenged, continues to mislead those on whose defence civilization depends, and to divide them. The responsibility of this tragic and possibly fatal division becomes ours if we hesitate to be outspoken in our criticism of what admittedly is a part of our intellectual heritage. By reluctance to criticize some of it, we may help to destroy it all."

"I see now more clearly than ever before that even our greatest troubles spring from something that is as admirable and sound as it is dangerous — from our impatience to better the lot of our fellows. For these troubles are the by-products of what is perhaps the greatest of all moral and spiritual revolutions of history, a movement which began three centuries ago. It is the longing of uncounted unknown men to free themselves and their minds from the tutelage of authority and prejudice. It is their attempt to build up an open society which rejects the absolute authority to preserve, to develop, and to establish traditions, old or new, that measure up to their standards of freedom, of humaneness, and of rational criticism. It is their unwillingness to sit back and leave the entire responsibility for ruling the world to human or superhuman authority,and their readiness to share the burden of responsibility for avoidable suffering, and to work for its avoidance. This revolution has created powers of appalling destructiveness; but they may yet be conquered."

"This book raises issues that might not be apparent from the table of contents. It sketches some of the difficulties faced by our civilization — a civilization which might be perhaps described as aiming at humanness and reasonableness, at equality and freedom; a civilization which is still in its infancy, as it were, and which continues to grow in spite of the fact that it has been so often betrayed by so many of the intellectual leaders of mankind. It attempts to show that this civilization has not yet fully recovered from the shock of its birth — the transition from the tribal or "enclosed society," with its submission to magical forces, to the 'open society' which sets free the critical powers of man. It attempts to show that the shock of this transition is one of the factors that have made possible the rise of those reactionary movements which have tried, and still try, to overthrow civilization and to return to tribalism."

"I believe that the advance of science depends upon the free competition of thought, and thus upon freedom, and that it must come to an end if freedom is destroyed (though it may well continue for some time in some fields, especially in technology)."

"Science is not a system of certain, or well established, statements; nor is it a statement which steadily advances towards state of finality. Our science is not knowledge (epistēmē): it can newer claim to have attained truth, or even substitute for it, such as probability. . . . We do not know; we can only guess."

"It seems to me that I may be living too long. Indeed: my nearest relations have all died, and so have some of my best friends, and even some of my best pupils. However, I do not have a reason to complain. I am grateful and happy to be alive, and still be able to continue with my work, if only just. My work seems to me more important than ever."

"Science does not rest upon solid bedrock. The bold structure of its theories arises, as it were, above a swamp. It is like a building erected on piles. The piles are driven down from above into the swamp, but not down to any natural or 'given' base; and if we stop driving the piles deeper, it is not because we have reached firm ground. We simply stop when we are satisfied that the piles are firm enough to carry the structure, at least for the time being."

"...non-reproducible single occurrences are of no significance to science."

"Science may be described as the art of systematic over-simplification — the art of discerning what we may with advantage omit."

"Whenever a theory appears to you as the only possible one, take this as a sign that you have neither understood the theory nor the problem which it was intended to solve."

"Piecemeal social engineering resembles physical engineering in regarding the ends as beyond the province of technology. (All that technology may say about ends is whether they are compatible with each other or realizable.)"

"The game of science is, in principle, without end. He who decides one day that scientific statements do not call for any further test, and that they can be regarded as finally verified, retires from the game."

"There is no history of mankind, there is only an indefinite number of histories of all kinds of aspects of human life. And one of these is the history of political power. This is elevated into the history of the world. But this, I hold, is an offence against every decent conception of mankind. It is hardly better than to treat the history of embezzlement or of robbery or of poisoning as the history of mankind. For the history of power politics is nothing but the history of international crime and mass murder (including it is true, some of the attempts to suppress them). This history is taught in schools, and some of the greatest criminals are extolled as heroes."

"A principle of induction would be a statement with the help of which we could put inductive inferences into a logically acceptable form. In the eyes of the upholders of inductive logic, a principle of induction is of supreme importance for scientific method: "... this principle", says Reichenbach, "determines the truth of scientific theories. To eliminate it from science would mean nothing less than to deprive science of the power to decide the truth or falsity of its theories. Without it, clearly, science would no longer have the right to distinguish its theories from the fanciful and arbitrary creations of the poet's mind." Now this principle of induction cannot be a purely logical truth like a tautology or an analytic statement. Indeed, if there were such a thing as a purely logical principle of induction, there would be no problem of induction; for in this case, all inductive inferences would have to be regarded as purely logical or tautological transformations, just like inferences in inductive logic. Thus the principle of induction must be a synthetic statement; that is, a statement whose negation is not self-contradictory but logically possible. So the question arises why such a principle should be accepted at all, and how we can justify its acceptance on rational grounds."

"There is an almost universal tendency, perhaps an inborn tendency, to suspect the good faith of a man who holds opinions that differ from our own opinions. ... It obviously endangers the freedom and the objectivity of our discussion if we attack a person instead of attacking an opinion or, more precisely, a theory."

"... the fight against suffering must be considered a duty, while the right to care for the happiness of others must be considered a privilege confined to the close circle of their friends. ... Pain, suffering, injustice, and their prevention, these are the eternal problems of public morals, the 'agenda' of public policy ..."

"Not only do I hate violence, but I firmly believe that the fight against it is not hopeless. I realize that the task is difficult. I realize that, only too often in the course of history, it has happened that what appeared at first to be a great success in the fight against violence was followed by a defeat. I do not overlook the fact that the new age of violence which was opened by the two World wars is by no means at an end. Nazism and Fascism are thoroughly beaten, but I must admit that their defeat does not mean that barbarism and brutality have been defeated. On the contrary, it is no use closing our eyes to the fact that these hateful ideas achieved something like a victory in defeat. I have to admit that Hitler succeeded in degrading the moral standards of our Western world, and that in the world of today there is more violence and brutal force than would have been tolerated even in the decade after the first World war. And we must face the possibility that our civilization may ultimately be destroyed by those new weapons which Hitlerism wished upon us, perhaps even within the first decade after the second World war; for no doubt the spirit of Hitlerism won its greatest victory over us when, after its defeat, we used the weapons which the threat of Nazism had induced us to develop."

"A rationalist, as I use the word, is a man who attempts to reach decisions by argument and perhaps, in certain cases, by compromise, rather than by violence. He is a man who would rather be unsuccessful in convincing another man by argument than successful in crushing him by force, by intimidation and threats, or even by persuasive propaganda."

"We all remember how many religious wars were fought for a religion of love and gentleness; how many bodies were burned alive with the genuinely kind intention of saving souls from the eternal fire of hell. Only if we give up our authoritarian attitude in the realm of opinion, only if we establish the attitude of give and take, of readiness to learn from other people, can we hope to control acts of violence inspired by piety and duty."

"There are many difficulties impeding the rapid spread of reasonableness. One of the main difficulties is that it always takes two to make a discussion reasonable. Each of the parties must be ready to learn from the other. You cannot have a rational discussion with a man who prefers shooting you to being convinced by you."

"Do not allow your dreams of a beautiful world to lure you away from the claims of men who suffer here and now. Our fellow men have a claim to our help; no generation must be sacrificed for the sake of future generations, for the sake of an ideal of happiness that may never be realised."

"The true Enlightenment thinker, the true rationalist, never wants to talk anyone into anything. No, he does not even want to convince; all the time he is aware that he may be wrong. Above all, he values the intellectual independence of others too highly to want to convince them in important matters. He would much rather invite contradiction, preferably in the form of rational and disciplined criticism. He seeks not to convince but to arouse — to challenge others to form free opinions."

"... the attempt to make heaven on earth invariably produces hell. It leads to intolerance. It leads to religious wars, and to the saving of souls through the inquisition. And it is, I believe, based on a complete misunderstanding of our moral duties. It is our duty to help those who need help; but it cannot be our duty to make others happy, since this does not depend on us, and since it would only too often mean intruding on the privacy of those towards whom we have such amiable intentions."