Mathematicians From The United States

"Truth and falsity are the most fundamental terms of rational criticism, and any adequate philosophy must give some account of these, or failing that, show that they can be dispensed with."

"In summary, then, the set theoretic 'needs' of physics are surprisingly similar to the set theoretic needs of pure logic. Both disciplines need some set theory to function at all. Both disciplines can 'live' - but live badly - on the meager diet of only predicative sets. Both can live extremely happily on the rich diet of impredicative sets. Insofar, then, as the indispensability of quantification over sets is any argument for their existence - and we will discuss why it is in the next section - we may say that it is a strong argument for the existence of at least predicative sets, and a pretty strong, but not as strong, argument for the existence of impredicative sets."

"To me it seems clear that the descriptions of human life we find in the novels of Tolstoy or George Eliot are not mere entertainment; they teach us to perceive what goes on in social and individual life. And such descriptions require the many subtle distinctions that ordinary language has made available to us. The question of the relevance or irrelevance of “how we speak” is not just a question for philosophers, although it is that too. It is a question for philosophers because once ordinary language is laughed out of the room, philosophical theories are no longer held responsible at all to the ways we actually speak and actually live; but it is a question for more than just philosophers because, at bottom, contempt for ordinary language is contempt for all the humanities."

"My own view is that philosophy at its best has always, in every period, included some philosophers who brilliantly represent the moral face of the subject and some philosophers who brilliantly represent the theoretical face, as well as some geniuses whose insights span and unite both sides of the subject. To renounce either the moral ambitions of philosophy or its theoretical ambitions is not just to kill the subject of philosophy; it is to commit intellectual and spiritual suicide."

"The similarity between Marxism and neoconservativism might be expressed in the following way: both perspectives say that certain injustices can't be cured under our present system of political democracy and mixed economy. The Marxist concludes that we have to overthrow the present system and the neoconservative concludes that we have to live with the injustices. But they are both wrong."

"A few years ago I had occasion to visit Peru, and I got to know a fine philosopher and a truly wonderful human being-Francisco Miro Casada. Miro Casada has been an idealist all his life, while being, at the same time, a man of great experience (a former member of several governments and a former Ambassador to France). I found him a man who represents the social democratic vision in its purest form. Talking to him, and to my other friends in Peru (who represented quite a spectrum of political opinion), I heard something that was summed up in a remark he, Miro Casada, made to me, "Whenever you have a Republican president, we get a wave of military dictatorships in Latin America"."

"To successfully adjudicate ethical problems, as opposed to 'solving' them, it is necessary that the members of the society have a sense of community. A compromise that cannot pretend to be the last word on an ethical question, that cannot pretend to derive from binding principles in an unmistakeably constraining way, can only derive its force from a shared sense of what is and is not reasonable, from loyalties to one another, and a commitment to 'muddling through' together."

"Part of what makes moral philosophy an anachronistic field is that its practitioners continue to argue in this very traditional and aprioristic way even though they themselves do not claim that one can provide a systematic and indubitable 'foundation' for the subject. Most of them rely on what are supposed to be 'intuitions' without claiming that those intuitions deliver uncontroversial ethical premises, on the one hand, or that they have an ontological or epistemological explanation of the reliability of those intuitions, on the other."

"Philosophers today are as fond as ever of apriori arguments with ethical conclusions. One reason such arguments are always unsatisfying is that they always prove too much; when a philosopher 'solves' an ethical problem for one, one feels as if one had asked for a subway token and been given a passenger ticket valid for the first interplanetary passenger-carrying space ship instead."

"What I am saying, then, is that elements of what we call "language" or "mind" penetrate so deeply into what we call "reality" that the very project of representing ourselves as being "mappers" of something "language-independent" is fatally compromised from the very start. Like Relativism, but in a different way, Realism is an impossible attempt to view the world from Nowhere. In this situation it is a temptation to say, "So we make the world," or "our language makes up the world," or "our culture makes up the world"; but this is just another form of the same mistake. If we succumb, once again we view the world-the only world we know-as a product. One kind of philosopher views it as a product from a raw material: Unconceptualized Reality. The other views it as a creation ex nihilo. But the world isn't a product. It's just the world."

"No sane person should believe that something is 'subjective' merely because it cannot be settled beyond controversy."

"Now, moral philosophers generally prefer to talk about virtues, or about (specific) duties, rights, and so on, rather than about moral images of the world. There are obvious reasons for this; nevertheless, I think that it is a mistake, and that Kant is profoundly right. What we require in moral philosophy is, first and foremost, a moral image of the world, or rather--since, here again, I am more of a pluralist than Kant--a number of complementary moral images of the world."

"The problem with all this--the problem I discussed in the first lecture--is that if the causes/background conditions distinction is fundamentally subjective, not descriptive of the world in itself, then current philosophical explanations of the metaphysical nature of reference are bankrupt."

"To require that all of these must be reducible to a single version is to make the mistake of supposing that 'Which are the real objects?' is a question that makes sense independently of our choice of concepts."

"If reason is both transcendent and immanent, then philosophy, as culture-bound reflection and argument about eternal questions, is both in time and eternity. We don't have an Archimedean point; we always speak the language of a time and place; but the rightness and wrongness of what we say is not just for a time and a place."

"Let me close with a picture. My picture of our situation is not the famous Neurath picture of science as the enterprise of reconstructing a boat while the boat floats on the open ocean, but it is a modification of it. I would change Neurath's picture in two ways. First, I would put ethics, philosophy, in fact the whole culture, in the boat, and not just 'science', for I believe all the parts of the culture are inter- dependent. And, second, my image is not of a single boat but of a fleet of boats. The people in each boat are trying to reconstruct their own boat without modifying it so much at anyone time that the boat sinks, as in the Neurath image. In addition, people are passing supplies and tools from one boat to another and shouting advice and encouragement (or discouragement) to each other. Finally, people sometimes decide they don't like the boat they're in and move to a different boat altogether. (And sometimes a boat sinks or is abandoned.) It's all a bit chaotic; but since it is a fleet, no one is ever totally out of signalling distance from all the other boats. There is, in short, both collectivity and individual responsibility. If we hanker for more, is that not our old and unsatisfiable yearning for Absolutes?"

"[Oddly enough, Putnam believes part of the attraction of some of these formalisms is their obscurity]. "I think part of the appeal of mathematical logic is that the formulas look mysterious - you write backward Es!""

"These papers are all written from what is called a realist perspective. The statements of science are in my view either true or false (although it is often the case that we don't know which) and their truth or falsity does not consist in their being highly derived ways of describing regularities in human experience. Reality is not a part of the human mind; rather the human mind is a part - and a small part at that - of reality."

"It was Rudolf Carnap’s dream for the last three decades of his life to show that science proceeds by a formal syntactic method; today no one to my knowledge holds out any hope for that project."

"Carnap was driven from Germany by Hitler, and his position has been condemned in the Soviet Union as 'subjective idealism*. Even in the United States, there have been a great many who could not understand this attempt to turn philosophy into a scientific discipline with substantive scientific results, and who have been led to extreme and absurd misinterpretations of the work I have been reporting to you here. Few, perhaps, would have expected traditional empiricism to lead to the development of a speculative theory of ' universal learning machines'; and yet, in retrospect, a concern with systematizing inductive logic has been the oldest concern of empiricist philosophers from Bacon on. No one can yet predict the outcome of this speculative scientific venture. But it is amply clear, whether this particular venture succeeds or fails, that the toleration of philosophical and scientific speculation brings rich rewards and that its suppression leads to sterility."

"In sum, a theory is only accepted if the theory has substantial, non-ad hoc, explanatory successes. This is in accordance with Popper; unfortunately, it is in even better accordance with the 'inductivist' accounts that Popper rejects, since these stress support rather than falsification."

"To sum up: we have seen that of the three notions of 'partial interpretation' discussed, each is either unsuitable for Carnap's purposes (starting with observation terms), or incompatible with a rather minimal scientific realism; and, in addition, the second notion depends upon gross and misleading changes in our use of language. Thus in none of these senses is 'a partially interpreted calculus in which only the observation terms are directly interpreted' an acceptable model for a scientific theory."

"I might try to save the view that 'future contingents' have no truth value by saying that even present-tense statements have no truth value if they refer to the outcome of events that are so far away that a causal signal informing me of the outcome could not have reached me-now without traveling faster than light. In other words, I might attempt saying that statements about events that are in neither the upper half nor the lower half of my light-cone have no truth value. In addition, statements about events in the upper half of my light-cone have no truth value, since they are in my future according to every coordinate system. So only statements about events in the lower half of my light-cone have a truth value; only events that are in 'my past* according to all observers are determined."

"Analytic philosophers - both in the 'constructivist' camp and in the camp that studies 'the ordinary use of words' - are disturbingly unanimous in regarding 2-valued logic as having a privileged position: privileged, not just in the sense of corresponding to the way we do speak, but in the sense of having no serious rival for logical reasons. If the foregoing analysis is correct, this is a prejudice of the same kind as the famous prejudice in favor of a privileged status for Euclidean geometry (a prejudice that survives in the tendency to cite 'space has three dimensions' as some kind of 'necessary' truth). One can go over from a 2-valued to a 3-valued logic without totally changing the meaning of 'true' and 'false'; and not just in silly ways, like the ones usually cited (e.g. equating truth with high probability, falsity with low probability, and middlehood with 'in between' probability)."

"In conclusion, then, no satisfactory interpretation of quantum mechanics exists today. The questions posed by the confrontation between the Copenhagen interpretation and the hidden variable theorists go to the very foundations of microphysics, but the answers given by hidden variable theorists and Copenhagenists are alike unsatisfactory. Human curiosity will not rest until those questions are answered, but whether they will be answered by conceptual innovations within the framework of the present theory or only within the framework of an as yet unforeseen theory is unknown. The first step toward answering them has been attempted here. It is the modest but essential step of becoming clear on the nature and magnitude of the difficulties."

"The philosophy of physics is continuous with physics itself. Just as certain issues in the Foundations of Mathematics have been discussed by both mathematicians and by philosophers of mathematics, so certain issues in the philosophy of physics have been discussed by both physicists and by philosophers of physics. And just as there are issues of a more epistemological kind that tend to concern philosophers of mathematics more than they do working mathematicians, so there are issues that concern philosophers of physics more than they do working physicists."

"The physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction."

"The real significance of the Russell paradox, from the standpoint of the modal-logic picture, is this: it shows that no concrete structure can be a standard model for the naive conception of the totality of all sets; for any concrete structure has a possible extension that contains more 'sets'. (If we identify sets with the points that represent them in the various possible concrete structures, we might say: it is not possible for all possible sets to exist in any one world!) Yet set theory does not become impossible. Rather, set theory becomes the study of what must hold in, e.g. any standard model for Zermelo set theory."

"In closing, I can only apologize for not having given any positive account of either mathematical truth or mathematical necessity. I can only say that I have not given such an account because I think that the search for such an account is a fundamental mistake. It is not that there is nothing special about mathematics; it is that, in my opinion, the investigation of mathematics must presuppose and not seek to account for the truth of mathematics. But this is the beginning of another paper and not the end of this one."

"If the importance of science does not lie in its constituting the whole of human knowledge, even less does it lie, in my view, in its technological applications. Science at the best is a way of coming to know, and hopefully a way of acquiring some reverence for, the wonders of nature. The philosophical study of science, at the best, has always been a way of coming to understand both some of the nature and some of the limitations of human reason. These seem to me to be sufficient grounds for taking science and philosophy of science seriously; they do not justify science worship."

"You ought to try to make significant contributions to humanity rather than just get along through life comfortably... the life of trying to achieve excellence in some area is in itself a worthy goal... A life without a struggle on your part to make yourself excellent is hardly a life worth living. ...a life without such a goal... is merely existing..."

"Formal proofs, where there is deliberately no meaning, can convince only formalists, and of the results they themselves seem to deny any meaning. Is that to be the mathematics we are to use to understand the world we live in?"

"I know that the great Hilbert said, "We will not be driven out of the paradise Cantor has created for us," and I reply, "I see no reason for walking in!" Indeed, in time, as more and more people get used to computers, I am inclined to believe that we here on this Earth will decide that the computable numbers are enough."

"Perhaps thinking should be measured not by what you do but by how you do it."

"We constantly use the word "simplify", but its meaning depends on what you are going to do next, and there is no uniform definition."

"The feeling of having free will is deep in us and we are reluctant to give it up for ourselves—but we are often willing to deny it to others!"

"This is the type of AI I am interested in—what can the human and machine do together, and not in the competition which can arise. ...There are doubts as to what fraction of the population can compete with computers, even with nice interactive prompting menus."

"We are beginning to find not only is intelligence not adequately defined so arguments can be settled scientifically, but a lot of other associated words like, computer, learning, information, ideas, decisions, expert behavior—all are a bit fuzzy..."

"When you take a course in Euclidean geometry is not the teacher putting a... learning program into you? ...You enter the course and cannot do problems; the teacher puts into you a program and at the end of the course you can solve such problems. ...Are you sure you are not merely "programmed" in life by what by chance events happens to you?"

"The fundamentals of language are not understood to this day. ...Until we understand languages of communication involving humans as they are then it is unlikely many of our software problems will vanish."

"Unfortunately... the ADA language was designed by experts, and it shows all the non-humane features you can expect from them. It is... a typical Computer Science hacking job—do not try and understand what you are doing, just get it running. As a result of this poor psychological design... although a government contract may specify the programming be in ADA, probably over 90% will be done in FORTRAN, debugged, tested, and then painfully, by hand, be converted to a poor ADA program, with a high probability of errors!"

"The Buddha told his disciples, "Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense". I say the same to you—you must assume the responsibility for what you believe."

"An idea which arises in the field, based on the direct experience of the people doing the job, cannot get going in a centrally controlled system... The not invented here (NIH) syndrome is one of the major curses of our society..."

"The people at the bottom do not have the larger, global view, but at the top they do not have the local view of all the details, many of which can often be very important, so either extreme gets poor results."

"We must not forget, in all the enthusiasm for computer simulations, occasionally we must look at Nature as She is."

"Somewhere in the mid-to-late 1950s in an address to the President and V.Ps of Bell Laboratories I said, "At present we are doing 1 out of 10 experiments on the computers and 9 in the labs, but before I leave it will be 9 out of 10 on the machines". They did not believe me... by now we do somewhere between 90% to 99% on the machines... And this trend will go on!"

"The more complex the designed system the more field maintenance must be central to the final design. Only when field maintenance is part of the original design can it be safely controlled... This applies to both mechanical things and to human organizations."

"Transmission through space (typically signaling) is the same as transmission through time (storage)."

"The standard process of organizing knowledge into departments, and subderpartments, and further breaking it up into separate courses, tends to conceal the homogeneity of knowledge, and at the same time to omit much which falls between the courses."

"In forming your plan for your future you need to distinguish three different questions: What is possible? What is likely to happen? What is desirable to have happen? In a sense the first is Science... The second is Engineering.. The third is ethics, morals, or... value judgements. ...you will probably have an idea of how to alter things to make the more desirable future occur ...having a vision is what tends to separate the leaders from the followers."