Mathematicians From The United States

"So high, so low, so many things to know."

"Interesting problem, pain. So helpful, so obnoxious."

"So even in hell, there are clowns."

"Politics may come and go, but trade goes on forever."

"Being alone was something he was very good at. There was so much to learn."

"The essence of real creativity is a certain playfulness, a flitting from idea to idea without getting bogged down by fixated demands. Of course, you don’t always get what you thought you were asking for. From this era on, I think invention will be the parent of necessity—and not the other way around."

"Didn’t Underhill understand? All decent societies agreed on basic issues, things that meant the healthy survival of their people. Things might be changing, but it was self-serving nonsense to throw the rules overboard."

"Funny how the least attempt at deception always seems to make life more complicated."

"Yet, even with the greatest care, a technological civilization carried the seeds of its own destruction. Sooner or later, it ossified and politics carried it into a fall."

"Civilizations rise and fall, but all technical civilizations know the greatest secrets now. They know which social mechanisms normally work, and which ones quickly fail. They know the means to postpone disaster and evade the most foolish catastrophes. They know that even so, each civilization must inevitably fall."

"Pham was overwhelmed by the other things that Gunnar Larson had to say, the advice that might be worthless but that had the stench of wisdom."

"There had been an era of ubiquitous law enforcement, and some kind of distributed terror."

"There was an old Qeng Ho saying, “You know you’ve stayed too long when you start using the locals’ calendar.”"

"“Do you think she really believes what she’s saying?” “Sure she believes it. That’s what makes her so funny.”"

"There is a deepness in the sky, and it extends forever."

"This was the sort of desperate hallucination he must guard against. If you raise your desires high enough, certainty can grow out of the background noise."

"Viki was awed by all the incredible quackery. Daddy thought such things were amusing—“like religion but not so deadly.”"

"“We think the governance has opted for ubiquitous law enforcement.” Pham whistled softly. Now every embedded computing system, down to a child’s rattle, was a governance utility. It was the most extreme form of social control ever invented. “So now they have to run everything.” The notion was terribly seductive to the authoritarian mind."

"What do you do when your dream dies?"

"What do you do when your dream dies? Dreams die in every life. Everyone gets old. There is promise in the beginning when life seems so bright. The promise fades when the years get short."

"So what do you do when your dream dies? When your dream dies, you give it up."

"The hard facts are extraordinary, without adding superstitious mumbo-jumbo."

"Yes, we did quite well. But then, telling people what they want to believe is an easy job."

"The more you realized what the stars really were, the more you realized what the universe must really be."

"“But see, your theory ‘explains’ all sorts of things without helping to do anything, much less providing tests for itself.”"

"So much technology, so little talent."

"Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulations of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?""

"I don't like the nonsense that passes for rational discourse so often in our society. I am very much bothered by the inaccuracies, ambiguities, code words, slogans, catch phrases, public relation devices, sweeping generalizations, and stereotypes, which are used (consciously or otherwise) to influence people."

"I am not here concerned with intent, but with scientific standards, especially the ability to tell the difference between a fact, an opinion, a hypothesis, and a hole in the ground."

"The tool that is so dull that you cannot cut yourself on it is not likely to be sharp enough to be either useful or helpful."

"The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data."

"Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise."

"Origami helps in the study of mathematics and science in many ways. … Using origami anyone can become a scientific experimenter with no fuss."

"No longer was light analogous to the discharge of a blunderbuss, but rather to the pulsating flight of birds."

"Every new discovery in science brings with it a host of new problems, just as the invention of the automobile brought with it gas stations, roads, garages, mechanics, and a thousand other subsidiary details."

"Philosophy was never just ontotheology, and even when philosophers were concerned with ontotheology, they were concerned with much more than that. That is the first reason that the idea of a fundamental “crisis” in philosophy and of the “end of philosophy” is deeply mistaken. And if the questions of philosophy are indeed “unsettleable,” in the sense that they will always be with us, that is a wonderful thing, not something to be regretted."

"He’s a quantum philosopher. I can’t understand him and his position at the same time."

"What we are left with, if what I have said so far is right, is a conclusion that I initially found very distressing: either GRW or some successor, or else Bohm or some successor, is the correct interpretation—or, to include a third possibility to please Itamar Pitowski, we will just fail to find a scientific realist interpretation which is acceptable. (And the ghost of Bohr will laugh, and say, ‘I told you all along that the human mind cannot produce a realist interpretation of quantum mechanics’!)"

"Even though the model referred to satisfies the theory, etc., it is 'unintended'; and we recognize that it is unintended from the description through which it is given (as in the intuitionist case). Models are not lost noumenal waifs looking for someone to name them; they are constructions within our theory itself. and they have names from birth."

"To me, believing that some correspondence intrinsically just is reference (not as a result of our operational and theoretical constraints, or our intentions, but as an ultimate metaphysical fact) amounts to a magical theory of reference. Reference itself becomes what Locke called a 'substantial form' (an entity which intrinsically belongs with a certain name) on such a view. Even if one is willing to contemplate such unexplainable metaphysical facts, the epistemological problems that accompany such a metaphysical view seem insuperable. For, assuming a world of mind- independent, discourse-independent entities (this is the presupposition of the view we are discussing), there are, as we have seen, many different 'correspondences' which represent possible or candidate reference relations (infinitely many, in fact, if there are infinitely many things in the universe)."

"What we have is a device for producing sentences in response to sentences. But none of these sentences is at all connected to the real world. If one coupled two of these machines and let them play the Imitation Game with each other, then they would go on 'fooling' each other forever, even if the rest of the world disappeared! There is no more reason to regard the machine's talk of apples as referring to real world apples than there is to regard the ant's 'drawing' as referring to Winston Churchill."

"Even if we consider not words by themselves but rules deciding what words may appropriately be produced in certain contexts — even if we consider, in computer jargon, programs for using words — unless those programs themselves refer to something extra-linguistic there is still no determinate reference that those words possess. This will be a crucial step in the process of reaching the conclusion that the Brain-in-a-Vat Worlders cannot refer to anything external at all (and hence cannot say that they are Brain-in-a-Vat Worlders)."

"Before I give the argument, let us consider why it seems so strange that such an argument can be given (at least to philosophers who subscribe to a 'copy' conception of truth). We conceded that it is compatible with physical law that there should be a world in which all sentient beings are brains in a vat. As philosophers say, there is a 'possible world' in which all sentient beings are brains in a vat. (This 'possible world' talk makes it sound as if there is a place where any absurd supposition is true, which is why it can be very misleading in philosophy.) The humans in that possible world have exactly the same experiences that we do. They think the same thoughts we do (at least, the same words, images, thought-forms, etc., go through their minds). Yet, I am claiming that there is an argument we can give that shows we are not brains in a vat. How can there be? And why couldn't the people in the possible world who really are brains in a vat give it too? The answer is going to be (basically) this: although the people in that possible world can think and 'say' any words we can think and say, they cannot (I claim) refer to what we can refer to. In particular, they cannot think or say that they are brains in a vat {even by thinking 'we are brains in a vat)."

"On another possible world or another planet a word might be associated with much the same stereotype and much the same criteria as our term 'water', but it might designate XYZ and not H₂O. At least this could happen in a prescientific era. And it would not follow that XYZ was water; it would only follow that XYZ could look like water, taste like water, etc. What 'water' refers to depends on the actual nature of the paradigms, not just on what is in our heads."

"Why, then, is semantics so hard? In terms of the foregoing, I want to suggest that semantics is a typical social science. The sloppiness, the lack of precise theories and laws, the lack of mathematical rigor, are all characteristic of the social sciences today. A general and precise theory which answers the questions (i) why do words have the different sorts of functions they do? and (2) exactly how does conveying core facts enable one to learn the use of a word? is not to be expected until one has a general and precise model of a language-user; and that is still a long way off. But the fact that Utopia is a long way off does not mean that daily life should come to a screeching halt. There is plenty for us to investigate, in our sloppy and impressionistic fashion, and there are plenty of real results to be obtained. The first step is to free ourselves from the oversimplifications foisted upon us by the tradition, and to see where the real problems lie. I hope this paper has been a contribution to that first step."

"It is evident that Feyerabend is misusing the term 'meaning.' He is not alone in such misuse: in the last thirty years, misusing the term 'meaning' has been one of the most common, if least successful, ways of 'establishing' philosophical propositions. But how did this distressing state of affairs come to be? The blame must be placed squarely upon the Logical Positivists. The 'Verifiability Theory of Meaning' ('the meaning of a sentence is its method of verification') was, from the first, nothing but a persuasive redefinition. If to call metaphysical propositions 'meaningless' were only to assert that these propositions are empirically untestable, it would be harmless (the metaphysicians always said that their assertions were neither empirically testable nor tautologies); but, of course, it is not harmless, because the Positivist hopes that we will accept his redefinition of the term 'meaning,' while retaining the pejorative connotations of being 'meaningless' in the customary {linguistic) sense, i.e. being literally without sense."

"In short, analytic statements are statements which we all accept and for which we do not give reasons. This is what we mean when we say that they are true by 'implicit convention'. The problem is then to distinguish them from other statements that we accept, and do not give reasons for, in particular from the statements that we unreasonably accept. To resolve this difficulty, we have to point out some of the crucial distinguishing features of analytic statements (e.g. the fact that the subject concept is not a law-cluster concept), and we have to connect these features with what, in the preceding section, was called the 'rationale' of the analytic-synthetic distinction. Having done this, we can see that the acceptance of analytic statements is rational, even though there are no reasons (in the sense of' evidence') in connection with them."

"I do not agree with Quine, that there is no analytic-synthetic distinction to be drawn at all. But I do believe that his emphasis on the monolithic character of our conceptual system and his negative emphasis on the silliness of regarding mathematics as consisting in some sense of 'rules of language', represent exceedingly important theoretical insights in philosophy. I think that what we have to do now is to settle the relatively trivial question concerning analytic statements properly so called ('All bachelors are unmarried'). We have to take a fresh look at the framework principles so much discussed by philosophers, disabusing ourselves of the idea that they are 'rules of language' in any literal or lexicographic sense; and above all, we have to take a fresh look at the nature of logical and mathematical truths. With Quine's contribution, we have to face two choices: We can ignore it and go on talking about the 'logic' of individual words. In that direction lies sterility and more, much more, of what we have already read. The other alternative is to face and explore the insight achieved by Quine, trying to reconcile the fact that Quine is overwhelmingly right in his critique of what other philosophers have done with the analytic-synthetic distinction with the fact that Quine is wrong in his literal thesis, namely, that the distinction itself does not exist at all. In the latter direction lies philosophic progress. For philosophic progress is nothing if it is not the discovery of new areas for dialectical exploration."

"The techniques employed by philosophers of physics are usually the very ones being employed by philosophers of a less specialized kind (especially empiricist philosophers) at the time. Thus Mill's philosophy of science largely reflects Hume's associationism; Reichenbach's philosophy of science reflects Viennese positivism with its conventionalism, its tendency to identify (or confuse) meaning and evidence, and its sharp dichotomy between 'the empirical facts' and 'the rules of the language'; and (coming up to the present time) Toulmin's philosophy of science is an attempt to give an account of what scientists do which is consonant with the linguistic philosophy of Wittgenstein. For this reason, errors in general philosophy can have a far-reaching effect on the philosophy of science. The confusion of meaning with evidence is one such error whose effects are well known: it is the contention of the present paper that overworking of the analytic-synthetic distinction is another root of what is most distorted in the writings of conventional philosophers of science."

"I have not attempted in these papers to put forward any grand view of the nature of philosophy; nor do I have any such grand view to put forward if I would. It will be obvious that I do not agree with those who see philosophy as the history of 'howlers', and progress in philosophy as the debunking of howlers. It will also be obvious that I do not agree with those who see philosophy as the enterprise of putting forward a priori truths about the real world (since, for one thing, there are no a priori truths, in my view). I see philosophy as a field which has certain central questions, for example, the relation between thought and reality, and, to mention some questions about which I have not written, the relation between freedom and responsibility, and the nature of the good life. It seems obvious that in dealing with these questions philosophers have formulated rival research programs, that they have put forward general hypotheses, and that philosophers within each major research program have modified their hypotheses by trial and error, even if they sometimes refuse to admit that that is what they are doing. To that extent philosophy is a 'science'. To argue about whether philosophy is a science in any more serious sense seems to me to be hardly a useful occupation. The important thing is that in spite of the stereotypes of science and philosophy that have become blinkers inhibiting the view of laymen, scientists, and philosophers, science and philosophy are interdependent activities; philosophers have always found it essential to draw upon the scientific knowledge of the time, and scientists have always found it essential to do a certain amount of philosophy in their very scientific work, even if they denied that that was what they were doing. It does not seem to me important to decide whether science is philosophy or philosophy is science as long as one has a conception of both that makes both essential to a responsible view of the real world and of man's place in it."