Logicians From The United States

"It struck me only recently that nineteenth century painters must have believed that visual truth was defined by photography, however alien to human vision what the camera reproduced often was. A good example of this would have been Eadweard Muybridge’s photographs of horses in motion. Painters decided that Muybridge’s images showed what horses really look like when they run, and in effect copied Muybridge’s photographs in their paintings of horses, even though that is not at all the way we see horses when they run. We really don’t see animals move the way Muybridge shows them moving, or else there would have been no need for the photographs in the first place: Muybridge hit upon his awkward but seemingly authoritative experiments that were really designed to answer such questions as whether all four of a horse’s hooves ever touch the ground at the same time—in other words, phenomena the human eye could not perceive."

"Any movement can be a dance movement and hence achieve the dreamlike. The same may be true of acting, as when, for example, an actress serves cocktails that are actually glasses filled with just water. To taste the tasteless is a kind of bad dream. It is not possible to catalog all the different ways artists have found to dream-ify. I’ll take a flier at Michelangelo’s masterpiece, the great decoration of the Sistine Chapel’s vault, with the scenes of a narrative in which, when I first saw it, figures move in and out of an enveloping dark."

"As a philosopher, I would cherish an argument which demonstrates that the mind cannot be mapped onto the brain any better than the Sistine ceiling can be mapped onto the brushstrokes—and that Eliminativists are as misled as Colalucci. It would be great if the analogy itself were accepted, even if we did not know where to go from there."

"The body that feels thirst and hunger, passion, desire, and love. The body that we understand when we read the ancients describing men in battle, men and women in love and in grief. The body, I would say, that our artistic tradition dealt with so gloriously for so many centuries, and somewhat less gloriously in a certain kind of performance art today."

"My thought is that if some art is imitation and some art is not, neither term belongs to the definition of art as philosophically understood. A property is part of the definition only if it belongs to every work of art there is."

"It is true that art today is pluralistic. Pluralism was noticed by certain followers of Ludwig Wittgenstein. What makes art so powerful a force as it appears to be in song and story is due to what makes it art to begin with. There is really nothing like it when it comes to stirring the spirit."

"There were limits to what art—composed of such genres as portraiture, landscape, still life, and historical painting (the latter of which, in royal academies, enjoyed the highest esteem)—could do to show movement."

"Thanks to Descartes and Plato, I will define art as “wakeful dreams.” One wants to explain the universality of art. My sense is that everyone, everywhere, dreams. Usually this requires that we sleep. But wakeful dreams require of us that we be awake. Dreams are made up of appearances, but they have to be appearances of things in their world. True, the different arts in the encyclopedic museum are made by different cultures."

"To him mathematics was a unified subject that one could master broadly. He had a deep understanding of most areas, and he taught advanced courses in logic, analysis, differential equations, algebra, topology, Lie theory, and number theory on a regular basis. He felt that good mathematics should be easy to understand and that it is always based on simple ideas once you get to the bottom of the issue. This attitude extended to a strong belief that the well-recognized unsolved problems in mathematics are the heart of the subject and have clear and transparent solutions once the right new ideas and viewpoints are found. This belief gave him the courage to work on notoriously difficult problems throughout his career."

"When the Löwenheim-Skolem theorem is applied to particular formal systems, we obtain as special cases: Every group, field, ordered field, etc., has a countable subsystem of the same type. A more spectacular result follows from applying the theorem to set theory (a system which we shall later formalize): There is a countable collection of sets, such that if restrict the membership relation to these sets alone, they form a model for set theory (more precisely all the true statements of set theory are true in this model). In particular, within this model which we may denote by M, there must be an uncountable set. This paradox, that a countable model can contain an uncountable set, is explained by noting that to say a set is uncountable merely asserts the nonexistence of a one-one mapping of the set with the set of integers. The "uncountable" set in M set actually has only countably many members in M, but there is no one-one correspondence \underline {within} M of this set with the set of integers."

"In 1963 P. J. Cohen completed Gödel's linguistic attack on set theory by introducing the immensely valuable, syntactic notion of forcing, and by using it to demonstrate that the axioms of set theory were not powerful enough to prove Cantor's continuum hypothesis. Thus the presently existing axioms of set theory leave the most celebrated hypothesis of set theory shrouded in uncertainty."

"To the average mathematician who merely wants to know that his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency."

"However, in all honesty, I must say that one must essentially forget that all proofs are transcribed in this formal language. In order to think productively, one must use all the intuitive and informal methods of reasoning at one's disposal. At the very end one must check that no errors have been committed; but in practice set theory is treated as any other branch of mathematics. The reason that we can do this is that we will never speak about proofs but only about models."

"The theorem of Löwenheim–Skolem was the first truly important discovery about formal systems in general, and it remains probably the most basic. It is not a negative result at all, but plays an important role in many situations. For example, in Gödel's proof of the consistency of the Continuum Hypothesis, the fact that the hypothesis holds in the universe of constructible sets is essentially an application of the theorem."

"It is now known that the truth or falsity of the continuum hypothesis and other related conjectures cannot be determined by set theory as we know it today. This state of affairs regarding a classical and presumably well-posed problem must certainly appear rather unsatisfactory to the average mathematician. One is tempted to look more closely and perhaps more critically at the foundations of mathematics. Although our present "Cantorian" mathematics is highly successful in its treatment of abstractions, one must not overlook the fact that from the very beginning the use of infinite processes was regarded with suspicion by many people."

"The object of mathematics is to discover "true" theorems. We shall use the term "valid" to describe statements formed according to certain rules and then shall discuss how this notion compares with the intuitive idea of "true"."

"Problem: Name a book that combines mathematical history, philosophy, more than a whiff of theology, personal palaver, and brilliant insights, along with evidence of a Borges-like imagination, eyebrow-raising mathematical constructions, breathtaking excitement, grandiose ruminations, and some bosh. Solution: The book under review. ...Ω leads me to turn 180 degrees away from these classical philosophical questions—discussed ad nauseam and with diminishing profit—of why mathematics is true, whether its objects and constructs have ontological validity, whether it is the only mode of inference, what its limitations are, whether it is the unique language in which theoretical physics must be formulated. If we could focus instead on mathematical pragmatics—why mathematics throughout the millennia has been useful or deleterious to society—then I believe that we might be able to illuminate an aspect of mathematics that is often ignored: mathematics as a enterprise, for that is surely what it is."

"I'm interested in the computer as a new idea, a new and fundamental philosophical concept that changes mathematics, that solves old problems better and suggests new problems, that changes our way of thinking and helps us to understand things better, that gives us radically new insights..."

"[A]ccording to Weyl, complexity is essential in understanding the concept of a law of nature. If laws of nature may be arbitrarily complex, he argued, the very concept... becomes vacuous. What difference would remain... if the laws meant to explain them were as complex as the phenomena they are meant to explain? Laws of nature must be simple."

"Mathematicians are coming up with s that they use in their actual mathematical research. And these, like , I think is the name of one of them ... these are actually like s that have been engineered in a way that they can actually be used by working mathematicians to check the work they're doing."

"...Once you entomb mathematics in an artificial language à la Hilbert, once you set up a completely formal axiomatic system, then you can forget that it has any meaning and just look at it as a game that you play with marks on paper that enable you to deduce theorems from axioms. You can forget about the meaning of the game, the game of mathematical reasoning, it's just combinatorial play with symbols! There are certain rules, and you can study these rules and forget that they have any meaning!"

"At first it might seem that quantum mechanics (QM), which began with Einstein's photon as the explanation for the photoelectric effect in 1905, goes further in the direction of discreteness. But the wave-particle duality discovered by de Broglie in 1925 is at the heart of QM, which means that this theory is profoundly ambiguous regarding the question of discreteness vs. continuity. QM can have its cake and eat it too, because discreteness is modeled via standing waves (eigenfunctions) in a continuous medium."

"Are there mathematical propositions for which there is a considerable amount of computational evidence, evidence that is so persuasive that a physicist would regard them as experimentally verified?"

"Why do I think that Turing's paper "On computable numbers" is so important? Well, in my opinion it's a paper on epistemology, because we only understand something if we can program it, as I will explain in more detail later. And it's a paper on physics, because what we can actually compute depends on the laws of physics in our particular universe and distinguishes it from other possible universes. And it's a paper on ontology, because it shows that some real numbers are uncomputable, which I shall argue calls into question their very existence, their mathematical and physical existence."

"Grace is not opposed to effort. It is opposed to earning. Effort is action. Earning is attitude. You have never seen people more active than those who have been set on fire by the grace of God."

"Morgenbesser described ethics as entailing “ought implies can” while in Jewish ethics “can implies don’t.”"

"When challenged why he had written so little, he fired back: "Moses wrote one book. Then what did he do?""

"Morgenbesser in response to B.F. Skinner: "Are you telling me it's wrong to anthropomorphize people?" (quoted by Daniel Dennett)"

"When asked his opinion of pragmatism, Morgenbesser replied "It's all very well in theory but it doesn't work in practice.""

"A proposed response to Heidegger's ontological query "Why is there something rather than nothing?" – "And if there were nothing? You'd still be complaining!""

"During a lecture, the Oxford linguistic philosopher J. L. Austin made the claim that although a in English implies a positive meaning, there is no language in which a double positive implies a negative. Morgenbesser responded in a dismissive tone, "Yeah, yeah.""

"A few weeks before his death, he asked another Columbia philosopher, David Albert, about God. "Why is God making me suffer so much?" he asked. "Just because I don't believe in him?""

"On the : Morgenbesser, ordering dessert, is told by the waitress that he can choose between apple pie and blueberry pie. He orders the apple pie. Shortly thereafter, the waitress comes back and says that cherry pie is also an option; Morgenbesser says "In that case I'll have the blueberry pie.""

"A student once interrupted him and said, "I just don't understand." He responded, "Why should you have the advantage over me?""

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

"Asked to prove a questioner's existence, Morgenbesser shot back, "Who's asking?""

"Morgenbesser said the following of George Santayana: “There’s a guy who asserted both p and not-p, and then drew out all the consequences…”"

"Interrogated by a student whether he agreed with Chairman Mao’s view that a statement can be both true and false at the same time, Morgenbesser replied “Well, I do and I don’t.”"

"During campus protests of the 1960s, Sidney Morgenbesser was hit over the head by police. When asked whether he had been treated unfairly or unjustly, he responded that it was "unfair but not unjust. It was unfair because they hit me over the head, but not unjust because they hit everyone else over the head.”"

"Morgenbesser was leaving a subway station in New York City and put his pipe in his mouth as he was ascending the steps. A police officer told him that there was no smoking on the subway. Morgenbesser pointed out that he was leaving the subway, not entering it, and hadn't lit up yet anyway. The cop again said that smoking was not allowed in the subway, and Morgenbesser repeated his comment. The cop said, "If I let you do it, I'd have to let everyone do it." To which Morgenbesser, in a much misunderstood line, retorted: "Who do you think you are, Kant?" He was then hauled off to the police station, where The Categorical Imperative had to be explained to the police officers. (Kant, as pronounced in American English, sounds similar to cunt, which is what he was mistaken for having said)."

"Logic is the last scientific ingredient of Philosophy; its extraction leaves behind only a confusion of non-scientific, pseudo problems."

"Science is a system of statements based on direct experience, and controlled by experimental verification. Verification in science is not, however, of single statements but of the entire system or a sub-system of such statements."

"In order to determine whether or not one sentence is a consequence of another, no reference need be made to the meaning of the sentences. The mere statement of the truth-values is certainly too little; but the statement of the meaning is, on the other hand, too much. It is sufficient that the syntactical design of the sentences be given."

"Thinking is most mysterious, and by far the greatest light upon it that we have is thrown by the study of language. This study shows that the forms of a person's thoughts are controlled by inexorable laws of pattern of which he is unconscious. These patterns are the unperceived intricate systematizations of his own language--shown readily enough by a candid comparison and contrast with other languages, especially those of a different linguistic family. His thinking itself is in a language—in English, in Sanskrit, in Chinese. And every language is a vast pattern-system, different from others, in which are culturally ordained the forms and categories by which the personality not only communicates, but also analyzes nature, notices or neglects types of relationship and phenomena, channels his reasoning, and builds the house of his consciousness."

"Whorf became increasingly concerned about the supposed conflict between science and religion... He wrote a 130,000-word manuscript on the subject, described as a book of religious philosophy in the form of a novel... Completed in 1925, [it] was submitted to several publishers and as promptly rejected by them... Another, briefer manuscript prepared about this time [was]... “Why I have discarded evolution.” An eminent geneticist to whom it was submitted for comment made a very courteous reply, starting with the admission that, although the manuscript at first appeared to be the work of a crank, its skill and perceptiveness soon marked it as otherwise, but continuing with a point-by-point rebuttal of Whorf’s arguments... Whorf’s reading led him to believe that the key to the apparent discrepancy between the Biblical and the scientific accounts of cosmology and evolution might lie in a penetrating linguistic exegesis of the Old Testament. For this reason, in 1924 he turned his mind to the study of Hebrew."

"By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language -- the systematic statement of the formal rules which govern it together with the development of the consequences which follow from these rules. A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for examples, the words) or to the sense of the expressions (e.g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are constructed."

"According to this view, the sentences of metaphysics are pseudo-sentences which on logical analysis are proved to be either empty phrases or phrases which violate the rules of syntax. Of the so-called philosophical problems, the only questions which have any meaning are those of the logic of science. To share this view is to substitute logical syntax for philosophy."

"The main purpose of this book is the development of a new method for the semantical analysis of meaning, that is, a new method for analyzing and describing the meanings of linguistic expressions. This method, called the method of extension and intension, is developed by modifying and extending certain customary concepts, especially those of class and property. The method will be contrasted with various other semantical methods used in traditional philosophy or by contemporary authors."

"Western culture has made, through language, a provisional analysis of reality and, without correctives, holds resolutely to that analysis as final. The only correctives lie in all those other tongues which by aeons of independent evolution have arrived at different, but equally logical, provisional analyses."

"It needs but half an eye to see in these latter days that science, the Grand Revelator of modern Western culture, has reached, without having intended to, a frontier. Either it must bury its dead, close its ranks, and go forward into a landscape of increasing strangeness, replete with things shocking to a culture-trammeled understanding, or it must become, in Claude Houghton's expressive phrase, the plagiarist of its own past. The frontier was foreseen in principle very long ago, and given a name that has descended to our day clouded with myth. That name is Babel. For science's long and heroic effort to be strictly factual has at last brought it into entanglement with the unsuspected facts of the linguistic order. These facts the older classical science had never admitted, confronted, or understood as facts. Instead they had entered its house by the back door and had been taken for the substance of Reason itself."