Mathematicians From The United Kingdom

"The symbol of the University is the iron statue outside the Rathskeller of a barefoot goose girl that every student kisses at graduation. The University is a Mecca to which students come with something less than perfect faith. It is important that students bring a certain ragamuffin, barefoot irreverence to their studies; they are not here to worship what is known but to question it."

"The Principle of Uncertainty is a bad name. In science, or outside of it, we are not uncertain; our knowledge is merely confined, within a certain tolerance. We should call it the Principle of Tolerance. And I propose that name in two senses. First, in the engineering sense: Science has progressed, step by step, the most successful enterprise in the ascent of man, because it has understood that the exchange of information between man and nature, and man and man, can only take place with a certain tolerance. But second, I also use the word, passionately, about the real world. All knowledge – all information between human beings – can only be exchanged within a play of tolerance. And that is true whether the exchange is in science, or in literature, or in religion, or in politics, or in any form of thought that aspires to dogma. It's a major tragedy of my lifetime and yours that scientists were refining, to the most exquisite precision, the Principle of Tolerance – and turning their backs on the fact that all around them, tolerance was crashing to the ground beyond repair. The Principle of Uncertainty or, in my phrase, the Principle of Tolerance, fixed once for all the realization that all knowledge is limited. It is an irony of history that at the very time when this was being worked out, there should rise, under Hitler in Germany and other tyrants elsewhere, a counter-conception: a principle of monstrous certainty. When the future looks back on the 1930's, it will think of them as a crucial confrontation of culture as I have been expounding it – the ascent of man against the throwback to the despots' belief that they have absolute certainty."

"It's said that science will dehumanize people and turn them into numbers. That's false, tragically false. Look for yourself. This is the concentration camp and crematorium at Auschwitz. This is where people were turned into numbers. Into this pond were flushed the ashes of some four million people. And that was not done by gas. It was done by arrogance, it was done by dogma, it was done by ignorance. When people believe that they have absolute knowledge, with no test in reality, this is how they behave. This is what men do when they aspire to the knowledge of gods.Science is a very human form of knowledge. We are always at the brink of the known; we always feel forward for what is to be hoped. Every judgment in science stands on the edge of error and is personal. Science is a tribute to what we can know although we are fallible. In the end, the words were said by Oliver Cromwell: "I beseech you in the bowels of Christ: Think it possible you may be mistaken."I owe it as a scientist to my friend Leo Szilard, I owe it as a human being to the many members of my family who died here, to stand here as a survivor and a witness. We have to cure ourselves of the itch for absolute knowledge and power. We have to close the distance between the push-button order and the human act. We have to touch people."

"Fifty years from now, if an understanding of man's origins, his evolution, his history, his progress is not in the common place of the school books, we shall not exist."

"And I am infinitely saddened to find myself suddenly surrounded in the west by a sense of terrible loss of nerve, a retreat from knowledge into—into what? Into Zen Buddhism; into falsely profound questions about, Are we not really just animals at bottom; into extra-sensory perception and mystery. They do not lie along the line of what we are able to know if we devote ourselves to it: an understanding of man himself. We are nature’s unique experiment to make the rational intelligence prove itself sounder than the reflex. Knowledge is our destiny. Self-knowledge, at last bringing together the experience of the arts and the explanations of science, waits ahead of us."

"The world can only be grasped by action, not by contemplation."

"We are all afraid - for our confidence, for the future, for the world. That is the nature of the human imagination. Yet every man, every civilization, has gone forward because of its engagement with what it has set itself to do. The personal commitment of a man to his skill, the intellectual commitment and the emotional commitment working together as one, has made the Ascent of Man."

"Since the word "knowledge" occurs in my general title... I am going to be talking about epistemology, although I prefer to use the eighteenth-century, indeed, medieval phrase, "natural philosophy." ...that enterprise of the human mind which attempts to trace lawfulness to nature, dead and living, but which is not directed to specific inquiries into how this or that law works. Philosophy in the sense in which I practice it, natural philosophy, is concerned with lawfulness rather than with laws and the general nature of laws rather than with the specific structure of this or that law. Natural philosophy was one of the three topics (moral philosophy and metaphysical philosophy were the others) to which one graduated in medieval universities after having studied the seven liberal arts. I believe that we need to review the whole of our natural philosophy in the light of scientific knowledge that has arisen in the last fifty years."

"What we really mean by free will... is the visualizing of alternatives and making a choice between them. ...the central problem of human consciousness depends on this ability to imagine."

"I believe that the world is totally connected: that is to say, that there are no events anywhere in the universe which are not tied to every other event in the universe. ...It is... an essential part of the methodology of science to divide the world for any experiment into... relevant and... irrelevant. We make a cut. We put the experiment... into a box. ...the moment we do that, we do violence to the connections ...I get a set of answers which I try to decode in this context. ...I am certainly not going to get the world right, because the basic assumption that I have made about the world is a lie. ...it is bound to give me only an approximation to what goes inside the fence. Therefore, when we practice science (and this is true of all our experience) we are always decoding a part of nature which is not complete. We simply cannot get out of our own finiteness."

"Let me close by reminding you of what Newton actually did on the day that he conceived G = k \frac{mm'}{r^2}. ...Newton did not have any subsidies, grants, funds, Secret Service money. But he had the moon. He said, "... I cannot throw a ball round the world, but let me picture the moon as if it were a ball which has been flung around the world... How long will it take to go round the world?" ...He knew the value of gravity at the earth's surface ...but he did not know the value of the earth's gravity for the moon. He said, "Let us suppose that it is given by an inverse square law. Now, how long will it take the moon to go around?" It comes out at twenty-eight days. As Newton said, "They agreed pretty nearly.""

"Progress is the exploration of our own error. Evolution is a consolidation of what have always begun as errors. And errors are of two kinds: errors that turn out to be true and errors that turn out to be false (which are most of them). But they both have the same character of being an imaginative speculation. ...it seems to me terribly important to say this in an age in which most nonscientists are feeling a kind of loss of nerve. ...by the time science becomes a closed—that is, computerizable—project, it is not science anymore. It is not in the area of the exploration of errors."

"[A]ll our symbols have the same purpose; words are merely the symbols we use most commonly. The function of words in human thought is to stand for things which are not present to the senses, and allow the mind to manipulate them—things, concepts, ideas, everything that does not have a physical reality in front of us now."

"The language of ideas creates a different universe: a universe which has multiplied the monkey's vocabulary of forty words to the million words in the English dictionary."

"Human beings can imagine situations which are different from those in front of their eyes... because they make and hold in their minds images for absent things."

"When a child... discovers his own imagination, he suddenly walks into a new life. ...seeing situations that do not exist. ...in part as fantasy, and in part as a quite rational exploration of future experiences. ...play ...frolics in the fantasy world, and it experiments in the rational world... They project themselves into all worlds, possible and impossible, and discover for themselves the knife-edge boundary between them."

"The ability... to experiment with imaginary situations, gives man a freedom... the pleasure in trying out and exploring imaginary situations. A child's play is concerned with this pleasure; and so is much of art, and much of science... [P]ure science... is a form of play, in this sense."

"He first became familiar to the British public through appearances on the BBC television version of The Brains Trust in the late 1950s, but is better known for his... series The Ascent of Man (1973). This was an inspiration for Carl Sagan to make Cosmos in 1980. During the making of The Ascent of Man he was interviewed by Michael Parkinson, and Bronowski's description of a visit to Auschwitz—he had lost many family members during the Nazi era—was described by Parkinson as one of his most memorable interviews. ...Bronowski died of a heart attack in East Hampton, New York a year after The Ascent of Man was completed, and was buried in the western side of London's ..."

"[[Richard von Mises|[V]on Mises]] and raised the objection that in Reichenbach's theory a theory was said to be "valid with a probability of 70%," if 70 per cent of the conclusions of the theory were confirmed by experiment. However, we know... a theory which is in disagreement with experience in 30% of... experiments is called "false" or "wrong." We come to the same conclusion if we apply Carnap's "." An attempt to avoid these difficulties and to advance a radically different approach... was made by Jacob Bronowski. His purpose was to formalize the criterion for the validity of a theory that was advocated by men like John Frederick Herschel and William Whewell. ...these scientists and philosophers saw the main achievement of a theory in its unifying and simplifying power. The simpler the theory... the more probable the theory. If the theory consists in a complete enumeration of all observable facts, the "theory" would have a very high probability... However, von Mises and Bronowski rejected this... If we have two theories which yield the same observable facts, the scientist prefers the... more economical or just simpler. Bronowski compares the scientific theory with a code... We prefer the code which is more practical, more efficient. ...to improve the code, we try systematically... "to break down the code into its constituent symbols and their laws of arrangement." ...If we break down [chemical] elements into... elementary particles (protons, neutrons, and electrons) and the forces acting between them, we have a code which describes... the interaction between hydrogen and oxygen... from which can be derived much more information than from any theory in which "oxygen" and "hydrogen"...occur as primitive symbols. ...Bronowski calls a theory the more probable, the more the code... is broken down into constituent symbols and laws of arrangement."

"Every acceptance of a debatable theory is due to a compromise between Reichenbach's and Bronowski's criteria: agreement with facts and efficiency as a code."

"The criteria of Reichenbach and Carnap, which are based, like John Stuart Mill's inductive logic, upon agreement with observations, have to be complemented by the criterion of economy and simplicity which was advanced in the history of science by men like William Ockham, Isaac Newton, and Ernst Mach. In out twentieth century, the importance of criteria other than mere agreement with observation was stressed by von Mises and Bronowski."

"The ashes of millions of people were flushed into the pond at Auschwitz, but on that sunny day... I stood at its edge... twenty-four years earlier I sat in front of our television set and watched... Bronowski... as a scientist, a human being, a survivor and a witness at the edge of this pond. Many members of his family had died at Auschwitz. He reminded us that it is said that science will turn people into numbers and told... passionately that this is '...tragically false', and that it was here... that people were turned into numbers and murdered not by the gas... but by the arrogance, the dogma and ignorance... and by the murderers'... belief that they possessed absolute knowledge... never—unlike scientists—tested... against objective reality. ...Bronowski ...reminded us that science is a very human form of knowledge in which every judgement stands on the edge of error and is personal. Quoting the words of Oliver Cromwell—'I beseech you, in the bowels of Christ, think it possible you may be mistaken'—he reached into the water... telling us that we must close the distance between the push-button order and the human act... ending with the words: 'We have to touch people.' ...Bronowski's active defiance of failings such as despair, failing of nerve, fashionable pessimism and irresponsibility had made a permanent and indelible impression on me..."

"Here is the ultimate hope of saving ourselves from extinction. We must learn to understand that the contents of all knowledge is empirical; that its test is whether it works; and we must learn to act on that understanding in the world as well as in the laboratory."

"Science is a great many things, and I have called them a great many names; but in the end they all return to this: science is the acceptance of what works and the rejection of what does not. That needs more courage than we might think."

"A good prediction is one which defines its area of uncertainty; a bad prediction ignores it."

"If any ideas have a claim to be called creative, because they have created something, then certainly it is the ideas of science."