Mathematicians From Poland

"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."

"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..."

"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."

"These are the moments when the powerful mind or the forceful character feels the ferment of the times, when his thoughts quicken, and when he can inject into the uncertainties of others the creative ideas which will strengthen them with purpose. At such a moment the man who can direct others, in thought or in action, can remake the world."

"The air in a man's lungs contains 10,000,000,000,000,000,000,000 atoms, so that sooner or later every one of us breathes an atom that has been breathed before by anyone you can think of who has ever lived — Michelangelo or George Washington or Moses."

"I grew up to be indifferent to the distinction between literature and science, which in my teens were simply two languages for experience that I learned together."

"The most modest research worker at his bench, pushing a probe into a neuron to measure the electric response when a light is flashed, is enmeshed in a huge and intertwined network of theories that he carries into his work from the whole field of science, all the way from Ohm’s law to Avogadro's number. He is not alone; he is sustained and held and in some sense imprisoned by the state of scientific theory in every branch. And what he finds is not a single fact either: it adds a thread to the network, ties a knot here and another there, and by these connections at once binds and enlarges the whole system."

"The progress of science is the discovery at each step of a new order which gives unity to what had long seemed unlike. Faraday did this when he closed the link between electricity and magnetism. Clerk Maxwell did it when he linked both with light. Einstein linked time with space, mass with energy, and the path of light past the sun with the flight of a bullet; and spent his dying years in trying to add to these likenesses another, which would find a single imaginative order between the equations between Clerk Maxwell and his own geometry of gravitation When Coleridge tried to define beauty, he returned always to one deep thought: beauty he said, is "unity in variety." Science is nothing else than the search to discover unity in the wild variety of nature — or more exactly, in the variety of our experience."

"It has been one of the most destructive modern prejudices that art and science are different and somehow incompatible interests. We have fallen into the habit of opposing the artistic to the scientific tempers; we even identify them with a creative and a critical approach."

"The sneer that science is only critical came from others. It was made by the timid and laboured artists of the nineties in order that they might by comparison appear to be creative and intuitive. Yet this finesse could not hide their own knowledge that the best minds were already being drawn to the more adventurous practice of the new sciences."

"Science is not a special sense. It is as wide as the literal meaning of its name: knowledge."

"Many people affect to believe that science has progressively strangled the arts, or distorted them into some unpleasant “modern” form; and therefore that the arts can be revived only by throwing over science. Often of course this is merely an elderly sentiment in favour of the art of our younger days, and the real scapegoat is not science but change."

"Science changes our values in two ways. It injects new ideas into the familiar culture. And it subjects it to the pressure of technical change, in the way I have just been describing, until the whole basis of our culture has imperceptibly been remade."

"Science and the arts shared the same language at the Restoration. They no longer seem to do so today. But the reason is that they share the same silence: they lack the same language. And it is the business of each of us to make that one universal language which alone can unite art and science, and layman and scientist, in a common understanding."

"There are three creative ideas which, each in its turn, have been central to science. They are the idea of order, the idea of causes, and the idea of chance."

"Unhappily, common sense has no recorded history."

"Newton was born during Cromwell’s revolution in the troubled 1640s; he was eighteen at the Restoration in 1660; and he published the Principia during the intrigues which ended by bringing William of Orange to England in the revolution of 1688. These are the moments when the powerful or the forceful character feels the ferment of the times, when his thoughts quicken and when he can inject into the uncertainties of others the creative ideas which will strengthen them with purpose. At such a moment the man who can direct others, in thought or in action, can remake the world."

"There has never been another moment in English history to equal the promise of that moment in the 1660’s when the Royal Society was formally founded. And though it was less dramatic elsewhere, it was a high moment throughout Europe."

"About 1660 therefore, Europe was in the course of a great revolution in thought. This was the Scientific Revolution, and it reached into all forms of culture. We sometimes speak as if science has step by step squeezed other interests out of our culture, and is slowly strangling the traditional ways of thinking. Nothing of the kind. The Scientific Revolution in the seventeenth century was a universal revolution. Indeed it could not have begun unless there had already been a deep change in the attitude to everything natural and super-natural among thoughtful men."

"The great flood was the seventeenth century. That was the time of change, the hanging moment of instability in which men like Cromwell and Newton could remake the world."

"The whole structure of thought in the Middle Ages is one which we find hard to grasp today. It was an orderly structure, but the principles by which it was ordered seem to us now outlandish and meaningless."

"Nature does not provide identical objects; on the contrary, these are always human creations. What nature provides is a tree full of apples which are all recognisably alike and yet are not identical, small apples and large ones, red ones and pale ones, apples with maggots and apples without. To make a statement about all these apples together, and about crab-apples, Orange Pippins, and Beauties of Bath, is the whole basis of reasoning."

"This ability to order things into likes and unlikes is, I think, the foundation of human thought."

"On the one hand, all science, and indeed all thinking starts from and rests upon notions of order; what marks the Middle Ages is that their order was always a hierarchy. And on the other hand what marks the scientific view is not that it turned to the mechanism of causes, but that it saw the world as a mechanism at all—a machine of events."

"The Scientific Revolution revolution was a change from a world of things ordered according to their ideal nature, to a world of events running in a steady mechanism of before and after."

"Of these massive achievements I shall single out two. One is Newton’s working out of the concept of the cause, by making it over from its scholastic form in, say, St. Thomas Aquinas, to the modern form which now seems so obvious to us. This is one theme in this chapter. But I shall go to it by way of a related achievement, and to my mind one which is as remarkable: the marriage of the logical with the empirical method."

"In order to act in a scientific manner, in order to act in a human manner at all, two things are necessary: fact and thought. Science does not consist only a finding the facts; nor is it enough only to think, however rationally. The processes of science are characteristic of human action in that they move by the union of empirical fact and rational thought, in a way which cannot be disentangled."

"In Whitehead’s view, the Middle Ages were quite as logical in their speculations about nature as we are. It is not as rationalists that we have the advantage of them; our material successes stem from joining to their logic a ruthless appeal, at each bold deductive step, back to the hard empirical facts."

"The outlook before the Scientific Revolution was content with scholastic logic applied to a nature of hierarchies. The Scientific Revolution ended that: it linked the rational and the empirical, thought and fact, theory and practical experiment. And this has remained the content of science ever since."

"The [eighteenth] century settled down in two camps of Philistines: the literary Philistines and, largely to spite them, the scientific Philistines. It was the beginning of the mistaken opposition between them from which we still suffer."

"Order is the selection of one set of appearances rather than another because it gives a better sense of the reality behind the appearances."

"We cannot change our character, we can only enlarge it. If we are wise, then we go on learning all through life, and go on fitting what is new to what we have learnt before, piece by piece."

"The purpose of science is to describe the world in an orderly scheme or language which will help us to look ahead. We want to forecast what we can of the future behaviour of the world; particularly we want to forecast how it would behave under several alternative actions of our own between which we are usually trying to choose."

"Science is a way of describing reality; it is therefore limited by the limits of observation; and it asserts nothing which is outside observation. Anything else is not science; it is scholastics."

"Of course, everyone is free to prefer his favourite article of faith to the scientific, that is the empirical method. But do not let us imagine that his faith is then anything except a piece of comfortable and customary superstition. To try to make a nice distinction between what science can predict and what is somehow supernaturally determined is a piece of elegant but really quite shameless self-deception. Science is a practical study of what can be observed, and the prediction from that of what will be observed. To say that causes are somehow getting under this observable world, when anything under it is essentially unobservable, is neither helpful nor meaningful; it is just a piece of faithful comfort. We might as well say that the electrons are really pushed about by blue fairies with red noses who know exactly what they are doing, only it happens that every time we look in their direction these fairies instantly hide. If they are essentially unobservable, beyond all hope of future unravelling, then it simply does not make sense to bring them into any system, logical, metaphysical, or even religious."

"Life is not an examination; we do not get marks for the steps; what matters is getting the right answer. So it is perfectly possible to base a system of prediction on no principle except trying to get the right answer. This is exactly what all plants and animals do. The bat avoids obstacles by shouting at them that shrill cry just beyond my hearing, and then listens for the echo. Whatever system it has for translating the echo into a prediction it has found by evolution, and evolution has found it by trial and error."

"The laws of science have two functions, to be true and to be helpful; probably each of these functions includes the other. If the statistical law does both, that is all that can be asked of it. We may persuade ourselves that it is intellectually less satisfying than a causal law, and fails somehow to give us the same feeling of understanding the process of nature. But this is an illusion of habit."

"Science as we know it is indeed a creation of the last three hundred years. It has been made in and by the world that took its settled shape about 1660, when Europe at last shook off the long nightmare of religious wars and settled into a life of inquisitive trade and industry. Science is embodied in those new societies; it has been made by them and has helped to make them."

"The mastery and the greatness of science rests in the end on this, that here the rational and the empirical are knotted together. Science is fact and thought giving strength to one another."

"We do not construct the world from our experiences; we are aware of the world in our experiences."

"We are troubled by a two-sidedness in our own behaviour, where one side is what we have long been taught to value, and the other is worldly success. We are faced every day with actions of which our own code of conduct makes us ashamed, but which we find compelling if we are to battle with the hard facts of society. We do not consciously blame science for this rift until it throws out some unavoidable challenge, such as in our time has been set by the atomic bomb. But that sharp issue is merely a symbol. Beyond all our actions stands the larger shadow: how are we to choose between that which we have been taught to think right and something else which manifestly succeeds?"