Mathematicians From The United Kingdom

"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."

"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 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."

"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."

"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 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."

"[T]he most important images for human beings are simply words, which are abstract symbols. ...[E]volution has greatly enlarged the front lobes of the human brain, which govern the sense of the past and the future; and... they are probably the seat of our other images."

"I am using the word image in a wide meaning, which does not restrict it to the mind's eye as a visual organ. An image in my usage is what Charles Pierce called a sign..."

"With the... symbolic memory we spell out the future—not one but many futures, which we weigh one against another."

"To imagine means to make images and to move them about inside one's head in new arrangements."

"To imagine is the characteristic act, not of the poet's mind, or the painter's, or the scientist's, but of the mind of man."

"A fact is discovered, a theory is invented; is any theory ever deep enough for it to be truly called a creation? Most scientists would answer: no! Science, they would say, engages only part of the mind - the rational intellect - but creation must engage the whole mind. Science demands none of that ground swell of emotion, none of the rich bottom of personality, which fills out the work of art..."

"Christopher Columbus discovered the West Indies, and Alexander Graham Bell invented the telephone. We do not call their achievements creations because they are not personal enough. The West Indies were there all the time; and as for the telephone, we feel that Bell's ingenious thought was somehow not fundamental. The groundwork was there, and if not Bell then someone else would have stumbled on the telephone almost as accidently as on the West Indies."

"The most remarkable discovery made by scientists is science itself. The discovery must be compared in importance with the invention of cave-painting and of writing. Like these earlier human creations, science is an attempt to control our surroundings by entering into them and understanding them from inside. And like them, science has surely made a critical step in human development which cannot be reversed. We cannot conceive a future society without science. I have used three words to describe these far - reaching changes : discovery, invention and creation. There are contexts in which one of these words is more appropriate than the others."

"There is today almost no scientific theory which was held when, say, the Industrial Revolution began about 1760. Most often today's theories flatly contradict those of 1760; many contradict those of 1900. In cosmology, in quantum mechanics, in genetics, in the social sciences, who now holds the beliefs that seemed firm sixty years ago? Yet the society of scientists has survived these changes without a revolution, and honors the men whose beliefs it no longer shares. No one has recanted abjectly at a trial before his colleagues. The whole structure of science has been changed and no one has been either disgraced or deposed. Through all the changes of science, the society of scientists is flexible and single-minded together, and evolves and rights itself. In the language of science, it is a stable society."

"Dissent is the mark of freedom."

"The force that makes the winter grow Its feathered hexagons of snow, and drives the bee to match at home Their calculated honeycomb, Is abacus and rose combined.An icy sweetness fills my mind, A sense that under thing and wing Lies, taut yet living, coiled, the spring."

"The painter's portrait and the physicist's explanation are both rooted in reality, but they have been changed by the painter or the physicist into something more subtly imagined than the photographic appearance of things."

"Nature is more subtle, more deeply intertwined and more strangely integrated than any of our pictures of her — than any of our errors. It is not merely that our pictures are not full enough; each of our pictures in the end turns out to be so basically mistaken that the marvel is that it worked at all."

"Tolerance among scientists cannot be based on indifference, it must be based on respect. Respect as a personal value implies, in any society, the public acknowledgements of justice and of due honor. These are values which to the layman seem most remote from any abstract study. Justice, honor, the respect of man for man: What, he asks, have these human values to do with science? [...] Those who think that science is ethically neutral confuse the findings of science, which are, with the activity of science, which is not."

"Has there ever been a society which has died of dissent? Several have died of conformity in our lifetime."

"There is a social injunction implied in the positivist and analyst methods. This social axiom is that We OUGHT to act in such a way that what IS true can be verified to be so."

"Positivists and analysts alike believe that the words is and ought belong to different worlds, so that sentences which are constructed with is usually have verifiable meaning, but sentences constructed with ought never have. This is because Ludwig Wittgenstein's unit, and Bertrand Russell's unit, is one man; all British empiricist philosophy is individualist. And it is of course clear that if the only criterion of true and false which a man accepts is that man's, then he has no base for social agreement. The question of how man ought to behave is a social question, which always involves several people; and if he accepts no evidence and no judgment except his own, he has no tools with which to frame an answer."

"No fact in the world is instant, infinitesimal and ultimate, a single mark. There are, I hold, no atomic facts. In the language of science, every fact is a field — a crisscross of implications, those that lead to it and those that lead from it. … We condense the laws around concepts. Science takes its coherence, its intellectual and imaginative strength together, from the concepts at which its laws cross, like knots in a mesh."

"In effect what Luther said in 1517 was that we may appeal to a demonstrable work of God, the Bible, to override any established authority. The Scientific Revolution begins when Nicolaus Copernicus implied the bolder proposition that there is another work of God to which we may appeal even beyond this: the great work of nature. No absolute statement is allowed to be out of reach of the test, that its consequence must conform to the facts of nature. The habit of testing and correcting the concept by its consequences in experience has been the spring within the movement of our civilization ever since. In science and in art and in self-knowledge we explore and move constantly by turning to the world of sense to ask, Is this so? This is the habit of truth, always minute yet always urgent, which for four hundred years has entered every action of ours; and has made our society and the value it sets on man."

"The symbol and the metaphor are as necessary to science as to poetry."

"Mass, time, magnetic moment, the unconscious: we have grown up with these symbolic concepts, so that we are startled to be told that man had once to create them for himself. He had indeed, and he has: for mass is not an intuition in the muscle, and time is not bought ready-made at the watchmaker's."

"We re-make nature by the act of discovery, in the poem or in the theorem. And the great poem and the deep theorem are new to every reader, and yet are his own experience, because he himself re-creates them."

"Science, like art, is not a copy of nature but a re-creation of her."

"The discoveries of science, the works of art are explorations — more, are explosions, of a hidden likeness. The discoverer or the artist presents in them two aspects of nature and fuses them into one. This is the act of creation, in which an original thought is born, and it is the same act in original science and original art."

"Can science heal that neurotic flaw in us? If science cannot, then nothing can. Let us stop pretending. There is no cure in high moral precepts. We have preached them too long to men who are forced to live how they can: that makes the strain which they have not been able to bear. We need an ethic which is moral and which works. It is often said that science has destroyed our values and put nothing in their place. What has really happened of course is that science has shown in harsh relief the division between our values and our world. We have not begun to let science get into our heads; where then was it supposed to create these values?"

"I have seen in my lifetime an abyss open in the human mind: a gulf between the endeavour to be man, and the relish in being brute. The scientist has indeed had a hand in this, and every other specialist too, with his prim detachment and his oracular airs. But of course, the large strain which has opened this fault is social. We have made men live in two halves, a Sunday half and a workday one. We have ordered them to love their neighbor and to turn the other cheek, in a society which has constantly compelled them to shoulder their neighbor aside and to turn their backs. So we have created a savage sense of failure which, as we now know to our cost, can be tapped with an ease which is frightening; and which can thrust up, with explosive force, a symbol to repeat to an unhappy people its most degrading dream."

"The images play out for us events which are not present in our senses, and... create the future—a future that... may never come to exist in that form."

"Almost everything that we do that is worth doing is done in the first place in the mind's eye."

"The richness of human life is that we have many lives, we live the events that do not happen (and some that cannot) as vividly as those that do, and if thereby we die a thousand deaths, that is the price we pay..."

"Imagination is the manipulation of images in one's head... the rational manipulation... as well as the literary and artistic manipulation."

"When a child begins to play games... he enters the gateway to reason and imagination together."

"[T]he human reason discovers new relations between things not by deduction, but by that unpredictable blend of speculation and insight... induction, which—like other forms of imagination—cannot be formalized."

"The strength of the imagination, its enriching power and excitement, lies in its interplay with reality—physical and emotional."

"All great scientists have used their imaginations freely, and let it ride them to outrageous conclusions without crying "Halt!""

"The symbol is the tool which gives man his power, and it is the same tool whether the symbols are images or words, mathematical signs or s."

"[S]ymbols have a reach and a roundness that goes beyond their literal and practical meaning. They are the rich concepts under which the mind gathers many particulars into one name, and many instances into one general induction."

"When a man counts one, two, three, he is not only doing mathematics, he is on the path to the mysticism of numbers in Pythagoras and Vitruvius and Kepler, to the Trinity and the signs of the Zodiac."

"[I]magination... is the faculty... the common root from which science and literature... spring and grow and flourish together. ...the great ages of science are the great ages of all the arts... [P]owerful minds have taken fire from one another... without asking... to tie their imagination to falling balls or a haunted island. ...When Galileo was looking through his telescope at the moon, Shakespeare was writing The Tempest and all Europe was in ferment, from Johannes Kepler to Peter Paul Rubens, and from the first table of logarithms by John Napier to the Authorized version of the Bible."

"Let me end with... man's ageless fantasy, to fly to the moon. ...Plutarch and Lucian, Ariosto and Ben Jonson wrote about it, before the days of Jules Verne and H. G. Wells and science fiction. The seventeenth century was heady with... fables about voyages to the moon. Kepler wrote one full of deep scientific ideas... wrote... '... wrote... The Discovery of a New World. They did not draw a line between science and fancy... they all tried to guess where... earth's gravity would stop. Only Kepler understood that gravity has no boundary, and put a law to it—... the wrong law. All this was a few years before Isaac Newton was born, and it was all in his head that day in 1666, when he... came to conceive... that the moon is like a ball... thrown so hard that it falls exactly as fast as the horizon... he went on to calculate how long... the distant moon would take to round the earth... [T]he imagination that day chimed with nature, and made a harmony."

"In every age there is a turning point, a new way of seeing and asserting the coherence of the world. It is frozen in the statues of Easter Island that put a stop to time—and in the medieval clocks of Europe that once also seemed to say the last word about the heavens for ever. Each culture tries to fix its visionary moment, when it was transformed by a new conception either of nature or of man. But in retrospect, what commands our attention as much are the continuities—the thoughts that run or recur from one civilization to another."

"Man is not the most majestic of the creatures; long before the mammals even, the dinosaurs were far more splendid. But he has what no other animal possesses: a jigsaw of faculties, which alone, over three thousand million years of life, made him creative. Every animal leaves traces of what he was. Man alone leaves traces of what he created."

"In a parched African landscape like this at Omo, man first put his foot to the ground. That seems a pedestrian way to begin the ascent of man."

"Of course it's tempting to close one's eyes to history, and instead speculate about the roots of war in some possible animal instinct: as if, like the tiger, we still had to kill to live, or, like the robin redbreast, to defend a nesting territory. But war, organized war, is not a human instinct. It is a highly planned and cooperative form of theft. And that form of theft began 10,000 years ago when the harvesters of wheat accumulated a surplus and the nomads rose out of the desert to rob them of what they themselves could not provide. The evidence for that, we saw, in the walled city of Jericho and its prehistoric tower... That is the beginning of war."

"One aim of the physical sciences has been to give an exact picture of the material world. One achievement of physics in the twentieth century has been to prove that that aim is unattainable. There is no absolute knowledge and those who claim it, whether they are scientist or dogmatist, open the door to tragedy. All knowledge, all information is imperfect. We have to treat it with humility."