Mathematicians From Poland

"There has never been a great book or a powerful work of art which has not been thought immoral by those with an older tradition."

"There is indeed no system of morality which does not set a high value on truth and on knowledge, above all on a conscious knowledge of oneself. It is therefore at least odd that science should be called amoral, and this by people who in their own lives set a high value on being truthful. For whatever else may be held against science, this cannot be denied, that it takes for ultimate judgment one criterion alone, that it shall be truthful."

"At bottom, we have remain individually too greedy to distribute our surplus, and collectively too stupid to pile it up in any more useful form than the traditional mountains of arms."

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

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

"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."

"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."

"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."

"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?"

"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."

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

"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."

"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."

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

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

"Dissent is the mark of freedom."

"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."

"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."

"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."

"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..."

"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."

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

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

"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..."

"[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."

"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."

"If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians to whom these our labors, if I am not mistaken, will appear to contribute something even to the ecclesiastical state the headship of which your Holiness now occupies.(Author's preface to de revolutionibus)"

"Copernicus made his revolutionary suggestion on purely theoretical grounds. The invention of the telescope came after and not before, the publication of his great work. This is a fact upon which we may well ponder, for it shows that the insight into natural phenomena has come as much from the genius of the theoretical as from that of the experimental and observational investigator."

"Germans and Poles have bitterly disputed the question of Copernicus's ethnic origin, each national group claiming the distinguished astronomer for its own. Where does the truth lie in this controversy? Politically, Copernicus was a subject of the king of Poland; he remained loyal to the Roman Catholic church ; and he wrote chiefly in Latin, but a few of his private letters were composed in German."

"It was not only... with classical antiquity that Copernicus showed himself a man of the Renaissance. He also strove to achieve the many-sided accomplishments of that humanistic ideal, the universal man. He was competent in canon law; he practiced medicine; he wrote a tract on coinage; he served his cathedral chapter as an administrator and diplomatic representative; he painted his own portrait; he made many of his own astronomical instruments; and he established the heliocentric system on a firm basis."

"Poles and Germans have a common history of great scientists: Today we no longer perceive Copernicus, Hevelius, Schopenhauer and Fahrenheit as the property of one nation but as representatives of one transnational culture."