Mathematicians From The United Kingdom

"Take a unit, halve it, halve the result, and so on continually. This gives—1 1⁄2 1⁄4 1⁄8 1⁄16 1⁄32 1⁄64 1⁄128 &c.Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c... We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2 is always equal to the last term added. ...We say that—1, 1 + 1⁄2, 1 + 1⁄2 + 1⁄4, 1 + 1⁄2 + 1⁄4 + 1⁄8, &c. &c.is a series of quantities which continually approximate to the limit 2. Now the truth is, these several quantities are fixed, and do not approximate to 2. ...it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, "let x be a quantity which continually approximates to the limit 2," we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for x which shall be as near to 2 as we please."

"When... we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And if the series of values increase in succession, so that name any quantity we may, however great, all after a certain point will be greater, then the series is said to increase without limit. It is also frequently said, when a quantity diminishes without limit, that it has nothing, zero or 0, for its limit: and that when it increases without limit it has infinity or ∞ or 1⁄0 for its limit."

"In every age of the world there has been an established system, which has been opposed from time to time by isolated and dissentient reformers. The established system has sometimes fallen, slowly and gradually: it has either been upset by the rising influence of some one man, or it has been sapped by gradual change of opinion in the many."

"During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical."

"A great many individuals ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. ...I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: ...something which is apart from general opinion, either in subject-matter, method, or conclusion. ...Thus in the sixteenth century many spoke of the earth's motion as the paradox of Copernicus, who held the ingenuity of that theory in very high esteem, and some, I think, who even inclined towards it. In the seventeenth century, the depravation of meaning took place... Phillips says paradox is "a thing which seemeth strange"—here is the old meaning...—"and absurd, and is contrary to common opinion," which is an addition due to his own time."

"Spinoza's Philosophia Scripturæ Interpres, Exercitatio Paradoxa, printed anonymously ...is properly paradox, though also heterodox. It supposes, contrary to all opinion, orthodox and heterodox, that philosophy can... explain the Athanasian doctrine so as to be at least compatible with orthodoxy. The author would stand almost alone, if not quite; and this is what he meant."

"The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself."

"Aspiring to lead others, they have never given themselves the fair chance of being first led by other others into something better than they can start for themselves; and that they should first do this is what both those classes of others have a fair right to expect. New knowledge... must come by contemplation of old knowledge... mechanical contrivance sometimes, not very often, escapes this rule."

"All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors, and learned in what had been before them. There is not one exception. I do not say that every man has made direct acquantance with the whole of his mental ancestry... But... it is remarkable how many of the greatest names in all departments of knowledge have been real antiquaries in their several subjects. I may cite among those... in science, Aristotle, Plato, Ptolemy, Euclid, Archimedes, Roger Bacon, Copernicus, Francis Bacon, Ramus, Tycho Brahe, Galileo, Napier, Descartes, Leibnitz, Newton, Locke."

"I will not, from henceforward, talk to any squarer of the circle, trisector of the angle, duplicator of the cube, constructor of perpetual motion, subverter of gravitation, stagnator of the earth, builder of the universe, etc."

"Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in nature."

"‘European science could never have reached its present height had it not been fertilised by successive wafts from the […] knowledge stored up in the East.’ ‘Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan and George Boole on the mathematical atmosphere of 1830–1865.’ ‘I do as George Boole and De Morgan did: I bow my head inreverent thankfulness to that mysterious East, whence come to us wafts of some transcendent power the nature of which we ourselves can hardly state in words.’"

"A very interesting detailed account of the peculiarities of the circle squarer, and of the futility of the attempts on the part of the Mathematicians to convince him of his errors, will be found in Augustus De Morgan's Budget of Paradoxes."

"The fact is known that having very thoroughly worked at the generalisations of Mathematics in theory and practice, Mr. De Morgan was enabled to establish with perfect precision the most highly generalised conception of Logic, perhaps, which it is possible to entertain. It is no new doctrine that Logic deals with the necessary laws of action of thought, and that Mathematics apply these laws to necessary matter of thought; but by showing that these laws can and must be applied with equal precision and equal necessity to all kinds of relations, and not only to those which the Aristotelian theory takes account of, he so enlarged the scope and intensified the power of Logic as an instrument, that we may hope for coming generations, as he must have hoped... another instalment of the kind... Mathematics are, meanwhile, and perhaps will always remain, the completest and most accurate example of the generalised Logic. At any rate, in the mind of the author, Logic and Mathematics as 'the two great branches of exact science, the study of the necessary laws of thought, the study of the necessary matter of thought, were always viewed in connection and antithesis."

"Dr. George Boole, author of The Laws of Thought had introduced himself in the year 1842 to Mr. De Morgan by a letter on the Differential and Integral Calculus then recently published. His character and pursuits were in many points like those of the author who found great pleasure in his correspondence and friendship. ...In 1847, his attention having been drawn to the subject by the publication of Mr. De Morgan's Formal Logic, he published the Mathematical Analysis of Logic and in the following year communicated... a paper on the Calculus of Logic. His great work, An Investigation into the Laws of Thought... was a development of the principle laid down in the Calculus..."

"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."

"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."

"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."

"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?"

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

"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."

"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 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."

"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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"Unhappily, common sense has no recorded history."

"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."

"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."

"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."

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