Logicians From The United Kingdom

"The absolute requisites for the study of this work... are a knowledge of algebra to the binomial at least, plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry. ...A. De Morgan. London July 1, 1836"

"The student of the Differential Calculus may... be brought to think it possible that the terms and ideas which that science requires may exist in his own mind in the same rude form as that of a straight line in the conceptions of a beginner in geometry. ...he must be prepared to stop his course until he can form exact notions, acquire precise ideas, both of resemblance between those things which have appeared most distinct, and of distinction between those which have appeared most alike. To do this... formal definitions would be useless; for he cannot be supposed to have one single notion in that precise form which would make it worth while to attach it to a word. One reason of the great difficulty which is found in treatises on this subject... the tacit assumption that nothing is necessary previously to actually embodying the terms and rules of the science, as if mere statement of definitions could give instantaneous power of using terms rightly. We shall here attempt... a wider degree of verbal explanation than is usual with the view of enabling the student to come to the definitions in some state of previous preparation."

"Find a fraction which, multiplied by itself, shall give 6, or... find the square root of 6. This can be shown to be an impossible problem; for it can be shown that no fraction whatsoever multiplied by itself, can give a whole number, unless it be itself a whole number disguised in a fractional form, such as 4⁄2 or 21⁄3. To this problem, then, there is but one answer, that it is self-contradictory. But if we propose the following problem,—to find a fraction which, multiplied by itself, shall give a product lying between 6 and 6 + a; we find that this problem admits of solution in every case. It therefore admits of solution however small a may be... as small as you please. ...there is such a thing as the square root of 6, and it is denoted by √6. But we do not say we actually find this, but that we approximate to it."

"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..."

"No rational argument will have a rational effect on a man who does not want to adopt a rational attitude."

"We must plan for freedom, and not only for security, if for no other reason than that only freedom can make security secure."

"I believe that there is, from the ethical point of view, no symmetry between suffering and happiness, or between pain and pleasure. Both the greatest happiness principle of the Utilitarians and Kant's principle 'Promote other people's happiness ..' seem to me (at least in their formulations) wrong on this point which, however, is not completely decidable by rational argument. (For the irrational aspect of ethical beliefs, see note 11 to the present chapter, and for the rational aspect, sections II and especially III of chapter 24). In my opinion (cp. note 6 (2) to chapter 5) human suffering makes a direct moral appeal, namely, the appeal for help, while there is no similar call to increase the happiness of a man who is doing well anyway. (A further criticism of the Utilitarian formula 'Maximize pleasure' is that it assumes, in principle, a continuous pleasure-pain scale which allows us to treat degrees of pain as negative degrees of pleasure. But, from the moral point of view, pain cannot be outweighed by pleasure, and especially not one man's pain by another man's pleasure. Instead of the greatest happiness for the greatest number, one should demand, more modestly, the least amount of avoidable suffering for all; and further, that unavoidable suffering—such as hunger in times of an unavoidable shortage of food—should be distributed as equally as possible.)"

"Although my own position is, I believe, clearly enough implied in the text, I may perhaps briefly formulate what seems to me the most important principles of humanitarian and equalitarian ethics. (1) Tolerance towards all who are not intolerant and who do not propagate intolerance. ... This implies, especially, that the moral decisions of others should be treated with respect, as long as such decisions do not conflict with the principle of tolerance. (2) The recognition that all moral urgency has its basis in the urgency of suffering or pain. I suggest, for this reason, to replace the utilitarian formula 'Aim at the greatest amount of happiness for the greatest number', or briefly, 'Maximize happiness' by the formula 'The least amount of avoidable suffering for all', or briefly, 'Minimize suffering'. Such a simple formula can, I believe, be made one of the fundamental principles (admittedly not the only one) of public policy. (The principle 'Maximize happiness', in contrast, seems to be apt to produce a benevolent dictatorship.) We should realize that from the moral point of view suffering and happiness must not be treated as symmetrical; that is to say, the promotion of happiness is in any case much less urgent than the rendering of help to those who suffer, and the attempt to prevent suffering. (The latter task has little to do with 'matters of taste', the former much.)"

"The so-called paradox of freedom is the argument that freedom in the sense of absence of any constraining control must lead to very great restraint, since it makes the bully free to enslave the meek. The idea is, in a slightly different form, and with very different tendency, clearly expressed in Plato. Less well known is the paradox of tolerance: Unlimited tolerance must lead to the disappearance of tolerance. If we extend unlimited tolerance even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them. — In this formulation, I do not imply, for instance, that we should always suppress the utterance of intolerant philosophies; as long as we can counter them by rational argument and keep them in check by public opinion, suppression would certainly be unwise. But we should claim the right to suppress them if necessary even by force; for it may easily turn out that they are not prepared to meet us on the level of rational argument, but begin by denouncing all argument; they may forbid their followers to listen to rational argument, because it is deceptive, and teach them to answer arguments by the use of their fists or pistols. We should therefore claim, in the name of tolerance, the right not to tolerate the intolerant. We should claim that any movement preaching intolerance places itself outside the law, and we should consider incitement to intolerance and persecution as criminal, in the same way as we should consider incitement to murder, or to kidnapping, or to the revival of the slave trade, as criminal."

"In speaking of sociological laws or natural laws of social life I have in mind such laws as are formulated by modern economic theories, for instance, the theory of international trade, or the theory of the trade cycle. These and other important sociological laws are connected with the functioning of social institutions. These laws play a role in our social life corresponding to the role played in mechanical engineering by, say, the principle of the lever. For institutions, like levers, are needed if we want to achieve anything which goes beyond the power of our muscles. Like machines, institutions multiply our power for good or evil. Like machines, they need intelligent supervision by someone who understands their way of functioning and, most of all, their purpose, since we cannot build them so that they work entirely automatically."

"The open society is one in which men have learned to be to some extent critical of taboos, and to base decisions on the authority of their own intelligence."

"What a monument of human smallness is this idea of the philosopher king. What a contrast between it and the simplicity of humaneness of Socrates, who warned the statesmen against the danger of being dazzled by his own power, excellence, and wisdom, and who tried to teach him what matters most — that we are all frail human beings. What a decline from this world of irony and reason and truthfulness down to Plato's kingdom of the sage whose magical powers raise him high above ordinary men; although not quite high enough to forgo the use of lies, or to neglect the sorry trade of every shaman — the selling of spells, of breeding spells, in exchange for power over his fellow-men."

"We may become the makers of our fate when we have ceased to pose as its prophets."

"This book raises issues that might not be apparent from the table of contents. It sketches some of the difficulties faced by our civilization — a civilization which might be perhaps described as aiming at humanness and reasonableness, at equality and freedom; a civilization which is still in its infancy, as it were, and which continues to grow in spite of the fact that it has been so often betrayed by so many of the intellectual leaders of mankind. It attempts to show that this civilization has not yet fully recovered from the shock of its birth — the transition from the tribal or "enclosed society," with its submission to magical forces, to the 'open society' which sets free the critical powers of man. It attempts to show that the shock of this transition is one of the factors that have made possible the rise of those reactionary movements which have tried, and still try, to overthrow civilization and to return to tribalism."

"I see now more clearly than ever before that even our greatest troubles spring from something that is as admirable and sound as it is dangerous — from our impatience to better the lot of our fellows. For these troubles are the by-products of what is perhaps the greatest of all moral and spiritual revolutions of history, a movement which began three centuries ago. It is the longing of uncounted unknown men to free themselves and their minds from the tutelage of authority and prejudice. It is their attempt to build up an open society which rejects the absolute authority to preserve, to develop, and to establish traditions, old or new, that measure up to their standards of freedom, of humaneness, and of rational criticism. It is their unwillingness to sit back and leave the entire responsibility for ruling the world to human or superhuman authority,and their readiness to share the burden of responsibility for avoidable suffering, and to work for its avoidance. This revolution has created powers of appalling destructiveness; but they may yet be conquered."

"If in this book harsh words are spoken about some of the greatest among the intellectual leaders of mankind, my motive is not, I hope, the wish to belittle them. It springs rather from my conviction that, if our civilization is to survive, we must break with the habit of deference to great men. Great men may make great mistakes; and as the book tries to show, some of the greatest leaders of the past supported the perennial attack on freedom and reason. Their influence, too rarely challenged, continues to mislead those on whose defence civilization depends, and to divide them. The responsibility of this tragic and possibly fatal division becomes ours if we hesitate to be outspoken in our criticism of what admittedly is a part of our intellectual heritage. By reluctance to criticize some of it, we may help to destroy it all."

"Bold ideas, unjustified anticipations, and speculative thought, are our only means for interpreting nature: our only organon, our only instrument, for grasping her. And we must hazard them to win our prize. Those among us who are unwilling to expose their ideas to the hazard of refutation do not take part in the scientific game."

"I believe that the advance of science depends upon the free competition of thought, and thus upon freedom, and that it must come to an end if freedom is destroyed (though it may well continue for some time in some fields, especially in technology)."

"Science is not a system of certain, or well established, statements; nor is it a statement which steadily advances towards state of finality. Our science is not knowledge (epistēmē): it can newer claim to have attained truth, or even substitute for it, such as probability. . . . We do not know; we can only guess."

"Science does not rest upon solid bedrock. The bold structure of its theories arises, as it were, above a swamp. It is like a building erected on piles. The piles are driven down from above into the swamp, but not down to any natural or 'given' base; and if we stop driving the piles deeper, it is not because we have reached firm ground. We simply stop when we are satisfied that the piles are firm enough to carry the structure, at least for the time being."

"...non-reproducible single occurrences are of no significance to science."

"The game of science is, in principle, without end. He who decides one day that scientific statements do not call for any further test, and that they can be regarded as finally verified, retires from the game."

"A principle of induction would be a statement with the help of which we could put inductive inferences into a logically acceptable form. In the eyes of the upholders of inductive logic, a principle of induction is of supreme importance for scientific method: "... this principle", says Reichenbach, "determines the truth of scientific theories. To eliminate it from science would mean nothing less than to deprive science of the power to decide the truth or falsity of its theories. Without it, clearly, science would no longer have the right to distinguish its theories from the fanciful and arbitrary creations of the poet's mind." Now this principle of induction cannot be a purely logical truth like a tautology or an analytic statement. Indeed, if there were such a thing as a purely logical principle of induction, there would be no problem of induction; for in this case, all inductive inferences would have to be regarded as purely logical or tautological transformations, just like inferences in inductive logic. Thus the principle of induction must be a synthetic statement; that is, a statement whose negation is not self-contradictory but logically possible. So the question arises why such a principle should be accepted at all, and how we can justify its acceptance on rational grounds."

"...no matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white."

"Good tests kill flawed theories; we remain alive to guess again."

"Philosophers should consider the fact that the greatest happiness principle can easily be made an excuse for a benevolent dictatorship. We should replace it by a more modest and more realistic principle — the principle that the fight against avoidable misery should be a recognized aim of public policy, while the increase of happiness should be left, in the main, to private initiative."

"Those who promise us paradise on earth never produced anything but a hell."

"When I speak of reason or rationalism, all I mean is the conviction that we can learn through criticism of our mistakes and errors, especially through criticism by others, and eventually also through self-criticism. A rationalist is simply someone for whom it is more important to learn than to be proved right; someone who is willing to learn from others — not by simply taking over another's opinions, but by gladly allowing others to criticize his ideas and by gladly criticizing the ideas of others. The emphasis here is on the idea of criticism or, to be more precise, critical discussion. The genuine rationalist does not think that he or anyone else is in possession of the truth; nor does he think that mere criticism as such helps us achieve new ideas. But he does think that, in the sphere of ideas, only critical discussion can help us sort the wheat from the chaff. He is well aware that acceptance or rejection of an idea is never a purely rational matter; but he thinks that only critical discussion can give us the maturity to see an idea from more and more sides and to make a correct judgement of it."

"If the many, the specialists, gain the day, it will be the end of science as we know it - of great science. It will be a spiritual catastrophe comparable in its consequences to nuclear armament."

"The answer to this problem is: as implied by Hume, we certainly are not justified in reasoning from an instance to the truth of the corresponding law. But to this negative result a second result, equally negative, may be added: we are justified in reasoning from a counterinstance to the falsity of the corresponding universal law (that is, of any law of which it is a counterinstance). Or in other words, from a purely logical point of view, the acceptance of one counterinstance to 'All swans are white' implies the falsity of the law 'All swans are white' - that law, that is, whose counterinstance we accepted. Induction is logically invalid; but refutation or falsification is a logically valid way of arguing from a single counterinstance to - or, rather, against - the corresponding law.This shows that I continue to agree with Hume's negative logical result; but I extend it.This logical situation is completely independent of any question of whether we would, in practice, accept a single counterinstance - for example, a solitary black swan - in refutation of a so far highly successful law. I do not suggest that we would necessarily be so easily satisfied; we might well suspect that the black specimen before us was not a swan.""

"I appeal to the philosophers of all countries to unite and never again mention Heidegger or talk to another philosopher who defends Heidegger. This man was a devil. I mean, he behaved like a devil to his beloved teacher, and he has a devilish influence on Germany. ... One has to read Heidegger in the original to see what a swindler he was."

"I think so badly of philosophy that I don't like to talk about it. ... I do not want to say anything bad about my dear colleagues, but the profession of teacher of philosophy is a ridiculous one. We don't need a thousand of trained, and badly trained, philosophers — it is very silly. Actually most of them have nothing to say."

"It seems to me that I may be living too long. Indeed: my nearest relations have all died, and so have some of my best friends, and even some of my best pupils. However, I do not have a reason to complain. I am grateful and happy to be alive, and still be able to continue with my work, if only just. My work seems to me more important than ever."

"Science may be described as the art of systematic over-simplification — the art of discerning what we may with advantage omit."

"There is an almost universal tendency, perhaps an inborn tendency, to suspect the good faith of a man who holds opinions that differ from our own opinions. ... It obviously endangers the freedom and the objectivity of our discussion if we attack a person instead of attacking an opinion or, more precisely, a theory."

"From Plato to Karl Marx and beyond, the fundamental problem has always been: who should rule the state? (One of my main points will be that this problem must be replaced by a totally different one.)"