Mathematicians From England

"Never mind what two tons refers to. What is it? How has it entered in so definite a way into our exprerience? Two tons is the reading of the pointer when the elephant was placed on a weighing machine. Let us pass on. … And so we see that the poetry fades out of the problem, and by the time the serious application of exact science begins we are left only with pointer readings."

"Schrödinger's wave-mechanics is not a physical theory, but a dodge — and a very good dodge too."

"If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly more favourable than the chance of the molecules returning to one half of the vessel."

"The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation."

"Shuffling is the only thing which Nature cannot undo."

"Motion with respect to the universal ocean of aether eludes us. We say, "Let V be the velocity of a body through the aether", and form the various electromagnetic equations in which V is scattered liberally. Then we insert the observed values, and try to eliminate everything which is unknown except V. The solution goes on famously; but just as we have got rid of all the other unknowns, behold! V disappears as well, and we are left with the indisputable but irritating conclusion —"

"The quest of the absolute leads into the four-dimensional world."

"There was just one alternative, the earth’s true velocity through space might happen to have been nil."

"In physics we have outgrown archer and apple-pie definitions of the fundamental symbols. To a request to explain what an electron really is supposed to be we can only answer, "It is part of the A B C of physics". The external world of physics has thus become a world of shadows. In removing our illusions we have removed the substance, for indeed we have seen that substance is one of the greatest of our illusions. Later perhaps we may inquire whether in our zeal to cut out all that is unreal we may not have used the knife too ruthlessly. Perhaps, indeed, reality is a child which cannot survive without its nurse illusion. But if so, that is of little concern to the scientist, who has good and sufficient reasons for pursuing his investigations in the world of shadows and is content to leave to the philosopher the determination of its exact status in regard to reality. In the world of physics we watch a shadowgraph performance of the drama of familiar life. The shadow of my elbow rests on the shadow table as the shadow ink flows over the shadow paper. It is all symbolic, and as a symbol the physicist leaves it. Then comes the alchemist Mind who transmutes the symbols. The sparsely spread nuclei of electric force become a tangible solid; their restless agitation becomes the warmth of summer; the octave of aethereal vibrations becomes a gorgeous rainbow. Nor does the alchemy stop here. In the transmuted world new significances arise which are scarcely to be traced in the world of symbols; so that it becomes a world of beauty and purpose — and, alas, suffering and evil. The frank realisation that physical science is concerned with a world of shadows is one of the most significant of recent advances."

"Science aims at constructing a world which shall be symbolic of the world of commonplace experience. It is not at all necessary that every individual symbol that is used should represent something in common experience or even something explicable in terms of common experience. The man in the street is always making this demand for concrete explanation of the things referred to in science; but of necessity he must be disappointed. It is like our experience in learning to read. That which is written in a book is symbolic of a story in real life. The whole intention of the book is that ultimately a reader will identify some symbol, say BREAD, with one of the conceptions of familiar life. But it is mischievous to attempt such identifications prematurely, before the letters are strung into words and the words into sentences. The symbol A is not the counterpart of anything in familiar life."

"The idealistic tinge in my conception of the physical world arose out of mathematical researches on the relativity theory. In so far as I had any earlier philosophical views, they were of an entirely different complexion. From the beginning I have been doubtful whether it was desirable for a scientist to venture so far into extra-scientific territory. The primary justification for such an expedition is that it may afford a better view of his own scientific domain."

"There is, then, in this analysis of variance no indication of any other than innate and heritable factors at work. (Coining of the phrase ‘analysis of variance’.)"

"I believe that no one who is familiar, either with mathematical advances in other fields, or with the range of special biological conditions to be considered, would ever conceive that everything could be summed up in a single mathematical formula, however complex."

"Fairly large print is a real antidote to stiff reading."

"(Coining the phrase ‘test of significance’): Critical tests of this kind may be called tests of significance, and when such tests are available we may discover whether a second sample is or is not significantly different from the first."

"I am genuinely sorry for scientists of the younger generation who never knew Fisher personally. So long as you avoided a handful of subjects like inverse probability that would turn Fisher in the briefest possible moment from extreme urbanity into a boiling cauldron of wrath, you got by with little worse than a thick head from the port which he loved to drink in the evening. And on the credit side you gained a cherished memory of English spoken in a Shakespearean style and delivered in the manner of a Spanish grandee."

"As a child, R. A. Fisher was fascinated by mathematics, statistics, and astronomy. His vision was severely impaired, but this was no impediment to his scholarship: He had tutors who would read mathematics to him, and young Fisher would picture the problems in his head. He eventually became so good at visualizing the solutions that he would frequently present his answers with absolutely no proof, saying that the solution was so obvious it didn’t need to be proven. Gosset remarked that these ‘obvious’ solutions to problems were generally not obvious to anyone else until they’d slogged through the math and could see on paper what Fisher saw in his head."

"This is perhaps the most important book on evolutionary genetics ever written."

"A book that I rate only second in importance in evolution theory to Darwin's Origin (this as joined with its supplement Of Man), and also rate as undoubtedly one of the greatest books of the twentieth century."

"He has made contributions to many areas of science; among them are agronomy, anthropology, astronomy, bacteriology, botany, economics, forestry, meteorology, psychology, public health, and-above all-genetics, in which he is recognized as one of the leaders. Out of this varied scientific research and his skill in mathematics, he has evolved systematic principles for the interpretation of empirical data; and he has founded a science of experimental design. On the foundations he has laid down, there has been erected a structure of statistical techniques that are used whenever people attempt to learn about nature from experiment and observation."

"A very able "manipulative" mathematician, Fisher enjoys a real mastery in evaluating complicated multiple integrals. In addition, he has a remarkable talent in the most difficult field of approaching problems of empirical research. As a result, his lifework includes a series of valuable contributions giving exact distributions of a variety of statistics, such as the correlation coefficient, the central χ2 with due allowance for degrees of freedom, the noncentral χ2, the quotient of two χ2, etc., etc."

"Before deciding anything I wrote to his tutor at Caius college... about his mathematical ability. The answer was that he could have been a first class mathematician had he "stuck to the ropes" but he would not... It took me a very short time to realize that he was more than a man of great ability, he was in fact a genius who must be retained."

"[...] the uncontrolled causes which may influence the result are always strictly innumerable."

"The million, million, million … to one chance happens once in a million, million, million … times no matter how surprised we may be that it results in us."

"More attention to the History of Science is needed, as much by scientists as by historians, and especially by biologists, and this should mean a deliberate attempt to understand the thoughts of the great masters of the past, to see in what circumstances or intellectual milieu their ideas were formed, where they took the wrong turning or stopped short on the right track."

"… the best causes tend to attract to their support the worst arguments, which seems to be equally true in the intellectual and in the moral sense."

"Faith Is Not Credulity."

"...it was Darwin’s chief contribution, not only to Biology but to the whole of natural science, to have brought to light a process by which contingencies a priori improbable, are given, in the process of time, an increasing probability, until it is their non-occurrence rather than their occurrence which becomes highly improbable. … Let the reader … attempt to calculate the prior probability that a hundred generations of his ancestry in the direct male line should each have left at least one son. The odds against such a contingency as it would have appeared to his hundredth ancestor (about the time of King Solomon) would require for their expression forty- four figures of the decimal notation; yet this improbable event has certainly happened."

"Natural selection is a mechanism for generating an exceedingly high degree of improbability."

"Available scientific knowledge provides a firm basis for believing that the groups of mankind differ in their innate capacity for intellectual and emotional development."

"… the so-called co-efficient of heritability, which I regard as one of those unfortunate short-cuts, which have often emerged in biometry for lack of a more thorough analysis of the data."

"After all, it is a common weakness of young authors to put too much into their papers."

"No, I cannot believe in the light of this speech that the reward of Lysenko's triumphant career is the advance of scientific knowledge; nor that it is the prosperity of poor peasants. The reward he is so eagerly grasping is Power, power for himself, power to threaten, power to torture, power to kill."

"He is an ambitious politician of a type likely to become prevalent in a system avowedly guided by a rigid ideology, who hopes to use the ideological dogmas to which he finds his colleagues committed as levers for his own advancement in the party and in the state."

"In organisms of all kinds the young are launched upon their careers endowed with a certain amount of biological capital derived from their parents. This varies enormously in amount in different species, but, in all, there has been, before the offspring is able to lead an independent existence, a certain expenditure of nutriment in addition, almost universally, to some expenditure of time or activity, which the parents are induced by their instincts to make for the advantage of their young. Let us consider the reproductive value of these offspring at the moment when this parental expenditure on their behalf has just ceased. If we consider the aggregate of an entire generation of such offspring it is clear that the total reproductive value of the males in this group is exactly equal to the total value of all the females, because each sex must supply half the ancestry of all future generations of the species. From this it follows that the sex ratio will so adjust itself, under the influence of Natural Selection, that the total parental expenditure incurred in respect of children of each sex, shall be equal; for if this were not so and the total expenditure incurred in producing males, for instance, were less than the total expenditure incurred in producing females, then since the total reproductive value of the males is equal to that of the females, it would follow that those parents, the innate tendencies of which caused them to produce males in excess, would, for the same expenditure, produce a greater amount of reproductive value; and in consequence would be the progenitors of a larger fraction of future generations than would parents having a congenital bias towards the production of females. Selection would thus raise the sex-ratio until the expenditure upon males became equal to that upon females."

"[We are now] in a position to judge of the validity of the objection which has been made, that the principle of Natural Selection depends on a succession of favourable chances. The objection is more in the nature of an innuendo than of a criticism, for it depends for its force upon the ambiguity of the word chance, in its popular uses. The income derived from a Casino by its proprietor may, in one sense, be said to depend upon a succession of favourable chances, although the phrase contains a suggestion of improbability more appropriate to the hopes of the patrons of his establishment. It is easy without any very profound logical analysis to perceive the difference between a succession of favourable deviations from the laws of chance, and on the other hand, the continuous and cumulative action of these laws. It is on the latter that the principle of Natural Selection relies."

"Professor Eddington has recently remarked that 'The law that entropy always increases — the second law of thermodynamics — holds, I think, the supreme position among the laws of nature'. It is not a little instructive that so similar a law [the fundamental theorem of natural selection] should hold the supreme position among the biological sciences."

"We may consequently state the fundamental theorem of Natural Selection in the form : The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time."

"No efforts of mine could avail to make the book easy reading."

"No practical biologist interested in sexual reproduction would be led to work out the detailed consequences experienced by organisms having three or more sexes; yet what else should he do if he wishes to understand why the sexes are, in fact, always two?"

"Natural Selection is not evolution."

"To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination. He can perhaps say what the experiment died of."

"The academic mind, as we know, is sometimes capable of assuming an aggressive attitude. The official mind, on the contrary, is and has to be, expert in the art of self-defence."

"Apart from sex linkage, we know almost nothing at present of linkage in man. Yet it is certain that every defect determined by a single factor must be located in one or other of twenty-three [sic] linkage groups. Each defect must therefore be linked in inheritance with numerous other observable traits, and with some of them is probably linked closely. The search for such linkage will certainly be lengthy, and at first, disappointing."

"In scientific subjects, the natural remedy for dogmatism has been found in research. By temperament and training, the research worker is the antithesis of the pundit. What he is actively and constantly aware of is his ignorance, not his knowledge; the insufficiency of his concepts, of the terms and phrases in which he tries to excogitate his problems: not their final and exhaustive sufficiency. He is, therefore, usually only a good teacher for the few who wish to use their mind as a workshop, rather than a warehouse.*"

"(Coining phrase "null hypothesis") In relation to any experiment we may speak of this hypothesis as the “null hypothesis,” and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation. Every experiment may be said to exist only in order to give the facts a chance of disproving the null hypothesis."

"The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic."

"However, perhaps the main point is that you are under no obligation to analyse variance into its parts if it does not come apart easily, and its unwillingness to do so naturally indicates that one’s line of approach is not very fruitful."

"A precise language awaits a completed metaphysics."

"The purpose of thinking is so our thoughts die instead of us."