Mathematicians From England

"[M]y axiom runs as follows: "The whole is not identical with a part." This axiom leads us at once to a problem. What relation has the part to the whole?"

"[D]efinition... Religion is the relation of the finite to the infinite. ...is the relation. ...[T]here is only one relation, there can be only one religion. ...only so far true as it actually explains the relation of the finite to the infinite. In so far as it builds up an imaginary relation between finite and infinite it is false. ...[S]ince no existing religion lays out before us fully the relation of finite and infinite, all systems of religion are of necessity but half truths. ...not whole falsehoods, for many... may have made... small advance towards the solution..."

"I simply assert that the universe alters, is "becoming;" what it is becoming I will not venture to say. ...the individual too is altering, is not only a "being" but also a "becoming." These alterations... I shall—merely for convenience—term life."

"If the reader questions whether there is still war between science and dogma, I must reply that there always will be as long as knowledge is opposed to ignorance. To know requires exertion, and it is intellectually easiest to shirk effort altogether by accepting phrases which cloak the unknown in the undefinable."

"The great danger... not content with our real knowledge of the relation of the finite to the infinite, they slur over our vast ignorance by the help of the imagination. Myth supplies the place of true knowledge where we are ignorant of the connection between finite and infinite. Hence... most concrete systems of religion present us with a certain amount of knowledge but a great deal of myth."

"[O]ur [small] knowledge of the relation of finite to infinite... is... continually increasing—science and philosophy are continally presenting... broader views of the relation of man to nature and of individual thought to abstract thought."

"It follows... that... in every concrete religion the knowledge element ought to increase and the myth element decrease or... every concrete religion ought to be in a state of development."

"Heredity. Given any organ in a parent and the same or any other organ in its offspring, the mathematical measure of heredity is the correlation of these organs for pairs of parent and offspring... The word organ here must be taken to include any characteristic which can be quantitatively measured."

"[T]his deficient organisation will not only in time be perceived by the enemy, but... has already had a very discouraging influence both on scientific recruits and on intelligent laymen."

"That all science is description and not explanation, that the mystery of change in the inorganic world is just as great and just as omnipresent as in the organic world, are statements which will appear platitudes to the next generation."

"The purpose of the mathematical theory of statistics is to deal with the relationship between 2 or more variable quantities without assuming that one is a single-valued mathematical function of the rest. The statistician does not think a certain x will produce a single-valued y; not a causative relation but a correlation. The relationship between x and y will be somewhere within a zone and we have to work out the probability that the point (x,y) will lie in different parts of that zone. The physicist is limited and shrinks the zone into a line. Our treatment will fit all the vagueness of biology, sociology, etc. A very wide science."

"Anything more hopelessly illogical than the statements with regard to force and matter current in elementary text-books of science, it is difficult to imagine; and the author, as a result of some ten years' teaching and examining, has been forced to the conclusion that these works possess little, if any, educational value; they neither encourage the growth of logical clearness nor form any exercise in scientific method."

"Even the fathers of statistical science forgot that a random series is bound to exhibit some pattern... Professor Pearson was among the first scholars interested in creating artificial random number generators, tables one could use as inputs for... simulations (precursors of our Monte Carlo simulator). ...[T]hey did not want these tables to exhibit... regularity. Yet real randomness does not look random!...A single random run is bound to exhibit some pattern ..."

"I wish I could communicate to you, and especially to those of you who are just now beginning your professional careers in a world of statistics incredibly more sophisticated than that of Karl Pearson's day, something of the thrill in meeting in person and studying under a man of Pearson's immense reputation. Author of the Grammar of Science, perfector of simple linear correlations; inventor of multiple and partial correlation, of curvilinear correlation, of tetrachoric and bi- serial correlation; discoverer of the \chi^2 function for summarizing multinomial data with magnificent simplicity; builder of a beautiful system of frequency curves derived from a single differential equation which in turn harked back to the hypergeometric series; founder of ' and author or co-author of a prolific literature applying these new statistics to biological and sociological data - Karl Pearson was a hero of to an American boy vouchsafed a visit to the home of the gods. Indeed, Pearson was Thor himself - for the thunderbolts with which he attacked unsparingly those who dared oppose him were echoing and reechoing."

"Professor Karl Pearson... devised the first test of nonrandomness (...in reality, deviation from normality ...for all intents and purposes, the same thing). He examined millions of runs of what was called a Monte Carlo (the old name for a roulette wheel) during... July 1902... He discovered... with a high degree of statistical significance... the runs were not purely random. ...Pearson was greatly surprised ...Philosophers of statistics call this the reference case problem to explain that there is no true attainable randomness in practice, only in theory."

"[T]his work is not written to gain a public, but piam memoriam prodere conditoris nostri and is intended especially for those who have known and loved Francis Galton in the past, or who may in the future desire to understand and honour him."

"The statistician Karl Pearson analyzed a large number of outcomes at certain roulette tables and suggested that the wheels were biased. He wrote in 1894: Clearly, since the Casino does not serve the valuable end of huge laboratory for the preparation of probability statistics, it has no scientific raison d’être... [E]arly experiments were suggestive and led to important discoveries in probability and statistics. They led Pearson to the ', which is of great importance in testing whether observed data fit a given ."

"Before meeting with Weldon... Pearson had grown into a social Darwinist anxious to provide his particular form of Darwinism with a proper scientific basis, and to show that Darwin's ideas and socialism were complementary, and not opposed, as had been maintained by several leading thinkers of the nineteenth century. Biometry offered him the chance of pursuing these ends. Moreover... Pearson's conception of 'properly scientific'... was one that made it probable that the development of biometry... would yield a harvest of statistical methods. Statistics, thus formed, embodied the central tenets of Pearson's philosophy of science, and, as such, was to be universally recommended."

"[T]his work... is intended fundamentally as a permanent memorial to the Founder of the , and embraces material which may easily perish or be ultimately lost sight of. ...My object is... to issue a volume to some extent worthy of the name of the man it bears,—which may be studied hereafter by those who wish to understand him, his origin and his aims..."

"If my view be correct, Erasmus Darwin planted the seed of suggestion in questioning whether adaptation meant no more to man than illustration of creative ingenuity; the one grandson, Charles Darwin, collected the facts which had to be dealt with and linked them together by wide-reaching hypotheses; the other grandson, Francis Galton, provided the methods by which they could be tested..."

"Those who know the real history of the one occasion on which Galton and Darwin disagreed know how loyal Galton was to Darwin—loyal with a loyalty far rarer to-day. Galton would not have wished me to put him in the same rank as his master, but the reader who follows my story to the end may possibly see that the ramifications of Galton's methods are producing a renascence in innumerable branches of science..."

"During the first two terms of 1881 he deputised for W. H. Drew, Professor of Mathematics at King's College, London, taking the senior mathematical teaching. It was in 1882... that he was engaged in his first considerable piece of mathematical work, according to his friend W. H. Macaulay "a theory of pulsating spheres in a fluid, forming an Atomic Theory,... a thing in spherical harmonics of the Clerk-Maxwell type." This work, or part of it, was probably not published till 1887. In 1883 two papers were printed "On the Motion of Spherical and Ellipsoidal Bodies in Fluid Media", and another "Note on Twists in an Infinite Elastic Solid". It was a natural step from such research to the completion of Clifford's Common Sense of the Exact Sciences and Todhunter's History of the Theory of Elasticity. We see from the subject-matter of these papers how Pearson's mind was already at work puzzling over the laws of the physical universe, which in terms of the Grammar of Science describe the "how" rather than the "why.""

"[I]t was soon clear to me that I was collecting as much information bearing on the family history of Charles Darwin as on that of Francis Galton. It seemed desirable to place the two men to some extent in contrast in my volume, showing in ancestry, in methods of work and in outlook on life what they had in common and how they differed."

"[I]t is better to be content with the fraction of a right solution than to beguile ourselves with the whole of a wrong solution."

"Step by step [aesthetic] judgment, restless under the growth of positive knowledge, has discarded creed after creed and philosophic system after philosophic system."

"It was Karl Pearson, a man with an unquenchable ambition for scholarly recognition and the kind of drive and determination that had taken Hannibal over the Alps and Marco Polo to China, who recognized the power in Edgeworth's formulations of Galton's ideas. Pearson lacked Galton's originality and Edgeworth's depth of understanding, but it was his zeal, with a vital assist from G. Udny Yule, that created the methodology and sold it to the world."

"[O]nly little by little, slowly... man, by the aid of organised observation and careful reasoning, can hope to reach knowledge of the truth... science... is the sole gateway to a knowledge which can harmonise with our past as well as with our... future... As Clifford puts it, "Scientific thought is not an accompaniment or condition of human progress, but human progress itself.""

"[T]he laws of science are products of the human mind rather than factors of the external world."

"It was only the feeling that, at least in one or two aspects of Francis Galton's later life and of his scientific work, I could perhaps put his contributions to human knowledge more adequately than possibly one or another who might take up the task, if I resigned it, and who would hardly grasp the bearing of that long and intimate scientific correspondence between Galton, Weldon and myself, that I persevered in my endeavour to give some account of a life, wherein an important chapter of personal development must remain largely unrecorded."

"[W]e are frequently told that the growth of science is destroying the beauty and poetry of life. It is undoubtedly rendering many of the old interpretations of life meaningless, because it demonstrates that they are false to the facts which they profess to describe. It does not follow from this, however, that the aesthetic and scientific judgments are opposed; the fact is, that with the growth of our scientific knowledge the basis of the aesthetic judgment is changing and must change. There is more real beauty [satisfaction,.. permanent delight] in what science has to tell us of the chemistry of a distant star, or in the life history of a protozoon than in any produced by the creative imagination of a pre-scientific age."

"Each one of us is... called upon to give a judgment upon an immense variety of problems, crucial for our social existence. If that judgment confirms measures and conduct tending to the increased welfare of society, then it may be termed a moral, or, better, a social judgment. It follows, then, that to ensure a judgement's being moral, method and knowledge are essential to its formation. ...[T]he formation of a moral judgment—that is, one which the individual is reasonably certain will tend to social welfare—does not depend solely on the readiness to sacrifice individual gain or comfort, or on the impulse to act unselfishly: it depends in the first place on knowledge and method. The first demand of the state upon the individual is not for self-sacrifice but for self-development. ...[T]he man who gives a vote... in the choice of a representative, after forming a judgement based upon knowledge, is... acting socially, and is fulfilling a higher standard of citizenship."

"All great scientists have, in a certain sense, been great artists; the man with no imagination may collect facts, but he cannot make great discoveries."

"The single statement, the brief formula, the few words of which replace in our minds a wide range of relationships between isolated phenomena, is what we term a scientific law. Such a law, relieving our memory from the burden of individual sequences, enables us, with the minimum of intellectual fatigue, to grasp a vast complexity of natural or social phenomena. The discovery of law is... the peculiar function of the creative imagination. But this imagination has to be a disciplined one."

"Although science claims the whole universe as its field... it confesses that its ignorance is more widely extended than its knowledge. In this very confession... it finds a safeguard for future progress. Science cannot... allow theologian or metaphysician... to the foreshore of our present ignorance, and so hinder the development in due time..."

"Now it will, I think, be found that the fields of inquiry where science has not yet penetrated and where the scientist still confesses ignorance, are very like... alchemy astrology and witchcraft... Either they involve facts which are in themselves unreal—conceptions which are self-contradictory and absurd, and therefore incapable of analysis by the scientific or any other method,—or, on the other hand, our ignorance arises from an inadequate classification and a neglect of scientific method."

"[S]uppose that the Emperor Karl V. had said to the learned of his day: "I want a method by which I can send a message in a few seconds to that new world, which my mariners take weeks in reaching. ..." ...It required centuries spent in the discovery and classification of new facts before the Atlantic cable became a possibility. It may require the like or even a longer time to unriddle... psychical and biological enigmas... but he who declares that they can never be solved by the scientific method is... as rash as the man of the early sixteenth century would have been had he declared it utterly impossible that the problem of talking across the Atlantic Ocean should ever be solved."

"[W]hen the law is reached... it must be tested and criticised by its discoverer in every conceivable way, till he is certain that the imagination has not played him false, and that his law is in real agreement with the whole group of phenomena..."

"[M]ediaeval ...alchemy, astrology, witchcraft. ...Do we now know how the stars influence human lives or how witches turn milk blue? Not in the least. We have learnt to look upon the facts themselves as unreal, as vain imaginings of the untrained human mind; we have learnt that they could not be described scientifically because they involved notions which were in themselves contradictory and absurd. ...So soon as science entered the field of alchemy with a true classification and a true method, alchemy was converted into chemistry and became an important branch of human knowledge."

"Wherever there is the slightest possibility for the human mind to know, there is a legitimate problem of science. Outside the field of actual knowledge can only lie a region of the vaguest opinion and imagination, to which... men too often... pay higher respect than to knowledge."

"The ignorance of science means the enforced ignorance of mankind."

"Who can give us the assurance that the fields already occupied by science are alone those in which knowledge is possible? Who, in the words of Galilei, is willing to set limits to the human intellect?"

"[I]n the seventeenth century... the system-mongers were the theologians who declared that cosmical problems were not the legitimate problems of science. It was vain for Galilei to assert that the theologians' classification of facts was hopelessly inadequate. ...[T]hey settled that:— "The doctrine that the earth is neither the centre of the universe nor immovable, but moves even with a daily rotation, is absurd, and both philosophically and theologically false, and at the least an error of faith." It took nearly two hundred years to convince the whole theological world that cosmical problems were the legitimate problems of science and science alone, for in 1819 the books of Galilei, Copernicus, and Keppler were still upon the index of forbidden books, and not till 1822 was a decree issued allowing books teaching the motion of the earth about the sun to be printed and published in Rome!"

"Science can only answer to the great majority of "metaphysical" problems "I am ignorant." Meanwhile, it is idle to be impatient or to indulge in system-making."

"[T]he philosophical method seems based upon an analysis which does not start with the classification of facts, but reaches its judgments by some obscure process of internal cogitation. It is therefore dangerously liable to the influence of individual bias; it results... in an endless number of competing and contradictory systems. It is because the so-called philosophical method does not, when different individuals approach the same range of facts, lead, like the scientific, to practical unanimity of judgment, that science, rather than philosophy, offers the better training for modern citizenship."

"Professor Huxley has invented the term Agnostic... for those who limit the possibility of knowledge in certain fields. ...Professor E. du Bois-Reymond has raised the cry "Ignorabimus" ("We shall be ignorant") and... undertaken the difficult task of demonstrating that with regard to certain problems human knowledge is impossible. ...Now ...there is great danger in this cry. ...[T]he attempt to demonstrate an endless futurity of ignorance ...approaches despair. ...Evolution has taught us of the continual growth of man's intellectual powers."

"Hundreds of men have allowed their imagination to solve the universe, but the men who have contributed to our real understanding of natural phenomena have been those who were unstinting in their application of criticism to the product of their imaginations. It is such criticism which is the essence of the scientific use of the imagination, which is... the very life-blood of science."

"[T]he universe is largely the construction of each individual mind."

"To say that there are certain fields—for example, metaphysics—from which science is excluded, wherein its methods have no application, is merely to say that the rules of methodical observation and the laws of logical thought do not apply to the facts, if any, which lie within such fields. These fields, if indeed such exist, must lie outside any intelligible definition which can be given of the word knowledge."

"Each metaphysician has his own system, which to a large extent excludes that of his predecessors and colleagues. Hence... metaphysics are built either on air or on quicksands—either they start from no foundation in facts at all, or the superstructure has been raised before a basis has been found in the accurate classification facts."

"The universe is a variable quantity, which depends upon the keenness and structure of our organs of sense, and upon the fineness of our powers and instruments of observation."