Mathematicians From England

"There is the definition [of mathematics], boldly proposed by Pierce that 'Mathematics is the science which draws necessary conclusions', and more explicitly formulated by Russel that 'Pure Mathematics is the class of all propositions of the form "p implies q"... it was... the purpose of Russell's treatise to provide a complete, exact and convincing justification of this definition... instead, he and Whitehead collaborated to give a magisterial account of the Principia Mathematica."

"Pure mathematics is much more than an armoury of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations."

"The concept of 'number' in its most elementary sense as the signless integer appears to be an immediate abstraction from quantitative reality subjected to processes of counting and measurement. Vulgar fractions arise from division of a quantity into equal parts. But in what sense is zero a number? Are there negative numbers? Are there numbers corresponding to incommensurable ratios? Each question requires for its solution a fresh exercise of that kind of creative imagination which we call mathematical abstraction."

"The 'language theory' is inadequate as a description of the nature of mathematics."

"At each stage of in the advance of mathematical thought the outstanding characteristics are novelty and originality. That is why mathematics is such a delight to study, such a challenge to practise and such a puzzle to define."

"The brilliant summaries by Bourbaki (1969) and Weyl (1951)... set a high standard of exposition, but... there is room for a history of mathematical ideas which will demand less mathematical expertise and offer a more detailed account of the motivation of research."

"This book... is primarily and essentially an account of the discovery or invention of mathematical concepts, and the historical material is... divided and classified, neither chronologically nor biographically, but philosophically."

"From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics."

"The subject matter of mathematics has increased so rapidly and extensively that there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life. We turn, then, from the attempt to characterize the material object of mathematics to an attempt to determine its formal object, i.e., its methodology."

"SEND + MORE = MONEY"

"He was an extraordinarily clear thinker: no-one could avoid more easily... the sort of confusions of thought to which even the best philosophers are liable, and he was capable of apprehending clearly... the subtlest distinctions. He had... an exceptional power of drawing conclusions from a complicated set of facts. ...his subtlety and ingenuity did not lead him, as it seems to have led some philosophers, to deny obvious facts. ...he could see which problems were the most fundamental, and it was these... which he was most anxious to solve. ...I almost always felt, with regard to any subject which we discussed, that he understood it much better than I did."

"Ramsey... had a most uncommon power of explaining clearly to others what he thought and why... But sometimes... he fails to explain things as clearly... simply because... he does not realize that what to him seems perfectly clear and straightforward may to others, less gifted, offer many puzzles. ...But even where you cannot understand him completely you can often understand him enough to find him extraordinarily interesting. ...even if he was wrong, [he] had very good reasons for the opinions at which he had arrived."

"With Ramsey’s young death, the world of learning was robbed of one of its most glittering stars. It is now time that he receive his due. What is needed is a thorough biography that would describe and place in intellectual history his important contributions to economics, mathematics, and philosophy, while keeping an eye out for what Virginia Woolf called the “fertile facts” that would reveal to us not only the impressive mind, but also the somewhat elusive personality of this extraordinary man."

"When he did descend from his accustomed stony heights, he still lived without effort in a rarer atmosphere than most economists care to breathe, and handled the technical apparatus of our science with the easy grace of someone accustomed to something far more difficult. But he has left behind him in print only two witnesses to his power - his papers published in The Economic Journal on 'A contribution to the Theory of Taxation' in March, 1927, and on 'A Mathematical Theory of Saving' in December, 1928. The latter of these is, I think, one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject, the power and elegance of the technical methods employed, and the clear purity of the illumination with which the writer's mind is felt by the reader to play about its subject. The article is terribly difficult reading for an economist, but it is not difficult to appreciate how scientific and aesthetic qualities are combined in it together."

"The loss of Ramsey is... to his friends, for whom his personal qualities joined most harmoniously with his intellectual powers, one which it will take them long to forget. His bulky Johnsonian frame, his spontaneous gurgling laugh, the simplicity of his feelings and reactions... his honesty of mind and heart, his modesty, and the amazing, easy efficiency of the intellectual machine which ground away behind his wide temples and broad smiling face, have been taken from us at the height of their excellence and before their harvest of work and life could be gathered in."

"But what we can't say we can't say, and we can't whistle it either."

"[M]y teacher Frank Ramsey... showed that if a scientific system was so completely precise that you could replace every word in it, such as "electrons," by the totality of all observations on the electron, then you could never discover anything new. ...Ramsey's theorem is really equivalent to all the Tarski-Turing theorems in essence because it says that if you push the symbolism even in a word like "mass" so that you say, as operationalists do... mass is everything you do when you weigh the mass, you are never going to discover that mass and energy are interchangeable. You have closed the system to new discoveries."

"[A]lthough my attempted reconstruction of the view of Whitehead and Russell overcomes, I think, many of the difficulties, it is impossible to regard it as altogether satisfactory."

"The assimilation of tautologies and contradictions with true and false propositions respectively results from the fact that tautologies and contradictions can be taken as truth-functions just like ordinary propositions, and for determining the truth of falsity of the truth-function, tautologies and contradictions among its arguments must be counted as true or false respectively. ...Are the propositions of symbolic logic and mathematics tautologies in Mr Wittgenstein's sense?"

"The death... is a heavy loss—though his primary interests were in Philosophy and Mathematical Logic—to the pure theory of economics. ...If he had followed the easier path of mere inclination, I am not sure that he would not have exchanged the tormenting exercises of the foundations of thought, where the mind tries to catch its own tail, for the delightful paths of our own most agreeable branch of the moral sciences, in which theory and fact, intuitive imagination and practical judgement, are blended in a manner comfortable to the human intellect."

"[W]e shall be concerned with the general nature of pure mathematics, and how it is distinguished from other sciences. Here there are... two distinct categories of things of which an account must be given—the ideas or concepts of mathematics, and the propositions of mathematics. ...the great majority of writers on the subject have concentrated their attention on the explanation of one or the other... and erroneously supposed that a satisfactory explanation of the other would immediately follow."

"[T]he formalist school, of whom the most eminent representative is Hilbert, have concentrated on the propositions of mathematics, such as '2 + 2 = 4'. They have pronounced these to be meaningless formulae to be manipulated according to arbitrary rules, and they hold that mathematical knowledge consists in knowing what formulae can be derived from what others consistently with the rules. ...for example...'2' is a meaningless mark occurring in these meaningless formulae. But... '2' occurs not only in '2 + 2 = 4', but also in 'It is 2 miles to the station', which is not a meaningless formulae, but a significant proposition, in which '2' cannot conceivably be a meaningless mark."

"The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with the general method of Frege, Whitehead and Russell. Following these authorities, I hold that mathematics is part of logic, and so belong to what may be called the logical school as opposed to the formalist and intuitionist schools. I have therefore taken Principia Mathematica as a basis for discussion and ammendment; and believe myself to have discovered how, by using the work of Mr Ludwig Wittgenstein, it can be rendered free from the serious objections which have caused its rejection by the majority of German authorities, who have deserted altogether its line of approach."

"It is worth pausing for a moment to consider how far our conclusions are affected by considerations which our simplifying assumptions have forced us to neglect."

"Philosophy must be of some use and we must take it seriously; it must clear our thoughts and so our actions. Or else it is a disposition that we have to check, and an inquiry to see that this is so; i.e. the chief proposition of philosophy is that philosophy is nonsense. And again we must then take seriously that it is nonsense, and not pretend, as Wittgenstein does, that it is important nonsense!"

"The formalists neglected the content altogether and made mathematics meaningless, the logicians neglected the form and made mathematics consist of any true generalizations; only by taking account of both sides and regarding it as composed of tautologous generalizations can we obtain an adequate theory."

"Tautologies and contradictions are not real propositions, but degenerate cases. ...Clearly, by negating a contradiction we get a tautology, and by negating a tautology a contradiction. ...A genuine proposition asserts something about reality, and it is true if reality is as it is asserted to be. But a tautology is a symbol constructed so as to say nothing whatever about reality, but to express total ignorance by agreeing with every possibility."

"The first problem I propose to tackle is this: how much of its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows. The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment."

"Just as his four-volume History is an indispensable aid to our discipline, his chemistry papers, his Higher Mathematics for Chemical Students, his Thermodynamics, his Specific Heats of Gases, and his huge Advanced Physical Chemistry remain monuments to the development of physical chemistry since the 1900s."

"Partington's method presupposes... that Greek fire and gunpowder represent premodern forms of "scientific" knowledge. ...Partington's second presumptuous belief is that the history of Greek fire and gunpowder is primarily to be understood through chemistry... These beliefs were what justified Partington's biographical approach; he was interested in those who wrote texts and what was written in them, mainly recipes and formulas. Perhaps the most familiar practitioner of this method is the founder of the modern founder of the history of science, George Sarton, whose Introduction to the History of Science (1927-47) Partington's work closely resembles."

"In 1919 he was appointed sole Professor of Chemistry at the East London College (renamed Queen Mary College in 1934). ...Partington chose to lecture exclusively on inorganic and physical chemistry. A compulsory one-term course on the history of chemistry that he introduced in 1919 was soon abandoned, though he revived it as an elective from 1945 onwards."

"The Greek chemical treatises contain... a great amount of practical chemical information... fusion, calcination, solution, filtration, crystallization, sublimation and especially distillation; and methods of heating include the open fire, lamps, and the sand and water baths. Nearly all this practical knowledge... the Arabs... derived... from the very source we are now considering."

"The Alexandrian chemists were very near to a recognition of gases."

"In Alexandria two streams of knowledge met and fused together... The ancient Egyptian industrial arts of metallurgy, dyeing and glass-making... and... the philosophical speculations of ancient Greece, now tinged with ancient mysticism, and partly transformed into that curious fruit of the tree of knowledge which we call Gnosticism. ...the result was the "divine" or "sacred" art (...also means sulphur) of making gold of silver. ...during the first four centuries a considerable body of knowledge came into existence. The treatises written in Greek... in Alexandria, are the earliest known books on chemistry. ...The treatises also contain much of an allegorical nature... sometimes described as "obscure mysticism." ...the Neoplatonism which was especially studied in Alexandria... is not so negligible as has sometimes been supposed. ...The study of astrology was connected with that of chemistry in the form of an association of the metals with the planets on a supposed basis of "sympathy". This goes back to early Chaldean sources but was developed by the Neoplatonists."

"A great number of our common ideas and ways of looking at the world were really shaped for us by the Greeks of antiquity, and... incorporated into the scientific knowledge of today. Such ideas as those of matter, force, element, number, space, time, etc., came to us from the ancient Greeks."

"We find Theophrastus (315 B.C.) describing... the manufacture of white lead... "lead is placed in an earthen vessel over sharp vinegar, and after it has acquired some thickness of a kind of rust... they open the vessels and scrape it off. ...repeating over and over again... til it is wholly gone. What has been scraped off they then beat to a powder and boil with water for a long time, and what at last settles to the bottom is white lead."

"The Chinese early learned to work in metals; bronze occurs in the 11th-10th centuries B.C., useful iron from about 500 B.C. At a later period they made brass... True porcelain was first made about A.D. 600. They were probably in possession of mercury at an early date, and learnt how to decompose cinnabar into mercury and sulphur, and recompose it from these materials."

"Side by side with the production of metals, the Egyptians and the inhabitants of Mesopotamia perfected the arts of making glazed pottery... and the production of glass. ...vessels were baked in tall closed furnaces. "Egyptian blue" was made in Egypt by heating silica with malachite and lime... applied with soda as a blue glaze on faience, and the blue glass is also colored with copper. Some early... Egyptian and Babylonian blue glass are coloured with cobalt."

"The blue dye indigo was obtained from the indigo plant by the Egyptians more than 4000 years ago. ...The famous and valuable "purple of Tyre" was perhaps first made in Crete in very early times... obtained at great cost... from tiny marine molluscs. ...The scarlet dye mentioned in the Bible was obtained from the kermes insect (hence the name "crimson")."

"The earliest applications of chemical processes were concerned with the extraction and working of metals and the manufacture of pottery. ...The irruption of an iron using race or races into Mediterranean sites ...introduced the Iron Age... but many of the oldest arts still survived in almost their original form. The potter, for example, still used nearly the same materials and appliances as Neolithic man."

"One of the most brilliant students we have had during the last thirty years."

"In early physical systems we have optics dealing with phenomena perceived by the eye; acoustics treating of auditory percepts, and so on. The subjective concepts of "tone" and "colour" have now been replaced by the objectified concepts of frequency of vibration; and wave-length. The object of this process of elimination is, according to Planck, the striving towards a unification of the whole theoretical system, so that it shall be equally significant for all intelligent beings."

"The first clear expression of the idea of an element occurs in the teachings of the Greek philosophers. ...Aristotle ...who summarized the theories of earlier thinkers, developed the view that all substances were made of a primary matter... On this, different forms could be impressed... so the idea of the transmutation of the elements arose. Aristotle's elements are really fundamental properties of matter... hotness, coldness, moistness, and dryness. By combining these in pairs, he obtained what are called the four elements, fire, air, earth and water... a fifth, immaterial, one was added, which appears in later writings as the quintessence. This corresponds with the ether. The elements were supposed to settle out naturally into the earth (below), water (the oceans), air (the atmosphere), fire and ether (the sky and heavenly bodies)."

"It is necessary to guard against a possible danger... of submitting too readily to the result of a so-called "crucial experiment". Very few experiments can, in the nature of things, be really crucial. One so-called "crucial experiment" which decided between Newton's corpuscular theory of light and Huyghens' wave-theory, viz. the relation between the law of refraction and the velocity of light, was not at all decisive."

"We perceive clearly that theories and hypotheses are not accepted or rejected outright; they have their periods of activity, and then lie dormant for a time, only to be revived in a new form later on."

"Disagreement between theory and experiment has proved a most potent agent in broadening theoretical views, and in making clear the necessity for new concepts or hypotheses."

"The philosopher Comte has made the statement that chemistry is a non-mathematical science. He also told us that astronomy had reached a stage when further progress was impossible. These remarks, coming after Dalton's atomic theory, and just before Guldberg and Waage were to lay the foundations of chemical dynamics, Kirchhoff to discover the reversal of lines in the solar spectrum, serve but to emphasize the folly of having "recourse to farfetched and abstracted Ratiocination," and should teach us to be "very far from the litigious humour of loving to wrangle about words or terms or notions as empty"."

"An explanation of a phenomenon is regarded, apparently instinctively, as the most general possible when it is a mechanical explanation. The "mechanism" of the process is the ultimate goal of experiment. Now this mechanism in general lies beyond the range of the senses; either by reason of their limitations, as in the case of the atomic structure of matter, or by the very nature of the supposed mechanism, as in the theory of the ether. The only way to bridge the gap between the machinery of the physical process and the world of sense-impressions is to think out some consequence of that mechanism. This we will call the hypothesis. The hypothesis, resting still on the mechanical basis, is yet beyond the range of direct experimental investigation; but if, by mathematical reasoning, a consequence of the hypothesis can be deduced, this will often lie within the range of experimental inquiry, and thus a test of the soundness of the original mechanical conception may be instituted."

"The fundamental materials from which we construct our picture of the universe may appear in different shapes, but there is really very little discontinuity between what seem at first sight very different views."

"As an instance of the remarkably far-reaching effect which a single mathematico-physical concept has had upon the development of chemical theory, one has but to recall the state of chemistry just before the revival of Avogadro's law by Cannizzaro, to be impressed by its confusion. Relying solely upon their "chemical instinct," the leaders of the various schools of chemical thought had developed each his own theoretical system. ...a host of ...conceptions strove for supremacy. The strife was stilled, order and unity were restored, as soon as Avogadro's great idea was seen in its true light, and the concept of the molecule was introduced into chemistry. A formula which had required pages of reasoning from a purely chemical standpoint to establish, and that insecurely, was fixed by a single numerical result."