University Of Cambridge Faculty

"[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."

"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 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?"

"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."

"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 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."

"[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."

"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."

"[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 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."

"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."

"Though a skilled mathematician, he used mathematics sparingly. He saw that excessive reliance on this instrument might lead us astray in pursuit of intellectual toys, imaginary problems not conforming to the conditions of real life: and further, might distort our sense of proportion by causing us to neglect factors that could not easily be worked up in the mathematical machine."

"Marshall did something much more effective than changing the answer. He changed the question. For Ricardo the Theory of Value was a means of studying the distribution of total output between wages, rent and profit, each considered as a whole. This is a big question. Marshall turned the meaning of Value into a little question: Why does an egg cost more than a cup of tea? It may be a small question but it is a very difficult and complicated one. It takes a lot of time and algebra to work out the theory of it. So it kept all Marshall’s pupils preoccupied for fifty years. They had no time to think about the big question, or even to remember that there was a big question, because they had to keep their noses right down to the grindstone, working out the theory of the price of a cup of tea."

"One of the most important skills of the economist, therefore, is that of simplification of the model. Two important methods of simplification have been developed by economists. One is the method of partial equilibrium analysis (or microeconomics), generally associated with the name of Alfred Marshall and the other is the method of aggregation (or macro-economics), associated with the name of John Maynard Keynes."

"[Y]our letter concerning my paper... has told me much... concerning the ideas current in philosophical subjects in Cambridge. I was not aware that Marshall had so long entertained notions of a quantitative theory of political economy, and think it a pity that he has so long delayed publishing something on the subject."

"The study of economics does not seem to require any specialized gifts of an unusually high order. Is it not, intellectually regarded, a very easy subject compared with the higher branches of philosophy and pure science? Yet good, or even competent, economists are the rarest of birds. An easy subject, at which very few excel! The paradox finds its explanation, perhaps, in that the master-economist must possess a rare combination of gifts. He must reach a high standard in several different directions and must combine talents not often found together. He must be mathematician, historian, statesman, philosopher – in some degree. He must understand symbols and speak in words. He must contemplate the particular in terms of the general, and touch abstract and concrete in the same flight of thought. He must study the present in the light of the past for the purposes of the future. No part of man's nature or his institutions must lie entirely outside his regard. He must be purposeful and disinterested in a simultaneous mood; as aloof and incorruptible as an artist, yet sometimes as near the earth as a politician. Much, but not all, of this many-sidedness Marshall possessed. But chiefly his mixed training and divided nature furnished him with the most essential and fundamental of the economist's necessary gifts – he was conspicuously historian and mathematician, a dealer in the particular and the general, the temporal and the eternal, at the same time."

"Our understanding of how markets and businesses operate was passed down to us more than a century ago by a handful of European economists — Alfred Marshall in England and a few of his contemporaries on the continent. It is an understanding based squarely upon the assumption of diminishing returns: products or companies that get ahead in a market eventually run into limitations, so that a predictable equilibrium of prices and market shares is reached. The theory was roughly valid for the bulk-processing, smokestack economy of Marshall’s day. And it still thrives in today’s economics textbooks. But steadily and continuously in this century, Western economies have undergone a transformation from bulk - material manufacturing to design and use of technology — from processing of resources to processing of information, from application of raw energy to application of ideas. As this shift has occurred, the underlying mechanisms that determine economic behavior have shifted from ones of diminishing to ones of increasing returns."

"If we compare one country of the civilized world with another, or one part of England with another, or one trade in England with another, we find that the degradation of the working-classes varies almost uniformly with the amount of rough work done by women. The most valuable of all capital is that invested in human beings; and of that capital the most precious part is the result of the care and influence of the mother, so long as she retains her tender and unselfish instincts, and has not been hardened by the strain and stress of unfeminine work."

"Jevons saw the kettle boil and cried out with the delighted voice of a child; Marshall too had seen the kettle boil and sat down silently to build an engine."

"I have not been able to lay my hands on any notes as to Mathematico-economics that would be of any use to you. I have very indistinct memories of what I used to think on the subject. I never read mathematics now: in fact I have even forgotten how to integrate a good many things. But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very well unlikely to be good economics: and I went more and more on the rules—(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in (4), burn (3). This last I do often."

"There has always been a temptation to classify economic goods in clearly defined groups, about which a number of short and sharp propositions could be made, to gratify at once the student’s desire for logical precision, and the popular liking for dogmas that have the air of being profound and are yet easily handled. But great mischief seems to have been done by yielding to this temptation, and drawing broad artificial lines of division where Nature has made none. The more simple and absolute an economic doctrine is, the greater will be the confusion which it brings into attempts to apply economic doctrines to practice, if the dividing lines to which it refers cannot be found in real life. There is not in real life a clear line of division between things that are and are not Capital, or that are and are not Necessaries, or again between labour that is and is not Productive."

"Political Economy or Economics is a study of mankind in the ordinary business of life; it examines that part of individual and social action which is most closely connected with the attainment and with the use of the material requisites of wellbeing."

"I am myself an uncompromising anti-Jingoe, a peace-at-almost-any-price man. Chamberlain is the only Unionist public man whom I have ever thoroughly distrusted. Excepting Napoleon, I believe that England's true greatness has had no such dangerous enemy since Lord North. When a radical, he delighted to dish his colleagues even more than to flout his opponents. He is now engaged in dishing his new colleagues and flouting his old friends."

"[E]ach several want is limited, and... with every increase in the amount of a thing which a man has, the eagerness of his desire to obtain more of it diminishes; until it yields place to the desire for some other thing, of which perhaps he hardly thought, so long as his more urgent wants were still unsatisfied. There is an endless supply of wants, but there is a limit to each separate want. This familiar and fundamental law of human nature may pass by the name of the Law of Satiable Wants or the Law of Diminishing Utility. It may be written thus:— The Total Utility of a commodity to a person (...the total benefit or satisfaction yielded ...by it) increase with every increase in his stock of it, but not as fast as his stock increases. ...[i.e.,] the additional benefit ...from an additional increment of his stock of anything, diminishes with every increase in the stock ..."

"The increment of the commodity which he is only just induced to acquire (whether by... direct labor or by purchase) may be called its Marginal Increment; because he is on the margin of doubt whether it is worth his while to incur the outlay required to obtain it. And the benefit-giving power, or Utility, of that increment to him may be called the Marginal Utility of the commodity to him."

"And thus the law may be worded:— The of a commodity to anyone diminishes with every increase in the amount of it he already has."

"Fermat... adopted Kepler's notion of the increment of the variable becoming evanescent near a maximum or minimum value, and upon it based his method of drawing tangents. Fermat's method of finding the maximum or minimum... involved the differentiation of any explicit algebraic function, in the form that appears in any beginner's text book of today (though Fermat does not seem to have the "function" idea); that is, the maximum or minimum values of f(x) are the roots of f'(x) = 0, where f'(x) is the limiting value of [f(x+h) - f(x)]/h; only Fermat uses the letter e or E instead of h."

"Here then we have all the essentials for the calculus; but only for explicit integral algebraic functions, needing the binomial expansion of Newton, or a general method of rationalization which did not impose too great algebraic difficulties, for their further development; also, on the authority of Poisson, Fermat is placed out of court, in that he also only applied his method to certain special cases. Following the lead of Roberval, Newton subsequently used the third definition of a tangent, and the idea of time as the independent variable, although this was only to insure that one at least of his working variables should increase uniformly. This uniform increase of the independent variable would seem to have been usual for mathematicians of the period and to have persisted for some time; for later we find with Leibniz and the Bernoullis that d(dy/dx) = (d2y/dx2)dx. Barrow also used time as the independent variable in order that, like Newton, he might insure that one of his variables, a moving point or line or superficies, should proceed uniformly; ...Barrow... chose his own definition of a tangent, the second of those given above; and to this choice is due in great measure his advance over his predecessors. For his areas and volumes he followed the idea of Cavalieri and Roberval."

"Thus we see that in the time of Barrow, Newton, and Leibniz the ground had been surveyed, and in many directions levelled; all the material was at hand, and it only wanted the master mind to "finish the job." This was possible in two directions, by geometry or by analysis; each method wanted a master mind of a totally different type, and the men were forthcoming. For geometry, Barrow; for analysis, Newton, and Leibniz with his inspiration in the matter of the application of the simple and convenient notation of his calculus of s to s and to geometry. With all due honour to these three mathematical giants, however, I venture to assert that their discoveries would have been well-nigh impossible... if they had lived a hundred years earlier; with the possible exception of Barrow, who, being a geometer, was more dependent on the ancients and less on the moderns of his time than were the two analysts, they would have been sadly hampered but for the preliminary work of Descartes and the others I have mentioned (and some I have not—such as Oughtred), but especially Descartes."

"Galileo... would appear to have led the way, by the introduction of the theory of composition of motions into mechanics; he also was one of the first to use infinitesimals in geometry, and from... what is equivalent to "virtual velocities" it is... inferred that the idea of time as the independent variable is due to him. Kepler... was the first to introduce... infinity into geometry and to note that the increment of a variable was evanescent for values of the variable in the immediate neighbourhood of a maximum or minimum; in 1613, an abundant vintage drew his attention to the defective methods in use for estimating the cubical contents of vessels, and his essay on the subject (Nova Stereometria Doliorum) entitles him to rank amongst those who made the discovery of the infinitesimal calculus possible."

"In 1635 Cavalieri published a theory of "indivisibles," in which he considered a line as made up of an infinite number of points, a superficies as composed of a succession of lines, and a solid as a succession of superficies, thus laying the foundation for the "aggregations" of Barrow. Roberval seems... first, or... an independent, inventor of the method; but he lost credit... because he did not publish it, preferring to keep the method... for his own use... a usual thing... of that time, due perhaps to... professional jealousy. The method was severely criticized... especially by Guldin, but Pascal... showed that the method of indivisibles was as rigorous as... exhaustions... they were practically identical. ...[T]he progress... is much indebted to this defence by Pascal. Since this method is... analogous to... integration, Cavalieri and Roberval have... claim... as... inventors of... one branch of the calculus; if it were not for the fact that they only applied it to special cases, and seem... unable to generalize... owing to cumbrous algebraical notation, or to have failed to perceive the inner meaning... concealed under a geometrical form. Pascal... applied the method with great success, but also to special cases only; such as his work on the ."

"The next step was... more analytical... [B]y the method of indivisibles, Wallis... reduced... many areas and volumes to... the series (0^m + 1^m + 2^m +... n^m) / (n + 1)n^m, i.e. the ratio of the mean of all the terms to the last term, for integral values of n; and later he extended his method, by a theory of , to fractional values of n. Thus the idea of the Integral Calculus was in a fairly advanced stage in the days immediately antecedent to Barrow."

"The beginnings of the Infinitesimal Calculus... arose from determinations of areas and volumes, and the finding of tangents to plane curves. The ancients attacked the problems in a strictly geometrical manner, making use of the "s." ... This was the method by means of which Archimedes proved most of his discoveries. But there seems to have been some distrust of the method, for we find... many... discoveries... proved by a '..."

"I have to all intents rewritten Barrow’s book; although throughout I... adhered... to Barrows own words. I have only retained those parts which seemed... essential for the purpose in hand. This was necessary... that room might be found for... critical notes on the theorems.., proofs omitted by Barrow, which when given in Barrow’s style, and afterwards translated into analysis, had an important bearing on the point as to how he found out the more difficult of his constructions; and lastly for deductions therefrom that point steadily, one after the other, to the fact that Barrow was writing a calculus and knew that he was inventing a great thing."

"I have used three distinct kinds of type: the most widely spaced type has been used for Barrow’s own words; only very occasionally have I inserted anything of my own in this, and then it will be found enclosed in heavy square brackets, that the reader will have no chance of confusing my explanations with the text... Barrow makes use of parentheses very frequently, so that the reader must understand that only remarks in heavy square brackets are mine... The small type is used for footnotes only. In the notes I... use the Leibniz notation, because it will... convey my meaning better; but there was really no absolute necessity for this, Barrow’s a and e, or its modern equivalent, h and k, would have done quite as well."

"What Cavalieri and Roberval did for the integral calculus, Descartes... accomplished for the differential branch by his... application of algebra to geometry. Cartesian coordinates made possible the extension of... drawing... tangents to... curves of any kind. ...[H]e habitually used the index notation ...this had a very great deal to do with ...Newton's discovery of the general binomial expansion and of many other infinite series. Descartes failed, however, to make... great progress... in... drawing of s, owing to... an unfortunate choice of a definition for a tangent to a curve in general. Euclid's circle-tangent definition being more or less hopeless in the general case, Descartes had the choice of three:—"

"During the next year Newton began to "reflect on his method of fluxions," and actually did produce his Analysis per Æquationes. This, though composed in 1666, was not published until 1711."

"Leibniz bought a copy of Barrow’s work in 1673, and was able "to communicate a candid account of his calculus to Newton" in 1677. In... the face of Leibniz’ persistent denial that he received any assistance whatever from Barrow’s book... bear... in mind Leibniz’ twofold idea of the "calculus":— (i) the freeing of the matter from geometry, (ii) the adoption of a convenient notation. ...[O]n these two points ...he derived not the slightest assistance from Barrow’s work; for the first ...would be dead against Barrow’s practice and instinct, and of the second Barrow had no knowledge whatever. ...[F]or [these points] ...the world has to thank Leibniz; but their inception does not mean the invention of the infinitesimal calculus. This, the epitome of the work of his predecessors, and its completion by his own discoveries until it formed a perfected method of dealing with the problems of tangents and areas for any curve in general, i.e. ...the differentiation and integration of any function whatever (such as were known in Barrow’s time), must be ascribed to Barrow."

"My attention was arrested by a theorem in which Barrow... rectified the cycloid, which... has usually been ascribed to Sir C. Wren. ...What I found induced me to treat a number of the theorems ...I came to the conclusion that Barrow had got the calculus; but I queried even then whether Barrow himself recognized the fact."

"He further developed... [infinitesimal calculus] in the years 1662-3-4, and in the latter year probably had it fairly complete. In this year he communicated to Newton the great secret of his geometrical constructions... and it was probably this that set Newton to... attempt to express everything as a sum of powers of the variable."

"Isaac Barrow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow’s work, obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Barrow’s book that he purchased in 1673."

"By the "Infinitesimal Calculus," I intend "a complete set of standard forms for both the differential and integral sections of the subject, together with rules for their combination, such as for a product, a quotient, or a power of a function; and also a recognition and demonstration of the fact that differentiation and integration are inverse operations.""

"Only on completing my annotation of the last chapter of this volume, Lect. XII, App. III, did I come to the conclusion that is given as the opening sentence of this Preface; for I then found that a batch of theorems.., on careful revision, turned out to be the few missing standard forms, necessary for completing the set for integration; and that one of his problems was a practical rule for finding the area under any curve, such as would not yield to the theoretical rules he had given, under the guise of an "inverse-tangent" problem."

"Barrow was familiar with the paraboliforms, and s and areas connected with them, in from 1655 to 1660 at the very latest; hence he could... differentiate and integrate by his own method any rational positive power of a variable, and thus also a sum of such powers."

"As a Line, I say, is looked upon to be the Trace of a Point moving forward, being in some sort divisible by a Point, and may be divided by Motion one Way, viz. as to Length; so Time may be conceiv'd as the Trace of a Moment continually flowing, having some Kind of Divisibility from an Instant, and from a successive Flux, inasmuch as it can be divided some how or other. And like as the Quantity of a Line consists of but one Length following the Motion; so the Quantity of Time pursues but one Succession stretched out as it were in Length, which the Length of the Space moved over shews and determines. We therefore shall always express Time by a right Line; first, indeed, taken or laid down at Pleasure, but whose Parts will exactly answer to the proportionable Parts of Time, as its Points do to the respective Instants of Time, and will aptly serve to represent them. Thus much for Time."

"J. M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Paul Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a differential and integral calculus is to do violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz."

"[R]ecent work... removes him from a major role in the development of the calculus or in Newton's early mathematical thinking. ...[T]he new symbolic algebra... was essential to calculus. Barrow accepted neither the art nor the notion, and... avoided the technique wherever possible. The theorems in the Geometrical Lectures that historians juxtapose to form the belong to... different lines of inquiry, one... characterized by its freedom from... "tediousness of calculation" and divorced in Barrow's mind from... s or limits. The central lectures contain a program of research, but... not... the program that led Newton and Leibniz to the calculus."

"It cannot be justly inferr'd... We do not perceive the Thing, therefore there is no such Thing, that is a false Illusion, a deceitful Dream, that wou'd cause us to join together two remote Instants of Time. But nevertheless this is very True... That is, for as much Motion as there was, so much Time seems to have been elapsed; nor, when we mention such a Quantity of Time, do we merely mean any Thing else, than the Performance of so much Motion, to the continued successive Extension of which we imagine the Permanency as Things is co-extended."

"[T]he conclusion is the effect of a gradual accumulation of evidence... I have given a wholly inadequate account of the work of Barrow’s immediate predecessors; but... to a sufficiency for... showing... the time was... ripe for the work of Barrow, Newton, and Leibniz."

", "Barrow's Mathematics: between ancients and moderns" Before Newton: the Life and Times of Isaac Barrow (1990) ed., Mordechai Feingold."