University Of Cambridge Faculty

"In general, the nightmare quality of Marx's thought gives it, in this bedevilled age, an air of greater reality than the gentle complacency of the orthodox academics. Yet he, at the same time, is more encouraging than they, for he releases hope as well as terror from Pandora's box, while they preach only the gloomy doctrine that all is for the best in the best of all possible worlds."

"The fundamental differences between Marxian and traditional orthodox economics are, first, that the orthodox economists accept the capitalist system as part of the eternal order of Nature, while Marx regards it as a passing phase in the transition from the feudal economy of the past to the socialist economy of the future."

"Until recently, Marx used to be treated in academic circles with contemptuous silence, broken only by an occasional mocking footnote. But modern developments in academic theory, forced by modern developments in economic life — the analysis of monopoly and the analysis of unemployment — have shattered the structure of orthodox doctrine and destroyed the complacency with which economists were wont to view the working of laissez-faire capitalism. Their attitude to Marx, as the leading critic of capitalism, is therefore much less cocksure than it used to be. In my belief, they have much to learn from him."

"I began to read Capital, just as one reads any book, to see what was in it; I found a great deal that neither its followers nor its opponents had prepared me to expect."

"Progress is slow partly from mere intellectual inertia. In a subject where there is no agreed procedure for knocking out errors, doctrines have a long life. A professor teaches what he was taught, and his pupils, with a proper respect and reverence for teachers, set up a resistance against his critics for no other reason than that it was he whose pupils they were."

"The misery of being exploited by capitalists is nothing compared to the misery of not being exploited at all."

"It is the business of economists, not to tell us what to do, but show why what we are doing anyway is in accord with proper principles."

"Whatever you can rightly say about India, the opposite is also true."

"If I knew I was going to die today, I think I should still want to hear the cricket scores."

"Hardy in his thirties held the view that the late years of a mathematician's life were spent most profitably in writing books; I remember a particular conversation about this, and though we never spoke of the matter again it remained an understanding."

"To illustrate to what extent Hardy and Littlewood in the course of the years came to be considered as the leaders of recent English mathematical research, I may report what an excellent colleague once jokingly said: 'Nowadays, there are only three really great English mathematicians: Hardy, Littlewood, and Hardy-Littlewood.'"

"In 1933 Landau was dismissed from his [University of Göttingen] chair on the grounds of his race. An important colleague... Ludwig Bieberbach ...wrote the following lines in a treatise on Personality structure and mathematical creativity: "In this way... the ultimate reason behind the courageous rejection which the students at Göttingen University meted out to a great mathematician, Edmund Landau, was that his un-German style in research and teaching had become intolerable to German sensitivities. A people which has seen how alien desires for dominion are gnawing at its identity, how enemies of the people are working to impose their alien ways on it, must reject teachers of a type alien to it." The English mathematician Godfrey H. Hardy... responded to Bieberbach... "There are many of us, many English and many Germans, who said things during the (First) War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all costs not to be outdone, may be natural if not particularly heroic excuses. Prof. Bieberbach's reputation excludes such explanation for his utterances; and I find myself driven to the more uncharitable conclusion that he really believes them true.""

"Hardy was a great internationalist who worked with foreign mathematicians, visiting them, encouraging them to visit him and settling some, including Besicovitch, in England. There were some major probability figures in Hardy’s network: George Pólya (1887-1985) of Zürich, Norbert Wiener (1894-1964) of MIT and Harald Cramér (1893-1985) of Stockholm-appropriately Wiener and Cramér first met when visiting Hardy in 1920."

"It is never worth a first-class man's time to express a majority opinion. By definition, there are plenty of others to do that. (pg 46)"

"Bradman is a whole class above any batsman who has ever lived: if Archimedes, Newton and Gauss remain in the Hobbs class, I have to admit the possibility of a class above them, which I find difficult to imagine. They had better be moved from now on into the Bradman class. (pg 28)"

"No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

"I still say to myself when I am depressed and find myself forced to listen to pompous and tiresome people "Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms.""

"No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years."

"We have still one more question to consider. We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not"

"Pure mathematics is on the whole distinctly more useful than applied. [...] For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics."

"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way."

"...there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, [...] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real', but [...] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics."

"[T]here is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this temptation, he will be abandoning his purely mathematical position."

"The play is independent of the pages on which it is printed, and 'pure geometries' are independent of lecture rooms, [rough blackboard drawings] or of any other detail of the physical world. This is the point of view of a pure mathematician. Applied mathematicians, mathematical physicists... take a different view... preoccupied with the physical world itself, which also has its structure or pattern. ...We may be able to trace a ...resemblance between the two sets of relations, and then the pure geometry will become interesting to physicists; it will give us ...a map which 'fits the facts' ...The geometer offers ...a whole set of maps from which to choose."

"There is the science of pure geometry, in which there are many geometries, , , non-Euclidean geometry... [etc.]. Each... is a , a pattern of ideas... judged by the interest and beauty of... pattern. It is a map or picture, the... product of many hands, a partial and imperfect copy (yet exact so far as it extends) of a section of mathematical reality. But... there is one thing... of which pure geometries are not pictures, and that is the spatio-temporal reality of the physical world. ...[T]hey cannot be, since earthquakes and eclipses are not mathematical concepts."

"[M]athematical reality lies outside us ...our function is to discover or observe it, and ...the theorems ...we prove, and ...describe grandiloquently as our 'creations', are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards [...]"

"A man who could give a convincing account of mathematical reality would have solved very many of the most difficult problems of metaphysics. If he could include physical reality in his account, he would have solved them all."

"Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts. The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers."

"It is... natural to suppose that there is a great difference in utility between 'pure' and 'applied' mathematics. This is a delusion..."

"[S]cience works for evil as well as for good (...particularly ...in time of war); and... mathematicians may be justified in rejoicing that there is one science... their own, whose ...remoteness from ordinary human activities should keep it gentle and clean."

"[A] good deal of elementary mathematics... 'elementary' in the sense in which professional mathematicians use it... [e.g.,] knowledge of the differential and integral calculus—has considerable practical utility. These... are... rather dull... the parts which have least aesthetic value. The 'real' mathematics of the 'real' mathematicians... of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly 'useless' (...as true of 'applied' as of 'pure' mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of... 'utility'..."

"It is... astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility. ...We live either by or on other people's professional knowledge."

"I am interested in mathematics only as a creative art."

"A chess problem is genuine mathematics, but it is in some way 'trivial' mathematics. However ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful – important if you like, but the word is very ambiguous, and 'serious' expresses what I mean much better."

"Chess problems are the hymn-tunes of mathematics."

"', which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."

"The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."

"A painter makes patterns with shapes and colours, a poet with words. A painting may embody an ‘idea’, but the idea is usually commonplace and unimportant. In poetry, ideas count for a good deal more; but, [...] the importance of ideas in poetry is habitually exaggerated: '... Poetry is not the thing said but a way of saying it.' [In poetry,] the poverty of the ideas seems hardly to affect the beauty of the verbal pattern."

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean."

"Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself."

"If a man has any genuine talent, he should be ready to make almost any sacrifice in order to cultivate it to the full."

"… there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds."

"I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof."

"He could remember the idiosyncrasies of numbers in an almost uncanny way. It was Littlewood who said that every positive integer was one of Ramanujan's personal friends. I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.""

"If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof."

"We can see that, the constant in the law of gravitation being fixed, there may be some upper limit to the amount of matter possible; as more and more matter is added in the distant parts, space curves round and ultimately closes; the process of adding more matter must stop, because there is no more space, and we can only return to the region already dealt with. But there seems nothing to prevent a defect of matter, leaving space unclosed. Some mechanism seems to be needed, whereby either gravitation creates matter, or all the matter in the universe conspires to define a law of gravitation."

"Not only is the universe stranger than we imagine, it is stranger than we can imagine."

"With regard to the Newtonian concept of absolute rotation, Eddington admitted that Einstein's plenum does in fact provide a world-wide inertial frame, with respect to which it can be measured. Nevertheless, Eddington believed that Einstein attributed too important a role to matter, for in his universe it appears that not only the metrical properties, as in General Relativity, but the very existence of space depends on the existence of matter. Eddington preferred to regard matter as a manifestation of the "structure" of space-time."

"The solution … was found only after the rise of nuclear physics, and, strange to relate, was not known to Eddington when he developed his celebrated theory of stellar structure between 1916 and 1924. Indeed, it is one of the most intriguing facts in the history of science that the two most influential theories concerning the stars — Newton's theory of gravitation and Eddington's theory of stellar construction — were each developed so successfully although Newton was ignorant of the origin of gravitation and Eddington of the origin of stellar energy."