Fellows Of The Royal Society

"I have regarded it as the highest goal of programming language design to enable good ideas to be elegantly expressed."

"I have learned more from my failures than can ever be revealed in the cold print of a scientific article. [...] [Failures] are much more fun to hear about afterwards; they are not so funny at the time."

"[About Ada] For none of the evidence we have so far can inspire confidence that this language has avoided any of the problems that have afflicted other complex language projects of the past. [...] It is not too late! I believe that by careful pruning of the ADA language, it is still possible to select a very powerful subset that would be reliable and efficient in implementation and safe and economic in use."

"[About Pascal] That is the great strength of PASCAL, that there are so few unnecessary features and almost no need for subsets. That is why the language is strong enough to support specialized extensions--Concurrent PASCAL for real time work, PASCAL PLUS for discrete event simulation, UCSD PASCAL for microprocessor work stations."

"[About PL/I] At first I hoped that such a technically unsound project would collapse but I soon realized it was doomed to success. Almost anything in software can be implemented, sold, and even used given enough determination. There is nothing a mere scientist can say that will stand against the flood of a hundred million dollars. But there is one quality that cannot be purchased in this way — and that is reliability. The price of reliability is the pursuit of the utmost simplicity. It is a price which the very rich find most hard to pay."

"[About Algol 68] The best we could do was to send with it a minority report, stating our considered view that, "… as a tool for the reliable creation of sophisticated programs, the language was a failure.""

"[About Algol W] It was not only a worthy successor of ALGOL 60, it was even a worthy predecessor of PASCAL […] I was astonished when the working group, consisting of all the best known international experts of programming languages, resolved to lay aside the commissioned draft on which we had all been working and swallow a line with such an unattractive bait."

"[About Algol 60 subset implementation] [E]very occurrence of every subscript of every subscripted variable was on every occasion checked at run time against both the upper and the lower declared bounds of the array. Many years later we asked our customers whether they wished us to provide an option to switch off these checks in the interests of efficiency on production runs. Unanimously, they urged us not to - they already knew how frequently subscript errors occur on production runs where failure to detect them could be disastrous. I note with fear and horror that even in 1980, language designers and users have not learned this lesson. In any respectable branch of engineering, failure to observe such elementary precautions would have long been against the law."

"[About Algol 60] Due credit must be paid to the genius of the designers of ALGOL 60 who included recursion in their language and enabled me to describe my invention [Quicksort] so elegantly to the world."

"There are two ways of constructing a software design: One way is to make it so simple that there are obviously no deficiencies, and the other way is to make it so complicated that there are no obvious deficiencies. The first method is far more difficult. It demands the same skill, devotion, insight, and even inspiration as the discovery of the simple physical laws which underlie the complex phenomena of nature."

"The most important property of a program is whether it accomplishes the intentions of its user."

"The real value of tests is not that they detect bugs in the code, but that they detect inadequacies in the methods, concentration, and skills of those who design and produce the code."

"Programming languages on the whole are very much more complicated than they used to be: object orientation, inheritance, and other features are still not really being thought through from the point of view of a coherent and scientifically well-based discipline or a theory of correctness. My original postulate, which I have been pursuing as a scientist all my life, is that one uses the criteria of correctness as a means of converging on a decent programming language design—one which doesn’t set traps for its users, and ones in which the different components of the program correspond clearly to different components of its specification, so you can reason compositionally about it. [...] The tools, including the compiler, have to be based on some theory of what it means to write a correct program."

"[About Algol 60] Here is a language so far ahead of its time, that it was not only an improvement on its predecessors, but also on nearly all its successors."

"I think it is a sad reflection on our civilisation that while we can and do measure the temperature in the atmosphere of Venus, we do not know what goes on inside our souffles."

"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."

"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."

"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."

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

"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."

"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"