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SNP addresses are a pathogen typing method based on whole-genome sequences (WGSs), assigning groups at seven different levels of genetic similarity. Public health surveillance uses it for several gastro-intestinal infections; this work trialled its use in veterinary surveillance for salmonella outbreak detection. Comparisons were made between temporal and spatio-temporal cluster detection models that either defined cases by their SNP address or by phage type, using historical data sets. Clusters of SNP incidents were effectively detected by both methods, but spatio-temporal models consistently detected these clusters earlier than the corresponding temporal models. Unlike phage type, SNP addresses appeared spatially and temporally limited, which facilitated the differentiation of novel, stable, or expanding clusters in spatio-temporal models. Furthermore, these models flagged spatio-temporal clusters containing only two to three cases at first detection, compared with a median of seven cases in phage-type models. The large number of SNP addresses will require automated methods to implement these detection models routinely. Further work is required to explore how temporal changes and different host species may impact the sensitivity and specificity of cluster detection. In conclusion, given validation with more sequencing data, SNP addresses are likely to be a valuable addition to early warning systems in veterinary surveillance.
G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician … the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his idiosyncrasies and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times.
The notion that our society, its education system and its intellectual life, is characterised by a split between two cultures – the arts or humanities on one hand and the sciences on the other – has a long history. But it was C. P. Snow's Rede lecture of 1959 that brought it to prominence and began a public debate that is still raging in the media today. This fiftieth anniversary printing of The Two Cultures and its successor piece, A Second Look (in which Snow responded to the controversy four years later) features an introduction by Stefan Collini, charting the history and context of the debate, its implications and its afterlife. The importance of science and technology in policy run largely by non-scientists, the future for education and research, and the problem of fragmentation threatening hopes for a common culture are just some of the subjects discussed.
We present a high precision frequency determination method for digitized NMR FID signals. The method employs high precision numerical integration rather than simple summation as in many other techniques. With no independent knowledge of the other parameters of a NMR FID signal (phase ф, amplitude A, and transverse relaxation time T2) this method can determine the signal frequency f0 with a precision of if the observation time T ≫ T2. The method is especially convenient when the detailed shape of the observed FT NMR spectrum is not well defined. When T2 is +∞ and the signal becomes pure sinusoidal, the precision of the method is which is one order more precise than the ±1 count error induced precision of a typical frequency counter. Analysis of this method shows that the integration reduces the noise by bandwidth narrowing as in a lock-in amplifier, and no extra signal filters are needed. For a pure sinusoidal signal we find from numerical simulations that the noise-induced error in this method reaches the Cramer-Rao Lower Band (CRLB) on frequency determination. For the damped sinusoidal case of most interest, the noise-induced error is found to be within a factor of 2 of CRLB when the measurement time T is 2 or 3 times larger than T2. We discuss possible improvements for the precision of this method.
The notion that our society, its education system and its intellectual life, is characterised by a split between two cultures – the arts or humanities on one hand, and the sciences on the other – has a long history. But it was C. P. Snow's Rede lecture of 1959 that brought it to prominence and began a public debate that is still raging in the media today. This 50th anniversary printing of The Two Cultures and its successor piece, A Second Look (in which Snow responded to the controversy four years later) features an introduction by Stefan Collini, charting the history and context of the debate, its implications and its afterlife. The importance of science and technology in policy run largely by non-scientists, the future for education and research, and the problem of fragmentation threatening hopes for a common culture are just some of the subjects discussed.
G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician … the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940 as his mathematical powers were declining, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his aphorisms and idiosyncrasies, and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times.
I must return to my Oxford apology, and examine a little more carefully some of the points which I postponed in § 6. It will be obvious by now that I am interested in mathematics only as a creative art. But there are other questions to be considered, and in particular that of the ‘utility’ (or uselessness) of mathematics, about which there is much confusion of thought. We must also consider whether mathematics is really quite so ‘harmless’ as I took for granted in my Oxford lecture.
A science or an art may be said to be ‘useful’ if its development increases, even indirectly, the material well-being and comfort of men, if it promotes happiness, using that word in a crude and commonplace way. Thus medicine and physiology are useful because they relieve suffering, and engineering is useful because it helps us to build houses and bridges, and so to raise the standard of life (engineering, of course, does harm as well, but that is not the question at the moment). Now some mathematics is certainly useful in this way; the engineers could not do their job without a fair working knowledge of mathematics, and mathematics is beginning to find applications even in physiology.
There is still one point remaining over from §11, where I started the comparison between ‘real mathematics’ and chess. We may take it for granted now that in substance, seriousness, significance, the advantage of the real mathematical theorem is overwhelming. It is almost equally obvious, to a trained intelligence, that it has a great advantage in beauty also; but this advantage is much harder to define or locate, since the main defect of the chess problem is plainly its ‘triviality’, and the contrast in this respect mingles with and disturbs any more purely aesthetic judgement. What ‘purely aesthetic’ qualities can we distinguish in such theorems as Euclid's and Pythagoras's? I will not risk more than a few disjointed remarks.
In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency.
I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics.
We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time that he was forty that his great creative days were over. His greatest ideas of all, fluxions and the law of gravitation, came to him about 1666, when he was twenty-four—‘in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since’. He made big discoveries until he was nearly forty (the ‘elliptic orbit’ at thirtyseven), but after that he did little but polish and perfect.
Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty.
It may be objected that my concept of ‘utility’ has been too narrow, that I have defined it in terms of ‘happiness’ or ‘comfort’ only, and have ignored the general ‘social’ effects of mathematics on which recent writers, with very different sympathies, have laid so much stress. Thus Whitehead (who has been a mathematician) speaks of ‘the tremendous effect of mathematical knowledge on the lives of men, on their daily avocations, on the organization of society’; and Hogben (who is as unsympathetic to what I and other mathematicians call mathematics as White-head is sympathetic) says that ‘without a knowledge of mathematics, the grammar of size and order, we cannot plan the rational society in which there will be leisure for all and poverty for none’ (and much more to the same effect).
I cannot really believe that all this eloquence will do much to comfort mathematicians. The language of both writers is violently exaggerated, and both of them ignore very obvious distinctions. This is very natural in Hogben's case, since he is admittedly not a mathematician; he means by ‘mathematics’ the mathematics which he can understand, and which I have called ‘school’ mathematics. This mathematics has many uses, which I have admitted, which we can call ‘social’ if we please, and which Hogben enforces with many interesting appeals to the history of mathematical discovery.
There is another misconception against which we must guard. It is quite natural to suppose that there is a great difference in utility between ‘pure’ and ‘applied’ mathematics. This is a delusion: there is a sharp distinction between the two kinds of mathematics, which I will explain in a moment, but it hardly affects their utility.
How do pure and applied mathematics differ from one another? This is a question which can be answered definitely and about which there is general agreement among mathematicians. There will be nothing in the least unorthodox about my answer, but it needs a little preface.
My next two sections will have a mildly philosophical flavour. The philosophy will not cut deep, or be in any way vital to my main theses; but I shall use words which are used very frequently with definite philosophical implications, and a reader might well become confused if I did not explain how I shall use them.
I have often used the adjective ‘real’, and as we use it commonly in conversation. I have spoken of ‘real mathematics’ and ‘real mathematicians’, as I might have spoken of ‘real poetry’ or ‘real poets’, and I shall continue to do so. But I shall also use the word ‘reality’, and with two different connotations.
All this is very comforting for dons, and especially for professors of mathematics. It is sometimes suggested, by lawyers or politicians or business men, that an academic career is one sought mainly by cautious and unambitious persons who care primarily for comfort and security. The reproach is quite misplaced. A don surrenders something, and in particular the chance of making large sums of money—it is very hard for a professor to make £2000 a year; and security of tenure is naturally one of the considerations which make this particular surrender easy. That is not why Housman would have refused to be Lord Simon or Lord Beaverbrook. He would have rejected their careers because of his ambition, because he would have scorned to be a man to be forgotten in twenty years.
Yet how painful it is to feel that, with all these advantages, one may fail. I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about a.d. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down book after book, glancing at them, restoring them to the shelves or dumping them into the bucket.
It is about three years since I made a sketch in print of a problem which had been on my mind for some time. It was a problem I could not avoid just because of the circumstances of my life. The only credentials I had to ruminate on the subject at all came through those circumstances, through nothing more than a set of chances. Anyone with similar experience would have seen much the same things and I think made very much the same comments about them. It just happened to be an unusual experience. By training I was a scientist: by vocation I was a writer. That was all. It was a piece of luck, if you like, that arose through coming from a poor home.
But my personal history isn't the point now. All that I need say is that I came to Cambridge and did a bit of research here at a time of major scientific activity. I was privileged to have a ringside view of one of the most wonderful creative periods in all physics. And it happened through the flukes of war—including meeting W. L. Bragg in the buffet on Kettering station on a very cold morning in 1939, which had a determining influence on my practical life—that I was able, and indeed morally forced, to keep that ringside view ever since. So for thirty years I have had to be in touch with scientists not only out of curiosity, but as part of a working existence.
At a few minutes past five o'clock in the afternoon of 7 May 1959, a bulky, shambling figure approached the lectern at the western end of the Senate House in Cambridge. In the body of the ornately plastered neoclassical building sat a large gathering of dons and students, together with a number of distinguished guests, who had assembled for one of Cambridge's show-piece public occasions, the annual Rede lecture. The figure who was about to address them was C.P. Snow (then more formally styled Sir Charles, soon to be Lord Snow, but known throughout the world by his initials). Snow had been a research scientist; he had high-level administrative experience in the Civil Service and in private industry; he was a successful novelist and prominent reviewer; and he had now achieved the indefinable status of a ‘public figure’, licensed to announce his opinions on all manner of topics. By the time he sat down over an hour later, Snow had done at least three things: he had launched a phrase, perhaps even a concept, on an unstoppably successful international career; he had formulated a question (or, as it turned out, several questions) which any reflective observer of modern societies needs to address; and he had started a controversy which was to be remarkable for its scope, its duration, and, at least at times, its intensity.
The title of Snow's lecture was ‘The Two Cultures and the Scientific Revolution’.
I am indebted for many valuable criticisms to Professor C. D. Broad and Dr C. P. Snow, each of whom read my original manuscript. I have incorporated the substance of nearly all of their suggestions in my text, and have so removed a good many crudities and obscurities.
In one case I have dealt with them differently. My § 28 is based on a short article which I contributed to Eureka (the journal of the Cambridge Archimedean Society) early in the year, and I found it impossible to remodel what I had written so recently and with so much care. Also, if I had tried to meet such important criticisms seriously, I should have had to expand this section so much as to destroy the whole balance of my essay. I have therefore left it unaltered, but have added a short statement of the chief points made by my critics in a note at the end.
If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of gratifying them than a mathematician. His subject is the most curious of all—there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.
We can see this even in semi-historic civilizations. The Babylonian and Assyrian civilizations have perished; Hammurabi, Sargon, and Nebuchadnezzar are empty names; yet Babylonian mathematics is still interesting, and the Babylonian scale of 60 is still used in astronomy. But of course the crucial case is that of the Greeks.
The Greeks were the first mathematicians who are still ‘real’ to us to-day. Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand; as Littlewood said to me once, they are not clever schoolboys or ‘scholarship candidates’, but ‘Fellows of another college’.
A ‘serious’ theorem is a theorem which contains ‘significant’ ideas, and I suppose that I ought to try to analyse a little more closely the qualities which make a mathematical idea significant. This is very difficult, and it is unlikely that any analysis which I can give will be very valuable. We can recognize a ‘significant’ idea when we see it, as we can those which occur in my two standard theorems; but this power of recognition requires a rather high degree of mathematical sophistication, and of that familiarity with mathematical ideas which comes only from many years spent in their company. So I must attempt some sort of analysis; and it should be possible to make one which, however inadequate, is sound and intelligible so far as it goes. There are two things at any rate which seem essential, a certain generality and a certain depth; but neither quality is easy to define at all precisely.
A significant mathematical idea, a serious mathematical theorem, should be ‘general’ in some such sense as this. The idea should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds. The theorem should be one which, even if stated originally (like Pythagoras's theorem) in a quite special form, is capable of considerable extension and is typical of a whole class of theorems of its kind.