Mathematicians From England

"If God holds all mankind guilty for the sin of Adam, if he has visited upon the innocent the punishment of the guilty, if he is to torture any single soul for ever, then it is wrong to worship him."

"A mervaylous newtrality have these thinges Mathematicall, and also a straunge participatió betwene thynges supernaturall, immortall, intellectual, simple and indivisible: and thynges naturall, mortall, sensible, compounded and divisible."

"Dee goes so far as to assert that, although he called the work hieroglyphic, it is endowed with a clarity and rigour almost mathematical; yet at the same time he leaves it to the reader even to guess that the subject of the elaborate display, which he is asked to view in such dim light, is the hermetic quest. The semblance of clarity is achieved by discussing the dark subject under the guise of a symbolic sign invented by Dee, which is his monad. This symbol indeed lends itself easily to digressive secondary interpretations of a numerological, cabbalistic, astrological, cosmological, or mathematical nature, all which, however, are without any doubt given so as to establish significant connexions with the all-embracing central theme, alchemy, which is barely mentioned."

"Such people, the real pioneers of the seaborne empire, were practical men: Captains Ralph Lane, John Smith and other soldiers of early Virginia, the merchant-commanders of the East Indiamen, the Arctic explorers from the Borough brothers to Baffin and the sea-captains of the Atlantic. Yet for success the movement needed also the moving power of ideas propagated by such men as Dee, Hakluyt, Ralegh and Drake. For who can doubt the inspiring force of Dee's Faustian genius, which cast a spell on the minds of mariners and intellectuals alike?"

"[T]he idea of a maritime empire did not immediately seize the imagination of the English people, nor did it arise spontaneously. It had to be propagated. John Dee, who invented the phrase "British Impire", was the most diligent and influential of the first generation of imperialists. He it was who sketched out the English claim to an empire of the North Atlantic, embracing the Arctic to the east and the west, the northern islands, including mythical Frisland, conquered allegedly by King Arthur and King Malgo, and Atlantis (the continent of North America), first discovered by the Welsh prince Owen Madoc in the twelfth century. All this Dee set down in 1578 in a statement for the queen of "Her Majesty's Title Royal" to the lands in question, depicted in his 1580 map of Atlantis. Thus the necessary underpinning of myth was provided, to be duly repeated by Peckham, Hakluyt and many more."

"O comfortable allurement, O ravishing persuasion to deal with a science whose subject is so ancient, so pure, so excellent, so surrounding all creatures, so used of the almighty and incomprehensible wisdom of the Creator, in distinct creation of all creatures: in all their distinct parts, properties, natures, and virtues, by order, and most absolute number, brought from nothing to the formality of their being and state."

"There is (gentle reader) nothing (the works of God only set apart) which so much beautifies and adorns the soul and mind of man as does knowledge of the good arts and sciences. Many arts there are which beautify the mind of man; but of all none do more garnish and beautify it than those arts which are called mathematical, unto the knowledge of which no man can attain, without perfect knowledge and instruction of the principles, grounds, and Elements of Geometry."

"We finish the brief hieroglyphic consideration of our Monad, which we would sum up in one only hieroglyphic context: The Sun and the Moon of this Monad desire that the Elements in which the tenth proportion will flower, shall be separated, and this is done by the application of Fire."

"Although the semicircle of the Moon is placed above the circle of the Sun and would appear to be superior, nevertheless we know that the Sun is ruler and King. We see that the Moon in her shape and her proximity rivals the Sun with her grandeur, which is apparent to ordinary men, yet the face, or a semi-sphere of the Moon, always reflects the light of the Sun."

"Therefore, the central point which we see in the centre of the hieroglyphic Monad produces the Earth, round which the Sun, the Moon, and the other planets follow their respective paths. The Sun has the supreme dignity, and we represent him by a circle having a visible centre."

"Neither the circle without the line, nor the line without the point, can be artificially produced. It is, therefore, by virtue of the point and the Monad that all things commence to emerge in principle. That which is affected at the periphery, however large it may be, cannot in any way lack the support of the central point."

"It is by the straight line and the circle that the first and most simple example and representation of all things may be demonstrated, whether such things be either non-existent or merely hidden under Nature's veils."

"Cut that in Three, which Nature hath made One, Then strengthen hyt, even by it self alone, Wherewith then Cutte the poudred Sonne in twayne, By length of tyme, and heale the woonde againe. The self same Sunne twys yet more, ye must wounde, Still with new Knives, of the same kinde, and grounde; Our Monas trewe thus use by natures Law, Both binde and lewse, only with rype and rawe, And ay thanke God who only is our Guyde, All is ynugh, no more then at this Tyde."

"The derivates theorem enables one to reject certain parts of the thing one wants to tend to zero. One day I was playing round with this, and a ghost of an idea entered my mind of making r, the number of differentiations, large. At that moment the spring cleaning that was in progress reached the room I was working in, and there was nothing for it but to go walking for 2 hours, in pouring rain. The problem seethed violently in my mind: the material was disordered and cluttered up with irrelevant complications cleared away in the final version, and the 'idea' was vague and elusive. Finally I stopped, in the rain, gazing blankly for minutes on end over a little bridge into a stream (near Kenwith wood), and presently a flooding certainty came into my mind that the thing was done. The 40 minutes before I got back and could verify were none the less tense."

"In 1908 Herman quoted to my father an extract from the main report on my first Fellowship thesis. 'He is to be a great mathematician, and his work is singularly mature.' (Herman, on the evidence of my Minor Scholarship and perhaps a first term's work, broke it to my father that he couldn't expect much of me. I'm not sure when this was reversed, and my father knew the ropes enough not to build too much on a Senior Wranglership. I remember, to my surprise, as being a prodigy of 14 in South Africa, his implying that I might expect at Cambridge to be nearly as good as himself, or possibly a shade better. The Fellowship report was something of a surprise; he said 'I feel like a hen that has hatched an eagle.')"

"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.'"

"There is much to be said for being a mathematician. To begin with, he has to be completely honest in his work, not from any superior morality, but because he simply cannot get away with a fake. ... A mathematician's normal day contains hours of close concentration, and leaves him jaded in the evening. ... This is why we tend to relax either on mild nonfiction like biographies, or - to be crude, and to the derision of arts people - on trash. There is, of course, good trash and bad trash. ... Minor depressions will occur, and most of a mathematician's life is spent in frustration, punctuated with rare inspirations. A beginner can't expect quick results; if they are quick they are pretty sure to be poor. ... When one has finished a substantial paper there is commonly a mood in which it seems that there is really nothing in it. Do not worry, later on you will be thinking 'At least I could do something good then.' At the end of a particularly long and exacting work there can be a strange melancholy. This, however, is romantic, and mildly pleasant, like some other melancholies."

"I had been struggling for two months to prove a result I was pretty sure was true. When I was walking up a Swiss mountain, fully occupied by the effort, a very odd device emerged - so odd that, though it worked, I could not grasp the resulting proof as a whole. But not only so; I had a sense that my subconscious was saying, 'Are you never going to do it, confound you; try this.'"

"'Always verify references.' This is so absurd in mathematics that I used to say provocatively: 'never...' When I began writing I innocently adopted the French habit of M. (Monsieur) in front of any surname. I thus created a ghost mathematician M. Landau (to whom some 'non-verified' references were made)."

"G. H. Hardy said he thought on paper ('with my pen'). He wrote everything out (in his invariably admirable handwriting), scrapping and copying whenever a page got into a mess. When I am thinking about a difficult problem everything goes onto a single page - all over the place with odd equations, diagrams, rings. However appalling the mess, I feel that to scrap this page would somehow break threads in the unconscious."

"[For an unproved Lemma] I had what looked like a promising idea for this, but it was fallacious. In the middle of a three week holiday - mathematics completely below the horizon - the idea came again out of the blue when I was in bed. I forgot it had been rejected as fallacious, and this forgetting did the trick; because this time I noticed that it did prove something, and, by what was nearly, or quite, automatic writing, a proof got written down which deduced the Lemma from the something. ... I did experience automatic writing again. ... my pencil wrote down the formula [omitted] for no reason at all, and almost unattended by consciousness. On the face of it the formula has no apparent connexion with the problem, but it turns out to be an essential key to the proof. ... as if my subconsciousness knew the thing all the time, and finally got impatient."

"It is possible for a mathematician to be 'too strong' for a given occasion. He forces through, where another might be driven to a different, and possibly more fruitful, approach. (So a rock climber might force a dreadful crack, instead of finding a subtle and delicate route.)"

"The Astronomer's fallacy. It is very hard to make a random selection of stars. If, for example, you see a star (with the naked eye) it is probably bright (as stars go). A lecturer was once making the point that middle class families were smaller than lower class ones. As a test he asked everyone to write down the number of children in his family. The average was larger than the lower class average. The obvious point he overlooked were that zero families were unrepresented in the audience. But further, families of n have a probability of being represented proportional to n; with all this, the result is to be expected."

"Besicovitch and Harry Williams asked me what God was doing before the Creation. I said: 'Millions of words must have been written on this; but he was doing Pure Mathematics and thought it would be a pleasant change to do some Applied.'"

"Improbabilities are apt to be overestimated. It is true that I should have been surprised in the past to learn that Professor Hardy had joined the Oxford Group. But one could not say the adverse chance was 10⁶ : 1. Mathematics is a dangerous profession; an appreciable proportion of us go mad, and then this particular event would be quite likely. ... There must exist a collection of well-authenticated coincidences, and I regret that I am not better acquainted with them. ... I sometimes ask the question: what is the most remarkable coincidence you have experienced, and is it, for the most remarkable one, remarkable? (With a lifetime to choose from, 10⁶ : 1 is a mere trifle.) ... Eddington once told me that information about a new (newly visible, not necessarily unknown) comet was received by an Observatory in misprinted form; they looked at the place indicated (no doubt sweeping a square degree or so), and saw a new comet. ..."

"My research began, naturally, in the Long Vacation of my 3rd year, 1906. My director of studies (and tutor) E. W. Barnes suggested the subject of integral functions of order 0... [After success,] Barnes was now encouraged to suggest a new problem: 'Prove the Riemann Hypothesis'."

"The difference when you do know (when, for example, we are looking for a new proof) is enormous. Like a Bridge problem 'if the thing is possible it must be necessary to lead the Ace and trump with the Ace in Dummy.'"

"I began on a question on elementary theory of numbers, in which I felt safe in my school days. It did not come out, nor did it on a later attack. I had occasion to fetch more paper; when passing a desk my eye lit on a heavy mark against the question. The candidate was not one of the leading people, and I half unconsciously inferred that I was making unnecessarily heavy weather; the question then came out fairly easily. The perfectly high-minded man would no doubt have abstained from further attack; I wish I had done so, but the offence does not lie very heavily on my conscience."

"If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: 'If this is what you want, try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician'. In practice he is apt to say: 'try this; if it works that will justify it'. But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence."

"I recall once saying that when I had given the same lecture several times I couldn't help feeling that they really ought to know it by now."

"I read in the proof-sheets of Hardy on Ramanujan: 'As someone said, each of the positive integers was one of his personal friends.' My reaction was, 'I wonder who said that; I wish I had.' In the next proof-sheets I read (what now stands): 'It was Littlewood who said...' (What had happened was that Hardy had received the remark in silence and with a poker face, and I wrote it off as a dud....)"

"Landau kept a printed form for dealing with proofs of Fermat's last theorem. 'On page blank, lines blank to blank, you will find there is a mistake.'"

"'The surprising thing about this paper is that a man who could write it--would.'"

"(A. S. Besicovitch) A mathematician's reputation rests on the number of bad proofs he has given. (Pioneer work is clumsy.)"

"A good mathematical joke is better, and better mathematics, than a dozen mediocre papers."

"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"."

"It was Mr. Littlewood (I believe) who remarked that "every positive integer was one of his personal friends.""

"I, your Flaccus, am busy carrying out your wishes and instructions at St. Martin's, giving some the honey of the holy scriptures, making others drunk on the old wine of ancient learning."

"The Northmen are often said to have burst out of their coastal settlements in what is now Sweden, Norway, and Denmark at the end of the eighth century. The most famous account of their arrival into the Christian realms of the west comes from Britain. In 793 warriors appeared off the coast of Northumbria, leaped from their ships, and robbed the island of Lindisfarne, desecrating the monastery and murdering its brothers. This ferocious raid sent shock waves rippling out from Britain. When the news reached Charlemagne’s court in Aachen, Alcuin of York wrote to the king of Northumbria, deploring the fact that “the church of St Cuthbert is spattered with the blood of the priests of God, stripped of all its furnishing, exposed to the plundering of pagans.” He suggested to the king that he and his noblemen might mend their ways, starting by adopting more Christian haircuts and clothing styles. But it was too late for any of that. The Northmen had announced themselves as a major power in the western world. The next year, 794, raiders appeared on the other side of the British Isles, in the Hebrides. In 799 Vikings raided the monastery of Saint-Philibert at Noirmoutier, just to the south of the river Loire."

"Quapropter potius animam curare memento, quam carnem, quoniam haec manet, illa perit"

"What makes bitter things sweet? Hunger."

"Nec audiendi qui solent dicere, vox populi, vox Dei, quum tumultuositas vulgi semper insaniae proxima sit."

"What is there in Paradise Lost to elevate and astonish like Herschel or Somerville?"

"If ever this theory of the Sun-Force being the primal cause of all life on earth, and of all motion in heaven, is accepted, and if that other far bolder theory of Herschel, about certain organisms in the Sun, is accepted even as a provisional hypothesis, then will our teachings be vindicated, and Esoteric allegory will be shown to have anticipated Modern Science by millions of years, probably, for such are the Archaic Teachings."

"In whatever state of knowledge we may conceive man to be placed, his progress towards a yet higher state need never fear a check, but must continue till the last existence of society."

"Not to feel elevated on an occasion like the present, by this noble, this magnificent testimony of approbation of my friends, is not in human nature — at all events, it is not in my nature; but if any overweaning self-complacency might arise, and mix itself with my feelings at this moment, there is one consideration which would suffice to set it at rest for ever. . . The assembly, magnificent as it is, comprising in itself, as it does, the elite of everything that is illustrious in rank, talent, wealth, in the metropolis — this very assembly is a proof of the justice to which I have adverted I should be weak indeed, if I supposed that all this glorious array has reference to myself. No; it has reference to a far higher and more dignified object. I am but as one drop in the ocean. Every man of science will feel quite as much a sharer in the honors of the day, will feel quite as much distinguished by this assembly as I can be; for when, ere this, would it have been possible to collect together such an assembly as is around me to do honor to science, place it n preeminence, and crown it with distinction? This is, indeed, a new era — this is a memorable day for science, and every man who regards truth for its own sake will feel that on this occasion the eyes of the country are on him, and that England expects every man to do his duty! By that I have been able to accomplish in Africa, I have been amply rewarded; but I stand here not so much for anything of this nature, but as the representative of a class that is distinguished — of a principle which is triumphant; and I hope we shall never allow ourselves to forget the infinitely higher and more important circumstance, that it is the great truths of science, that it is the interpretation of God's great book of nature, and not the men who interpret these pages, that are the ultimate objects of all this praise."

"Unfortunately... the philosophy of Aristotle laid it down as a principle, that the celestial motions were regulated by laws proper to themselves, and bearing no affinity to those which prevail on earth. By thus drawing a broad and impassable line of separation between celestial and terrestrial mechanics, it placed the former altogether out of the pale of experimental research, while it at the same time impeded the progress of the latter by the assumption of principles respecting natural and unnatural motions, hastily adopted from the most superficial and cursory and remark, undeserving even the name of observation. Astronomy therefore continued for ages a science of mere record, in which theory had no part, except in so far as it attempted to conciliate the inequalities of the celestial motions with that assumed law of uniform circular revolution which was alone considered consistent with the perfection of the heavenly mechanism."

"The question "cui bono" to what practical end and advantage do your researches tend? is one which the speculative philosopher who loves knowledge for its own sake, and enjoys, as a rational being should enjoy, the mere contemplation of harmonious and mutually dependent truths, can seldom hear without a sense of humiliation. He feels that there is a lofty and disinterested pleasure in his speculations which ought to exempt them from such questioning; communicating as they do to his own mind the purest happiness (after the exercise of the benevolent and moral feelings) of which human nature is susceptible, and tending to the injury of no one, he might surely allege this as a sufficient and direct reply to those who, having themselves little capacity, and less relish for intellectual pursuits, are constantly repeating upon him this enquiry."

"Time! Time! Time! — we must not impugn the Scripture Chronology, but we must interpret it in accordance with whatever shall appear on fair enquiry to be the truth for there cannot be two truths. And really there is scope enough: for the lives of the Patriarchs may as reasonably be extended to 5000 or 50000 years apiece as the days of Creation to as many thousand millions of years."

"Science is the knowledge of many, orderly and methodically digested and arranged, so as to become attainable by one."