Mathematicians From England

"I have lately spent some Thoughts relative to the Nature of Light, whether it be subject to the common Laws of Motion. In this particular Newton seems to contradict himself. For in his Principia Sect. 14th he applies the common Laws of Motion to account for Reflection and Refraction, as he does also in one Part of his Optics where he proves the Sine of Incid. to Sine Refr, in a given in a given Ratio. But in another Part he says, “nothing more is requisite for producing all the Variety of Colours and Degrees of Refrangibility than that the Rays of Light be Bodies of different Sizes, the least of which may make Violet, and the Greatest the Red"; this manifestly is not consistent with the Theory of Motion applied to Bodies, where the Magnitude of the Bodies is of no Consequence. Now it is evident that if the common Theory of Motion can be applied to Light, the Red Light must have had the greatest Velocity before Incidence, as it suffers the least Deviation, for if the Vels of all the Difft colour'd Light were equal before Incidence, they must by Newton's Principia Sect. Sect. 8. Prop. 1. have continued equal after, and therefore must have suffered the same Deviation. The Determination of this Point seems to be of considerable Importance, as we so often apply the Theory of Motion to Light."

"St. Paul says, that God commendeth his love to us in that while we were yet sinners, Christ died for us: and that when we were enemies we were reconciled to God by the death of his Son. St. Paul was in a peculiar manner a child of grace: with gratitude, therefore, he honours and extols its efficacy in all his epistles; and particularly in his epistle to the Romans throughout he defends his doctrines with great precision and copiousness. 'Every mouth,' says he, 'must be stopped, and all the world become guilty before God. By the deeds of the law no flesh can be justified. Man must be justified freely by his grace. By grace are ye saved through faith, and that not of yourselves, it is the gift of God: not of works, lest any man should boast.'"

"If I were to grasp once, in emulation, work of the absolute, origin-creating mind, its opus est, conclusive otherness, the veil of certitude discovered as itself that which is to be resolved, I should hold for my own, my self-giving, my retort upon Emerson's 'alienated majesty', the De Causa Dei of Thomas Bradwardine."

"Bradwardine was solicitous to employ the many talents now entrusted to him to the glory of his divine Master. It was his care to mitigate as far as possible the impetuosity of the king's temper, when immoderately fired with warlike rage, or unbecomingly elated with the advantages of victory. And so much meekness and persuasive eloquence mingled with his addresses to the army, that the soldiers, wrought upon by his earnest admonitions, were more than ordinarily restrained from practising the excesses attendant upon war. In fact, so truly did Bradwardine sustain amid arms the character of an ambassador of peace, and so influential was the spirit which evidently swayed him, that some of the writers of that time do not hesitate to attribute the conquests of the English king rather to the virtues and holiness of his chaplain than to his own conduct, or the prowess of his troops."

"In 1525 he filled the office of proctur in the university [Oxford]. His great work "concerning the cause of God against the Pelagians," was first delivered in the form of lectures at Oxford. These he afterwards, while chancellor of the diocese of London, at the request of the students of Merton, enlarged and polished. The publication of this book earned for its author the highest reputation. It was speedily in the hands of nil the learned men both in England and on the continent; and Bradwardine was thenceforward known by the title of "the profound doctor." His subject was treated with a mathematical accuracy, and his reasoning was pursued in one connected series of arguments, very different from the discursive remarks of the divines who had preceded him."

"Undoubtedly, such expressions as, 'Turn yourselves,' &c. relate to the free power which every man has to will; but if Pelagius had half an eye, he might see that God, in giving the precept which directs us to turn unto him, influences also the human will, and excites it to action; not, indeed, in opposition our free choice, but the reverse, as I have all along maintained. Hence it is written, 'Without me ye do nothing.' And again, 'I laboured more abundantly than they all, yet not I, but the grace of God which with me.' And lastly, 'I do not this for your sakes, O house of Israel, but for mine holy name's sake. Then will I sprinkle clean water upon you, and shall be clean; and I will cleanse you from your idols. A new heart also will I give you, and a new spirit will put within you; and I will take away the stony heart, and will give you a heart of flesh.'"

"The Pelagians produce such Scriptures as these: 'The Lord is with you, while ye be with him; and if ye seek him, he will be found of you' (2 Chron. xv. 2). 'Turn ye, ...and I will turn unto you' (Zech. i. 3) From which they would infer, that the grace of God is proportioned to the merits of men. But all this would be to no purpose, if they would but compare one Scripture with another; for example: 'Turn us, O God of our salvation' (Ps lxxxv. 4); and, 'after that I was turned, I repented' (Jer. xxxi. 19); and, 'turn us unto thee, O Lord, and we shall be turned' (Lam v. 21)"

"The word grace evidently implies that there is no antecedent merit. And in this way the apostle to the Romans appears to argue, when he says, 'And if by grace, then is it no more of works: otherwise grace is no more grace. Now to him that worketh is the reward not reckoned of grace, but of debt.' All this is perfectly intelligible, even in the conduct of liberal and magnificent human characters. They frequently bestow their gifts from a pure spirit of liberality, without the smallest previous claim on the score of merit. And shall not God, whose perfections are infinite, do more than this?"

"God gives his grace freely, in the strictest sense of the word, and without merit on the part of man. For if God did not bestow his grace in this perfectly gratuitous manner, but on account of some subordinate contingent uncertain cause, he could not possibly foresee how he should bestow his free gifts."

"St. Augustine confesses that he himself had been formerly in a similar mistake. 'I was once,' says he, 'a Pelagian in my principles; I thought that faith towards God was not the gift of God, but that we procured it by our own powers, and that then, through the use of it, we obtain the gifts of God; I never supposed that the preventing grace of God was the proper cause of our faith, till my mind was struck in a particular manner by the apostle's argument and testimony: What hast thou that thou hast not received, and if thou hast received it, why dost thou glory as if thou hadst not received it?' In this whole business I follow the steps of Augustine."

"When I heard those parts of the Scriptures read in the Church which extol the grace of God, and lower the free-will of man, for example, 'It is not of him that willeth, or of him that runneth, but of God that sheweth mercy,' and many similar passages,—this doctrine of grace was very disagreeable to my ungrateful mind. But afterwards I began to perceive some few distant rays of light respecting this matter. I seemed to see, but by no means clearly, that the grace of God is prior, both in nature and in time, to any good actions that men can possibly perform; and I return thanks to God, from whom proceeds every good thing, for thus freely enlightening my understanding."

"The mischievous Pelagians maintain that this sort of grace is not given freely by God but is to be obtained by preceding merits. I myself was once so foolish and empty, when I first applied myself to the study of philosophy, as to be seduced by this error."

"O great and wonderful Lord our God, thou only light of the eyes, open, I implore thee, the eyes of my heart, and of others my fellow-creatures, that we may truly understand and contemplate thy wondrous works. And the more thoroughly we comprehend them, the more may our minds be affected in the contemplation with pious reverence and profound devotion. Who is not struck with awe in beholding thy all-powerful will completely efficacious throughout every part of the creation? It is by this same sovereign and irresistible will, that whom and when thou pleasest thou bringest low and liftest up, killest and makest alive. How intense and how unbounded is thy love to me, O Lord! whereas my love, how feeble and remiss! my gratitude, how cold and inconstant! Far be it from thee that thy love should even resemble mine; for in every kind of excellence thou art consummate. O thou who fillest heaven and earth, why fillest thou not this narrow heart? O human soul, low, abject, and miserable, whoever thou art, if thou be not fully replenished with the love of so great a good, why dost thou not open all thy doors, expand all thy folds, extend all thy capacity, that, by the sweetness of love so great, thou mayest be wholly occupied, satiated, and ravished; especially since, little as thou art, thou canst not be satisfied with the love of any good inferior to the One supreme? Speak the word, that thou mayest become my God and most enviable in mine eyes, and it shall instantly be so, without the possibility of failure. What can be more efficacious to engage the affection than preventing love? Most gracious Lord, by thy love thou hast prevented me, wretch that I am, who had no love for thee, but was at enmity with my Maker and Redeemer. I see, Lord, that it is easy to say and to write these things, but very difficult to execute them. Do thou, therefore, to whom nothing is difficult, grant that I may more easily practise these things with my heart than utter them with my lips. Open thy liberal hand, that nothing may be easier, sweeter, or more delightful to me, than to be employed in these things. Thou, who preventest thy servants with thy gracious love, whom dost thou not elevate with the hope of finding thee?"

"The impulse to understand, and not merely to know and to act, is an impulse characteristic of man and apparently not shared by other animals. I am not concerned here with the origin and nature of this impulse, but with its implications that there is something to be understood and that understanding is not reducible to knowledge and action."

"John G. Bennett was a research scientist who discovered more efficient methods for burning coal, thereby enhancing productivity and reducing pollution. He also was an intellectual who in the four-volume The Dramatic Universe formulated a "cosmic context" for integrating the discussion of environmental ethics."

"John G. Bennett was a distinguished scientist, mathematician and linguist. In the course of his researches and travels all over the world, Bennett made contact with many remarkable men. He devoted his life to the study, practice and teaching of the theory and techniques for the development of the latent powers of man: the widening of the intellect, the discipline of the body, and the steadying of the emotions."

"John Godolphin Bennett was a skilled player of the game, one who kept his mind open and was always ready to experiment. He had charisma. He had personal power. He had a way with people, especially young people. I had debated at length with myself and others whether his influence on his students had been beneficial or disastrous. No answer came. It was perhaps too early to tell. In any case, he was dead. One more link in the old chain was destroyed. Soon that particular chain would vanish entirely."

"Structure is a primary element of experience and not something that is added by the mind. In this respect, it can be said that the techniques of understanding call for a drastic revision of the usual modes of thought that treat being and understanding as independent or at least as separable from one another."

"Facts, that are no more than facts, are atomic and unrelated except by general laws. That is how the world was studied until the middle of the present century."

"We do not know structures, but we know because of structures."

"Every evening after dinner, a new life began. There was no hurry. Some walked in the garden. Others smoked. About nine o’clock we made our way alone or in twos and threes to the Study House. Outdoor shoes came off and soft shoes or moccasins were put on. We sat quietly, each on his or her own cushion, round the floor in the centre. Men sat on the right, women on the left; never together. Some went straight on to the stage and began to practice the rhythmic exercises. On our first arrival, each of us had the right to choose his own teacher for the movements. I had chosen Vasili Ferapontoff, a young Russian, tall, with a sad studious face. He wore pince-nez, and looked the picture of the perpetual student, Trofimov, in The Cherry Orchard. He was a conscientious instructor, though not a brilliant performer. I came to value his friendship, which continued until his premature death ten years later. He told me in one of our first conversations that he expected to die young. The exercises were much the same as those I had seen in Constantinople three years before. The new pupils, such as myself, began with the series called Six Obligatory Exercises. I found them immensely exciting, and worked hard to master them quickly so that I could join in the work of the general class."

"Gurdjieff said, “Change depends on you, and it will not come about through study. You can know everything and yet remain where you are. It is like a man who knows all about money and the laws of banking, but has no money of his own in the bank. What does all his knowledge do for him?” Here Gurdjieff suddenly changed his manner of speaking, and looking at me very directly he said: “You have the possibility of changing, but I must warn you that it will not be easy. You are still full of the idea that you can do what you like. In spite of all your study of free will and determinism, you have not yet understood that so long as you remain in this place, you can do nothing at all. Within this sphere there is no freedom. Neither your knowledge nor all your activity will give you freedom. This is because you have no …” Gurdjieff found it difficult to express what he wanted in Turkish. He used the word varlik, which means roughly the quality of being present. I thought he was referring to the experience of being separated from one’s body. Neither I nor the Prince [Sabaheddin] could understand what Gurdjieff wished to convey. I felt sad, because his manner of speaking left me in no doubt that he was telling me something of great importance. I answered, rather lamely, that I knew that knowledge was not enough, but what else was there to do but study?..."

"There is no need in these mathematical days to defend the use of symbolism. It is regarded by many schools of modern thought as the only safe form of language. Wittgenstein treats symbols as something more than conventional signs, and regards them as corresponding in some way to the reality to which they refer. He would probably accept Gurdjieff’s dictum that: Symbols not only transmit knowledge but show the way to it. Even though other thinkers deny any objective reference to symbols, no one questions that symbolism has a power beyond that of ordinary language. It is different with the language of myth. This is despised by superficial thinkers, but the greatest philosophers have known its value."

"Ouspensky records a conversation in St. Petersburg during the summer of 1916 in which Gurdjieff discussed the problem of communication, and the impossibility of conveying in our ordinary language ideas which are intelligible and obvious only for a higher state of consciousness. Speaking of the unity between man, the Universe, and God, he said that the objective knowledge by which alone this unity is to be understood can never be expressed in words or logical forms. At this point, Gurdjieff made a statement which is a key to the understanding of his own subsequent writings. He said: Realising the imperfection and weakness of ordinary language, the people who have possessed objective knowledge have tried to express the idea of unity in ‘myths,’ in ‘symbols,’ and in particular ‘verbal formulas,’ which, having been transmitted without alteration, have carried on the idea from one school to another, often from one epoch to another. In All and Everything Gurdjieff makes extensive use of these three forms, that is, symbol, myth, and verbal formula...."

"Since we tend to see ourselves primarily in the light of our intentions, which are invisible to others, while we see others mainly in the light of their actions, which are visible to us, we have a situation in which misunderstanding and injustice are the order of the day"

"I must warn you that Gurdjieff is far more of an enigma than you can imagine. I am certain that he is deeply good, and that he is working for the good of mankind. But his methods are often incomprehensible. For example, he uses disgusting language, especially to ladies who are likely to be squeamish about such things. He has the reputation of behaving shamelessly over money matters, and with women also. At his table we have to drink spirits, often to the point of drunkenness. People have said that he is a magician, and that he uses his powers for his own ends... I do not believe that the scandalous tales told of Gurdjieff are true: but you must take into account that they may be true and act accordingly."

"G. I. Gurdjieff's sexual life was strange in its unpredictability. At certain times he led a strict, almost ascetic life, having no relation with women at all. At other times, his sex life seemed to go wild and it must be said that his unbridled periods were more frequent than the ascetic. At times, he had sexual relationships not only with almost any woman who happened to come within the sphere of his influence, but also with his own pupils. Quite a number of his women pupils bore him children and some of them remained closely connected with him all their lives. Others were just as close to him, as far as one could tell, without a sexual relationship."

"True sensitivity is the beginning of what Gurdjieff calls Objective Reason and which he says, cannot be in this body and can only belong to the Second, or Kesdjanian Body, and when it is formed it can begin to acquire this direct perception of how things are, combined with experience that gives this vision a practical and realistic application. Out of this comes what he calls Objective Reason"

"The systematic principle is based upon the hypothesis that there is a structure in the real world that transcends the distinctions of subjective and objective experience."

"I have had considerable experience in dealing with minds of low logical power, and have found that studies may be made so easy and mechanical as to render thought almost superfluous."

"Five hundred years ago, when faced with an eclipse, many of us would have believed it was the work of an angry god. But as we've unearthed the language of the Code, we've discovered that the apparent mysteries of our world can be understood without invoking the supernatural. And this for me is what's so remarkable. That despite the incredible complexity of the world we live in, it can all, ultimately, be explained by numbers. Just like the orbit of the planets, life too follows a pattern. And it can all be reduced to cause and effect.In the end, even the flip of a coin is determined by how fast it's spinning and how long it takes to hit the ground. The ultimate symbol of chance isn't random at all. It only appears that way. When we don't understand the Code, the only way we can make sense of our world is to make up stories. But the truth is far more extraordinary. Everything has mathematics at its heart. When everything is stripped away all that remains is the Code."

"I would teach the world how the Greeks proved, more than 2,000 years ago, that there are infinitely many prime numbers. In my mind, this discovery is the beginning of mathematics – when humankind realised that, by pure thought alone, it could prove eternal truths of the universe. Prime numbers are the indivisible numbers, numbers that can be divided only by themselves and one. They are the most important numbers in mathematics, because every number is built by multiplying prime numbers together – for example, 60 = 2 x 2 x 3 x 5. They are like the atoms of arithmetic, the hydrogen and oxygen of the world of numbers."

"It is with Bernhard Riemann's work that we finally have the mathematical glasses to explore such worlds of the mind. And now my journey through the abstract world of 20th century mathematics has revealed that maths is the true language that the universe is written in. They key to understanding the world around us. Mathematicians aren't motivated by money and material gain, or even by practical applications of their work. For us it's the glory of solving one of the great unsolved problems that have outwitted previous generations of mathematicians. David Hilbert was right; it’s the unsolved problems of mathematics which make it a living subject. Which obsess each new generation of mathematicians. Despite all the things we've discovered over the last 7 millennia, there are still many things we don't understand. And its Hilbert’s call of "We must know, we will know" which drives mathematics."

"We take as given the idea of distinction and the idea of indication, and that we cannot make an indication without drawing a distinction. We take, therefore, the form of distinction for the form."

"George Spencer-Brown—born in Grimsby, Lincolnshire, England 1923—has held a number of occupational roles, such as a mathematician, consulting engineer, psychologist, pilot, educational consultant, author and poet, adviser in military communications (coding and code-breaking), football correspondent to the Daily Express; he even practised car racing with Gavin Maxwell."

"To teach pride in knowledge is to put up an effective barrier against any advance upon what is already known, since it makes one ashamed to look beyond the bounds imposed by one's own ignorance."

"Let us consider, for a moment, the world as described by the physicist. It consists of a number of fundamental particles which, if shot through their own space, appear as waves, and are thus... of the same laminated structure as pearls or onions, and other wave forms called electromagnetic which it is convenient, by Occam’s razor, to consider as travelling through space with a standard velocity. All these appear bound by certain natural laws which indicate the form of their relationship."

"On Mondays, Wednesdays, and Fridays we use the wave theory; on Tuesdays, Thursdays, and Saturdays we think in streams of flying energy quanta or corpuscles."

"Waves from moving sources: Adagio. Andante. Allegro moderato."

"... the present time is the age of communication ... communication engineering began with Gauss, Wheatstone, and the first telegraphers. It received its first reasonably scientific treatment at the hands of Lord Kelvin, after the failure of the first transatlantic cable in the middle of the last century, and from the eighties on, it was perhaps Heaviside who did the most to bring into a modern shape."

"General circuit theory, the theory of transmission lines and wave propagation, vector analysis, even the four famous Maxwell's equations—all had flowed from the pen of this amazing man."

"Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latter Bold and Broad. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical inquiries. Am I to refuse to eat because I do not fully understand the mechanism of digestion?"

"However absurd it may seem, I do in all seriousness hereby declare that I am animated mainly by philanthropic motives. I desire to do good to my fellow creatures, even to the Cui bonos."

"My own introduction to quaternionics took place in quite a different manner. Maxwell exhibited his main results in quaternionic form in his treatise. I went to Prof Tait's treatise to get information, and to learn how to work them. I had the same difficulties as the deceased youth, but by skipping them, was able to see that quaternionics could be employed consistently in vectorial work. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectorial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalar and vectors, using a very simple vectorial algebra in my papers from 1883 onward. The paper at the beginning of vol. 2 of my Electrical Papers may be taken as a developed specimen; the earlier work is principally concerned with the vector differentiator ∇ and its applications, and physical interpretations of the various operations. Up to 1888 I imagined that I was the only one doing vectorial work on positive physical principles; but then I received a copy of Prof. Gibbs's Vector Analysis (unpublished, 1881-4)."

"More than a third part of a century ago, in the library of an ancient town, a youth might have been seen tasting the sweets of knowledge to see how he liked them. He was of somewhat unprepossessing appearance, carrying on his brow the heavy scowl that the "mostly-fools" consider to mark a scoundrel. In his father's house were not many books, so it was like a journey into strange lands to go book-tasting. Some books were poison; theology and metaphysics in particular they were shut up with a bang. But scientific works were better; there was some sense in seeking the laws of God by observation and experiment, and by reasoning founded thereon. Some very big books bearing stupendous names, such as Newton, Laplace, and so on, attracted his attention. On examination, he concluded that he could understand them if he tried, though the limited capacity of his head made their study undesirable. But what was Quaternions? An extraordinary name! Three books; two very big volumes called Elements, and a smaller fat one called Lectures. What could quaternions be? He took those books home and tried to find out. He succeeded after some trouble, but found some of the properties of vectors professedly proved were wholly incomprehensible. How could the square of a vector be negative? And Hamilton was so positive about it. After the deepest research, the youth gave it up, and returned the books. He then died, and was never seen again. He had begun the study of Quaternions too soon."

"Electric and magnetic forces. May they live for ever, and never be forgot, if only to remind us that the science of electromagnetics, in spite of the abstract nature of its theory, involving quantities whose nature is entirely unknown at the present, is really and truly founded on the observations of real Newtonian forces, electric and magnetic respectively."

"The following story is true. There was a little boy, and his father said, “Do try to be like other people. Don’t frown.” And he tried and tried, but could not. So his father beat him with a strap; and then he was eaten up by lions. Reader, if young, take warning by his sad life and death. For though it may be an honour to be different from other people, if Carlyle’s dictum about the 30 million be still true, yet other people do not like it. So, if you are different, you had better hide it, and pretend to be solemn and wooden-headed. Until you make your fortune. For most wooden-headed people worship money; and, really, I do not see what else they can do. In particular, if you are going to write a book, remember the wooden-headed. So be rigorous; that will cover a multitude of sins. And do not frown."

"We do not dwell in the Palace of Truth. But, as was mentioned to me not long since, "There is a time coming when all things shall be found out." I am not so sanguine myself, believing that the well in which Truth is said to reside is really a bottomless pit."

"That axiom of Metaphysicians which is termed the principle of contradiction and which affirms that it is impossible for anything to possess a quality, and in the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x²=x."

"All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations."