Mathematicians From England

"Images and pictures... have played a key role in shaping our scientific picture of the world. ...Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking."

"There is a good deal more to nothing than meets the eye."

"Ultimate explanation no longer means only a story that encompasses everything."

"By interlinking causes, by searching always for unity in the face of superficial diversity, modern scientific explanations prize depth above breadth. A deep and narrow theory can, and often does, graduate to become a deep and broad one. A broad and shallow theory never does."

"The legacy of the great monotheistic religions is the expectation of a single over-arching explanation for the Universe. ...There is no logical reason why the Universe should not contain surds of arbitrary elements that do not relate to the rest."

"Scanning the past millennia of human achievement reveals just how much has been achieved during the last three hundred years since Newton set in motion the effective mathematization of Nature. We found that the world is curiously adapted to a simple mathematical description. It is enigma enough that the world is described by mathematics; but by simple mathematics, of the sort that a few years energetic study now produces familiarity with, this is an enigma within an enigma."

"Assuming the simulators, or at least the early generations of them, have a very advanced knowledge of the laws of Nature, it's likely that they would still have incomplete knowledge of them. ...gradually the little flaws will begin to build up. ...The only escape is if their creatures intervene to patch up the problems one by one as they arise."

"...logical contradictions will inevitably arise and the laws of the simulations will appear to break down now and again. The inhabitants of the simulation—especially the simulated scientists—will occasionally be puzzled by the experimental results they obtain. The simulated astronomers, might, for instance, make observations that show that their so-called constants of Nature are very slowly changing."

"We say that the string is 'random' if there is no other representation of the string which is shorter than itself. But we will say that it is 'non-random' if there does exist such an abbreviated representation. ...In general, the shorter the possible representation... the less random... On this view we recognize science to be the search for algorithmic compressions."

"Gods reappear in unlimited numbers in the guise of the simulators who have the power of life and death over the simulated realities that they bring into being. The simulators determine the laws, and can change the laws, that govern their worlds. They can engineer anthropic fine-tunings. They can pull the plug on the simulation at any moment, intervene or distance themselves from their simulation; watch as the simulated creatures argue about whether there is a god who controls or intervenes, work miracles or impose their ethical principles upon the simulated reality."

"The Universe has imposed aspects of its structure upon us by the inevitability of the forces of Nature... In a world where adapters succeed, but non-adapters fail, one expects to find vestigial remnants... Many of these adaptations... give rise to a suite of curious byproducts, some of which have played a role in determining our aesthetic sense. We are products of a past world where sensitivities to certain things were a matter of life or death."

"The various creation stories of ancient time were not scientific theories in any modern sense. They did not attempt to reveal anything new about the structures of the world; they aimed simply to remove the specter of the unknown from human imaginings."

"The secret of how galaxies came into being may well be fathomed by the study of the most elementary particles of matter in particle detectors buried deep underground; the identity of those elementary particles may be revealed by observations of distant starlight. ...by the coming together of the largest and the smallest aspects of the physical world our appreciation of the unity of the universe becomes more impressive and complete."

"If musical appreciation is a by-product of a more general pattern-processing propensity of the brain, then why are our senses heightened by pink noises? It is significant that the world around us is full of variations with 1/f spectra. Benoit Mandelbrot has pioneered the study of natural and computer-generated patterns that are scale-free. (He calls them fractals...) Mandelbrot draws attention to the fact that there is a pattern to the noise spectrum displayed by the human nervous system. At the extremities of the body... it tends to be of white-noise form; but, as one approaches closer to the central nervous system and the brain, these variations become 1/f-like. Our nervous system may act as a spectral filter to prevent the brain from being swamped with uninteresting white background noise about the world..."

"What Hubble discovered was the expansion of the universe. ...We are not expanding. Nor is Brooklyn. Nor is earth. Nor is the solar system. Nor, in fact, is the Milky Way galaxy. Nor even those aggregates of thousands of galaxies that we call "galaxy clusters." These collections of matter are bound together by chemical and gravitational forces between their constituents—forces that are stronger than the force of expansion."

"Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers."

"Highly correlated brown and black noise patterns do not seem to have seem to have attractive counterparts in the visual arts. There, over-correlation is the order of the day, because it creates the same dramatic associations that we find in attractive natural scenery, or in the juxtaposition of symbols. Somehow, it is tediously predictable when cast in a one-dimensional medium, like sound."

"It is simplest to think of mathematics as the catalogue of all possible patterns. ...When viewed in this way, it is inevitable that the world is described by mathematics. ...In many ways the search for a Theory of Everything is a manifestation of a faith that this compression goes all the way down to the bedrock of reality..."

"Ancient belief in a cosmos composed of spheres, producing music as angels guided them through the heavens, was still flourishing in Elizabethan times. ...There is a good deal more to Pythagorean musical theory than celestial harmony. Besides the music of the celestial spheres (musica mundana), two other varieties of music were distinguished: the sound of instruments...(musica instrumentalis), and the continuous unheard music that emanated from the human body (musica humana), which arises from a resonance between the body and the soul. ...In the medieval world, the status of music is revealed by its position within the Quadrivium—the fourfold curriculum—alongside arithmetic, geometry, and astronomy. Medieval students... believed all forms of harmony to derive from a common source. Before Boethius' studies in the ninth century, the idea of musical harmony was not considered independently of wider matters of celestial or ethical harmony."

"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."

"A less inflexible picture of mathematics is one that focuses on the fact that it is an open-ended human activity. Inventionism is the belief that mathematics is nothing more than what mathematicians do. ...We invent mathematics; we do not discover it. ...The independent discovery of the same mathematical theorems by different mathematicians from totally different economic, cultural, and political backgrounds—often at widely separated times in history—argues against such a simple view. The inventionist could respond by pointing to the universality of human languages. ...One might expect that those aspects of this universal grammar that share features of logic, and hence counting, would also make counting appear instinctive. In fact, although simple counting... is fairly universal in ancient and primitive cultures, virtually none of them went on to carry out mathematical operations more sophisticated than counting. This suggests that these higher mathematical operations are not genetically programmed into the human brain... They are more likely to be by-products of multi-purpose pattern-recognition capabilities."

"Hundreds of years ago, natural theologians tried to impress their readers with stories of the wondrous symmetries of Nature; now we see that, ironically, it is the departures from those symmetries that makes life possible. It is upon the flaws of Nature, not the laws of Nature, that the possibility of our existence hinges. ...The laws and constants of Nature are features that enforce uniformity and simplicity, while initial conditions and symmetry breakings permit complexity and diversity."

"Unfortunately, truth is neither a listable nor a decidable property; nor is the truth of a statement of arithmetic. The American logician John Myhill has used the term 'prospective' to characterize those attributes of the world that are neither listable nor decidable. They are properties that cannot be recognized by the application of some formula, made to conform to a rule, or generated by some computer program. They are characterized by incessant novelty that cannot be encompassed by any finite set of rules. 'Beauty', 'ugliness', 'truth', 'harmony', simplicity', and 'poetry' are names we give to some of the attributes of this sort. There is no way of listing all examples of beauty or ugliness, nor any procedure for saying whether or not something possesses either of those attributes, without redefining them in some more restrictive fashion that kills their prospective character."

"Life, like science and art, is a theory about the world: a theory that in our case takes bodily form. By a succession of adaptations, most of which are favourable and none of which are lethal, living things have invested in particular expectations about the future course of their environments. If those theories are good enough, then life will prosper and multiply; but if they are outmoded by changing conditions, their embodiments will dwindle and perish."

"The laws of Nature are based upon the existence of a pattern, linking one state of affairs to another; and where there is pattern, there is symmetry. Yet...the symmetries that the laws enshrine are broken in... outcomes. Suppose that we balance a needle on its point and then release it. The law of gravity, which governs its subsequent motion, is perfectly democratic. It has no preference for any particular direction in the Universe: it is symmetrical in this respect. Yet, when the needle falls, it must fall in a particular direction. The directional symmetry of the underlying law is broken, therefore... By the same token, the fallen needle hides the symmetry of the law... Such 'symmetry-breaking' governs much of what we see in the Universe... It allows a Universe governed by a small number of symmetrical laws to manifest an infinite diversity of complex, asymmetrical states. This is how the Universe can be at once, simple and complicated."

"Our sensitivity to changes of pitch ... is underused in musical sound. Western music, in particular, is based on scales that use pitch changes that are at least twenty times bigger than the smallest changes that we could perceive. If we used our discriminatory power to full, we could generate an undulating sea of sound that displayed continuously changing frequency rather like the undersea sonic songs of dolphins and whales."

"It is not hard to see why the Eastern holistic perspective made scientific progress so difficult. It denies the intuition that one can study the parts of the world in isolation from the rest—that one can analyze the world..."

"The living world is not a marble palace. It is a higgledy-piggledy outcome of natural selection and the competition between many interacting factors. The outcome is often neither elegant nor symmetrical."

"This principle, which is thus made the foundation of the operations and results of Symbolical Algebra, has been called "The principle of the permanence of equivalent forms", and may be stated as follows: "Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form.""

"I have separated arithmetical from symbolical algebra, and I have devoted the present volume entirely to the exposition of the principles of the former science and their application to the theory of numbers and of arithmetical processes: the second volume, which is now in the press, will embrace the principles of symbolical algebra: it will be followed, if other and higher duties should allow me the leisure to complete them, by other works, embracing all the more important departments of analysis, with the view of presenting their principles in such a form, as may make them component parts of one uniform and connected system."

"In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a - b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and \frac{a}{b} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science."

"Symbolical algebra adopts the rules of arithmetical algebra, but removes altogether their restrictions: thus symbolical subtraction differs from the same operation in arithmetical algebra in being possible for all relations of value of the symbols or expressions employed... all the results of arithmetical algebra which are deduced by the application of its rules, and which are general in form, though particular in value, are results likewise of symbolical algebra, where they are general in value as well as in form: thus the product of a^{m} and a^{n}, which is a^{m+n} when m and n are whole numbers, and therefore general in form though particular in value, will be their product likewise when m and n are general in value as well as in form: the series for (a+b)^{n}, determined by the principles of arithmetical algebra, when n is any whole number, if it be exhibited in a general form, without reference to a final term, may be shewn, upon the same principle, to the equivalent series for (a+b)^n, when n is general both in form and value."

"This work... was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1830, and which has been long out of print; but I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise."

"I Gladly avail myself of the opportunity of inscribing to you, for a second time, a work of mine on Algebra, as a sincere tribute of my respect, affection and gratitude. I trust that I shall not be considered as derogating from the higher duties which, (in common with you), I owe to my station in the Church, if I continue to devote some portion of the leisure at my command, to the completion of an extensive Treatise, embracing the more important departments of Analysis, the execution of which I have long contemplated, and which, in its first volume I now offer to the public, under the auspices of one of my best and dearest friends."

"I have endeavoured... to present the principles and applications of Symbolical, in immediate sequence to those of Arithmetical, Algebra, and at the same time to preserve that strict logical order and simplicity of form and statement which is essential to an elementary work. This is a task of no ordinary difficulty, more particularly when the great generality of the language of Symbolical Algebra and the wide range of its applications are considered, and this difficulty has not been a little increased, in the present instance, by the wide departure of my own views of its principles from those which have been commonly entertained."

"I assure you that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. I am nearly certain of being nominated to the office of Moderator in the year 1818-1819, and as I am an examiner in virtue of my office, for the next year I shall pursue a course even more decided than hitherto, since I shall feel that men have been prepared for the change, and will then be enabled to have acquired a better system by the publication of improved elementary books. I have considerable influence as a lecturer, and I will not neglect it. It is by silent perseverance only, that we can hope to reduce the many-headed monster of prejudice and make the University answer her character as the loving mother of good learning and science."

"It is now more than twenty years since I somewhat rashly undertook to write the Life of Dr. Young. For many years, however, after making this engagement, I found myself so much occupied by the duties of a very laborious college office, that I had no leisure to commence the work; and when the possession of leisure would have enabled me to have done so, my health became so seriously deranged that I felt myself unequal to any continued and severe literary labour. The undertaking was consequently abandoned, and it was proposed to transfer it to other hands; but it was not found easy to secure the services of a person who possessed sufficient scientific knowledge to enable him to write the life of an author whose works were so various in their character and not unfrequently so difficult to understand and analyse, as those of Dr. Young."

"Geomancy was a natural art, drawing on the inborn powers of the human soul to glean information from the larger soul of the world."

"At the beginning of the 17th century many learned heads in England were occupied with Theosophy, Cabbalism, and Alchemy: amongst the proofs of this... may be cited the works of , of Norbert, of Thomas and Samuel Norton, but above all (in reference to our present inquiry) of Robert Fludd. Fludd it was, or whosoever was the author of the Summum Bonum 1629, that must be considered as the immediate father of Freemasonry, as Andrea was its remote father. What was the particular occasion of his own first acquaintance with , is not recorded: all the books of Alchemy or other occult knowledge, published in Germany, were at that time immediately carried over to England — provided they were written in Latin; and, if written in German, were soon translated for the benefit of English students. He may therefore have gained his knowledge immediately from the three Rosicrucian books. But it is more probable that he acquired his knowledge on this head from his friend Maier (mentioned in the preceding chapter) who was intimate with Fludd during his stay in England, and corresponded with him after he left it. At all events he must have been initiated into Rosicrucianism at an early period, having published his apology for it in the year 1617."

"THE central figure of Rosicrucian literature, towering as an intellectual giant above the crowd of souffleurs, theosophists, and charlatanic professors of the magnum opus, who, directly or otherwise, were connected with the mysterious Brotherhood, is Robertus de Fluctibus, the great English mystical philosopher of the seventeenth century, a man of immense erudition, of exalted mind, and, to judge by his writings, of extreme personal sanctity. Ennemoser describes him as one of the most distinguished disciples of Paracelsus, but refuses to number him with "those consecrated theosophists who draw all wisdom from the fountain of eternal light." He does not state his reasons for this depreciatory judgment, and the brief and inadequate notice which he gives of Fludd's system displays such a cursory acquaintance with the works in which it is developed, that it is doubtful whether he had taken pains to understand his author."

"The art, also, of alchemy or chemistry is surrounded with such insoluble enigmas that we can scarcely gain anything but ignorance therefrom, and ignotum per ignotius."

"But absolutely nothing remains of that one and only medicament of which Hippocrates makes mention (darkly and mystically, I admit) in several places, and still less are its operations understood, inasmuch as no one now searches with lynx-like eyes into the profound depths of true natural philosophy, to gain an accurate knowledge of its composition and its virtues."

"Which of us has, at this day, the ability to discover those true and vivific numbers whereby the elements are united and bound to one another?"

"Fludd's apology for the Brotherhood of the Rosy Cross was first issued in 1616, being printed in Leyden. It "entitles him to be regarded as the high priest of their mysteries." It is said that Maier visited Fludd in London in 1615, and the result of his visit was, we know, the publication of his 'Apologia,' written in Latin, and published in Leyden in 1616."

"But, good God, what is this when compared with that deep and true music of the wise, whereby the proportions of natural things are investigated, the harmonical concord and the qualities of the whole world are revealed, by which also connected things are bound together, peace established between conflicting elements, and whereby each star is perpetually suspended in its appointed place by its weight and strength, and by the harmony of its lucent spirit"

"Particulars are frequently fallible, but universals never. Occult philosophy lays bare Nature in her complete nakedness, and alone contemplates the wisdom of universals by the eyes of intelligence. Accustomed to partake of the rivers which flow from the Fountain of Life, it is unacquainted with grossness and with clouded waters."

"We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves."

"Our family are an alternate stratification of poetry and mathematics."

"Circumstances have been such, that I have lived almost entirely secluded for some time. Those who are much in earnest and with single minds devoted to any great object in life, must find this occasionally inevitable.... You will wonder at having heard nothing from me; but you have experience and candour enough to perceive and know that God has not given to us (in this state of existence) more than very limited powers of expression of one's ideas and feelings... I shall be very desirous of again seeing you. You know what that means from me, and that it is no form, but the simple expression and result of the respect and attraction I feel for a mind that ventures to read direct in God's own book, and not merely thro' man's translation of that same vast and mighty work."

"[The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine. … Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent."