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Cusp, as a Singularity at (0,0)

'Singularity' may refer to a mathematical, gravitational, mechanical, meteorological, technological or sociological singularity, or with the state of being singular, distinct, peculiar, uncommon or unusual. In mathematics and science the term is most often associated with the concept of infinity. This article is a collection of quotes with reference to any or all of the meanings enveloped within the use of the word 'singularity.'


  • Many futurists are concerned with what are termed singularities, periods of such rapid and profound change due to technology that the state of the world is forever altered. ...these phase transitions happen so quickly that they can forever alter humanity's relationship with its surroundings. The quintessential singularity that futurists dwell on is that of the potential creation of superhuman machine intelligence.
    • Samuel Arbesman, The Half-life of Facts: Why Everything We Know has an Expiration Date (2012)
  • To challenge and to cope with this paradoxical state of things, we need a paradoxical way of thinking; since the world drifts into delirium, we must adopt a delirious point of view. We must no longer assume any principle of truth, of causality, or any discursive norm. Instead, we must grant both the poetic singularity of events and the radical uncertainty of events. It is not easy. We usually think that holding to the protocols of experimentation and verification is the most difficult thing. But in fact the most difficult thing is to renounce the truth and the possibility of verification, to remain as long as possible on the enigmatic, ambivalent, and reversible side of thought.
    • Jean Baudrillard in the lecture "The Murder of the Real" (May 1999); published in The Vital Illusion (2000)
  • York: You're that freak from TV!
Static: You say freak, I say unique!
  • Static Shock Aftershock written by Stan Berkowitz & Alan Burnett
  • Strive not for singularity in dress;
    Fools have the more and men of sense the less.
    To look original is not worth while,
    But be in mind a little out of style.
  • You, who know better than any one the motley world of cosmopolites, understand why I have confined myself to painting here only a fragment of it. That world, indeed, does not exist, it can have neither defined customs nor a general character. It is composed of exceptions and of singularities.
  • Throughout history, the circumpunct has been all things to all people — it is the sun god Ra, alchemical gold, the all-seeing eye, the singularity point before the Big Bang...
  • "There is no security"--to quote his own words--"against the ultimate development of mechanical consciousness, in the fact of machines possessing little consciousness now. A mollusc has not much consciousness. Reflect upon the extraordinary advance which machines have made during the last few hundred years, and note how slowly the animal and vegetable kingdoms are advancing. The more highly organized machines are creatures not so much of yesterday, as of the last five minutes, so to speak, in comparison with past time.
“Either,” he proceeds, “a great deal of action that has been called purely mechanical and unconscious must be admitted to contain more elements of consciousness than has been allowed hitherto (and in this case germs of consciousness will be found in many actions of the higher machines)—Or (assuming the theory of evolution but at the same time denying the consciousness of vegetable and crystalline action) the race of man has descended from things which had no consciousness at all. In this case there is no à priori improbability in the descent of conscious (and more than conscious) machines from those which now exist, except that which is suggested by the apparent absence of anything like a reproductive system in the mechanical kingdom.
  • But the machines which reproduce machinery do not reproduce machines after their own kind. A thimble may be made by machinery, but it was not made by, neither will it ever make, a thimble. Here, again, if we turn to nature we shall find abundance of analogies which will teach us that a reproductive system may be in full force without the thing produced being of the same kind as that which produced it. Very few creatures reproduce after their own kind; they reproduce something which has the potentiality of becoming that which their parents were. Thus the butterfly lays an egg, which egg can become a caterpillar, which caterpillar can become a chrysalis, which chrysalis can become a butterfly; and though I freely grant that the machines cannot be said to have more than the germ of a true reproductive system at present, have we not just seen that they have only recently obtained the germs of a mouth and stomach? And may not some stride be made in the direction of true reproduction which shall be as great as that which has been recently taken in the direction of true feeding?
It is possible that the system when developed may be in many cases a vicarious thing. Certain classes of machines may be alone fertile, while the rest discharge other functions in the mechanical system, just as the great majority of ants and bees have nothing to do with the continuation of their species, but get food and store it, without thought of breeding. One cannot expect the parallel to be complete or nearly so; certainly not now, and probably never; but is there not enough analogy existing at the present moment, to make us feel seriously uneasy about the future, and to render it our duty to check the evil while we can still do so? Machines can within certain limits beget machines of any class, no matter how different to themselves. Every class of machines will probably have its special mechanical breeders, and all the higher ones will owe their existence to a large number of parents and not to two only.
Complex now, but how much simpler and more intelligibly organised may it not become in another hundred thousand years? or in twenty thousand? For man at present believes that his interest lies in that direction; he spends an incalculable amount of labour and time and thought in making machines breed always better and better; he has already succeeded in effecting much that at one time appeared impossible, and there seem no limits to the results of accumulated improvements if they are allowed to descend with modification from generation to generation.
  • “Herein lies our danger. For many seem inclined to acquiesce in so dishonourable a future. They say that although man should become to the machines what the horse and dog are to us, yet that he will continue to exist, and will probably be better off in a state of domestication under the beneficent rule of the machines than in his present wild condition. We treat our domestic animals with much kindness. We give them whatever we believe to be the best for them; and there can be no doubt that our use of meat has increased their happiness rather than detracted from it. In like manner there is reason to hope that the machines will use us kindly, for their existence will be in a great measure dependent upon ours; they will rule us with a rod of iron, but they will not eat us; they will not only require our services in the reproduction and education of their young, but also in waiting upon them as servants; in gathering food for them, and feeding them; in restoring them to health when they are sick; and in either burying their dead or working up their deceased members into new forms of mechanical existence.
  • The power of custom is enormous, and so gradual will be the change, that man's sense of what is due to himself will be at no time rudely shocked; our bondage will steal upon us noiselessly and by imperceptible approaches; nor will there ever be such a clashing of desires between man and the machines as will lead to an encounter between them. Among themselves the machines will war eternally, but they will still require man as the being through whose agency the struggle will be principally conducted. In point of fact there is no occasion for anxiety about the future happiness of man so long as he continues to be in any way profitable to the machines; he may become the inferior race, but he will be infinitely better off than he is now. Is it not then both absurd and unreasonable to be envious of our benefactors? And should we not be guilty of consummate folly if we were to reject advantages which we cannot obtain otherwise, merely because they involve a greater gain to others than to ourselves?
“With those who can argue in this way I have nothing in common. I shrink with as much horror from believing that my race can ever be superseded or surpassed, as I should do from believing that even at the remotest period my ancestors were other than human beings. Could I believe that ten hundred thousand years ago a single one of my ancestors was another kind of being to myself, I should lose all self-respect, and take no further pleasure or interest in life. I have the same feeling with regard to my descendants, and believe it to be one that will be felt so generally that the country will resolve upon putting an immediate stop to all further mechanical progress, and upon destroying all improvements that have been made for the last three hundred years. I would not urge more than this. We may trust ourselves to deal with those that remain, and though I should prefer to have seen the destruction include another two hundred years, I am aware of the necessity for compromising, and would so far sacrifice my own individual convictions as to be content with three hundred. Less than this will be insufficient.”
  • Because we are not in a civilization which believes strongly in oracles or sacred places, we see the full frenzy of those who killed themselves to find the sepulchre of Christ. But being in a civilization which does believe in this dogma of fact for facts' sake, we do not see the full frenzy of those who kill themselves to find the North Pole. I am not speaking of a tenable ultimate utility which is true both of the Crusades and the polar explorations. I mean merely that we do see the superficial and aesthetic singularity, the startling quality, about the idea of men crossing a continent with armies to conquer the place where a man died. But we do not see the aesthetic singularity and startling quality of men dying in agonies to find a place where no man can live — a place only interesting because it is supposed to be the meeting-place of some lines that do not exist.
  • Men will seem to see new destructions in the sky. The flames that fall from it will seem to rise in it and to fly from it with terror. They will hear every kind of animals speak in human language. They will instantaneously run in person in various parts of the world, without motion. They will see the greatest splendour in the midst of darkness. O! marvel of the human race! What madness has led you thus! You will speak with animals of every species and they with you in human speech. You will see yourself fall from great heights without any harm and torrents will accompany you, and will mingle with their rapid course.
    • Leonardo da Vinci, The Notebooks of Leonardo Da Vinci (1938), XX Humorous Writings, as translated by Edward MacCurdy.
  • Mathematicians call the infinite curvature limit of spacetime a singularity. In this picture, then, the big bang emerges from a singularity. The best way to think about singularities is as boundaries or edges of spacetime. In this respect they are not, technically, part of spacetime itself... So the first moment of the universe — in this highly simplified picture — is not a moment or a place at all, but a boundary to moments and places. signals a "no further" warning.
    • Paul Davies, Cosmic Jackpot: Why Our Universe Is Just Right for Life (2007)
  • If the alternative that's being offered to what physicists now talk about - a big bang, a spontaneous singularity which gave rise to the origin of the universe - if the alternative to that is a divine intelligence, a creator, which would have to have been complicated, statistically improbable, the very kind of thing which scientific theories such as Darwin's exists to explain, then immediately we see that however difficult and apparently inadequate the theory of the physicists is, the theory of the theologians - that the first course was a complicated intelligence - is even more difficult to accept.
  • Caustics are the brightest places in an optical field. They are the singularities of geometrical optics. The most familiar caustic is the rainbow, a grossly distorted image of the Sun in the form of a giant arc in the skyspace of directions, formed by the angular focusing of sunlight that has been twice refracted and once reflected in raindrops.
  • Current theories about matter are valid only up to a finite temperature, and hence valid only after the first fraction of a second following the Singularity, the initial moment entailed by general relativity. … Closer to the Singularity there comes a moment, presumably the 'Planck Time'... when general relativity must be replaced by a quantum theory of gravity. Theories of space and time applicable before Planck Time are not known; even the meaningfulness of 'space' and 'time' is uncertain. ...The intitial Singularity itself is a third limit, at least if there is such an initial Singularity. The Big Bang theory results in unreallistic numbers, like infinite density and so on. has been shown that such peculiarities are unavoidable in general relativity, once certain very general assumptions are made, like causality and positive energy density (Hawking and Penrose 1970).
  • In the initial world-making singularity, "The Weyl tensor was exactly zero at the Big Bang itself, while the Ricci Curvature diverged to infinity. By contrast, in the singularity of a terminal black hole or collapse universe, the reverse is true: Weyl curvature is infinite and dominates over Ricci curvature. This explains why entropy or disorder is low in the initial Big Bang singularity, and why the Second Law of Thermodynamics would not be reversed in a closed collapsing universe. An initial singularity is infinitely ordered, completely free from quantum disortions or irregularities in space-time geometry with respect to Weyl curvature; and a terminal singularity is infinitely disordered in this respect, if Penrose is right.
    • Rem Blanchard Edwards, What Caused the Big Bang? (2001)
  • this is not to say, we repeat, that the conditions of labor and production are becoming the same throughout the world or throughout the different sectors ot the economy. The claim is rather is that many singular instances of labor processes, productive conditions, local situations, and lived experiences coexist with a "becoming common", at a different level of abstraction, of the forms of labor and the general relations of production and exchange - and that there is no contradiction between this singularity and commonality.
  • The idea, therefore, that religious faith is somehow a sacred human convention—distinguished, as it is, both by the extravagance of its claims and by the paucity of its evidence—is really too great a monstrosity to be appreciated in all its glory. Religious faith represents so uncompromising a misuse of the power of our minds that it forms a kind of perverse, cultural singularity—a vanishing point beyond which rational discourse proves impossible.
  • Until recently, it might have seemed that we were an unhappy bit of mildew on a heavenly body whirling in space among many that have no mildew on them at all. this was something that classical science could explain. Yet, the moment it begins to appear that we are deeply connected to the entire universe, science reaches the outer limits of its powers. Because it is founded on the search for universal laws, it cannot deal with singularity, that is, with uniqueness. The universe is a unique event and a unique story, and so far we are the unique point of that story. But unique events and stories are the domain of poetry, not science. With the formulation of the Anthropic Cosmological Principle, science has found itself on the border between formula and story, between science and myth. In that, however, science has paradoxically returned, in a roundabout way, to man, and offers him — in new clothing — his lost integrity. It does so by anchoring him once more in the cosmos.
    • Václav Havel, "The Need for Transcendence in the Postmodern World" (1994)
  • The subject of this book is the structure of space-time on length-scales from 10-13cm, the radius of an elementary particle, up to 1028cm, the radius of the universe. ...we base our treatment on Einstein's General Theory of Relativity. This theory leads to two remarkable predictions about the universe: first, that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly, that there is a singularity in our past which constitutes, in some sense, a beginning to the universe.
  • Your reason is now mature enough to examine this object [religion]. In the first place divest yourself of all bias in favour of novelty & singularity of opinion. Indulge them in any other subject rather than that of religion. It is too important, & the consequences of error may be too serious.
  • He was under the middle height; and his lower limbs were small in comparison with the upper, but neat and well-turned. His shoulders were very broad for his size; he had a face, in which energy and sensibility were remarkably mixed-up, an eager power checked and made patient by ill-health. Every feature was at once strongly cut, and delicately alive. If there was any faulty expression, it was in the mouth, which was not without something of a character of pugnacity... The head was a particular puzzle for the phrenologist, being remarkably small in the skull; a singularity he has in common with Lord Byron and Mr Shelley, none of whose hats I could get on.
  • In Poetry I have a few axioms, and you will see how far I am from their centre. I think Poetry should surprise by a fine excess and not by singularity — it should strike the reader as a wording of his own highest thoughts, and appear almost a remembrance — Its touches of Beauty should never be halfway thereby making the reader breathless instead of content: the rise, the progress, the setting of imagery should like the Sun come natural to him — shine over him and set soberly although in magnificence leaving him in the luxury of twilight — but it is easier to think what Poetry should be than to write it — and this leads me on to another axiom. That if Poetry comes not as naturally as the leaves to a tree it had better not come at all.
    • John Keats, Letter to John Taylor (February 27, 1818) Letters (1817-1820)
  • "Turn on" meant go within to activate your neural and genetic equipment. Become sensitive to the many and various levels of consciousness and the specific triggers that engage them. Drugs were one way to accomplish this end. "Tune in" meant interact harmoniously with the world around you — externalize, materialize, express your new internal perspectives. Drop out suggested an elective, selective, graceful process of detachment from involuntary or unconscious commitments. "Drop Out" meant self-reliance, a discovery of one's singularity, a commitment to mobility, choice, and change. Unhappily my explanations of this sequence of personal development were often misinterpreted to mean "Get stoned and abandon all constructive activity."
  • An orator whose purpose is to persuade men must speak the things they wish to hear; an orator, whose purpose is to move men, must also avoid disturbing the emotional effect by any obtrusion of intellectual antagonism; but an author whose purpose is to instruct men, who appeals to the intellect, must be careless of their opinions, and think only of truth. It will often be a question when a man is or is not wise in advancing unpalateable opinions, or in preaching heresies; but it can never be a question that a man should be silent if unprepared to speak the truth as he conceives it. Deference to popular opinion is one great source of bad writing, and is all the more disastrous because the deference is paid to some purely hypothetical requirement. When a man fails to see the truth of certain generally accepted views, there is no law compelling him to provoke animosity by announcing his dissent. He may be excused if he shrink from the lurid glory of martyrdom; he may be justified in not placing himself in a position of singularity. He may even be commended for not helping to perplex mankind with doubts which he feels to be founded on limited and possibly erroneous investigation. But if allegiance to truth lays no stern command upon him to speak out his immature dissent, it does lay a stern command not to speak out hypocritical assent.
  • We agree with Vinge's suggestion for naming events that are "capable of rupturing the fabric of human history" (or leading to profound societal changes) as a "singularity". This is a useful terminology especially since a mathematically rigorous singularity seems impossible for technological and related societal change. ...a wide variety of methods have been used and almost all point to singularities in the present century, particularly in the middle of the century.
  • It is hard to understand how this infinitely dense singularity can evaporate into nothing. For matter inside the black hole leak out into the universe requires that it travel faster than the speed of light.
    • John Moffat, Reinventing Gravity (2008) Ch.5, "Conventional Black Holes"
  • Where does one start with a theory of man if the theory of man as an organism in an environment doesn't work and all the attributes of man which were accepted in the old modern age are now called into question: his soul, mind, freedom, will, Godlikeness?
    There is only one place to start: the place where man's singularity is there for all to see and cannot be called into question, even in a new age in which everything else is in dispute.
    That singularity is language...
  • ...the stereographic projection of the spherical surface. From the north pole P we draw radial lines to project every point of the surface of the sphere upon the horizontal plane [below, perpendicular to a line joining it to P and the sphere's center]. In general this transformation is unique and continuous , although the metrical relations are distorted; for the point P, however, it shows a singularity. Point P is mapped upon the infinite; i.e., no finitely located point of the plane corresponds to it. It can be shown that every transformation possesses a singularity in at least one point. The surface of the sphere is therefore called topologically different from the plane. Only a "sphere without a north pole" [point] would be topologically equivalent to a plane. ...such a sphere has a point-shaped hole without a boundary and is no longer a closed surface.
  • Each word is a singularity, or is connected with a singularity, in our way of understanding existence.
    • Léon Rosenfeld (1904–74) As quoted in A Question of Physics: Conversations in Physics and Biology (1979), Paul Buckley and F. David Peat, Routledge and Kegan Paul, London, p. 29.
  • On the night of the exodus, the people met God, had a rendezvous with Him, and made His acquaintance for the first time. On Yom Kippur night, man gets very close to his Father in heaven, again meets Him, talks to Him, cries before and implores Him. The grandeur and singularity of these two nights lie in the God-man confrontation.
  • Welcome to the early twenty-first century, human.
    It’s night in Milton Keynes, sunrise in Hong Kong. Moore’s Law rolls inexorably on, dragging humanity toward the uncertain future. The planets of the solar system have a combined mass of approximately 2 x 1027 kilograms. Around the world, laboring women produce forty-five thousand babies a day, representing 1023 MIPS of processing power. Also around the world, fab lines casually churn out thirty million microprocessors a day, representing 1023 MIPS. In another ten months, most of the MIPS being added to the solar system will be machine-hosted for the first time. About ten years after that, the solar system’s installed processing power will nudge the critical 1 MIPS per gram threshold—one million instructions per second per gram of matter. After that, singularity—a vanishing point beyond which extrapolating progress becomes meaningless. The time remaining before the intelligence spike is down to single-digit years ...
  • Celibate, like the Fly in the Heart of an apple, dwells in a perpetual sweetness, but sits alone, and is confin'd and dies in Singularity; but Marriage, like the useful Bee, builds a house and gathers sweetness from every flower, and labours and unites into Societies and Republicks, and sends out Colonies, and feeds the World with delicacies, and obeys their King, and keeps order, and exercises many Vertues, and promotes the Interest of mankind, and is that state of good things to which God hath designed the present constitution of the world.
  • My grand mother, who was very religious, and to whom I was much attached — my master, who belonged to the church, and other religious persons who visited the house, and whom I often saw at prayers, noticing the singularity of my manners, I suppose, and my uncommon intelligence for a child, remarked I had too much sense to be raised – and if I was, I would never be of any service to any one – as a slave.
    • Nat Turner, The Confessions of Nat Turner (1831)
  • Quite aware that the criteria of value in mathematical work are, to some extent, purely aesthetic, he [von Neumann] once expressed an apprehension that the values put on abstract scientific achievement in out present civilization might diminish: "The interests of humanity might change, the present curiosities in science may cease, and entirely different things may occupy the human mind in the future." One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.
  • Here I had tried a straightforward extrapolation of technology, and found myself precipitated over an abyss. It’s a problem we face every time we consider the creation of intelligences greater than our own. When this happens, human history will have reached a kind of singularity — a place where extrapolation breaks down and new models must be applied — and the world will pass beyond our understanding.
  • Developments that before were thought might only happen in "a million years" (if ever) will likely happen in the next century. ...I think it's fair to call this event a singularity ("the Singularity" for the purposes of this paper). It is a point where our models must be discarded and a new reality rules. As we move closer and closer to this point, it will loom vaster and vaster over human affairs till the notion becomes a commonplace. Yet when it finally happens it may still be a great surprise and a greater unknown.
  • I have argued above that we cannot prevent the Singularity, that its coming is an inevitable consequence of the humans' natural competitiveness and the possibilities inherent in technology. And yet … we are the initiators. Even the largest avalanche is triggered by small things. We have the freedom to establish initial conditions, make things happen in ways that are less inimical than others. Of course (as with starting avalanches), it may not be clear what the right guiding nudge really is...
  • The "hard" science-fiction writers are the ones who try to write specific stories about all that technology may do for us. More and more, these writers felt an opaque wall across the future. Once, they could put such fantasies millions of years in the future. Now they saw that their most diligent extrapolations resulted in the unknowable … soon.
    • Vernor Vinge, "The Coming Technological Singularity" (1993)
  • We all have our little solipsistic delusions, ghastly intuitions of utter singularity: that we are the only one in the house who ever fills the ice-cube tray, who unloads the clean dishwasher, who occasionally pees in the shower, whose eyelid twitches on first dates; that only we take casualness terribly seriously; that only we fashion supplication into courtesy; that only we hear the whiny pathos in a dog's yawn, the timeless sigh in the opening of the hermetically-sealed jar, the splattered laugh in the frying egg, the minor-D lament in the vacuum's scream; that only we feel the panic at sunset the rookie kindergartner feels at his mother's retreat. That only we love the only-we. That only we need the only-we. Solipsism binds us together, J.D. knows. That we feel lonely in a crowd; stop not to dwell on what's brought the crowd into being. That we are, always, faces in a crowd.
    • David Foster Wallace, "Westward The Course Of Empire Takes Its Way," Girl With Curious Hair (1989)
  • The beekeeper and the Sherpa, one from a remote former colony of the Crown on the edge of the Pacific Ocean, the other from the edge of the heavens. They affirmed the power of humble determination and, placing themselves firmly with the mythic paradigms of their respective cultures, won one for the underdogs. ...On this lonely planet of freeze dried food, computer generated fabrics and commercialised mountain climbing, it is almost impossible to imagine the earth-shaking impact that Hillary and Norgay’s achievement had in 1953. For many it represented the last of the earth’s great challenges. It placed Hillary in the lineage of great terrestrial explorers. … His achievement as one of mankind’s great accomplishments came at one of the last times in history when such a feat could still be recognised as a distinctly human one, and not technological. … Hillary’s near-mythical status puts him on a plateau above sporting heroes, for he has distinguished himself well beyond the singularity of a mountain. From a feat that would have been the crowning achievement of many careers, he has gone on to become a humanitarian, an ambassador and elder statesman, never giving up, never giving in to either despair or complacency, always planning the next goal.
  • With Mie's view of matter there is contrasted another, according to which matter is a limiting singularity of the field, but charges and masses are force-fluxes in the field. This entails a new and more cautious attitude towards the whole problem of matter.
    • Hermann Weyl, Space—Time—Matter (1952) Author's Preface to Fourth Edition (1920)
  • The true human being … is the meaning of the universe. He is a dancing star. He is the exploding singularity pregnant with infinite possibilities.

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