On Literature: On Radio-Television (From Fiction to Science-Fiction)

"There is nothing wrong with your television set. Do not attempt to adjust the picture. We are controlling transmission. We will control the horizontal, we will control the vertical. We can change the focus to a soft blur, or sharpen it to crystal clarity. For the next hour, sit quietly and we will control all that you see and hear. You are about to participate in a great adventure. You are about to experience the awe and mystery which reaches from the inner mind to... The Outer Limits." (Leslie Stevens, The Outer Limits, [TV series, opening narration] 1963)

"Do you see, then, that the important prediction is not the automobile, but the parking problem; not radio, but the soap-opera; not the income tax but the expense account; not the Bomb but the nuclear stalemate? Not the action, in short, but the reaction?" (Isaac Asimov, "Future? Tense!", 1965)

"Even the vagal flushes that seized at my chest seemed extensions of that real world of violence calmed and tamed within our television programmes and the pages of news magazines." (J. G Ballard, "Crash", 1973)

"Her grandparents, believers in progress, had always told her things were better now. Human minds had been darker when people couldn’t read late at night, their prejudices greater when they had lacked television’s images of other places, their work harder without the appliances many took for granted. Nina was not so sure; technical civilization had isolated people from the basics of life, and had fooled them into believing that they controlled the world." (Pamela Sargent, "The Old Darkness", 1983)

"The sky above the port was the color of television, tuned to a dead channel." (William Gibson, "Neuromancer", 1984)

"A computer terminal is not some clunky old television with a typewriter in front of it. It is an interface where the mind and body can connect with the universe and move bits of it about." (Douglas N Adams, "Mostly Harmless", 1992)

On Literature: On Matrices (From Fiction to Science-Fiction)

"Man himself, at the very least, is music, a brave theme that makes music also of its vast accompaniment, its matrix of storms and stars. Man himself in his degree is eternally a beauty in the eternal form of things. It is very good to have been man. And so we may go forward together with laughter in our hearts, and peace, thankful for the past, and for our own courage. For we shall make after all a fair conclusion to this brief music that is man." (Olaf Stapledon, "Last and First Men: A Story of the Near and Far Future", 1930)

"Literacy remains even now the base and model of all programs of industrial mechanization; but, at the same time, locks the minds and senses of its users in the mechanical and fragmentary matrix that is so necessary to the maintenance of mechanized society." (Marshall McLuhan, "Understanding Media, 1964)

"Deception, in a system of this sort, can be defined simply as entropy [...] And of course, entropy, or degradation of order, is avoided by all civilized beings, since no local increase in complexity can offset entropic effects in the larger matrix." (James Tiptree Jr, "Faithful to Thee, Terra, in Our Fashion" ["Parimutuel Planet"], Galaxy, 1969) 

"Thought is a matrix which engenders its own reality. The ideas, concepts, belief-systems that your ancestors trapped have become your trap." (Alfred A Attanasio, "Radix", 1981)

"As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and, try as we may, we cannot always decipher them precisely in the clearer light of our own day." (Margaret Atwood, "The Handmaid’s Tale", 1986)

"Metaphor has traditionally been regarded as the matrix and pattern of the figures of speech." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

William Gibson - Collected Quotes

"Cyberspace. A consensual hallucination experienced daily by billions of legitimate operators, in every nation, by children being taught mathematical concepts. [...] A graphic representation of data abstracted from banks of every computer in the human system. Unthinkable complexity. Lines of light ranged in the nonspace of the mind, clusters and constellations of data." (William Gibson, "Neuromancer", 1984)

"The sky above the port was the color of television, tuned to a dead channel." (William Gibson, "Neuromancer", 1984)

"It was one of those nights, I quickly decided, when you slip into an alternate continuum, a city that looks exactly like the one where you live, except for the peculiar difference that it contains not one person you love or know or have even spoken to before." (William Gibson, "The Winter Market", 1985)

"He’d used decks in school, toys that shuttled you through the infinite reaches of that space that wasn’t space, mankind’s unthinkably complex consensual hallucination, the matrix, cyberspace, where the great corporate hotcores burned like neon novas, data so dense you suffered sensory overload if you tried to apprehend more than the merest outline." (William Gibson, "Count Zero", 1986)

"Some things you teach yourself to remember to forget." (William Gibson, "Count Zero", 1986)

"If we envision the entire System of Mathematics as a great Engine for proving theorems, then we must say, through the agency of the Modus, that such an Engine lives, and could indeed prove its own life, should it develop the capacity to look upon itself." (William Gibson & Bruce Sterling, "The Difference Engine", 1991)

"It is a concatenation of synergistic interactions; the whole system is on the period-doubling route to Chaos! [...] in layman’s terms, it means that everything gets twice as bad, twice as fast, until everything falls completely apart!" (William Gibson & Bruce Sterling, "The Difference Engine", 1991)

"Storms of Cataclysm lashed the Cretaceous earth, vast fires raged, and cometary grit sifted through the roiling atmosphere, to blight and kill the wilting foliage, till the mighty Dinosauria, adapted to a world now shattered, fell in massed extinction, and the leaping machineries of Evolution were loosed in chaos, to re-populate the stricken Earth with strange new orders of being." (William Gibson & Bruce Sterling, "The Difference Engine", 1991)

On Literature: On Communities (From Fiction to Science-Fiction)

"In such a universe as this what significance could there be in our fortuitous, our frail, our evanescent community?" (Olaf Stapledon, "Star Maker", 1937)

"A man is morally responsible to his community. That’s a good idea. But his community is also morally responsible to him." (Philip K Dick, "The Man Who Japed", 1956)

"We spread among the community and we became a part of them, sharing in their consciousness and directing them in their total integration. [. . .] They had integrated their group personality on a level that we could perceive and understand. This is the natural evolution of men and truly their one salvation in the total hostile universe." (Thomas N. Scortia, "The Armageddon Tapes - Tape 1", 1974)

"Part of human life is the need to reassure ourselves about the future that we may never live to see, rather than fool ourselves, as many did in the last century, that there won’t be any future and they might as well lie down and die." (George Zebrowski, "Macrolife: A Mobile Utopia", 1979)

"Science fiction is the branch of literature that deals with the effects of change on people in the real world as it can be projected into the past, the future, or to distant places. It often concerns itself with scientific or technological change, and it usually involves matters whose importance is greater than the individual or the community; often civilization or the race itself is in danger." (James Gunn, "Road to Science Fiction" Vol. 2, 1979)

"I think that most of us, anyway, read these stories that we know are not 'true' because we're hungry for another kind of truth: the mythic truth about human nature in general, the particular truth about those life-communities that define our own identity, and the most specific truth of all: our own self-story. Fiction, because it is not about someone who lived in the real world, always has the possibility of being about oneself." (Orson Scott Card, "Ender’s Game", [introduction] 1985)

"Species at this stage of their development tend to be territorial, Memory reminded Drill. Their political mentality is based around the concept of borders. The idea of a borderless community of species may be perceived as a threat." (Walter J Williams, "Dinosaurs", 1987)

"Universities are meant to pass the torch of civilization, not just download data into student skulls, and the values of the academic community are strongly at odds with those of all would-be information empires. Teachers at all levels, from kindergarten up, have proven to be shameless and persistent software and data pirates. Universities do not merely ‘'leak information' but vigorously broadcast free thought." (Bruce Sterling, "The Hacker Crackdown", 1992)

"Here's the secret that every successful software company is based on: You can domesticate programmers the way beekeepers tame bees. You can't exactly communicate with them, but you can get them to swarm in one place and when they're not looking, you can carry off the honey." (Orson Scott Card, "How Software Companies Die", Windows Sources: The Magazine for Windows Experts, 1995)

"Here is the truth. What human life is, what it’s for, what we do, is create communities." (Orson Scott Card, "Pastwatch: The Redemption of Christopher Columbus", 1996)

"Human evolution is driven by community needs […]. How is that possible, since genetic information is passed only by and to individuals?" (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"Most human communities demand anti-survival behavior from large numbers of their members." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"That would be true […] and indeed is true in the main, except that there are only a few types of human communities that actually survive long enough to improve the chances of individual survival." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

"There are community traits that promote survival of the individual, and individual traits that promote the survival of the community." (Orson Scott Card, "First Meetings in Ender's Universe", 2002)

On Literature: On Writers (From Fiction to Science-Fiction)

"No one for a moment can pretend that printing is so great a discovery as writing, or algebra as a language." (Benjamin Disraeli, "Lothair", 1870)

"A writer must make up his mind to the possible rough treatment of the critics, who swarm like bacteria whenever there is any literary material on which they can feed." (Oliver W Holmes, "Over the Teacups", 1891)

"The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." (Herbert G Wells, "Mankind in the Making", 1903)

"The 'romance' of space - drivel written in the old days. When you’re not blasting, you float in a cramped hotbox, crawl through dirty mazes of greasy pipe and cable to tighten a lug, scratch your arms and bark your shins, get sick and choked up because no gravity helps your gullet get the food down." (Walter M Miller Jr., "Death of a Spaceman", 1954)

"Science fiction is no more written for scientists than ghost stories are written for ghosts." (Brian Aldiss, Penguin Science Fiction, 1961)

"The role of the writer today has totally changed - he is now merely one of a huge army of people filling the environment with fictions of every kind. To survive, he must become far more analytic, approaching his subject matter like a scientist or engineer. If he is to produce fiction at all, he must outimagine everyone else, scream louder, whisper more quietly. For the first time in the history of narrative fiction, it will require more than talent to become a writer." (James G Ballard, "Fictions of Every Kind", 1971)

"A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects." (Robert A Heinlein, "Time Enough for Love", 1973)

"If a book were written all in numbers, it would be true. It would be just. Nothing said in words ever came out quite even. Things in words got twisted and ran together, instead of staying straight and fitting together. But underneath the words, at the center, like the center of the Square, it all came out even. Everything could change, yet nothing would be lost. If you saw the numbers you could see that, the balance, the pattern. You saw the foundations of the world. And they were solid." (Ursula K Le Guin, "The Dispossessed: An Ambiguous Utopia", 1974)

"Science fiction writers foresee the inevitable, and although problems and catastrophes may be inevitable, solutions are not." (Isaac Asimov, "How Easy to See the Future", Natural History magazine, 1975)

"To write fiction, one needs a whole series of inspirations about people in an actual environment, and then a whole lot of work on the basis of those inspirations." (Aldous Huxley, "Moksha", 1977)

"Human brains back then had become such copious and irresponsible generators of suggestions as to what might be done with life, that they made acting for the benefit of future generations seem one of many arbitrary games which might be played by narrow enthusiasts - like poker or polo or the bond market, or the writing of science-fiction novels." (Kurt Vonnegut Jr, "Galapagos" 1985)

"A science fiction writer is - or should be - constrained by what is, or logically might be. That can mean simple fidelity to facts (which, in science, are always more important than theories - though Lord knows the two help shape each other, undermining the convenient, complacent separation of observer and observed). To me it also means heeding the authentic, the actual and concrete. Bad fiction uses the glossy generality; good writing needs the smattering of detail, the unrelenting busy mystery of the real." (Gregory Benford, "Afterword to Exposures", [in Alien Flesh] 1986)

"Science fiction offers its writers chances of embarrassment that no other form of fiction does." (Isaac Asimov, "Robot Dreams" [introduction] 1986)

"If poets are the unacknowledged legislators of the world, science-fiction writers are its court jesters." (Bruce Sterling, "Burning Chrome", [preface] 1986)

"Science fiction rarely is about scientists doing real science, in its slowness, its vagueness, the sort of tedious quality of getting out there and digging amongst rocks and then trying to convince people that what you're seeing justifies the conclusions you're making. The whole process of science is wildly under-represented in science fiction because it's not easy to write about. There are many facets of science that are almost exactly opposite of dramatic narrative. It's slow, tedious, inconclusive, it's hard to tell good guys from bad guys [...]" (Kim S Robinson, [interview] 1997)

"Do you know why teachers use me? Because I speak in tongues. I write metaphors. Every one of my stories is a metaphor you can remember. The great religions are all metaphor. We appreciate things like Daniel and the lion's den, and the Tower of Babel. People remember these metaphors because they are so vivid you can't get free of them and that's what kids like in school." (Ray Bradbury, The Paris Review, [interview] 2010)

"Science fiction these days is only half a step ahead of science. Astrophysicists and scientists are working in the same way as science fiction writers. They’re working things out in their imagination based on the slim scientific facts that they know. Hawking imagines a black hole and then discovers the mathematics that support his theory, and new possibilities come to light. That’s the imaginative flair that scientists have to have. For me as a sci-fi writer, spinning those ideas in your mind brings you to the point where you dream in science fiction. Suddenly you think of something in the middle of the night, and it’s so vivid you don’t need to write it down because you know you’ll remember it in the morning. That’s what these books, Zero G, reflect: a vivid imagination." (William Shatner, "William Shtner on Sci-Fi, Aging and the Environment", Saturday Evening Post, [interview] 2017)

"Of course all children's literature is not fantastic, so all fantastic books need not be children's books. It is still possible, even in an age so ferociously anti-romantic as our own, to write fantastic stories for adults: though you will usually need to have made a name in some more fashionable kind of literature before anyone will publish them." (Clive S Lewis)

On Numbers: On Units (-1199)

"Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two. […] Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one." (Aristotle, "Physics", cca. 350 BC)

"A prime number is one" (which is) measured by a unit alone." (Euclid, "Elements" Book VII, cca. 300 BC)

"If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect." (Euclid, "Elements", cca 300 BC)

"Numbers prime to one another are those which are measured by a unit alone as a common measure." (Euclid, "Elements" Book VII, cca 300 BC)

"There exists an elegant and sure method of generating these numbers, which does not leave out any perfect numbers and which does not include any that are not; and which is done in the following way. First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096; and then they must be totalled each time there is a new term, and at each totaling examine the result, if you find that it is prime and non-composite, you must multiply it by the quantity of the last term that you added to the line, and the product will always be perfect. If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term. If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity." (Nicomachus of Gerasa,"Introductio Arithmetica", cca. 100 AD)

On Numbers: On Units (1200-1699)

"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"There can only be one wisdom. For if it were possible that there be several wisdoms, then these would have to be from one. Namely, unity is prior to all plurality." (Nicholas of Cusa, "De Pace Fidei" ["The Peace of Faith"], 1453)

"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)

"The Fractions which represent the Probabilities of happening and failing, being added together, their Sum will always be equal to Unity, since the Sum of their Numerators will be equal to their common Denominator : now it being a certainty that an Event will either happen or fail, it follows that Certainty, which may be conceived under the notion of an infinitely great degree of Probability, is fitly represented by Unity." (Abraham de Moivre, "The Doctrine of Chances", 1718)

"The probability of an Event is greater, or less, according to the number of Chances by which it may Happen, compar’d with the number of all the Chances, by which it may either Happen or Fail. […] Therefore, if the Probability of Happening and Failing are added together, the Sum will always be equal to Unit." (Abraham De Moivre,"The Doctrine of Chances", 1718)

"Statics is the science of the equilibrium of forces. In general, force or power is the cause, whatever it may be, which induces or tends to impart motion to the body to which it is applied. The force or power must be measured by the quantity of motion produced or to be produced. In the state of equilibrium, the force has no apparent action. It produces only a tendency for motion in the body it is applied to. But it must be measured by the effect it would produce if it were not impeded. By taking any force or its effect as unity, the relation of every other force is only a ratio, a mathematical quantity, which can be represented by some numbers or lines. It is in this fashion that forces must be treated in mechanics." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

"Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number and thus producing a positive number, of a number being imaginary. Hence they talk of two roots to every equation of the second order, and the learner is to try which will succeed in a given equation: they talk of solving an equation which requires two impossible roots to make it solvable: they can find out some impossible numbers, which, being multiplied together, produce unity. This is all jargon, at which common sense recoils; but, from its having been once adopted, like many other figments, it finds the most strenuous supporters among those who love to take things upon trust, and hate the labour of a serious thought." (William Frend, "The Principles of Algebra", 1796)

On Numbers: On Units (1800-1899)

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If we call +1, -1, and √-1 had been called direct, inverse and lateral units, instead of positive, negative, and imaginary"" (or impossible) units, such an obscurity would have been out of the question." (Carl F Gauss, "Theoria residuorum biquadraticum. Commentatio secunda", Göttingische gelehrte Anzeigen 23 (4), 1831)

"I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative." (Hermann G Grassmann, "Ausdehnungslehre", 1844)

"If they [mathematicians] find a quantity greater than any finite number of the assumed units, they call it infinitely great; if they find one so small that its every finite multiple is smaller than the unit, they call it infinitely small; nor do they recognise any other kind of infinitude than these two, together with the quantities derived from them as being infinite to a higher order of greatness or smallness, and thus based after all on the same idea." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"The general equations are next applied to the case of a magnetic disturbance propagated through a non-conductive field, and it is shown that the only disturbances which can be so propagated are those which are transverse to the direction of propagation, and that the velocity of propagation is the velocity v, found from experiments such as those of Weber, which expresses the number of electrostatic units of electricity which are contained in one electromagnetic unit. This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws." (James C Maxwell, "A Dynamical Theory of the Electromagnetic Field", 1865)

"Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but" (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it ‘mind-mass’. All thinking is, accordingly, formation of new mind masses." (Bernhard Riemann,"Gesammelte Mathematische Werke", 1876)

"What is commonly called the geometrical representation of complex numbers has at least this advantage […] that in it 1 and i do not appear as wholly unconnected and different in kind: the segment taken to represent i stands in a regular relation to the segment which represents 1. […] A complex number, on this interpretation, shows how the segment taken as its representation is reached, starting from a given segment" (the unit segment), by means of operations of multiplication, division, and rotation." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)

"Any system in stable chemical equilibrium, subjected to the influence of an external cause tends to change either its temperature or its condensation"" (pressure, concentration, number of molecules in unit volume), either as a whole or in some of its parts, can only undergo such internal modifications as would, if produced alone, bring about a change of temperature or of condensation of opposite sign to that resulting from the external cause." (Henri L Le Chatelier, "A General Statement of the Laws of Chemical Equilibrium", Comptes rendus Vol. 99, 1884)

On Numbers: On Units (2000-)

"√-1 is take for granted today. No serious mathematician would deny that it is a number. Yet it took centuries for √-1 to be officially admitted to the pantheon of numbers. For almost three centuries, it was controversial; mathematicians didn't know what to make of it; many of them worked with it successfully without admitting its existence. […] Primarily for cognitive reasons. Mathematicians simply could not make it fit their idea of what a number was supposed to be. A number was supposed to be a magnitude. √-1 is not a magnitude comparable to the magnitudes of real numbers. No tree can be √-1 units high. You cannot owe someone √-1 dollars. Numbers were supposed to be linearly ordered. √-1 is not linearly ordered with respect to other numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Descartes’ idea to use numbers to describe points in space involves the choice of a coordinate system or coordinate frame: an origin, together with axes and units of length along the axes. A recurring theme of all the different geometries [...] is the question of what a coordinate frame is, and what I can get out of it. While coordinates provide a convenient framework to discuss points, lines, and so on, it is a basic requirement that any meaningful statement in geometry is independent of the choice of coordinates. That is, coordinate frames are a humble technical aid in determining the truth, and are not allowed the dignity of having their own meaning." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Lie groups describe finite symmetries or symmetries which smoothly depend on a finite number of real parameters. Lie algebras are the linearization of Lie groups at the unit element. The passage from Lie groups to Lie algebras simplifies considerably the approach. Lie algebras are frequently called infinitesimal symmetries." (Eberhard Zeidler, "Quantum Field Theory III: Gauge Theory", 2006)

"The correlation coefficient has two fabulously attractive characteristics. First, for math reasons that have been relegated to the appendix, it is a single number ranging from –1 to 1. A correlation of 1, often described as perfect correlation, means that every change in one variable is associated with an equivalent change in the other variable in the same direction. A correlation of –1, or perfect negative correlation, means that every change in one variable is associated with an equivalent change in the other variable in the opposite direction. The closer the correlation is to 1 or –1, the stronger the association. […] The second attractive feature of the correlation coefficient is that it has no units attached to it. […] The correlation coefficient does a seemingly miraculous thing: It collapses a complex mess of data measured in different units" (like our scatter plots of height and weight) into a single, elegant descriptive statistic." (Charles Wheelan, "Naked Statistics: Stripping the Dread from the Data", 2012)

"At any rate, long before the curvature of space was first detected, Beltrami’s construction of the hyperbolic plane showed that more than one kind of geometry is possible. Beltrami assumed that Euclidean space exists, and constructed a non-Euclidean plane inside it, with nonstandard definitions of 'line' and 'distance' (namely, line segments in the unit disk and pseudodistance). This shows that the geometry of Bolyai and Lobachevsky is logically as valid as the geometry of Euclid: if there is a space in which 'lines' and 'distance' behave as Euclid thought they do, then there is also a surface in which 'lines' and 'distance' behave as Bolyai and Lobachevsky thought they might." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)

"The higher the dimension, in other words, the higher the number of possible interactions, and the more disproportionally difficult it is to understand the macro from the micro, the general from the simple units. This disproportionate increase of computational demands is called the curse of dimensionality." (Nassim N Taleb, "Skin in the Game: Hidden Asymmetries in Daily Life", 2018)

"Quaternions are not actual extensions of imaginary numbers, and they are not taking complex numbers into a multi-dimensional space on their own. Quaternion units are instances of some number-like object type, identified collectively, but they are not numbers (be it real or imaginary). In other words, they form a closed, internally consistent set of object instances; they can of course be plotted visually on a multi-dimensional space but this only is a visualization within their own definition." (Huseyin Ozel, "Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects", 2018)

"Consider for example the complex numbers x + iy, where you of course ask what is i = √ −1 when you first encounter this mathematical construction. But that uncomfortable feeling of what this strange imaginary unit really is fades away as you get more experienced and learn that C is a field of numbers that is extremely useful, to say the least. You no longer care what kind of object i is but are satisfied only to know that i^2 = −1, which is how you calculate with i." (Andreas Rosén,"Geometric Multivector Analysis: From Grassmann to Dirac", 2019)

On Numbers: On Units (1950-1999)

"A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit. If these or similar statements were set down as definitions, then it could be objected with good reason that they define idem per idemi or even obscurum per obscurius. However, we can consider them as expository, as references to a primitive concept, familiar to us all, whose resolution into more fundamental concepts would perhaps be neither competent nor necessary." (Felix Hausdorff, "Set Theory", 1962)

"We may state as characteristic of modern science that this scheme of isolable units acting in one-way causality has proven to be insufficient. Hence the appearance, in all fields of science, of notions like wholeness, holistic, organismic, gestalt, etc., which all signify that, in the last resort, we must think in terms of systems of elements in mutual interaction […]." (Ludwig von Bertalanffy, "General System Theory", 1968)

"But the laws of addition and multiplication (the associative laws, for example) are not a human invention. They are unintended consequences of human invention, and they were discovered. And the existence of prime numbers - indivisible numbers that are the product only of themselves and unity - is also a discovery, no doubt quite a late one. The prime numbers were discovered in the series of natural numbers, not by everyone but by people who studied these numbers and their special peculiarities - by real mathematicians." (Karl R Popper, "Notes of a Realist on the Body-Mind Problem", [in "All Life is Problem Solving", 1999] 1972)

"Concepts are inventions of the human mind used to construct a model of the world. They package reality into discrete units for further processing, they support powerful mechanisms for doing logic, and they are indispensable for precise, extended chains of reasoning." (John Sown,"Conceptual Structures - Information Processing in Mind and Machine, 1984)

"In everyday language, the words 'pattern' and 'symmetry' are used almost interchangeably, to indicate a property possessed by a regular arrangement of more-or-less identical units […]" (Ian Stewart & Martin Golubitsky, "Fearful Symmetry: Is God a Geometer?", 1992)