Applied Mathematics Quotes

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It was all so very businesslike that one watched it fascinated. It was pork-making by machinery, pork-making by applied mathematics. And yet somehow the most matter-of-fact person could not help thinking of the hogs; they were so innocent, they came so very trustingly; and they were so very human in their protests - and so perfectly within their rights! They had done nothing to deserve it; and it was adding insult to injury, as the thing was done here, swinging them up in this cold-blooded, impersonal way, without pretence at apology, without the homage of a tear.
Upton Sinclair
Where there was nature and earth, life and water, I saw a desert landscape that was unending, resembling some sort of crater, so devoid of reason and light and spirit that the mind could not grasp it on any sort of conscious level and if you came close the mind would reel backward, unable to take it in. It was a vision so clear and real and vital to me that in its purity it was almost abstract. This was what I could understand, this was how I lived my life, what I constructed my movement around, how I dealt with the tangible. This was the geography around which my reality revolved: it did not occur to me, ever, that people were good or that a man was capable of change or that the world could be a better place through one’s own taking pleasure in a feeling or a look or a gesture, of receiving another person’s love or kindness. Nothing was affirmative, the term “generosity of spirit” applied to nothing, was a cliche, was some kind of bad joke. Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire- meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface, was all that anyone found meaning in…this was civilization as I saw it, colossal and jagged…
Bret Easton Ellis (American Psycho)
Only someone who doesn’t understand art tells an artist their art somehow failed. How the fuck can art fail? Art can’t be graded, because it’s going to mean something different to everyone. You can’t apply a mathematical absolute to art because there is no one formula for self-expression.
Kevin Smith (Tough Shit: Life Advice from a Fat, Lazy Slob Who Did Good)
Mathematics becomes very odd when you apply it to people. One plus one can add up to so many different sums
Michael Frayn (Copenhagen)
I would say that the five most important skills are of course, reading, writing, arithmetic, and then as you’re adding in, persuasion, which is talking. And then finally, I would add computer programming just because it’s an applied form of arithmetic that just gets you so much leverage for free in any domain that you operate in. If you’re good with computers, if you’re good at basic mathematics, if you’re good at writing, if you’re good at speaking, and if you like reading, you’re set for life.
Naval Ravikant
It was as though applied mathematics was my spouse, and pure mathematics was my secret lover.
Edward Frenkel (Love and Math: The Heart of Hidden Reality)
Although in principle we know the equations that govern the whole of biology, we have not been able to reduce the study of human behavior to a branch of applied mathematics.
Stephen Hawking (Black Holes and Baby Universes)
A famous Japanese Zen master, Hakuun Yasutani Roshi, said that unless you can explain Zen in words that a fisherman will comprehend, you don’t know what you’re talking about. Some fifty years ago a UCLA professor told me the same thing about applied mathematics. We like to hide from the truth behind foreign-sounding words or mathematical lingo. There’s a saying: The truth is always encountered but rarely perceived. If we don’t perceive it, we can’t help ourselves and we can’t much help anyone else.
Jeff Bridges (The Dude and the Zen Master)
There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
Nicholas Lobachevsky
In the early universe—when the universe was small enough to be governed by both general relativity and quantum theory—there were effectively four dimensions of space and none of time. That means that when we speak of the “beginning” of the universe, we are skirting the subtle issue that as we look backward toward the very early universe, time as we know it does not exist! We must accept that our usual ideas of space and time do not apply to the very early universe. That is beyond our experience, but not beyond our imagination, or our mathematics.
Stephen Hawking (The Grand Design)
They had chains which they fastened about the leg of the nearest hog, and the other end of the chain they hooked into one of the rings upon the wheel. So, as the wheel turned, a hog was suddenly jerked off his feet and borne aloft. At the same instant the ear was assailed by a most terrifying shriek; the visitors started in alarm, the women turned pale and shrank back. The shriek was followed by another, louder and yet more agonizing--for once started upon that journey, the hog never came back; at the top of the wheel he was shunted off upon a trolley and went sailing down the room. And meantime another was swung up, and then another, and another, until there was a double line of them, each dangling by a foot and kicking in frenzy--and squealing. The uproar was appalling, perilous to the ear-drums; one feared there was too much sound for the room to hold--that the walls must give way or the ceiling crack. There were high squeals and low squeals, grunts, and wails of agony; there would come a momentary lull, and then a fresh outburst, louder than ever, surging up to a deafening climax. It was too much for some of the visitors--the men would look at each other, laughing nervously, and the women would stand with hands clenched, and the blood rushing to their faces, and the tears starting in their eyes. Meantime, heedless of all these things, the men upon the floor were going about their work. Neither squeals of hogs nor tears of visitors made any difference to them; one by one they hooked up the hogs, and one by one with a swift stroke they slit their throats. There was a long line of hogs, with squeals and life-blood ebbing away together; until at last each started again, and vanished with a splash into a huge vat of boiling water. It was all so very businesslike that one watched it fascinated. It was pork-making by machinery, pork-making by applied mathematics. And yet somehow the most matter-of-fact person could not help thinking of the hogs; they were so innocent, they came so very trustingly; and they were so very human in their protests--and so perfectly within their rights! They had done nothing to deserve it; and it was adding insult to injury, as the thing was done here, swinging them up in this cold-blooded, impersonal way, without a pretence at apology, without the homage of a tear. Now and then a visitor wept, to be sure; but this slaughtering-machine ran on, visitors or no visitors. It was like some horrible crime committed in a dungeon, all unseen and unheeded, buried out of sight and of memory.
Upton Sinclair (The Jungle)
Mr Baley", said Quemot, "you can't treat human emotions as though they were built about a positronic brain". "I'm not saying you can. Robotics is a deductive science and sociology an inductive one. But mathematics can be made to apply in either case.
Isaac Asimov (The Naked Sun (Robot, #2))
You simply need to know where to look for the questions. An easy mathematical formula applied to Homo sapiens. And behold! Science reigns over nature once more. No emotions needed.
Kerri Maniscalco (Stalking Jack the Ripper (Stalking Jack the Ripper, #1))
You never say what I wish you’d say, and you frequently say nothing at all when it’s clear you should say something, so it’s not entirely fantastical that you’d say a certain thing when you mean something else entirely.” He opened his mouth, shut it, and considered the ground briefly before responding. “I remember studying Fleet Admiral Starcrest’s Mathematical Probabilities Applied to Military Strategies as a young boy and finding that less confusing than what you just said.” Now it was her turn for a stunned pause before answering. “Sicarius?” She laid a tentative hand on his shoulder. “Was that a joke?” “A statement of fact.
Lindsay Buroker (Dark Currents (The Emperor's Edge, #2))
As he soars, he thinks, suddenly, of Dr. Kashen. Or not of Dr. Kashen, necessarily, but the question he had asked him when he was applying to be his advisee: What's your favorite axiom? (The nerd pickup line, CM had once called it.) "The axiom of equality," he'd said, and Kashen had nodded, approvingly. "That's a good one," he'd said. The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for all time, that its very elementalness can never be altered. But it is impossible to prove. Always, absolutes, nevers: these are the words, as much as numbers, that make up the world of mathematics. Not everyone liked the axiom of equality––Dr. Li had once called it coy and twee, a fan dance of an axiom––but he had always appreciated how elusive it was, how the beauty of the equation itself would always be frustrated by the attempts to prove it. I was the kind of axiom that could drive you mad, that could consume you, that could easily become an entire life. But now he knows for certain how true the axiom is, because he himself––his very life––has proven it. The person I was will always be the person I am, he realizes. The context may have changed: he may be in this apartment, and he may have a job that he enjoys and that pays him well, and he may have parents and friends he loves. He may be respected; in court, he may even be feared. But fundamentally, he is the same person, a person who inspires disgust, a person meant to be hated. And in that microsecond that he finds himself suspended in the air, between ecstasy of being aloft and the anticipation of his landing, which he knows will be terrible, he knows that x will always equal x, no matter what he does, or how many years he moves away from the monastery, from Brother Luke, no matter how much he earns or how hard he tries to forget. It is the last thing he thinks as his shoulder cracks down upon the concrete, and the world, for an instant, jerks blessedly away from beneath him: x = x, he thinks. x = x, x = x.
Hanya Yanagihara (A Little Life)
The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics.
G.H. Hardy (A Mathematician's Apology)
If your wish is to become really a man of science and not merely a petty experimentalist, I should advise you to apply to every branch of natural philosophy, including mathematics.
Mary Wollstonecraft Shelley (Frankenstein)
It is not easy to become an educated person.
Richard Hamming (Methods of Mathematics Applied to Calculus, Probability, and Statistics (Dover Books on Mathematics))
You can't apply mathematical formulas to people, Cresswell. There's no equation for human emotion, there are too many variables.
Kerri Maniscalco (Stalking Jack the Ripper (Stalking Jack the Ripper, #1))
To be an engineer, and build a marvelous machine, and to see the beauty of its operation is as valid an experience of beauty as a mathematician's absorption in a wondrous theorem. One is not "more" beautiful than the other. To see a space shuttle standing on the launch pad, the vented gases escaping, and witness the thunderous blast-off as it climbs heavenward on a pillar of flame - this is beauty. Yet it is a prime example of applied mathematics.
Calvin C. Clawson (Mathematical Mysteries: The Beauty and Magic of Numbers)
The applications of knowledge, especially mathematics, reveal the unity of all knowledge. In a new situation almost anything and everything you ever learned might be applicable, and the artificial divisions seem to vanish.
Richard Hamming (Methods of Mathematics Applied to Calculus, Probability, and Statistics (Dover Books on Mathematics))
I don't mind nothing happening in a book, but nothing happening in a phony way--characters saying things people never say, doing jobs that don't fit, the whole works--is simply asking too much of a reader. Something happening in a phony way must beat nothing happening in a phony way every time, right? I mean, you could prove that, mathematically, in an equation, and you can't often apply science to literature.
Nick Hornby (The Polysyllabic Spree)
It is in the world of things and places, times and troubles and turbid processes, that mathematics is not so much applied as illustrated.
David Berlinski
The first man to understand the extraordinary magical power of applying mathematical calculation to things in nature was an Italian called Galileo Galilei.
E. H. Gombrich (A Little History of the World (Little Histories))
Poincaré [was] the last man to take practically all mathematics, pure and applied, as his province. ... Few mathematicians have had the breadth of philosophic vision that Poincaré had, and none in his superior in the gift of clear exposition.
Eric Temple Bell (Men of Mathematics)
If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes.
Alfred Tarski
A painter, who finds no satisfaction in mere representation, however artistic, in his longing to express his inner life, cannot but envy the ease with which music, the most non-material of the arts today, achieves this end. He naturally seeks to apply the methods of music to his own art. And from this results that modern desire for rhythm in painting, for mathematical, abstract construction, for repeated notes of colour, for setting colour in motion.
Wassily Kandinsky (Concerning the Spiritual in Art)
The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholely ‘useless’ (and this is true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.… The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. It
Andrew Hodges (Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game)
You can tell if a discipline is BS if the degree depends severely on the prestige of the school granting it. I remember when I applied to MBA programs being told that anything outside the top ten or twenty would be a waste of time. On the other hand a degree in mathematics is much less dependent on the school (conditional on being above a certain level, so the heuristic would apply to the difference between top ten and top two thousand schools). The same applies to research papers. In math and physics, a result posted on the repository site arXiv (with a minimum hurdle) is fine. In low-quality fields like academic finance (where papers are usually some form of complicated storytelling), the “prestige” of the journal is the sole criterion.
Nassim Nicholas Taleb (Skin in the Game: Hidden Asymmetries in Daily Life)
One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. ... For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.
G.H. Hardy (A Mathematician's Apology)
Nothing was affirmative, the term "generosity of spirit" applied to nothing, was a cliché, was some kind of bad joke. Sex is mathematics. Individuality no longer and issue. What does intelligence signify? Define reason. Desire-meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface was all that anyone found meaning in...this was civilization as I saw it, colossal and jagged.
Bret Easton Ellis (American Psycho)
If Henry Adams, whom you knew slightly, could make a theory of history by applying the second law of thermodynamics to human affairs, I ought to be entitled to base one on the angle of repose, and may yet. There is another physical law that teases me, too: the Doppler Effect. The sound of anything coming at you -- a train, say, or the future -- has a higher pitch than the sound of the same thing going away. If you have perfect pitch and a head for mathematics you can compute the speed of the object by the interval between its arriving and departing sounds. I have neither perfect pitch nor a head for mathematics, and anyway who wants to compute the speed of history? Like all falling bodies, it constantly accelerates. But I would like to hear your life as you heard it, coming at you, instead of hearing it as I do, a sober sound of expectations reduced, desires blunted, hopes deferred or abandoned, chances lost, defeats accepted, griefs borne. I don't find your life uninteresting, as Rodman does. I would like to hear it as it sounded while it was passing. Having no future of my own, why shouldn't I look forward to yours.
Wallace Stegner
I do not think the division of the subject into two parts - into applied mathematics and experimental physics a good one, for natural philosophy without experiment is merely mathematical exercise, while experiment without mathematics will neither sufficiently discipline the mind or sufficiently extend our knowledge in a subject like physics.
Balfour Stewart
For the purposes of science, information had to mean something special. Three centuries earlier, the new discipline of physics could not proceed until Isaac Newton appropriated words that were ancient and vague—force, mass, motion, and even time—and gave them new meanings. Newton made these terms into quantities, suitable for use in mathematical formulas. Until then, motion (for example) had been just as soft and inclusive a term as information. For Aristotelians, motion covered a far-flung family of phenomena: a peach ripening, a stone falling, a child growing, a body decaying. That was too rich. Most varieties of motion had to be tossed out before Newton’s laws could apply and the Scientific Revolution could succeed. In the nineteenth century, energy began to undergo a similar transformation: natural philosophers adapted a word meaning vigor or intensity. They mathematicized it, giving energy its fundamental place in the physicists’ view of nature. It was the same with information. A rite of purification became necessary. And then, when it was made simple, distilled, counted in bits, information was found to be everywhere.
James Gleick (The Information: A History, a Theory, a Flood)
If you’d like to see how to apply these ideas directly to memorizing formulas, try out the SkillsToolbox .com website for a list of easy-to-remember visuals for mathematical symbols.7
Barbara Oakley (A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra))
This was the geography around which my reality revolved: it did not occur to me, ever, that people were good or that a man was capable of change or that the world could be a better place through one’s taking pleasure in a feeling or a look or a gesture, of receiving another person’s love or kindness. Nothing was affirmative, the term “generosity of spirit” applied to nothing, was a cliche, was some kind of bad joke. Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire—meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface was all that anyone found meaning in … this was civilization as I saw it, colossal and jagged …
Bret Easton Ellis (American Psycho)
Well, regular math, or applied math, is what I suppose you could call practical math," he said. "It's used to solve problems, to provide solutions, whether it's in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn't exist to provide immediate, or necessarily obvious, practical applications. It's purely an expression of form, if you will - the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.
Hanya Yanagihara (A Little Life)
The astonishing fact is that similar mathematics applies so well to planets and to clocks. It needn’t have been this way. We didn’t impose it on the Universe. That’s the way the Universe is. If this is reductionism, so be it.
Carl Sagan (The Demon-Haunted World: Science as a Candle in the Dark)
For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to convey just what the subject is about. Instead, the focus is on learning and applying various procedures to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers to get the ball into the goal. Both accurately describe various key features, but they miss the \what?" and the \why?" of the big picture.
Keith Devlin (Introduction to Mathematical Thinking)
...where there was nature and earth, life and water, I saw a desert landscape that was unending, resembling some sort of crater, so devoid of reason and light and spirit that the mind could not grasp it on any sort of conscious level and if you came close the mind would reel backward, unable to take it in. It was a vision so clear and real and vital to me that in its purity it was almost abstract. This was what I could understand, this was how I lived my life, what I constructed my movement around, how I dealt with the tangible. This was the geography around which my reality revolved: it did not occur to me, ever, that people were good or that a man was capable of change or that the world could be a better place through one's taking pleasure in a feeling or a look or a gesture, of receiving another person's love or kindness. Nothing was affirmative, the term "generosity of spirit" applied to nothing, was a cliche, was some kind of bad joke. Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire - meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface was all that anyone found meaning in... this was civilization as I saw it, colossal and jagged...
Bret Easton Ellis (American Psycho)
In the post-Covid world, the mathematics of chaos theory will experience a greater relevancy as it is applied across a broader set of science disciplines, especially epidemiology, precision medicine and climate science. - Tom Golway
Tom Golway
[It] wants you to believe there are foreseeable trends and forces. When in fact it's all random phenomena. You apply mathematics and other disciplines, yes. But in the end you're dealing with a system that's out of control. Hysteria at high speeds, day to day, minute to minute. "People in free societies don't have to fear the pathology of the state. We create our own frenzy, our own mass convulsions, driven by thinking machines that we have no final authority over. The frenzy is barely noticeable most of the time. It's simply how we live.
Don DeLillo (Cosmopolis)
Kepler and Newton represent a critical transition in human history, the discovery that fairly simple mathematical laws pervade all of Nature; that the same rules apply on Earth as in the skies; and that there is a resonance between the way we think and the way the world works.
Carl Sagan (Cosmos)
There is such a thing as nonnerdy applied mathematics: find a problem first, and figure out the math that works for it (just as one acquires language), rather than study in a vacuum through theorems and artificial examples, then change reality to make it look like these examples.
Nassim Nicholas Taleb (Antifragile: Things That Gain From Disorder)
Her husband once said that he believed some sort of mathematical equation could be applied to life - since the longer you lived, the greater its seeming velocity. She always attributed this to familiarity. If you kept the same habits - and if you lived in the same place, worked in the same place - then you no longer spent a lot of time noticing. Noticing things - and trying to make sense of them - is what makes time remarkable. Otherwise, life blurs by, as it does now, so that she has difficulty keeping track of time at all, one day evaporating into the next.
Benjamin Percy (Red Moon)
When the lessons of symbolic or philosophical mathematics seen in nature, which were designed into religious architecture or art, are applied functionally (not just intellectually) to facilitate the growth and transformation of consciousness, then mathematics may rightly be called “sacred.
Michael S. Schneider (A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science)
As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer...
Oliver Heaviside (Electromagnetic Theory (Volume 1))
Mathematics occupies exactly the same logical space as God. No proof of God has ever relied on observation or experiment. The existence of God, in philosophy, is defended via rational and logical arguments. All of these same arguments can be applied to mathematics. Mathematics and God go together with consummate ease. God is mathematics.
Steve Madison (Ultrahuman)
The science of mathematics applies to the clouds; the radiance of starlight nourishes the rose; no thinker will dare say that the scent of hawthorn is valueless to the constellations... The cheese-mite has its worth; the smallest is large and the largest is small... Light does not carry the scents of earth into the upper air without knowing what it is doing with them; darkness confers the essence of the stars upon the sleeping flowers... Where the telescope ends the microscope begins, and which has the wider vision? You may choose. A patch of mould is a galaxy of blossom; a nebula is an antheap of stars. There is the same affinity, if still more inconceivable, between the things of the mind and material things.
Victor Hugo
If you want a branch of mathematics, you develop it in the physical context, not as something separate, which you then try to apply, rather than integrating it from the beginning.
Jagdish Mehra (Climbing the Mountain: The Scientific Biography of Julian Schwinger)
Ever since his first ecstasy or vision of Christminster and its possibilities, Jude had meditated much and curiously on the probable sort of process that was involved in turning the expressions of one language into those of another. He concluded that a grammar of the required tongue would contain, primarily, a rule, prescription, or clue of the nature of a secret cipher, which, once known, would enable him, by merely applying it, to change at will all words of his own speech into those of the foreign one. His childish idea was, in fact, a pushing to the extremity of mathematical precision what is everywhere known as Grimm's Law—an aggrandizement of rough rules to ideal completeness. Thus he assumed that the words of the required language were always to be found somewhere latent in the words of the given language by those who had the art to uncover them, such art being furnished by the books aforesaid.
Thomas Hardy (Jude the Obscure)
Richard Charnin is an author and quantitative software developer with advanced degrees in applied mathematics and operations research. He paints a very clear portrait of the JFK witness deaths in the context of the mathematical landscape: I have proved mathematically what many have long suspected: The scores of convenient JFK unnatural witness deaths cannot be coincidental.15
Richard Belzer (Hit List: An In-Depth Investigation Into the Mysterious Deaths of Witnesses to the JFK Assassination)
In the history of ideas, it's repeatedly happened that an idea, developed in one area for one purpose, finds an unexpected application elsewhere. Concepts developed purely for philosophy of mathematics turned out to be just what you needed to build a computer. Statistical formulae for understanding genetic change in biology are now applied in both economics and in programming.
Patrick Grim
If passion drives you, let mathematical functioning hold the reins. As an eye is meant to see things, a soul is here for its own joy. Memorize the formulas and apply them to life. Passion is energy.
Anisa Claire West
Magic is like a lot of other disciplines that people have recently begun developing, in historic terms. Working with magic is a way of understanding the universe and how it functions. You can approach it from a lot of different angles, applying a lot of different theories and mental models to it. You can get to the same place through a lot of different lines of theory and reasoning, kind of like really advanced mathematics. There's no truly right or wrong way to get there, either--there are just different ways, some more or less useful than others for a given application. And new vistas of thought, theory, and application open up on a pretty regular basis, as the Art develops and expands through the participation of multiple brilliant minds. But that said, once you have a good grounding in it,you get a pretty solid idea of what's possible and what isn't. No matter how much circumlocution you do with your formulae, two plus two doesn't equal five. (Except maybe very, very rarely, sometimes, in extremely specific and highly unlikely circumstances.)
Jim Butcher (Cold Days (The Dresden Files, #14))
I want economists to quit concerning themselves with allocation problems, per se, with the problem, as it has been traditionally defined. The vocabulary of science is important here, and as T. D. Weldon once suggested, the very word "problem" in and of itself implies the presence of "solution." Once the format has been established in allocation terms, some solution is more or less automatically suggested. Our whole study becomes one of applied maximization of a relatively simple computational sort. Once the ends to be maximized are provided by the social welfare function, everything becomes computational, as my colleague, Rutledge Vining, has properly noted. If there is really nothing more to economics than this, we had as well turn it all over to the applied mathematicians. This does, in fact, seem to be the direction in which we are moving, professionally, and developments of note, or notoriety, during the past two decades consist largely in improvements in what are essentially computing techniques, in the mathematics of social engineering. What I am saying is that we should keep these contributions in perspective; I am urging that they be recognized for what they are, contributions to applied mathematics, to managerial science if you will, but not to our chosen subject field which we, for better or for worse, call "economics.
James M. Buchanan
Winfree came from a family in which no one had gone to college. He got started, he would say, by not having proper education. His father, rising from the bottom of the life insurance business to the level of vice president, moved family almost yearly up and down the East Coast, and Winfree attended than a dozen schools before finishing high school. He developed a feeling that the interesting things in the world had to do with biology and mathematics and a companion feeling that no standard combination of the two subjects did justice to what was interesting. So he decided not to take a standard approach. He took a five-year course in engineering physics at Cornell University, learning applied mathematics and a full range of hands-on laboratory styles. Prepared to be hired into military-industrial complex, he got a doctorate in biology, striving to combine experiment with theory in new ways.
James Gleick (Chaos: Making a New Science)
You cannot transform a domain unless you first thoroughly understand how it works. Which means that one has to acquire the tools of mathematics, learn the basic principles of physics, and become aware of the current state of knowledge. But the old Italian saying seems to apply: Impara l’arte, e mettila da parte (learn the craft, and then set it aside). One cannot be creative without learning what others know, but then one cannot be creative without becoming dissatisfied with that knowledge and rejecting it (or some of it) for a better way.
Mihály Csíkszentmihályi (Creativity: Flow and the Psychology of Discovery and Invention)
It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill. Time may change all this. No one foresaw the applications of matrices and groups and other purely mathematical theories to modern physics, and it may be that some of the 'highbrow' applied mathematics will become 'useful' in as unexpected a way; but the evidence so far points to the conclusion that, in one subject as in the other, it is what is commonplace and dull that counts for practical life.
G.H. Hardy (A Mathematician's Apology)
Today we possess science precisely to the extent to which we have decided to accept the testimony of the senses—to the extent to which we sharpen them further, arm them, and have learned to think them through. The rest is miscarriage and not-yet-science—in other words, metaphysics, theology, psychology, epistemology—or formal science, a doctrine of signs, such as logic and that applied logic which is called mathematics. In them reality is not encountered at all, not even as a problem—no more than the question of the value of such a sign convention as logic.
Friedrich Nietzsche (Twilight of the Idols)
Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical conceptsãthe four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable.
Eugene Paul Wigner (The Unreasonable Effectiveness of Mathematics in the Natural Sciences)
Outsiders sometimes have an impression that mathematics consists of applying more and more powerful tools to dig deeper and deeper into the unknown, like tunnelers blasting through the rock with ever more powerful explosives. And that's one way to do it. But Grothendieck, who remade much of pure mathematics in his own image in the 1960's and 70's, had a different view: "The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration...the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it...yet it finally surrounds the resistant substance." The unknown is a stone in the sea, which obstructs our progress. We can try to pack dynamite in the crevices of rock, detonate it, and repeat until the rock breaks apart, as Buffon did with his complicated computations in calculus. Or you can take a more contemplative approach, allowing your level of understanding gradually and gently to rise, until after a time what appeared as an obstacle is overtopped by the calm water, and is gone. Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite.
Jordan Ellenberg (How Not to Be Wrong: The Power of Mathematical Thinking)
I like the term “stretch” for describing what deliberate practice feels like, as it matches my own experience with the activity. When I’m learning a new mathematical technique—a classic case of deliberate practice—the uncomfortable sensation in my head is best approximated as a physical strain, as if my neurons are physically re-forming into new configurations. As any mathematician will admit, this stretching feels much different than applying a technique you’ve already mastered, which can be quite enjoyable. But this stretching, as any mathematician will also admit, is the precondition to getting better.
Cal Newport (So Good They Can't Ignore You)
Nothing was affirmative, the term 'generosity of spirit,' applied to nothing, was a cliche, was some kind of bad joke. Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire--meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface was all that anyone found meaning in...this was civilization as I saw it, colossal and jagged...
Bret Easton Ellis (American Psycho)
However, in 1930 (published in 1931), Godel produced his bombshell, which eventually showed that the formalists' dream was unattainable! He demonstrated that there could be no formal system F, whatever, that is both consistent (in a certain 'strong' sense that I shall describe in the next section) and complete-so long as F is taken to be powerful enough to contain a formulation of the statements of ordinary arithmetic together with standard logic. Thus, Godel's theorem would apply to systems F for which arithmetical statements such as Lagrange's theorem and Goldbach's conjecture, as described in 2.3, could be formulated as mathematical statements.
Roger Penrose
Equality (outside mathematics) is a purely social conception. It applies to man as a political and economic animal. It has no place in the world of the mind. Beauty is not democratic; she reveals herself more to the few than to the many, more to the persistent and disciplined seekers than to the careless. Virtue is not democratic; she is achieved by those who pursue her more hotly than most men. Truth is not democratic; she demands special talents and special industry in those to whom she gives her favours. Political democracy is doomed if it tries to extend its demand for equality into these higher spheres. Ethical, intellectual, or aesthetic democracy is death.
C.S. Lewis (Present Concerns: Journalistic Essays)
There is one thing only which a Muslim can profitably learn from the west, the exact sciences in their pure and applied form. Only natural sciences and mathematics should be taught in Muslim schools, while tuition of European philosophy, literature and history should lose the position of primacy which today it holds on the curriculum.
Muhammad Asad
The people’s right to know”—the people’s right to know what? Daniel Shipstone, having first armed himself with great knowledge of higher mathematics and physics, went down into his basement and patiently suffered seven lean and weary years and thereby learned an applied aspect of natural law that let him construct a Shipstone. Any and all of “the people” are free to do as he did—he did not even take out a patent. Natural laws are freely available to everyone equally, including flea-bitten Neanderthals crouching against the cold. In this case, the trouble with “the people’s right to know” is that it strongly resembles the “right” of someone to be a concert pianist—but who does not
Robert A. Heinlein (Friday)
Pedantry and mastery are opposite attitudes toward rules. 1. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. Some pedants are poor fools; they never did understand the rule which they apply so conscientiously and so indiscriminately. Some pedants are quite successful; they understood their rule, at least in the beginning (before they became pedants), and chose a good one that fits in many cases and fails only occasionally. To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.
George Pólya (How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library Book 34))
It was all so very businesslike that one watched it fascinated. It was porkmaking by machinery, porkmaking by applied mathematics. And yet somehow the most matter-of-fact person could not help thinking of the hogs; they were so innocent, they came so very trustingly; and they were so very human in their protests—and so perfectly within their rights! They had done nothing to deserve it; and it was adding insult to injury, as the thing was done here, swinging them up in this cold-blooded, impersonal way, without a pretense of apology, without the homage of a tear. Now and then a visitor wept, to be sure; but this slaughtering machine ran on, visitors or no visitors. It was like some horrible crime committed in a dungeon, all unseen and unheeded, buried out of sight and of memory. One could not stand and watch very long without becoming philosophical, without beginning to deal in symbols and similes, and to hear the hog squeal of the universe. Was it permitted to believe that there was nowhere upon the earth, or above the earth, a heaven for hogs, where they were requited for all this suffering? Each one of these hogs was a separate creature. Some were white hogs, some were black; some were brown, some were spotted; some were old, some young; some were long and lean, some were monstrous. And each of them had an individuality of his own, a will of his own, a hope and a heart’s desire; each was full of self-confidence, of self-importance, and a sense of dignity. And trusting and strong in faith he had gone about his business, the while a black shadow hung over him and a horrid Fate waited in his pathway. Now suddenly it had swooped upon him, and had seized him by the leg. Relentless, remorseless, it was; all his protests, his screams, were nothing to it—it did its cruel will with him, as if his wishes, his feelings, had simply no existence at all; it cut his throat and watched him gasp out his life. And now was one to believe that there was nowhere a god of hogs, to whom this hog personality was precious, to whom these hog squeals and agonies had a meaning? Who would take this hog into his arms and comfort him, reward him for his work well done, and show him the meaning of his sacrifice?
Upton Sinclair (The Jungle)
The question as to which of these two theories applies to the actual world is, like all questions concerning the actual world, in itself irrelevant to pure mathematics.* But the argument against absolute position usually takes the form of maintaining that a space composed of points is logically inadmissible, and hence issues are raised which a philosophy of mathematics must discuss. In what follows, I am concerned only with the question: Is a space composed of points self-contradictory? It is true that, if this question be answered in the negative, the sole ground for denying that such a space exists in the actual world is removed; but this is a further point, which, being irrelevant to our subject, will be left entirely to the sagacity of the reader.
Bertrand Russell (Principles of Mathematics (Routledge Classics))
At the same time, Kaufmann discovered that in developing his genetic networks, he had reinvented some of the most avant-garde work in physics and applied mathematics-albeit in a totally new context. The dynamics of his genetic regulatory networks turned out to be a special case of what the physicists were calling "nonlinear dynamics." From the nonlinear point of view, in fact, it was easy to see why his sparsely connected networks could organize themselves into stable cycles so easily: mathematically, their behavior was equivalent to the way all the rain falling on the hillsides around a valley will flow into a lake at the bottom of the valley. In the space of all possible network behaviors, the stable cycles were like basins-or as the physicists put it, "attractors.
M. Mitchell Waldrop (Complexity: The Emerging Science at the Edge of Order and Chaos)
According to our estimates, the optimal top tax rate in the developed countries is probably above 80 percent.50 Do not be misled by the apparent precision of this estimate: no mathematical formula or econometric estimate can tell us exactly what tax rate ought to be applied to what level of income. Only collective deliberation and democratic experimentation can do that. What is certain, however, is that our estimates pertain to extremely high levels of income, those observed in the top 1 percent or 0.5 percent of the income hierarchy. The evidence suggests that a rate on the order of 80 percent on incomes over $500,000 or $1 million a year not only would not reduce the growth of the US economy but would in fact distribute the fruits of growth more widely while imposing reasonable limits on economically useless (or even harmful) behavior.
Thomas Piketty (Capital in the Twenty-First Century)
Emotion is not a defect in an otherwise perfect reasoning machine. Reason, unfettered from human feeling, has led to as many horrors as any crusader’s zeal. What use is pity in a world devoted to maximizing efficiency and productivity? Scientific husbandry tells us to weed out the sick, the infirm, the weak. The ruthless efficiency of euthanasia initiatives and ethnic cleansing are but the programmatic application of Nietzsche’s point: from any quantifiable cost-benefit analysis, the principles of animal husbandry should apply to the human race. Charles Darwin himself acknowledged that strict obedience to “hard reason” rather than sympathy for fellow humans would represent a sacrifice of “the noblest part of our nature.”6 It is the human heart resonating with empathy, not the logical brain attuned to the mathematics of efficiency, that revolts at cruelty and inhumanity. p15
Terryl L. Givens (The Crucible of Doubt: Reflections On the Quest for Faith)
More generally, we underestimate the share of randomness in about everything, a point that may not merit a book—except when it is the specialist who is the fool of all fools. Disturbingly, science has only recently been able to handle randomness (the growth in available information has been exceeded only by the expansion of noise). Probability theory is a young arrival in mathematics; probability applied to practice is almost nonexistent as a discipline. In addition we seem to have evidence that what is called “courage” comes from an underestimation of the share of randomness in things rather than the more noble ability to stick one’s neck out for a given belief. In my experience (and in the scientific literature), economic “risk takers” are rather the victims of delusions (leading to overoptimism and overconfidence with their underestimation of possible adverse outcomes) than the opposite. Their “risk taking” is frequently randomness foolishness.
Nassim Nicholas Taleb (Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets (Incerto Book 1))
gene plays a role, are quite tractable, but anything entailing higher dimensionality falls apart. Understanding the genetic makeup of a unit will never allow us to understand the behavior of the unit itself. A reminder that what I am writing here isn’t an opinion. It is a straightforward mathematical property. The mean-field approach is when one uses the average interaction between, say, two people, and generalizes to the group—it is only possible if there are no asymmetries. For instance, Yaneer Bar-Yam has applied the failure of mean-field to evolutionary theory of the selfish-gene narrative trumpeted by such aggressive journalistic minds as Richard Dawkins and Steven Pinker, with more mastery of English than probability theory. He shows that local properties fail and the so-called mathematics used to prove the selfish gene are woefully naive and misplaced. There has been a storm around work by Martin Nowack and his colleagues (which include the biologist E. O. Wilson) about the terminal flaws in the selfish gene theory.fn2
Nassim Nicholas Taleb (Skin in the Game: Hidden Asymmetries in Daily Life)
Feynman said, “If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied” Our sentence would be: “The Monadology asserts that the fundamental units of existence are INFINITE, dimensionless, living, thinking points – monads, ZEROS, souls – each of which has INFINITE energy content, all controlled by a single equation – Euler’s Formula – and the collective energy of this universe of mathematical points creates a physical universe of which every objective value is ZERO, but, through a self-solving, self-optimizing, dialectical, evolving process, the universe generates a final, subjective value of INFINITY – divinity, perfection, the ABSOLUTE.” For ours is the religion of zero and infinity, the two numbers that define the soul and the whole of existence. As above, so below.
Mike Hockney (The God Equation)
We have a problem and a a problem demands a solution. Problem and solution—these two terms are inseparably connected to each other. It means if a person has a problem there is a certain method—it could be mathematical, algebra—that you can apply to this problem. What you get out of that is the solution. With that the problem is over. But it is not like that in human life. It is not like that with—now I am going to use the word myself—problems. Human problems, social problems, societal problems will never be solved... It is an illusion that some kind of problem will be solved.
Joseph Weizenbaum (Islands in the Cyberstream: Seeking Havens of Reason in a Programmed Society)
...computer technology functions more as a new mode of transportation than as a new means of substantive communication. It moves information—lots of it, fast, and mostly in a calculating mode. The computer, in fact, makes possible the fulfillment of Descartes’ dream of the mathematization of the world. Computers make it easy to convert facts into statistics and to translate problems into equations. And whereas this can be useful (as when the process reveals a pattern that would otherwise go unnoticed), it is diversionary and dangerous when applied indiscriminately to human affairs.
Neil Postman (Technopoly: The Surrender of Culture to Technology)
was once asked to give a talk to a group of science journalists who were meeting in my hometown. I decided to talk about the design of bridges, explaining how their form does not derive from a set of equations expressing the laws of physics but rather from the creative mind of the engineer. The first step in designing a bridge is for the engineer to conceive of a form in his mind’s eye. This is then translated into words and pictures so that it can be communicated to other engineers on the team and to the client who is commissioning the work. It is only when there is a form to analyze that science can be applied in a mathematical and methodical way. This is not to say that scientific principles might not inform the engineer’s conception of a bridge, but more likely they are embedded in the engineer’s experience with other, existing bridges upon which the newly conceived bridge is based. The journalists to whom I was speaking were skeptical. Surely science is essential to design, they insisted. No, it is not. And it is not a chicken-and-egg paradox. The design of engineering structures is a creative process in the same way that paintings and novels are the products of creative minds.
Henry Petroski (The Essential Engineer)
There are some mysteries in this world," Yukawa said suddenly, "that cannot be unraveled with modern science. However, as science develops, we will one day be able to understand them. The question is, is there a limit to what science can know? If so, what creates that limit?" Kyohei looked at Yukawa. He couldn't figure out why the professor was telling him this, except he had a feeling it was very important. Yukawa pointed a finger at Kyohei's forehead. "People do." he said. "People's brains, to be more precise. For example, in mathematics, when somebody discovers a new theorem, they may have other mathematicians verify it to see if it's correct. The problem is, the theorems getting discovered are becoming more and more complex. That limits the number of mathematicians who can properly verify them. What happens when someone comes up with a theorem so hard to understand that there isn't anyone else who can understand it? In order for that theorem to be accepted as fact, they have to wait until another genius comes along. That's the limit the human brain imposes on the progress of scientific knowledge. You understand?" Kyohei nodded, still having no idea where he was going with this. "Every problem has a solution," Yukawa said, staring straight at Kyohei through his glasses. "But there's no guarantee that the solution will be found immediately. The same holds true in our lives. We encounter several problems to which the solutions are not immediately apparent in life. There is value to be had in worrying about those problems when you get to them. But never feel rushed. Often, in order to find the answer, you need time to grow first. That's why we apply ourselves, and learn as we go." Kyohei chewed on that for a moment, then his mouth opened a little and he looked up with sudden understanding. "You have questions now, I know, and until you find your answers, I'll be working on those questions too, and worrying with you. So don't forget, you're never alone.
Keigo Higashino (A Midsummer's Equation (Detective Galileo #3))
You are from alone in the community of scientists, and here is a professional secret to encourage you: many of the most successful scientists in the world today are mathematically no more than semiliterate. A metaphor will clarify the paradox in this statement. Where elite mathematicians often serve as architects of theory in the expanding realm of science, the remaining large majority of basic and applied scientists map the terrain, scout the frontier, cut the pathways, and raise the first buildings along the way. They define the problems that mathematicians, on occasion, may help solve. They think primarily in images and facts, and only marginally in mathematics.
Edward O. Wilson (Letters to a Young Scientist)
A.N. Kolmogorov and Yasha Sinai had worked out some illuminating mathematics for the way a system's "entropy per unit time" applies to the geometric pictures of surfaces stretching and folding in phase space. The conceptual core of the technique was a matter of drawing some arbitrarily small box around some set of initial conditions, as one might draw a small square on the side of a balloon, then calculating the effect of various expressions or twists on the box. It might stretch in one direction, for example, while remaining narrow in the other. The change in area corresponded to an introduction of uncertainty about the system's past, a gain or loss of information.
James Gleick (Chaos: Making a New Science)
Galileo showed that the same physical laws that govern the movements of bodies on earth apply aloft , to the celestial spheres; and our astronauts, as we have all now seen, have been transported by those earthly laws to the moon. They will soon be on Mars and beyond. Furthermore, we know that the mathematics of those outermost spaces will already have been computed here on earth by human minds. There are no laws out there that are not right here; no gods out there that are not right here, and not only here, but within us, in our minds. So what happens now to those childhood images of the ascent of Elijah, Assumption of the Virgin, Ascension of Christ - all bodily - into heaven?
Joseph Campbell (Myths to Live By)
A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult. What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.
Ellen Kaplan (Out of the Labyrinth: Setting Mathematics Free)
… where there was nature and earth, life and water, I saw a desert landscape that was unending, resembling some sort of crater, so devoid of reason and light and spirit that the mind could not grasp it on any sort of conscious level and if you came close the mind would reel backward, unable to take it in. It was a vision so clear and real and vital to me that in its purity it was almost abstract. This was what I could understand, this was how I lived my life, what I constructed my movement around, how I dealt with the tangible. This was the geography around which my reality revolved: it did not occur to me, ever, that people were good or that a man was capable of change or that the world could be a better place through one’s taking pleasure in a feeling or a look or a gesture, of receiving another person’s love or kindness. Nothing was affirmative, the term “generosity of spirit” applied to nothing, was a cliché, was some kind of bad joke. Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire—meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface was all that anyone found meaning in … this was civilization as I saw it, colossal and jagged …
Bret Easton Ellis (American Psycho (Vintage Contemporaries))
Taking least squares is no longer optimal, and the very idea of ‘accuracy’ has to be rethought. This simple fact is as important as it is neglected. This problem is easily illustrated in the Logistic Map: given the correct mathematical formula and all the details of the noise model – random numbers with a bell-shaped distribution – using least squares to estimate α leads to systematic errors. This is not a question of too few data or insufficient computer power, it is the method that fails. We can compute the optimal least squares solution: its value for α is too small at all noise levels. This principled approach just does not apply to nonlinear models because the theorems behind the principle of least squares repeatedly assume bell-shaped distributions.
Leonard A. Smith (Chaos: A Very Short Introduction (Very Short Introductions))
What you learn after a long time in math-and I think the lesson applies much more broadly-is that there's always somebody ahead of you, whether they're right there in class with you or not. People just starting out look to people with good theorems, people with some good theorems look to people with lots of good theorems, people with lots of good theorems look to people with Fields Medals, people with Fields Medals look to the "inner circle" Medalists, and those people can always look toward the dead. Nobody ever looks in the mirror and says, "Let's face it, I'm smarter than Gauss." And yet, in the last hundred years, the joined effort of all these dummies-compared-to-Gauss has produced the greatest flowering of mathematical knowledge the world has ever seen.
Jordan Ellenberg (How Not to Be Wrong: The Power of Mathematical Thinking)
The beauty of the principle idea of string theory is that all the known elementary particles are supposed to represent merely different vibration modes of the same basic string. Just as a violin or a guitar string can be plucked to produce different harmonics, different vibrational patterns of a basic string correspond to distinct matter particles, such as electrons and quarks. The same applies to the force carriers as well. Messenger particles such as gluons or the W and Z owe their existence to yet other harmonics. Put simply, all the matter and force particles of the standard model are part of the repertoire that strings can play. Most impressively, however, a particular configuration of vibrating string was found to have properties that match precisely the graviton-the anticipated messenger of the gravitational force. This was the first time that the four basic forces of nature have been housed, if tentatively, under one roof.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
But the whole theory rests, if I am not mistaken, upon neglect of the fundamental distinction between an idea and its object. Misled by neglect of being, people have supposed that what does not exist is nothing. Seeing that numbers, relations, and many other objects of thought, do not exist outside the mind, they have supposed that the thoughts in which we think of these entities actually create their own objects. Every one except a philosopher can see the difference between a post and my idea of a post, but few see the difference between the number 2 and my idea of the number 2. Yet the distinction is as necessary in one case as in the other. The argument that 2 is mental requires that 2 should be essentially an existent. But in that case it would be particular, and it would be impossible for 2 to be in two minds, or in one mind at two times. Thus 2 must be in any case an entity, which will have being even if it is in no mind.* But further, there are reasons for denying that 2 is created by the thought which thinks it. For, in this case, there could never be two thoughts until some one thought so; hence what the person so thinking supposed to be two thoughts would not have been two, and the opinion, when it did arise, would be erroneous. And applying the same doctrine to 1; there cannot be one thought until some one thinks so. Hence Adam’s first thought must have been concerned with the number 1; for not a single thought could precede this thought. In short, all knowledge must be recognition, on pain of being mere delusion; Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. The number 2 is not purely mental, but is an entity which may be thought of. Whatever can be thought of has being, and its being is a precondition, not a result, of its being thought of.
Bertrand Russell (Principles of Mathematics (Routledge Classics))
What we know today, if we know anything at all, is that every individual is unique and that the laws of his life will not be those of any other on this earth. We also know that if divinity is to be found anywhere, it will not be “out there,” among or beyond the planets. Galileo showed that the same physical laws that govern the movements of bodies on earth apply aloft, to the celestial spheres; and our astronauts, as we have all now seen, have been transported by those earthly laws to the moon. They will soon be on Mars and beyond. Furthermore, we know that the mathematics of those outermost spaces will already have been computed here on earth by human minds. There are no laws out there that are not right here; no gods out there that are not right here, and not only here, but within us, in our minds. So what happens now to those childhood images of the ascent of Elijah, Assumption of the Virgin, Ascension of Christ - all bodily - into heaven?
Joseph Campbell (Myths to Live By)
Supersymmetry was (and is) a beautiful mathematical idea. The problem with applying supersymmetry is that it is too good for this world. We simply do not find particles of the sort it predicts. We do not, for example, see particles with the same charge and mass as electrons, but a different amount of spin. However, symmetry principles that might help to unify fundamental physics are hard to come by, so theoretical physicists do not give up on them easily. Based on previous experience with other forms of symmetry, we have developed a fallback strategy, called spontaneous symmetry breaking. In this approach, we postulate that the fundamental equations of physics have the symmetry, but the stable solutions of these equations do not. The classic example of this phenomenon occurs in an ordinary magnet. In the basic equations that describe the physics of a lump of iron, any direction is equivalent to any other, but the lump becomes a magnet with some definite north-seeking pole.
Frank Wilczek (The Lightness of Being: Mass, Ether, and the Unification of Forces)
The spirit of revolution and the power of free thought were Percy Shelley's biggest passions in life.” One could use precisely the same words to describe Galois. On one of the pages that Galois had left on his desk before leaving for that fateful duel, we find a fascinating mixture of mathematical doodles, interwoven with revolutionary ideas. After two lines of functional analysis comes the word "indivisible," which appears to apply to the mathematics. This word is followed, however, by the revolutionary slogans "unite; indivisibilite de la republic") and "Liberte, egalite, fraternite ou la mort" ("Liberty, equality, brotherhood, or death"). After these republican proclamations, as if this is all part of one continuous thought, the mathematical analysis resumes. Clearly, in Galois's mind, the concepts of unity and indivisibility applied equally well to mathematics and to the spirit of the revolution. Indeed, group theory achieved precisely that-a unity and indivisibility of the patterns underlying a wide range of seemingly unrelated disciplines.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
There were also many cases of feedback between physics and mathematics, where a physical phenomenon inspired a mathematical model that later proved to be the explanation of an entirely different physical phenomenon. An excellent example is provided by the phenomenon known as Brownian motion. In 1827, British botanist Robert Brown (1773-1858) observed that wen pollen particles are suspended in water, they get into a state of agitated motion. This effect was explained by Einstein in 1905 as resulting from the collisions that the colloidal particles experience with the molecules of the surrounding fluid. Each single collision has a negligible effect, because the pollen grains are millions of times more massive than the water molecules, but the persistent bombardment has a cumulative effect. Amazingly, the same model was found to apply to the motions of stars in star clusters. There the Brownian motion is produced by the cumulative effect of many stars passing by any given star, with each passage altering the motion (through gravitational interaction) by a tiny amount.
Mario Livio (The Golden Ratio: The Story of Phi, the World's Most Astonishing Number)
The language of mathematics differs from that of everyday life, because it is essentially a rationally planned language. The languages of size have no place for private sentiment, either of the individual or of the nation. They are international languages like the binomial nomenclature of natural history. In dealing with the immense complexity of his social life man has not yet begun to apply inventiveness to the rational planning of ordinary language when describing different kinds of institutions and human behavior. The language of everyday life is clogged with sentiment, and the science of human nature has not advanced so far that we can describe individual sentiment in a clear way. So constructive thought about human society is hampered by the same conservatism as embarrassed the earlier naturalists. Nowadays people do not differ about what sort of animal is meant by Cimex or Pediculus, because these words are used only by people who use them in one way. They still can and often do mean a lot of different things when they say that a mattress is infested with bugs or lice. The study of a man's social life has not yet brought forth a Linnaeus. So an argument about the 'withering away of the State' may disclose a difference about the use of the dictionary when no real difference about the use of the policeman is involved. Curiously enough, people who are most sensible about the need for planning other social amenities in a reasonable way are often slow to see the need for creating a rational and international language.
Lancelot Hogben (Mathematics for the Million: How to Master the Magic of Numbers)
Bohr advanced a heavyhanded remedy: evolve probability waves according to Schrodinger's equation whenever you're not looking or performing any kind of measurement. But when you do look, Bohr continued, you should throw Schrodinger's equation aside and declare that your observation has caused the wave to collapse. Now, not only is this prescription ungainly, not only is it arbitrary, not only does it lack a mathematical underpinning, it's not even clear. For instance, it doesn't precisely define "looking" or "measuring." Must a human be involved? Or, as Einstein once asked, will a sidelong glance from a mouse suffice? How about a computer's probe, or even a nudge from a bacterium or virus? Do these "measurements" cause probability waves to collapse? Bohr announced that he was drawing a line in the sand separating small things, such as atoms and their constituents, to which Schrodinger's equation would apply, and big things, such as experimenters and their equipment, to which it wouldn't. But he never said where exactly that line would be. The reality is, he couldn't. With each passing year, experimenters confirm that Schrodinger's equation works, without modification, for increasingly large collections of particles, and there's every reason to believe that it works for collections as hefty as those making up you and me and everything else. Like floodwaters slowly rising from your basement, rushing into your living room, and threatening to engulf your attic, the mathematics of quantum mechanics has steadily spilled beyond the atomic domain and has succeeded on ever-larger scales.
Brian Greene (The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos)
To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The definition of the Golden Ratio emerged originally from the axioms of Euclidean geometry; the definition of the Fibonacci sequence from the axioms of the theory of numbers. Yet the fact that the ratio of successive Fibonacci numbers converges to the Golden Ratio was imposed on us-humans had not choice in the matter. Therefore, mathematical objects, albeit imaginary, do have real properties. Second, the explanation of the unreasonable power of mathematics cannot be based entirely on evolution in the restricted sense. For example, when Newton proposed his theory of gravitation, the data that he was trying to explain were at best accurate to three significant figures. Yet his mathematical model for the force between any two masses in the universe achieved the incredible precision of better than one part in a million. Hence, that particular model was not forced on Newton by existing measurements of the motions of planets, nor did Newton force a natural phenomenon into a preexisting mathematical pattern. Furthermore, natural selection in the common interpretation of that concept does not quite apply either, because it was not the case that five competing theories were proposed, of which one eventually won. Rather, Newton's was the only game in town!
Mario Livio (The Golden Ratio: The Story of Phi, the World's Most Astonishing Number)
That such a surprisingly powerful philosophical method was taken seriously can be only partially explained by the backwardness of German natural science in those days. For the truth is, I think, that it was not at first taken really seriously by serious men (such as Schopenhauer, or J. F. Fries), not at any rate by those scientists who, like Democritus2, ‘would rather find a single causal law than be the king of Persia’. Hegel’s fame was made by those who prefer a quick initiation into the deeper secrets of this world to the laborious technicalities of a science which, after all, may only disappoint them by its lack of power to unveil all mysteries. For they soon found out that nothing could be applied with such ease to any problem whatsoever, and at the same time with such impressive (though only apparent) difficulty, and with such quick and sure but imposing success, nothing could be used as cheaply and with so little scientific training and knowledge, and nothing would give such a spectacular scientific air, as did Hegelian dialectics, the mystery method that replaced ‘barren formal logic’. Hegel’s success was the beginning of the ‘age of dishonesty’ (as Schopenhauer3 described the period of German Idealism) and of the ‘age of irresponsibility’ (as K. Heiden characterizes the age of modern totalitarianism); first of intellectual, and later, as one of its consequences, of moral irresponsibility; of a new age controlled by the magic of high-sounding words, and by the power of jargon. In order to discourage the reader beforehand from taking Hegel’s bombastic and mystifying cant too seriously, I shall quote some of the amazing details which he discovered about sound, and especially about the relations between sound and heat. I have tried hard to translate this gibberish from Hegel’s Philosophy of Nature4 as faithfully as possible; he writes: ‘§302. Sound is the change in the specific condition of segregation of the material parts, and in the negation of this condition;—merely an abstract or an ideal ideality, as it were, of that specification. But this change, accordingly, is itself immediately the negation of the material specific subsistence; which is, therefore, real ideality of specific gravity and cohesion, i.e.—heat. The heating up of sounding bodies, just as of beaten or rubbed ones, is the appearance of heat, originating conceptually together with sound.’ There are some who still believe in Hegel’s sincerity, or who still doubt whether his secret might not be profundity, fullness of thought, rather than emptiness. I should like them to read carefully the last sentence—the only intelligible one—of this quotation, because in this sentence, Hegel gives himself away. For clearly it means nothing but: ‘The heating up of sounding bodies … is heat … together with sound.’ The question arises whether Hegel deceived himself, hypnotized by his own inspiring jargon, or whether he boldly set out to deceive and bewitch others. I am satisfied that the latter was the case, especially in view of what Hegel wrote in one of his letters. In this letter, dated a few years before the publication of his Philosophy of Nature, Hegel referred to another Philosophy of Nature, written by his former friend Schelling: ‘I have had too much to do … with mathematics … differential calculus, chemistry’, Hegel boasts in this letter (but this is just bluff), ‘to let myself be taken in by the humbug of the Philosophy of Nature, by this philosophizing without knowledge of fact … and by the treatment of mere fancies, even imbecile fancies, as ideas.’ This is a very fair characterization of Schelling’s method, that is to say, of that audacious way of bluffing which Hegel himself copied, or rather aggravated, as soon as he realized that, if it reached its proper audience, it meant success.
Karl Popper (The Open Society and Its Enemies)
At this point, the cautious reader might wish to read over the whole argument again, as presented above, just to make sure that I have not indulged in any 'sleight of hand'! Admittedly there is an air of the conjuring trick about the argument, but it is perfectly legitimate, and it only gains in strength the more minutely it is examined. We have found a computation Ck(k) that we know does not stop; yet the given computational procedure A is not powerful enough to ascertain that facet. This is the Godel(-Turing) theorem in the form that I require. It applies to any computational procedure A whatever for ascertaining that computations do not stop, so long as we know it to be sound. We deduce that no knowably sound set of computational rules (such as A) can ever suffice for ascertaining that computations do not stop, since there are some non-stopping computations (such as Ck(k)) that must elude these rules. Moreover, since from the knowledge of A and of its soundness, we can actually construct a computation Ck(k) that we can see does not ever stop, we deduce that A cannot be a formalization of the procedures available to mathematicians for ascertaining that computations do not stop, no matter what A is. Hence: (G) Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth. It seems to me that this conclusion is inescapable. However, many people have tried to argue against it-bringing in objections like those summarized in the queries Q1-Q20 of 2.6 and 2.10 below-and certainly many would argue against the stronger deduction that there must be something fundamentally non-computational in our thought processes. The reader may indeed wonder what on earth mathematical reasoning like this, concerning the abstract nature of computations, can have to say about the workings of the human mind. What, after all, does any of this have to do with the issue of conscious awareness? The answer is that the argument indeed says something very significant about the mental quality of understanding-in relation to the general issue of computation-and, as was argued in 1.12, the quality of understanding is something dependent upon conscious awareness. It is true that, for the most part, the foregoing reasoning has been presented as just a piece of mathematics, but there is the essential point that the algorithm A enters the argument at two quite different levels. At the one level, it is being treated as just some algorithm that has certain properties, but at the other, we attempt to regard A as being actually 'the algorithm that we ourselves use' in coming to believe that a computation will not stop. The argument is not simply about computations. It is also about how we use our conscious understanding in order to infer the validity of some mathematical claim-here the non-stopping character of Ck(k). It is the interplay between the two different levels at which the algorithm A is being considered-as a putative instance of conscious activity and as a computation itself-that allows us to arrive at a conclusion expressing a fundamental conflict between such conscious activity and mere computation.
Roger Penrose (Shadows of the Mind: A Search for the Missing Science of Consciousness)
*There is only one God*. Whatever exists is *ipso facto* individual; to be one it needs no extra property and calling it one merely denies that it is divided. Simple things are neither divided nor divisible; composite things do not exist when their parts are divided. So existence stands or falls with individuality, and things guard their unity as they do their existence. But what is simply speaking one can yet in certain respects be many: an individual thing, essentially undivided, can have many non-essential properties; and a single whole, actually undivided, can have potentially many parts. Only when one is used to count with does it presuppose in what it counts some extra property over and above existence, namely, quantity. The one we count with contrasts with the many it counts in the way a unity of measurement contrasts with what it measures; but the individual unity common to everything that exists contrasts with plurality simply by lacking it, as undividedness does division. A plurality is however *a* plurality: though simply speaking many, inasmuch as it exists, it is, incidentally, one. A continuum is homogeneous: its parts share the form of the whole (every bit of water is water); but a plurality is heterogeneous: its parts lack the form of the whole (no part of the house is a house). The parts of a plurality are unities and non-plural, though they compose the plurality not as non-plural but as existing; just as the parts of a house compose the house as material, not as not houses. Whereas we define plurality in terms of unity (many things are divided things to each of which is ascribed unity), we define unity in terms of division. For division precedes unity in our minds even if it doesn’t really do so, since we conceive simple things by denying compositeness of them, defining a point, for example, as lacking dimension. Division arises in the mind simply by negating existence. So the first thing we conceive is the existent, then―seeing that this existent is not that existent―we conceive division, thirdly unity, and fourthly plurality. There is only one God. Firstly, God and his nature are identical: to be God is to be this individual God. In the same way, if to be a man was to be Socrates there would only be one man, just as there was only one Socrates. Moreover, God’s perfection is unlimited, so what could differentiate one God from another? Any extra perfection in one would be lacking in the other and that would make him imperfect. And finally, the world is one, and plurality can only produce unity incidentally insofar as it too is somehow one: the primary and non-incidental source of unity in the universe must himself be one. The one we count with measures only material things, not God: like all objects of mathematics, though defined without reference to matter, it can exist only in matter. But the unity of individuality common to everything that exists is a metaphysical property applying both to non-material things and to God. But what in God is a perfection has to be conceived by us, with our way of understanding things, as a lack: that is why we talk of God as lacking a body, lacking limits and lacking division.
Thomas Aquinas (Summa Theologiae: A Concise Translation)