Applied Mathematics Quotes

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.

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…

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.

Mathematics becomes very odd when you apply it to people. One plus one can add up to so many different sums

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.

It was as though applied mathematics was my spouse, and pure mathematics was my secret lover.

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.

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.

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

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.

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.

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.

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.

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.

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.

It is not easy to become an educated person.

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.

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.

You can't apply mathematical formulas to people, Cresswell. There's no equation for human emotion, there are too many variables.

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.

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.

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.

It is in the world of things and places, times and troubles and turbid processes, that mathematics is not so much applied as illustrated.

The first man to understand the extraordinary magical power of applying mathematical calculation to things in nature was an Italian called Galileo Galilei.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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 …

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.

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.

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.

...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...

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

In Philosophy structures and systems are useless (one wants to be struck by direct insight). Systems have value only when applied in the struggle with an enemy; philosophy should not be applied. Philosophy cannot work mathematically.

[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.

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.

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.

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.

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.

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...

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.

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.

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.

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.

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.

I used to have a bumper sticker that read "Black Holes are out of sight" on the door of my office in DAMTP [Department of Applied Mathematics and Theoretical Physics, Cambridge]. This so irritated the head of the department that he engineered my election to the Lucasian Professorship, moved me to a better office on the strength of it, and personally tore the offending notice off the door of the old office.

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.)

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.

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.

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.

Johannes Kepler, who was one of the first to apply mathematics to the motion of the planets, was an imperial adviser to Emperor Rudolf Il and perhaps escaped persecution by piously including religious elements in his scientific work.

Pythagoras had the insight to apply a mathematical description to worldly phenomena like music. According to legend, he noticed similarities between the sound of plucking a lyre string and the resonances made by hammering a metal bar.

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.

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.

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.

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.

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.

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...

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.

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.

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

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.

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?

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.

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.

In mathematical physics, quantum field theory and statistical mechanics are characterized by the probability distribution of exp(−βH(x)) where H(x) is a Hamiltonian function. It is well known in [12] that physical problems are determined by the algebraic structure of H(x). Statistical learning theory can be understood as mathematical physics where the Hamiltonian is a random process defined by the log likelihood ratio function.

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

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.

Johannes Kepler, who was one of the first to apply mathematics to the motion of the planets, was an imperial adviser to Emperor Rudolf Il and perhaps escaped persecution by piously including religious elements in his scientific work. The former monk Giordano Bruno was not so lucky. In 1600, he was tried and sentenced to death for heresy. He was gagged, paraded naked in the streets of Rome, and finally burned at the stake. His chief crime? Declaring that life may exist on planets circling other stars.

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

While I enjoy the work because of my love of mathematics, I luckily realized that this career path was simply designed to exploit inefficiencies in markets in order to extract profits from others. This financial realm known as trading is a zero-sum game where for every dollar you make, someone else loses a dollar, and I know I’m not destined to become such an obvious parasite on society. I only aspire to lead a meaningful, impactful life where I can apply my skills as an extremely analytical individual toward the benefit of humanity. I’m

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.

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.

But we don’t correct for the difference in science, medicine, and mathematics, for the same reasons we didn’t pay attention to iatrogenics. We are suckers for the sophisticated. In institutional research, one can selectively report facts that confirm one’s story, without revealing facts that disprove it or don’t apply to it—so the public perception of science is biased into believing in the necessity of the highly conceptualized, crisp, and purified Harvardized methods. And statistical research tends to be marred with this one-sidedness. Another reason one should trust the disconfirmatory more than the confirmatory.

Pure math,” he replied. “How is that different from”—she laughed—“regular math?” Gillian asked. “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.

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?

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.

… 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 …

Pythagoras had the insight to apply a mathematical description to worldly phenomena like music. According to legend, he noticed similarities between the sound of plucking a lyre string and the resonances made by hammering a metal bar. He found that they created musical frequencies that vibrated with certain ratios. So something as aesthetically pleasing as music has its origin in the mathematics of resonances. This, he thought, might show that the diversity of the objects we see around us must obey these same mathematical rules. So at least two great theories of our world emerged from ancient Greece: the idea that everything consists of invisible, indestructible atoms and that the diversity of nature can be described by the mathematics of vibrations.

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.

Base two especially impressed the seventeenth-century religious philosopher and mathematician Gottfried Wilhelm Leibniz. He observed that in this base all numbers were written in terms of the symbols 0 and 1 only. Thus eleven, which equals 1 · 23 + 0 · 22 + 1 · 2 + 1, would be written 1011 in base two. Leibniz saw in this binary arithmetic the image and proof of creation. Unity was God and zero was the void. God drew all objects from the void just as the unity applied to the zero creates all numbers. This conception, over which the reader would do well not to ponder too long, delighted Leibniz so much that he sent it to Grimaldi, the Jesuit president of the Chinese tribunal for mathematics, to be used as an argument for the conversion of the Chinese emperor to Christianity.

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.

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.

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?

Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

When he applied this approach to a gas of quantum particles, Einstein discovered an amazing property: unlike a gas of classical particles, which will remain a gas unless the particles attract one another, a gas of quantum particles can condense into some kind of liquid even without a force of attraction between them. This phenomenon, now called Bose-Einstein condensation,* was a brilliant and important discovery in quantum mechanics, and Einstein deserves most of the credit for it. Bose had not quite realized that the statistical mathematics he used represented a fundamentally new approach. As with the case of Planck’s constant, Einstein recognized the physical reality, and the significance, of a contrivance that someone else had devised.49 Einstein’s method had the effect of treating particles as if they had wavelike traits, as both he and de Broglie had suggested. Einstein even predicted that if you did Thomas Young’s old double-slit experiment (showing that light behaved like a wave by shining a beam through two slits and noting the interference pattern) by using a beam of gas molecules, they would interfere with one another as if they were waves. “A beam of gas molecules which passes through an aperture,” he wrote, “must undergo a diffraction analogous to that of a light ray.

Among much else, Einstein’s general theory of relativity suggested that the universe must be either expanding or contracting. But Einstein was not a cosmologist, and he accepted the prevailing wisdom that the universe was fixed and eternal. More or less reflexively, he dropped into his equations something called the cosmological constant, which arbitrarily counterbalanced the effects of gravity, serving as a kind of mathematical pause button. Books on the history of science always forgive Einstein this lapse, but it was actually a fairly appalling piece of science and he knew it. He called it “the biggest blunder of my life.” Coincidentally, at about the time that Einstein was affixing a cosmological constant to his theory, at the Lowell Observatory in Arizona, an astronomer with the cheerily intergalactic name of Vesto Slipher (who was in fact from Indiana) was taking spectrographic readings of distant stars and discovering that they appeared to be moving away from us. The universe wasn’t static. The stars Slipher looked at showed unmistakable signs of a Doppler shift‖—the same mechanism behind that distinctive stretched-out yee-yummm sound cars make as they flash past on a racetrack. The phenomenon also applies to light, and in the case of receding galaxies it is known as a red shift (because

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.

I believe that the clue to his mind is to be found in his unusual powers of continuous concentrated introspection. A case can be made out, as it also can with Descartes, for regarding him as an accomplished experimentalist. Nothing can be more charming than the tales of his mechanical contrivances when he was a boy. There are his telescopes and his optical experiments, These were essential accomplishments, part of his unequalled all-round technique, but not, I am sure, his peculiar gift, especially amongst his contemporaries. His peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. I fancy his pre-eminence is due to his muscles of intuition being the strongest and most enduring with which a man has ever been gifted. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one's mind and apply all one's powers of concentration to piercing through it, and how it will dissolve and escape and you find that what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret. Then being a supreme mathematical technician he could dress it up, how you will, for purposes of exposition, but it was his intuition which was pre-eminently extraordinary - 'so happy in his conjectures', said De Morgan, 'as to seem to know more than he could possibly have any means of proving'. The proofs, for what they are worth, were, as I have said, dressed up afterwards - they were not the instrument of discovery.

One way to distinguish philosophy from other disciplines is to see that the problems posed by philosophy are distinct from those of other disciplines. 카톡【AKR331】텔레【RDH705】위커【SPR705】라인【98K33】 Even until the 18th century, mathematics and physics were perceived as natural philosophy rather than philosophy and independent science. 수면제,무조건 피하지 마라… 복용법 지키면 먹는데 도움이 되는 수면제 졸피뎀 스틸녹스 복용방법 제품정보 소개해드리겠습니다 정품수면제 추천해드릴테니 위 카톡 텔레 라인등으로 추가해서 구입문의주세요 The inherent problems of philosophy can be summed up by the four questions of Manuel Kant as an 18th century philosopher. What do I know ?: The main problem of epistemology. How is the external object recognized? Is the external thing real? Is there a real existence that exists independently of human perception ability? How can human perception respond to reality in "out there"? How is awareness formed? What are the criteria by which one consciousness can be true? And how can we acquire knowledge from true awareness? On the other hand, the problem posed by metaphysics can not be solved by most human recognition methods. Does God exist? Does the beginning and end of the universe exist? Is time and space continuous? 수면제는 불면증 초기에 일주일에 3일 이상 잠을 제대로 못자 피로와 스트레스가 심하다면 불면증이라고 생각하고 수면제를 복용을 고려해봐야한다 What should I do?: Ethics major problems. Is there a difference between right and wrong? If so, how can we prove it? In real situations, how do we apply theoretical ideas to right and wrong? What do I want?: The main problem of art philosophy (aesthetics). What kind of pleasure does art give to humans? What is beauty? Where is the value of a work of art? What is human ?: The main problem of social philosophy. How does man make society? How is the state established and how does it operate?

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.

In many fields—literature, music, architecture—the label ‘Modern’ stretches back to the early 20th century. Philosophy is odd in starting its Modern period almost 400 years earlier. This oddity is explained in large measure by a radical 16th century shift in our understanding of nature, a shift that also transformed our understanding of knowledge itself. On our Modern side of this line, thinkers as far back as Galileo Galilei (1564–1642) are engaged in research projects recognizably similar to our own. If we look back to the Pre-Modern era, we see something alien: this era features very different ways of thinking about how nature worked, and how it could be known. To sample the strange flavour of pre-Modern thinking, try the following passage from the Renaissance thinker Paracelsus (1493–1541): The whole world surrounds man as a circle surrounds one point. From this it follows that all things are related to this one point, no differently from an apple seed which is surrounded and preserved by the fruit … Everything that astronomical theory has profoundly fathomed by studying the planetary aspects and the stars … can also be applied to the firmament of the body. Thinkers in this tradition took the universe to revolve around humanity, and sought to gain knowledge of nature by finding parallels between us and the heavens, seeing reality as a symbolic work of art composed with us in mind (see Figure 3). By the 16th century, the idea that everything revolved around and reflected humanity was in danger, threatened by a number of unsettling discoveries, not least the proposal, advanced by Nicolaus Copernicus (1473–1543), that the earth was not actually at the centre of the universe. The old tradition struggled against the rise of the new. Faced with the news that Galileo’s telescopes had detected moons orbiting Jupiter, the traditionally minded scholar Francesco Sizzi argued that such observations were obviously mistaken. According to Sizzi, there could not possibly be more than seven ‘roving planets’ (or heavenly bodies other than the stars), given that there are seven holes in an animal’s head (two eyes, two ears, two nostrils and a mouth), seven metals, and seven days in a week. Sizzi didn’t win that battle. It’s not just that we agree with Galileo that there are more than seven things moving around in the solar system. More fundamentally, we have a different way of thinking about nature and knowledge. We no longer expect there to be any special human significance to natural facts (‘Why seven planets as opposed to eight or 15?’) and we think knowledge will be gained by systematic and open-minded observations of nature rather than the sorts of analogies and patterns to which Sizzi appeals. However, the transition into the Modern era was not an easy one. The pattern-oriented ways of thinking characteristic of pre-Modern thought naturally appeal to meaning-hungry creatures like us. These ways of thinking are found in a great variety of cultures: in classical Chinese thought, for example, the five traditional elements (wood, water, fire, earth, and metal) are matched up with the five senses in a similar correspondence between the inner and the outer. As a further attraction, pre-Modern views often fit more smoothly with our everyday sense experience: naively, the earth looks to be stable and fixed while the sun moves across the sky, and it takes some serious discipline to convince oneself that the mathematically more simple models (like the sun-centred model of the solar system) are right.