Differential Calculus Quotes

All the world's a differential equation, and the men and women are merely variables.

Alongside the liberating relief of the veteran who tells us his story, I now felt in the writing a complex, intense, and new pleasure, similar to that I felt as a student when penetrating the solemn order of differentials calculus. It was exalting to search and find, or create, the right word, that is, commensurate, concise, and strong; to dredge up events from my memory and describe them with the greatest rigor and the least clutter.

At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, “out of a curiosity to see what there was in it.” He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

Without warning, she asked me, “Hey, Watanabe, can you explain the difference between the English subjunctive present and the subjunctive past?” “I think I can,” I said. “Let me ask you, then, what purpose does stuff like that serve in daily life?” “None at all,” I said. “It may not serve any concrete purpose, but it does give you some kind of training to help you grasp things in general more systematically.” Midori took a moment to give that some serious thought. “You’re amazing,” she said. “That never occurred to me before. I always thought of things like the subjunctive case and differential calculus and chemical symbols as totally useless. A pain in the neck. So I’ve always ignored them. Now I have to wonder if my whole life has been a mistake.

Winston in his own words found himself in an "Alice in Wonderland world at the portals of which stood a quadratic equation followed by the dim chambers inhabited by the differential calculus, and then a strange corridor of sines and cosines in a highly square rooted condition.

Twenty minutes later, Three Body’s Von Neumann architecture human-formation computer had begun full operations under the Qin 1.0 operating system. “Run solar orbit computation software ‘Three Body 1.0’!” Newton screamed at the top of his lungs. “Start the master computing module! Load the differential calculus module! Load the finite element analysis module! Load the spectral method module! Enter initial condition parameters … and begin calculation!” The motherboard sparkled as the display formation flashed with indicators in every color. The human-formation computer began the long computation.

We know, however, that the mind is capable of understanding these matters in all their complexity and in all their simplicity. A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball's spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball. People who call this "instinct" are merely giving the phenomenon a name, not explaining anything. I think that the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts - it has no meaning or purpose other than to be itself. Every single aspect of a piece of music can be represented by numbers. From the organization of movements in a whole symphony, down through the patterns of pitch and rhythm that make up the melodies and harmonies, the dynamics that shape the performance, all the way down to the timbres of the notes themselves, their harmonics, the way they change over time, in short, all the elements of a noise that distinguish between the sound of one person piping on a piccolo and another one thumping a drum - all of these things can be expressed by patterns and hierarchies of numbers. And in my experience the more internal relationships there are between the patterns of numbers at different levels of the hierarchy, however complex and subtle those relationships may be, the more satisfying and, well, whole, the music will seem to be. In fact the more subtle and complex those relationships, and the further they are beyond the grasp of the conscious mind, the more the instinctive part of your mind - by which I mean that part of your mind that can do differential calculus so astoundingly fast that it will put your hand in the right place to catch a flying ball- the more that part of your brain revels in it. Music of any complexity (and even "Three Blind Mice" is complex in its way by the time someone has actually performed it on an instrument with its own individual timbre and articulation) passes beyond your conscious mind into the arms of your own private mathematical genius who dwells in your unconscious responding to all the inner complexities and relationships and proportions that we think we know nothing about. Some people object to such a view of music, saying that if you reduce music to mathematics, where does the emotion come into it? I would say that it's never been out of it.

Fifteen thousand years ago, my prowess as a hunter of woolly mammoths would probably have accorded me more status in the culture than my ability to handle the kinds of abstract mathematical concepts involved, for example, in twelfth grade differential calculus. I need to see: one is not inherently more valuable than another; I am not inherently worth more or less dependent on these abilities. If I can see this, I open a door to a more natural sense of self-worth, and to a degree of freedom.

The ultimate goal of a meteorologist is to set up differential equations of the movements of the air and to obtain, as their integral, the general atmospheric circulation, and as particular integrals the cyclones, anticyclones, tornados, and thunderstorms.

It was as if she had just discovered the irreversible process. It astonished her to think that so much could be lost, even the quantity of hallucination belonging just to the sailor that the world would bear no further trace of. She knew, because she had held him, that he suffered DT’s. Behind the initials was a metaphor, a delirium tremens, a trembling unfurrowing of the mind’s plowshare. The saint whose water can light lamps, the clairvoyant whose lapse in recall is the breath of God, the true paranoid for whom all is organized in spheres joyful or threatening about the central pulse of himself, the dreamer whose puns probe ancient fetid shafts and tunnels of truth all act in the same special relevance to the word, or whatever it is the word is there, buffering, to protect us from. The act of metaphor then was a thrust at truth and a lie, depending where you were: inside, safe, or outside, lost. Oedipa did not know where she was. Trembling, unfurrowed, she slipped sidewise, screeching back across grooves of years, to hear again the earnest, high voice of her second or third collegiate love Ray Glozing bitching among “uhs” and the syncopated tonguing of a cavity, about his freshman calculus; “dt,” God help this old tattooed man, meant also a time differential, a vanishingly small instant in which change had to be confronted at last for what it was, where it could no longer disguise itself as something innocuous like an average rate; where velocity dwelled in the projectile though the projectile be frozen in midflight, where death dwelled in the cell though the cell be looked in on at its most quick. She knew that the sailor had seen worlds no other man had seen if only because there was that high magic to low puns, because DT’s must give access to dt’s of spectra beyond the known sun, music made purely of Antarctic loneliness and fright. But nothing she knew of would preserve them, or him.

Indeed, much of Newton’s intellectual development can be attributed to this tension between rationalism and mysticism. At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, ‘out of a curiosity to see what there was in it.’ He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).

There is a mathematical underpinning that you must first acquire, mastery of each mathematical subdiscipline leading you to the threshold of the next. In turn you must learn arithmetic, Euclidian geometry, high school algebra, differential and integral calculus, ordinary and partial differential equations, vector calculus, certain special functions of mathematical physics, matrix algebra, and group theory. For most physics students, this might occupy them from, say, third grade to early graduate school—roughly 15 years. Such a course of study does not actually involve learning any quantum mechanics, but merely establishing the mathematical framework required to approach it deeply.

Rain is the last thing you want when you're chasing someone in Miami. They drive shitty enough as it is, but on top of that, snow is a foreign concept, which means they never got the crash course in traction judgment for when pavement slickness turns less than ideal. And because of the land-sea temperature differential, Florida has regular afternoon rain showers. Nothing big, over in a jiff. But minutes later, all major intersections in Miami-Dade are clogged with debris from spectacular smash-ups. In Northern states, snow teaches drivers real fast about the Newtonian physics of large moving objects. I haven't seen snow either, but I drink coffee, so the calculus of tire-grip ratio is intuitive to my body.

Newton had invented the calculus, which was expressed in the language of "differential equations," which describe how objects smoothly undergo infinitesimal changes in space and time. The motion of ocean waves, fluids, gases, and cannon balls could all be expressed in the language of differential equations. Maxwell set out with a clear goal, to express the revolutionary findings of Faraday and his force fields through precise differential equations. Maxwell began with Faraday's discovery that electric fields could turn into magnetic fields and vice versa. He took Faraday's depictions of force fields and rewrote them in the precise language of differential equations, producing one of the most important series of equations in modern science. They are a series of eight fierce-looking differential equations. Every physicist and engineer in the world has to sweat over them when mastering electromagnetism in graduate school. Next, Maxwell asked himself the fateful question: if magnetic fields can turn into electric fields and vice versa, what happens if they are constantly turning into each other in a never-ending pattern? Maxwell found that these electric-magnetic fields would create a wave, much like an ocean wave. To his astonishment, he calculated the speed of these waves and found it to be the speed of light! In 1864, upon discovering this fact, he wrote prophetically: "This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself...is an electromagnetic disturbance.

In 1604, at the height of his scientific career, Galileo argued that for a rectilinear motion in which speed increases proportionally to distance covered, the law of motion should be just that (x = ct^2) which he had discovered in the investigation of falling bodies. Between 1695 and 1700 not a single one of the monthly issues of Leipzig’s Acta Eruditorum was published without articles of Leibniz, the Bernoulli brothers or the Marquis de l'Hôpital treating, with notation only slightly different from that which we use today, the most varied problems of differential calculus, integral calculus and the calculus of variations. Thus in the space of almost precisely one century infinitesimal calculus or, as we now call it in English, The Calculus, the calculating tool par excellence, had been forged; and nearly three centuries of constant use have not completely dulled this incomparable instrument.

pure mathematics, but these were very great indeed, and were indispensable to much of the work in the physical sciences. Napier published his invention of logarithms in 1614. Co-ordinate geometry resulted from the work of several seventeenth-century mathematicians, among whom the greatest contribution was made by Descartes. The differential and integral calculus was invented independently by Newton and Leibniz; it is the instrument for almost all higher mathematics. These are only the most outstanding achievements in pure mathematics; there were innumerable others of great importance.

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.

Mathematical analysis and computer modelling are revealing to us that the shapes and processes we encounter in nature -the way that plants grow, the way that mountains erode or rivers flow, the way that snowflakes or islands achieve their shapes, the way that light plays on a surface, the way the milk folds and spins into your coffee as you stir it, the way that laughter sweeps through a crowd of people — all these things in their seemingly magical complexity can be described by the interaction of mathematical processes that are, if anything, even more magical in their simplicity. Shapes that we think of as random are in fact the products of complex shifting webs of numbers obeying simple rules. The very word “natural” that we have often taken to mean ”unstructured” in fact describes shapes and processes that appear so unfathomably complex that we cannot consciously perceive the simple natural laws at work.They can all be described by numbers. We know, however, that the mind is capable of understanding these matters in all their complexity and in all their simplicity. A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball's spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball. People who call this "instinct" are merely giving the phenomenon a name, not explaining anything. I think that the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts - it has no meaning or purpose other than to be itself.

An essential pedagogic step here is to relegate the teaching of mathematical methods in economics to mathematics departments. Any mathematical training in economics, if it occurs at all, should come after students have at the very least completed course work in basic calculus, algebra and differential equations (the last being one about which most economists are woefully ignorant). This simultaneously explains why neoclassical economists obsess too much about proofs and why non-neoclassical economists, like those in the Circuit School, experience such difficulties in translating excellent verbal ideas about credit creation into coherent dynamic models of a monetary production economy.

The description given earlier of the relationship between integrating a 2-form over the surface of a sphere and integrating its derivative over the solid sphere can be thought of as a generalization of the fundamental theorem of calculus, and can itself be generalized considerably: Stokes’s theorem is the assertion that for any oriented manifold S and form ω, where ∂ S is the oriented boundary of S (which we will not define here). Indeed one can view this theorem as a definition of the derivative operation ω → dω; thus, differentiation is the adjoint of the boundary operation. (For instance, the identity (11) is dual to the geometric observation that the boundary ∂s of an oriented manifold itself has no boundary: ∂(∂S) = ∅.) As a particular case of Stokes’s theorem, we see that ∫s dω = 0 whenever S is a closed manifold, i.e., one with no boundary. This observation lets one extend the notions of closed and exact forms to general differential forms, which (together with (11)) allows one to fully set up de Rham cohomology.

Already uneasy over the foundations of their subject, mathematicians got a solid dose of ridicule from a clergyman, Bishop George Berkeley (1685-1753). Bishop Berkeley, in his caustic essay 'The Analyst, or a Discourse addressed to an Infidel Mathematician,' derided those mathematicians who were ever ready to criticize theology as being based upon unsubstantiated faith, yet who embraced the calculus in spite of its foundational weaknesses. Berkeley could not resist letting them have it: 'All these points [of mathematics], I say, are supposed and believed by certain rigorous exactors of evidence in religion, men who pretend to believe no further than they can see... But he who can digest a second or third fluxion, a second or third differential, need not, methinks, be squeamish about any point in divinity.' As if that were not devastating enough, Berkeley added the wonderfully barbed comment: 'And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, not yet nothing. May we not call them the ghosts of departed quantities...?' Sadly, the foundations of the calculus had come to this - to 'ghosts of departed quantities.' One imagines hundreds of mathematicians squirming restlessly under this sarcastic phrase. Gradually the mathematical community had to address this vexing problem. Throughout much of the eighteenth century, they had simply been having too much success - and too much fun - in exploiting the calculus to stop and examine its underlying principles. But growing internal concerns, along with Berkeley's external sniping, left them little choice. The matter had to be resolved. Thus we find a string of gifted mathematicians working on the foundational questions. The process of refining the idea of 'limit' was an excruciating one, for the concept is inherently quite deep, requiring a precision of thought and an appreciation of the nature of the real number system that is by no means easy to come by. Gradually, though, mathematicians chipped away at this idea. By 1821, the Frenchman Augustin-Louis Cauchy (1789-1857) had proposed this definition: 'When the values successively attributed to a particular variable approach indefinitely a fixed value, so as to end by differing from it by as little as one wishes, this latter is called the limit of all the others.

As the arguments of this book have shown, mathematical understanding is something different from computation and cannot be completely supplanted by it. Computation can supply extremely valuable aid to understanding, but it never supplies actual understanding itself. However, mathematical understanding is often directed towards the finding of algorithmic procedures for solving problems. In this way, algorithmic procedures can take over and leave the mind free to address other issues. A good notation is something of this nature, such as is supplied by the differential calculus, or the ordinary 'decimal' notation for numbers. Once the algorithm for multiplying numbers together has been mastered, for example, the operations can be performed in an entirely mindless algorithmic way, rather than 'understanding' having to be invoked as to why those particular algorithmic rules are being adopted, rather than something else.

Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus.

But in order to make this new theory echo Newton’s laws and conform to the rigours of differential calculus, Jevons, Walras and their fellow mathematical pioneers had to make some heroically simplifying assumptions about how markets and people work. Crucially, the nascent theory hinged on assuming that, for any given mix of preferences that consumers might have, there was just one price at which everyone who wanted to buy and everyone who wanted to sell would be satisfied, having bought or sold all that they wanted for that price. In other words, each market had to have one single, stable point of equilibrium, just as a pendulum has only one point of rest. And for that condition to hold, the market’s buyers and sellers all had to be ‘price-takers’—no single actor being big enough to have sway over prices—and they had to be following the law of diminishing returns. Together these assumptions underpin the most widely recognised diagram in all of microeconomic theory,

It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all. “If you could write down the solution to a differential equation,” Yorke said, “then necessarily it’s not chaotic, because to write it down, you must find regular invariants, things that are conserved, like angular momentum. You find enough of these things, and that lets you write down a solution. But this is exactly the way to eliminate the possibility of chaos.

Spengler's book is rich in these "morphological relationships" between dissimilar activities that prove the coherent spirit of each culture and epoch. So there was a common spirit int eh ancient Greek polis and in Euclidean geometry, as there was also between the differential calculus and the state of Louis XIV. Chronological "contemporaneity" was misleading. It should be replaced by an understanding of how different events play similar roles in expressing the culture spirit. Thus he sees his own kind of "contemporaneity" in the Trojan War and the Crusades, in Homer and the songs of the Nibelungs.

Who amongst them realizes that between the Differential Calculus and the dynastic principle of politics in the age of Louis XIV, between the Classical city-state and the Euclidean geometry, between the space perspective of Western oil painting and the conquest of space by railroad, telephone and long range weapon, between contrapuntal music and credit economics, there are deep uniformities?

For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it’s the only way a universe with us in it could be, because nonmathematical universes can’t harbor life intelligent enough to ask the question. In any case, it’s a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations.

Reimprinting the third semantic circuit can now follow easily. The human brain is capable of mastering any symbol-system if sufficiently motivated. Some people can even play Beethoven’s late piano music, although to me this is as “miraculous” as any feat alleged by psychic researchers; people can learn French, Hindustani, differential calculus, Swahili, etc. ad. infinitum — if motivated. When the first circuit security needs have been reimprinted and second-circuit ego-needs have been hooked to mastering a new semantic reality-tunnel, that tunnel will be imprinted.

Furthermore, what you can say about what you saw depends on the structure of your symbolism — whether you describe it in English, Persian, Chinese, Euclidean geometry, non-Euclidean geometry, differential calculus or quaternions. This explains why, in Dr. Jones's words, "whatever we are describing, the human mind cannot be parted from it.

Duration” tells you how risky a bond is. The greater the duration, the greater the risk. For example, a ten-year bond has greater duration—and greater risk—than a one-year bond. That’s it. Mathematically, of course, duration is more complicated than this. It’s the length of time until you receive the average present value-weighted cash flow, and is itself a derivative (in calculus terms), of the partial differential equation that describes the price behavior of a bond.

Life is poetry no poet can write, life is science no scientist can grasp. Life is to be lived for life's sake, not autopsied as differential calculus.

Distribution theory was one of the two great revolutions in mathematical analysis in the 20th century. It can be thought of as the completion of differential calculus, just as the other great revolution, measure theory (or Lebesgue integration theory), can be thought of as the completion of integral calculus. There are many parallels between the two revolutions. Both were created by young, highly individualistic French mathematicians (Henri Lebesgue and Laurent Schwartz). Both were rapidly assimilated by the mathematical community, and opened up new worlds of mathematical development. Both forced a complete rethinking of all mathematical analysis that had come before, and basically altered the nature of the questions that mathematical analysts asked.

Distribution theory was one of the two great revolutions in mathematical analysis in the 20th century. It can be thought of as the completion of differential calculus, just as the other great revolution, measure theory (or Lebesgue integration theory), can be thought of as the completion of integral calculus. There are many parallels between the two revolutions. Both were created by young, highly individualistic French mathematicians (Henri Lebesgue and Laurent Schwartz). Both were rapidly assimilated by the mathematical community, and opened up new worlds of mathematical development. Both forced a complete rethinking of all mathematical analysis that had come before, and basically altered the nature of the questions that mathematical analysts asked.

Even at the cutting edge of modern physics, partial differential equations still provide the mathematical infrastructure. Consider Einstein’s general theory of relativity. It reimagines gravity as a manifestation of curvature in the four-dimensional fabric of space-time. The standard metaphor invites us to picture space-time as a stretchy, deformable fabric, like the surface of a trampoline. Normally the fabric is pulled taut, but it can curve under the weight of something heavy placed on it, say a massive bowling ball sitting at its center. In much the same way, a massive celestial body like the sun can curve the fabric of space-time around it. Now imagine something much smaller, say a tiny marble (which represents a planet), rolling on the trampoline’s curved surface. Because the surface sags under the bowling ball’s weight, it deflects the marble’s trajectory. Instead of traveling in a straight line, the marble follows the contours of the curved surface and orbits around the bowling ball repeatedly. That, says Einstein, is why the planets go around the sun. They’re not feeling a force; they’re just following the paths of least resistance in the curved fabric of space-time.

Pedagogically speaking, a good share of physics and mathematics was—and is—writing differential equations on a blackboard and showing students how to solve them. Differential equations represent reality as a continuum, changing smoothly from place to place and from time to time, not broken in discrete grid points or time steps. As every science student knows, solving differential equations is hard. But in two and a half centuries, scientists have built up a tremendous body of knowledge about them: handbooks and catalogues of differential equations, along with various methods for solving them, or “finding a closed-form integral,” as a scientist will say. It is no exaggeration to say that the vast business of calculus made possible most of the practical triumphs of post-medieval science; nor to say that it stands as one of the most ingenious creations of humans trying to model the changeable world around them. So by the time a scientist masters this way of thinking about nature, becoming comfortable with the theory and the hard, hard practice, he is likely to have lost sight of one fact. Most differential equations cannot be solved at all.

Run solar orbit computation software ‘Three Body 1.0’!” Newton screamed at the top of his lungs. “Start the master computing module! Load the differential calculus module! Load the finite element analysis module! Load the spectral method module! Enter initial condition parameters … and begin calculation!” The motherboard sparkled as the display formation flashed with indicators in every color. The human-formation computer began the long computation. “This is really interesting,” Qin Shi Huang said, pointing to the spectacular sight. “Each individual’s behavior is so simple, yet together, they can produce such a complex, great whole! Europeans criticize me for my tyrannical rule, claiming that I suppress creativity. But in reality, a large number of men yoked by severe discipline can also produce great wisdom when bound together as one.” “Great First Emperor, this is just the mechanical operation of a machine, not wisdom. Each of these lowly individuals is just a zero. Only when someone like you is added to the front as a one can the whole have any meaning.” Newton’s smile was ingratiating. “Disgusting philosophy!” Von Neumann said as he glanced at Newton. “If, in the end, the results computed in accordance with your theory and mathematical model don’t match reality, then you and I aren’t even zeroes.” “Indeed. If that turns out to be the case, you will be nothing!” Qin Shi Huang turned and left the scene.

I mean the poetry of differential calculus. Calculating points in time and place. The infinite movement we make in time, moving forward, but never eating zero. Integral caucus, where we don't head to zero -- we head to fucking infinity, man. Get as close as we can. But again we never get there. Because in nature we never get an absolute. Absolutes are bullshit.

How hard can it be to follow five black SUVs?” Serge leaned over the steering wheel. “Except we’re in Miami.” “So?” “Miami drivers are a breed unto their own. Always distracted.” He uncapped a coffee thermos and chugged. “Quick on the gas and the horn. No separation between vehicles, every lane change a new adventure. The worst of both worlds: They race around as if they are really good, but they’re really bad, like if you taught a driver’s-ed class with NASCAR films.” He watched the first few droplets hit the windshield. “Oh, and worst of all, most of them have never seen snow.” “But it’s not snow,” said Felicia. “It’s rain. And just a tiny shower.” “That’s right.” Serge hit the wipers and took another slug from the thermos. “Rain is the last thing you want when you’re chasing someone in Miami. They drive shitty enough as it is, but on top of that, snow is a foreign concept, which means they never got the crash course in traction judgment for when pavement slickness turns less than ideal. And because of the land-sea temperature differential, Florida has regular afternoon rain showers. Nothing big, over in a jiff. But minutes later, all major intersections in Miami-Dade are clogged with debris from spectacular smash-ups. In Northern states, snow teaches drivers real fast about the Newtonian physics of large moving objects. I haven’t seen snow either, but I drink coffee, so the calculus of tire-grip ratio is intuitive to my body. It feels like mild electricity. Sometimes it’s pleasant, but mostly I’m ambivalent. Then you’re chasing someone in the rain through Miami, and your pursuit becomes this harrowing slalom through wrecked traffic like a disaster movie where everyone’s fleeing the city from an alien invasion, or a ridiculous change in weather that the scientist played by Dennis Quaid warned about but nobody paid attention.” Serge held the mouth of the thermos to his mouth. “Empty. Fuck it—

But the point is that a story is exciting because it has in it so strong an element of will, of what theology calls free-will. You cannot finish a sum how you like. But you can finish a story how you like. When somebody discovered the Differential Calculus there was only one Differential Calculus he could discover. But when Shakespeare killed Romeo he might have married him to Juliet’s old nurse if he had felt inclined. And Christendom has excelled in the narrative romance exactly because it has insisted on the theological free-will.

Berkeley explained that by finding the tangent by means of differentials, one first assumes increments; but these determine the secant, not the tangent. One undoes this error, however, by neglecting higher differentials, and thus "by virtue of a twofold mistake you arrive, though not at science, yet at the truth.

Most of his predecessors had considered the differential calculus as bound up with geometry, but Euler made the subject a formal theory of functions which had no need to revert to diagrams or geometrical conceptions.

I took 17 computer science classes and made an A in 11 of them. 1 point away from an A in 3 of them and the rest of them didn't matter. Math is a tool for physics,chemistry,biology/basic computation and nothing else. CS I(Pascal Vax), CS II(Pascal Vax), Sr. Software Engineering, Sr. Distributed Systems, Sr. Research, Sr. Operating Systems, Sr. Unix Operating Systems, Data Structures, Sr. Object Oriented A&D, CS (perl/linux), Sr. Java Programming, Information Systems Design, Jr. Unix Operating Systems, Microprocessors, Programming Algorithms, Calculus I,II,III, B Differential Equations, TI-89 Mathematical Reasoning, 92 C++ Programming, Assembly 8086, Digital Computer Organization, Discrete Math I,II, B Statistics for the Engineering & Sciences (w/permutations & combinatorics) -- A-American Literature A-United States History 1865 CLEP-full year english CLEP-full year biology A-Psychology A-Environmental Ethics

A few books that I've read.... Pascal, an Introduction to the Art and Science of Programming by Walter Savitch Programming algorithms Introduction to Algorithms, 3rd Edition (The MIT Press) Data Structures and Algorithms in Java Author: Michael T. Goodrich - Roberto Tamassia - Michael H. Goldwasser The Algorithm Design Manual Author: Steven S Skiena Algorithm Design Author: Jon Kleinberg - Éva Tardos Algorithms + Data Structures = Programs Book by Niklaus Wirth Discrete Math Discrete Mathematics and Its Applications Author: Kenneth H Rosen Computer Org Structured Computer Organization Andrew S. Tanenbaum Introduction to Assembly Language Programming: From 8086 to Pentium Processors (Undergraduate Texts in Computer Science) Author: Sivarama P. Dandamudi Distributed Systems Distributed Systems: Concepts and Design Author: George Coulouris - Jean Dollimore - Tim Kindberg - Gordon Blair Distributed Systems: An Algorithmic Approach, Second Edition (Chapman & Hall/CRC Computer and Information Science Series) Author: Sukumar Ghosh Mathematical Reasoning Mathematical Reasoning: Writing and Proof Version 2.1 Author: Ted Sundstrom An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Author: Peter J. Eccles Differential Equations Differential Equations (with DE Tools Printed Access Card) Author: Paul Blanchard - Robert L. Devaney - Glen R. Hall Calculus Calculus: Early Transcendentals Author: James Stewart And more....

From the old elements of earth, air, fire, and water to the latest in electrons, quarks, black holes, and superstrings, every inanimate thing in the universe bends to the rule of differential equations. I bet this is what Feynman meant when he said that calculus is the language God talks.

Differential calculus has to do with rates of change. Integral calculus has to do with sums of many tiny incremental quantities. It’s not immediately obvious that these have anything to do with each other, but they do.