Limit Calculus Quotes

She gave me the same smile I'd seen on Saturday night. That type of smile caused men to write those pussy-ass songs that Isaiah and I made fun of. I'd sit in Mrs. Collins's office for hours and wake my ass up early to go to calculus in order to see that smile again.

If slavery persists as an issue in the political life of black America, it is not because of an antiquarian obsession with bygone days or the burden of a too-long memory, but because black lives are still imperiled and devalued by a racial calculus and a political arithmetic that were entrenched centuries ago. This is the afterlife of slavery--skewed life chances, limited access to health and education, premature death, incarceration, and impoverishment.

As a student, frustrated by the limitations of conventional mathematics, he invented an entirely new form, the calculus, but then told no-one about it for twenty-seven years5.

We are finite creatures, bound to this place and this time, and helpless before an endless expanse. It is within the calculus that for the first time the infinite is charmed into compliance, its luxuriance subordinated to the harsh concept of a limit.

Then he smiled into her eyes and asked, in the dry academic tones of an astronomer discussing a theoretical point with a colleague, 'How long do you suppose I can go on loving you more every day?' And he devised for her a calculus of love, which approached infinity as a limit, and made her smile again.

I think I know what will help you chill.” The way his eyes devoured me hinted I shouldn’t take the bait, but I did anyhow. “And what would that be?” Noah pressed his body into mine, pushing me against the lockers. “Kissing.” I held my books close to my chest and fought the urge to drop them and pull him close. But that would only encourage his behavior, and good God, bring on his fantastic kissing. Fantastic or not, kissing in public would definitely mean detention and a tardy slip. I ducked underneath his arm and breathed in fresh air, welcoming any scent that didn’t remind me of him. Noah caught up to me, slowing his pace to mine. “You know, you may have never noticed, but we have calculus together,” he said. “You could have waited for me.” “And give you the chance to drag me into the janitor’s closet?

The Greeks couldn't do this neat little mathematical trick. They didn't have the concept of a limit because they didn't believe in zero. The terms in the infinite series didn't have a limit or a destination; they seemed to get smaller and smaller without any particular end in sight. As a result, the Greeks couldn't handle the infinite. They pondered the concept of the void but rejected zero as a number, and they toyed with the concept of the infinite but refused to allow infinity-numbers that are infinitely small and infinitely large-anywhere near the realm of numbers. This is the biggest failure in Greek mathematics, and it is the only thing that kept them from discovering calculus.

He flashed his wicked grin and lowered his voice. “Mrs. Frost always runs late. I could kiss you now and give the crowd what they’re looking for.” That would be an awesome way to start class. I licked my lips and whispered, “You are going to get me in so much trouble.” “Damn straight.” Noah caressed my cheek before heading to his seat in the back. I settled in my seat and spent the entire hour trying to keep my mind focused on calculus and not on kissing Noah Hutchins.

“Echo and I have class together.” Ashley brightened and pressed a hand to her belly. “Really? Which one?” “Calculus.” “Physics,” I added. “And business technology.” “Español.” Had he purposely made his voice all deep and sexy? His hand moved up a fraction of an inch and squeezed my leg, exerting delicious pressure on my inner thigh. I twisted my hair away from my neck to release some of the heat. Noah either choked on his own spit or stifled a laugh. Thankfully, my father missed the show.

Dating apps open up vast pools of potential mates. Living in a small town with a limited mating market allows a 10 to be happily mated with an 8, as long as there exist no other 9s or 10s in town. Living in a cyberworld containing millions of potential mates opens the floodgates to thousands of 9s and 10s. In the cold calculus of relative mate value, if a more desirable potential mate than my current partner is interested and within reach, I may become dissatisfied with my current partner, which may motivate me to switch.

It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true.

The overall structure of the calculus is simple. The subject is defined by a fantastic leading idea, one basic axiom, a calm and profound intellectual invention, a deep property, two crucial definitions, one ancillary definition, one major theorem, and the fundamental theorem of the calculus. The fantastic leading idea: the real world may be understood in terms of the real numbers. The basic axiom: brings the real numbers into existence. The calm and profound invention: the mathematical function. The deep property: continuity. The crucial definitions: instantaneous speed and the area underneath a curve. The ancillary definition: a limit The major theorem: the mean value theorem. The fundamental theorem of the calculus is the fundamental theorem of the calculus. These are the massive load-bearing walls and buttresses of the subject.

to conceive of the Infinite quantitatively is not only to limit it, but in addition it is to conceive of it as subject to increase and decrease, which is no less absurd; with similar considerations one quickly finds oneself envisaging not only several infinites that coexist without confounding or excluding one another, but also infinites that are larger or smaller than others; and finally, the infinite having become so relative under these conditions that it no longer suffices, the ‘transfinite’ is invented, that is, the domain of quantities greater than the infinite. Here, indeed, it is properly a matter of ‘invention, for such conceptions correspond to no reality. So many words, so many absurdities, even regarding simple, elementary logic, yet this does not prevent one from finding among those responsible some who even claim to be ‘specialists’ in logic, so great is the intellectual confusion of our times!

Mathematics has, of course, given the solution of the difficulties in terms of the abstract concept of converging infinite series. In a certain metaphysical sense this notion of convergence does not answer Zeno’s argument, in that it does not tell how one is to picture an infinite number of magnitudes as together making up only a finite magnitude; that is, it does not give an intuitively clear and satisfying picture, in terms of sense experience, of the relation subsisting between the infinite series and the limit of this series.

It is important to realize that ideas about infinity are not abstract scholastic thoughts that plague absentminded professors in the ivy-covered towers of academia. Rather, all of calculus is based on the modern notions of infinity mentioned in this chapter. Calculus, in turn, is the basis of all of the modern mathematics, physics, and engineering that make our advanced technological civilization possible. The reason the counterintuitive ideas of infinity are central to modern science is that they work. We cannot simply ignore them.

Querencia requires the willingness to limit short-term profits for long-term sustainability and is reflective of an orientation to the world that is tied more to a love of place and people than any form of economic calculus.

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.

C’est ici qu’intervient, pour rectifier cette fausse notion, ou plutôt pour la remplacer par une conception vraie des choses (7), l’idée de l’indéfini, qui est précisément l’idée d’un développement de possibilités dont nous ne pouvons atteindre actuellement les limites ; et c’est pourquoi nous regardons comme fondamentale, dans toutes les questions où apparaît le prétendu infini mathématique, la distinction de l’Infini et de l’indéfini. C’est sans doute à cela que répondait, dans l’intention de ses auteurs, la distinction scolastique de l’infinitum absolutum et de l’infinitum secundum quid ; il est certainement fâcheux que Leibnitz, qui pourtant a fait par ailleurs tant d’emprunts à la scolastique, ait négligé ou ignoré celle-ci, car, tout imparfaite que fût la forme sous laquelle elle était exprimée, elle eût pu lui servir à répondre assez facilement à certaines des objections soulevées contre sa méthode.

What Zeno is forcing us to do is to ask the question of whether space (which is not made of atoms) can be infinitely divvied up. If it can be, the slacker will not reach his goal. If it cannot be, there must be discrete "space atoms," and continuous real-number mathematics is not a proper model for space. We cannot, however be so flippant about asserting that space is discrete and not continuous. The world certainly does not look discrete. Movement has the feel of being continuous. Much of mathematical physics is based on calculus, which assumes that the real world is infinitely divisible. Outside of some quantum theory and Zeno, the continuous real number make a good model for the physical world. We build rockets and bridges using mathematics that assumes that the world is continuous. Let us not be so quick to abandon it.

If we assume the world is discrete, the mathematics needed to build rockets and bridges is far more complicated than calculus. Perhaps calculus is simply an easy approximation of the true mathematics that has to be done to concretely model the discrete world in which we live.

[...] These observations will allow us to understand more precisely in what sense one can say, as we did at the beginning, that the limits of the indefinite can never be reached through any analytical procedure, or, in other words, that the indefinite, while not absolutely and in every way inexhaustible, is at least analytically inexhaustible. In this regard, we must naturally consider those procedures analytical which ,in order to reconstitute a whole, consist in taking its elements distinctly and successively; such is the procedure for the formation of an arithmetical sum, and it is precisely in this regard that it differs essentially from integration. This is particularly interesting from our point of view, for one can see in it, as a very clear example, the true relationship between analysis and synthesis: contrary to current opinion, accordng to which analysis is as it were a preparation for synthesis, or again something leading to it, so much so that one must always begin with analysis, even when one does not intend to stop there, the truth is that one can never actually arrive at synthesis through analysis. All synthesis, in the true sense of the word, is something immediate, so to speak, something that is not preceded by any analysis and is entirely indfependent of it, just as integration is an operation carried out in a single stroke, by no means presupposing the consideration of elements comparable to those of an arithmetical sum; and as this arithmetical sum can yield no means of attaining and exhausting the indefinite, this latter must, in every domain, be one of those things that by their very nature resist analysis and can be known only through synthesis.[3] [3]Here, and in what follows, it should be understood that we take the terms 'analysis' and 'synthesis' in their true and original sense, and one must indeed take care to distinguish this sense from the completely different and quite improper sense in which one currently speaks of 'mathematical analysis', according to which integration itself, despite its essentially synthetic character, is regarded as playing a part in what one calls 'infinitesimal analysis'; it is for this reason, moreorever, that we prefer to avoid using this last expression, availing ourselves only of those of 'the infinitesimal calculus' and 'the infinitesimal method', which lead to no such equivocation.

P3 - ten minutes of that movie, or indeed of any movie whose message is similarly dystopian about a post-aging world (Blade Runner), you will see that they set it up by insinuating, with exactly no justification and also no attempt at discussion (which is how they get away with not justifying it), that the defeat of aging will self-evidently bring about some new problem that we will be unable to solve without doing more harm than good. The most common such problem, of course, is overpopulation - and I refer you to literally about 1000 interviews and hundreds of talks I have given on stage and camera over the past 20 years, of which several dozen are online, for why such a concern is misplaced. The reason there are 1000, of course, is that most people WANT to believe that aging is a blessing in disguise - they find it expedient to put aging out of their minds and get on with their miserably short lives, however irrational must be the rationalizations by which they achieve that. Aubrey has been asked on numerous occasions whether humans should use future tech to extend their lifespans. Aubrey opines, "I believe that humans should (and will) use (and, as a prerequisite, develop) future technologies to extend their healthspan, i.e. their healthy lifespan. But before fearing that I have lost my mind, let me stress that that is no more nor less than I have always believed. The reason people call me an “immortalist” and such like is only that I recognize, and am not scared to say, two other things: one, that extended lifespan is a totally certain side-effect of extended healthspan, and two, that the desire (and the legitimacy of the desire) to further extend healthspan will not suddenly cease once we achieve such-and-such a number of years." On what people can do to advance longevity research, my answer to this question has radically changed in the past year. For the previous 20 years, my answer would have been “make a lot of money and give it to the best research”, as it was indisputable that the most important research could go at least 2 or 3x times faster if not funding-limited. But in the past year, with the influx of at least a few $B, much of it non-profit (and much of it coming from tech types who did exactly the above), the calculus has changed: the rate-limiter now is personnel. It’s more or less the case now that money is no longer the main rate-limiter, talent is: we desperately need more young scientists to see longevity as the best career choice. As for how much current cryopreservation technology will advance in the next 10-20 years, and whether it enough for future reanimation? No question about the timeframe for a given amount of progress in any pioneering tech can be answered other than probabilistically. Or, to put it more simply, I don’t know - but I think there's a very good chance that within five years we will have cryo technology that inflicts only very little damage on biological tissue, such that yes, other advances in rejuvenation medicine that will repair the damage that caused the cryonaut to be pronounced dead in the first place will not be overwhelmed by cryopreservation damage, hence reanimation will indeed be possible. As of now, the people who have been cryopreserved(frozen) the best (i.e. w/ vitrification, starting very shortly immediately after cardiac arrest) may, just possibly, be capable of revival by rewarming and repair of damage - but only just possibly. Thus, the priority needs to be to improve the quality of cryopreservation - in terms of the reliability of getting people the best preservation that is technologically possible, which means all manner of things like getting hospitals more comfortable with cryonics practice and getting people to wear alarms that will alert people if they undergo cardiac arrest when alone, but even more importantly in terms of the tech itself, to reduce (greatly) the damage that is done to cells and tissues by the cryopreservation process.

Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process.

but most of them had never worked with captive populations before. They didn’t know how to cuff someone at the wrist and elbow so that the perp couldn’t get his hands out in front to strangle them. They didn’t know how to restrain someone with a length of cord around the neck so that the prisoner couldn’t choke himself to death, by accident or intentionally. Half of them didn’t even know how to pat someone down. Miller knew all of it like a game he’d played since childhood. In five hours, he found twenty hidden blades on the science crew alone. He hardly had to think about it. A second wave of transport ships arrived: personnel haulers that looked ready to spill their air out into the vacuum if you spat on them, salvage trawlers already dismantling the shielding and superstructure of the station, supply ships boxing and packing the precious equipment and looting the pharmacies and food banks. By the time news of the assault reached Earth, the station would be stripped to a skeleton and its people hidden away in unlicensed prison cells throughout the Belt. Protogen would know sooner, of course. They had outposts much closer than the inner planets. There was a calculus of response time and possible gain. The mathematics of piracy and war. Miller knew it, but he didn’t let it worry him. Those were decisions for Fred and his attachés to make. Miller had taken more than enough initiative for one day. Posthuman. It was a word that came up in the media every five or six years, and it meant different things every time. Neural regrowth hormone? Posthuman. Sex robots with inbuilt pseudo intelligence? Posthuman. Self-optimizing network routing? Posthuman. It was a word from advertising copy, breathless and empty, and all he’d ever thought it really meant was that the people using it had a limited imagination about what exactly humans were capable of. Now, as he escorted a dozen captives in Protogen uniforms to a docked transport heading God-knew-where, the word was taking on new meaning. Are you even human anymore? All posthuman meant, literally speaking, was what you were when you weren’t

Hence, the energy for independent thoughts is additive except for a term log[B(n1,n2)], the log of a binomial coefficient. Since binomial coefficients are always bigger than (or equal to) one, it follows that energy is super-additive. Combining thoughts demand more and more mental power as the sizes increase: the MIND is limited in the complexity of thoughts.

Economics operates legitimately and usefully within a 'given' framework which lies altogether outside the economic calculus. We might say that economics does not stand on its own feet, or that it is a 'derived' body of thought - derived from meta- economics. If the economist fails to study meta-economics, or, even worse. If he remains unaware of the fact that there are boundaries to the applicability of the economic calculus, he is likely to fall into a similar kind of error to that of certain medieval theologians who tried to settle questions of physics by means of biblical quotations. Every science is beneficial within its proper limits but becomes evil and destructive as soon as it transgresses them. The

No matter what room I was in, I always knew I was not the smartest person there. This was not false modesty. A D in freshman calculus and being in the presence of anyone who had mastered biochemistry, mathematics, or engineering—which I could never have done—were constant reminders to me of my limitations. What I brought into the room was a willingness to listen (I got better at that with every passing year), an ability to analyze and synthesize large and diverse amounts of information, opinions, and recommendations and come up with practical solutions to problems and proposals for reform. That, and a willingness to be bold.

Robins realized more clearly than did Jurin the nature of the limit concept. He recognized that the phrase "the ultimate ratio of vanishing quantities" was a figurative expression, referring , not to a last ratio, but to a "fixed quantity which some varying quantity, by a continual augmentation of diminuation shall perpetually approach,... provided the varying quantity can be made in its approach to the other to differ from it by less than by and quantity how minute soever, that can be assigned, .."though it can never be made absolutely equal to it.

In making the basis of the calculus more rigorously formal, Weierstrass also attacked the appeal to intuition of continuous motion which is implied in Cauchy's expression -- that a variable approaches a limit.

Thus the required rigor was found in the application of the concept of number, made formal by divorcing it from the idea of geometrical quantity

Mathematics is unable to specify whether motion is continuous, for it deals merely with hypothetical relations and can make its variable continuous or discontinuous at will. The paradoxes of Zeno are consequences of the failure to appreciate this fact and of the resulting lack of a precise specification of the problem. The former is a matter of scientific description a posteriori, whereas the latter is a matter solely of mathematical definition a priori. The former may consequently suggest that motion be defined mathematically in terms of continuous variable, but cannot, because of the limitations of sensory perception, prove that it must be so defined.

In making predictions and judgments under uncertainty,” they wrote, “people do not appear to follow the calculus of chance or the statistical theory of prediction. Instead, they rely on a limited number of heuristics which sometimes yield reasonable judgments and sometimes lead to severe and systematic error.