Integration Math Quotes

Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

The root of the word “integrity” is “integer.” It’s a math term - and it refers to whole numbers. The word itself implies “wholeness.” These are the questions we must ask ourselves frequently. “Am I whole?” “Are there parts of my character that are lacking?

The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon... when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms.

This is apparently a little promotional ¶ where we’re supposed to explain “how and why we came to” the subject of our GD series book (the stuff in quotations is the editor’s words). The overall idea is to humanize the series and make the books and their subjects seem warmer and more accessible. So that people will be more apt to buy the books. I’m pretty sure this is how it works. The obvious objection to such promotional ¶s is that, if the books are any good at all, then the writers’ interest and investment in their subjects will be so resoundingly obvious in the texts themselves that these little pseudo-intimate Why I Cared Enough About Transfinite Math and Where It Came From to Spend a Year Writing a Book About It blurblets are unnecessary; whereas, if the books aren’t any good, it’s hard to see how my telling somebody that as a child I used to cook up what amounted to simplistic versions of Zeno’s Dichotomy and ruminate on them until I literally made myself sick, or that I once almost flunked a basic calc course and have seethed with dislike for conventional higher-math education ever since, or that the ontology and grammar of abstractions have always struck me as one of the most breathtaking problems in human consciousness—how any such stuff will help. The logic of this objection seems airtight to me. In fact, the only way the objection doesn’t apply is if these ¶s are really nothing more than disguised ad copy, in which case I don’t see why anyone reading them should even necessarily believe that the books’ authors actually wrote them—I mean, maybe somebody in the ad-copy department wrote them and all we did was sort of sign off on them. There’d be a kind of twisted integrity about that, though—at least no one would be pretending to pretend.

Fairy tales, fantasy, legend and myth...these stories, and their topics, and the symbolism and interpretation of those topics...these things have always held an inexplicable fascination for me," she writes. "That fascination is at least in part an integral part of my character — I was always the kind of child who was convinced that elves lived in the parks, that trees were animate, and that holes in floorboards housed fairies rather than rodents. You need to know that my parents, unlike those typically found in fairy tales — the wicked stepmothers, the fathers who sold off their own flesh and blood if the need arose — had only the best intentions for their only child. They wanted me to be well educated, well cared for, safe — so rather than entrusting me to the public school system, which has engendered so many ugly urban legends, they sent me to a private school, where, automatically, I was outcast for being a latecomer, for being poor, for being unusual. However, as every cloud does have a silver lining — and every miserable private institution an excellent library — there was some solace to be found, between the carved oak cases, surrounded by the well–lined shelves, among the pages of the heavy antique tomes, within the realms of fantasy. Libraries and bookshops, and indulgent parents, and myriad books housed in a plethora of nooks to hide in when I should have been attending math classes...or cleaning my room...or doing homework...provided me with an alternative to a reality I didn't much like. Ten years ago, you could have seen a number of things in the literary field that just don't seem to exist anymore: valuable antique volumes routinely available on library shelves; privately run bookshops, rather than faceless chains; and one particular little girl who haunted both the latter two institutions. In either, you could have seen some variation upon a scene played out so often that it almost became an archetype: A little girl, contorted, with her legs twisted beneath her, shoulders hunched to bring her long nose closer to the pages that she peruses. Her eyes are glued to the pages, rapt with interest. Within them, she finds the kingdoms of Myth. Their borders stand unguarded, and any who would venture past them are free to stay and occupy themselves as they would.

When you’re carefully paying attention to what your teacher or boss is saying, that’s your frontal lobe at work. Doing math? Frontal lobe. Crossword puzzle? Frontal lobe. Trying to figure out how to handle a former friend who has lately been talking behind your back? The integration of all those feelings, memories, and possible responses requires the quarterbacking of the frontal lobe.

In 1994, Karl Sims was doing experiments on simulated organisms, allowing them to evolve their own body designs and swimming strategies to see if they would converge on some of the same underwater locomotion strategies that real-life organisms use.5, 6, 7 His physics simulator—the world these simulated swimmers inhabited—used Euler integration, a common way to approximate the physics of motion. The problem with this method is that if motion happens too quickly, integration errors will start to accumulate. Some of the evolved creatures learned to exploit these errors to obtain free energy, quickly twitching small body parts and letting the math errors send them zooming through the water.

A single strum of the strings or even one pluck is too complex, too complete in itself to admit any theory. Between this complex sound—so strong that it can stand alone—and that point of intense silence preceding it, called ma, there is a metaphysical continuity that defies analysis. In its complexity and integrity this single sound can stand alone. To the Japanese listener who appreciates this refined sound, the unique idea of ma—the unsounded part of this experience—has at the same time a deep, powerful, and rich resonance that can stand up to the sound. …the Japanese sound ideal: sound, in its ultimate expressiveness, being constantly refined, approaches the nothingness of that wind in the bamboo grove.

Many people who celebrate the arts and the humanities, who applaud vigorously the tributes to their importance in our schools, will proclaim without shame (and sometimes even joke) that they don’t understand math or physics. They extoll the virtues of learning Latin, but they are clueless about how to write an algorithm or tell BASIC from C++, Python from Pascal. They consider people who don’t know Hamlet from Macbeth to be Philistines, yet they might merrily admit that they don’t know the difference between a gene and a chromosome, or a transistor and a capacitor, or an integral and a differential equation. These concepts may seem difficult. Yes, but so, too, is Hamlet. And like Hamlet, each of these concepts is beautiful.

But empires of old kept their colonies at a distance: Rome conquered the Gauls across the Alps. France ruled Algeria from across the Mediterranean. King George III dispatched troops across the Atlantic to administer the new world. In the United States in 2016 such distance does not exist: the “rough” part of Ferguson is maybe a thousand yards from the “nice” neighborhoods. And so the maintenance of the Nation’s integrity requires constant vigilance. The borders must be enforced without the benefit of actual walls and checkpoints. This requires an ungodly number of interactions between the sentries of the state and those the state views as the disorderly class. The math of large numbers means that with enough of these interactions and enough fear and suspicion on the part of the officers who wield the gun, hundreds of those who’ve been marked for monitoring will die.

Montessori believed that if children were exposed to a safe, experiential learning environment (as opposed to a structured classroom), with access to specific learning materials and supplies, and if they were supervised by a gentle and attentive teacher, they would become self-motivated to learn. She discovered that, in this environment, older children readily worked with younger children, helping them to learn from, and cooperate with, each other. Montessori advocated teaching practical skills, like cooking, carpentry, and domestic arts, as an integrated part of a classical education in literature, science, and math. To her surprise, teenagers seemed to benefit from this approach the most; it built confidence, and the students became less resistant to traditional educational goals. Through this method, each child could reach his or her potential, regardless of age and intellectual ability.

Two mathematicians were having dinner. One was complaining: ‘The average person is a mathematical idiot. People cannot do arithmetic correctly, cannot balance a checkbook, cannot calculate a tip, cannot do percents, …’ The other mathematician disagreed: ‘You’re exaggerating. People know all the math they need to know.’ Later in the dinner the complainer went to the men’s room. The other mathematician beckoned the waitress to his table and said, ‘The next time you come past our table, I am going to stop you and ask you a question. No matter what I say, I want you to answer by saying “x squared.”‘ She agreed. When the other mathematician returned, his companion said, ‘I’m tired of your complaining. I’m going to stop the next person who passes our table and ask him or her an elementary calculus question, and I bet the person can solve it.’ Soon the waitress came by and he asked: ‘Excuse me, Miss, but can you tell me what the integral of 2x with respect to x is?’ The waitress replied: ‘x squared.’ The mathematician said, ‘See!’ His friend said, ‘Oh … I guess you were right.’ And the waitress said, ‘Plus a constant.

What Cantor's Diagonal Proof does is generate just such a number, which let's call R. The proof is both ingenious and beautiful-a total confirmation of art's compresence in pure math. First, have another look at the above table. We can let the integral value of R be whatever X we want; it doesn't matter. But now look at the table's very first row. We're going to make sure R's first post-decimal digit, a, is a different number from the table's a1. It's easy to do this even though we don't know what particular number a1 is: let's specify that a=(a1-1) unless a1 happens to be 0, in which case a=9. Now look at the table's second row, because we're going to do the same thing for R's second digit b: b=(b2-1), or b=9 if b2=0. This is how it works. We use the same procedure for R's third digit c and the table's c3, for d and d4, for e and e5, and so on, ad inf. Even though we can't really construct the whole R (just as we can't really finish the whole infinite table), we can still see that this real number R=X.abcdefhi... is going to be demonstrably different from every real number in the table. It will differ from the table's 1st Real in its first post-decimal digit, from the 2nd Real in its second digit, from the 3rd Real in its third digit,...and will, given the Diagonal Method here, differ from the table's Nth Real in its nth digit. Ergo R is not-cannot be-included in the above infinite table; ergo the infinite table is not exhaustive of all the real numbers; ergo (by the rules of reductio) the initial assumption is contradicted and the set of all real numbers is not denumerable, i.e. it's not 1-1 C-able with the set of integers. And since the set of all rational numbers is 1-1C-able with the integers, the set of all reals' cardinality has got to be greater than the set of all rationals' cardinality. Q.E.D.*

The most important pillar behind innovation and opportunity—education—will see tremendous positive change in the coming decades as rising connectivity reshapes traditional routines and offers new paths for learning. Most students will be highly technologically literate, as schools continue to integrate technology into lesson plans and, in some cases, replace traditional lessons with more interactive workshops. Education will be a more flexible experience, adapting itself to children’s learning styles and pace instead of the other way around. Kids will still go to physical schools, to socialize and be guided by teachers, but as much, if not more, learning will take place employing carefully designed educational tools in the spirit of today’s Khan Academy, a nonprofit organization that produces thousands of short videos (the majority in science and math) and shares them online for free. With hundreds of millions of views on the Khan Academy’s YouTube channel already, educators in the United States are increasingly adopting its materials and integrating the approach of its founder, Salman Khan—modular learning tailored to a student’s needs. Some are even “flipping” their classrooms, replacing lectures with videos watched at home (as homework) and using school time for traditional homework, such as filling out a problem set for math class. Critical thinking and problem-solving skills will become the focus in many school systems as ubiquitous digital-knowledge tools, like the more accurate sections of Wikipedia, reduce the importance of rote memorization. For children in poor countries, future connectivity promises new access to educational tools, though clearly not at the level described above. Physical classrooms will remain dilapidated; teachers will continue to take paychecks and not show up for class; and books and supplies will still be scarce. But what’s new in this equation—connectivity—promises that kids with access to mobile devices and the Internet will be able to experience school physically and virtually, even if the latter is informal and on their own time.

The importance of others' opinion in your life should be same as importance of arbitrary constant at the end of an indefinite integration, for a non-serious mathematician.

Einstein further explained that the pull of gravity actually slows time down. So if you were an astronaut on a long interstellar trip and your spacecraft passed close to a black hole (where the gravitational force is massive), time would slow down significantly. When you got back to Earth you might have aged several years, but your spouse and your friends would have already lived into old age. We can observe this effect in a much smaller way right here on Earth. If you lived in Dubai on the top floor of Burj Khalifa, the world’s highest tower, time would pass slightly faster for you than it would for someone living on the ground floor, just because gravity affects each of you differently. While a variance like this is too small for the human body to detect, it’s measurable with today’s technology. It gets even more bizarre. The math indicates that in space-time, past, present, and future are all part of an integrated four-dimensional structure in which all of space and all of time exist perpetually.

Art assignments are not a novel concept. They are an integral part of early childhood education in many parts of the world and are often the way we first learn about color, pattern, math, science, feelings, and how to be a communicative being in the world.

Double-entry accounting was popularized in Europe toward the end of the fifteenth century, and most scholars believe it set the table for the flowering of the Renaissance and the emergence of modern capitalism. What is far less well understood is the why. Why was something as dull as bookkeeping so integral to a complete cultural revolution in Europe? Over nearly seven centuries, “the books” have become something that, in our collective minds, we equate with truth itself—even if only subconsciously. When we doubt a candidate’s claims of wealth, we want to go to his bank records—his personal balance sheet. When a company wants to tap the public markets for capital, they have to open their books to prospective investors. To remain in the market, they need accountants to verify those books regularly. Well-maintained and clear accounting is sacrosanct. The ascendance of bookkeeping to a level equal to truth itself happened over many centuries, and began with the outright hostility European Christendom had to lending before double-entry booking came along. The ancients were pretty comfortable with debt. The Babylonians set the tone in the famous Code of Hammurabi, which offered rules for handling loans, debts, and repayments. The Judeo-Christian tradition, though, had a real ax to grind against the business of lending. “Thou shalt not lend upon usury to thy brother,” Deuteronomy 23:19–20 declares. “In thee have they taken gifts to shed blood; thou hast taken usury and increase, and thou hast greedily gained of thy neighbors by extortion, and hast forgotten me, saith the Lord God,” Ezekiel 22:12 states. As Christianity flourished, this deep anti-usury culture continued for more than a thousand years, a stance that coincided with the Dark Ages, when Europe, having lost the glories of ancient Greece and Rome, also lost nearly all comprehension of math. The only people who really needed the science of numbers were monks trying to figure out the correct dates for Easter.

Quantum computing is not only faster than conventional computing, but its workload obeys a different scaling law—rendering Moore’s Law little more than a quaint memory. Formulated by Intel founder Gordon Moore, Moore’s Law observes that the number of transistors in a device’s integrated circuit doubles approximately every two years. Some early supercomputers ran on around 13,000 transistors; the Xbox One in your living room contains 5 billion. But Intel in recent years has reported that the pace of advancement has slowed, creating tremendous demand for alternative ways to provide faster and faster processing to fuel the growth of AI. The short-term results are innovative accelerators like graphics-processing unit (GPU) farms, tensor-processing unit (TPU) chips, and field-programmable gate arrays (FPGAs) in the cloud. But the dream is a quantum computer. Today we have an urgent need to solve problems that would tie up classical computers for centuries, but that could be solved by a quantum computer in a few minutes or hours. For example, the speed and accuracy with which quantum computing could break today’s highest levels of encryption is mind-boggling. It would take a classical computer 1 billion years to break today’s RSA-2048 encryption, but a quantum computer could crack it in about a hundred seconds, or less than two minutes. Fortunately, quantum computing will also revolutionize classical computing encryption, leading to ever more secure computing. To get there we need three scientific and engineering breakthroughs. The math breakthrough we’re working on is a topological qubit. The superconducting breakthrough we need is a fabrication process to yield thousands of topological qubits that are both highly reliable and stable. The computer science breakthrough we need is new computational methods for programming the quantum computer.

Scott Eastman told me that he “never completely fit in one world.” He grew up in Oregon and competed in math and science contests, but in college he studied English literature and fine arts. He has been a bicycle mechanic, a housepainter, founder of a housepainting company, manager of a multimillion-dollar trust, a photographer, a photography teacher, a lecturer at a Romanian university—in subjects ranging from cultural anthropology to civil rights—and, most unusually, chief adviser to the mayor of Avrig, a small town in the middle of Romania. In that role, he did everything from helping integrate new technologies into the local economy to dealing with the press and participating in negotiations with Chinese business leaders. Eastman narrates his life like a book of fables; each experience comes with a lesson. “I think that housepainting was probably one of the greatest helps,” he told me. It afforded him the chance to interact with a diverse palette of colleagues and clients, from refugees seeking asylum to Silicon Valley billionaires whom he would chat with if he had a long project working on their homes. He described it as fertile ground for collecting perspectives. But housepainting is probably not a singular education for geopolitical prediction. Eastman, like his teammates, is constantly collecting perspectives anywhere he can, always adding to his intellectual range, so any ground is fertile for him.

Statistical malfeasance has very little to do with bad math. Judgement an integrity turn out to be surprisingly important. A detailed knowledge of statistics does not deter wrongdoing any more than a detailed knowledge of the law averts criminal behavior.

Now, of course, having just told you that your inner teacher is indescribable, I’m going to try to describe it. When we think, hear, or understand something that’s deeply true for us, our inner teachers rise in us as a delicious, lucid resonance. When we grasp truth—any truth, from the correct solution to a math problem to the capacity for love—all of our ways of knowing align. We recognize this alignment as our ideal state of being. It feels calm, clear, still, open. That feeling is the inner teacher saying yes.

The body’s reaction to recognizing truth is relaxation, a literal, involuntary release of muscle tension. When we surrender to the truth, even difficult truth, our bodies may go almost limp and we begin breathing more deeply. This may have happened to you when you read the statements at the end of Chapter 1, such as “I don’t know what to do” or “I need help.” When our minds recognize truth, we experience that invisible cartoon light bulb going on in our heads, the feeling of a riddle being solved. “Aha!” we think, or “I get it!” or “Of course!” All the puzzle pieces fit. The math works. Everything makes logical sense. To our heart, the ring of truth feels like a flower opening up. In total integrity, we’re completely available to all emotion: overwhelming love, deep grief, terrible anger, sharp fear. This emotion may be painful, but it doesn’t cause the intense, dull suffering we feel in the dark wood of error. The emotional pain of a hard truth is eased by our soul’s response to aligning with reality. Around and beyond mere emotion, we feel a sense of freedom, a vast openness that includes all aspects of our experience. We connect with an unalterable stillness around and within us. There’s space for pain. There’s space for joy. And the space in which all sensation happens is made up of absolute well-being. It is (we are) a perfect, fertile no-thing-ness in which everything, even pain, has a useful place.