Math Measurement Quotes

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.

When you can measure what you are speaking about, and express it in numbers, you know something about it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarely, in your thoughts advanced to the stage of science.

Music and Dancing, not only give great pleasure but have the honour of depending on Mathematics, for they consist in number and in measure.....Therefore, whatever the old doctors may say, to employ oneself at all this is to be a Philosopher and a Mathematician.

The importance of C.F. Gauss for the development of modern physical theory and especially for the mathematical fundament of the theory of relativity is overwhelming indeed; also his achievement of the system of absolute measurement in the field of electromagnetism. In my opinion it is impossible to achieve a coherent objective picture of the world on the basis of concepts which are taken more or less from inner psychological experience.

Yes. It was Tatiana, not specter but matter. She was measurable. His little Newton had mass and occupied space. A small finite matter in infinite space. That is what math gave him—principles of design that tied together the boundless universe. That is why he measured her. Because she was order.

The probability of an event is the reason we have to believe that it has taken place, or that it will take place. The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible.

I like doing math that involves measuring the lengths of numbers written out on the page (which is really just a way of loosely estimating log10x). It works, but it feels so wrong.

I am sorry that I cannot make it okay. I am sorry that I cannot save you -- but not that sorry. Part of me thinks that your very vulnerability brings you closer to the meaning of life, just as for others, the quest to believe oneself white divides them from it. The fact is that despite their dreams, their lives are also not inviolable. When their own vulnerability becomes real -- when the police decide that tactics for the ghetto should enjoy wider usage, when their armed society shoots down their children, when nature sends hurricanes against their cities -- they are shocked in a way that those of us who were born and bred to understand cause and effect can never be. And I would not have you like them. You have been cast into a race in which the wind is always at your face and the hounds are always at your heels. And to varying degrees this is true of all life. The difference is that you do not have the privilege of living in ignorance of this essential fact. I am speaking to you as I always have -- as the sober and serious man I have always wanted you to be, who does not apologize for his human feelings, who does not make excuses for his height, his long arms, his beautiful smile. You are growing into consciousness, and my wish for you is that you feel no need to constrict yourself to make other people comfortable. None of that can change the math anyway. I never wanted you to be twice as good as them, so much as I have always wanted you to attack every day of your brief bright life in struggle. The people who must believe they are white can never be your measuring stick. I would not have you descend into your own dream. I would have you be a conscious citizen of this terrible and beautiful world.

Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there. [Having himself spent a lifetime unsuccessfully trying to prove Euclid's postulate that parallel lines do not meet, Farkas discouraged his son János from any further attempt.]

When you fear you will confirm a negative stereotype, it can become a self-fulfilling prophecy not because the stereotype is true, but because you can't stop worrying that you could become an example proving it. This self-fulfilling prophecy, being only a matter of perception, can be easily sublimated. Another study by Steele measured the math abilities of men versus women. When the questions were easy, the women and men performed the same. When they were difficult, the women's scores plummeted lower than did those of their male peers. When they ran the tests again with new participants, but this time before handing out the problems told the subjects that men and women tended to perform equally on the exam, the scores leveled out. The women performed just as well as did the men. The power of the stereotype--women are bad at math--was nullified.

Before an experiment can be performed, it must be planned—the question to nature must be formulated before being posed. Before the result of a measurement can be used, it must be interpreted—nature's answer must be understood properly. These two tasks are those of the theorist, who finds himself always more and more dependent on the tools of abstract mathematics. Of course, this does not mean that the experimenter does not also engage in theoretical deliberations. The foremost classical example of a major achievement produced by such a division of labor is the creation of spectrum analysis by the joint efforts of Robert Bunsen, the experimenter, and Gustav Kirchhoff, the theorist. Since then, spectrum analysis has been continually developing and bearing ever richer fruit.

Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems—general and specific statements—can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.

For as long as we can remember,” [Griff] said, “I mean literally our whole conscious lives, time has been neatly divided into semesters and years. Each year completely distinguishable and unique. First grade, third grade. We didn’t measure by age, we measured by grade. Like I know I broke my arm in sixth grade but I’d have to do the math to figure out what year that was, or how old I was.

A distinguished writer [Siméon Denis Poisson] has thus stated the fundamental definitions of the science: 'The probability of an event is the reason we have to believe that it has taken place, or that it will take place.' 'The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen). From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.

Anything you try to quantify can be divided into any number of "anythings," or become the thing - the unit - itself. And what is any number, itself, but just another unit of measurement? What is a 'six' but two 'threes', or three 'twos'...half a 'twelve', or just six 'ones' - which are what? (attrib: F.L. Vanderson)

You are growing into consciousness, and my wish for you is that you feel no need to constrict yourself to make other people comfortable. None of that can change the math anyway. I never wanted you to be twice as good as them, so much as I have always wanted you to attack every day of your brief bright life in struggle. The people who must believe they are white can never be your measuring stick. I would not have you descend into your own dream. I would have you be a conscious citizen of this terrible and beautiful world. —

Your role is to be in your life, not cut and measure it. If we are always doing the math, we are never part of the equation.

Going through old papers I came across the transcript of a university debate on Rublyov. God, what a level. Abysmal, pathetic. But there is one remarkable contribution by a maths professor called Manin, Lenin Prize winner, who can hardly be more than thirty. I share his views. Not that one should say that about oneself. But it's exactly what I felt when I was making Andrey. And I'm grateful to Manin for that. "Almost every speaker has asked why they have to be made to suffer all through the three hours of the film. I'll try to reply to that question. It is because the twentieth century has seen the rise of a kind of emotional inflation. When we read in a newspaper that two million people have been butchered in Indonesia, it makes as much impression on us as an account of our hockey team winning a match. The same degree of impression! We fail to notice the monstrous discrpancy between these two events. The channels of our perception have been smoothed out to the point where we are no longer aware. However, I don't want to preach about this. It may be that without it life would be impossible. Only the point is that there are some artists who do make us feel the true measure of things. It is a burden which they carry throughout their lives, and we must be thankful to them.

Things happen. Things that physics and math and crap that gets measured in a lab can't explain. People aren't just laws and rules, Claire. They're... sparks. Sparks of something beautiful and huge. And some sparks glow brighter

We're not doing maths again?' Harriet lamented. Lord Winstead looked at Anne with unconcealed curiosity. 'Mathematics? On Rotten Row?' 'We have been studying measurement,' she informed him. 'They have already measured the average length of their strides. Now they will count their steps and compute the length of the path.' 'Very nice,' he said approvingly. 'And it keeps them busy and quiet as they count.' 'You have not heard them count,' Anne told him.

In my opinion, defining intelligence is much like defining beauty, and I don’t mean that it’s in the eye of the beholder. To illustrate, let’s say that you are the only beholder, and your word is final. Would you be able to choose the 1000 most beautiful women in the country? And if that sounds impossible, consider this: Say you’re now looking at your picks. Could you compare them to each other and say which one is more beautiful? For example, who is more beautiful— Katie Holmes or Angelina Jolie? How about Angelina Jolie or Catherine Zeta-Jones? I think intelligence is like this. So many factors are involved that attempts to measure it are useless. Not that IQ tests are useless. Far from it. Good tests work: They measure a variety of mental abilities, and the best tests do it well. But they don’t measure intelligence itself.

The solution to a math problem is not a number: it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying.

{Before the time of Benjamin Peirce it never occurred to anyone that mathematical research} was one of the things for which a mathematical department existed. Today it is a commonplace in all the leading universities. Peirce stood alone—a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.

There have been many authorities who have asserted that the basis of science lies in counting or measuring, i.e. in the use of mathematics. Neither counting nor measuring can however be the most fundamental processes in our study of the material universe—before you can do either to any purpose you must first select what you propose to count or measure, which presupposes a classification.

NAEP is central to any discussion of whether American students and the public schools they attend are doing well or badly. It has measured reading and math and other subjects over time. It is administered to samples of students; no one knows who will take it, no one can prepare to take it, no one takes the whole test. There are no stakes attached to NAEP; no student ever gets a test score.

Maybe you’ve noticed what I’ve noticed, and thought it strange, or dismissed it as youthful foolishness or that you were missing some critical piece of information that would reveal itself with age and wisdom – that is: every single teacher believes feverishly in the importance of the content of their class, and furthermore, believes that their assessment of you in their class is a direct measure of your capacity for future success, while simultaneously not having a clue as to the content of virtually any other discipline in the school. They will boldly state things like, That’s math, I’m an English teacher or That’s literature, I’m a biology teacher, practically admitting out loud that nothing learned in school is important (except, of course, the course they are teaching).

In 2005, a survey was conducted in thirty-four countries measuring the percentage of adults who accept evolution. The United States ranked thirty-third, just above Turkey. Meanwhile, high school students in the United States test below those of every European and Asian nation in their understanding of science and math. These data are unequivocal: we are building a civilization of ignorance.

Among other things, HeartMath research tests theories about the electromagnetic field of the human heart using machines that measure faint magnetic fields, such as those that are often used in MRIs and cardiologic tests. Remarkably, the heart’s toroidally shaped electrical field is sixty times greater than that of the brain, and its magnetic field is 5,000 times greater than that of the brain. The heart generates the strongest electromagnetic field in the body, and its pumping action transmits powerful rhythmic information patterns containing neurological, hormonal, and electromagnetic data to the brain and throughout the rest of the body. The heart actually sends more information to the brain than the brain sends to the heart. In other words, the heart has a mind of its own. Studies reveal this electromagnetic field seems to pick up information in the surrounding environment and also broadcasts one’s emotional state out from the body. Their measurements reveal that the field is large enough to extend several feet (or more) outside our bodies. Positive moods such as gratitude, joy, and happiness correlate to a larger, more expanded heart field, while emotions such as greed, anger, or sadness correlate to a constricted heart field.

The Measure of America, a report of the Social Science Research Council, ranks every state in the United States on its “human development.” Each rank is based on life expectancy, school enrollment, educational degree attainment, and median personal earnings. Out of the 50 states, Louisiana ranked 49th and in overall health ranked last. According to the 2015 National Report Card, Louisiana ranked 48th out of 50 in eighth-grade reading and 49th out of 50 in eighth-grade math. Only eight out of ten Louisianans have graduated from high school, and only 7 percent have graduate or professional degrees. According to the Kids Count Data Book, compiled by the Annie E. Casey Foundation, Louisiana ranked 49th out of 50 states for child well-being. And the problem transcends race; an average black in Maryland lives four years longer, earns twice as much, and is twice as likely to have a college degree as a black in Louisiana. And whites in Louisiana are worse off than whites in Maryland or anywhere else outside Mississippi. Louisiana has suffered many environmental problems too: there are nearly 400 miles of low, flat, subsiding coastline, and the state loses a football field–size patch of wetland every hour. It is threatened by rising sea levels and severe hurricanes, which the world’s top scientists connect to climate change.

Contrary to the earlier belief that a steady heart rate was an indicator of health, we now know that a loss of the naturally occurring variability in heart rate is actually a sign of disease and a strong predictor of future health problems. [23] Because heart rate variability declines as we get older, it’s one way to measure our physiological aging. [24] In essence, HRV is a measure of the flexibility of our heart and nervous system, and as such reflects our health and fitness.

Let us wonder, Carole, at the genius of hyperbolic geometry, where the sum of the angles adds up to tales than 180 degrees Let us wonder at how the ancient Egyptians worked out how to measure an irregularly-shaped field Let us wonder how X was just a rare letter until algebra came along and made it something special that can be unraveled to reveal its inner value You see, maths is a process of discovery, Carole, it is like the exploration of space, the planets were always there, it just took us a long time to find them

The distance between my lips and yours cannot be deciphered from the square root of the sum of the days we have spent wondering what to do with three minutes and ten seconds. The distance between my lips and yours cannot be deduced by the difference in the circumferences of our necks or in how many minutes we can sit in the noon sun. The distance between my lips and yours can only be measured in poems. Tell me, how many are there? Were there? Will there be? (But who knows what to call a poem and what to call a conversation? And who knows whether to call at all?)

If you didn’t find some way of stopping it, people would go on asking questions. The teachers were useful there. Bands of them wandered through the mountains, along with the tinkers, portable blacksmiths, miracle medicine men, cloth peddlers, fortune-tellers, and all the other travelers who sold things the people didn’t need every day but occasionally found useful. They went from village to village delivering short lessons on many subjects. They kept apart from the other travelers and were quite mysterious in their ragged robes and strange square hats. They used long words, like corrugated iron. They lived rough lives, surviving on what food they could earn from giving lessons to anyone who would listen. When no one would listen, they lived on baked hedgehog. They went to sleep under the stars, which the math teachers would count, the astronomy teachers would measure, and the literature teachers would name. The geography teachers got lost in the woodsand fell into bear traps. People were usually quite pleased to see them. They taught children enough to shut them up, which was the main thing, after all. But they always had to be driven out of the villages by nightfall in case they stole chickens.

I suddenly remember being about seven, riding beside him in the car, and asking him how grown-ups found their way to places. After all, I had never seen him pull out a map. "I guess we just get used to taking the same turns," he said, but I wasn't satisfied. "Then what about the first time you go somewhere?" "Well," he said, "we get directions." But what I want to know is who got them the very first time? What if no one's ever been where you're going? "Dad?" I ask, "is it true that you can use stars like a map?" "Yeah, if you understand celestial navigation." "Is it hard?" I'm thinking maybe I should learn. A backup plan, for all those times I feel like I'm just wandering in circles. "It's pretty jazzy math—you have to measure the altitude of a star, figure out its position using a nautical almanac, figure out what you think the altitude should be and what direction the star should be in based on where you think you are, and compare the altitude you measured with the one you calculated. Then you plot this on a chart, as a line of position. You get several lines of position to cross, and that's where you go." My father takes one look at my face and smiles. "Exactly," he laughs. "Never leave home without your GPS.

Watching violent, arousing shows may actually contribute to suppressing your immune system. As you identify with the anger you see on the screen or read about, stress chemicals called catecholamine and cortisone are released that can adversely affect your immune system. The effect of exposure to both anger and love on the immune system was shown in research by Harvard scientist David Mclelland, and later reproduced by the Heart Math Institute in California (Bhat 1995). Watching an anger-provoking movie suppressed the immune system (as measured by chemicals in the saliva) for five to six hours in study subjects. However, watching a movie about the compassionate work of Mother Teresa caused elevation of the immune level in the participants.

I am speaking to you as I always have—as the sober and serious man I have always wanted you to be, who does not apologize for his human feelings, who does not make excuses for his height, his long arms, his beautiful smile. You are growing into consciousness, and my wish for you is that you feel no need to constrict yourself to make other people comfortable. None of that can change the math anyway. I never wanted you to be twice as good as them, so much as I have always wanted you to attack every day of your brief bright life in struggle. The people who must believe they are white can never be your measuring stick. I would not have you descend into your own dream. I would have you be a conscious citizen of this terrible and beautiful world.

When the subjects arrived at the psychology lab, they were sent into individual dressing rooms with full-length mirrors. Half of the dressing rooms contained bathing suits (one-piece for the women, trunks for the men) and half contained sweaters, all of which were available in a wide range of sizes. Once the subjects put on the assigned clothing, they were told to hang out in the dressing room for fifteen minutes before they filled out a questionnaire about whether or not they would want to purchase the item. While they waited, they were asked, in order to help the researchers use the time efficiently, to complete a math test “for an experimenter in the Department of Education.” As you’ve already guessed, the psychologists weren’t helping their colleagues in the Department of Education. They were measuring whether taking a math test while wearing a bathing suit would affect the women’s scores.

Even as our world is being daily transformed by breathtaking innovations in science and technology, many people continue to imagine that math and science are mostly a matter of memorizing formulas to get “the right answer.” Even engineering, which is in fact the process of creating something from scratch or putting things together in novel and non-self-evident ways, is perplexingly viewed as a mechanical or rote subject. This viewpoint, frankly, could only be held by people who never truly learned math or science, who are stubbornly installed on one side of the so-called Two Culture divide. The truth is that anything significant that happens in math, science, or engineering is the result of heightened intuition and creativity. This is art by another name, and it’s something that tests are not very good at identifying or measuring. The skills and knowledge that tests can measure are merely warm-up exercises.

I have both some good news and some bad news for you: there is little that is unique or special about your problems. That’s why letting go is so liberating. There’s a kind of self-absorption that comes with fear based on an irrational certainty. When you assume that your plane is the one that’s going to crash, or that your project idea is the stupid one everyone is going to laugh at, or that you’re the one everyone is going to choose to mock or ignore, you’re implicitly telling yourself, “I’m the exception; I’m unlike everybody else; I’m different and special.” This is narcissism, pure and simple. You feel as though your problems deserve to be treated differently, that your problems have some unique math to them that doesn’t obey the laws of the physical universe. My recommendation: don’t be special; don’t be unique. Redefine your metrics in mundane and broad ways. Choose to measure yourself not as a rising star or an undiscovered genius. Choose to measure yourself not as some horrible victim or dismal failure. Instead, measure yourself by more mundane identities: a student, a partner, a friend, a creator. The narrower and rarer the identity you choose for yourself, the more everything will seem to threaten you. For that reason, define yourself in the simplest and most ordinary ways possible. This often means giving up some grandiose ideas about yourself: that you’re uniquely intelligent, or spectacularly talented, or intimidatingly attractive, or especially victimized in ways other people could never imagine. This means giving up your sense of entitlement and your belief that you’re somehow owed something by this world. This means giving up the supply of emotional highs that you’ve been sustaining yourself on for years. Like a junkie giving up the needle, you’re going to go through withdrawal when you start giving these things up. But you’ll come out the other side so much better.

Our two taco specials get shoved up on the serving counter, crispy, cheesy goodness in brown plastic baskets lined with parchment paper, sour cream and guacamole exactly where they should be. On the side. There is a perfect ratio of sour cream, guac, and salsa on a shredded chicken tostada. No one can make it happen for you. Many restaurants have tried. All have failed. Only the mouth knows its own pleasure, and calibration like Taco Heaven cannot be mass produced. It simply cannot. Taco Heaven is a sensory explosion of flavor that defies logic. First, you have to eye the amount of spiced meat, shredded lettuce, chopped tomatoes, and tomatillos. You must consider the size and crispiness of the shells. Some people–I call them blasphemers–like soft tacos. I am sitting across from Exhibit A. We won’t talk about soft tacos. They don’t make it to Taco Heaven. People who eat soft tacos live in Taco Purgatory, never fully understanding their moral failings, repeating the same mistakes again and again for all eternity. Like Perky and dating. Once you inventory your meat, lettuce, tomato, and shell quality, the real construction begins. Making your way to Taco Heaven is like a mechanical engineer building a bridge in your mouth. Measurements must be exact. Payloads are all about formulas and precision. One miscalculation and it all fails. Taco Death is worse than Taco Purgatory, because the only reason for Taco Death is miscalculation. And that’s all on you. “Oh, God,” Fiona groans through a mouthful of abomination. “You’re doing it, aren’t you?” “Doing what?” I ask primly, knowing damn well what she’s talking about. “You treat eating tacos like you’re the star of some Mythbusters show.” “Do not.” “Do too.” “Even if I do–and I am notconceding the point–it would be a worthwhile venture.” “You are as weird about your tacos as Perky is about her coffee.” “Take it back! I am not that weird.” “You are.” “Am not.” “This is why Perky and I swore we would never come here with you again.” Fiona grabs my guacamole and smears the rounded scoop all over the outside of her soft taco. I shriek. “How can you do that?” I gasp, the murder of the perfect ratio a painful, almost palpable blow. The mashed avocado has a death rattle that rings in my ears. Smug, tight lips give me a grimace. “See? A normal person would shout, ‘Hey! That’s mine!’ but you’re more offended that I’ve desecrated my inferior taco wrapping with the wrong amount of guac.” “Because it’s wrong.” “You should have gone to MIT, Mal. You need a job that involves nothing but pure math for the sake of calculating stupid shit no one else cares about.” “So glad to know that a preschool teacher holds such high regard for math,” I snark back. And MIT didn’t give me the kind of merit aid package I got from Brown, I don’t add. “Was that supposed to sting?” She takes the rest of my guacamole, grabs a spoon, and starts eating it straight out of the little white paper scoop container thing. “How can you do that? It’s like people who dip their french fries in mayonnaise.” I shudder, standing to get in line to buy more guac. “I dip my french fries in mayo!” “More evidence of your madness, Fi. Get help now. It may not be too late.” I stick my finger in her face. “And by the way, you and Perky talk about my taco habits behind my back? Some friends!” I hmph and turn toward the counter.

When I threw the stick at Jamie, I hadn't intended to hit him with it. But the moment it left my hand, I knew that's what was going to happen. I didn't yet know any calculus or geometry, but I was able to plot, with some degree of certainty, the trajectory of that stick. The initial velocity, the acceleration, the impact. The mathematical likelihood of Jamie's bloody cheek. It had good weight and heft, that stick. It felt nice to throw. And it looked damn fine in the overcast sky, too, flying end over end, spinning like a heavy, two-pronged pinwheel and (finally, indifferently, like math) connecting with Jamie's face. Jamie's older sister took me by the arm and she shook me. Why did you do that? What were you thinking? The anger I saw in her eyes. Heard in her voice. The kid I became to her then, who was not the kid I thought I was. The burdensome regret. I knew the word "accident" was wrong, but I used it anyway. If you throw a baseball at a wall and it goes through a window, that is an accident. If you throw a stick directly at your friend and it hits your friend in the face, that is something else. My throw had been something of a lob and there had been a good distance between us. There had been ample time for Jamie to move, but he hadn't moved. There had been time for him to lift a hand and protect his face from the stick, but he hadn't done that either. He just stood impotent and watched it hit him. And it made me angry: That he hadn't tried harder at a defense. That he hadn't made any effort to protect himself from me. What was I thinking? What was he thinking? I am not a kid who throws sticks at his friends. But sometimes, that's who I've been. And when I've been that kid, it's like I'm watching myself act in a movie, reciting somebody else's damaging lines. Like this morning, over breakfast. Your eyes asking mine to forget last night's exchange. You were holding your favorite tea mug. I don't remember what we were fighting about. It doesn't seem to matter any more. The words that came out of my mouth then, deliberate and measured, temporarily satisfying to throw at the bored space between us. The slow, beautiful arc. The spin and the calculated impact. The downward turn of your face. The heavy drop in my chest. The word "accident" was wrong. I used it anyway.

T'he dca& id, a bunch of ruunbeha,that mean nothing and everything. That pesky, dangerous box is a booby trap full of good and bad news math. I hate math, I hate the scale. Back in the bun-head ballerina days I would measure myself constantly and then punish or praise accordingly. On punishment days I'd yack. Tis true, tis pathetic, tis my history; a golden nugget that makes me wise and human. However, that box is just that, a small container to cage ourselves. Prison is not sexy. No matter how it looks in the movies, prison is not a place to voluntarily visit.

During the 1919 solar eclipse, people go out to measure the positions of the stars and they find exactly what Einstein predicted. Einstein gets a telegram saying this, and somebody asked him, Professor Einstein, what would you have said if the observations didn’t agree with what your prediction of general relativity said should be happening? And Einstein said, “I’d be sorry for the dear lord; the theory is correct.” What he meant by that is the math is just so elegant, so beautiful, so powerful, that almost seemingly it can’t possibly be wrong.

The point is that there’s something very mathematical about our Universe, and that the more carefully we look, the more math we seem to find. Apropos constants of nature, there are hundreds of thousands of pure numbers that have been measured across all areas of physics, ranging from ratios of masses of elementary particles to ratios of characteristic wavelengths of light emitted by different molecules, and using sufficiently powerful computers to solve the equations describing the laws of nature,

One thing we did was we had 2 titanium cups, and one large measuring jug (blue) and a then end of the day we would elect the ‘man of the day’ and they would have the jug for tea and breakfast. I know that although bigger, the jug only held the same amount of tea as the smaller cups but it always seemed to raise moral. What’s this got to do with food? Nothing, but never lose sight of the psychological boost that food and drink plays on a team, instead focusing only on the raw math of the calories.

context. Take 5 heel-to-toe steps in each of the four directions from a center point. From one “corner” to the next will be 7 heel-to-toe steps, and when you connect those four corners with straight lines, the resulting square will measure 28 footsteps around the edges.1 (See Figure 3-1.) Whether used as a square or curved outward into a circle, 28 reminds us of the Moon phases, a worthy underpinning to any magical space.

Personal measurements provide an opportunity to be wildly creative. If you’re doing a spell for mental clarity, find a way to incorporate your head measurement. If your spellwork is aimed at expressing your feelings more clearly, try incorporating the distance from your heart to your mouth. Talismans, charms, garments, and tools: When we personalize these items, we imbue them with powerful ties to our own imaginations, associations, bodies, and beliefs.

For example, if healthy 30-year-olds are sleep deprived for six days (averaging, in this study, about four hours of sleep per night), parts of their body chemistry soon revert to that of a 60-year-old. And if they are allowed to recover, it will take them almost a week to get back to their 30-year-old systems. Taken together, these studies show that sleep loss cripples thinking in just about every way you can measure thinking. Sleep loss hurts attention, executive function, working memory, mood, quantitative skills, logical reasoning ability, general math knowledge. Eventually, sleep loss affects manual dexterity, including fine motor control, and even gross motor movements, such as the ability to walk on a treadmill.

7 things every kid should master A noted Williams College psychologist argues standardized tests are useful, if they measure the abilities students really need. By Susan Engel | 2458 words In the past few years, parents, teachers, and policy makers have furiously debated whether standardized tests should be used to promote or hold back children, fire teachers, and withhold funds from schools. The debate has focused for the most part on whether the tests are being used in unfair ways. But almost no one has publicly questioned a fundamental assumption — that the tests measure something meaningful or predict something significant beyond themselves. I have reviewed more than 300 studies of K–12 academic tests. What I have discovered is startling. Most tests used to evaluate students, teachers, and school districts predict almost nothing except the likelihood of achieving similar scores on subsequent tests. I have found virtually no research demonstrating a relationship between those tests and measures of thinking or life outcomes. When you hear people debate the use of tests in schools, the talk usually assumes that the only alternative to the current approach is no testing at all. But nothing could be further from the truth. Ideally, everyone would benefit from objective measures of children’s learning in schools. The answer is not to abandon testing, but to measure the things we most value, and find good ways to do that. How silly to measure a child’s ability to parse a sentence or solve certain kinds of math problems if in fact those measures don’t predict anything important about the child or lead to better teaching practices. Why not test the things we value, and test them in a way that provides us with an accurate picture of what children really do, not what they can do under the most constrained circumstances after the most constrained test preparation? Nor should this be very difficult. After all, in the past 50 years economists and psychologists have found ways to measure things as subtle and dynamic as the mechanisms that explain when and why we give in to impulse, the forces that govern our moral choices, and the thought processes that underlie unconscious stereotyping.

France had recently switched to the metric system of measurement. This gave scientists a much-needed standardized system to measure and compare results, but it also required a whole new set of calculating tables. The sheer number of calculations was beyond what could be accomplished by all the mathematicians in France, so Riche established calculating 'factories' to manufacture logarithms the same way workers manufactured mercantile goods. Each factory employed between 60 and 80 human 'computers.' But they weren’t trained mathematicians; they were mostly out-of-work hairdressers who had found their skill at constructing elaborate pompadours for aristocrats much less in demand after so many former clients lost their heads at the height of the French Revolution. Riche had hit upon a rote system of compiling results based on a set of given values and formulas, and the workers just cranked out the answers in what must have been the world’s first mathematical assembly line. Babbage figured that if an army of untrained hairdressers could make the calculations, so could a computing 'engine.

One's most essential attributes cannot be defined by math, but instead, by opinion; and those that can be measured by a number, are less significant.

She said she learned it in three weeks—but somebody else told me she really did it in two.” These feats are routine at Meadowmount, in part because the teachers take the idea of chunking to its extreme. Students scissor each measure of their sheet music into horizontal strips, which are stuffed into envelopes and pulled out in random order. They go on to break those strips into smaller fragments by altering rhythms. For instance, they will play a difficult passage in dotted rhythm (the horses' hooves sound—da-dum, da-dum). This technique forces the player to quickly link two of the notes in a series, then grants them a beat of rest before the next two-note link. The goal is always the same: to break a skill into its component pieces (circuits), memorize those pieces individually, then link them together in progressively larger groupings (new, interconnected circuits).

Fibonacci’s new numbering system became a hit with the merchant class and for centuries was the preeminent source for mathematical knowledge in Europe. But something equally important also happened around this time: Europeans learned of double-entry bookkeeping, picking it up from the Arabians, who’d been using it since the seventh century. Merchants in Florence and other Italian cities began applying these new accounting measures to their daily businesses. Where Fibonacci gave them new measurement methods for business, double-entry accounting gave them a way to record it all. Then came a seminal moment: in 1494, two years after Christopher Columbus first set foot in the Americas, a Franciscan friar named Luca Pacioli wrote the first comprehensive manual for using this accounting system. Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalita, written in Italian rather than Latin so as to be more accessible to the public, would become the first popular work on math and accounting. Its section on accounting was so well received that the publisher eventually published it as its own volume. Pacioli offered access to the precision of mathematics. “Without double entry, businessmen would not sleep easily at night,” Pacioli wrote, mixing in the practical with the technical—Pacioli’s Summa would become a kind of self-help book for the merchant class.

Kill Yourself Buddhism argues that your idea of who “you” are is an arbitrary mental construction and that you should let go of the idea that “you” exist at all; that the arbitrary metrics by which you define yourself actually trap you, and thus you’re better off letting go of everything. In a sense, you could say that Buddhism encourages you to not give a fuck. It sounds wonky, but there are some psychological benefits to this approach to life. When we let go of the stories we tell about ourselves, to ourselves, we free ourselves up to actually act (and fail) and grow. When someone admits to herself, “You know, maybe I’m not good at relationships,” then she is suddenly free to act and end her bad marriage. She has no identity to protect by staying in a miserable, crappy marriage just to prove something to herself. When the student admits to himself, “You know, maybe I’m not a rebel; maybe I’m just scared,” then he’s free to be ambitious again. He has no reason to feel threatened by pursuing his academic dreams and maybe failing. When the insurance adjuster admits to himself, “You know, maybe there’s nothing unique or special about my dreams or my job,” then he’s free to give that screenplay an honest go and see what happens. I have both some good news and some bad news for you: there is little that is unique or special about your problems. That’s why letting go is so liberating. There’s a kind of self-absorption that comes with fear based on an irrational certainty. When you assume that your plane is the one that’s going to crash, or that your project idea is the stupid one everyone is going to laugh at, or that you’re the one everyone is going to choose to mock or ignore, you’re implicitly telling yourself, “I’m the exception; I’m unlike everybody else; I’m different and special.” This is narcissism, pure and simple. You feel as though your problems deserve to be treated differently, that your problems have some unique math to them that doesn’t obey the laws of the physical universe. My recommendation: don’t be special; don’t be unique. Redefine your metrics in mundane and broad ways. Choose to measure yourself not as a rising star or an undiscovered genius. Choose to measure yourself not as some horrible victim or dismal failure. Instead, measure yourself by more mundane identities: a student, a partner, a friend, a creator. The narrower and rarer the identity you choose for yourself, the more everything will seem to threaten you. For that reason, define yourself in the simplest and most ordinary ways possible. This often means giving up some grandiose ideas about yourself: that you’re uniquely intelligent, or spectacularly talented, or intimidatingly attractive, or especially victimized in ways other people could never imagine.

Government alone cannot restore the economy to health. Innovation is a primary driver of economic growth. One way of measuring inventive creativeness is through patent applications. Chetty, along with Alex Bell, Xavier Jaravel, Neviana Petkova, and John Van Reenen, studied the childhoods of more than a million patent holders, linking family income with elementary test scores and other key factors. Children at the top of their third-grade math class were the most likely to become inventors—but only if they also came from a high-income family. High-scoring children who were from low-income or minority families were no more likely to become inventors than affluent children with mediocre scores. Successful inventors were also less likely to be women, Black, Latino, or from the Southeast. Chetty called these failed inventors the “lost Einsteins.” “If women, minorities, and children from low-income families were to invent at the same rate as white men from high-income (top 20%) families, the rate of innovation in America would quadruple,” the authors said. The most ominous finding by Chetty and his colleagues was the effect of Covid-19 on educational progress. Using a popular math program called Zearn, the economists plotted the achievement of children from upper-income families versus those from lower incomes. When schools shut down and instruction switched to remote learning, children in the upper-income tier suffered a small drop in the lessons completed, but low-income children fell in a hole—a 60 percent drop in the rate of progress in learning math. The long-term economic prospects for those children are dire. “We’re likely to see further erosion of social mobility the longer this lasts,” Chetty said. The American dream was drifting farther out of reach for another generation.

What Dodger is doing isn’t math in the truest sense of the word, and at the same time, it’s a deeper, truer arithmetic than she’s done in years. It is the instinctive math of children weighing parental disapproval against the rapidly setting sun, and the heartbroken math of sailors measuring the holes in their boat against the distance to the shore. This is the math that moves the universe, the measure and countermeasure that dances over and around the numbers.

Humanity has always regarded math as a weird, unreal abstraction. Only a few geniuses here and there have realised otherwise. Mathematics is the most concrete thing of all – eternally concrete. Thoughts are numbers. What could be more bizarre to an ordinary human mind? Reality is just an enormous numerical system, relentlessly solving and optimising itself. Numbers = frequencies = energies. Why is math so good at describing the world? Because the world is mathematical! Duh! Why are numbers exactly the right things to measure the world? Because the world is made of numbers! As soon as you realise that thoughts, numbers, life, energy, motion, frequencies and waves are all ways of talking about the same thing, it’s obvious why mathematics is the explanation for everything.

Our decisions, Clyde Coombs thought, might be treated as a collection of judgments about the similarity between two things: the ideal in our head, and the object on offer. Amos was as fascinated as Coombs by questions of how to measure what couldn’t be observed (so interested that he taught himself the math he needed to do it). Years later, he also saw that the attempt to measure these preferences raised another question. If you were going to take as your (possibly unrealistic) working assumption the proposition that people made choices by comparing some ideal in their head and the real-world versions, you had to know how people made such judgments.

It's times like this when the age of the universe becomes a useful unit of measurement: 64-bit Unix time will last until twenty-one times the age of the universe from now - until (assuming we don't manage another upgrade in the meantime) December 4 in the year 292,277,026,596 CE, when all the computers will go down. On a Sunday.

Your role is to be in your life, not cut an measure it. If we are always doing the math, we are never part of the equation.

The customary units of measurement? They have practical applications, but learning them is mostly done through rote memorization— There aren’t any proofs or critical evaluations regarding REASONS why a foot equals twelve inches There are only facts to memorize, but not puzzles to solve, With algebra—YES! There isn’t just memorization— There is the rearrangement of jigsaw pieces— jigsaw pieces that you can solve, not just memorize!

What matters is that we're here for each other. True love which grows over time may not just be a psychological cognitive experience. There's something deeper in shared going on. There's a growing scientific literature on the physiological and energetic interactions between people. Rollin McCraty, HeartMaths director of research, has done work in this area. He first measured the electromagnetic field around individuals in different mood States and demonstrated that there are clear, reproducible patterns of electromagnetic energy that appear with different moods. If a person is angry or hostile, there's one clear pattern. If the same person is in a loving state, a very different pattern of electromagnetic energy is created. Is it possible that we can detect these changes radiating from another person and respond to this energy? He has every reason to believe that this is possible.

We intuitively know that the heart is the center of love and empathy, and studies are showing this to be true. In fact, empathy manifests in the electromagnetic field (EMF), which is generated by the heart in amounts greater than anywhere else in the body. The heart’s EMF emits fifty thousand femtoteslas (a measure of EMF), in contrast to the ten generated by the brain.37 Other research shows that when separated from the magnetic field, the heart’s electrical field is sixty times greater in amplitude than the brain’s field.38 Through this field, a person’s nervous system tunes in to and responds to the magnetic fields produced by the hearts of other people.39 The heart’s field is therefore one of the means by which a practitioner affects patients. This effect leads to the question, What do you want to share? To generate positive outcomes for a patient, a practitioner must hold positive feelings in his or her own heart. Not only does good will profit the client, but it also benefits the practitioner as a person. A set of studies by researcher Dr. Rollin McCraty of the HeartMath Institute in California, and described in his e-book, The Energetic Heart, helps explain the importance of positive energy.40 For decades, scientists have known that information is encoded in the nervous system in the time intervals between activities or in the pattern of electrical activity. Recent studies also reveal that information is captured in hormone pulses. Moreover, there is a hormone pulse that coincides with heart rhythms, which means that information is also shared in the interbeat intervals of the pressure and electromagnetic waves produced by the heart. Negative emotions such as anger, frustration, or anxiety disturb the heart rhythm. Positive emotions such as appreciation, love, or compassion produce coherent or functional patterns. Feelings, distributed throughout the body, produce chemical changes within the entire system. Do you want to be a healthy person? Be sincerely positive as often as you can. You thus “increase the probability of maintaining coherence and reducing stress, even during challenging situations.”41 What you as a practitioner believe will be shared—everywhere and with everyone you meet.

Other research establishes the ability of one person to affect another through these fields. For instance, studies at the Institute of HeartMath in California have shown that one person’s electrocardiograph (heart) signal can be registered in another person’s electroencephalogram (EEG, measuring brain activity) and elsewhere on the other person’s body. An individual’s cardiac signal can also be registered in another’s EEG recording when two people sit quietly opposite one another.89 This interconnectivity of fields and intention is a marriage of subtle energy theory and quantum physics. As Dr. Benor pointed out, Albert Einstein has already proven that matter and energy are interchangeable. For centuries, healers have been reporting the existence of interpenetrating, subtle energy fields around the physical body. Hierarchical in organization (and vibration), these fields affect every aspect of the human being.90 Studies show that healing states invoke at least the subtle biomagnetic fields. For example, one study employed a magnetometer to quantify biomagnetic fields coming from the hands of meditators and yoga and Qigong practitioners. These fields were a thousand times stronger than the strongest human biomagnetic field and were located in the same range as those being used in medical research labs for speeding the healing of biological tissues—even wounds that had not healed in forty years.91 Yet another study involving a superconducting quantum interference device (SQUID) showcased large frequency-pulsing biomagnetic fields emanating from the hands of therapeutic touch professionals during treatments

Flower of life: A figure composed of evenly-spaced, overlapping circles creating a flower-like pattern. Images of the Platonic solids and other sacred geometrical figures can be discerned within its pattern. FIGURE 3.14 FLOWER OF LIFE The Platonic solids: Five three-dimensional solid shapes, each containing all congruent angles and sides. If circumscribed with a sphere, all vertices would touch the edge of that sphere. Linked by Plato to the four primary elements and heaven. FIGURE 3.15 PENTACHORON The applications of these shapes to music are important to sound healing theory. The ancients have always professed a belief in the “music of the spheres,” a vibrational ordering to the universe. Pythagorus is famous for interconnecting geometry and math to music. He determined that stopping a string halfway along its length created an octave; a ratio of three to two resulted in a fifth; and a ratio of four to three produced a fourth. These ratios were seen as forming harmonics that could restore a disharmonic body—or heal. Hans Jenny furthered this work through the study of cymatics, discussed later in this chapter, and the contemporary sound healer and author Jonathan Goldman considers the proportions of the body to relate to the golden mean, with ratios in relation to the major sixth (3:5) and the minor sixth (5:8).100 Geometry also seems to serve as an “interdimensional glue,” according to a relatively new theory called causal dynamical triangulation (CDT), which portrays the walls of time—and of the different dimensions—as triangulated. According to CDT, time-space is divided into tiny triangulated pieces, with the building block being a pentachoron. A pentachoron is made of five tetrahedral cells and a triangle combined with a tetrahedron. Each simple, triangulated piece is geometrically flat, but they are “glued together” to create curved time-spaces. This theory allows the transfer of energy from one dimension to another, but unlike many other time-space theories, this one makes certain that a cause precedes an event and also showcases the geometric nature of reality.101 The creation of geometry figures at macro- and microlevels can perhaps be explained by the notion called spin, first introduced in Chapter 1. Everything spins, the term spin describing the rotation of an object or particle around its own axis. Orbital spin references the spinning of an object around another object, such as the moon around the earth. Both types of spin are measured by angular momentum, a combination of mass, the distance from the center of travel, and speed. Spinning particles create forms where they “touch” in space.

What Dodger is doing isn’t math in the truest sense of the word, and at the same time, it’s a deeper, truer arithmetic than she’s done in years. It is the instinctive math of children weighing parental disapproval against the rapidly setting sun, and the heartbroken math of sailors measuring the holes in their boat against the distance to the shore. This is the math that moves the universe, the measure and countermeasure that dances over and around the numbers. Dodger holds this kind of math in her bones, and her fingers dance through the blend of vegetable and mineral scattered in front of her, separating them from one another with almost thoughtless swipes.

I'm still just as slow… At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn't have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant. (Schwartz, 2001)

When you measure a single spiked wave, such as that in Figure 8.9, the device registers the spike's location. If it's spiked at Strawberry Fields, that's what the device reads; if you look at the result, your brain registers that location and you become aware of it. If it's spiked at Grant's Tomb, that's what the device registers; if you look, your brain registers that location and you become aware of it. When you measure the double spiked wave in Figure 8.10, Schrodinger's math tells you to combine the two results you just found. But, says Everett, be careful and precise when you combine them. The combined result, he argued, does not yield a meter and a mind each simultaneously registering two locations. That's sloppy thinking. Instead, proceeding slowly and literally, we find that the combined result is a device and a mind registering Strawberry Fields, and a device and a mind registering Grant's Tomb. And what does that mean? I'll use broad strokes in painting the general picture, which I'll refine shortly. To accommodate Everett's suggested outcome, the device and you and everything else must split upon measurement, yielding two devices, two yous, and two everything elses-the only difference between the two being that one device and one you registers Strawberry Fields, while the other device and the other you registers Grant's Tomb. As in Figure 8.12, this implies that we now have two parallel realities, two parallel worlds. To the you occupying each, the measurement and your mental impression of the result are sharp and unique and thus fell like life as usual. The peculiarity, of course, is that there are two of you who feel this way.

And so, whereas Bohr and the Copenhagen gang would argue that only one of these universes would exist (because the act of measurement, which they claim lies outside of Schrodinger's purview, would collapse away all the others), and whereas a first-pass attempt to go beyond Bohr and extend Schrodinger's math to all particles, including those constituting equipment and brains, yielded dizzying confusion (because a given machine or mind seemed to internalize all possible outcomes simultaneously), Everett found that a more careful reading of Schrodinger's math leads somewhere else: to a plentiful reality populated by an ever-growing collection of universes.

So the way to think about the problem is this. You and I and computers and bacteria and viruses and everything else material are made of molecules and atoms, which are themselves composed of particles like electrons and quarks. Schrodinger's equation works for electrons and quarks, and all evidence points to its working for things made of these constituents, regardless of the number of particles involved. This means that Schrodinger's equation should continue to apply during a measurement. After all, a measurement is just one collection of particles (the person, the equipment, the computer...) coming into contact with another (the particle or particles being measured). But if that's the case, if Schrodinger's math refuses to bow down, then Bohr is in trouble. Schrodinger's equation doesn't allow waves to collapse. An essential element of the Copenhagen approach would therefore be undermined. So the third question is this: If the reasoning just recounted is right and probability waves don't collapse, how do we pass from the range of possible outcomes that exist before a measurement to the single outcome the measurement reveals? Or to put it in more general terms, what happens to a probability wave during a measurement that allows a familiar, definite, unique reality to take hold? Everett pursued this question in his Princeton doctoral dissertation and came to an unforeseen conclusion.

the objection to using stats boils down to nothing more than an irrational fear of numbers causing some to believe math somehow detracts from understanding or appreciation

What he found was the geometry of the universe. Looking at the bubbles made by the Wego’s propellers, he recalled his boarding school math teachers, who had taught him to measure a sphere’s volume in terms of pi. He also remembered that pi was an irrational number, a decimal that never ended. He asked himself how nature could ever make bubbles in such circumstances. Did nature approximate? The rules his teachers had taught him must be mistaken. Spheres ought to be understood in terms of the forces that made them. At the age of twenty-one, Bucky determined that the universe had no objects. Geometry described forces. It was an insight bound to shape Bucky’s entire worldview—informing every future invention—but

Pyramidiots Some history books will tell you that the Egyptians were great at maths – even better than your teacher. Here’s why they think the Egyptians were such brainy boffins… People who believe the pyramids have magical secrets are called ‘pyramidologists’. They started investigating the pyramids in the 1850s and modern pyramidologists are still selling millions of books around the world. The Horrible Histories truth is … they are pyramidiots. The pyramids don’t measure 365.242 sacred cubits. The pyramidiots didn’t check their facts – they are as potty as the people who say…

In many companies, managers’ pay is contingent upon the efficiency of their staff as measured by revenue per employee hour. Scheduling software helps them boost these numbers and their own compensation. Even when executives tell managers to loosen up, they often resist. It goes against everything they’ve been taught. What’s more, at Starbucks, if a manager exceeds his or her “labor budget,” a district manager is alerted, said one employee. And that could lead to a write-up. It’s usually easier just to change someone’s schedule, even if it means violating the corporate pledge to provide one week’s notice.

The conclusion: homework was found to have little to no benefit before high school and diminishing returns for high school students as the hours spent doing it increased. Another study in 2011 reinforced this finding. In that experiment, a pair of economists crunched numbers from 25,000 eighth graders and found that a little extra math homework (amounting to about fifteen minutes per weekday) did boost students’ math scores slightly (just over 3 percent). But piling on an extra weekly hour of English, science, or history assignments, beyond average levels, made no impact whatsoever on scores in those subjects.

Math tip: Beans tend to triple in size when you cook them, so if you want to end up with 1½ cups of cooked beans (the standard can measurement), you want to start with ½ cup dried beans.

In mathematics, by placing our fingers on a given problem, no matter how trite or pedestrian it apparently seems, we may end up measuring the pulse of the universe. p. 119

This solution is incorrect, Miss Walker." I looked down at the formula and went back over it carefully. "No, sir. I believe that this is the correct answer. I'm sure I got it right." "No. It's wrong." "Could you tell me why?" "Because a mongoose doesn't mate with a chicken." "What? I'm sorry. I don't understand what that has to do with math." "Exactly. Perhaps you haven't been working hard enough. Maybe you got too many A's and not enough F's. Everyone in this class knows that a mongoose doesn't mate with a chicken." I looked around at the class. All the desks were occupied with... chickens. They all looked at me with beady red eyes and sharp yellow beaks, laughing their fool chicken heads off. Oh god, I was being mocked by a roomful of chickens who knew how to do math better than I did. "But they're all chickens. Of course, they would know the answer." "That's right, and you're not a chicken." "But I could be a chicken. I could study more, work harder." "I'm afraid not. Do you know what happens to you in this class if you get the problem wrong? If you don't measure up?" "No, sir." "It's the stewpot. We don't tolerate stupid chickens in here." "But...but I'm not a chicken." "No? Then you're just plain stupid." "No!" I cried. "I'll try harder. I'll be as good as I can." "I'll be the perfect chicken," I murmured, tossing and turning, kicking at the bed sheets. A pillow sailed across the room and struck me right in the head, drawing me out of that fitful dream. "Aubree, you're having the chicken dream again. If you don't shut up, I'm going to yank out all your feathers," Ashley grumbled.

The salient and undeniable truth about cryptography is that no measure of violence or proscriptive legislation will ever solve a math problem.

At the moment of measuring one electron, a collapse of the wave function of both electrons occurs regardless of the distance between them.

WMDs, by contrast, tend to favor efficiency. By their very nature, they feed on data that can be measured and counted. But fairness is squishy and hard to quantify. It is a concept. And computers, for all of their advances in language and logic, still struggle mightily with concepts.

You are growing into consciousness, and my wish for you is that you feel no need to constrict yourself to make other people comfortable. None of that can change the math anyway. I never wanted you to be twice as good as them, so much as I have always wanted you to attack every day of your brief bright life in struggle. The people who must believe they are white can never be your measuring stick. I would not have you descend into your own dream. I would have you be a conscious citizen of this terrible and beautiful world.

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.

When a friend introduced me to a Bach chaconne, he started by describing it by saying that it has 256 measures (256=2^8) divided into 4 sections of 64 measures (64=2^6), and I liked it even before I heard a single note.

Adam: Adam was a young man whose anxiety turned into a monster. Where Shelly had a very mild case of social anxiety, Adam’s case could only be called severe. Over a period of several years, his underlying social fears developed into a full-blown school phobia. A quiet, unassuming person, Adam had never stood out in the classroom. Through elementary school and on into high school, he neither excelled nor failed his subjects. By no means a discipline problem, the “shy” Adam kept to himself and seldom talked in class, whether to answer a teacher’s question or chat with his buddies. In fact, he really had no friends, and the only peers he socialized with were his cousins, whom he saw at weekly family gatherings. Though he watched the other kids working together on projects or playing sports together, Adam never approached them to join in. Maybe they wouldn’t let him, he thought. Maybe he wasn’t good enough. Being rejected was not a chance he was willing to take. Adam never tried hard in school either. If he didn’t understand something, he kept quiet, fearful that raising his hand would bring ridicule. When he did poorly on an exam or paper, it only confirmed to him what he was sure was true: He didn’t measure up. He became so apprehensive about his tests that he began to feel physically ill at the thought of each approaching reminder of his inadequacy. Even though he had studied hard for a math test, for example, he could barely bring himself to get out of bed on the morning it was to take place. His parents, who thought of their child as a reserved but obedient boy who would eventually grow out of this awkward adolescent stage, did not pressure him. Adam was defensive and withdrawn, overwrought by the looming possibility that he would fail. For the two class periods preceding the math test, Adam’s mind was awash with geometry theorems, and his stomach churning. As waves of nausea washed over him, he began to salivate and swallowed hard. His eyes burned and he closed them, wishing he could block the test from his mind. When his head started to feel heavy and he became short of breath, he asked for a hall pass and headed for the bathroom. Alone, he let his anxiety overtake him as he stared into the mirror, letting the cool water flow from the faucet and onto his sweaty palms. He would feel better, he thought, if he could just throw up. But even when he forced his finger down his throat, there was no relief. His dry heaves made him feel even weaker. He slumped to the cold tile and began to cry. Adam never went back to math class that day; instead, he got a pass from the nurse and went straight home. Of course, the pressure Adam was feeling was not just related to the math test. The roots of his anxiety went much deeper. Still, the physical symptoms of anxiety became so debilitating that he eventually quit going to school altogether. Naturally, his parents were extremely concerned but also uncertain what to do. It took almost a year before Adam was sufficiently in control of his symptoms to return to school.

I am speaking to you as I always have - as the sober and serious man I have always wanted you to be, who does not apologize for his human feelings, who does not make excuses for his height, his long arms, his beautiful smile. You are growing into consciousness, and my wish for you is that you feel no need to constrict yourself to make other people comfortable. None of that can change the math anyway. I never wanted you to be twice as good as them, so much as I have always wanted you to attack every day of your brief bright life in struggle. The people who must believe they are white can never be your measuring stick. I would not have you descend into your won dream. I would have you be a conscious citizen of this terrible and beautiful world.

Deepwater violated that agreement shockingly, manifesting a substance on which most modern human life depends but that few people encounter in the raw. After returning from Norway, I would learn that the Moskstraumen Maelstrom had become literally enabling of the oil industry. In the 1980s a man called Bjørn Gjevig – an antiquarian scholar, professional mathematician and amateur sailor, who seems as if he must have been invented by Poe, but truly exists – became fascinated by the hydrodynamics of the Maelstrom. Using data gathered in part while sailing close to the whirlpool, Gjevig began to model the maths of its currents. When oil was discovered off the Lofotens, he realized that his data had gained application: oil companies would need to understand such ocean forces in order to construct rigs that could withstand ‘destructive currents of the kind found in the Maelstrom’. At the climax of Poe’s story, the human body loses all volition and becomes a kind of drift-matter, helpless within the ‘destructive currents’. The fisherman and his brother are drawn steadily deeper into the vortex. The fisherman realizes that he has entered a giant grading-machine, which weighs and measures the objects that have been pulled into it – and moves the heaviest and most irregularly shaped items to destruction at its base.

Certainly she can't and won't measure what is measureless, what neither terminates nor repeats, what is beyond even the transcendental of π - though HE doesn't think so - what is beyond polynomials and quadratic formulas, beyond the rational and irrational, the humanist and the logical, beyond the minds of the Cantors and the Dedekinds, the Renaissance philosophers and the Indian Tantrists, what falls instead into the realm of gods and kinds, of myth, of dawn of man, of the mystery of mankind - that there is a space inside her designed solely for him and despite clear Euclidian impossibilities not only does everything, in plenary excess, cleave like it's meant to, but it makes her feel what math cannot explain, what science cannot explain. What nothing can explain.

One of the measures my best clients use when leading their critical projects is the “energy equation.” Are you ready for high math? During checkpoints for the project, the question is asked, “Is the energy you are getting out of doing this project greater than the energy you are putting in?

No, no I don’t believe the P.M. would hesitate, she thought. If only math could help. But the value of human life is immeasurable, and so neither option is morally acceptable. How do you measure and compare the quantity of x versus the quantity of y + z if you don’t know the values of any of them? How many angels can die on the head of a pin?

A few years ago, MIT researchers analyzed the behavior of call center employees for Bank of America to find out why some teams were more productive than others. They hung a so-called sociometric badge around each employee’s neck. The electronics in these badges tracked the employees’ location and also measured, every sixteen milliseconds, their tone of voice and gestures. It recorded when people were looking at each other and how much each person talked, listened, and interrupted. Four teams of call center employees—eighty people in total—wore these badges for six weeks. These employees’ jobs were highly regimented. Talking was discouraged because workers were supposed to spend as many of their minutes as possible on the phone, solving customers’ problems. Coffee breaks were scheduled one by one. The researchers found, to their surprise, that the fastest and most efficient call

Lift your eyes for a moment from the curriculum guide, teacher’s manual, math problems, and state-imposed guidelines. Look at the high goal of education: a virtuous man or woman who loves the Lord—an individual who thinks rightly and acts righteously. This is not the work of a day or a week or a year. Progress is not measured by checking off a finished assignment. Though we must give attention to the details of the journey, fix your eyes on the goal as often as possible, though you can do no more than take one step at a time.

The findings that were deemed believable enough to be published, however, revolutionized ethologists’ thinking. Ethologists began to speak less often of a chasm between man and ape; they began to speak instead of a dividing “line.” And it was a line that, in the words of Harvard primatologist Irven De Vore, was “a good deal less clear than one would ever have expected.” What makes up this line between us and our fellow primates? No longer can it be claimed to be tool use. Is it the ability to reason? Wolfgang Kohler once tested captive chimps’ reasoning ability by placing several boxes and a stick in an enclosure and hanging a banana from the high ceiling by a string. The animals quickly figured out that they could get to the banana by stacking the boxes one atop the other and then reaching to swat at the banana with a stick. (Once Geza Teleki found himself in exactly this position at Gombe. He had followed the chimpanzees down into a valley and around noon discovered he had forgotten to bring his lunch. The chimps were feeding on fruit in the trees at the time, and he decided to try to knock some fruit from nearby vines with a stick. For about ten minutes he leaped and swatted with his stick but didn’t manage to knock down any fruit. Finally an adolescent male named Sniff collected a handful of fruit, came down the tree, and dropped the fruit into Geza’s hands.) Some say language is the line that separates man from ape. But this, too, is being questioned. Captive chimpanzees, gorillas, and orangutans have been taught not only to comprehend, but also to produce language. They have been taught American Sign Language (ASL), the language of the deaf, as well as languages that use plastic chips in place of words and computer languages. One signing chimp, Washoe, often combined known signs in novel and creative ways: she had not been taught the word for swan, but upon seeing one, she signed “water-bird.” Another signing chimp, Lucy, seeing and tasting a watermelon for the first time, called it a “candy-drink”; the acidic radish she named “hurt-cry-food.” Lucy would play with toys and sign to them, much as human children talk to their dolls. Koko, the gorilla protegee of Penny Patterson, used sign language to make jokes, escape blame, describe her surroundings, tell stories, even tell lies. One of Biruté’s ex-captives, a female orangutan named Princess, was taught a number of ASL signs by Gary Shapiro. Princess used only the signs she knew would bring her food; because she was not a captive, she could not be coerced into using sign language to any ends other than those she found personally useful. Today dolphins, sea lions, harbor seals, and even pigeons are being taught artificial languages, complete with a primitive grammar or syntax. An African grey parrot named Alex mastered the correct use of more than one hundred spoken English words, using them in proper order to answer questions, make requests, do math, and offer friends and visitors spontaneous, meaningful comments until his untimely death at age 31 in 2007. One leading researcher, Ronald Schusterman, is convinced that “the components for language are present probably in all vertebrates, certainly in mammals and birds.” Arguing over semantics and syntax, psychologists and ethologists and linguists are still debating the definitions of the line. Louis Leakey remarked about Jane’s discovery of chimps’ use of tools that we must “change the definition of man, the definition of tool, or accept chimps as man.” Now some linguists have actually proposed, in the face of the ape language experiments, changing the definition of language to exclude the apes from a domain we had considered uniquely ours. The line separating man from the apes may well be defined less by human measurement than by the limits of Western imagination. It may be less like a boundary between land and water and more like the lines we draw on maps separating the domains of nations.