Further Maths Quotes

I had one class in the morning, the mysteriously named "Further Maths". It was two hours long and so deeply frightening that I think I went into a trance.

[The Old Astronomer to His Pupil] Reach me down my Tycho Brahe, I would know him when we meet, When I share my later science, sitting humbly at his feet; He may know the law of all things, yet be ignorant of how We are working to completion, working on from then to now. Pray remember that I leave you all my theory complete, Lacking only certain data for your adding, as is meet, And remember men will scorn it, 'tis original and true, And the obloquy of newness may fall bitterly on you. But, my pupil, as my pupil you have learned the worth of scorn, You have laughed with me at pity, we have joyed to be forlorn, What for us are all distractions of men's fellowship and smiles; What for us the Goddess Pleasure with her meretricious smiles. You may tell that German College that their honor comes too late, But they must not waste repentance on the grizzly savant's fate. Though my soul may set in darkness, it will rise in perfect light; I have loved the stars too fondly to be fearful of the night. What, my boy, you are not weeping? You should save your eyes for sight; You will need them, mine observer, yet for many another night. I leave none but you, my pupil, unto whom my plans are known. You 'have none but me,' you murmur, and I 'leave you quite alone'? Well then, kiss me, -- since my mother left her blessing on my brow, There has been a something wanting in my nature until now; I can dimly comprehend it, -- that I might have been more kind, Might have cherished you more wisely, as the one I leave behind. I 'have never failed in kindness'? No, we lived too high for strife,-- Calmest coldness was the error which has crept into our life; But your spirit is untainted, I can dedicate you still To the service of our science: you will further it? you will! There are certain calculations I should like to make with you, To be sure that your deductions will be logical and true; And remember, 'Patience, Patience,' is the watchword of a sage, Not to-day nor yet to-morrow can complete a perfect age. I have sown, like Tycho Brahe, that a greater man may reap; But if none should do my reaping, 'twill disturb me in my sleep So be careful and be faithful, though, like me, you leave no name; See, my boy, that nothing turn you to the mere pursuit of fame. I must say Good-bye, my pupil, for I cannot longer speak; Draw the curtain back for Venus, ere my vision grows too weak: It is strange the pearly planet should look red as fiery Mars,-- God will mercifully guide me on my way amongst the stars.

The day was ridiculous. In fact, the situation was so serious I thought they had to be joking - like maybe they staged a special first day to psych people out. I had one class in the morning, the mysteriously named "Further Maths." It was two hours long and so deeply frightening that I think I went into a trance.

Dylan Bennet Klebold was born brilliant. He started school a year early, and by third grade was enrolled in the CHIPS program: Challenging High Intellectual Potential Students. Even among the brains, Dylan stood out as a math prodigy. The early start didn’t impede him intellectually, but strained his shyness further.

Whereas Nature does not admit of more than three dimensions ... it may justly seem very improper to talk of a solid ... drawn into a fourth, fifth, sixth, or further dimension.

We tend to see math and science as a steady state of facts rather than as the accumulated knowledge of linear traditions. As Korzybski put it, we see further because we “stand on the shoulders”5 of the previous generation. The danger of such a position is that we can forget to put our own feet on the ground.

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.]

I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain ... But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further—then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty.

Students who are beginning to struggle in math and science often look at others who are intellectual racehorses and tell themselves they have to keep up. Then they don’t give themselves the extra time they need to truly master the material, and they fall still further behind. As a result of this uncomfortable and discouraging situation, students end up unnecessarily dropping out of math and science. Take a step back and look dispassionately at your strengths and weaknesses. If you need more time to learn math and science, that’s simply the reality. If you’re in high school, try to arrange your schedule to give yourself the time you need to focus on the more difficult materials, and limit these materials to manageable proportions. If you’re in college, try to avoid a full load of heavy courses, especially if you are working on the side. A lighter load of math and science courses can, for many, be the equivalent of a heavy load of other types of courses. Especially in the early stages of college, avoid the temptation to keep up with your peers.

Godly education means that the sovereignty of God is the baseline fact of all further knowledge, so that even elementary addition problems—two plus two equals four—follow not from any internal logic but...an acknowledgment that Jesus invented and rules over math, that He is in fact "the reason for" math.

Svetlana said that I thought of myself as a robot who could act only negatively. She said I had cynical ideas about language. “You think language is an end in itself. You don’t believe it stands for anything. No, it’s not that you don’t believe—it’s that you don’t care. For you, language itself is a self-sufficient system.” “But it is a self-sufficient system.” “Do you see what you’re saying? This is how you get yourself involved with the devil incarnate. Ivan sensed this attitude in you. He’s cynical in the same way you are only more so, because of math. It’s like you said: math is a language that started out so abstract, more abstract than words, and then suddenly it turned out to be the most real, the most physical thing there was. With math they built the atomic bomb. Suddenly this abstract language is leaving third-degree burns on your skin. Now there’s this special language that can control everything, and manipulate everything, and if you’re the elite who speaks it—you can control everything. “Ivan wanted to try an experiment, a game. It would never have worked with someone different, on someone like me. But you, you’re so disconnected from truth, you were so ready to jump into a reality the two of you made up, just through language. Naturally, it made him want to see how far he could go. You went further and further—and then something went wrong. It couldn’t continue in the same way. It had to develop into something else—into sex, or something else. But for some reason, it didn’t. The experiment didn’t work. But by now you’re so, so far from all the landmarks. You’re just drifting in space.

Psychologist Robert Zajonc takes this claim one step further: “For most decisions, it is extremely difficult to demonstrate that there has actually been any prior cognitive process whatsoever.”28 It isn’t that the decisions people make are irrational; it’s that the process by which decisions are made are utterly unlike the step-by-step rational process that might be used to solve, say, a math problem. Decisions are typically made in the unconscious mind, by means of some unknown process. Indeed,

Please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life. [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.]

Further, the same Arguments which explode the Notion of Luck, may, on the other side, be useful in some Cases to establish a due comparison between Chance and Design: We may imagine Chance and Design to be, as it were, in Competition with each other, for the production of some sorts of Events, and many calculate what Probability there is, that those Events should be rather be owing to the one than to the other.

world. Many children grow up thinking that either you can do math or you can’t. When they struggle, they assume they can’t. From that point on, any struggle is a further reminder of their perceived inadequacies. This affects millions of people. One study found that 48 percent of all young adults in a work-apprentice program had math anxiety;1 other studies have found that approximately 50 percent of students taking

though, will improve her math and reading skills, and every carefree summer day she spends puts her further and further behind Alex. Alex isn’t necessarily smarter than Katie. He’s just out-learning her: he’s putting in a few solid months of learning during the summer while she watches television and plays outside. What Alexander’s work suggests is that the way in which education has been discussed in the United States

I felt so lost again. Every time I started to find my way, I seemed to slip further down. How could I trust what Gus and I have? How could I trust my own feelings? People were complicated. They weren't math problems; they were collections of feelings and decisions and dumb luck. The world was complicated too, not a beautifully hazy French film, but a disastrous, horrible mess, speckled with brilliance and love and meaning.

A third positive result even further from the traditional tool kit of financial incentives comes from a recent randomized control trial conducted in the U.K., using the increasingly popular and low-cost method of text reminders. This intervention involved sending texts to half the parents in some school in advance of a major math test to let them know that their child had a test coming up in five days, then in three days, then in one day. The researchers call this approach “pre-informing.” The other half of parents did not receive the texts. The pre-informing texts increased student performance on the math test by the equivalent of one additional month of schooling, and students in the bottom quartile benefited most. These children gained the equivalent of two additional months of schooling, relative to the control group. Afterward, both parents and students said they wanted to stick with the program, showing that they appreciated being nudged. This program also belies the frequent claim, unsupported by any evidence, that nudges must be secret to be effective.

This success permits us to hope that after thirty or forty years of observation on the new Planet [Neptune], we may employ it, in its turn, for the discovery of the one following it in its order of distances from the Sun. Thus, at least, we should unhappily soon fall among bodies invisible by reason of their immense distance, but whose orbits might yet be traced in a succession of ages, with the greatest exactness, by the theory of Secular Inequalities. [Following the success of the confirmation of the existence of the planet Neptune, he considered the possibility of the discovery of a yet further planet.]

Philosopher Nick Bostrom takes things one step further by asking whether we are a “real” species or a simulation created by an intelligent civilization that came before. This idea comes from the simple fact that we will someday be able to create software simulations that believe they are real creatures. And when we achieve that level of technical proficiency, we’re unlikely to stop with one simulation of that type. In the long run, you could expect far more simulated realities than the real one that started it all. So the math of it says we are far more likely to be a simulation than an original species. The interesting thing here is that neither the real species nor the simulations would be in a position to know which one they are.

I occasionally return to some very basic, mathematical thoughts: There are only so many letters in our Roman alphabet. It is then by the power of the math—the infinite (or not strictly infinite, but huge) number of combinations of letters—that we have so many words. But then, further expanding the number of possibilities, truly to infinity, one word (one combination of letters) can have multiple meanings. Further multiplying the number of meanings, even within one given dictionary definition of a word, is the effect of its context, both immediate and larger, which can endow it with, again, an infinite number of subtly differing shades of meaning. Further enriching the single word, within and beyond its contexts, is one’s own personal associations with it, either from one’s reading or from one’s life experiences. And even in one’s own native language, there are a great many words, meanings, and shades of meaning that one simply doesn’t know and may never encounter.

Brunelleschi’s successor as a theorist of linear perspective was another of the towering Renaissance polymaths, Leon Battista Alberti (1404 –1472), who refined many of Brunelleschi’s experiments and extended his discoveries about perspective. An artist, architect, engineer, and writer, Alberti was like Leonardo in many ways: both were illegitimate sons of prosperous fathers, athletic and good-looking, never-married, and fascinated by everything from math to art. One difference is that Alberti’s illegitimacy did not prevent him from being given a classical education. His father helped him get a dispensation from the Church laws barring illegitimate children from taking holy orders or holding ecclesiastical offices, and he studied law at Bologna, was ordained as a priest, and became a writer for the pope. During his early thirties, Alberti wrote his masterpiece analyzing painting and perspective, On Painting, the Italian edition of which was dedicated to Brunelleschi. Alberti had an engineer’s instinct for collaboration and, like Leonardo, was “a lover of friendship” and “open-hearted,” according to the scholar Anthony Grafton. He also honed the skills of courtiership. Interested in every art and technology, he would grill people from all walks of life, from cobblers to university scholars, to learn their secrets. In other words, he was much like Leonardo, except in one respect: Leonardo was not strongly motivated by the goal of furthering human knowledge by openly disseminating and publishing his findings; Alberti, on the other hand, was dedicated to sharing his work, gathering a community of intellectual colleagues who could build on each other’s discoveries, and promoting open discussion and publication as a way to advance the accumulation of learning. A maestro of collaborative practices, he believed, according to Grafton, in “discourse in the public sphere.” When Leonardo was a teenager in Florence, Alberti was in his sixties and spending much of his time in Rome, so it is unlikely they spent time together. Alberti was a major influence nonetheless.

In this march through a virtual lifetime, we’ve visited school and college, the courts and the workplace, even the voting booth. Along the way, we’ve witnessed the destruction caused by WMDs. Promising efficiency and fairness, they distort higher education, drive up debt, spur mass incarceration, pummel the poor at nearly every juncture, and undermine democracy. It might seem like the logical response is to disarm these weapons, one by one. The problem is that they’re feeding on each other. Poor people are more likely to have bad credit and live in high-crime neighborhoods, surrounded by other poor people. Once the dark universe of WMDs digests that data, it showers them with predatory ads for subprime loans or for-profit schools. It sends more police to arrest them, and when they’re convicted it sentences them to longer terms. This data feeds into other WMDs, which score the same people as high risks or easy targets and proceed to block them from jobs, while jacking up their rates for mortgages, car loans, and every kind of insurance imaginable. This drives their credit rating down further, creating nothing less than a death spiral of modeling. Being poor in a world of WMDs is getting more and more dangerous and expensive.

The trouble was that the rankings were self-reinforcing. If a college fared badly in U.S. News, its reputation would suffer, and conditions would deteriorate. Top students would avoid it, as would top professors. Alumni would howl and cut back on contributions. The ranking would tumble further. The ranking, in short, was destiny.

PageRank, the algorithm that gave rise to Google, is itself a Markov chain. Larry Page’s idea was that web pages with many incoming links are probably more important than pages with few, and links from important pages should themselves count for more. This sets up an infinite regress, but we can handle it with a Markov chain. Imagine a web surfer going from page to page by randomly following links: the states of this Markov chain are web pages instead of characters, making it a vastly larger problem, but the math is the same. A page’s score is then the fraction of the time the surfer spends on it, or equivalently, his probability of landing on the page after wandering around for a long time. Markov chains turn up everywhere and are one of the most intensively studied topics in mathematics, but they’re still a very limited kind of probabilistic model. We can go one step further with a model like this: The states form a Markov chain, as before, but we don’t get to see them; we have to infer them from the observations. This is called a hidden Markov model, or HMM for short. (Slightly misleading, because it’s the states that are hidden, not the model.)

A third positive result even further from the traditional tool kit of financial incentives comes from a recent randomized control trial conducted in the U.K., using the increasingly popular and low-cost method of text reminders. This intervention involved sending texts to half the parents in some school in advance of a major math test to let them know that their child had a test coming up in five days, then in three days, then in one day. The researchers call this approach “pre-informing.” The other half of parents did not receive the texts. The pre-informing texts increased student performance on the math test by the equivalent of one additional month of schooling, and students in the bottom quartile benefited most. These children gained the equivalent of two additional months of schooling, relative to the control group. Afterward, both parents and students said they wanted to stick with the program, showing that they appreciated being nudged.

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.

That said, let’s talk about the 1933-1970 period itself. This period of “peak state” was real, but in overstated form it has become the basis for books like Mazzucato’s Entrepreneurial State — which I disagree with, and which Mingardi and McCloskey have rebutted at length in the Myth of the Entrepreneurial State. Here’s why I disagree with the thesis of the Entrepreneurial State: The name itself is oxymoronic. As macroeconomists never tire of telling us, governments aren’t households, because unlike actual entrepreneurs the state can seize funds and print money. So there is no financial risk, and hence nothing of “entrepreneurship” in the entrepreneurial state. The book doesn’t consider the fact that most math/physics/etc was invented prior to the founding of NSF, and therefore doesn’t need NSF to exist. It further doesn’t acknowledge that it was possible to do science and technology before the massive centralized state, through the distributed model of the “gentleman scientist,” and that this model is returning in the form of open source and (now) decentralized science. It doesn’t take into account the waxing and waning of centralized state capacity due to technology. It doesn’t contend with the state-caused slowdown in physical world innovation that happened during the post-1970 period, which Thiel, Cowen, and J Storrs Hall have all documented.

Judith Richards (1993), a teacher in Cambridge, Massachusetts, recounts her experience in a third/fourth grade classroom of children of white and African-American professionals and working-class Haitian immigrants. When she structured a traditional math problem-solving activity, the children of professionals invariably took the lead. However, when she embedded the same type of math problem-solving activity in a traditional Haitian folk tale, the Haitian children took the lead. It seems reasonable that culturally responsive pedagogy would positively affect learning. In both instances, the cognitive task facing children from cultures that were different from mainstream culture was simplified when they did not have to deal with both an unfamiliar speech event and instructional content. Further, one can imagine that using a familiar communication style could possibly reduce cultural dissonance, create a sense of membership, and symbolically affirm children who are members of racial minority groups (Erickson 1987).

Broadly speaking, it seems that teachers help you know and understand and coaches help you practise and get better. Returning to the notion of how we are useful to our students, which do your students need most from you? The following questionnaire can help you think about this further.   Your turn 2 Underlying philosophies Grade each statement from 1 (don’t agree) to 10 (agree completely). Make a note of your answers, if you wish, to discuss with a colleague. a) Learning a language means learning words and rules. b) Learning a language means repeatedly using it. c) People learn best by noticing and working things out. d) They learn best by communicating with other people. e) Mistakes show that students have not understood the grammar properly. f) They show that they have not practised enough. g) Speaking is a conscious, cognitive process. h) Speaking is an automatic habit.   Perhaps all of these statements are true for you. Still, you may have found that you favour some more than others. Notice how a), c), e) and g) promote the idea of language as knowledge to learn, a bit like maths or music theory, whereas b), d), f) and h) reflect the side of learning that involves practising a skill, more like tennis or playing the piano. If you think that the skill side of language learning is particularly important, you will probably feel comfortable thinking of yourself as a language coach, someone whose main job is to get students practising and improving. If you see language as knowledge, the question to ask yourself is how your students can best acquire that knowledge, from you teaching it to them or from other sources of reference and input?

She pulls me further down. More trapped souls reach out to us, dressed in clothes from decades past. The girl ignores them as we descend along the timeline – decade by decade – towards the birth of the island.

Will Bates was drummed out of Carter and reassigned to teach industrial arts in a middle school. He was given an unsatisfactory evaluation rating, placed on probation for a year, and had his salary frozen. And, of course, he was forbidden to teach math to prevent further threats to the sanctity of football.

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.

Annex students took few English classes and no math. They spent the day learning “homemaking” skills, such as simple sewing, and left school with scant qualifications for either employment or further education.

Every single cell in the human body replaces itself over a period of seven years. That means there’s not even the smallest part of you now that was part of you seven years ago. Everything is changing. In the early days of my second life I noticed how the shadow of a telegraph pole would inch between the gardens of two houses across the street – from 152 to the garden of 150 – over the course of several hours, from lunchtime into evening. After watching this a few times I did the maths: the shadow movement from one garden to the next meant that both houses, the telegraph pole, the street, all of us, had travelled one thousand, one hundred and sixty miles around the earth with the turning of the planet. We’d also travelled about seventy-six thousand miles through space around the sun in the same period and much much further as part of the wider spiralling of the galaxy. And nobody noticed a thing. There is no stillness, only change. Yesterday’s here is not today’s here. Yesterday’s here is somewhere in Russia, in a wilderness in Canada, a deep blue nowhere out in the middle of the Atlantic Ocean. It’s behind the sun, it’s in deep space, hundreds of thousands, millions of miles left behind. We can never wake up in the same place we went to sleep in. Our place in the universe, the universe itself, it all changes faster and faster by the second. Every one of us standing on this planet, we’re all moving forwards and we’re never ever coming back. The truth is, stillness is an idea, a dream. It’s the thought of friendly, welcoming lights still shining in all the places we’ve been forced to abandon.

Further Reading For the Children’s Sake: Foundations of Education for Home and School by Susan Schaeffer Macaulay The Brave Learner: Finding Everyday Magic in Homeschool, Learning, and Life by Julie Bogart The Read-Aloud Family: Making Meaningful and Lasting Connections with Your Kids by Sarah Mackenzie Rethinking School: How to Take Charge of Your Child’s Education by Susan Wise Bauer A Gracious Space: Daily Reflections to Sustain Your Homeschooling Commitment by Julie Bogart Teaching from Rest: A Homeschooler’s Guide to Unshakable Peace by Sarah Mackenzie Free to Learn: Why Unleashing the Instinct to Play Will Make Our Children Happier, More Self-Reliant, and Better Students for Life by Peter Gray Last Child in the Woods: Saving Our Children from Nature-Deficit Disorder by Richard Louv How to Raise a Wild Child: The Art and Science of Falling in Love with Nature by Scott D. Sampson Home Grown: Adventures in Parenting off the Beaten Path, Unschooling, and Reconnecting with the Natural World by Ben Hewitt Project-Based Homeschooling: Mentoring Self-Directed Learners by Lori Pickert Let’s Play Math: How Families Can Learn Math Together—and Enjoy It by Denise Gaskins The Art of Self-Directed Learning: 23 Tips for Giving Yourself an Unconventional Education by Blake Boles Gifts Differing: Understanding Personality Type by Isabel Briggs Meyers and Peter B. Myers

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.

The long belief in the universal validity of the principle of excluded third in mathematics is considered by intuitionism as a phenomenon of history of civilization of the same kind as the old-time belief in the rationality of π or in the rotation of the firmament on an axis passing through the earth. And intuitionism tries to explain the long persistence of this dogma by . . . the practical validity. . . of classical logic for an extensive group of simple everyday phenomena. [This] fact apparently made such a strong impression that. . . classical logic . . . became a deep-rooted habit of thought which was considered not only as useful but as a priori. I hope I have made clear that intuitionism on the one hand subtilizes logic, on the other hand denounces logic as a source of truth. Further that intuitionistic mathematics is inner architecture, and that research in the foundations of math- ematics is inner inquiry. . .

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.

None of those things, though, will improve her math and reading skills, and every carefree summer day she spends puts her further and further behind Alex.

Even with a legitimate MLM and product, the ultimate enemy is math. The structure of an MLM is inherently self-defeating. If, for example, the company starts with just six people, and each level has to recruit six people of their own, in just twelve levels you have over two billion people. Even without complete recruitment, you quickly saturate a population. Further, every person you recruit is now a competitor. They may share your social network, and you’ll both be recruiting from the same pool.

However, when first approaching the game, you might be scared, after all we’ve been taught that is a game for brainiacs of math geniuses. But nothing could be further than the truth, anyone can learn to play Chess and start to think like a Grandmaster! You will need a little help and a lot of practice, but if you just cracked open this book, so you are on the right path.

Arguments from beauty have failed us in the past, and I worry I am witnessing another failure right now. “So what?” you may say. “Hasn’t it always worked out in the end?” It has. But leaving aside that we could be further along had scientists not been distracted by beauty, physics has changed—and keeps on changing. In the past, we muddled through because data forced theoretical physicists to revise ill-conceived aesthetic ideals. But increasingly we first need theories to decide which experiments are most likely to reveal new phenomena, experiments that then take decades and billions of dollars to carry out. Data don’t come to us anymore—we have to know where to get them, and we can’t afford to search everywhere. Hence, the more difficult new experiments become, the more care theorists must take to not sleepwalk into a dead end while caught up in a beautiful dream. New demands require new methods. But which methods? I hope the philosophers have a plan.

I had been sitting in Joan Countryman’s living room for two hours talking about mathematics. My pulse was steady, the hand that held my note-taking pencil didn’t shake and wasn’t even clammy. Where was that old math anxiety? I hadn’t been at any loss for questions; they came to me naturally. Like the process of writing, the process of asking questions had been a form of learning, raising further questions and telling me what I wanted to know next. I was genuinely curious. It never occurred to me that this was a subject I wasn’t supposed to be any good at. What did occur to me was that mathematics was not some arcane system of numbers; it was a language, a way of putting thoughts together. I might never master the language—my checkbook might still go unbalanced—but at least I had begun to glimpse what the language was trying to say and how it could help people to understand the world around them.

It actually drove them further apart. Among those with weak math skills, subjects were 25 percentage points likelier to get the answer right when it bolstered their ideology. But partisans with strong math skills were 45 percentage points likelier to get the answer right when it fit their ideology. The smarter the person is, the dumber politics can make them.

It was an incredibly strong correlation and would have warranted further investigation, had there been any causal link. But there wasn’t. The finding was meaningless. And I can say that because I was that mathematician.

To further distract herself, Miranda started doing mental math. Dividing ten mpg into their total distance, then multiplying that number by the current gas price was a bit staggering. “Do you realize it’s going to cost you around a thousand bucks to make this trip?” “Sure. I have it all figured out. And that’s just gas. It’ll be about that much again for campsites and food.” “Two thousand bucks for a road trip?” Miranda shook her head. “It’s worth it, dear. This’ll be the trip of a lifetime.” She glanced over with a grin. “Besides, I got a real nice price for my house. I can afford this.” “Right.” Miranda pointed at the road. “Better keep focused, Joy.” Joy just laughed. “You’re just like George used to be. At first anyway. After a while he’d sit back and relax. That chair reclines. You can even put your feet up if you like.

...all this abstraction is also potentially distancing. We don't see the labor that went into building our railroads or the civilizations that were wiped out in order to clear the land. We don't see the millennia of dinosaurs or plankton that went into our oil, the Chinese repetitive stress injuries that went into our iPhones, or any of the other time-intensive processes we can spend in an instant today. We tend to see math and science as a steady state of facts rather than as the accumulated knowledge of linear traditions. As Korzybski put it, we see further because we "stand on the shoulders" of the previous generation.

Taking a superior variation always makes sense, but we would only take inferior ones when the die shows, say, a 2 or more. After a while, we’d cool it further by only taking a higher-price change if the die shows a 3 or greater—then 4, then 5. Eventually we’d be mostly hill climbing, making the inferior move just occasionally when the die shows a 6. Finally we’d start going only uphill, and stop when we reached the next local max. This approach, called Simulated Annealing, seemed like an intriguing way to map physics onto problem solving. But would it work? The initial reaction among more traditional optimization researchers was that this whole approach just seemed a little too … metaphorical. “I couldn’t convince math people that this messy stuff with temperatures, all this analogy-based stuff, was real,” says Kirkpatrick, “because mathematicians are trained to really distrust intuition.