Mathematical Educational Quotes

In real life, I assure you, there is no such thing as algebra.

I couldn't claim that I was smarter than sixty-five other guys--but the average of sixty-five other guys, certainly!

No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a "mixed number " while 5/2 is an "improper fraction." They're EQUAL for crying out loud. They are the exact same numbers and have the exact same properties. Who uses such words outside of fourth grade?

Reading list (1972 edition)[edit] 1. Homer – Iliad, Odyssey 2. The Old Testament 3. Aeschylus – Tragedies 4. Sophocles – Tragedies 5. Herodotus – Histories 6. Euripides – Tragedies 7. Thucydides – History of the Peloponnesian War 8. Hippocrates – Medical Writings 9. Aristophanes – Comedies 10. Plato – Dialogues 11. Aristotle – Works 12. Epicurus – Letter to Herodotus; Letter to Menoecus 13. Euclid – Elements 14. Archimedes – Works 15. Apollonius of Perga – Conic Sections 16. Cicero – Works 17. Lucretius – On the Nature of Things 18. Virgil – Works 19. Horace – Works 20. Livy – History of Rome 21. Ovid – Works 22. Plutarch – Parallel Lives; Moralia 23. Tacitus – Histories; Annals; Agricola Germania 24. Nicomachus of Gerasa – Introduction to Arithmetic 25. Epictetus – Discourses; Encheiridion 26. Ptolemy – Almagest 27. Lucian – Works 28. Marcus Aurelius – Meditations 29. Galen – On the Natural Faculties 30. The New Testament 31. Plotinus – The Enneads 32. St. Augustine – On the Teacher; Confessions; City of God; On Christian Doctrine 33. The Song of Roland 34. The Nibelungenlied 35. The Saga of Burnt Njál 36. St. Thomas Aquinas – Summa Theologica 37. Dante Alighieri – The Divine Comedy;The New Life; On Monarchy 38. Geoffrey Chaucer – Troilus and Criseyde; The Canterbury Tales 39. Leonardo da Vinci – Notebooks 40. Niccolò Machiavelli – The Prince; Discourses on the First Ten Books of Livy 41. Desiderius Erasmus – The Praise of Folly 42. Nicolaus Copernicus – On the Revolutions of the Heavenly Spheres 43. Thomas More – Utopia 44. Martin Luther – Table Talk; Three Treatises 45. François Rabelais – Gargantua and Pantagruel 46. John Calvin – Institutes of the Christian Religion 47. Michel de Montaigne – Essays 48. William Gilbert – On the Loadstone and Magnetic Bodies 49. Miguel de Cervantes – Don Quixote 50. Edmund Spenser – Prothalamion; The Faerie Queene 51. Francis Bacon – Essays; Advancement of Learning; Novum Organum, New Atlantis 52. William Shakespeare – Poetry and Plays 53. Galileo Galilei – Starry Messenger; Dialogues Concerning Two New Sciences 54. Johannes Kepler – Epitome of Copernican Astronomy; Concerning the Harmonies of the World 55. William Harvey – On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals 56. Thomas Hobbes – Leviathan 57. René Descartes – Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy 58. John Milton – Works 59. Molière – Comedies 60. Blaise Pascal – The Provincial Letters; Pensees; Scientific Treatises 61. Christiaan Huygens – Treatise on Light 62. Benedict de Spinoza – Ethics 63. John Locke – Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education 64. Jean Baptiste Racine – Tragedies 65. Isaac Newton – Mathematical Principles of Natural Philosophy; Optics 66. Gottfried Wilhelm Leibniz – Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology 67. Daniel Defoe – Robinson Crusoe 68. Jonathan Swift – A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal 69. William Congreve – The Way of the World 70. George Berkeley – Principles of Human Knowledge 71. Alexander Pope – Essay on Criticism; Rape of the Lock; Essay on Man 72. Charles de Secondat, baron de Montesquieu – Persian Letters; Spirit of Laws 73. Voltaire – Letters on the English; Candide; Philosophical Dictionary 74. Henry Fielding – Joseph Andrews; Tom Jones 75. Samuel Johnson – The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets

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.

Me, and thousands of others in this country like me, are half-baked, because we were never allowed to complete our schooling. Open our skulls, look in with a penlight, and you'll find an odd museum of ideas: sentences of history or mathematics remembered from school textbooks (no boy remembers his schooling like the one who was taken out of school, let me assure you), sentences about politics read in a newspaper while waiting for someone to come to an office, triangles and pyramids seen on the torn pages of the old geometry textbooks which every tea shop in this country uses to wrap its snacks in, bits of All India Radio news bulletins, things that drop into your mind, like lizards from the ceiling, in the half hour before falling asleep--all these ideas, half formed and half digested and half correct, mix up with other half-cooked ideas in your head, and I guess these half-formed ideas bugger one another, and make more half-formed ideas, and this is what you act on and live with.

When we are young, we spend much time and pains in filling our note-books with all definitions of Religion, Love, Poetry, Politics, Art, in the hope that, in the course of a few years, we shall have condensed into our encyclopaedia the net value of all the theories at which the world has yet arrived. But year after year our tables get no completeness, and at last we discover that our curve is a parabola, whose arcs will never meet.

What music is to the heart, mathematics is to the mind.

I mean that they (students) should not play life, or study it merely, while the community supports them at this expensive game, but earnestly live it from beginning to end. How could youths better learn to live than by at once trying the experiment of living? Methinks this would exercise their minds as much as mathematics.

In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant mathematical genius thus far produced since the higher education of women began.

A man may possess a profound knowledge of history and mathematics; he may be an authority in psychology, biology, or astronomy; he may know all the discovered truths pertaining to geology and natural science; but if he has not with this knowledge that nobility of soul which prompts him to deal justly with his fellow men, to practice virtue and holiness in his personal life, he is not truly an educated man. Character is the aim of true education; and science, history, and literature are but means used to accomplish the desired end. Character is not the result of chance work but of continuous right thinking and right acting.

Just as all things speak about God to those that know Him, and reveal Him to those that love Him, they also hide Him from all those that neither seek nor know Him.

Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles?

What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?” —David Eugene Smith, American mathematician and educator

I had been to school most all the time, and could spell, and read, and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics, anyway.

I needed to wander… whenever and wherever I wanted! I’d found myself at the end of my rope as far as school was concerned; there seemed no particular reason for me to stay. The teachers didn’t want to teach, and I didn’t want to learn—from them. I wanted my education to come from living life, getting out there in the world, seeing and doing and moving amongst the other vagabonds who had had the same sneaking suspicion that I did, that there would be no great need for high-end mathematics, nope… I was not going to be doing other people’s taxes and going home at 5:37 p.m. to pat my dog’s head and sit down to my one meat and two vegetable table waiting for Jeopardy to pop on the glass tit, the Pat Sajak of my own private game show, in the bellybutton of the universe, Miramar, Florida.

A totally new kind of education is needed in the world. The person who is born to be a poet is proving himself stupid in mathematics and the person who could have been a great mathematician is just cramming history and feeling lost. Everything is topsy-turvy because education is not according to your nature: it does not pay any respect to the individual. It forces everybody into a certain pattern.

She needs to be educated. She needs to know the contents of those books, there. She needs to understand the movements of the stars and the origins of the universe and the requirements of kindness. She needs to know mathematics and poetry. She must ask questions. She must seek to understand. She must understand the laws of cause and effect and unintended consequences. She must learn compassion and curiosity and awe. All of these things. We have to instruct her, Glerk. All three of us. It is a great responsibility.” The

what you learn from the intensity and the focus you had when under the influence of risk stays with you. You may lose the sharpness, but nobody can take away what you’ve learned. This is the principal reason I am now fighting the conventional educational system, made by dweebs for dweebs. Many kids would learn to love mathematics if they had some investment in it, and, more crucially, they would build an instinct to spot its misapplications.

[Math] curriculum is obsessed with jargon and nomenclature seemingly for no other purpose than to provide teachers with something to test the students on.

There isn't an education system on the planet that teaches dance everyday to children the way we teach them mathematics. Why? Why not?

How is it that there are so many minds that are incapable of understanding mathematics? ... the skeleton of our understanding, ... and actually they are the majority. ... We have here a problem that is not easy of solution, but yet must engage the attention of all who wish to devote themselves to education.

In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is concern for content.

Give a man a fish, he’ll eat for a day. Teach a man to fish, he’ll eat for a lifetime.”? It’s the same for mathematics education for twenty-first century life.

Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.

It is not easy to become an educated person.

Jesus probably studied this same information, in his youth. The apostle Paul probably studied this same information. How can I make such a bold assertion? Because, without this knowledge, much of the New Testament would make no sense. Many of the idioms used in the New Testament are the result of lessons learned from this ancient Hebrew education system. Unfortunately, what was common in their day, has become forgotten in ours. For a Hebrew, math doesn’t get in the way. It blazes the way. Other languages are disconnected from this mathematical relationship . . . and it shows.

A good expository paper will benefit far more people than most research papers. A good text is worth a thousand of the usual trifles that appear in research journals.

It struck me at some point that the whole basis of education was memory. A list of names, a column of numbers, a mathematical formula, a beautiful poem—to learn it you had to upload it to the part of the brain that stored stuff, but that was the same part of my brain I was resisting. My memory had been spotty since Mummy disappeared, by design, and I didn’t want to fix it, because memory equaled grief. Not remembering was balm.

(1) disengaging their minds; sabotaging their mental activities; providing a low-quality program of public education in mathematics, logic, systems design and economics; and discouraging technical creativity.

One of the best examples of a polymath is Leonardo da Vinci. Born in Italy in 1452, Leonardo was a sculptor, painter, architect, mathematician, musician, engineer, inventor, anatomist, botanist, geologist, cartographer and writer. Although he received an informal education that included geometry, Latin and mathematics, he was essentially an autodidact, or a self-taught individual.

Speaking of Newton but also commenting more broadly on education and the Enlightenment: "I have seen a professor of mathematics only because he was great in his vocation, buried like a king who had done well by his subjects.

Making mathematics accessible to the educated layman, while keeping high scientific standards, has always been considered a treacherous navigation between the Scylla of professional contempt and the Charybdis of public misunderstanding.

Morality has got nothing to do with the politicians. Simply, because the straight line is considered by them as the longest distance between the two points.

Education makes your maths better, not necessarily your manners.

As John Adams famously wrote during the American Revolution, “I must study politics and war, that our sons may have liberty to study mathematics and philosophy. Our sons ought to study mathematics and philosophy, geography, natural history and naval architecture, navigation, commerce and agriculture in order to give their children a right to study painting, poetry, music, architecture, statuary, tapestry and porcelain.” So maybe today they’re writing apps rather than studying poetry, but that’s an adjustment for the age.

Our schools will not improve if we continue to focus only on reading and mathematics while ignoring the other studies that are essential elements of a good education. Schools that expect nothing more of their students than mastery of basic skills will not produce graduates who are ready for college or the modern workplace. *** Our schools will not improve if we value only what tests measure. The tests we have now provide useful information about students' progress in reading and mathematics, but they cannot measure what matters most in education....What is tested may ultimately be less important that what is untested... *** Our schools will not improve if we continue to close neighborhood schools in the name of reform. Neighborhood schools are often the anchors of their communities, a steady presence that helps to cement the bond of community among neighbors. *** Our schools cannot improve if charter schools siphon away the most motivated students and their families in the poorest communities from the regular public schools. *** Our schools will not improve if we continue to drive away experienced principals and replace them with neophytes who have taken a leadership training course but have little or no experience as teachers. *** Our schools cannot be improved if we ignore the disadvantages associated with poverty that affect children's ability to learn. Children who have grown up in poverty need extra resources, including preschool and medical care.

The basic principle of the new education is to be that dunces and idlers must not be made to feel inferior to intelligent and industrious pupils. That would be ‘undemocratic’. These differences between the pupils—for they are obviously and nakedly individual differences—must be disguised. This can be done on various levels. At universities, examinations must be framed so that nearly all the students get good marks. Entrance examinations must be framed so that all, or nearly all, citizens can go to universities, whether they have any power (or wish) to profit by higher education or not. At schools, the children who are too stupid or lazy to learn languages and mathematics and elementary science can be set to doing the things that children used to do in their spare time. Let them, for example, make mud-pies and call it modelling. But all the time there must be no faintest hint that they are inferior to the children who are at work. Whatever nonsense they are engaged in must have—I believe the English already use the phrase—‘parity of esteem’. An even more drastic scheme is not impossible. Children who are fit to proceed to a higher class may be artificially kept back, because the others would get a trauma—Beelzebub, what a useful word!—by being left behind. The bright pupil thus remains democratically fettered to his own age-group throughout his school career, and a boy who would be capable of tackling Aeschylus or Dante sits listening to his coaeval’s attempts to spell out A CAT SAT ON THE MAT.

A remarkably consistent finding, starting with elementary school students, is that males are better at math than females. While the difference is minor when it comes to considering average scores, there is a huge difference when it comes to math stars at the upper extreme of the distribution. For example, in 1983, for every girl scoring in the highest percentile in the math SAT, there were 11 boys. Why the difference? There have always been suggestions that testosterone is central. During development, testosterone fuels the growth of a brain region involved in mathematical thinking and giving adults testosterone enhances their math skills. Oh, okay, it's biological. But consider a paper published in science in 2008. The authors examined the relationship between math scores and sexual equality in 40 countries based on economic, educational and political indices of gender equality. The worst was Turkey, United States was middling, and naturally, the Scandinavians were tops. Low and behold, the more gender equal the country, the less of a discrepancy in math scores. By the time you get to the Scandinavian countries it's statistically insignificant. And by the time you examine the most gender equal country on earth at the time, Iceland, girls are better at math than boys. Footnote, note that the other reliable sex difference in cognition, namely better reading performance by girls than by boys doesn't disappear in more gender equal societies. It gets bigger. In other words, culture matters. We carry it with us wherever we go.

What we mean when speaking of "myth" in general is story, the ability of story to explain ourselves to ourselves in ways that physics, philosophy, mathematics, chemistry—all very highly useful and informative in their own right—can't.

My lack of initiative was the root cause of all my troubles - of my inability to want something before having thought about it, of my inability to commit myself, of my inability to decide in the only way one can decide: by deciding, not by thinking. I'm like Buridan's donkey, dying at the mathematical midpoint between the water of emotion and the hay of action; if I didn't think, I might still die, but it wouldn't be from thirst or hunger.

Adaptability to change is itself a hallmark of successful education.

Teaching mathematics, like teaching any art, requires the ability to inspire the student. Inspiration requires marketing, and marketing requires stirring communication.

For the things of this world cannot be made known without a knowledge of mathematics.

Ignorance often makes a complicated thing seem simple, or vice versa.

The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It's that the world is full of non-proofs, and grown-ups need to know the difference. It's hard to settle for a non-proof once you've really familiarized yourself with the genuine article.

A man may possess a profound knowledge of history and mathematics; he may be an authority in psychology, biology, or astronomy; he may know all the discovered truths pertaining to geology and natural science; but if he has not with this knowledge that nobility of soul which prompts him to deal justly with his fellow men, to practice virtue and holiness in personal life, he is not a truly educated man. "Character is the aim of true education; and science, history, and literature are but means used to accomplish the desired end. Character is not the result of chance work but of continuous right thinking and right acting. "True education seeks, then, to make men and women not only good mathematicians, proficient linguists, profound scientists, or brilliant literary lights, but also honest men, combined with virtue, temperance, and brotherly love-men and women who prize truth, justice, wisdom, benevolence, and self-control as the choicest acquisitions of a successful life.

The habit of looking at life as a social relation — an affair of society — did no good. It cultivated a weakness which needed no cultivation. If it had helped to make men of the world, or give the manners and instincts of any profession — such as temper, patience, courtesy, or a faculty of profiting by the social defects of opponents — it would have been education better worth having than mathematics or languages; but so far as it helped to make anything, it helped only to make the college standard permanent through life.

The forms of mathematics, the harmonies of music, the motions of the planets, and the gods of the mysteries were all essentially related for Pythagoreans, and the meaning of that relation was revealed in an education that culminated in the human soul’s assimilation to the world soul, and thence to the divine creative mind of the universe.

I don't know which is worse—to have a bad teacher or no teacher at all. In any case, I believe the teacher's work should be largely negative. He can't put the gift into you, but if he finds it there, he can try to keep it from going in an obviously wrong direction. We can learn how not to write, but this is a discipline that does not simply concern writing itself but concerns the whole intellectual life. A mind cleared of false emotion and false sentiment and egocentricity is going to have at least those roadblocks removed from its path. If you don't think cheaply, then there at least won't be the quality of cheapness in your writing, even though you may not be able to write well. The teacher can try to weed out what is positively bad, and this should be the aim of the whole college. Any discipline can help your writing: logic, mathematics, theology, and of course and particularly drawing. Anything that helps you to see, anything that makes you look. The writer should never be ashamed of staring. There is nothing that doesn't require his attention.

The oldest problem in economic education is how to exclude the incompetent. A certain glib mastery of verbiage-the ability to speak portentously and sententiously about the relation of money supply to the price level-is easy for the unlearned and may even be aided by a mildly enfeebled intellect. The requirement that there be ability to master difficult models, including ones for which mathematical competence is required, is a highly useful screening device.

The only gain of civilisation for mankind is the greater capacity for variety of sensations--and absolutely nothing more. And through the development of this many-sidedness man may come to finding enjoyment in bloodshed. In fact, this has already happened to him. Have you noticed that it is the most civilised gentlemen who have been the subtlest slaughterers, to whom the Attilas and Stenka Razins could not hold a candle, and if they are not so conspicuous as the Attilas and Stenka Razins it is simply because they are so often met with, are so ordinary and have become so familiar to us. In any case civilisation has made mankind if not more bloodthirsty, at least more vilely, more loathsomely bloodthirsty. In old days he saw justice in bloodshed and with his conscience at peace exterminated those he thought proper. Now we do think bloodshed abominable and yet we engage in this abomination, and with more energy than ever. Which is worse? Decide that for yourselves. They say that Cleopatra (excuse an instance from Roman history) was fond of sticking gold pins into her slave-girls' breasts and derived gratification from their screams and writhings. You will say that that was in the comparatively barbarous times; that these are barbarous times too, because also, comparatively speaking, pins are stuck in even now; that though man has now learned to see more clearly than in barbarous ages, he is still far from having learnt to act as reason and science would dictate. But yet you are fully convinced that he will be sure to learn when he gets rid of certain old bad habits, and when common sense and science have completely re-educated human nature and turned it in a normal direction. You are confident that then man will cease from INTENTIONAL error and will, so to say, be compelled not to want to set his will against his normal interests. That is not all; then, you say, science itself will teach man (though to my mind it's a superfluous luxury) that he never has really had any caprice or will of his own, and that he himself is something of the nature of a piano-key or the stop of an organ, and that there are, besides, things called the laws of nature; so that everything he does is not done by his willing it, but is done of itself, by the laws of nature. Consequently we have only to discover these laws of nature, and man will no longer have to answer for his actions and life will become exceedingly easy for him. All human actions will then, of course, be tabulated according to these laws, mathematically, like tables of logarithms up to 108,000, and entered in an index; or, better still, there would be published certain edifying works of the nature of encyclopaedic lexicons, in which everything will be so clearly calculated and explained that there will be no more incidents or adventures in the world.

There are hundreds of miracles within a single machine. Americans calmly explain these with mathematical formulas. Our difficulty is to learn, theirs to appreciate.

Mathematics is the language of science-- but it is also the hidden structure behind art… and its basis is the invisible Logos of God.

This is a point I’ll be returning to in future chapters: we’ve seen time and again that mathematical models can sift through data to locate people who are likely to face great challenges, whether from crime, poverty, or education. It’s up to society whether to use that intelligence to reject and punish them—or to reach out to them with the resources they need.

There are two moments in the course of education where a lot of kids fall off the math train. The first comes in the elementary grades, when fractions are introduced. Until that moment, a number is a natural number, one of the figures 0, 1, 2, 3 . . . It is the answer to a question of the form “how many.”* To go from this notion, so primitive that many animals are said to understand it, to the radically broader idea that a number can mean “what portion of,” is a drastic philosophical shift. (“God made the natural numbers,” the nineteenth-century algebraist Leopold Kronecker famously said, “and all the rest is the work of man.”) The second dangerous twist in the track is algebra. Why is it so hard? Because, until algebra shows up, you’re doing numerical computations in a straightforwardly algorithmic way. You dump some numbers into the addition box, or the multiplication box, or even, in traditionally minded schools, the long-division box, you turn the crank, and you report what comes out the other side. Algebra is different. It’s computation backward. When you’re asked to solve

We’ve seen time and again that mathematical models can sift through data to locate people who are likely to face great challenges, whether from crime, poverty, or educations. It’s up to society whether to use that intelligence to reject and punish them—or to reach out to them with the resources they need. We can use the scale and efficiency that make WMDs so pernicious in order to help people. It all depends on the objective we choose.

In medieval Europe, logic, grammar and rhetoric formed the educational core, while the teaching of mathematics seldom went beyond simple arithmetic and geometry. Nobody studied statistics. The undisputed monarch of all sciences was theology.

We all love stories, even if they’re not true. As we grow up, one of the ways we learn about the world is through the stories we hear. Some are about particular events and personalities within our personal circles of family and friends. Some are part of the larger cultures we belong to—the myths, fables, and fairy tales about our own ways of life that have captivated people for generations. In stories that are told often, the line between fact and myth can become so blurred that we easily mistake one for the other. This is true of a story that many people believe about education, even though it’s not real and never really was. It goes like this: Young children go to elementary school mainly to learn the basic skills of reading, writing, and mathematics. These skills are essential so they can do well academically in high school. If they go on to higher education and graduate with a good degree, they’ll find a well-paid job and the country will prosper too.

The reason why I prioritized philosophy over law in my higher education is for my ambivalent relationship with rules. Mathematics compensates it where the mind becomes creative, conceptual and flexible but also logically organized, data and rules-driven.

Other countries whose educational systems achieve more than ours often do so in part by attempting less. While school children in Japan are learning science, mathematics, and a foreign language, American school children are sitting around in circles, unburdening their psyches and “expressing themselves” on scientific, economic and military issues for which they lack even the rudiments of competence. Worse than what they are not learning is what they are learning—presumptuous superficiality, taught by practitioners of it. The

Lots of people wrote to the magazine to say that Marilyn vos Savant was wrong, even when she explained very carefully why she was right. Of the letters she got about the problem, 92% said that she was wrong and lots of these were from mathematicians and scientists. Here are some of the things they said: 'I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error.' -Robert Sachs, Ph.D., George Mason University ... 'I am sure you will receive many letters from high school and college students. Perhaps you should keep a few addresses for future columns.' -W. Robert Smith, Ph.D., Georgia State University... 'If all those Ph.D.'s were wrong, the country would be in very serious trouble.' -Everett Harman, Ph.D., U.S. Army Research Institute

NAEP data show beyond question that test scores in reading and math have improved for almost every group of students over the past two decades; slowly and steadily in the case of reading, dramatically in the case of mathematics. Students know more and can do more in these two basic skills subjects now than they could twenty or forty years ago... So the next time you hear someone say that the system is "broken," that American students aren't as well educated as they used to be, that our schools are failing, tell that person the facts.

In adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics. Now, on the contrary, I enjoy life; I might almost say that with every year that passes I enjoy it more…very largely it is due to a diminishing preoccupation with myself. Like others who had a Puritan education, I had the habit of meditating on my sins, follies, and shortcomings. I seemed to myself - no doubt justly - a miserable specimen. Gradually I learned to be indifferent to myself and my deficiencies; I came to center my attention increasingly upon external objects: the state of the world, various branches of knowledge, individuals for whom I felt affection…And every external interest inspires some activity which, so long as the interest remains alive, is a complete preventive of ennui. Interest in oneself, on the contrary, leads to no activity of a progressive kind. It may lead to the keeping of a diary, to getting psychoanalyzed, or perhaps to becoming a monk. But the monk will not be happy until the routine of the monastery has made him forget his own soul. The happiness which he attributes to religion he could have obtained from becoming a crossing-sweeper, provided he were compelled to remain one. External discipline is the only road to happiness for those unfortunates whose self-absorption is too profound to be cured in any other way.

These forays into the real world sharpened his view that scientists needed the widest possible education. He used to say, “How can you design for people if you don’t know history and psychology? You can’t. Because your mathematical formulas may be perfect, but the people will screw it up. And if that happens, it means you screwed it up.” He peppered his lectures with quotations from Plato, Chaka Zulu, Emerson, and Chang-tzu. But as a professor who was popular with his students—and who advocated general education—Thorne found himself swimming against the tide. The academic world was marching toward ever more specialized knowledge, expressed in ever more dense jargon. In this climate, being liked by your students was a sign of shallowness; and interest in real-world problems was proof of intellectual poverty and a distressing indifference to theory.

The human mind is an incredible thing. It can conceive of the magnificence of the heavens and the intricacies of the basic components of matter. Yet for each mind to achieve its full potential, it needs a spark. The spark of enquiry and wonder. Often that spark comes from a teacher. Allow me to explain. I wasn’t the easiest person to teach, I was slow to learn to read and my handwriting was untidy. But when I was fourteen my teacher at my school in St Albans, Dikran Tahta, showed me how to harness my energy and encouraged me to think creatively about mathematics. He opened my eyes to maths as the blueprint of the universe itself. If you look behind every exceptional person there is an exceptional teacher. When each of us thinks about what we can do in life, chances are we can do it because of a teacher. [...] The basis for the future of education must lie in schools and inspiring teachers. But schools can only offer an elementary framework where sometimes rote-learning, equations and examinations can alienate children from science. Most people respond to a qualitative, rather than a quantitative, understanding, without the need for complicated equations. Popular science books and articles can also put across ideas about the way we live. However, only a small percentage of the population read even the most successful books. Science documentaries and films reach a mass audience, but it is only one-way communication.

A 2013 study by the National Center on Education and the Economy found that “the mathematics that most enables students to be successful in college courses is not high school mathematics, but middle school mathematics, especially arithmetic, ratio, proportion, expressions and simple equations.

If, redesigning our education system from scratch, it was suggested that we should attempt to teach Swahili to children but carry out those lessons in another foreign tongue, such as Swedish, this would rightly be derided as lunacy. Yet this is not so very far from what we are attempting to do. Take Coyne, for example. He is 14 now. His grasp of English is, at best, tenuous. Despite this, we are trying to teach him to speak French. Equally, his mathematical ability is next to nil; we are trying, in economics lessons, to explain concepts like inflation and money supply to a boy who can’t add..

He was imperfectly educated, and ignorant on many points; but he was aware of his deficiency, and regretted it in theory. He was awkward and ungainly in society, and so kept out of it as much as possible; and he was obstinate, violent-tempered, and dictatorial in his own immediate circle. On the other side, he was generous, and true as steel; the very soul of honour, in fact. He had so much natural shrewdness, that his conversation was always worth listening to, although he was apt to start by assuming entirely false premises, which he considered as incontrovertible as if they had been mathematically proved;

Ideological agendas in public schools absorb time, energy and resources that are especially needed in the education of young people from a cultural background often lacking in many of the things that youngsters in more fortunate circumstances can take for granted— such as highly educated parents, books in the home and a whole way of life that prepares them in childhood for achievements as adults. Propagandists in the classroom are a luxury that the poor can afford leas of all. While a mastery of mathematics and English can be a ticket out of poverty, a highly cultivated sense of grievance and resentment is not.

I was perplexed by the failure of teachers at school to address what seemed the most urgent matter of all: the bewildering, stomach-churning insecurity of being alive. The standard subjects of history, geography, mathematics, and English seemed perversely designed to ignore the questions that really mattered. As soon as I had some inkling of what 'philosophy' meant, I was puzzled as to why we were not taught it. And my skepticism about religion only grew as I failed to see what the vicars and priests I encountered gained from their faith. They struck me either as insincere, pious, and aloof or just bumblingly good-natured. (p. 10)

There is one thing only which a Muslim can profitably learn from the west, the exact sciences in their pure and applied form. Only natural sciences and mathematics should be taught in Muslim schools, while tuition of European philosophy, literature and history should lose the position of primacy which today it holds on the curriculum.

A study of kindergartens in Germany compared fifty play-based classes with fifty early-learning centers and found that the children who played excelled over the others in reading and mathematics and were better adjusted socially and emotionally in school. They also excelled in creativity and intelligence, oral expression, and industry.8

All through our education we are being taught a kind of reverse mindfulness. A kind of Future Studies where- via the guise of mathematics, or literature, or history, or computer programming, or French- we are being taught to think of a time different to the time we are in. Exam time. Job time. When-we-are-grown-up time. To see the act of learning as something not for its own sake but because of what it will get you reduces the wonder of humanity. We are thinking, feeling, art-making, knowledge-hungry, marvelous animals, who understand ourselves and our world through the act of learning. It is an end in itself. It has far more to offer than the things it lets us write on application forms. It is a way to love living right now.

Every now and then, I'm lucky enough to teach a kindergarten or first-grade class. Many of these children are natural-born scientists - although heavy on the wonder side, and light on skepticism. They're curious, intellectually vigorous. Provocative and insightful questions bubble out of them. They exhibit enormous enthusiasm. I'm asked follow-up questions. They've never heard of the notion of a 'dumb question'. But when I talk to high school seniors, I find something different. They memorize 'facts'. By and large, though, the joy of discovery, the life behind those facts has gone out of them. They've lost much of the wonder and gained very little skepticism. They're worried about asking 'dumb' questions; they are willing to accept inadequate answers, they don't pose follow-up questions, the room is awash with sidelong glances to judge, second-by-second, the approval of their peers. They come to class with their questions written out on pieces of paper, which they surreptitiously examine, waiting their turn and oblivious of whatever discussion their peers are at this moment engaged in. Something has happened between first and twelfth grade. And it's not just puberty. I'd guess that it's partly peer pressure not to excel - except in sports, partly that the society teaches short-term gratification, partly the impression that science or mathematics won't buy you a sports car, partly that so little is expected of students, and partly that there are few rewards or role-models for intelligent discussion of science and technology - or even for learning for it's own sake. Those few who remain interested are vilified as nerds or geeks or grinds. But there's something else. I find many adults are put off when young children pose scientific questions. 'Why is the Moon round?', the children ask. 'Why is grass green?', 'What is a dream?', 'How deep can you dig a hole?', 'When is the world's birthday?', 'Why do we have toes?'. Too many teachers and parents answer with irritation, or ridicule, or quickly move on to something else. 'What did you expect the Moon to be? Square?' Children soon recognize that somehow this kind of question annoys the grown-ups. A few more experiences like it, and another child has been lost to science.

The human mind is an incredible thing. It can conceive of the magnificence of the heavens and the intricacies of the basic components of matter. Yet for each mind to achieve its full potential, it needs a spark. The spark of enquiry and wonder. Often that spark comes from a teacher. Allow me to explain. I wasn’t the easiest person to teach, I was slow to learn to read and my handwriting was untidy. But when I was fourteen my teacher at my school in St Albans, Dikran Tahta, showed me how to harness my energy and encouraged me to think creatively about mathematics. He opened my eyes to maths as the blueprint of the universe itself. If you look behind every exceptional person there is an exceptional teacher. When each of us thinks about what we can do in life, chances are we can do it because of a teacher.

Winfree came from a family in which no one had gone to college. He got started, he would say, by not having proper education. His father, rising from the bottom of the life insurance business to the level of vice president, moved family almost yearly up and down the East Coast, and Winfree attended than a dozen schools before finishing high school. He developed a feeling that the interesting things in the world had to do with biology and mathematics and a companion feeling that no standard combination of the two subjects did justice to what was interesting. So he decided not to take a standard approach. He took a five-year course in engineering physics at Cornell University, learning applied mathematics and a full range of hands-on laboratory styles. Prepared to be hired into military-industrial complex, he got a doctorate in biology, striving to combine experiment with theory in new ways.

It is said that there comes a point in every mathematics student's education when he hears himself saying to the teacher, "I think I understand"-- and that's the point at which he has hit a wall. Making sure that all gifted students hit their own personal walls is crucial for developing the empathy with the rest of the world. When they see their less lucky peers struggle academically, they need to be able to say "I know how it feels,"-- and be telling the truth.

Yet, during the last forty years, its contributions have been obscured by the rise of ‘macro-economics’, which seeks causal connections between hypothetically measurable entities or statistical aggregates. These may sometimes, I concede, indicate some vague probabilities, but they certainly do not explain the processes involved in generating them. But because of the delusion that macro-economics is both viable and useful (a delusion encouraged by its extensive use of mathematics, which must always impress politicians lacking any mathematical education, and which is really the nearest thing to the practice of magic that occurs among professional economists) many opinions ruling contemporary government and politics are still based on naive explanations of such economic phenomena as value and prices, explanations that vainly endeavour to account for them as ‘objective’ occurrences independent of human knowledge and aims.

While the universality of the creative process has been noticed, it has not been noticed universally. Not enough people recognize the preverbal, pre-mathematical elements of the creative process. Not enough recognize the cross-disciplinary nature of intuitive tools for thinking. Such a myopic view of cognition is shared not only by philosophers and psychologists but, in consequence, by educators, too. Just look at how the curriculum, at every educational level from kindergarten to graduate school, is divided into disciplines defined by products rather than processes. From the outset, students are given separate classes in literature, in mathematics, in science, in history, in music, in art, as if each of these disciplines were distinct and exclusive. Despite the current lip service paid to “integrating the curriculum,” truly interdisciplinary courses are rare, and transdisciplinary curricula that span the breadth of human knowledge are almost unknown. Moreover, at the level of creative process, where it really counts, the intuitive tools for thinking that tie one discipline to another are entirely ignored. Mathematicians are supposed to think only “in mathematics,” writers only “in words,” musicians only “in notes,” and so forth. Our schools and universities insist on cooking with only half the necessary ingredients. By half-understanding the nature of thinking, teachers only half-understand how to teach, and students only half-understand how to learn.

There are hundreds of miracles within a single machine. Americans calmly explain these with mathematical formulas. Our difficulty is to learn, theirs to appreciate. We Latins, even the most intelligent of us, still count on our fingers and toes. But once we do learn, we shall surpass the Americano, because we understand the spiritual significance of a machine. We see the beauty of combining gas, grease and steel into a powerful, exact movement. We appreciate the material destiny of the universe.

She needs to be educated. She needs to know the contents of those books, there. She needs to understand the movements of the stars and the origins of the universe and the requirements of kindness. She needs to know mathematics and poetry. She must ask questions. She must seek to understand. She must understand the laws of cause and effect and unintended consequences. She must learn compassion and curiosity and awe. All of these things. We have to instruct her, Glerk. All three of us. It is a great responsibility

When he was not in class, Thorne often served as an expert witness in legal cases involving materials engineering. He specialized in explosions, crashed airplanes, collapsed buildings, and other disasters. These forays into the real world sharpened his view that scientists needed the widest possible education. He used to say, “How can you design for people if you don’t know history and psychology? You can’t. Because your mathematical formulas may be perfect, but the people will screw it up. And if that happens, it means you screwed it up.

He is a man of good birth and excellent education, endowed by nature with a phenomenal mathematical faculty. At the age of twenty-one he wrote a treatise upon the Binomial Theorem, which has had a European vogue. On the strength of it he won the Mathematical Chair at one of our smaller universities, and had, to all appearances, a most brilliant career before him. But the man had hereditary tendencies of the most diabolical kind. A criminal strain ran in his blood, which, instead of being modified, was increased and rendered infinitely more dangerous by his extraordinary mental powers.

What’s stopping me is I’m a Dahlite, a heatsinker on Dahl. I don’t have the money to get an education and I can’t get the credits to get an education. A real education, I mean. All they taught me was to read and cipher and use a computer and then I knew enough to be a heatsinker. But I wanted more. So I taught myself.” “In some ways, that’s the best kind of teaching. How did you do that?” “I knew a librarian. She was willing to help me. She was a very nice woman and she showed me how to use computers for learning mathematics. And she set up a software system that would connect me with other libraries.

Because much of the content of education is not cognitively natural, the process of mastering it may not always be easy and pleasant, notwithstanding the mantra that learning is fun. Children may be innately motivated to make friends, acquire status, hone motor skills, and explore the physical world, but they are not necessarily motivated to adapt their cognitive faculties to unnatural tasks like formal mathematics. A family, peer group, and culture that ascribe high status to school achievement may be needed to give a child the motive to persevere toward effortful feats of learning whose rewards are apparent only over the long term.

What Homer could never have foreseen is the double idiocy into which we now educate our children. We have what look like our equivalent to the Greek “assemblies”; we can watch them on cable television, as long as one can endure them. For they are charades of political action. They concern themselves constantly, insufferably, about every tiniest feature of human existence, but without slow deliberation, without balance, without any commitment to the difficult virtues. We do not have men locked in intellectual battle with other men, worthy opponents both, as Thomas Paine battled with John Dickinson, or Daniel Webster with Robert Hayne. We have men strutting and mugging for women nagging and bickering. We have the sputters of what used to be language, “tweets,” expressions of something less than opinion. It is the urge to join—something, anything—while remaining aloof from the people who live next door, whose names we do not know. Aristotle once wrote that youths should not study politics, because they had not the wealth of human experience to allow for it; all would become for them abstract and theoretical, like mathematics, which the philosopher said was more suitable for them. He concluded that men should begin to study politics at around the age of forty. Whether that wisdom would help us now, I don’t know.

The fundamental problem with learning mathematics is that while the number sense may be genetic, exact calculation requires cultural tools—symbols and algorithms—that have been around for only a few thousand years and must therefore be absorbed by areas of the brain that evolved for other purposes. The process is made easier when what we are learning harmonizes with built-in circuitry. If we can’t change the architecture of our brains, we can at least adapt our teaching methods to the constraints it imposes. For nearly three decades, American educators have pushed “reform math,” in which children are encouraged to explore their own ways of solving problems. Before reform math, there was the “new math,” now widely thought to have been an educational disaster. (In France, it was called les maths modernes and is similarly despised.) The new math was grounded in the theories of the influential Swiss psychologist Jean Piaget, who believed that children are born without any sense of number and only gradually build up the concept in a series of developmental stages. Piaget thought that children, until the age of four or five, cannot grasp the simple principle that moving objects around does not affect how many of them there are, and that there was therefore no point in trying to teach them arithmetic before the age of six or seven.

A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult. What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.

All through our education we are being taught a kind of reverse mindfulness. A kind of Future Studies where—via the guise of mathematics, or literature, or history, or computer programming, or French—we are being taught to think of a time different to the time we are in. Exam time. Job time. When-we-are-grown-up time. To see the act of learning as something not for its own sake but because of what it will get you reduces the wonder of humanity. We are thinking, feeling, art-making, knowledge-hungry, marvelous animals, who understand ourselves and our world through the act of learning. It is an end in itself. It has far more to offer than the things it lets us write on application forms. It is a way to love living right now.

Countries measured their success by the size of their territory, the increase in their population and the growth of their GDP – not by the happiness of their citizens. Industrialised nations such as Germany, France and Japan established gigantic systems of education, health and welfare, yet these systems were aimed to strengthen the nation rather than ensure individual well-being. Schools were founded to produce skilful and obedient citizens who would serve the nation loyally. At eighteen, youths needed to be not only patriotic but also literate, so that they could read the brigadier’s order of the day and draw up tomorrow’s battle plans. They had to know mathematics in order to calculate the shell’s trajectory or crack the enemy’s secret code. They needed a reasonable command of electrics, mechanics and medicine in order to operate wireless sets, drive tanks and take care of wounded comrades. When they left the army they were expected to serve the nation as clerks, teachers and engineers, building a modern economy and paying lots of taxes. The same went for the health system. At the end of the nineteenth century countries such as France, Germany and Japan began providing free health care for the masses. They financed vaccinations for infants, balanced diets for children and physical education for teenagers. They drained festering swamps, exterminated mosquitoes and built centralised sewage systems. The aim wasn’t to make people happy, but to make the nation stronger. The country needed sturdy soldiers and workers, healthy women who would give birth to more soldiers and workers, and bureaucrats who came to the office punctually at 8 a.m. instead of lying sick at home. Even the welfare system was originally planned in the interest of the nation rather than of needy individuals. When Otto von Bismarck pioneered state pensions and social security in late nineteenth-century Germany, his chief aim was to ensure the loyalty of the citizens rather than to increase their well-being. You fought for your country when you were eighteen, and paid your taxes when you were forty, because you counted on the state to take care of you when you were seventy.30 In 1776 the Founding Fathers of the United States established the right to the pursuit of happiness as one of three unalienable human rights, alongside the right to life and the right to liberty. It’s important to note, however, that the American Declaration of Independence guaranteed the right to the pursuit of happiness, not the right to happiness itself. Crucially, Thomas Jefferson did not make the state responsible for its citizens’ happiness. Rather, he sought only to limit the power of the state.

At the same time, it must be admitted that, unless words, to some extent, had fixed meanings, discourse would be impossible. Here again, however, it is easy to be too absolute. Words do change their meanings; take, for example, the word 'idea'. It is only by a considerable process of education that we learn to give to this word something like the meaning which Plato gave to it. It is necessary that the changes in the meanings of words should be slower than the changes that the words describe; but it is not necessary that there should be no changes in the meanings of words. Perhaps this does not apply to the abstract words of logic and mathematics, but these words, as we have seen, apply only to the form, not to the matter, of propositions. Here, again, we find that logic and mathematics are peculiar. Plato, under the influence of the Pythagoreans, assimilated other knowledge too much to mathematics. He shared this mistake with many of the greatest philosophers, but it was a mistake none the less.

There are two moments in the course of education where a lot of kids fall off the math train. The first comes in the elementary grades, when fractions are introduced. Until that moment, a number is a natural number, one of the figures 0, 1, 2, 3 . . . It is the answer to a question of the form “how many.”* To go from this notion, so primitive that many animals are said to understand it, to the radically broader idea that a number can mean “what portion of,” is a drastic philosophical shift. (“God made the natural numbers,” the nineteenth-century algebraist Leopold Kronecker famously said, “and all the rest is the work of man.”) The second dangerous twist in the track is algebra. Why is it so hard? Because, until algebra shows up, you’re doing numerical computations in a straightforwardly algorithmic way. You dump some numbers into the addition box, or the multiplication box, or even, in traditionally minded schools, the long-division box, you turn the crank, and you report what comes out the other side. Algebra is different. It’s computation backward.

unfairness can take many forms. It can take the form of the inheritance of property—bonds and stocks, houses, factories; it can also take the form of the inheritance of talent—musical ability, strength, mathematical genius. The inheritance of property can be interfered with more readily than the inheritance of talent. But from an ethical point of view, is there any difference between the two? Yet many people resent the inheritance of property but not the inheritance of talent. Look at the same issue from the point of view of the parent. If you want to assure your child a higher income in life, you can do so in various ways. You can buy him (or her) an education that will equip him to pursue an occupation yielding a high income; or you can set him up in a business that will yield a higher income than he could earn as a salaried employee; or you can leave him property, the income from which will enable him to live better. Is there any ethical difference among these three ways of using your property? Or again, if the state leaves you any money to spend over and above taxes, should the state permit you to spend it on riotous living but not to leave it to your children?

Let me return from history and draw my conclusion. What all this means to us at the present time is this: Our system has already passed its flowering. Some time ago it reached that summit of blessedness which the mysterious game of world history sometimes allows to things beautiful and desirable in themselves. We are on the downward slope. Our course may possible stretch out for a very long time, but in any case nothing finer, ore beautiful, and more desirable than what we have already had can henceforth be expected. The road leads downhill. Historically we are, I believe, ripe for dismantling. And there is no doubt that such will be our fate, not today or tomorrow, but the day after tomorrow. I do not draw this conclusion from any excessively moralistic estimate of our accomplishments and our abilities: I draw it far more from the movements which I see already on the way in the outside world. Critical times are approaching; the omens can be sensed everywhere; the world is once again about to shift its center of gravity. Displacements of power are in the offing. They will not take place without war and violence. From the Far East comes a threat not only to peace, but to life and liberty. Even if our country remains politically neutral, even if our whole nation unanimously abides by tradition (which is not the case) and attempts to remain faithful to Castalian ideals, that will be in vain. Some of our representatives in Parliament are already saying that Castalia is a rather expensive luxury for our country. The country may very soon be forced into a serious rearmament - armaments for defensive purposes only, of course - and great economies will be necessary. In spite of the government's benevolent disposition towards us, much of the economizing will strike us directly. We are proud that our Order and the cultural continuity it provides have cost the country as little as they have. In comparison with other ages, especially the early period of the Feuilletonistic Age with its lavishly endowed universities, its innumerable consultants and opulent institutes, this toll is really not large. It is infinitesimal compared with the sums consumed for war and armaments during the Century of Wars. But before too long this kind of armament may once again be the supreme necessity; the generals will again dominate Parliament; and if the people are confronted with the choice of sacrificing Castalia or exposing themselves to the danger of war and destruction, we know how they will choose. Undoubtedly a bellicose ideology will burgeon. The rash of propaganda will affect youth in particular. Then scholars and scholarship, Latin and mathematics, education and culture, will be considered worth their salt only to the extent that they can serve the ends of war.

Goodness and Reality being timeless, the best State will be the one which most nearly copies the heavenly model, by having a minimum of change and a maximum of static perfection, and its rulers should be those who best understand the eternal Good. In the second place: Plato, like all mystics, has, in his beliefs, a core of certainty which is essentially incommunicable except by a way of life. The Pythagoreans had endeavoured to set up a rule of the initiate, and this is, at bottom, what Plato desires. If a man is to be a good statesman, he must know the Good; this he can only do by a combination of intellectual and moral discipline. If those who have not gone through this discipline are allowed a share in the government, they will inevitably corrupt it. In the third place: much education is needed to make a good ruler on Plato's principles. It seems to us unwise to have insisted on teaching geometry to the younger Dionysius, tyrant of Syracuse, in order to make him a good king, but from Plato's point of view it was essential. He was sufficiently Pythagorean to think that without mathematics no true wisdom is possible. This view implies an oligarchy. In the fourth place: Plato, in common with most Greek philosophers, took the view that leisure is essential to wisdom, which will therefore not be found among those who have to work for their living, but only among those who have independent means or who are relieved by the State from anxieties as to their subsistence. This point of view is essentially aristocratic.

this I say,—we must never forget that all the education a man's head can receive, will not save his soul from hell, unless he knows the truths of the Bible. A man may have prodigious learning, and yet never be saved. He may be master of half the languages spoken round the globe. He may be acquainted with the highest and deepest things in heaven and earth. He may have read books till he is like a walking cyclopædia. He may be familiar with the stars of heaven,—the birds of the air,—the beasts of the earth, and the fishes of the sea. He may be able, like Solomon, to "speak of trees, from the cedar of Lebanon to the hyssop that grows on the wall, of beasts also, and fowls, and creeping things, and fishes." (1 King iv. 33.) He may be able to discourse of all the secrets of fire, air, earth, and water. And yet, if he dies ignorant of Bible truths, he dies a miserable man! Chemistry never silenced a guilty conscience. Mathematics never healed a broken heart. All the sciences in the world never smoothed down a dying pillow. No earthly philosophy ever supplied hope in death. No natural theology ever gave peace in the prospect of meeting a holy God. All these things are of the earth, earthy, and can never raise a man above the earth's level. They may enable a man to strut and fret his little season here below with a more dignified gait than his fellow-mortals, but they can never give him wings, and enable him to soar towards heaven. He that has the largest share of them, will find at length that without Bible knowledge he has got no lasting possession. Death will make an end of all his attainments, and after death they will do him no good at all. A man may be a very ignorant man, and yet be saved. He may be unable to read a word, or write a letter. He may know nothing of geography beyond the bounds of his own parish, and be utterly unable to say which is nearest to England, Paris or New York. He may know nothing of arithmetic, and not see any difference between a million and a thousand. He may know nothing of history, not even of his own land, and be quite ignorant whether his country owes most to Semiramis, Boadicea, or Queen Elizabeth. He may know nothing of the affairs of his own times, and be incapable of telling you whether the Chancellor of the Exchequer, or the Commander-in-Chief, or the Archbishop of Canterbury is managing the national finances. He may know nothing of science, and its discoveries,—and whether Julius Cæsar won his victories with gunpowder, or the apostles had a printing press, or the sun goes round the earth, may be matters about which he has not an idea. And yet if that very man has heard Bible truth with his ears, and believed it with his heart, he knows enough to save his soul. He will be found at last with Lazarus in Abraham's bosom, while his scientific fellow-creature, who has died unconverted, is lost for ever. There is much talk in these days about science and "useful knowledge." But after all a knowledge of the Bible is the one knowledge that is needful and eternally useful. A man may get to heaven without money, learning, health, or friends,—but without Bible knowledge he will never get there at all. A man may have the mightiest of minds, and a memory stored with all that mighty mind can grasp,—and yet, if he does not know the things of the Bible, he will make shipwreck of his soul for ever. Woe! woe! woe to the man who dies in ignorance of the Bible! This is the Book about which I am addressing the readers of these pages to-day. It is no light matter what you do with such a book. It concerns the life of your soul. I summon you,—I charge you to give an honest answer to my question. What are you doing with the Bible? Do you read it? HOW READEST THOU?

The goal was ambitious. Public interest was high. Experts were eager to contribute. Money was readily available. Armed with every ingredient for success, Samuel Pierpont Langley set out in the early 1900s to be the first man to pilot an airplane. Highly regarded, he was a senior officer at the Smithsonian Institution, a mathematics professor who had also worked at Harvard. His friends included some of the most powerful men in government and business, including Andrew Carnegie and Alexander Graham Bell. Langley was given a $50,000 grant from the War Department to fund his project, a tremendous amount of money for the time. He pulled together the best minds of the day, a veritable dream team of talent and know-how. Langley and his team used the finest materials, and the press followed him everywhere. People all over the country were riveted to the story, waiting to read that he had achieved his goal. With the team he had gathered and ample resources, his success was guaranteed. Or was it? A few hundred miles away, Wilbur and Orville Wright were working on their own flying machine. Their passion to fly was so intense that it inspired the enthusiasm and commitment of a dedicated group in their hometown of Dayton, Ohio. There was no funding for their venture. No government grants. No high-level connections. Not a single person on the team had an advanced degree or even a college education, not even Wilbur or Orville. But the team banded together in a humble bicycle shop and made their vision real. On December 17, 1903, a small group witnessed a man take flight for the first time in history. How did the Wright brothers succeed where a better-equipped, better-funded and better-educated team could not? It wasn’t luck. Both the Wright brothers and Langley were highly motivated. Both had a strong work ethic. Both had keen scientific minds. They were pursuing exactly the same goal, but only the Wright brothers were able to inspire those around them and truly lead their team to develop a technology that would change the world. Only the Wright brothers started with Why. 2.

Queen Anne of England established the Longitude Act in 1714, and offered a monetary prize of over a million in today’s dollars to anyone who invented a method to accurately calculate longitude at sea. Longitude is about determining one’s point in space. So one might ask what it has to do with clocks? Mathematically speaking, space (distance) is the child of time and speed (distance equals time multiplied by speed). Thus, anything that moves at a constant speed can be used to calculate distance, provided one knows for how long it has been moving. Many things have constant speeds, including light, sound, and the rotation of the Earth. Your brain uses the near constancy of the speed of sound to calculate where sounds are coming from. As we have seen, you know someone is to your left or right because the sound of her voice takes approximately 0.6 milliseconds to travel from your left to your right ear. Using the delays it takes any given sound to arrive to your left and right ears allows the brain to figure out if the voice is coming directly from the left, the right, or somewhere in between. The Earth is rotating at a constant speed—one that results in a full rotation (360 degrees) every 24 hours. Thus there is a direct correspondence between degrees of longitude and time. Knowing how much time has elapsed is equivalent to knowing how much the Earth has turned: if you sit and read this book for one hour (1/24 of a day), the Earth has rotated 15 degrees (360/24). Thus, if you are sitting in the middle of the ocean at local noon, and you know it is 16:00 in Greenwich, then you are “4 hours from Greenwich”—exactly 60 degrees longitude from Greenwich. Problem solved. All one needs is a really good marine chronometer. The greatest minds of the seventeenth and eighteenth centuries could not overlook the longitude problem: Galileo Galilei, Blaise Pascal, Robert Hooke, Christiaan Huygens, Gottfried Leibniz, and Isaac Newton all devoted their attention to it. In the end, however, it was not a great scientist but one of the world’s foremost craftsman who ultimately was awarded the Longitude Prize. John Harrison (1693–1776) was a self-educated clockmaker who took obsessive dedication to the extreme.

Our political system today does not engage the best minds in our country to help us get the answers and deploy the resources we need to move into the future. Bringing these people in—with their networks of influence, their knowledge, and their resources—is the key to creating the capacity for shared intelligence that we need to solve the problems we face, before it’s too late. Our goal must be to find a new way of unleashing our collective intelligence in the same way that markets have unleashed our collective productivity. “We the people” must reclaim and revitalize the ability we once had to play an integral role in saving our Constitution. The traditional progressive solution to problems that involve a lack of participation by citizens in civic and democratic processes is to redouble their emphasis on education. And education is, in fact, an extremely valuable strategy for solving many of society’s ills. In an age where information has more economic value than ever before, it is obvious that education should have a higher national priority. It is also clear that democracies are more likely to succeed when there is widespread access to high-quality education. Education alone, however, is necessary but insufficient. A well-educated citizenry is more likely to be a well-informed citizenry, but the two concepts are entirely different, one from the other. It is possible to be extremely well educated and, at the same time, ill informed or misinformed. In the 1930s and 1940s, many members of the Nazi Party in Germany were extremely well educated—but their knowledge of literature, music, mathematics, and philosophy simply empowered them to be more effective Nazis. No matter how educated they were, no matter how well they had cultivated their intellect, they were still trapped in a web of totalitarian propaganda that mobilized them for evil purposes. The Enlightenment, for all of its liberating qualities—especially its empowerment of individuals with the ability to use reason as a source of influence and power—has also had a dark side that thoughtful people worried about from its beginning. Abstract thought, when organized into clever, self-contained, logical formulations, can sometimes have its own quasi-hypnotic effect and so completely capture the human mind as to shut out the leavening influences of everyday experience. Time and again, passionate believers in tightly organized philosophies and ideologies have closed their minds to the cries of human suffering that they inflict on others who have not yet pledged their allegiance and surrendered their minds to the same ideology. The freedoms embodied in our First Amendment represented the hard-won wisdom of the eighteenth century: that individuals must be able to fully participate in challenging, questioning, and thereby breathing human values constantly into the prevailing ideologies of their time and sharing with others the wisdom of their own experience.