Mathematics In Nature Quotes

The science of government it is my duty to study, more than all other sciences; the arts of legislation and administration and negotiation ought to take the place of, indeed exclude, in a manner, all other arts. 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.

I am convinced that the act of thinking logically cannot possibly be natural to the human mind. If it were, then mathematics would be everybody's easiest course at school and our species would not have taken several millennia to figure out the scientific method.

But in my opinion, all things in nature occur mathematically.

Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

This most beautiful system of the sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being... This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont, to be called Lord God παντοκρατωρ or Universal Ruler.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.

The laws of nature are but the mathematical thoughts of God.

[Grief] is everything. It is the fabric of selfhood, and beautifully chaotic. It shares mathematical characteristics with many natural forms.

The law of gravity and gravity itself did not exist before Isaac Newton." ...and what that means is that that law of gravity exists nowhere except in people's heads! It 's a ghost!" Mind has no matter or energy but they can't escape its predominance over everything they do. Logic exists in the mind. numbers exist only in the mind. I don't get upset when scientists say that ghosts exist in the mind. it's that only that gets me. science is only in your mind too, it's just that that doesn't make it bad. or ghosts either." Laws of nature are human inventions, like ghosts. Law of logic, of mathematics are also human inventions, like ghosts." ...we see what we see because these ghosts show it to us, ghosts of Moses and Christ and the Buddha, and Plato, and Descartes, and Rousseau and Jefferson and Lincoln, on and on and on. Isaac Newton is a very good ghost. One of the best. Your common sense is nothing more than the voices of thousands and thousands of these ghosts from the past.

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

[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.

There’s no better guarantee of failure than convincing yourself that success is impossible, and therefore never even trying.

Why nature is mathematical is, again, a mystery.

It is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.

Where there was nature and earth, life and water, I saw a desert landscape that was unending, resembling some sort of crater, so devoid of reason and light and spirit that the mind could not grasp it on any sort of conscious level and if you came close the mind would reel backward, unable to take it in. It was a vision so clear and real and vital to me that in its purity it was almost abstract. This was what I could understand, this was how I lived my life, what I constructed my movement around, how I dealt with the tangible. This was the geography around which my reality revolved: it did not occur to me, ever, that people were good or that a man was capable of change or that the world could be a better place through one’s own taking pleasure in a feeling or a look or a gesture, of receiving another person’s love or kindness. Nothing was affirmative, the term “generosity of spirit” applied to nothing, was a cliche, was some kind of bad joke. Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire- meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface, was all that anyone found meaning in…this was civilization as I saw it, colossal and jagged…

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

God without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and everywhere, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing.

The Hitchhiker's Guide to the Galaxy's definition of "Universe": The Universe is a very big thing that contains a great number of planets and a great number of beings. It is Everything. What we live in. All around us. The lot. Not nothing. It is quite difficult to actually define what the Universe means, but fortunately the Guide doesn't worry about that and just gives us some useful information to live in it. Area: The area of the Universe is infinite. Imports: None. This is a by product of infinity; it is impossible to import things into something that has infinite volume because by definition there is no outside to import things from. Exports: None, for similar reasons as imports. Population: None. Although you might see people from time to time, they are most likely products of your imagination. Simple mathematics tells us that the population of the Universe must be zero. Why? Well given that the volume of the universe is infinite there must be an infinite number of worlds. But not all of them are populated; therefore only a finite number are. Any finite number divided by infinity is zero, therefore the average population of the Universe is zero, and so the total population must be zero. Art: None. Because the function of art is to hold a mirror up to nature there can be no art because the Universe is infinite which means there simply isn't a mirror big enough. Sex: None. Although in fact there is quite a lot, given the zero population of the Universe there can in fact be no beings to have sex, and therefore no sex happens in the Universe.

Naturally, we are inclined to be so mathematical and calculating that we look upon uncertainty as a bad thing...Certainty is the mark of the common-sense life. To be certain of God means that we are uncertain in all our ways, we do not know what a day may bring forth. This is generally said with a sigh of sadness; it should rather be an expression of breathless expectation.

I think that consciousness is the way information feels when being processed in certain complex ways.

I must study politics and war that my sons may have liberty to study mathematics and philosophy. My sons ought to study mathematics and philosophy, geography, natural history, naval architecture, navigation, commerce, and agriculture in order to give their children a right to study paintings, poetry, music, architecture, statuary, tapestry, and porcelain.

I should attempt to treat human vice and folly geometrically... the passions of hatred, anger, envy, and so on, considered in themselves, follow from the necessity and efficacy of nature... I shall, therefore, treat the nature and strength of the emotion in exactly the same manner, as though I were concerned with lines, planes, and solids.

Real knowledge is to know the extent of one’s ignorance. —Confucius The highest form of ignorance is when you reject something you don’t know anything about. —Wayne Dyer

Every formula which expresses a law of nature is a hymn of praise to God.

I wanted very much to learn to draw, for a reason that I kept to myself: I wanted to convey an emotion I have about the beauty of the world. It's difficult to describe because it's an emotion. It's analogous to the feeling one has in religion that has to do with a god that controls everything in the whole universe: there's a generality aspect that you feel when you think about how things that appear so different and behave so differently are all run "behind the scenes" by the same organization, the same physical laws. It's an appreciation of the mathematical beauty of nature, of how she works inside; a realization that the phenomena we see result from the complexity of the inner workings between atoms; a feeling of how dramatic and wonderful it is. It's a feeling of awe — of scientific awe — which I felt could be communicated through a drawing to someone who had also had this emotion. It could remind him, for a moment, of this feeling about the glories of the universe.

Why do we perceive the world as stable and ourselves as local and unique? Here’s my guess: because it’s useful.

Mathematics is not a science from our point of view, in the sense that it is not a natural science. The test of its validity is not experiment.

Seni sama pentingnya dengan matematika. Seni memanusiakan manusia. Seni menciptakan rasa empati

If nature leads us to mathematical forms of great simplicity and beauty—by forms, I am referring to coherent systems of hypotheses, axioms, etc.—to forms that no one has previously encountered, we cannot help thinking that they are “true,” that they reveal a genuine feature of nature…. You must have felt this too: the almost frightening simplicity and wholeness of the relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.

And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object--that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated--chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point!

The calculative exactness of practical life which the money economy has brought about corresponds to the ideal of natural science: to transform the world by mathematical formulas. Only money economy has filled the days of so many people with weighing, calculating, with numerical determinations, with a reduction of qualitative values to quantitative ones.

I feel that my main responsibility as a teacher isn’t to convey facts, but to rekindle that lost enthusiasm for asking questions.

This most beautiful system [The Universe] could only proceed from the dominion of an intelligent and powerful Being.

You know, people think mathematics is complicated. Mathematics is the simple bit. It's the stuff we can understand. It's cats that are complicated. I mean, what is it in those little molecules and stuff that make one cat behave differently than another, or that make a cat? And how do you define a cat? I have no idea.

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.

So I feel that the experimental verdict is in: the world is weird, and we just have to learn to live with it.

Eternal truths are ultimately invisible, and you won't find them in material things or natural phenomenon, or even in human emotions. Mathematics, however, can illuminate them, can give the expression – in fact, nothing can prevent it from doing so.

Codes and patterns are very different from each other,” Langdon said. “And a lot of people confuse the two. In my field, it’s crucial to understand their fundamental difference.” “That being?” Langdon stopped walking and turned to her. “A pattern is any distinctly organized sequence. Patterns occur everywhere in nature—the spiraling seeds of a sunflower, the hexagonal cells of a honeycomb, the circular ripples on a pond when a fish jumps, et cetera.” “Okay. And codes?” “Codes are special,” Langdon said, his tone rising. “Codes, by definition, must carry information. They must do more than simply form a pattern—codes must transmit data and convey meaning. Examples of codes include written language, musical notation, mathematical equations, computer language, and even simple symbols like the crucifix. All of these examples can transmit meaning or information in a way that spiraling sunflowers cannot.

The key point to keep in mind, however, is that symmetry is one of the most important tools in deciphering nature's design.

economics was largely a form of intellectual prostitution where you got rewarded for saying what the powers that be wanted to hear.

Our craving for generality has [as one] source … our preoccupation with the method of science. I mean the method of reducing the explanation of natural phenomena to the smallest possible number of primitive natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization. Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer in the way science does. This tendency is the real source of metaphysics, and leads the philosopher into complete darkness. I want to say here that it can never be our job to reduce anything to anything, or to explain anything. Philosophy really is “purely descriptive.

Some believers accuse skeptics of having nothing left but a dull, cold, scientific world. I am left with only art, music, literature, theatre, the magnificence of nature, mathematics, the human spirit, sex, the cosmos, friendship, history, science, imagination, dreams, oceans, mountains, love, and the wonder of birth. That’ll do for me.

Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.

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.

There cannot be a language more universal and more simple, more free from errors and obscurities...more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes

It is convenient for the old men to blame Eve. To insist we are damned because a country girl talked to the snake one afternoon long ago. Children must starve in Somalia for that, and old women be abandoned in our greatest cities. It’s why we will finally be thrown into the lakes of molten lead. Because she was confused by happiness that first time anyone said she was beautiful. Nevertheless, she must be the issue, so people won’t notice that rocks and galaxies, mathematics and rust are also created in His image.

The world of strict naturalism in which clever mathematical laws all by themselves bring the universe and life into existence, is pure (and, one might add, poor) fiction. To call it science-fiction would besmirch the name of science.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty— a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

The Golden Number is a mathematical definition of a proportional function which all of nature obeys, whether it be a mollusk shell, the leaves of plants, the proportions of the animal body, the human skeleton, or the ages of growth in man.

Mathematics, natural science, laws, arts, even morality, etc. do not completely fill the soul; there is always a space left over reserved for pure and speculative reason, the emptiness of which prompts us to seek in vagaries, buffooneries, and mysticism for what seems to be employment and entertainment, but what actually is mere pastime undertaken in order to deaden the troublesome voice of reason, which, in accordance with its nature, requires something that can satisfy it and does not merely subserve other ends or the interests of our inclinations.

This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centers of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One.

And, indeed, this is the odd thing that is continually happening: there are continually turning up in life moral and rational persons, sages and lovers of humanity who make it their object to live all their lives as morally and rationally as possible, to be, so to speak, a light to their neighbours simply in order to show them that it is possible to live morally and rationally in this world. And yet we all know that those very people sooner or later have been false to themselves, playing some queer trick, often a most unseemly one. Now I ask you: what can be expected of man since he is a being endowed with strange qualities? Shower upon him every earthly blessing, drown him in a sea of happiness, so that nothing but bubbles of bliss can be seen on the surface; give him economic prosperity, such that he should have nothing else to do but sleep, eat cakes and busy himself with the continuation of his species, and even then out of sheer ingratitude, sheer spite, man would play you some nasty trick. He would even risk his cakes and would deliberately desire the most fatal rubbish, the most uneconomical absurdity, simply to introduce into all this positive good sense his fatal fantastic element. It is just his fantastic dreams, his vulgar folly that he will desire to retain, simply in order to prove to himself--as though that were so necessary-- that men still are men and not the keys of a piano, which the laws of nature threaten to control so completely that soon one will be able to desire nothing but by the calendar. And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point!

Physics, mathematics, music, painting, my politics, my love for you, my work, the star-dust of my body, the spirit that impels it, clocks diurnal, time perpetual, the roll, rough, tender, swamping, liberating, breathing, moving, thinking nature, human nature and the cosmos are patterned together.

At night I would return home, set out a lamp before me, and devote myself to reading and writing. Whenever sleep overcame me or I became conscious of weakening, I would turn aside to drink a cup of wine, so that my strength would return to me. Then I would return to reading. And whenever sleep seized me I would see those very problems in my dream; and many questions became clear to me in my sleep. I continued in this until all of the sciences were deeply rooted within me and I understood them as is humanly possible. Everything which I knew at the time is just as I know it now; I have not added anything to it to this day. Thus I mastered the logical, natural, and mathematical sciences, and I had now reached the science.

When we find that God's ways always coincide with our own ways, it's time to question who we're really worshipping, God or ourselves. The latter moves the nature of godliness from the King to our servant to a slave, a deduction into the realm of selfhood and then the lower, slavehood. It's a spiritual mathematics in that men who need God in his godhood are humble yet strong and spiritually ambitious while men who need a slave in their selfhood are ultimately paralyzed and will remain paralyzed.

The mathematical likelihood that God exists is astronomical. But the idea that our limited human mentality can understand the true nature of God borders on the ridiculous. For this reason the religionist and the ahteist are equally naive.

Take this neat little equation here. It tells me all the ways an electron can make itself comfortable in or around an atom. That's the logic of it. The poetry of it is that the equation tells me how shiny gold is, how come rocks are hard, what makes grass green, and why you can't see the wind. And a million other things besides, about the way nature works.

I’ve come to believe that genius is an exceedingly common human quality, probably natural to most of us. I didn’t want to accept that notion — far from it: my own training in two elite universities taught me that intelligence and talent distributed themselves economically over a bell curve and that human destiny, because of those mathematical, seemingly irrefutable scientific facts, was as rigorously determined as John Calvin contended.

All nature has come to expect from God a sense of orderliness. Whatever God does carries with it His fingerprint. And in the world around us His fingerprint of orderliness is evident to anybody who is honest with the facts. If you look at nature, you will discover a mathematical exactness. Without this precision, the entire world would be in utter confusion. One plus one always equals two no matter what part of the universe you happen to be in. And the laws of nature operate in beautiful harmony, a harmony that is ordered by God Himself.

While ritual, emotion and reasoning are all significant aspects of human nature, the most nearly unique human characteristic is the ability to associate abstractly and to reason. Curiosity and the urge to solve problems are the emotional hallmarks of our species; and the most characteristically human activities are mathematics, science, technology, music and the arts--a somewhat broader range of subjects than is usually included under the "humanities." Indeed, in its common usage this very word seems to reflect a peculiar narrowness of vision about what is human. Mathematics is as much a "humanity" as poetry.

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.

The nature of a letter can also be revealed within its numeric value. All letters and numbers behave in a certain but recognizable way, from which we can deduce its nature. The number two is the only even prime. There is an inherent mathematical dilemma with, “one.” No matter how many times you multiply it, by itself, you still can’t get past “one” (1 x 1 x 1 x 1 = 1). So, how does “one” move beyond itself? How does the same, produce the different? Mathematically, “one” is forced to divide itself and work from that duality. Therein, hides the divine puzzle of bet (b). To become “two,” the second must revolt from wholeness—a separation. Yet, the second could not have existed without the benefit of the original wholeness. Also, the first wanted the second to exist, but the first doesn’t know what the second will become. Again, two contains potential badness, to a Hebrew. (Ge 25:24)

Seven is my favorite number," he said. "Why?" He nuzzled gently at her stomach. "There are seven colors in a rainbow, seven days of the week, and..." His voice lowered seductively, "...seven is the lowest natural number that can't be represented as the sum of the squares of three integers." "Mathematics," she exclaimed, laughing breathlessly. "How stirring.

Language as putative science. - The significance of language for the evolution of culture lies in this, that mankind set up in language a separate world beside the other world, a place it took to be so firmly set that, standing upon it, it could lift the rest of the world off its hinges and make itself master of it. To the extent that man has for long ages believed in the concepts and names of things as in aeternae veritates he has appropriated to himself that pride by which he raised himself above the animal: he really thought that in language he possessed knowledge of the world. The sculptor of language was not so modest as to believe that he was only giving things designations, he conceived rather that with words he was expressing supreame knowledge of things; language is, in fact, the first stage of occupation with science. Here, too, it is the belief that the truth has been found out of which the mightiest sources of energy have flowed. A great deal later - only now - it dawns on men that in their belief in language they have propagated a tremendous error. Happily, it is too late for the evolution of reason, which depends on this belief, to be put back. - Logic too depends on presuppositions with which nothing in the real world corresponds, for example on the presupposition that there are identical things, that the same thing is identical at different points of time: but this science came into existence through the opposite belief (that such conditions do obtain in the real world). It is the same with mathematics, which would certainly not have come into existence if one had known from the beginning that there was in nature no exactly straight line, no real circle, no absolute magnitude.

Of course, reading novels was just another form of escape. As soon as he closed their pages he had to come back to the real world. But at some point Tengo noticed that returning to reality from the world of a novel was not as devastating a blow as returning from the world of mathematics. Why should that have been? After much deep thought, he reached a conclusion. No matter how clear the relationships of things might become in the forest of story, there was never a clear-cut solution. That was how it differed from math. The role of a story was, in the broadest terms, to transpose a single problem into another form. Depending on the nature and direction of the problem, a solution could be suggested in the narrative. Tengo would return to the real world with that suggestion in hand. It was like a piece of paper bearing the indecipherable text of a magic spell. At times it lacked coherence and served no immediate practical purpose. But it would contain a possibility. Someday he might be able to decipher the spell. That possibility would gently warm his heart from within.

A painter, who finds no satisfaction in mere representation, however artistic, in his longing to express his inner life, cannot but envy the ease with which music, the most non-material of the arts today, achieves this end. He naturally seeks to apply the methods of music to his own art. And from this results that modern desire for rhythm in painting, for mathematical, abstract construction, for repeated notes of colour, for setting colour in motion.

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.

. . . we come astonishingly close to the mystical beliefs of Pythagoras and his followers who attempted to submit all of life to the sovereignty of numbers. Many of our psychologists, sociologists, economists and other latter-day cabalists will have numbers to tell them the truth or they will have nothing. . . . We must remember that Galileo merely said that the language of nature is written in mathematics. He did not say that everything is. And even the truth about nature need not be expressed in mathematics. For most of human history, the language of nature has been the language of myth and ritual. These forms, one might add, had the virtues of leaving nature unthreatened and of encouraging the belief that human beings are part of it. It hardly befits a people who stand ready to blow up the planet to praise themselves too vigorously for having found the true way to talk about nature.

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.

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.

The best thing for being sad,” replied Merlyn, beginning to puff and blow, “is to learn something. That is the only thing that never fails. You may grow old and trembling in your anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then—to learn. Learn why the world wags and what wags it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the thing for you. Look at what a lot of things there are to learn—pure science, the only purity there is. You can learn astronomy in a lifetime, natural history in three, literature in six. And then, after you have exhausted a milliard lifetimes in biology and medicine and theocriticism and geography and history and economics—why, you can start to make a cartwheel out of the appropriate wood, or spend fifty years learning to begin to learn to beat your adversary at fencing. After that you can start again on mathematics, until it is time to learn to plough.

Modern science was born through the Scientific Revolution in the 11th/17th century at a time when, as we saw earlier, European philosophy had itself rebelled against revelation and the religious world view. The background of modern science is a particular philosophical outlook which sees the parameters of the physical world, that is, space, time, matter and energy to be realities that are independent of higher orders of being and cut off from the power of God, at least during the unfolding of the history of the cosmos. It views the physical world as being primarily the subject of mathematicization and quatification and, in a sense, absolutizes the mathematical study of nature relegating the non-quantifiable aspects of physical existence to irrelevance.

Our experience teaches us that there are indeed laws of nature, regularities in the way things behave, and that these laws are best expressed using the language of mathematics. This raises the interesting possibility that mathematical consistency might be used to guide us, along with experimental observation, to the laws that describe physical reality, and this has proved to be the case time and again throughout the history of science. We will see this happen during the course of this book, and it is truly one of the wonderful mysteries of our universe that it should be so.

Like Molière’s M. Jourdain, who spoke prose all his life without knowing it, mathematicians have been reasoning for at least two millennia without being aware of all the principles underlying what they were doing. The real nature of the tools of their craft has become evident only within recent times A renaissance of logical studies in modern times begins with the publication in 1847 of George Boole’s 'The Mathematical Analysis of Logic'.

It is impossible to derive happiness from the company of those whom we deprive of happiness. To be happy in old age it is necessary that we accustom ourselves to objects than can accompany the mind all the way through life, and that we take the rest as good in their day. The mere man of pleasure is miserable in old age; and the mere drudge in business is but little better: whereas, natural philosophy, mathematical and mechanical science, are a continual source of tranquil pleasure, and in spite of the gloomy study of the true theology; it teaches man to know and to admire the Creator, for the principles of science are in the creation, and are unchangeable, and of divine origin.

From all this we concluded that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of the matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry (tr. Toomer, p. 6).

For the purposes of science, information had to mean something special. Three centuries earlier, the new discipline of physics could not proceed until Isaac Newton appropriated words that were ancient and vague—force, mass, motion, and even time—and gave them new meanings. Newton made these terms into quantities, suitable for use in mathematical formulas. Until then, motion (for example) had been just as soft and inclusive a term as information. For Aristotelians, motion covered a far-flung family of phenomena: a peach ripening, a stone falling, a child growing, a body decaying. That was too rich. Most varieties of motion had to be tossed out before Newton’s laws could apply and the Scientific Revolution could succeed. In the nineteenth century, energy began to undergo a similar transformation: natural philosophers adapted a word meaning vigor or intensity. They mathematicized it, giving energy its fundamental place in the physicists’ view of nature. It was the same with information. A rite of purification became necessary. And then, when it was made simple, distilled, counted in bits, information was found to be everywhere.

The best thing for being sad,’ replied Merlyn, beginning to puff and blow, ‘is to learn something. That is the only thing that never fails. You may grow old and trembling in you anatomies, you may lie awake at night listening to the disorder of your veins, you may miss your only love, you may see the world about you devastated by evil lunatics, or know your honour trampled in the sewers of baser minds. There is only one thing for it then – to learn. Learn why the world wags and what wags in it. That is the only thing which the mind can never exhaust, never alienate, never be tortured by, never fear or distrust, and never dream of regretting. Learning is the thing for you. Look at what a lot of things there are to learn – pure science, the only purity there is. You can learn astronomy in a lifetime, natural history in three, literature in six. And then, after you have exhausted a milliard lifetimes in biology and medicine and theocriticism and geography and history and economics – why, you can start to make a cartwheel out of the appropriate wood, or spend fifty years learning to begin to learn to beat your adversary at fencing. After that you can start again on mathematics, until it is time to learn to plough.

Kepler's laws, although not rigidly true, are sufficiently near to the truth to have led to the discovery of the law of attraction of the bodies of the solar system. The deviation from complete accuracy is due to the facts, that the planets are not of inappreciable mass, that, in consequence, they disturb each other's orbits about the Sun, and, by their action on the Sun itself, cause the periodic time of each to be shorter than if the Sun were a fixed body, in the subduplicate ratio of the mass of the Sun to the sum of the masses of the Sun and Planet; these errors are appreciable although very small, since the mass of the largest of the planets, Jupiter, is less than 1/1000th of the Sun's mass.

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

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.

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.

THE BOTTOM LINE •  Parallel universes are not a theory, but a prediction of certain theories. •  Eternal inflation predicts that our Universe (the spherical region of space from which light has had time to reach us during the 14 billion years since our Big Bang) is just one of infinitely many universes in a Level I multiverse where everything that can happen does happen somewhere. •  For a theory to be scientific, we need not be able to observe and test all its predictions, merely at least one of them. Inflation is the leading theory for our cosmic origins because it’s passed observational tests, and parallel universes seem to be a non-optional part of the package. •  Inflation converts potentiality into reality: if the mathematical equations governing

...where there was nature and earth, life and water, I saw a desert landscape that was unending, resembling some sort of crater, so devoid of reason and light and spirit that the mind could not grasp it on any sort of conscious level and if you came close the mind would reel backward, unable to take it in. It was a vision so clear and real and vital to me that in its purity it was almost abstract. This was what I could understand, this was how I lived my life, what I constructed my movement around, how I dealt with the tangible. This was the geography around which my reality revolved: it did not occur to me, ever, that people were good or that a man was capable of change or that the world could be a better place through one's taking pleasure in a feeling or a look or a gesture, of receiving another person's love or kindness. Nothing was affirmative, the term "generosity of spirit" applied to nothing, was a cliche, was some kind of bad joke. Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire - meaningless. Intellect is not a cure. Justice is dead. Fear, recrimination, innocence, sympathy, guilt, waste, failure, grief, were things, emotions, that no one really felt anymore. Reflection is useless, the world is senseless. Evil is its only permanence. God is not alive. Love cannot be trusted. Surface, surface, surface was all that anyone found meaning in... this was civilization as I saw it, colossal and jagged...

It must be this overarching commitment to what is really an abstraction, to one's children right or wrong, that can be even more fierce than the commitment to them as explicit, difficult people, and that can consequently keep you devoted to them when as individuals they disappoint. On my part it was this broad covenant with children-in-theory that I may have failed to make and to which I was unable to resort when Kevin finally tested my maternal ties to a perfect mathematical limit on Thursday. I didn't vote for parties, but for candidates. My opinions were as ecumenical as my larder, then still chock full of salsa verde from Mexico City, anchovies from Barcelona, lime leaves from Bangkok. I had no problem with abortion but abhorred capital punishment, which I suppose meant that I embraced the sanctity of life only in grown-ups. My environmental habits were capricious; I'd place a brick in our toilet tank, but after submitting to dozens of spit-in-the-air showers with derisory European water pressure, I would bask under a deluge of scalding water for half an hour. My closet wafter with Indian saris, Ghanaian wraparounds, and Vietnamese au dais. My vocabulary was peppered with imports -- gemutlich, scusa, hugge, mzungu. I so mixed and matched the planet that you sometimes worried I had no commitments to anything or anywhere, though you were wrong; my commitments were simply far-flung and obscenely specific. By the same token, I could not love a child; I would have to love this one. I was connected to the world by a multitude of threads, you by a few sturdy guide ropes. It was the same with patriotism: You loved the idea of the United States so much more powerfully than the country itself, and it was thanks to your embrace of the American aspiration that you could overlook the fact that your fellow Yankee parents were lining up overnight outside FAO Schwartz with thermoses of chowder to buy a limited release of Nintendo. In the particular dwells the tawdry. In the conceptual dwells the grand, the transcendent, the everlasting. Earthly countries and single malignant little boys can go to hell; the idea of countries and the idea of sons triumph for eternity. Although neither of us ever went to church, I came to conclude that you were a naturally religious person.

Moreover: then, you say, science itself will teach man (though this is really a luxury in my opinion) that in fact he has neither will nor caprice, and never did have any, and that he himself is nothing but a sort of piano key or a sprig in an organ;14 and that, furthermore, there also exist in the world the laws of nature; so that whatever he does is done not at all according to his own wanting, but of itself, according to the laws of nature. Consequently, these laws of nature need only be discovered, and then man will no longer be answerable for his actions, and his life will become extremely easy. Needless to say, all human actions will then be calculated according to these laws, mathematically, like a table of logarithms, up to 108,000, and entered into a calendar; or, better still, some well-meaning publications will appear, like the present-day encyclopedic dictionaries, in which everything will be so precisely calculated and designated that there will no longer be any actions or adventures in the world.

The peculiarity of the evidence of mathematical truths is, that all the argument is on one side. There are no objections, and no answers to objections. But on every subject on which difference of opinion is possible, the truth depends on a balance to be struck between two sets of confliting reasons. Even in natural philosophy, there is always some other explanation possible of the same facts; some geocentric theory instead of heliocentric, some phlogiston instead of oxygen; and it has to be shown why that other theory cannot be the true on: and until this is shown, and until we know how it is shown, we do not understand the grounds of our opinion. But when we turn to subjects infinitely more complicated, to morals, religion, politics, social relations, and the business of life, three-fourths of the arguments for every disputed opinion consist in dispelling the appearances which favour some opinion different from it.

And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object--that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated--chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! It may be at the cost of his skin, it may be by cannibalism! And this being so, can one help being tempted to rejoice that it has not yet come off, and that desire still depends on something we don't know?

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.

Now I ask you: what can be expected of man since he is a being endowed with strange qualities? Shower upon him every earthly blessing, drown him in a sea of happiness, so that nothing but bubbles of bliss can be seen on the surface; give him economic prosperity, such that he should have nothing else to do but sleep, eat cakes and busy himself with the continuation of his species, and even then out of sheer ingratitude, sheer spite, man would play you some nasty trick. He would even risk his cakes and would deliberately desire the most fatal rubbish, the most uneconomical absurdity, simply to introduce into all this positive good sense his fatal fantastic element. It is just his fantastic dreams, his vulgar folly that he will desire to retain, simply in order to prove to himself—as though that were so necessary—that men still are men and not the keys of a piano, which the laws of nature threaten to control so completely that soon one will be able to desire nothing but by the calendar. And that is not all: even if man really were nothing but a piano-key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of simple ingratitude, simply to gain his point. And if he does not find means he will contrive destruction and chaos, will contrive sufferings of all sorts, only to gain his point! He will launch a curse upon the world, and as only man can curse (it is his privilege, the primary distinction between him and other animals), may be by his curse alone he will attain his object—that is, convince himself that he is a man and not a piano-key! If you say that all this, too, can be calculated and tabulated—chaos and darkness and curses, so that the mere possibility of calculating it all beforehand would stop it all, and reason would reassert itself, then man would purposely go mad in order to be rid of reason and gain his point! I believe in it, I answer for it, for the whole work of man really seems to consist in nothing but proving to himself every minute that he is a man and not a piano-key! It may be at the cost of his skin, it may be by cannibalism! And this being so, can one help being tempted to rejoice that it has not yet come off, and that desire still depends on something we don’t know?

As always when he worked with this much concentration he began to feel a sense of introverting pressure. There was no way out once he was in, no genuine rest, no one to talk to who was capable of understanding the complexity (simplicity) of the problem or the approaches to a tentative solution. There came a time in every prolonged effort when he had a moment of near panic, or "terror in a lonely place," the original semantic content of the word. The lonely place was his own mind. As a mathematician he was free from subjection to reality, free to impose his ideas and designs on his own test environment. The only valid standard for his work, its critical point (zero or infinity), was the beauty it possessed, the deft strength of his mathematical reasoning. THe work's ultimate value was simply what it revealed about the nature of his intellect. What was at stake, in effect, was his own principle of intelligence or individual consciousness; his identity, in short. This was the infalling trap, the source of art's private involvement with obsession and despair, neither more nor less than the artist's self-containment, a mental state that led to storms of overwork and extended stretches of depression, that brought on indifference to life and at times the need to regurgitate it, to seek the level of expelled matter. Of course, the sense at the end of a serious effort, if the end is reached successfully, is one of lyrical exhilaration. There is air to breathe and a place to stand. The work gradually reveals its attachment to the charged particles of other minds, men now historical, the rediscovered dead; to the main structure of mathematical thought; perhaps even to reality itself, the so-called sum of things. It is possible to stand in time's pinewood dust and admire one's own veronicas and pavanes.

Human nature is not taken into account, it is excluded, it's not supposed to exist! They don't recognise that humanity, developing by a historical living process, will become at last a normal society, but they believe that a social system that has come out of some mathematical brain is going to organise all humanity at once and make it just and sinless in an instant, quicker than any living process! That's why they instinctively dislike history, 'nothing but ugliness and stupidity in it,' and they explain it all as stupidity! That's why they so dislike the living process of life; they don't want a living soul! The living soul demands life, the soul won't obey the rules of mechanics, the soul is an object of suspicion, the soul is retrograde! But what they want though it smells of death and can be made of india-rubber, at least is not alive, has no will, is servile and won't revolt! And it comes in the end to their reducing everything to the building of walls and the planning of rooms and passages in a phalanstery! The phalanstery is ready, indeed, but your human nature is not ready for the phalanstery--it wants life, it hasn't completed its vital process, it's too soon for the graveyard! You can't skip over nature by logic.

Q. Would you repeat, Dr. Seldon, your thoughts concerning the future of Trantor? A. I have said, and I say again, that Trantor will lie in ruins within the next three centuries. Q. You do not consider your statement a disloyal one? A. No, sir. Scientific truth is beyond loyalty and disloyalty." Q. You are sure that your statement represents scientific truth? A. I am. Q. On what basis? A. On the basis of the mathematics of psychohistory. Q. Can you prove that this mathematics is valid? A. Only to another mathematician. Q. ( with a smile) Your claim then is that your truth is of so esoteric a nature that it is beyond the understanding of a plain man. It seems to me that truth should be clearer than that, less mysterious, more open to the mind. A. It presents no difficulties to some minds. The physics of energy transfer, which we know as thermodynamics, has been clear and true through all the history of man since the mythical ages, yet there may be people present who would find it impossible to design a power engine. People of high intelligence, too. I doubt if the learned Commissioners— At this point, one of the Commissioners leaned toward the Advocate. His words were not heard but the hissing of the voice carried a certain asperity. The Advocate flushed and interrupted Seldon. Q. We are not here to listen to speeches, Dr. Seldon. Let us assume that you have made your point. Let me suggest to you that your predictions of disaster might be intended to destroy public confidence in the Imperial Government for purposes of your own! A. That is not so. Q. Let me suggest that you intend to claim that a period of time preceding the so-called ruin of Trantor will be filled with unrest of various types. A. That is correct. Q. And that by the mere prediction thereof, you hope to bring it about, and to have then an army of a hundred thousand available. A. In the first place, that is not so. And if it were, investigation will show you that barely ten thousand are men of military age, and none of these has training in arms. Q. Are you acting as an agent for another? A. I am not in the pay of any man, Mr. Advocate. Q. You are entirely disinterested? You are serving science? A. I am.

Bradley Headstone, in his decent black coat and waistcoat, and decent white shirt, and decent formal black tie, and decent pantaloons of pepper and salt, with his decent silver watch in his pocket and its decent hair-guard round his neck, looked a thoroughly decent young man of six-and-twenty. He was never seen in any other dress, and yet there was a certain stiffness in his manner of wearing this, as if there were a want of adaptation between him and it, recalling some mechanics in their holiday clothes. He had acquired mechanically a great store of teacher's knowledge. He could do mental arithmetic mechanically, sing at sight mechanically, blow various wind instruments mechanically, even play the great church organ mechanically. From his early childhood up, his mind had been a place of mechanical stowage. The arrangement of his wholesale warehouse, so that it might be always ready to meet the demands of retail dealers history here, geography there, astronomy to the right, political economy to the left—natural history, the physical sciences, figures, music, the lower mathematics, and what not, all in their several places—this care had imparted to his countenance a look of care; while the habit of questioning and being questioned had given him a suspicious manner, or a manner that would be better described as one of lying in wait. There was a kind of settled trouble in the face. It was the face belonging to a naturally slow or inattentive intellect that had toiled hard to get what it had won, and that had to hold it now that it was gotten. He always seemed to be uneasy lest anything should be missing from his mental warehouse, and taking stock to assure himself.

Among this bewildering multiplicity of ideals which shall we choose? The answer is that we shall choose none. For it is clear that each one of these contradictory ideals is the fruit of particular social circumstances. To some extent, of course, this is true of every thought and aspiration that has ever been formulated. Some thoughts and aspirations, however, are manifestly less dependent on particular social circumstances than others. And here a significant fact emerges: all the ideals of human behaviour formulated by those who have been most successful in freeing themselves from the prejudices of their time and place are singularly alike. Liberation from prevailing conventions of thought, feeling and behaviour is accomplished most effectively by the practice of disinterested virtues and through direct insight into the real nature of ultimate reality. (Such insight is a gift, inherent in the individual; but, though inherent, it cannot manifest itself completely except where certain conditions are fulfilled. The principal pre-condition of insight is, precisely, the practice of disinterested virtues.) To some extent critical intellect is also a liberating force. But the way in which intellect is used depends upon the will. Where the will is not disinterested, the intellect tends to be used (outside the non-human fields of technology, science or pure mathematics) merely as an instrument for the rationalization of passion and prejudice, the justification of self-interest. That is why so few even of die acutest philosophers have succeeded in liberating themselves completely from the narrow prison of their age and country. It is seldom indeed that they achieve as much freedom as the mystics and the founders of religion. The most nearly free men have always been those who combined virtue with insight. Now, among these freest of human beings there has been, for the last eighty or ninety generations, substantial agreement in regard to the ideal individual. The enslaved have held up for admiration now this model of a man, now that; but at all times and in all places, the free have spoken with only one voice. It is difficult to find a single word that will adequately describe the ideal man of the free philosophers, the mystics, the founders of religions. 'Non-attached* is perhaps the best. The ideal man is the non-attached man. Non-attached to his bodily sensations and lusts. Non-attached to his craving for power and possessions. Non-attached to the objects of these various desires. Non-attached to his anger and hatred; non-attached to his exclusive loves. Non-attached to wealth, fame, social position. Non-attached even to science, art, speculation, philanthropy. Yes, non-attached even to these. For, like patriotism, in Nurse Cavel's phrase, 'they are not enough, Non-attachment to self and to what are called 'the things of this world' has always been associated in the teachings of the philosophers and the founders of religions with attachment to an ultimate reality greater and more significant than the self. Greater and more significant than even the best things that this world has to offer. Of the nature of this ultimate reality I shall speak in the last chapters of this book. All that I need do in this place is to point out that the ethic of non-attachment has always been correlated with cosmologies that affirm the existence of a spiritual reality underlying the phenomenal world and imparting to it whatever value or significance it possesses.

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.

All this attempt to control... We are talking about Western attitudes that are five hundred years old... The basic idea of science - that there was a new way to look at reality, that it was objective, that it did not depend on your beliefs or your nationality, that it was rational - that idea was fresh and exciting back then. It offered promise and hope for the future, and it swept away the old medieval system, which was hundreds of years old. The medieval world of feudal politics and religious dogma and hateful superstitions fell before science. But, in truth, this was because the medieval world didn't really work any more. It didn't work economically, it didn't work intellectually, and it didn't fit the new world that was emerging... But now... science is the belief system that is hundreds of years old. And, like the medieval system before it, science is starting to not fit the world any more. Science has attained so much power that its practical limits begin to be apparent. Largely through science, billions of us live in one small world, densely packed and intercommunicating. But science cannot help us decide what to do with that world, or how to live. Science can make a nuclear reactor, but it can not tell us not to build it. Science can make pesticide, but cannot tell us not to use it. And our world starts to seem polluted in fundamental ways - air, and water, and land - because of ungovernable science... At the same time, the great intellectual justification of science has vanished. Ever since Newton and Descartes, science has explicitly offered us the vision of total control. Science has claimed the power to eventually control everything, through its understanding of natural laws. But in the twentieth century, that claim has been shattered beyond repair. First, Heisenberg's uncertainty principle set limits on what we could know about the subatomic world. Oh well, we say. None of us lives in a subatomic world. It doesn't make any practical difference as we go through our lives. Then Godel's theorem set similar limits to mathematics, the formal language of science. Mathematicians used to think that their language had some inherent trueness that derived from the laws of logic. Now we know what we call 'reason' is just an arbitrary game. It's not special, in the way we thought it was. And now chaos theory proves that unpredictability is built into our daily lives. It is as mundane as the rain storms we cannot predict. And so the grand vision of science, hundreds of years old - the dream of total control - has died, in our century. And with it much of the justification, the rationale for science to do what it does. And for us to listen to it. Science has always said that it may not know everything now but it will know, eventually. But now we see that isn't true. It is an idle boast. As foolish, and misguided, as the child who jumps off a building because he believes he can fly... We are witnessing the end of the scientific era. Science, like other outmoded systems, is destroying itself. As it gains in power, it proves itself incapable of handling the power. Because things are going very fast now... it will be in everyone's hands. It will be in kits for backyard gardeners. Experiments for schoolchildren. Cheap labs for terrorists and dictators. And that will force everyone to ask the same question - What should I do with my power? - which is the very question science says it cannot answer.

We know, however, that the mind is capable of understanding these matters in all their complexity and in all their simplicity. A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball's spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball. People who call this "instinct" are merely giving the phenomenon a name, not explaining anything. I think that the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts - it has no meaning or purpose other than to be itself. Every single aspect of a piece of music can be represented by numbers. From the organization of movements in a whole symphony, down through the patterns of pitch and rhythm that make up the melodies and harmonies, the dynamics that shape the performance, all the way down to the timbres of the notes themselves, their harmonics, the way they change over time, in short, all the elements of a noise that distinguish between the sound of one person piping on a piccolo and another one thumping a drum - all of these things can be expressed by patterns and hierarchies of numbers. And in my experience the more internal relationships there are between the patterns of numbers at different levels of the hierarchy, however complex and subtle those relationships may be, the more satisfying and, well, whole, the music will seem to be. In fact the more subtle and complex those relationships, and the further they are beyond the grasp of the conscious mind, the more the instinctive part of your mind - by which I mean that part of your mind that can do differential calculus so astoundingly fast that it will put your hand in the right place to catch a flying ball- the more that part of your brain revels in it. Music of any complexity (and even "Three Blind Mice" is complex in its way by the time someone has actually performed it on an instrument with its own individual timbre and articulation) passes beyond your conscious mind into the arms of your own private mathematical genius who dwells in your unconscious responding to all the inner complexities and relationships and proportions that we think we know nothing about. Some people object to such a view of music, saying that if you reduce music to mathematics, where does the emotion come into it? I would say that it's never been out of it.