Mathematical Intelligence Quotes

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

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.

Sex is mathematics. Individuality no longer an issue. What does intelligence signify? Define reason. Desire - meaningless. Intellect is not a cure. Justice is dead.

As an orangutan cannot embrace higher mathematics or comprehend the architecture and operation of a computer, we humans __ so good at loudly proclaiming our intelligence and applauding our own doltish displays of cerebral gymnastics __ cannot begin to understand the true structure and functioning of the Universe.

Infinite is a meaningless word: except – it states / The mind is capable of performing / an endless process of addition.

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

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…

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.

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.

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

We are not told, or not told early enough so that it sinks in, that mathematics is a language, and that we can learn it like any other, including our own. We have to learn our own language twice, first when we learn to speak it, second when we learn to read it. Fortunately, mathematics has to be learned only once, since it is almost wholly a written language.

There are patterns within the dimensions,” Paul insisted, never looking up again. “Mathematical parallels. It’s plausible to hypothesize that these patterns will be reflected in events and people in each dimension. That people who have met in one quantum reality will be likely to meet in another. Certain things that happen will happen over and over, in different ways, but more often than you could explain by chance alone.” “In other words,” I said, “you’re trying to prove the existence of fate.” I was joking, but Paul nodded slowly, like I’d said something intelligent. “Yes. That’s it exactly.

...it's really more intelligent to be able to simplify things than to complicate them. Even if some people think it makes you look stupid.

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, sympathizing, guilt, waste, failure, grief, were things, emotions, that no one really felt any more. 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 aspirations of democracy are based on the notion of an informed citizenry, capable of making wise decisions. The choices we are asked to make become increasingly complex. They require the longer-term thinking and greater tolerance for ambiguity that science fosters. The new economy is predicated on a continuous pipeline of scientific and technological innovation. It can not exist without workers and consumers who are mathematically and scientifically literate.

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

The contribution of mathematics, and of people, is not computation but intelligence.

One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.

Before you can ask 'Is Darwinian theory correct or not?', You have to ask the preliminary question 'Is it clear enough so that it could be correct?'. That's a very different question. One of my prevailing doctrines about Darwinian theory is 'Man, that thing is just a mess. It's like looking into a room full of smoke.' Nothing in the theory is precisely, clearly, carefully defined or delineated. It lacks all of the rigor one expects from mathematical physics, and mathematical physics lacks all the rigor one expects from mathematics. So we're talking about a gradual descent down the level of intelligibility until we reach evolutionary biology.

Blake’s colleagues viewed intelligence as a means to an end, and the end was always making more money. But in the mathematics department at Santa Monica College, no one expected to be rich. It was enough to know. She was lucky to spend her days like this, knowing.

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.

....young people unskilled in mathematics, addled by credit cards, and weaned on so-called intelligent design...will somehow retool American science for another generation of world industrial leadership.

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.

We often say that the earth is a sphere, but to be precise, the term sphere refers only to the surface. The correct mathematical term for the solid earth is a ball.

IQ is a statistical method for quantifying specific kinds of problem-solving ability, mathematically convenient but not necessarily corresponding to a real attribute of the human brain, and not necessarily representing whatever it is that we mean by ‘intelligence’.

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.

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.

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.

To be a scholar study math, to be a smart study magic.

Why a journey into space? Because science is now learning that the infinite reaches of our universe probably teem with as much life and adventure as Earth's own oceans and continents. Our galaxy alone is so incredibly vast that the most conservative mathematical odds still add up to millions of planets almost identical to our own — capable of life, even intelligence and strange new civilizations. Alien beings that will range from the fiercely primitive to the incredibly exotic intelligence which will far surpass Mankind. (The Hollywood Reporter, Sept. 8, 1966)

I discovered that the predisposition for languages is as mysterious as the inclination of certain people for mathematics or music and has nothing to do with intelligence or knowledge. It is something separate, a gift that some possess and others don’t.

In my short stay I realized that without a deep understanding of human psychology, without the acceptance that we are all crazy, irrational, impulsive, emotionally driven animals, all the raw intelligence and mathematical logic in the world is little help in the fraught, shifting interplay of two people negotiating.

Nothing was affirmative, the term "generosity of spirit" applied to nothing, was a cliché, was some kind of bad joke. Sex is mathematics. Individuality no longer and 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.

Deep Blue didn't win by being smarter than a human; it won by being millions of times faster than a human. Deep Blue had no intuition. An expert human player looks at a board position and immediately sees what areas of play are most likely to be fruitful or dangerous, whereas a computer has no innate sense of what is important and must explore many more options. Deep Blue also had no sense of the history of the game, and didn't know anything about its opponent. It played chess yet didn't understand chess, in the same way a calculator performs arithmetic bud doesn't understand mathematics.

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.

I love moving water, I love ships, I love the sharp definition, the concentrated humanity, the sublime solitude of life at sea. The dangers of it only make present to us the peril inherent in all existence, which the stupid, ignorant, untravelled land-worm never discovers; and the art of it, so mathematical, so exact, so rewarding to intelligence, appeals to courage and clears the mind of superstition, while filling it with humility and true religion.

Mathematics is one of the major modern mysteries. Perhaps it is the leading one, occupying a place in our society similar to the religious mysteries of another age. If we want to know something about what our age is all about, we should have some understanding of what mathematics is, and of how the mathematician operates and thinks.

I wasn't a genius in the end, but a girl could still hope.

Is it possible that the Pentateuch could not have been written by uninspired men? that the assistance of God was necessary to produce these books? Is it possible that Galilei ascertained the mechanical principles of 'Virtual Velocity,' the laws of falling bodies and of all motion; that Copernicus ascertained the true position of the earth and accounted for all celestial phenomena; that Kepler discovered his three laws—discoveries of such importance that the 8th of May, 1618, may be called the birth-day of modern science; that Newton gave to the world the Method of Fluxions, the Theory of Universal Gravitation, and the Decomposition of Light; that Euclid, Cavalieri, Descartes, and Leibniz, almost completed the science of mathematics; that all the discoveries in optics, hydrostatics, pneumatics and chemistry, the experiments, discoveries, and inventions of Galvani, Volta, Franklin and Morse, of Trevithick, Watt and Fulton and of all the pioneers of progress—that all this was accomplished by uninspired men, while the writer of the Pentateuch was directed and inspired by an infinite God? Is it possible that the codes of China, India, Egypt, Greece and Rome were made by man, and that the laws recorded in the Pentateuch were alone given by God? Is it possible that Æschylus and Shakespeare, Burns, and Beranger, Goethe and Schiller, and all the poets of the world, and all their wondrous tragedies and songs are but the work of men, while no intelligence except the infinite God could be the author of the Pentateuch? Is it possible that of all the books that crowd the libraries of the world, the books of science, fiction, history and song, that all save only one, have been produced by man? Is it possible that of all these, the bible only is the work of God?

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 …

...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...

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.

we’ve been redefining what it means to be human. Over the past 60 years, as mechanical processes have replicated behaviors and talents we thought were unique to humans, we’ve had to change our minds about what sets us apart. As we invent more species of AI, we will be forced to surrender more of what is supposedly unique about humans. Each step of surrender—we are not the only mind that can play chess, fly a plane, make music, or invent a mathematical law—will be painful and sad. We’ll spend the next three decades—indeed, perhaps the next century—in a permanent identity crisis, continually asking ourselves what humans are good for. If we aren’t unique toolmakers, or artists, or moral ethicists, then what, if anything, makes us special? In the grandest irony of all, the greatest benefit of an everyday, utilitarian AI will not be increased productivity or an economics of abundance or a new way of doing science—although all those will happen. The greatest benefit of the arrival of artificial intelligence is that AIs will help define humanity. We need AIs to tell us who we are.

My other teachers did not seem to care about the challenge of being human and instead they taught us to think about mathematics and analyze different chemicals and as the months went by I felt further from myself. And the only thing that seemed to make sense was Ben Sweet and the way he talked to us and urged something in the deeps of us to come out—the way he looked, and listened, as if he had no other place on this Earth to be except with us, as if there were nothing more important in his life than what we had to say at just that moment in time.

Geometry exist everywhere.It is necessary, however, to have eyes to see it, intelligence to understand it , and spirit to wonder at it.The wild Bedouin sees geometric forms but doesn't understand them ; the Sunni understands them but does not admire them; the artist, finally, perceives the perfection of figures, understands beauty, and admires order and harmony.God was the Great Geometer.He geometrized heaven and earth.

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.

Since all terms that are defined are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order to have a starting point for its definitions...[and] since human powers are finite, the definitions known to us must always begin somewhere, with terms undefined for the moment, though perhaps not permanently." - Introduction to Mathematical Philosophy

Believers in psychic phenomena... appear to have won a decisive victory and virtually silenced opposition.... This victory is the result of careful experimentation and intelligent argumentation. Dozens of experimenters have obtained positive results in ESP experiments, and the mathematical procedures have been approved by leading statisticians.... Against all this evidence, almost the only defense remaining to the skeptical scientist is ignorance.

It's solely thanks to mathematics that existence is rational, explicable, intelligible. Existence has an answer only because it's a mathematical organism. It isn't a mathematical AI. It's not a dead computer. It's a living, purposeful intelligence, a Hive Mind made of individual, living minds, of which each of us is one.

If one chooses to call tests that require the mastery of abstractions culturally biased, because some cultures put more emphasis on abstractions than others, that raises fundamental questions about what the tests are for. In a world where the ability to master abstractions is fundamental to mathematics, science and other endeavors, the measurement of that ability is not an arbitrary bias. A culture-free test might be appropriate in a culture-free society—but there are no such societies.

A man must be quite intelligent in order to accept that a woman is his intellectual equal.

People are better than computers when finding patterns in images, but computers are better than people when finding patterns in numbers.

Even though without a brain, Mother Nature is mathematically, biologically, mechanically, chemically, logically intelligent.

The Intelligence of Mathematics existed first before the Intelligence of the Mind.

Quantum attention function is deeply grounded on the mathematics of theoretical physics and penetrates deeply the world of the very small and the world of the very big.

is clear that human knowledge must always be content to accept some terms as intelligible without definition,

EXPECTATIONS ALSO SHAPE stereotypes. A stereotype, after all, is a way of categorizing information, in the hope of predicting experiences. The brain cannot start from scratch at every new situation. It must build on what it has seen before. For that reason, stereotypes are not intrinsically malevolent. They provide shortcuts in our never-ending attempt to make sense of complicated surroundings. This is why we have the expectation that an elderly person will need help using a computer or that a student at Harvard will be intelligent.* But because a stereotype provides us with specific expectations about members of a group, it can also unfavorably influence both our perceptions and our behavior. Research on stereotypes shows not only that we react differently when we have a stereotype of a certain group of people, but also that stereotyped people themselves react differently when they are aware of the label that they are forced to wear (in psychological parlance, they are “primed” with this label). One stereotype of Asian-Americans, for instance, is that they are especially gifted in mathematics and science. A common stereotype of females is that they are weak in mathematics. This means that Asian-American women could be influenced by both notions.

He possesses the minimum sensibility necessary for his intelligence not to be merely mathematical, the minimum a human being needs so that it can be proven with a thermometer that he's not dead.

One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.

When we use a measuring tape, we are using a system of numbers that is human invented. What are the birds using? And further, how are they storing it in memory? To be stored in memory the measurements have to be encoded in some form and that form has to have an internal consistency to it. In other words it has to possess the same kind of structural integrity as our system of mathematical measurement.

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.

Consider a cognitive scientist concerned with the empirical study of the mind, especially the cognitive unconscious, and ultimately committed to understanding the mind in terms of the brain and its neural structure. To such a scientist of the mind, Anglo-American approaches to the philosophy of mind and language of the sort discussed above seem odd indeed. The brain uses neurons, not languagelike symbols. Neural computation works by real-time spreading activation, which is neither akin to prooflike deductions in a mathematical logic, nor like disembodied algorithms in classical artificial intelligence, nor like derivations in a transformational grammar.

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.

Nobody, even in the provinces, should ever be allowed to ask an intelligent question about pure mathematics across a dinner table. A question of this kind is quite as bad as inquiring suddenly about the state of a man’s soul …

Ken appeared, was taller than she, wanted her, was acceptable and accepted on all sides; similarly, nagging mathematical problems abruptly crack open. Foxy could find no fault with him, and this challenged her, touched off her stubborn defiant streak. She felt between his handsomeness and intelligence a contradiction that might develop into the convoluted humour of her Jew. Ken looked lika a rich boy and worked like a poor one. From Farmington, he was the only son of a Hartford laywer who never lost a case. Foxy came to imagine his birth as cool and painless, without a tear or outcry. Nothing puzzled him. There were unknowns, but no mysteries. (...) He was better-looking, better-thinking, a better machine.

Unlike the laws of reality that scientist describe in complex mathematical equations, the laws that govern human behavior are imprecise and in constant flux. We are complex organisms because we possess the capacity to experience, recall, and imagine. We are self-constructed. How we think becomes our reality. The highest act of human intelligence is not building bombs and inventing poisons that can destroy the world, but engaging in acts of contemplation that expands human consciousness.

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.

I am sorry for your disappointment,’ he continued, glancing into her face. Their eyes having met, became, as it were, mutually locked together, and the single instant only which good breeding allows as the length of such a look, became trebled: a clear penetrating ray of intelligence had shot from each into each, giving birth to one of those unaccountable sensations which carry home to the heart before the hand has been touched or the merest compliment passed, by something stronger than mathematical proof, the conviction, ‘A tie has begun to unite us.’ Both faces also unconsciously stated that their owners had been much in each other’s thoughts of late. Owen had talked to the young architect of his sister as freely as to Cytherea of the young architect.

[W]hen food is placed at the start and end points of the maze, the slime mold withdraws from the dead-end corridors and shrinks its body to a tube spanning the shortest path between food sources. The single-celled slime solves the maze in this way each time it is tested.”23 Toshiyuki Nakagaki, the researcher conducting the study, commented that Even for humans it is not easy to solve a maze. But the plasmodium of true slime mold, an amoeba-like organism, has shown an amazing ability to do so. This implies that an algorithm and a high computing capacity are included in the unicellular organism.24 This capacity for mathematical differentiation and computation is wide spread. All self-organized biological systems possess it. One of the more amazing examples is the Clark’s Nutcracker.

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

मैं नहीं चाहता कि मेरा मन खंगाला जाए चाहे उसमें इस्तेमाल लायक कुछ भी न हो MAIN NAHIN CHAHTA KI MERA MANN KHANGALA JAYE CHAHE USMEIN ISTEMAL LAYAK KUCHH BHI NA HO I DON'T WANT THAT MY MIND BE SCRUTINIZED EVEN IF THERE WAS NO THING OF VALUE INSIDE 24 Dec National Mathematics Day

As Bertrand Russell, the famous British mathematical philosopher and Nobel laureate, famously lamented in an essay condemning the rise of Nazi Germany, “the fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt.

The brain is a statistical, probabilistic system, with logic and mathematics running as higher-level processes. The computer is a logical, mathematical system, upon which higher-level statistical, probabilistic systems, such as human language and intelligence, could possibly be built.

She always felt lucky to be in the presence of such brilliant people. Thinkers. Blake’s colleagues viewed intelligence as a means to an end, and the end was always making more money. But in the mathematics department at Santa Monica College, no one expected to be rich. It was enough to know.

Writing in 1921, the University of Chicago economist Frank Knight uttered strange words for a man of his profession: “There is much question as to how far the world is intelligible at all. . . . It is only in the very special and crucial cases that anything like a mathematical study can be made.

we must not forget that the restful experience of enjoyable beauty is not limited to the contemplation of sensible objects. We can experience it as well in the contemplation of purely intelligible objects—the contemplation of truths we understand. “Mathematics,” wrote Bertrand Russell, “rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere … without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music …” Or, as the poet Edna St. Vincent Millay wrote in the opening line of her sonnet on Euclid, “Euclid alone has looked on beauty bare.

Philosophy, a love of wisdom, is both a desire for a good and an appreciation of the admirable. The good is an object of desire and love, the admirable is an object of contemplation. If we focus too exclusively on what is useful or even on what is good, we lose the capacity for admiration: “We become blind to the beauty that completes the good.” The admirable manifests itself in all the works of intelligence: in the elegance of well-formed mathematical systems, in deeply moving political speeches, in a life well lived, and in a well-ordered city. What is admirable in all of these things is the way they have to be. Their forms express this necessity, not in the sense of something relentless and overpowering, but in the sense of a fullness that displays their perfection. Philosophy is to remind us of the necessity in things: not just the necessities to which we have to resign ourselves, but those we can find splendid.

Each triangulation is different, each a different measurement (or number) of distance. There are numerous implications in this. Here are three of them: 1) mathematical relationships that are inherent in Universe can be perceived, and utilized, by more organisms than the human; 2) numbering systems are arbitrary and are only metaphors for those mathematical relationships—they are not foundational; 3) organisms other than the human not only have the capacity to perceive distance but also differentials—they can add and subtract; 4) they possess a sense of congruency—they know when they have the right answer—and the wrong one.

It is both mysterious and miraculous that roughly the same intelligence necessary to flake a barbed spearpoint is sufficient to discover the theorems of mathematics. In a different universe, it might have been otherwise. And so human beings would have been spared the tragedy of existing half as ape and half as god.

A self, created out of mathematics, engineering, material science and all the rest. No history — not that I’d want a false one. Nothing before me. Self-aware existence. I’m lucky to have it, but there are times when I think that I ought to know better what to do with it. What it’s for. Sometimes it seems entirely pointless.

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...

An electronic machine can carry out mathematical calculations, remember historical facts, play chess and translate books from one language to another. It is able to solve mathematical problems more quickly than man and its memory is faultless. Is there any limit to progress, to its ability to create machines in the image and likeness of man? It seems the answer is no. It is not impossible to imagine the machine of future ages and millennia. It will be able to listen to music and appreciate art; it will even be able to compose melodies, paint pictures and write poems. Is there a limit to its perfection? Can it be compared to man? Will it surpass him? Childhood memories… tears of happiness … the bitterness of parting… love of freedom … feelings of pity for a sick puppy … nervousness … a mother’s tenderness … thoughts of death … sadness … friendship … love of the weak … sudden hope … a fortunate guess … melancholy … unreasoning joy … sudden embarrassment… The machine will be able to recreate all of this! But the surface of the whole earth will be too small to accommodate this machine – this machine whose dimensions and weight will continually increase as it attempts to reproduce the peculiarities of mind and soul of an average, inconspicuous human being. Fascism annihilated tens of millions of people.

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

I suspect gentlemen, that you're regarding me with pity; you keep repeating to me that an enlightened and cultured man -- such as, in short, as the man of the future will be -- cannot knowingly desire anything unprofitable for himself -- that that's mathematics. I agree totally that it really is mathematics. But I repeat to you for the hundredth time: there is only one case, only one, when a man can intentionally and consciously desire for himself even what is harmful and stupid, even what is extremely stupid: namely, in order to have the right to desire for himself even what is extremely stupid and not be constrained by the obligation to desire for himself only what is intelligent.

On another night, in a different dream I was asking a question. “How is it that you say all are equal, yet the obvious contradictions smack us in the face: inequalities in virtues, temperances, finances, rights, abilities and talents, intelligence, mathematical aptitude, ad infinitum?” The answer was a metaphor. “It is as if a large diamond were to be found inside each person. Picture a diamond a foot long. The diamond has a thousand facets, but the facets are covered with dirt and tar. It is the job of the soul to clean each facet until the surface is brilliant and can reflect a rainbow of colors. “Now, some have cleaned many facets and gleam brightly. Others have only managed to clean a few; they do not sparkle so. Yet, underneath the dirt, each person possesses within his or her breast a brilliant diamond with a thousand gleaming facets. The diamond is perfect, not one flaw. The only differences among people are the number of facets cleaned. But each diamond is the same, and each is perfect. “When all the facets are cleaned and shining forth in a spectrum of lights, the diamond returns to the pure energy that it was originally. The lights remain. It is as if the process that goes into making the diamond is reversed, all that pressure released. The pure energy exists in the rainbow of lights, and the lights possess consciousness and knowledge. “And all of the diamonds are perfect.” Sometimes

Although DNA does not convey information that is received, understood, or used by a conscious mind, it does have information that is received and used by the cell’s machinery to build the structures critical to the maintenance of life. DNA displays a property—functional specificity—that transcends the merely mathematical formalism of Shannon’s theory. Is

Byron had said teaching was an underpaid and underappreciated profession. He’d said it drained the life out of people. He’d said the system takes advantage of teachers and sets them up to fail, so why would any intelligent, reasonable person with an aptitude for science or mathematics or engineering ever willingly accept a teacher’s salary to do a teacher’s job?

We’re searching for intelligent, conscious, tool-making beings that have developed a language we’re capable of understanding. We’re searching for intelligent conscious, tool-making, communicative beings that live in social groups (so they can reap the benefits of civilization) and that develop the tools of science and mathematics. We’re searching for ourselves . . .

[...] Queste difficoltà possono essere risolte nel modo migliore facendo buon viso a cattivo gioco. I ritardi possono essere tollerati accettandoli ed elaborando una scansione temporale che li preveda. Si può poi tollerare una certa imprecisione nella risposta pensando in termini di <>. Così invece di dire: <>, noi diremo: <>. Le varie classi devono essere del tutto distinte e ben lontane dal sovrapporsi, cioé - topologicamente parlando - potremmo dire che devono avere tra loro una distanza finita. Con una decisione del genere avremo introdotto una ben definita divisione del lavoro tra il matematico e l'ingegnere, che permetterà a ognuno dei due di andare avanti senza preoccuparsi se le sue assunzioni siano in accordo con quelle dell'altro.

It may have something to do with intelligence, but I am certain it has nothing to do with knowledge - I mean that there are people who have an instinctive yet perfect moral judgment, who can perform the most complex ethical calculations as Indian peasants can sometimes perform astounding mathematical feats in a matter of seconds. Lily was such a person. And I craved her approval.

I always find it extraordinary that so many studies are made of price and volume behavior, the stuff of chartists. Can you imagine buying an entire business simply because the price of the business had been marked up substantially last week and the week before? Of course, the reason a lot of studies are made of these price and volume variables is that now, in the age of computers, there are almost endless data available about them. It isn’t necessarily because such studies have any utility; it’s simply that the data are there and academicians have worked hard to learn the mathematical skills needed to manipulate them. Once these skills are acquired, it seems sinful not to use them, even if the usage has no utility or negative utility. As a friend said, to a man with a hammer, everything looks like a nail.

that rotten feeling of antlike industry. There is really no need to belabor the point, since it is obvious to most of us these days that mathematics has taken possession, like a demon, of every aspect of our lives. Most of us may not believe in the story of a Devil to whom one can sell one’s soul, but those who must know something about the soul (considering that as clergymen, historians, and artists they draw a good income from it) all testify that the soul has been destroyed by mathematics and that mathematics is the source of an evil intelligence that while making man the lord of the earth has also made him the slave of his machines. The inner drought, the dreadful blend of acuity in matters of detail and indifference toward the whole, man’s monstrous abandonment in a desert of details, his restlessness, malice, unsurpassed callousness, money-grubbing, coldness, and violence, all so characteristic of our times, are by these accounts solely the consequence of damage done to the soul by keen logical thinking! Even back when Ulrich first turned to mathematics there were already those who predicted the collapse of European civilization because no human faith, no love, no simplicity, no goodness, dwelt any longer in man.

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’re 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 its own sake. Those few who remain interested are vilified as “nerds” or “geeks” or “grinds.

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. 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.

For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it’s the only way a universe with us in it could be, because nonmathematical universes can’t harbor life intelligent enough to ask the question. In any case, it’s a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations.

If, as I suggested before, the ability to tell right from wrong should turn out to have anything to do with the ability to think, then we must be able to “demand” its exercise from every sane person, no matter how erudite or ignorant, intelligent or stupid, he may happen to be. Kant—in this respect almost alone among the philosophers—was much bothered by the common opinion that philosophy is only for the few, precisely because of its moral implications, and he once observed that “stupidity is caused by a wicked heart.”21 This is not true: absence of thought is not stupidity; it can be found in highly intelligent people, and a wicked heart is not its cause; it is probably the other way round, that wickedness may be caused by absence of thought. In any event, the matter can no longer be left to “specialists” as though thinking, like higher mathematics, were the monopoly of a specialized discipline.

By our very nature, we humans are linear thinkers. We evolved to estimate a distance from the predator or to the prey, and advanced mathematics is only a recent evolutionary addition. This is why it’s so difficult even for a modern man to grasp the power of exponentials. 40 steps in linear progression is just 40 steps away; 40 steps in exponential progression is a cool trillion (with a T) – it will take you 3 times from Earth to the Sun and back to Earth.

Paul closed his eyes and turned his face to the sun. In spite of everything, it was hard not to take solace from the warmth flooding onto his skin. He stretched the muscles in his arms, his shoulders, his back -- and it felt like he was reaching out from the "self" in his virtual skull to all his mathematical flesh, imprinting the nebulous data with meaning; binding it all together, staking some kind of claim. He felt the stirrings of an erection. Existence was beginning to seduce him. He let himself surrender for a moment to a visceral sense of identity which drowned out all his pale mental images of optical processors, all his abstract reflections on the software's approximations and short-cuts. This body didn't want to evaporate. This body didn't want to bale out. It didn't much care that there was another -- "more real" -- version of itself elsewhere. It wanted to retain its wholeness. It wanted to endure.

The question, is it true? can be asked of anything we read. It is applicable to every kind of writing, in one or another sense of "truth" -- mathematical, scientific, philosophical, historial and poetical. No higher commendation can be given any work of the human mind than to praise it for the measure of truth it has achieved; by the same token, to criticize it adversely for its failure in this respect is to treat it with the seriousness that a serious work deserves.

Richard Lewontin observes . . . In Cladocera, small fresh-water arthropods, reproduction remains asexual as long as conditions of temperature, oxygen dissolved in the water, food availability, and degree of crowding remain constant. Then, if a sudden change in these conditions occurs . . . the Cladocera switch to sexual reproduction. . . . The organisms are detecting a rate of change of an input, not its absolute value. They are performing mathematical differentiation.22

there are now methods that produce unbiased results according to several plausible and desirable definitions of fairness.39 The mathematical analysis of these definitions of fairness shows that they cannot be achieved simultaneously and that, when enforced, they result in lower prediction accuracy and, in the case of lending decisions, lower profit for the lender. This is perhaps disappointing, but at least it makes clear the trade-offs involved in avoiding algorithmic bias.

Intelligence can also be broken down in terms of the skills that constitute it. One theory breaks it down into three separate skills: language ability, the speed and ability to perceive the world accurately, and the ability to manipulate spatial images in one’s head. Another theory goes even further, arguing that there are eight distinct dimensions that underlie intelligence: linguistic, logical-mathematical, spatial, musical, naturalist, bodily kinesthetic, interpersonal, and intrapersonal.

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.

There is no doubt that Earth Central, the planetary and sector AIs, and even some ship and drone AIs are capable, without acquiring additional processing space, of setting up synergetic systems within themselves that result in an exponential climb in intelligence (mathematically defined as climbing beyond all known scales within minutes). So why not? Ask then why a human, capable of learning verbatim the complete works of Shakespeare, instead drinks a bottle of brandy, then giggles a lot and falls over.

To understand what rationality is, why it seems scarce, and why it matters, we must begin with the ground truths of rationality itself: the ways an intelligent agent ought to reason, given its goals and the world in which it lives. These “normative” models come from logic, philosophy, mathematics, and artificial intelligence, and they are our best understanding of the “correct” solution to a problem and how to find it. They serve as an aspiration for those who want to be rational, which should mean everyone.

Hobbes's natural philosophy is of the type classically represented by Democritean-Epicurean physics. Yet he regarded, not Epicurus or Democritus, but Plato, as "the best of the ancient philosophers." What he learned from Plato's natural philosophy was not that the universe cannot be understood if it is not ruled by divine intelligence. Whatever may have been Hobbes's private thoughts, his natural philosophy is as atheistic as Epicurean physics. What he learned from Plato's natural philosophy was that mathematics is "the mother of all natural science." By being both mathematical and materialistic-mechanistic, Hobbes's natural philosophy is a combination of Platonic physics and Epicurean physics. From his point of view, premodern philosophy or science as a whole was "rather a dream than science" precisely because it did not think of that combination. His philosophy as a whole may be said to be the classic example of the typically modern combination of political idealism with a materialistic and atheistic view of the whole.

In mathematics, students are at the mercy of rigidly applied algorithms. They learn to use certain formalisms in certain ways, often effectively, if provided with a pre-arranged signal that a particular formalism is wanted. In social studies and the humanities, the enemies of understanding are scripts and stereotypes. Students readily believe that events occur in typical ways, and they evoke these scripts even inappropriately. For example, they regard struggles between two parties in a dispute as a "good guy versus bad guy" movie script.

Distributions can only be based on measurements, but as in the case of measuring intelligence, the nature of measurement is often complicated and troubled by ambiguities. Consider the problem of noise, or what is known as luck in human affairs. Since the rise of the new digital economy, around the turn of the century, there has been a distinct heightening of obsessions with contests like American Idol, or other rituals in which an anointed individual will suddenly become rich and famous. When it comes to winner-take-all contests, onlookers are inevitably fascinated by the role of luck. Yes, the winner of a singing contest is good enough to be the winner, but even the slightest flickering of fate might have changed circumstances to make someone else the winner. Maybe a different shade of makeup would have turned the tables. And yet the rewards of winning and losing are vastly different. While some critics might have aesthetic or ethical objections to winner-take-all outcomes, a mathematical problem with them is that noise is amplified. Therefore, if a societal system depends too much on winner-take-all contests, then the acuity of that system will suffer. It will become less reality-based.

That Marxism is not a science is entirely clear to intelligent people in the Soviet Union. One would even feel awkward to refer to it as a science. Leaving aside the exact sciences, such as physics, mathematics, and the natural sciences, even the social sciences can predict an event—when, in what way and how an event might occur. Communism has never made any such forecasts. It has never said where, when, and precisely what is going to happen. Nothing but declamations. Rhetoric to the effect that the world proletariat will overthrow the world bourgeoisie and the most happy and radiant society will then arise.

In a representative statement from 1963, he claimed, “Man does not know most of the rules on which he acts; and even what we call his intelligence is largely a system of rules which operate on him but which he does not know.”60 This deference to the precognitive or the pre-rational is what separated him from the rational choice and rational expectations models of Chicago School economists, who professed much more faith in the possibility of both formal mathematical modeling and forecasting. As he explained in his Nobel speech, Hayek saw such efforts as not only presumptuous but misleading. The best one could hope for was pattern prediction.

121. George Bernard Shaw – Plays and Prefaces 122. Max Planck – Origin and Development of the Quantum Theory; Where Is Science Going?; Scientific Autobiography 123. Henri Bergson – Time and Free Will; Matter and Memory; Creative Evolution; The Two Sources of Morality and Religion 124. John Dewey – How We Think; Democracy and Education; Experience and Nature; Logic; the Theory of Inquiry 125. Alfred North Whitehead – An Introduction to Mathematics; Science and the Modern World; The Aims of Education and Other Essays; Adventures of Ideas 126. George Santayana – The Life of Reason; Skepticism and Animal Faith; Persons and Places 127. Vladimir Lenin – The State and Revo

… 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 cliché, 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 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.

Computational models of the mind would make sense if what a computer actually does could be characterized as an elementary version of what the mind does, or at least as something remotely like thinking. In fact, though, there is not even a useful analogy to be drawn here. A computer does not even really compute. We compute, using it as a tool. We can set a program in motion to calculate the square root of pi, but the stream of digits that will appear on the screen will have mathematical content only because of our intentions, and because we—not the computer—are running algorithms. The computer, in itself, as an object or a series of physical events, does not contain or produce any symbols at all; its operations are not determined by any semantic content but only by binary sequences that mean nothing in themselves. The visible figures that appear on the computer’s screen are only the electronic traces of sets of binary correlates, and they serve as symbols only when we represent them as such, and assign them intelligible significances. The computer could just as well be programmed so that it would respond to the request for the square root of pi with the result “Rupert Bear”; nor would it be wrong to do so, because an ensemble of merely material components and purely physical events can be neither wrong nor right about anything—in fact, it cannot be about anything at all. Software no more “thinks” than a minute hand knows the time or the printed word “pelican” knows what a pelican is. We might just as well liken the mind to an abacus, a typewriter, or a library. No computer has ever used language, or responded to a question, or assigned a meaning to anything. No computer has ever so much as added two numbers together, let alone entertained a thought, and none ever will. The only intelligence or consciousness or even illusion of consciousness in the whole computational process is situated, quite incommutably, in us; everything seemingly analogous to our minds in our machines is reducible, when analyzed correctly, only back to our own minds once again, and we end where we began, immersed in the same mystery as ever. We believe otherwise only when, like Narcissus bent above the waters, we look down at our creations and, captivated by what we see reflected in them, imagine that another gaze has met our own.

This one is ready to risk his scientific credibility—and his adviser’s, no matter what he says—by arguing that destiny is real.” “Destiny?” That sounded weirdly . . . romantic from a guy like Paul. “There are patterns within the dimensions,” Paul insisted, never looking up again. “Mathematical parallels. It’s plausible to hypothesize that these patterns will be reflected in events and people in each dimension. That people who have met in one quantum reality will be likely to meet in another. Certain things that happen will happen over and over, in different ways, but more often than you could explain by chance alone.” “In other words,” I said, “you’re trying to prove the existence of fate.” I was joking, but Paul nodded slowly, like I’d said something intelligent. “Yes. That’s it exactly.

One day during the 1930s, Einstein invited Saint-John Perse to Princeton to find out how the poet worked. “How does the idea of a poem come?” Einstein asked. The poet spoke of the role played by intuition and imagination. “It’s the same for a man of science,” Einstein responded with delight. “It is a sudden illumination, almost a rapture. Later, to be sure, intelligence analyzes and experiments confirm or invalidate the intuition. But initially there is a great forward leap of the imagination.”16 There was an aesthetic to Einstein’s thinking, a sense of beauty. And one component to beauty, he felt, was simplicity. He had echoed Newton’s dictum “Nature is pleased with simplicity” in the creed he declared at Oxford the year he left Europe for America: “Nature is the realization of the simplest conceivable mathematical ideas.

Both Samuel Butler and Olaf Stapledon saw that mind, once given a taste of time, would never rest until eternity lay within its grasp. Thus we pursue those relations between sequence and structure that allow us to escape time’s surface, venturing into that ocean that separates eternity from the instant in which we exist. Mathematics and music are two of the vehicles that assist us in our escape. Mathematics is available to a few; music is available to all. Mathematics allows us to assemble mental structures by which we comprehend entire sequences of logical implication at once. Music allows us to assemble temporal sequences into mental scaffolding that transcends the thinness of time in which we live. Through music, we are able to share four-dimensional structures we are otherwise only able to observe in slices, one moment at a time.

[Scarlett] knew how to smile so that her dimples leaped, how to walk pigeon-toed so that her wide hoop skirts swayed entrancingly, how to look up into a man's face and then drop her eyes and bat the lids rapidly so that she seemed a-tremble with gentle emotion. Most of all she learned how to conceal from men a sharp intelligence beneath a face as sweet and bland as a baby's. Ellen, by soft admonition, . . . labored to inculcate in her the qualities that would make her truly desirable as a wife. "You must be more gentle, dear, more sedate," Ellen told her daughter. "You must not interrupt gentlemen when they are speaking, even if you do think you know more about matters than they do. Gentlemen do not like forward girls." [Ellen] taught her all that a gentlewoman should know, but she learned only the outward signs of gentility. The inner grace from which these signs should spring, she never learned nor did she see any reason for learning it. Appearances were enough, for the appearances of ladyhood won her popularity and that was all she wanted. . . . At sixteen, thanks to Mammy and Ellen, she looked sweet, charming and giddy, but she was, in reality, self-silled, vain and obstinate. She had the easily stirred passions of her Irish father and nothing except the thinnest veneer of her mother's unselfish and forbearing nature. . . It was not that these two loving mentors deplored Scarlett's high spirits, vivacity and charm. These were traits of which Southern women were proud. It was Gerald's headstrong and impetuous nature in her that gave them concern, and they sometimes feared they would not be able to conceal her damaging qualities until she had made a good match. But Scarlett intended to marry-and marry Ashley-and she was willing to appear demure, pliable and scatterbrained, if those were the qualities that attracted men. Just why men should be this way, she did not know. She only knew that such methods worked. It never interested her enough to try to think out the reason for it, for she knew nothing of the inner workings of any human being's mind, not even her own. She knew only that if she did or said thus-and-so, men would unerringly respond with the complementary thus-and-so. It was like a mathematical formula and no more difficult . . . If she knew little about men's minds, she knew even less about the minds of women, for they interested her less. She had never had a girl friend, and she never felt any lack on that account. To her, all women, including her two sisters, were natural enemies in pursuit of the same prey-man.

common sense observations of human behavior support a similar dissociation in reasoning abilities which cuts in both directions. We all know persons who are exceedingly clever in their social navigation, who have an unerring sense of how to seek advantage for themselves and for their group, but who can be remarkably inept when trusted with a nonpersonal, nonsocial problem. The reverse condition is just as dramatic: We all know creative scientists and artists whose social sense is a disgrace, and who regularly harm themselves and others with their behavior. The absent-minded professor is the benign variety of the latter type. At work, in these different personality styles, are the presence or absence of what Howard Gardner has called “social intelligence,” or the presence or absence of one or the other of his multiple intelligences such as the “mathematical.

A fact, once discovered, leads an existence of its own, and enters into relations with other facts of which their discoverers have never dreamt. Apollonius of Perga discovered the laws of the useless curves which emerge when a plane intersects a cone at various angles: these curves proved, centuries later, to represent the paths followed by planets, comets, rockets, and satellites. One cannot escape the feeling [wrote Heinrich Herz] that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them. This confession of the discoverer of radio-waves sounds suspiciously like an echo of Kepler, echoing Plato, echoing Pythagoras: 'Methinks that all of nature and the graceful sky are set into symbols in geomatriam.

As a criticism of philosophies and general conclusions based on physics, one could point to the exclusivity accorded to mathematical logic as if this were the only form of logic. What is mathematically satisfactory is considered to be true even if it violates the principles of intelligence and the logic connected with the imaginative faculty. But there is no reason whatsoever to limit all the intellectual faculties to mathematical logic and overlook the demands of the rest. So much of modern philosophy that relies on physics, and so many generalizations within physics itself, are based on this unconscious mathematicism which Cartesian philosophy bestowed upon mathematical physics, and which has become accentuated in contemporary science. In the domains of both micro- and astrophysics direct contact with objective reality has been removed, leaving only an abstract mathematical model as the means of analysing the structure of matter.

What’s more, AI researchers have begun to realize that emotions may be a key to consciousness. Neuroscientists like Dr. Antonio Damasio have found that when the link between the prefrontal lobe (which governs rational thought) and the emotional centers (e.g., the limbic system) is damaged, patients cannot make value judgments. They are paralyzed when making the simplest of decisions (what things to buy, when to set an appointment, which color pen to use) because everything has the same value to them. Hence, emotions are not a luxury; they are absolutely essential, and without them a robot will have difficulty determining what is important and what is not. So emotions, instead of being peripheral to the progress of artificial intelligence, are now assuming central importance. If a robot encounters a raging fire, it might rescue the computer files first, not the people, since its programming might say that valuable documents cannot be replaced but workers always can be. It is crucial that robots be programmed to distinguish between what is important and what is not, and emotions are shortcuts the brain uses to rapidly determine this. Robots would thus have to be programmed to have a value system—that human life is more important than material objects, that children should be rescued first in an emergency, that objects with a higher price are more valuable than objects with a lower price, etc. Since robots do not come equipped with values, a huge list of value judgments must be uploaded into them. The problem with emotions, however, is that they are sometimes irrational, while robots are mathematically precise. So silicon consciousness may differ from human consciousness in key ways. For example, humans have little control over emotions, since they happen so rapidly and because they originate in the limbic system, not the prefrontal cortex of the brain. Furthermore, our emotions are often biased.

More recently, mathematical script has given rise to an even more revolutionary writing system, a computerised binary script consisting of only two signs: 0 and 1. The words I am now typing on my keyboard are written within my computer by different combinations of 0 and 1. Writing was born as the maidservant of human consciousness, but is increasingly becoming its master. Our computers have trouble understanding how Homo sapiens talks, feels and dreams. So we are teaching Homo sapiens to talk, feel and dream in the language of numbers, which can be understood by computers. And this is not the end of the story. The field of artificial intelligence is seeking to create a new kind of intelligence based solely on the binary script of computers. Science-fiction movies such as The Matrix and The Terminator tell of a day when the binary script throws off the yoke of humanity. When humans try to regain control of the rebellious script, it responds by attempting to wipe out the human race.

The three main mediaeval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism. Realism, as the word is used in connection with the mediaeval controversy over universals, is the Platonic doctrine that universals or abstract entities have being independently of the mind; the mind may discover them but cannot create them. Logicism, represented by Frege, Russell, Whitehead, Church, and Carnap, condones the use of bound variables to refer to abstract entities known and unknown, specifiable and unspecifiable, indiscriminately. Conceptualism holds that there are universals but they are mind-made. Intuitionism, espoused in modern times in one form or another by Poincaré, Brouwer, Weyl, and others, countenances the use of bound variables to refer to abstract entities only when those entities are capable of being cooked up individually from ingredients specified in advance. As Fraenkel has put it, logicism holds that classes are discovered while intuitionism holds that they are invented—a fair statement indeed of the old opposition between realism and conceptualism. This opposition is no mere quibble; it makes an essential difference in the amount of classical mathematics to which one is willing to subscribe. Logicists, or realists, are able on their assumptions to get Cantor’s ascending orders of infinity; intuitionists are compelled to stop with the lowest order of infinity, and, as an indirect consequence, to abandon even some of the classical laws of real numbers. The modern controversy between logicism and intuitionism arose, in fact, from disagreements over infinity. Formalism, associated with the name of Hilbert, echoes intuitionism in deploring the logicist’s unbridled recourse to universals. But formalism also finds intuitionism unsatisfactory. This could happen for either of two opposite reasons. The formalist might, like the logicist, object to the crippling of classical mathematics; or he might, like the nominalists of old, object to admitting abstract entities at all, even in the restrained sense of mind-made entities. The upshot is the same: the formalist keeps classical mathematics as a play of insignificant notations. This play of notations can still be of utility—whatever utility it has already shown itself to have as a crutch for physicists and technologists. But utility need not imply significance, in any literal linguistic sense. Nor need the marked success of mathematicians in spinning out theorems, and in finding objective bases for agreement with one another’s results, imply significance. For an adequate basis for agreement among mathematicians can be found simply in the rules which govern the manipulation of the notations—these syntactical rules being, unlike the notations themselves, quite significant and intelligible.

On another night, in a different dream I was asking a question. “How is it that you say all are equal, yet the obvious contradictions smack us in the face: inequalities in virtues, temperances, finances, rights, abilities and talents, intelligence, mathematical aptitude, ad infinitum?” The answer was a metaphor. “It is as if a large diamond were to be found inside each person. Picture a diamond a foot long. The diamond has a thousand facets, but the facets are covered with dirt and tar. It is the job of the soul to clean each facet until the surface is brilliant and can reflect a rainbow of colors. “Now, some have cleaned many facets and gleam brightly. Others have only managed to clean a few; they do not sparkle so. Yet, underneath the dirt, each person possesses within his or her breast a brilliant diamond with a thousand gleaming facets. The diamond is perfect, not one flaw. The only differences among people are the number of facets cleaned. But each diamond is the same, and each is perfect.

Intelligence is a capacity so godlike, so protean that it must be contained and disciplined. This is the work of politics—understood as the ordering of society and the regulation of power to permit human flourishing while simultaneously restraining the most Hobbesian human instincts. There could be no greater irony: For all the sublimity of art, physics, music, mathematics and other manifestations of human genius, everything depends on the mundane, frustrating, often debased vocation known as politics (and its most exacting subspecialty—statecraft). Because if we don’t get politics right, everything else risks extinction. We grow justly weary of our politics. But we must remember this: Politics—in all its grubby, grasping, corrupt, contemptible manifestations—is sovereign in human affairs. Everything ultimately rests upon it. Fairly or not, politics is the driver of history. It will determine whether we will live long enough to be heard one day. Out there. By them, the few—the only—who got it right.

Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

If Bob envies Alice, he derives unhappiness from the difference between Alice’s well-being and his own; the greater the difference, the more unhappy he is. Conversely, if Alice is proud of her superiority over Bob, she derives happiness not just from her own intrinsic well-being but also from the fact that it is higher than Bob’s. It is easy to show that, in a mathematical sense, pride and envy work in roughly the same way as sadism; they lead Alice and Bob to derive happiness purely from reducing each other’s well-being, because a reduction in Bob’s well-being increases Alice’s pride, while a reduction in Alice’s well-being reduces Bob’s envy.31 Jeffrey Sachs, the renowned development economist, once told me a story that illustrated the power of these kinds of preferences in people’s thinking. He was in Bangladesh soon after a major flood had devastated one region of the country. He was speaking to a farmer who had lost his house, his fields, all his animals, and one of his children. “I’m so sorry—you must be terribly sad,” Sachs ventured. “Not at all,” replied the farmer. “I’m pretty happy because my damned neighbor has lost his wife and all his children too!

Lovelace defined as an ‘operation’ the control of material and symbolic entities beyond the second-order language of mathematics (like the idea, discussed in chapter 1, of an algorithmic thinking beyond the boundary of computer science). In a visionary way, Lovelace seemed to suggest that mathematics is not the universal theory par excellence but a particular case of the science of operations. Following this insight, she envisioned the capacity of numerical computers qua universal machines to represent and manipulate numerical relations in the most diverse disciplines and generate, among other things, complex musical artefacts: [The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine … Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.

When separated at birth and reunited as adults, they say they feel they have known each other all their lives. Testing confirms that identical twins, whether separated at birth or not, are eerily alike (though far from identical) in just about any trait one can measure. They are similar in verbal, mathematical, and general intelligence, in their degree of life satisfaction, and in personality traits such as introversion, agreeableness, neuroticism, conscientiousness, and openness to experience. They have similar attitudes toward controversial issues such as the death penalty, religion, and modern music. They resemble each other not just in paper-and-pencil tests but in consequential behavior such as gambling, divorcing, committing crimes, getting into accidents, and watching television. And they boast dozens of shared idiosyncrasies such as giggling incessantly, giving interminable answers to simple questions, dipping buttered toast in coffee, and—in the case of Abigail van Buren and Ann Landers—writing indistinguishable syndicated advice columns. The crags and valleys of their electroencephalograms (brainwaves) are as alike as those of a single person recorded on two occasions, and the wrinkles of their brains and distribution of gray

The legendary inscription above the Academy's door speaks loudly about Plato's attitude toward mathematics. In fact, most of the significant mathematical research of the fourth century BC was carried out by people associated in one way or another with the Academy. Yet Plato himself was not a mathematician of great technical dexterity, and his direct contributions to mathematical knowledge were probably minimal. Rather, he was an enthusiastic spectator, a motivating source of challenge, an intelligent critic, an an inspiring guide. The first century philosopher and historian Philodemus paints a clear picture: "At that time great progress was seen in mathematics, with Plato serving as the general architect setting out problems, and the mathematicians investigating them earnestly." To which the Neoplatonic philosopher and mathematician Proclus adds: "Plato...greatly advanced mathematics in general and geometry in particular because of his zeal for these studies. It is well known that his writings are thickly sprinkled with mathematical terms and that he everywhere tries to arouse admiration for mathematics among students of philosophy." In other words, Plato, whose mathematical knowledge was broadly up to date, could converse with the mathematicians as an equal and as a problem presenter, even though his personal mathematical achievements were not significant.

We remark, first, that in all ages, and especially in primitive philosophy, words such as being, essence, unity, good, have exerted an extraordinary influence over the minds of men. The meagreness or negativeness of their content has been in an inverse ratio to their power. They have become the forms under which all things were comprehended. There was a need or instinct in the human soul which they satisfied; they were not ideas, but gods, and to this new mythology the men of a later generation began to attach the powers and associations of the elder deities. The idea of good is one of those sacred words or forms of thought, which were beginning to take the place of the old mythology. It meant unity, in which all time and all existence were gathered up. It was the truth of all things, and also the light in which they shone forth, and became evident to intelligences human and divine. It was the cause of all things, the power by which they were brought into being. It was the universal reason divested of a human personality. It was the life as well as the light of the world, all knowledge and all power were comprehended in it. The way to it was through the mathematical sciences, and these too were dependent on it. To ask whether God was the maker of it, or made by it, would be like asking whether God could be conceived apart from goodness, or goodness apart from God.

The general intelligence which is the faculty of arranging concepts “reasonably” and handling words suitably, must therefore aid in the social life just as intelligence in the narrower sense of the word, which is the mathematical function of the mind, presides over the knowledge of matter. It is the first of these we have in mind when we say of a man that he is intelligent. By that we mean that he has the ability and the facility for combining the ordinary concepts and for drawing probable conclusions from them. One can hardly take issue with him on that account, as long as he confines himself to things of every-day life, for which the concepts were made. But one would hardly admit of a man who was merely intelligent undertaking to speak with authority on scientific questions seeing that the intellect, made precise in science, becomes a mathematical, physical and biological attitude of mind, and substitutes for words more appropriate signs. All the more should one forbid him to meddle in philosophy when the questions raised are no longer in the domain of the intelligence alone. But no, it is agreed that the intelligent man is on this point a competent man. Against this I protest most vigorously. I hold the intelligence in high esteem, but I have a very mediocre opinion of the “intelligent man,” whose cleverness consists in talking about all things with a show of truth.

Identical twins think and feel in such similar ways that they sometimes suspect they are linked by telepathy. When separated at birth and reunited as adults, they say they feel they have known each other all their lives. Testing confirms that identical twins, whether separated at birth or not, are eerily alike (though far from identical) in just about any trait one can measure. They are similar in verbal, mathematical, and general intelligence, in their degree of life satisfaction, and in personality traits such as introversion, agreeableness, neuroticism, conscientiousness, and openness to experience. They have similar attitudes toward controversial issues such as the death penalty, religion, and modern music. They resemble each other not just in paper-and-pencil tests but in consequential behavior such as gambling, divorcing, committing crimes, getting into accidents, and watching television. And they boast dozens of shared idiosyncrasies such as giggling incessantly, giving interminable answers to simple questions, dipping buttered toast in coffee,and-in the case of Abigail van Buren and Ann Landers-writing indistinguishable syndicated advice columns. The crags and valleysof their electroencephalograms (brainwaves) are as alike as those of a single person recorded on two occasions, and the wrinkles of their brains and distribution of gray matter across cortical areas are also similar.

When a high IQ-test score is accompanied by subpar performance in some other domain, this is thought "surprising," and a new disability category is coined to name the surprise. So, similarly, the diagnostic criterion for mathematics disorder (sometimes termed dyscalculia) in DSM IV is that "Mathematical ability that falls substantially below that expected for the individual's chronological age, measured intelligence, and age-appropriate education" (p. 50)- The logic of discrepancy-based classification based on IQ-test performance has created a clear precedent whereby we are almost obligated to create a new disability category when an important skill domain is found to be somewhat dissociated from intelligence. It is just this logic that I exploited in creating a new category of disability- dysrationalia.T he proposed definition of the disability was as follows: Dysrationalia is the inability to think and behave rationally despite adequate intelligence. It is a general term that refers to a heterogeneous group of disorders manifested by significant difficulties in belief formation, in the assessment of belief consistency, and/or in the determination of action to achieve one's goals. Although dysrationalia may occur concomitantly with other handicapping conditions (e.g., sensory impairment), dysrationalia is not the result of those conditions. The key diagnostic criterion for dysrationalia is a level of rationality, as demonstrated in thinking and behavior, that is significantly below the level of the individual's intellectual capacity (as determined by an individually administered IQ test).

That such a surprisingly powerful philosophical method was taken seriously can be only partially explained by the backwardness of German natural science in those days. For the truth is, I think, that it was not at first taken really seriously by serious men (such as Schopenhauer, or J. F. Fries), not at any rate by those scientists who, like Democritus2, ‘would rather find a single causal law than be the king of Persia’. Hegel’s fame was made by those who prefer a quick initiation into the deeper secrets of this world to the laborious technicalities of a science which, after all, may only disappoint them by its lack of power to unveil all mysteries. For they soon found out that nothing could be applied with such ease to any problem whatsoever, and at the same time with such impressive (though only apparent) difficulty, and with such quick and sure but imposing success, nothing could be used as cheaply and with so little scientific training and knowledge, and nothing would give such a spectacular scientific air, as did Hegelian dialectics, the mystery method that replaced ‘barren formal logic’. Hegel’s success was the beginning of the ‘age of dishonesty’ (as Schopenhauer3 described the period of German Idealism) and of the ‘age of irresponsibility’ (as K. Heiden characterizes the age of modern totalitarianism); first of intellectual, and later, as one of its consequences, of moral irresponsibility; of a new age controlled by the magic of high-sounding words, and by the power of jargon. In order to discourage the reader beforehand from taking Hegel’s bombastic and mystifying cant too seriously, I shall quote some of the amazing details which he discovered about sound, and especially about the relations between sound and heat. I have tried hard to translate this gibberish from Hegel’s Philosophy of Nature4 as faithfully as possible; he writes: ‘§302. Sound is the change in the specific condition of segregation of the material parts, and in the negation of this condition;—merely an abstract or an ideal ideality, as it were, of that specification. But this change, accordingly, is itself immediately the negation of the material specific subsistence; which is, therefore, real ideality of specific gravity and cohesion, i.e.—heat. The heating up of sounding bodies, just as of beaten or rubbed ones, is the appearance of heat, originating conceptually together with sound.’ There are some who still believe in Hegel’s sincerity, or who still doubt whether his secret might not be profundity, fullness of thought, rather than emptiness. I should like them to read carefully the last sentence—the only intelligible one—of this quotation, because in this sentence, Hegel gives himself away. For clearly it means nothing but: ‘The heating up of sounding bodies … is heat … together with sound.’ The question arises whether Hegel deceived himself, hypnotized by his own inspiring jargon, or whether he boldly set out to deceive and bewitch others. I am satisfied that the latter was the case, especially in view of what Hegel wrote in one of his letters. In this letter, dated a few years before the publication of his Philosophy of Nature, Hegel referred to another Philosophy of Nature, written by his former friend Schelling: ‘I have had too much to do … with mathematics … differential calculus, chemistry’, Hegel boasts in this letter (but this is just bluff), ‘to let myself be taken in by the humbug of the Philosophy of Nature, by this philosophizing without knowledge of fact … and by the treatment of mere fancies, even imbecile fancies, as ideas.’ This is a very fair characterization of Schelling’s method, that is to say, of that audacious way of bluffing which Hegel himself copied, or rather aggravated, as soon as he realized that, if it reached its proper audience, it meant success.

In 1933 things were still being taught in the higher educational establishments which had been proven by science to be false as long ago as 1899. The young man who wishes to keep abreast of the times, therefore, had to accept a double load on his unfortunate brain. In a hundred years' time, the number of people wearing spectacles, and the size of the human brain, will both have increased considerably; but the people will be none the more intelligent. What they will look like, with their enormous, bulging heads, it is better not to try to imagine; they will probably be quite content with their own appearance, but if things continue in the manner predicted by the scientists, I think we can count ourselves lucky that we shall not live to see them! When I was a schoolboy, I did all I could to get out into the open air as much as possible—my school reports bear witness to that ! In spite of this, I grew up into a reasonably intelligent young man, I developed along very normal lines, and I learnt a lot of things of which my schoolfellows learnt nothing. In short, our system of education is the exact opposite of that practised in the gymnasia of ancient days. The Greek of the golden age sought a harmonious education; we succeed only in producing intellectual monsters. Without the introduction of conscription, we should have fallen into complete decadence, and it is thanks to this universal military service that the fatal process has been arrested. This I regard as one of the greatest events in history. When I recall my masters at school, I realise that half of them were abnormal; and the greater the distance from which I look back on them, the stronger is my conviction that I am quite right. The primary task of education is to train the brain of the young. It is quite impossible to recognise the potential aspirations of a child of ten. In old days teachers strove always to seek out each pupil's weak point, and by exposing and dwelling on it, they successfully killed the child's self-confidence. Had they, on the contrary, striven to find the direction in which each pupil's talents lay, and then concentrated on the development of those talents, they would have furthered education in its true sense. Instead, they sought mass-production by means of endless generalisations. A child who could not solve a mathematical equation, they said, would do no good in life. It is a wonder that they did not prophesy that he would come to a bad and shameful end! Have things changed much to-day, I wonder? I am not sure, and many of the things I see around me incline me to the opinion that they have not.

This makes a mockery of real science, and its consequences are invariably ridiculous. Quite a few otherwise intelligent men and women take it as an established principle that we can know as true only what can be verified by empirical methods of experimentation and observation. This is, for one thing, a notoriously self-refuting claim, inasmuch as it cannot itself be demonstrated to be true by any application of empirical method. More to the point, though, it is transparent nonsense: most of the things we know to be true, often quite indubitably, do not fall within the realm of what can be tested by empirical methods; they are by their nature episodic, experiential, local, personal, intuitive, or purely logical. The sciences concern certain facts as organized by certain theories, and certain theories as constrained by certain facts; they accumulate evidence and enucleate hypotheses within very strictly limited paradigms; but they do not provide proofs of where reality begins or ends, or of what the dimensions of truth are. They cannot even establish their own working premises—the real existence of the phenomenal world, the power of the human intellect accurately to reflect that reality, the perfect lawfulness of nature, its interpretability, its mathematical regularity, and so forth—and should not seek to do so, but should confine themselves to the truths to which their methods give them access. They should also recognize what the boundaries of the scientific rescript are. There are, in fact, truths of reason that are far surer than even the most amply supported findings of empirical science because such truths are not, as those findings must always be, susceptible of later theoretical revision; and then there are truths of mathematics that are subject to proof in the most proper sense and so are more irrefutable still. And there is no one single discourse of truth as such, no single path to the knowledge of reality, no single method that can exhaustively define what knowledge is, no useful answers whose range has not been limited in advance by the kind of questions that prompted them. The failure to realize this can lead only to delusions of the kind expressed in, for example, G. G. Simpson’s self-parodying assertion that all attempts to define the meaning of life or the nature of humanity made before 1859 are now entirely worthless, or in Peter Atkins’s ebulliently absurd claims that modern science can “deal with every aspect of existence” and that it has in fact “never encountered a barrier.” Not only do sentiments of this sort verge upon the deranged, they are nothing less than violent assaults upon the true dignity of science (which lies entirely in its severely self-limiting rigor).

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.

To a theoretician, all these criticisms are troublesome but not fatal. But what does cause problems for a theoretician is that the model seems to predict a multiverse of parallel universes, many of which are crazier than those in the imagination of a Hollywood scriptwriter. String theory has an infinite number of solutions, each describing a perfectly well-behaved finite theory of gravity, which do not resemble our universe at all. In many of these parallel universes, the proton is not stable, so it would decay into a vast cloud of electrons and neutrinos. In these universes, complex matter as we know it (atoms and molecules) cannot exist. They only consist of a gas of subatomic particles. (Some might argue that these alternate universes are only mathematical possibilities and are not real. But the problem is that the theory lacks predictive power, since it cannot tell you which of these alternate universes is the real one.) This problem is actually not unique to string theory. For example, how many solutions are there to Newton’s or Maxwell’s equations? There are an infinite number, depending on what you are studying. If you start with a light bulb or a laser and you solve Maxwell’s equations, you find a unique solution for each instrument. So Maxwell’s or Newton’s theories also have an infinite number of solutions, depending on the initial conditions—that is, the situation you start with. This problem is likely to exist for any theory of everything. Any theory of everything will have an infinite number of solutions depending on the initial conditions. But how do you determine the initial conditions of the entire universe? This means you have to input the conditions of the Big Bang from the outside, by hand. To many physicists this seems like cheating. Ideally, you want the theory itself to tell you the conditions that gave rise to the Big Bang. You want the theory to tell you everything, including the temperature, density, and composition of the original Big Bang. A theory of everything should somehow contain its own initial conditions, all by itself. In other words, you want a unique prediction for the beginning of the universe. So string theory has an embarrassment of riches. Can it predict our universe? Yes. That is a sensational claim, the goal of physicists for almost a century. But can it predict just one universe? Probably not. This is called the landscape problem. There are several possible solutions to this problem, none of them widely accepted. The first is the anthropic principle, which says that our universe is special because we, as conscious beings, are here to discuss this question in the first place. In other words, there might be an infinite number of universes, but our universe is the one that has the conditions that make intelligent life possible. The initial conditions of the Big Bang are fixed at the beginning of time so that intelligent life can exist today. The other universes might have no conscious life in them.

We are living now, not in the delicious intoxication induced by the early successes of science, but in a rather grisly morning-after, when it has become apparent that what triumphant science has done hitherto is to improve the means for achieving unimproved or actually deteriorated ends. In this condition of apprehensive sobriety we are able to see that the contents of literature, art, music—even in some measure of divinity and school metaphysics—are not sophistry and illusion, but simply those elements of experience which scientists chose to leave out of account, for the good reason that they had no intellectual methods for dealing with them. In the arts, in philosophy, in religion men are trying—doubtless, without complete success—to describe and explain the non-measurable, purely qualitative aspects of reality. Since the time of Galileo, scientists have admitted, sometimes explicitly but much more often by implication, that they are incompetent to discuss such matters. The scientific picture of the world is what it is because men of science combine this incompetence with certain special competences. They have no right to claim that this product of incompetence and specialization is a complete picture of reality. As a matter of historical fact, however, this claim has constantly been made. The successive steps in the process of identifying an arbitrary abstraction from reality with reality itself have been described, very fully and lucidly, in Burtt’s excellent “Metaphysical Foundations of Modern Science"; and it is therefore unnecessary for me to develop the theme any further. All that I need add is the fact that, in recent years, many men of science have come to realize that the scientific picture of the world is a partial one—the product of their special competence in mathematics and their special incompetence to deal systematically with aesthetic and moral values, religious experiences and intuitions of significance. Unhappily, novel ideas become acceptable to the less intelligent members of society only with a very considerable time-lag. Sixty or seventy years ago the majority of scientists believed—and the belief often caused them considerable distress—that the product of their special incompetence was identical with reality as a whole. Today this belief has begun to give way, in scientific circles, to a different and obviously truer conception of the relation between science and total experience. The masses, on the contrary, have just reached the point where the ancestors of today’s scientists were standing two generations back. They are convinced that the scientific picture of an arbitrary abstraction from reality is a picture of reality as a whole and that therefore the world is without meaning or value. But nobody likes living in such a world. To satisfy their hunger for meaning and value, they turn to such doctrines as nationalism, fascism and revolutionary communism. Philosophically and scientifically, these doctrines are absurd; but for the masses in every community, they have this great merit: they attribute the meaning and value that have been taken away from the world as a whole to the particular part of the world in which the believers happen to be living.

The mathematical exposition is extremely concise and occasionally awkward. Laplace was interested in results, not in how he got them. To avoid condensing a complicated mathematical argument to a brief, intelligible form he frequently omits everything but the conclusion, with the optimistic remark “Il est aisé à voir” (It is easy to see). He himself would often be unable to restore the reasoning by which he had “seen” these easy things without hours—sometimes days—of hard labor.

There are no such things as material particles (enduring “things”). There are no forces in the sense of things that can be transferred from one thing to another. What actually exists is information. This is defined mathematically. Information is intelligible; “things” are sensible. The evolving cosmic wavefunction is an information wavefunction. It’s made of mathematical information. Every part of it reflects information. It’s this information that is mathematically interpreted by minds as matter, force, energy, sensory things, and so on. Because humans interpret information non-mathematically (i.e. empirically, not rationally), they are astounded by the assertion that the universe is entirely mathematical. Our own interpretations are what conceal the Truth from us. We must transcend our empirical viewpoint if we ever wish to attain the divine – rational – perspective. Science, as pure empiricism, is anti-divinity. It locks us into human sensory delusion. Mathematics frees us.

he had made such and such a move, then I had such and such a winning combination in mind.’ But the ‘great game’ of chess is primarily psychological, a conflict between one trained intelligence and another, and not a mere collection of small mathematical theorems.

So what precisely is it that we don’t understand about consciousness? Few have thought harder about this question than David Chalmers, a famous Australian philosopher rarely seen without a playful smile and a black leather jacket—which my wife liked so much that she gave me a similar one for Christmas. He followed his heart into philosophy despite making the finals at the International Mathematics Olympiad—and despite the fact that his only B grade in college, shattering his otherwise straight As, was for an introductory philosophy course. Indeed, he seems utterly undeterred by put-downs or controversy, and I’ve been astonished by his ability to politely listen to uninformed and misguided criticism of his own work without even feeling the need to respond.

Ontological mathematics is based on light. Light is eternal (it does not experience time and it does not experience space and is therefore indestructible); light is mental (it is massless and immaterial), light is absolute (it provides the absolute reference frame – the ether – for all spacetime reference frames). Light corresponds exactly to the immaterial, unextended mind posited by Descartes. Have you seen the light? Once you realize that light is nothing but sinusoidal waves, as per Euler’s Formula, you have the means to understand the whole of reality. Light is God. Light is the substance of an intelligent, living, thinking mathematical organism, calculating its own perfection. All of the great ancients understood this type of picture of reality. No modern scientist does.

Anthropic models propose that life and intelligence are developmentally destined to emerge in our particular universe, and range from the mathematical (the apparent fine tuning of fundamental universal parameters, e.g., Rees 1999), to the empirical (special universal chemistry that promotes precursors to biogenesis, e.g., Henderson 1913, 1917; Miller 1953; Lazcano 2004), to the teleological (analogies and arguments for systemic function or purpose to cosmic intelligence, e.g., this paper). Today, as acknowledged by even their most adept practitioners (Barrow and Tipler 1986; Krauss et. al. 2008), anthropic universe models proceed more from ignorance and assumption than from knowledge.

It’s time to replace the scientific method with the mathematical method. It’s time to recognize that true reality is intelligible, not sensible; noumenal, not phenomenal; unobservable, not observable; metaphysical, not physical; hidden, not manifest; rationalist, not empiricist; necessary, not contingent.

Why do people that ask for “evidence” never ask for rational explanation? What is more reliable? – analytic reason, or the unreliable, fallible, limited, frequently delusional human senses where it is guaranteed that they are showing us only phenomena and never noumena (i.e. things in themselves). You cannot understand reality as a phenomenon, although this is in fact exactly what science tries to do. You can understand reality only as a noumenon – as an intelligible thing in itself – and that’s exactly what ontological mathematics is all about. Anyone that obsesses over phenomenal evidence is an opponent of noumenal truth, which is never subject to phenomenal evidence.

Many of the mathematical models for how a trait will spread in a population have failed—they don’t tell you this. No, I don’t talk about miracles, whatever words you put them under. And the “design” is there, but it is by no means benevolent or intelligent, nor comprehensible. You see in the spider’s web a creature of rudimentary nervous system and little intelligence “design” something beautiful and complex, and this is key to understanding also all of nature. There is an inherent “intelligence” inside things, uncanny, silent and demonic. Its workings and aims are obscure to us. Our own intelligence is only a crude deviation of it, an approximation. There is an “intelligence” in all things, and inborn in our bodies before anything to do with the brain or the nervous system. And all “adaptations,” no matter how much natural or unnatural selection may have gone to spreading them within a population, occur not by random but by a spontaneous correspondence of some kind between the organism and the environment.

Humanitarian Science 101 (Sonnet 1202) BRAIN means Benevolent Reformer Applying Information Nobly. DATA means Determined Action of Transformative Awareness. Information Technology is primitive IT, Civilized IT means Informed Transformation. Heuristic and holistic can never go together, Shortcuts only obstruct the rise of realization. Electronics means electron artistry. Chemistry is an art of bonding. Mathematics is the art of numbers, Evolution is the art of correcting. Society without science dumps the world in stoneage, Science devoid of society shoves the mind into iceage.

Electronics means electron artistry. Chemistry is an art of bonding. Mathematics is the art of numbers, Evolution is the art of correcting.

Unhappy is he who assumes the burden of a debt whose worth cannot be measured by the simple means of his own intelligence.

Ironically, GAAP rules have a requirement that governs the reporting of non-GAAP numbers. Companies usually show how they got, mathematically, from the GAAP number to the non-GAAP number. This is often called a bridge statement. We aren’t going to get into that here—too many details!—but feel free to look into companies’ financial statement notes or supplemental documents if you’re interested.

The intercept slips are pulled into the hut on the wooden tray and then move from chair to chair according to some highly organized scheme that Waterhouse can only vaguely grasp at this point. Someone explains to him that the bombes just broke the day’s codes around sundown, and so the entire day’s load of intercepts has just come down the tunnel from Hut 6 during the last couple of hours. He decides to think of the hut as a mathematical black box for the time being—that is, he’ll concentrate only on its inputs and outputs of information and ignore the internal details. Bletchley Park, taken in its entirety, is a black box of sorts: random letters stream into it, strategic intelligence streams out, and the internal particulars are of no interest to most of the people on the Ultra distribution list. The question that Waterhouse is here to figure out is: is there another vector of information coming out of this place, hidden subliminally in the teletype signals and the behaviors of the Allied commanders?

Our study of the various methods has led us to suggest a foreshortened and quite simple formula for the valuation of growth stocks, which is intended to produce figures fairly close to those resulting from the more refined mathematical calculations. Our formula is: Value = Current (Normal) Earnings × (8.5 plus twice the expected annual growth rate) The growth figure should be that expected over the next seven to ten years.

Turing proved, mathematically, that if you choose the right set of rules for the CPU and give it an indefinitely long tape to work with, it can perform any definable set of operations in the universe. It would be one of many equivalent machines now called Universal Turing Machines.

Of the components of IQ tests, Ashkenazim do well on verbal and mathematical questions but score lower than average on visuospatial questions. In most people, these two kinds of ability are highly correlated. This suggests that some specific force has been at work in shaping the nature of Ashkenazi intelligence, as if the population were being adapted not to hunting, which requires excellent visuospatial skills, but to more urban occupations served by the ability to manipulate words and numerals. So it’s striking to find that Ashkenazim, almost from the moment their appearance in Europe was first recorded, around 900 AD, were heavily engaged in moneylending. This was the principal occupation of Jews in England, France and Germany. The trade required a variety of high level skills, including the ability to read and write contracts and to do arithmetic. Literacy was a rare ability in medieval Europe. As late as 1500, only 10% of the population of most European countries was literate, whereas almost all Jews were.7 As for arithmetic, it may be simple enough with the Arabic numerals in use today. But Arabic numerals did not become widespread in Europe until the mid-16th century. Before that, people used Roman numerals, a notation system that has no zero. Calculating interest rates and currency swaps without the use of zero is not a straightforward computation.

Concerning intelligence, "the capability to identify what to carefully examine—often a decision driven by mathematical analysis—has become as essential as the capacity to gather the intelligence itself." -- K. Lee Lerner. Cornwall, U.K. May, 2003.

The fact that I solved a problem is a purely subjective fact that in itself has as yet no general interest. However, the logical and mathematical problems possess the special property of the general validity of their solutions: If I have solved a logical or mathematical problem, then I can present this solution in a way that is intelligible to all and it is necessary that it be recognised as correct solution although this necessity has to a certain extent an ideal character, for it presupposes a sufficient intelligence on the part of the listener.

Is intelligence a formal (or mathematically definable) system? Is life a recursive (or mechanically calculable) function? What happens when you replicate discrete-state microprocessors by the billions and run these questions the other way? (Are formal systems intelligent? Are recursive functions alive?) Life and intelligence have learned to operate on any number of different scales: larger, smaller, slower, and faster than our own. Biology and technology evidence parallel tendencies toward collective, hierarchical processes based on information exchange. As information is distributed, it tends to be represented (encoded) by increasingly economical (meaningful) forms. This evolutionary process, whereby the most economical or meaningful representation wins, leads to a hierarchy of languages, encoding meaning on levels that transcend comprehension by the system’s individual components—whether genes, insects, microprocessors, or human minds.

Mathematics is that portion of our intellectual activity which transcends our biology and our environment. The principles of biology as we know them may apply to life forms on other worlds, yet there is no necessity for this to be so. The principles of physics should be more universal, yet it is easy to imagine another universe governed by different physical laws. Mathematics, a creation of mind, is less arbitrary than biology or physics, creations of nature; the creatures we imagine inhabiting another world in another universe, with another biology and another physics, will develop a mathematics which in essence is the same as ours. In believing this we may be falling into a trap: Mathematics being a creation of our mind, it is, of course, difficult to imagine how mathematics could be otherwise without actually making it so, but perhaps we should not presume to predict the course of the mathematical activities of all possible types of intelligence. On the other hand, the pragmatic content of our belief in the transcendence of mathematics has nothing to do with alien forms of life. Rather it serves to give a direction to mathematical investigation, resulting from the insistence that mathematics be born of an inner necessity.

Is it enough to live in a universe whose laws spontaneously create life? Or do you prefer ... God?” She paused, looking embarrassed. “Sorry, after all we’ve been through tonight, I know that’s a strange question.” “Well,” Langdon said with a laugh, “I think my answer would benefit from a decent night’s sleep. But no, it’s not strange. People ask me all the time if I believe in God.” “And how do you reply?” “I reply with the truth,” he said. “I tell them that, for me, the question of God lies in understanding the difference between codes and patterns.” Ambra glanced over. “I’m not sure I follow you.” “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.” Ambra grasped the concept, but not how it related to God. “The other difference between codes and patterns,” Langdon continued, “is that codes do not occur naturally in the world. Musical notation does not sprout from trees, and symbols do not draw themselves in the sand. Codes are the deliberate inventions of intelligent consciousnesses.” Ambra nodded. “So codes always have an intention or awareness behind them.” “Exactly. Codes don’t appear organically; they must be created.” Ambra studied him a long moment. “What about DNA?” A professorial smile appeared on Langdon’s lips. “Bingo,” he said. “The genetic code. That’s the paradox.” Ambra felt a rush of excitement. The genetic code obviously carried data — specific instructions on how to build organisms. By Langdon’s logic, that could mean only one thing. “You think DNA was created by an intelligence!” Langdon held up a hand in mock self-defense. “Easy, tiger!” he said, laughing. “You’re treading on dangerous ground. Let me just say this. Ever since I was a child, I’ve had the gut sense that there’s a consciousness behind the universe. When I witness the precision of mathematics, the reliability of physics, and the symmetries of the cosmos, I don’t feel like I’m observing cold science; I feel as if I’m seeing a living footprint ... the shadow of some greater force that is just beyond our grasp.

This creation is not run by blind forces. It operates according to an intelligent plan. […] It is unreasonable to suppose that this world is just a chance result of different combinations of atoms, with no guiding intelligence behind those atoms. On the contrary, is evident that there is law and order in the universe. Your life, and all life, is governed with mathematical precision by God’s intelligently framed cosmic laws. So by the divine law of action or karma, cause and effect, everything that you do is recorded in your soul. Thus, according to the measure of your work, whatever you accomplish through will power and creativity will be your passport after death to the heavenly regions earned by dutiful souls. And when you reincarnate in this world, you will be born with those mental powers developed by your previous efforts.

Concerning intelligence, "the capability to identify what to carefully examine—often a decision driven by mathematical analysis—has become as essential as the capacity to gather the intelligence itself." K. Lee Lerner. Cornwall, U.K. May, 2003. intelligence, "the capability to identify what to carefully examine—often a decision driven by mathematical analysis—has become as essential as the capacity to gather the intelligence itself." -- K. Lee Lerner. Cornwall, U.K. May, 2003.

However, when value is recognised as objective, individual preference (though it need not be utterly eradicated) is relativised, one’s private perspective is shown to be relative to the real value of beautiful things, and therefore one’s appreciation of beauty can be more or less intelligent, more or less attuned to the structures that inhere in nature and that can even sometimes be mathematically discerned (revealing the presence of such measurables as Pi, the Golden Ratio, and the Fibonacci Sequence).

Mr. Bentham would answer, that the knowledge which carries virtue along with it, is the knowledge how to take care of number one—a clear appreciation of what is pleasurable, what painful, and what promotes the one and prevents the other. An uneducated man is ever mistaking his own interest, and standing in the way of his own true enjoyments. Useful Knowledge is that which tends to make us more useful to ourselves;—a most definite and intelligible account of the matter, and needing no explanation. But it would be a great injustice, both to Lord Brougham and to Sir Robert, to suppose, when they talk of Knowledge being Virtue, that they are Benthamizing. Bentham had not a spark of poetry in him; on the contrary, there is much of high aspiration, generous sentiment, and impassioned feeling in the tone of Lord Brougham and Sir Robert. They speak of knowledge as something "pulchrum," fair and glorious, exalted above the range of ordinary humanity, and so little connected with the personal interest of its votaries, that, though Sir Robert does obiter talk of improved modes of draining, and the chemical properties of manure, yet he must not be supposed to come short of the lofty enthusiasm of Lord Brougham, who expressly panegyrizes certain ancient philosophers who gave up riches, retired into solitude, or embraced a life of travel, smit with a sacred curiosity about physical or mathematical truth.

The question of the intelligibility of the universe is a theological one: Why an intelligible world? Why a universe whose structures correspond to the human capacity to understand? The physicist and theologian John Polkinghorne gives a theological answer to the question: “Science is possible, and mathematics so remarkably effective, because the world is a creation and we creatures are made in the image of the Creator. Fundamental physics reveals a universe shot through with signs of mind, and it is an attractive understanding that it is indeed the Mind of God that lies behind the wonderful cosmic order.

Everything exists because other things cause it to exist, otherwise, it exists and doesn't exist at all.

After all, when one asks if a person is being rational, we aren’t asking very much: really, just whether they are capable of making basic logical connections. The matter rarely comes up unless one suspects someone might actually be crazy or perhaps so blinded by passion that their arguments make no sense. Consider, in contrast, what’s entailed when one asks if someone is being "reasonable." The standard here is much higher. Reasonableness implies a much more sophisticated ability to achieve a balance between different perspectives, values, and imperatives, non of which, usually, could possibly be reduced to mathematical formulae. It means coming up with a compromise between positions that are, according to formal logic, incommensurable, just as there’s no formal way, when deciding what to cook for dinner, to measure the contrasting advantages of ease of preparation, healthiness, and taste. But of course we make such decisions all the time. Most of life--particularly life with others--consists of making reasonable compromises that could never be reduced to mathematical models. Another way to put this is that political theorists tend to assume actors who are operating on the intellectual level of an eight-year-old. Developmental psychologists have observed that children begin to make logical arguments not to solve problems, but when coming up with reasons for what they already wan to think. Anyone who deals with small children on a regular basis will immediately recognize that this is true. The ability to compare and coordinate contrasting perspectives on the other hand comes later and is the very essence of mature intelligence. It’s also precisely what those used to the power of command rarely have to do. (p. 200-201)

In 1971, Cattell published a book entitled Abilities: Their Structure, Growth, and Action. In it, he posited that there were two types of intelligence that people possess, but at greater abundance at different points in life. The first is fluid intelligence, which Cattell defined as the ability to reason, think flexibly, and solve novel problems. It is what we commonly think of as raw smarts, and researchers find that it is associated with both reading and mathematical ability.[4] Innovators typically have an abundance of fluid intelligence. Cattell, who specialized in intelligence testing, observed that it was highest relatively early in adulthood and diminished rapidly starting in one’s thirties and forties.[

Comprehensibility Theorem is the first mathematical theorem implying the impossibility of any AI agent or natural agent—including a not-necessarily infallible human agent—satisfying a rigorous and deductive interpretation of the self-comprehensibility challenge. … Self-comprehensibility in some form might be essential for a kind of self-reflection useful for self-improvement that might enable some agents to increase their success”. It is reasonable to conclude that a system which doesn’t comprehend itself would not be able to explain itself.

The fact that I have solved a problem is a purely subjective fact that in itself has as yet no general interest. However, the logical and mathematical problems possess the special property of the general validity of their solutions: If I have solved a logical or a mathematical problem, then I can present this solution in a way that is intelligible to all and it is necessary that it be recognized as a correct solution although this necessity has to a certain extent an ideal character, for it presupposes a sufficient intelligence on the part of the listener.

Max Tegmark, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality

He doesn’t have to think about it to do it well. It is my experience that in some areas Charley is more intelligent than I am, but in others he is abysmally ignorant. He can’t read, can’t drive a car, and has no grasp of mathematics. But in his own field of endeavor, which he was now practicing, the slow, imperial smelling over and anointing of an area, he has no peer. Of course his horizons are limited, but how wide are mine?

1. The clear and quantitative physical differences among people in size, strength, speed, agility, coordination, and other physical attributes that translates into some being more successful than others, and that at least half of these differences are inherited. 2. The clear and quantitative intellectual differences among people in memory, problem solving ability, cognitive speed, mathematical talent, spatial reasoning, verbal skills, emotional intelligence, and other mental attributes that translates into some being more successful than others, and that at least half of these differences are inherited.

Military expediency aside, how did the new emperor appear to his subjects? Experience, inclination and natural intelligence had made him a polymath, though the demands of his role as emperor, and the infinite resources available to him, left him open to accusations of dilettantism. This charge was unfair; he was unusual in that he genuinely wanted to become adept in many areas himself, rather than simply be served or amused by the ability of others. Throughout his reign his understanding was gained either by direct observation or by the development of skills that he admired in others. Poetry, architecture, music, philosophy and mathematics all intrigued him and he was patron of them all, surrounding himself with men of genius: the poet and satirist Juvenal, the architect Apollodorus, the historians Tacitus, Suetonius and Arrian, the writers Pliny the Younger, Pausanias and Plutarch.

About the Author After a misspent youth doing mathematical and medical research, Stuart Armstrong was blown away by the idea that people would actually pay him to work on the most important problems facing humanity. He hasn’t looked back since, and has been focusing mainly on existential risk, anthropic probability, AI, decision theory, moral uncertainty, and long-term space exploration. He also walks the dog a lot, and was recently involved in the coproduction of the strange intelligent agent that is a human baby.

Let us say, in a word, that the correlation between the laws of mathematics and of physics is the evidence of the rational character of nature. Nature may be reduced to motions; and motions can be understood only as force, activity. But the laws which connect motions are fundamentally mathematical laws,- laws of reason. Hence force, activity, can be understood only as rational, as spiritual. Nature is thus seen to mean Activity, and Activity is seen to mean Intelligence

Teachers greatly influence how students perceive and approach struggle in the mathematics classroom. Even young students can learn to value struggle as an expected and natural part of learning, as demonstrated by the class motto of one first-grade math class: If you are not struggling, you are not learning. Teachers must accept that struggle is important to students' learning of mathematics, convey this message to students, and provide time for them to try to work through their uncertainties. Unfortunately, this may not be enough, since some students will still simply shut down in the face of frustration, proclaim, 'I don't know,' and give up. Dweck (2006) has shown that students with a fixed mindset--that is, those who believe that intelligence (especially math ability) is an innate trait--are more likely to give up when they encounter difficulties because they believe that learning mathematics should come naturally. By contrast, students with a growth mindset--that is, those who believe that intelligence can be developed through effort--are likely to persevere through a struggle because they see challenging work as an opportunity to learn and grow.

Adelard of Bath (ca. 1075-1160) disguised himself as a Muslim and studied at Cordoba; he translated Euclid's Elements from the Arabic translation into Latin, and Ptolemy's Almagest from Greek into Latin. When Alfonso VI of Castile captured Toledo from the Moors in 1085, he did not burn their libraries, containing a wealth of Muslim manuscripts. Under the encouragement of the Archbishop of Toledo, a veritable intelligence evaluation center was set up. A large number of translators, the best known of whom was Gerard of Cremona (1114-1187), translated from Arabic, Greek and Hebrew into Latin, at last acquainting Europe not only with classical Greek mathematics, but also with contemporary Arab algebra, trigonometry and astronomy. Before the Toledo leak opened, mediaeval Europe did not have a mathematician who was not a Moor, Greek or a Jew.

Jack Dongarra, a researcher at Tennessee’s Oak Ridge National Lab and part of a team that tracks supercomputer speed, determined that Apple’s best-selling tablet, the iPad 2, is as fast as a circa 1985 Cray 2 supercomputer. In fact, running at over 1.5 gigaflops (one gigaflop equals one billion mathematical operations, or calculations per second) the iPad 2 would have made the list of the world’s five hundred fastest supercomputers as late as 1994. In

FATHER OF THE COMPUTER Alan Turing was sneered at for not being a tough guy, a he-man with hair on his chest. He whined, croaked, stuttered. He used an old necktie for a belt. He rarely slept and went without shaving for days. And he raced from one end of the city to the other all the while concocting complicated mathematical formulas in his mind. Working for British intelligence, he helped shorten the Second World War by inventing a machine that cracked the impenetrable military codes used by Germany’s high command. At that point he had already dreamed up a prototype for an electronic computer and had laid out the theoretical foundations of today’s information systems. Later on, he led the team that built the first computer to operate with integrated programs. He played interminable chess games with it and asked it questions that drove it nuts. He insisted that it write him love letters. The machine responded by emitting messages that were rather incoherent. But it was flesh-and-blood Manchester police who arrested him in 1952 for gross indecency. At the trial, Turing pled guilty to being a homosexual. To stay out of jail, he agreed to undergo medical treatment to cure him of the affliction. The bombardment of drugs left him impotent. He grew breasts. He stayed indoors, no longer went to the university. He heard whispers, felt stares drilling into his back. He had the habit of eating an apple before going to bed. One night, he injected the apple with cyanide.

Modern culture has disenchanted the world by disenchanting numbers. For us, numbers are about quantity and control, not quality and contemplation. After Bacon, knowledge of numbers is a key to manipulation, not meditation. Numbers are only meaningful (like all raw materials that comprise the natural world) when we can do something with them. When we read of twelve tribes and twelve apostles and twelve gates and twelve angels, we typically perceive something spreadsheet-able. By contrast, in one of Caldecott’s most radical claims, he insists, “It is not simply that numbers can be used as symbols. Numbers have meaning—they are symbols. The symbolism is not always merely projected onto them by us; much of it is inherent in their nature” (p. 75). Numbers convey to well-ordered imaginations something of (in Joseph Cardinal Ratzinger’s metaphor) the inner design of the fabric of creation. The fact that the words “God said” appear ten times in the account of creation and that there are ten “words” in the Decalogue is not a random coincidence. The beautiful meaningfulness of a numberly world is most evident in the perception of harmony, whether in music, architecture, or physics. Called into being by a three-personed God, creation’s essential relationality is often evident in complex patterns that can be described mathematically. Sadly, as Caldecott laments, “our present education tends to eliminate the contemplative or qualitative dimension of mathematics altogether” (p. 55). The sense of transcendence that many (including mathematicians and musicians) experience when encountering beauty is often explained away by materialists as an illusion. Caldecott offers an explanation rooted in Christology. Since the Logos is love, and since all things are created through him and for him and are held together in him, we should expect the logic, the rationality, the intelligibility of the world to usher in the delight that beauty bestows. One

We’re talking about fundamentals here; the fundamental physical laws pertaining to the day-to-day running of the universe. Physicists call them the fundamental constants—things like the masses of atomic particles, the speed of light, the electric charges of electrons, the strength of gravitational force.… They’re beginning to realize just how finely balanced they are. One flip of a decimal point either way and things would start to go seriously wrong. Matter wouldn’t form, stars wouldn’t twinkle, the universe as we know it wouldn’t exist and, if we insist on taking the selfish point of view in the face of such spectacular, epic, almighty destruction, nor would we. The cosmic harmony that made life possible exists at the mercy of what appear, on the face of it, to be unlikely odds. Who or what decided at the time of the Big Bang that the number of particles created would be 1 in 1 billion more than the number of antiparticles, thus rescuing us by the width of a whisker from annihilation long before we even existed (because when matter and antimatter meet, they cancel each other out)? Who or what decided that the number of matter particles left behind after this oversize game of cosmic swapping would be exactly the right number to create a gravitational force that balanced the force of expansion and didn’t collapse the universe like a popped balloon? Who decided that the mass of the neutron should be just enough to make the formation of atoms possible? That the nuclear force that holds atomic nuclei together, in the face of their natural electromagnetic desire to repulse each other, should be just strong enough to achieve this, thus enabling the universe to move beyond a state of almost pure hydrogen? Who made the charge on the proton exactly right for the stars to turn into supernovas? Who fine-tuned the nuclear resonance level for carbon to just delicate enough a degree that it could form, making life, all of which is built on a framework of carbon, possible? The list goes on. And on. And as it goes on—as each particularly arrayed and significantly defined property, against all the odds, and in spite of billions of alternative possibilities, combines exquisitely, in the right time sequence, at the right speed, weight, mass, and ratio, and with every mathematical quality precisely equivalent to a stable universe in which life can exist at all—it adds incrementally in the human mind to a growing sense, depending on which of two antithetical philosophies it chooses to follow, of either supreme and buoyant confidence, or humble terror. The first philosophy says this perfect pattern shows that the universe is not random; that it is designed and tuned, from the atom up, by some supreme intelligence, especially for the purpose of supporting life. The other says it’s a one in a trillion coincidence.

We’re searching for intelligent, conscious, tool-making beings that have developed a language we’re capable of understanding.We’re searching for intelligent conscious, tool-making, communicative beings that live in social groups (so they can reap the benefits of civilization) and that develop the tools of science and mathematics. We’re searching for ourselves . . .

This concludes the review of three commonly cited prototypical confluence theories of creativity, namely the systems approach (Csikszentmihalyi, 2000), which suggests that creativity is a sociocultural process involving the interaction between the individual, domain, and field; Gruber & Wallace’s (2000) model that treats each individual case study as a unique evolving system of creativity; and investment theory (Sternberg & Lubart, 1996), which suggests that creativity is the result of the convergence of six elements (intelligence, knowledge, thinking styles, personality, motivation, and environment).

Besides revealing the difficulty of describing mental imagery, all the mathematicians reported that they did not use computers in their work. This characteristic of the pure mathematician's work is echoed in Poincaré's (1948) use of the “choice” metaphor and Ervynck's (1991) use of the term “nonalgorithmic decision making.” The doubts expressed by the mathematicians about the incapability of machines to do their work brings to mind the reported words of Garrett Birkhoff, one of the great applied mathematicians of our time. In his retirement presidential address to the Society for Industrial and Applied Mathematics, Birkhoff (1969) addressed the role of machines in human creative endeavors. In particular, part of this address was devoted to discussing the psychology of the mathematicians (and hence of mathematics). Birkhoff (1969) said: The remarkable recent achievements of computers have partially fulfilled an old dream. These achievements have led some people to speculate that tomorrow's computers will be even more "intelligent" than humans, especially in their powers of mathematical reasoning...the ability of good mathematicians to sense the significant and to avoid undue repetition seems, however, hard to computerize; without it, the computer has to pursue millions of fruitless paths avoided by experienced human mathematicians. (pp. 430-438)

Investment theory claims that the convergence of six elements constitutes creativity. The six elements are intelligence, knowledge, thinking styles, personality, motivation, and environment. It is important that the reader not mistake the word intelligence for an IQ score. On the contrary, Sternberg (1985) suggests a triarchic theory of intelligence that consists of synthetic (ability to generate novel, task appropriate ideas), analytic, and practical abilities. Knowledge is defined as knowing enough about a particular field to move it forward. Thinking styles are defined as a preference for thinking in original ways of one’s choosing, the ability to think globally as well as locally, and the ability to distinguish questions of importance from those that are not important. Personality attributes that foster creative functioning are the willingness to take risks, overcome obstacles, and tolerate ambiguity. Finally, motivation and an environment that is supportive and rewarding are essential elements of creativity (Sternberg, 1985).

The investment theory model Bharath Sriraman 25 suggests that creativity is more than a simple sum of the attained level of functioning in each of the six elements. Regardless of the functioning levels in other elements, a certain level or threshold of knowledge is required without which creativity is impossible. High levels of intelligence and motivation can positively enhance creativity, and compensations can occur to counteract weaknesses in other elements. For example, one could be in an environment that is non-supportive of creative efforts, but a high level of motivation could possibly overcome this and encourage the pursuit of creative endeavors.

Bucky was unfailingly logical and he used that logic to recognize that this universe exists only because of a Universal Intelligence, which in this culture we call God. I love his reference to the use of the word “God.” “It is mathematically hypothesizable that all of the truths are potentially integratable and that the resulting integral truth constitutes the cosmic integrity that humans intuitively sense to be in governance of Universe and speak of to one another with the inadequate sound-word god.”(1) This

I suspect that if we can build such ultra-intelligent machines, then the first one will be severely limited by the software we’ve written for it, and that we’ll have compensated for our lack of understanding about how to optimally program intelligence by building hardware with significantly more computing power than our brains have. After all, our neurons are no better or more numerous than those of dolphins, just differently connected, suggesting that software can sometimes be more important than hardware.

space has. If the Mathematical Universe Hypothesis is correct, then our Universe is a mathematical structure, and from its description, an infinitely intelligent mathematician should be able to derive all these physics theories.

If the hub AI has callously arranged for this white dwarf to be extremely close to its Chandrasekhar limit, the guard AI could be effective even if it were extremely dumb (indeed, largely because it was so dumb): it could be programmed to simply verify that the subjugated civilization had delivered its monthly quota of cosmic bitcoins, mathematical proofs or whatever other taxes were stipulated, and if not, toss enough mass onto the white dwarf to ignite the supernova and blow the entire region to smithereens.

Remarkably, physicists have since discovered that all laws of classical physics can be mathematically reformulated in an analogous way: out of all ways that nature could choose to do something, it prefers the optimal way, which typically boils down to minimizing or maximizing some quantity.

I agree with John Holt that the true test of intelligence is not what you know or can regurgitate from memory on an exam. It’s not what you know how to do, but “what you do when you don’t know what to do.” Harold Gardner has convincingly shown that we have eight or nine different kinds of intelligence. Unfortunately we only measure literacy and mathematical intelligence for our IQ.

it is clear that in the study of beings this aim can be fulfilled by us perfectly only through successive examinations of them by one man after another,41 the later ones seeking the help of the earlier in that task, on the model of what has happened in the mathematical sciences. For if we suppose that the art of geometry did not exist in this age of ours, and likewise the art of astronomy, and a single person wanted to ascertain by himself the sizes of the 15 heavenly bodies, their shapes, and their distances from each other, that would not be possible for him—e.g. to know the proportion of the sun to the earth or other facts about the sizes of the stars—even though he were the most intelligent of men by nature, unless by a revelation or something resembling revelation.42 Indeed if he were told that the sun is about 150 or 160 times43 as great as the earth, he would think this statement madness on the part of the speaker, although this is a fact which has been demonstrated in 20 astronomy so surely that no one who has mastered that science doubts it.

computer programs to forecast listeners’ habits. A company named Polyphonic HMI—a collection of artificial intelligence experts and statisticians based in Spain—had created a program called Hit Song Science that analyzed the mathematical characteristics of a tune and predicted its popularity. By comparing the tempo, pitch, melody, chord progression, and other factors of a particular song against the thousands of hits stored in Polyphonic HMI’s database, Hit Song Science could deliver a score that forecasted if a tune was likely to succeed.7.14

we find hints of how he rose from modest intelligence to genius, when he talks about his compulsion to tear down important papers and mathematical concepts until he could understand the concepts from the bottom up.

In trying to explain life we have reduced it to a series of chemical reactions, whether it be the burning of glucose in mitochondria to create energy, or the folding of proteins to make bile, or pollen, or blood. Zoom out to where we perceive things, the titanic mathematics of it all is silent. We have twisted our thoughts and feelings into all sorts of psychological origami about whether these things are a result of evolution, intelligent design, or creation ex nihilo, and for all we know, our little planet is the only place that holds all of this wonder in a void that is too staggeringly huge to conceive.