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.

...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...

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.

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.

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.

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

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’.

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.

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.

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.

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)

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.

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.

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...

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.

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.

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

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

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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...

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

[...] 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.

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.

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.

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.

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 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.

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.

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.

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.

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 …

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.

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.

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.

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!

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

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.

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.

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.

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.