Logical Mathematical Intelligence Quotes

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.

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

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.

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.

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.

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

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.

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.

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.

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.

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

THIRD EMENDED VERSION , SOME OMISSIONS HAVING BEEN ADDED TO MY LAST '' PUBLICATION '' TO KEEP THE LOGIC MORE LUCID SORRY FOR SETTING EVERYTHING DOWN SO QUICKLY - ''SCALE THE HUMAN MOUNTAIN OF SUMLESS LIES UNTIL YOU LABORIOUSLY REACH THE SUMMIT THEN CAUSE IT TO CRUMBLE BY YOUR EQUALLY SUMLESS BURDEN OF VERITY THAT NO HUMAN MAY FAVOUR YOU WITH A GLANCE ANY MORE AND THOSE WHO DO ARE NO LONGER HUMAN HAVING DIVESTED THEMSELVES OF THEIR HUMANITY AS YOU DID BY VIRTUE OF THE FACT OF WHAT MAN HAS DONE TO HIMSELF BESIDES , YOU ARE ABLE TO ASCERTAIN HOW MANY '' FRIENDS '' YOU HAVE WHICH IS THE EMPTY SET CONTAINING ONE ELEMENT ONLY : VERITY ! , TO WHICH YOU PERTAIN AS WELL IT IS WHY IT IS THE HARDEST THING TO FIND THE PATH LEADING TO YOURSELF AND IT IS BY THE EMPTY SET THAT ALL OF MATHEMATICS HAS BEEN MADE AN EGREGIOUS LIE TOO IT IS MORE FACILE TO KILL SOMEONE OR , IF YOU ARE UNABLE TO , YOURSELF THAN IT IS TO LIVE ! DO YOU SEE THE POPLAR AND THE ROBIN THAT IS PERCHED ON IT ? ASK THEM ! THEY KNOW HOW TO LIVE YOU DON'T BECAUSE YOU ARE HUMAN AND INTELLIGENT : MAN IS ENDUED WITH HIS SPIRIT OF INVENTION WHICH HAS REDUCED LIFE TO ABSURDITY AS ALL THOSE THEORIES AND TEACHINGS SPRINGING FROM IT HAVE NEVER BENEFITED LIFE , ON THE CONTRARY , DESTROYED IT ! AN APPRECIATION OF THE MAJESTY OF VERITY ALSO ENTAILS THE INEVITABLE CATASTROPHE OF '' BEING '' AND HENCE THE INFELICITY OF YOURSELF WHICH HAS TO BE ASCRIBED TO THOSE PROFOUND TEACHINGS OF MAN AND THE IMPRECATIONS WHICH THEY HEAPED UPON LIFE AND BEHIND WHICH EVERYONE STRIVES TO CONCEAL HIMSELF AS SOMETHING SUBLIME , BROTHERLY , CUNNING , INGENIOUS CONVINCED OF THE '' SUCCESS '' OF SUCH BEING ! INGENUITY AND SUCCESS , DO THOSE TWO WORDS DIFFER ? , AS MAN IS DETREMINED BY THOSE CRITERIA AND HENCE LIFE !... WHAT ALSO COMES TO MIND HERE IS THIS - THERE IS SOMETHING VASTLY ABOMINABLE ABOUT SOCIETY : ITS MEMBERS ARE EVER SO FOND OF ALL THOSE MOVIE STARS AND ALL THOSE OTHER LUMINARIES AND WHAT IS LUMINOUS ABOUT THEM I DO NOT KNOW ! YET THEY ARE IN THE HABIT OF TREATING THOSE VERY SIGNIFICANT PEOPLE DIFFERENTLY FROM ORDINARY PEOPLE SUCH AS A HOUSEMAID OR A GROCER OR A SALESMAN AND SO FORTH , THEREBY CREATING SOMETHING UTTERLY CORRUPT : A FALSE IDEALISM ! THEY NEED THOSE LUMINARIES AS THEY LACK ANY IDEALISM THEMSELVES IN THEIR EVERYDAY REALITY WHICH HAS DEPRAVED THEM OF IT , OVERLOOKING HOWEVER , HOW TRULY ORDINARY IN TRUTH ALL THOSE STARS ARE ! AND ALLOWING THEIR LACK OF IDEALISM TO BE SUPERSEDED BY OTHER PEOPLE'S NONPRESENT IDEALISM ON ACCOUNT OF THEIR PROMINENCE MAKES EVERYTHING LOOK EVEN DARKER IN LIFE , AS THOUGH LIFE CONSISTED IN FAME ! IS THIS WHY IT IS SO DARK IN THE HUMAN WORLD ? AM I THE ONLY PERSON TO APPREHEND DARKNESS IN THEIR LIGHTNESS ? OR WHY IS SO DARK IN THIS WORLD ? SOMETHING LIKE THAT NEEDS TO BE SHRUGGED OFF AS SOMETHING INEXPLICABLY RATIONAL , WHENCE I HAVE ALWAYS THOUGHT MYSELF IRRATIONAL IN NOT GROVELLING BEFORE THOSE WHO ARE EVEN MORE ORDINARY THAN ALL THE OTHER ORDINARY NON-FAMOUS PEOPLE ARE ! IT IS IN PARTICULAR THOSE ALL-IMPORTANT DIGNITARIES WHO TASTE OF METHYLATED SPIRITS IN A MOST ACRID AND NAUSEATING FASHION ! SO MUCH FOR CLEANLINESS !... VENERABLE ANCIENT SHADES HOVERING OVER THIS LAKE THAT IS NO MORE AND OF WHICH I AM PART THE WORLD AROUND ME FADES I DISPEL ALL THOSE BLANK AND GRAINED IDEAS MAKING UP HUMAN EXISTENCE I AM NO MORE I DREAM AND HOPEFULLY I WILL NEVER TURN BACK SO AS TO SEE THAT BLANK AND GRAINED HUMAN EXISTENCE AGAIN WHICH CAUSES LIFE TO BLUR SO MUCH THAT I AM NO LONGER IN A POSITION TO SUFFER FOR THIS MUCH GUILT , WHAT IS LIFE ? AMEN !...

[S]uitably complicated recursive systems might be strong enough to break out of any predetermined patterns. And isn't this one of the defining properties of intelligence? Instead of just considering programs composed of procedures which can recursively

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.

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)

Thus arises the second tendency, which consists in regarding logical and mathematical relations as irreducible, and in making an analysis of the higher intellectual functions depend on an analysis of them. But it is questionable whether logic, regarded as something eluding the attempts of experimental psychology to explain it, can in its turn legitimately explain anything in psychological experience.

The IQ test was supposed to measure your capacity to think and learn and therefore to predict your success in school. However, contemporary psychologists have debunked this whole idea of a single capacity called intelligence. You have not one but at least seven intelligences, according to Harvard psychologist Howard Gardner. • Linguistic intelligence • Logical-mathematical intelligence • Spatial intelligence • Musical intelligence • Bodily-kinesthetic intelligence • Intrapersonal intelligence (knowing yourself) • Interpersonal intelligence (knowing other people)

Theoretical Science employs assumptions, abstractions and mathematical models to explain possibilities or outcomes.

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

Perhaps we are being a bit presumptuous in calling our species “intelligent.” After all, this species has waged numerous inane wars where millions of their own were slaughtered. As a whole, this species spends trillions of hours a year watching insipid television shows. And “intelligent” is not the right name for a species that invented spam e-mails and encourages narcissistic pastimes like Facebook. Nevertheless, over the millennia, this species produced many shining lights that make us worthy of the lofty title: Blaise Pascal, Isaac Newton, David Hume, Marie Curie, Albert Einstein, Arthur Stanley Eddington, Emmy Noether, Andrew Lloyd Webber, Meryl Streep, and, of course, tiramisu.

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.

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.

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.

The Marland definition of giftedness (page 499) broadened the view of giftedness from one based strictly on IQ to one encompassing six areas of outstanding or potentially outstanding performance. The passage of Public Law 94–142, the Education for All Handicapped Children Act, in 1975 led to an increased interest in and awareness of individual differences and exceptionalities. PL 94–142, however, was a missed opportunity for gifted children, as there was no national mandate to serve them. Mandates to provide services for children and youth who are gifted and talented are the result of state rather than federal legislation. The 1980s and 1990s: The Field Matures and Provides Focus for School Reform Building on Guilford’s multifaceted view of intelligence, Howard Gardner and Robert Sternberg advanced their own theories of multiple intelligences in the 1980s. Gardner (1983) originally identified seven intelligences—linguistic, logical-mathematical, spatial, bodily-kinesthetic, musical, interpersonal, and intrapersonal (see Table 15.2). Describing these intelligences as relatively independent of one another, he later added naturalistic as an eighth intelligence (Gardner, 1993). Sternberg (1985) presented a triarchic view of “successful intelligence,” encompassing practical, creative, and executive intelligences. Using these models, the field of gifted education has expanded its understanding of intelligence while not abandoning IQ as a criterion for identifying intellectually gifted children. A Nation at Risk (National Commission on Excellence in Education, 1983) described the state of education in U.S. schools as abysmal. The report made a connection between the education of children who are gifted and our country’s future. This commission found that 50 percent of the school-age gifted population was not performing to full potential and that mathematics and science were in deplorable conditions in the schools. The message in this report percolated across the country and was responsible for a renewed interest in gifted education as well as in massive education reform that occurred nationally and state by state.

Each person has a unique mixture of intelligences, or ways of understanding the world—linguistic, logical, mathematical, spatial, musical, physical

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.