Mathematical Symbols Quotes

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

No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.

Too large a proportion of recent "mathematical" economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.

It appeared that way, Lawrence, but this raised the question of was mathematics really true or was it just a game played with symbols? In other words—are we discovering Truth, or just wanking?

A circle in a straight line is the mathematical symbol of miracle.

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.

Our mathematics is the symbolic counterpart of the universe we perceive, and its power has been continuously enhanced by human exploration.

The goal of mathematics is the symbolic comprehension of the infinite with human, that is finite, means.

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

If you’d like to see how to apply these ideas directly to memorizing formulas, try out the SkillsToolbox .com website for a list of easy-to-remember visuals for mathematical symbols.7

that man is a reality, mankind an abstraction; that men cannot be treated as units in operations of political arithmetic because they behave like the symbols for zero and the infinite, which dislocate all mathematical operations; that the end justifies the means only within very narrow limits; that ethics is not a function of social utility, and charity not a petty bourgeois sentiment but the gravitational force which keeps civilization in its orbit.

Coincidence, when raised to a symbol, occurs with mathematical precision at the most crucial moment, even for the squarest of minds. A moment the rest of us call higher will, Fate’s gesture, something like that.

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

Even the most carefully defined philosophical or mathematical concept, which we are sure does not contain more than we have put into it, is nevertheless more than we assume. It is a psychic event and as such partly unknowable. The very numbers you use in counting are more than you take them to be. They are at the same time mythological elements (for the Pythagoreans, they were even divine); but you are certainly unaware of this when you use numbers for a practical purpose.

But that would mean it was originally a sideways number eight. That makes no sense at all. Unless..." She paused as understanding dawned. "You think it was the symbol for infinity?" "Yes, but not the usual one. A special variant. Do you see how one line doesn't fully connect in the middle? That's Euler's infinity symbol. Absolutus infinitus." "How is it different from the usual one?" "Back in the eighteenth century, there were certain mathematical calculations no one could perform because they involved series of infinite numbers. The problem with infinity, of course, is that you can't come up with a final answer when the numbers keep increasing forever. But a mathematician named Leonhard Euler found a way to treat infinity as if it were a finite number- and that allowed him to do things in mathematical analysis that had never been done before." Tom inclined his head toward the date stone. "My guess is whoever chiseled that symbol was a mathematician or scientist." "If it were my date stone," Cassandra said dryly, "I'd prefer the entwined hearts. At least I would understand what it means." "No, this is much better than hearts," Tom exclaimed, his expression more earnest than any she'd seen from him before. "Linking their names with Euler's infinity symbol means..." He paused, considering how best to explain it. "The two of them formed a complete unit... a togetherness... that contained infinity. Their marriage had a beginning and end, but every day of it was filled with forever. It's a beautiful concept." He paused before adding awkwardly, "Mathematically speaking.

Mass psychology is not simply a summation of individual psychologies; that is a prime theorem of social psychodynamics—not just my opinion; no exception has ever been found to this theorem. It is the social mass-action rule, the mob-hysteria law, known and used by military, political, and religious leaders, by advertising men and prophets and propagandists, by rabble rousers and actors and gang leaders, for generations before it was formulated in mathematical symbols. It works. It is working now.

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated ... The importance of this invention is more readily appreciated when one considers that it was beyod the two greatest men of antiquity, Archimedes and Apollonius.

Away and away the aeroplane shot, till it was nothing but a bright spark; an aspiration; a concentration; a symbol (so it seemed to Mr. Bentley, vigorously rolling his strip of turf at Greenwich) of man's soul; of his determination, thought Mr. Bentley, sweeping round the cedar tree, to get outside his body, beyond his house, by means of thought, Einstein, speculation, mathematics, the Mendelian theory––away the aeroplane shot.

Nature itself rests on an internal foundation of archetypal principles symbolized by numbers, shapes, and their arithmetic and geometric relationships.

Do mathematics have a relation to reality or are they only a mathematical symbol?

What is the largest possible number you can write using only 2 numbers - just 2 numbers, no other mathematical symbols?

In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]

The truth, he thought, has never been of any real value to any human being- it is a symbol for mathematicians and philosophers to pursue. I human relations kindness and lies are worth a thousand truths.

hurry” was not a concept that could be symbolized in the Martian language and therefore must be presumed to be unthinkable. Speed, velocity, simultaneity, acceleration, and other mathematical abstractions having to do with the pattern of eternity were part of Martian mathematics, but not of Martian emotion.

The world of physics is essentially the real world construed by mathematical abstractions, and the world of sense is the real world construed by the abstractions which the sense-organs immediately furnish. To suppose that the "material mode" is a primitive and groping attempt at physical conception is a fatal error in epistemology.

Some people gain their understanding of the world by symbols and mathematics. Others gain their understanding by pure geometry and space. There are some others that find an acceleration in the muscular effort that is brought to them in understanding, in feeling the force of objects moving through the world. What they want are words of power that stir their souls like the memory of childhood. For the sake of persons of these different types, whether they want the paleness and tenuity of mathematical symbolism, or they want the robust aspects of this muscular engagement, we should present all of these ways. It’s the combination of them that give us our best access to truth

These three tools of light, energy, and mass by which the Divine geometer constructs the cosmos and by which the symbolic geometer approximates archetypal patterns are also mirrored in us. What scientists call “light, energy, and mass” are the traditional “spirit, soul, and body” described by Plutarch as nous (divine intellect), psyche (soul), and soma (body).

Congratulations,” he says. Uh? “On your Ph.D.” “W-what?” “A noteworthy accomplishment,” he continues, serious, calm, “given that less than twenty-four hours ago you weren’t even working on one.” I exhale deeply. “Listen, it’s not what you—” “Will you be leaving your post at the library, or are you planning on a dual career? I’d be worried for your schedule, but I hear that theoretical physics often consists of staring into the void and jotting down the occasional mathematical symbol—

We treat ourselves both as objects of language and as speakers of language, both as objects of the symbolism and as symbols in it. And all the difficult paradoxes which go right back to Greek times and reappear in modern mathematics depend essentially on this.

When the lessons of symbolic or philosophical mathematics seen in nature, which were designed into religious architecture or art, are applied functionally (not just intellectually) to facilitate the growth and transformation of consciousness, then mathematics may rightly be called “sacred.

Central to all these interlinked themes was that curious irrational, phi, the Golden Section. Schwaller de Lubicz believed that if ancient Egypt possessed knowledge of ultimate causes, that knowledge would be written into their temples not in explicit texts but in harmony, proportion, myth and symbol.

The semanticists maintained that everything depends on how you interpret the words “potato,” “is” and “moving.” Since the key here is the operational copula “is,” one must examine “is” rigorously. Whereupon they set to work on an Encyclopedia of Cosmic Semasiology, devoting the first four volumes to a discussion of the operational referents of “is.” The neopositivists maintained that it is not clusters of potatoes one directly perceives, but clusters of sensory impressions. Then, employing symbolic logic, they created terms for “cluster of impressions” and “cluster of potatoes,” devised a special calculus of propositions all in algebraic signs and after using up several seas of ink reached the mathematically precise and absolutely undeniable conclusion that 0=0.

The lesson provided by Morse’s code is that it matters profoundly how one translates a message into electrical signals. This matter is at the very heart of communication theory.

In fact, network theory might have been developed to explain the behavior of mechanical systems,

Theories become more mathematical or abstract when they deal with an idealized class of phenomena or with only certain aspects of phenomena.

communication theory grew out of the study of electrical communication,

Only women could bleed without injury or death; only they rose from the gore each month like a phoenix; only their bodies were in tune with the ululations of the universe and the timing of the tides. Without this innate lunar cycle, how could men have a sense of time, tides, space, seasons, movement of the universe, or the ability to measure anything at all? How could men mistress the skills of measurement necessary for mathematics, engineering, architecture, surveying—and so many other professions? In Christian churches, how could males, lacking monthly evidence of Her death and resurrection, serve the Daughter of the Goddess? In Judaism, how could they honor the Matriarch without the symbol of Her sacrifices recorded in the Old Ovariment? Thus insensible to the movements of the planets and the turning of the universe, how could men become astronomers, naturalists, scientists—or much of anything at all?

Thus, an increase in entropy means a decrease in our ability to change thermal energy, the energy of heat, into mechanical energy. An increase of entropy means a decrease of available energy.

A branch of electrical theory called network theory deals with the electrical properties of electrical circuits, or networks, made by interconnecting three sorts of idealized electrical structures:

But one cannot rely solely on games and art to improve the quality of life. To achieve control over what happens in the mind, one can draw upon an almost infinite range of opportunities for enjoyment—for instance, through the use of physical and sensory skills ranging from athletics to music to Yoga (chapter 5), or through the development of symbolic skills such as poetry, philosophy, or mathematics

They were aware that the symbols of mythology and the the symbols of mathematical science were different aspects of the same, indivisible Reality. They did not live in a 'divided house of faith and reason'; the two were interlocking, like ground-plan and elevation on an architect's drawing. It is a state of mind very difficult for twentieth-century man to imagine- or even to believe that it could ever have existed. It may help to remember though, that some of the greatest pre-Socratic sages formulated their philosophies in verse; the unitary source of inspiration of prophet, poet, and philosopher was still taken for granted.

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.

Think of mathematical symbols as mere labels without intrinsic meaning. It doesn’t matter whether you write, “Two plus two equals four,” “2 + 2 = 4,” or “Dos más dos es igual a cuatro.” The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don’t invent mathematical structures—we discover them, and invent only the notation for describing them.

But Miss Ferguson preferred science over penmanship. Philosophy over etiquette. And, dear heavens preserve them all, mathematics over everything. Not simply numbering that could see a wife through her household accounts. Algebra. Geometry. Indecipherable equations made up of unrecognizable symbols that meant nothing to anyone but the chit herself. It was enough to give Miss Chase hives. The girl wasn’t even saved by having any proper feminine skills. She could not tat or sing or draw. Her needlework was execrable, and her Italian worse. In fact, her only skills were completely unacceptable, as no one wanted a wife who could speak German, discuss physics, or bring down more pheasant than her husband.

the Godelian strange loop that arises in formal systems in mathematics (i.e., collections of rules for churning out an endless series of mathematical truths solely by mechanical symbol-shunting without any regard to meanings or ideas hidden in the shapes being manipulated) is a loop that allows such a system to "perceive itself", to talk about itself, to become "self-aware", and in a sense it would not be going too far to say that by virtue of having such a loop, a formal system acquires a self.

In other words, our conscious representations are sometimes ordered (or arranged in a pattern) before they have become conscious to us. The 18th-century German mathematician Karl Friedrich Gauss gives an example of an experience of such an unconscious order of ideas: He says that he found a certain rule in the theory of numbers "not by painstaking research, but by the Grace of God, so to speak. The riddle solved itself as lightning strikes, and I myself could not tell or show the connection between what I knew before, what I last used to experiment with, and what produced the final success." The French scientist Henri Poincare is even more explicit about this phenomenon; he describes how during a sleepless night he actually watched his mathematical representations colliding in him until some of them "found a more stable connection. One feels as if one could watch one's own unconscious at work, the unconscious activity partially becoming manifest to consciousness without losing its own character. At such moments one has an intuition of the difference between the mechanisms of the two egos.

But what is the use of the humanities as such? Admittedly they are not practical, and admittedly they concern themselves with the past. Why, it may be asked, should we engage in impractical investigations, and why should we be interested in the past? The answer to the first question is: because we are interested in reality. Both the humanities and the natural sciences, as well as mathematics and philosophy, have the impractical outlook of what the ancients called vita contemplativa as opposed to vita activa. But is the contemplative life less real or, to be more precise, is its contribution to what we call reality less important, than that of the active life? The man who takes a paper dollar in exchange for twenty-five apples commits an act of faith, and subjects himself to a theoretical doctrine, as did the mediaeval man who paid for indulgence. The man who is run over by an automobile is run over by mathematics, physics and chemistry. For he who leads the contemplative life cannot help influencing the active, just as he cannot prevent the active life from influencing his thought. Philosophical and psychological theories, historical doctrines and all sorts of speculations and discoveries, have changed, and keep changing, the lives of countless millions. Even he who merely transmits knowledge or learning participates, in his modest way, in the process of shaping reality - of which fact the enemies of humanism are perhaps more keenly aware than its friends. It is impossible to conceive of our world in terms of action alone. Only in God is there a "Coincidence of Act and Thought" as the scholastics put it. Our reality can only be understood as an interpenetration of these two.

Music can be appreciated from several points of view: the listener, the performer, the composer. In mathematics there is nothing analogous to the listener; and even if there were, it would be the composer, rather than the performer, that would interest him. It is the creation of new mathematics, rather than its mundane practice, that is interesting. Mathematics is not about symbols and calculations. These are just tools of the tradequavers and crotchets and five-finger exercises. Mathematics is about ideas. In particular it is about the way that different ideas relate to each other. If certain information is known, what else must necessarily follow? The aim of mathematics is to understand such questions by stripping away the inessentials and penetrating to the core of the problem. It is not just a question of getting the right answer; more a matter of understanding why an answer is possible at all, and why it takes the form that it does. Good mathematics has an air of economy and an element of surprise. But, above all, it has significance.

The point being that everything emerges from the same collection of ingredients governed by the same physical principles. And those principles, as attested to by a few hundred years of observation, experimentation, and theorizing, will likely be expressed by a handful of symbols arranged in a small collection of mathematical equations. That is an elegant universe.

No one is without Christianity, if we agree on what we mean by the word. It is every individual's individual code of behavior, by means of which he makes himself a better human being than his nature wants to be, if he followed his nature only. Whatever its symbol—cross or crescent or whatever—that symbol is man's reminder of his duty inside the human race. Its various allegories are the charts against which he measures himself and learns to know what he is. It cannot teach man to be good as the textbook teaches him mathematics. It shows him how to discover himself, evolve for himself a moral code and standard within his capacities and aspirations, by giving him a matchless example of suffering and sacrifice and the promise of hope.

[We] cannot and should not expect to rediscover the full body of ancient wisdom by studying dusty monuments and myths full of idioms and subtle references understood only by those who lived at the time. The perennial wisdom requires each individual and age to discover it anew in external mathematics, expressing it in ways and symbols suitable for those times and cultures.

Even in the realm of pure mathematics, the mathematician may use any set of symbols he desires within any given region of space-time; he may even go so far as to maintain that any one set of symbols fits the scheme as well as any other, but to erect this method into a philosophy and confuse independence of any one special meaning with independence of all meaning is unjustified and unwarranted.

What we can imagine as plausible is a narrow band in the middle of a much broader spectrum of what is actually possible. [O]ur eyes are built to cope with a narrow band of electromagnetic frequencies. [W]e can't see the rays outside the narrow light band, but we can do calculations about them, and we can build instruments to detect them. In the same way, we know that the scales of size and time extend in both directions far outside the realm of what we can visualize. Our minds can't cope with the large distances that astronomy deals in or with the small distances that atomic physics deals in, but we can represent those distances in mathematical symbols. Our minds can't imagine a time span as short as a picosecond, but we can do calculations about picoseconds, and we can build computers that can complete calculations within picoseconds. Our minds can't imagine a timespan as long as a million years, let alone the thousands of millions of years that geologists routinely compute. Just as our eyes can see only that narrow band of electromagnetic frequencies that natural selection equipped our ancestors to see, so our brains are built to cope with narrow bands of sizes and times. Presumably there was no need for our ancestors to cope with sizes and times outside the narrow range of everyday practicality, so our brains never evolved the capacity to imagine them. It is probably significant that our own body size of a few feet is roughly in the middle of the range of sizes we can imagine. And our own lifetime of a few decades is roughly in the middle of the range of times we can imagine.

That is, for a mathematical Platonist, what the C.H. proofs really show is that set theory needs to find a better set of core axioms than classical ZFS, or at least it will need to add some further postulates that are-like the Axiom of Choice-both "self-evident" and Consistent with classical axioms. If you're interested, Godel's own personal view was that the Continuum Hypothesis is false, that there are actually a whole (Infinity Symbol) of Zeno-type (Infinity Symbol)s nested between (Aleph0) and c, and that sooner or later a principle would be found that proved this. As of now no such principle's ever been found. Godel and Cantor both died in confinement, bequeathing a world with no finite circumference. One that spins, now, in a new kind of all-formal Void. Mathematics continues to get out of bed.

On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,' ... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.

Alan Turing appears to be becoming a symbol of the shift towards computing, not least because of his attitude of open-minded defiance of convention and conventional thinking. Not only did he conceptualise the modern computer – imagining a simple machine that could use different programmes – but he put his thinking into practice in the great code breaking struggle with the Nazis in World War II, and followed it up with pioneering early work in the mathematics of biology and chaos.

“Mathematics isn’t just science, it is poetry—our efforts to crystallize the unglimpsed connections between things. Poetry that bridges and magnifies the mysteries of the galaxy. But the signs and symbols and equations sentients employ to express these connections are not discoveries but the teasing out of secrets that have always existed. All our theories belong to nature, not to us. As in music, every combination of notes and chords, every melody has already been played and sung, somewhere, by someone—”

But in this story, as in so many others, what we really discern is the deceptive, ambiguous and giddy riddle of violence, passion, poetry and symbolism that lies at the heart of Greek myth and refuses to be solved. An algebra too unstable properly to be computed, it is human-shaped and god-shaped, not pure and mathematical. It is fun trying to interpret such symbols and narrative turns, but the substitutions don't quite work and the answers yielded are usually no clearer than those of an equivocating oracle.

The actual issue is that someone a hundred thousand years ago sat up in his robes and said Holy Shit. Sort of. He didn’t have a language yet. But what he had just understood is that one thing can be another thing. Not look like it or act upon it. Be it. Stand for it. Pebbles can be goats. Sounds can be things. The name for water is water. What seems inconsequential to us by reason of usage is in fact the founding notion of civilization. Language, art, mathematics, everything. Ultimately the world itself and all in it.

The essence of Hilbert's program was to find a decision process that would operate on symbols in a purely mechanical fashion, without requiring any understanding of their meaning. Since mathematics was reduced to a collection of marks on paper, the decision process should concern itself only with the marks and not with the fallible human intuitions out of which the marks were reduced. In spite of the prolonged efforts of Hilbert and his disciples, the Entscheidungsproblem was never solved. Success was achieved only in highly restricted domains of mathematics, excluding all the deeper and more interesting concepts. Hilbert never gave up hope, but as the years went by his program became an exercise in formal logic having little connection with real mathematics. Finally, when Hilbert was seventy years old, Kurt Godel proved by a brilliant analysis that the Entscheindungsproblem as Hilbert formulated it cannot be solved. Godel proved that in any formulation of mathematics, including the rules of ordinary arithmetic, a formal process for separating statements into true and false cannot exist. He proved the stronger result which is now known as Godel's theorem, that in any formalization of mathematics including the rules of ordinary arithmetic there are meaningful arithmetical statements that cannot be proved true or false. Godel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.

What governs what we choose to notice? The first (which we shall have to qualify later) is whatever seems advantageous or disadvantageous for our survival, our social status, and the security of our egos. The second, again working simultaneously with the first, is the pattern and the logic of all the notation symbols which we have learned from others, from our society and our culture. It is hard indeed to notice anything for which the languages available to us (whether verbal, mathematical, or musical) have no description. This is why we borrow words from foreign languages.

The authors’ prior experience in clinical research4 had amply convinced us of the possibility of long-term performance enhancement using psychedelic agents in a safe, supportive setting. Though not deliberately sought, there were numerous spontaneous incidents of what appeared to be temporarily enhanced performance during the drug experience itself. These observations led us to postulate the following: Any human function can be performed more effectively. We do not function at our full capacity. Psychedelics appear to temporarily inhibit censors that ordinarily limit what is available to conscious awareness. Participants may, for example, discover a latent ability to form colorful and complex imagery, to recall forgotten experiences of early childhood, or to generate meaningful symbolic presentations. By leading participants to expect enhancement of other types of performance—creative problem solving, learning manual or verbal skills, manipulating logical or mathematical symbols, acquiring sensory or extrasensory perception, memory, and recall—and by providing favorable preparatory and environmental conditions, it may be possible to improve any desired aspect of mental functioning.

The amount of information conveyed by the message increases as the amount of uncertainty as to what message actually will be produced becomes greater. A message which is one out of ten possible messages conveys a smaller amount of information than a message which is one out of a million possible messages. The entropy of communication theory is a measure of this uncertainty and the uncertainty, or entropy, is taken as the measure of the amount of information conveyed by a message from a source. The more we know about what message the source will produce, the less uncertainty, the less the entropy, and the less the information.

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

It was all so very businesslike that one watched it fascinated. It was porkmaking by machinery, porkmaking by applied mathematics. And yet somehow the most matter-of-fact person could not help thinking of the hogs; they were so innocent, they came so very trustingly; and they were so very human in their protests—and so perfectly within their rights! They had done nothing to deserve it; and it was adding insult to injury, as the thing was done here, swinging them up in this cold-blooded, impersonal way, without a pretense of apology, without the homage of a tear. Now and then a visitor wept, to be sure; but this slaughtering machine ran on, visitors or no visitors. It was like some horrible crime committed in a dungeon, all unseen and unheeded, buried out of sight and of memory. One could not stand and watch very long without becoming philosophical, without beginning to deal in symbols and similes, and to hear the hog squeal of the universe. Was it permitted to believe that there was nowhere upon the earth, or above the earth, a heaven for hogs, where they were requited for all this suffering? Each one of these hogs was a separate creature. Some were white hogs, some were black; some were brown, some were spotted; some were old, some young; some were long and lean, some were monstrous. And each of them had an individuality of his own, a will of his own, a hope and a heart’s desire; each was full of self-confidence, of self-importance, and a sense of dignity. And trusting and strong in faith he had gone about his business, the while a black shadow hung over him and a horrid Fate waited in his pathway. Now suddenly it had swooped upon him, and had seized him by the leg. Relentless, remorseless, it was; all his protests, his screams, were nothing to it—it did its cruel will with him, as if his wishes, his feelings, had simply no existence at all; it cut his throat and watched him gasp out his life. And now was one to believe that there was nowhere a god of hogs, to whom this hog personality was precious, to whom these hog squeals and agonies had a meaning? Who would take this hog into his arms and comfort him, reward him for his work well done, and show him the meaning of his sacrifice?

Wherever the relevance of speech is at stake, matters become political by definition, for speech is what makes man a political being. If we would follow the advice, so frequently urged upon us, to adjust our cultural attitudes to the present status of scientific achievement, we would in all earnest adopt a way of life in which speech is no longer meaningful. For the sciences today have been forced to adopt a “language” of mathematical symbols which, though it was originally meant only as an abbreviation for spoken statements, now contains statements that in no way can be translated back into speech. The reason why it may be wise to distrust the political judgment of scientists qua scientists is not primarily their lack of “character”—that they did not refuse to develop atomic weapons—or their naïveté—that they did not understand that once these weapons were developed they would be the last to be consulted about their use—but precisely the fact that they move in a world where speech has lost its power. And whatever men do or know or experience can make sense only to the extent that it can be spoken about. There may be truths beyond speech, and they may be of great relevance to man in the singular, that is, to man in so far as he is not a political being, whatever else he may be. Men in the plural, that is, men in so far as they live and move and act in this world, can experience meaningfulness only because they can talk with and make sense to each other and to themselves. Closer

Base two especially impressed the seventeenth-century religious philosopher and mathematician Gottfried Wilhelm Leibniz. He observed that in this base all numbers were written in terms of the symbols 0 and 1 only. Thus eleven, which equals 1 · 23 + 0 · 22 + 1 · 2 + 1, would be written 1011 in base two. Leibniz saw in this binary arithmetic the image and proof of creation. Unity was God and zero was the void. God drew all objects from the void just as the unity applied to the zero creates all numbers. This conception, over which the reader would do well not to ponder too long, delighted Leibniz so much that he sent it to Grimaldi, the Jesuit president of the Chinese tribunal for mathematics, to be used as an argument for the conversion of the Chinese emperor to Christianity.

I think of myth and magic as the hieroglyphics of the human psyche. They are a special language that circumvents conscious thought and goes straight to the subconscious. Non-fiction uses the medium of information. It tells us what we need to know. Science fiction primarily uses the medium of physics and mathematics. It tells us how things work, or could work. Horror taps into the darker imagery of the psychology, telling us what we should fear. Fantasy, magic and myth, however, tap into the spiritual potential of the human life. Their medium is symbolism, truth made manifest in word pictures, and they tell us what things mean on a deep, internal level. I have always been a meaning-maker. I have always been someone who strives to make sense of everything and perhaps that is where my life as a storyteller first began. Life doesn't always make sense, but story must. And so I write stories, and the world comes right again.

That words are not things. (Identification of words with things, however, is widespread, and leads to untold misunderstanding and confusion.) That words mean nothing in themselves; they are as much symbols as x or y. That meaning in words arises from context of situation. That abstract words and terms are especially liable to spurious identification. The higher the abstraction, the greater the danger. That things have meaning to us only as they have been experienced before. “Thingumbob again.” That no two events are exactly similar. That finding relations and orders between things gives more dependable meanings than trying to deal in absolute substances and properties. Few absolute properties have been authenticated in the world outside. That mathematics is a useful language to improve knowledge and communication. That the human brain is a remarkable instrument and probably a satisfactory agent for clear communication. That to improve communication new words are not needed, but a better use of the words we have. (Structural improvements in ordinary language, however, should be made.) That the scientific method and especially the operational approach are applicable to the study and improvement of communication. (No other approach has presented credentials meriting consideration.) That the formulation of concepts upon which sane men can agree, on a given date, is a prime goal of communication. (This method is already widespread in the physical sciences and is badly needed in social affairs.) That academic philosophy and formal logic have hampered rather than advanced knowledge, and should be abandoned. That simile, metaphor, poetry, are legitimate and useful methods of communication, provided speaker and hearer are conscious that they are being employed. That the test of valid meaning is: first, survival of the individual and the species; second, enjoyment of living during the period of survival.

Eternity, in the sense of the pools, manifests as an enigma within the mathematical fabric of existence. It represents a fractal realm in which the notion of endless duration deviates from conventional human experience. Far beyond the finite bounds of what we call ‘time,’ eternity morphs into a disorienting continuum of perpetual recurrence and unbounded expansion. The cyan merely acts as a catalyst to understanding. Within this eerie realm, space dissolves into a concept, and the usual arithmetic constraints fail to hold sway. The rooms become a ceaseless amalgamation of symbolic sequences and iterations, where infinite series relentlessly converge and diverge, oscillating in rhythm to the waves. The wave function collapses when th//Цијан цијан цијан цијан цијан цијан цијан цијан цијан HELP ME цијан цијан цијан цијан цијан цијан цијан цијан цијан цијан Цијан цијан цијан цијан цијан цијан цијан цијан цијан HELP ME цијан цијан цијан цијан цијан цијан цијан

[W]e can calculate our way into regions of miraculous improbability far greater than we can imagine as plausible. Let's look at this matter of what we think is plausible. What we can imagine as plausible is a narrow band in the middle of a much broader spectrum of what is actually possible. Sometimes it is narrower than what is actually there. There is a good analogy with light. Our eyes are built to cope with a narrow band of electromagnetic frequencies (the ones we call light), somewhere in the middle of the spectrum from long radio waves at one end to short X-rays at the other. We can't see the rays outside the narrow light band, but we can do calculations about them, and we can build instruments to detect them. In the same way, we know that the scales of size and time extend in both directions far outside the realm of what we can visualize. Our minds can't cope with the large distances that astronomy deals in or with the small distances that atomic physics deals in, but we can represent those distances in mathematical symbols. Our minds can't imagine a time span as short as a picosecond, but we can do calculations about picoseconds, and we can build computers that can complete calculations within picoseconds. Our minds can't imagine a timespan as long as a million years, let alone the thousands of millions of years that geologists routinely compute. Just as our eyes can see only that narrow band of electromagnetic frequencies that natural selection equipped our ancestors to see, so our brains are built to cope with narrow bands of sizes and times. Presumably there was no need for our ancestors to cope with sizes and times outside the narrow range of everyday practicality, so our brains never evolved the capacity to imagine them. It is probably significant that our own body size of a few feet is roughly in the middle of the range of sizes we can imagine. And our own lifetime of a few decades is roughly in the middle of the range of times we can imagine.

One winter she grew obsessed with a fashionable puzzle known as Solitaire, the Rubik’s Cube of its day. Thirty-two pegs were arranged on a board with thirty-three holes, and the rules were simple: Any peg may jump over another immediately adjacent, and the peg jumped over is removed, until no more jumps are possible. The object is to finish with only one peg remaining. “People may try thousands of times, and not succeed in this,” she wrote Babbage excitedly. I have done it by trying & observation & can now do it at any time, but I want to know if the problem admits of being put into a mathematical Formula, & solved in this manner.… There must be a definite principle, a compound I imagine of numerical & geometrical properties, on which the solution depends, & which can be put into symbolic language. A formal solution to a game—the very idea of such a thing was original. The desire to create a language of symbols, in which the solution could be encoded—this way of thinking was Babbage’s, as she well knew.

It occurred to me that we now as a culture, as a people have legitimately become the progeny of the Digital Age. Ostensibly, we subsist within a dehumanized frontier--a computational, compartmentalized, mathematized collectivist-grid. Metrics have prohibitively supplanted ethics. Alternately, the authentic aesthetic experience has been sacrificed and transposed by the new breed of evangelicals: the purveyors of the advertising industry. Thus the symbolic euphoria induced by the infomercial is celebrated as the new Delphic Oracle. Alas, we've transitioned from a carbon-based life form into an information-based, bio-mechanical, heuristically deprived and depleted entity best described as "a self-balancing 28-jointed adaptor-based biped, an electro-chemical reduction plant integral with segregated stowages of special energy extracts." Consequently, we exist under the tyranny of hyper-specialization, which dislodges and disposes our sense of logic, proportion and humanity from both our cognitive and synaptic ballet.

the Game of games had developed into a kind of universal language through which the players could express values and set these in relation to one another. Throughout its history the Game was closely allied with music, and usually proceeded according to musical or mathematical rules. One theme, two themes, or three themes were stated, elaborated, varied, and underwent a development quite similar to that of the theme in a Bach fugue or a concerto movement. A Game, for example, might start from a given astronomical configuration, or from the actual theme of a Bach fugue, or from a sentence out of Leibniz or the Upanishads, and from this theme, depending on the intentions and talents of the player, it could either further explore and elaborate the initial motif or else enrich its expressiveness by allusions to kindred concepts. Beginners learned how to establish parallels, by means of the Game’s symbols, between a piece of classical music and the formula for some law of nature. Experts and Masters of the Game freely wove the initial theme into unlimited combinations.

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.

As you know, there was a famous quarrel between Max Planck and Einstein, in which Einstein claimed that, on paper, the human mind was capable of inventing mathematical models of reality. In this he generalized his own experience because that is what he did. Einstein conceived his theories more or less completely on paper, and experimental developments in physics proved that his models explained phenomena very well. So Einstein says that the fact that a model constructed by the human mind in an introverted situation fits with outer facts is just a miracle and must be taken as such. Planck does not agree, but thinks that we conceive a model which we check by experiment, after which we revise our model, so that there is a kind of dialectic friction between experiment and model by which we slowly arrive at an explanatory fact compounded of the two. Plato-Aristotle in a new form! But both have forgotten something- the unconscious. We know something more than those two men, namely that when Einstein makes a new model of reality he is helped by his unconscious, without which he would not have arrived at his theories...But what role DOES the unconscious play?...either the unconscious knows about other realities, or what we call the unconscious is a part of the same thing as outer reality, for we do not know how the unconscious is linked with matter.

In 1906, the year after Einstein’s annus mirabilis, Kurt Gödel was born in the city of Brno (now in the Czech Republic). Kurt was both an inquisitive child—his parents and brother gave him the nickname der Herr Warum, “Mr. Why?”—and a nervous one. At the age of five, he seems to have suffered a mild anxiety neurosis. At eight, he had a terrifying bout of rheumatic fever, which left him with the lifelong conviction that his heart had been fatally damaged. Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians. In the philosophical world of 1920s Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.

Death Vision I think it’s a multiplication of sight, Like after a low hovering autumn rain When the invisible web of funnel weaves And sheetweb weavers all at once are seen Where they always were, spread and looping The grasses, every strand, waft and leaf- Crest elucidated with water-light and frost, completing the fullest aspect of field. Or maybe the grace of death is split-second Transformation of knowledge, an intricate, Turning realization, as when a single Sperm-embracing deep ovum transforms, In an instant, from stasis to replicating, Star-shifting shimmer, rolls, reaches, Alters its plane of intentions, becomes A hoofing, thumping host of purpose. I can imagine not merely The falling away of blank walls And blinds in that moment, not merely A shutter flung open for the first time Above a valley of interlocking forests And constellations but a sweeping, Penetrating circumference of vision Encompassing both knotweed bud And its seed simultaneously, seeing Blood bone and its ash as one, The repeated light and fall and flight Of hawk-owl and tundra vole As a union of origin and finality. A mathematics of flesh and space might Take hold if we ask for it in that last Moment, might appear as if it had always Existed within the eyes, translucent, Jewel-like in stained glass patterns Of globes and measures, equations, Made evident by a revelation of galaxies In the knees, spine, fingers, all The ceasings, all the deaths within deaths That compose the body becoming at once Their own symbolic perception and praise Of river salt, blooms and breaths, strings, Strains, sun-seas of gravels and gills; This one expression breaking, this same Expression healing.

Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof. Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did! Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.

In many fields—literature, music, architecture—the label ‘Modern’ stretches back to the early 20th century. Philosophy is odd in starting its Modern period almost 400 years earlier. This oddity is explained in large measure by a radical 16th century shift in our understanding of nature, a shift that also transformed our understanding of knowledge itself. On our Modern side of this line, thinkers as far back as Galileo Galilei (1564–1642) are engaged in research projects recognizably similar to our own. If we look back to the Pre-Modern era, we see something alien: this era features very different ways of thinking about how nature worked, and how it could be known. To sample the strange flavour of pre-Modern thinking, try the following passage from the Renaissance thinker Paracelsus (1493–1541): The whole world surrounds man as a circle surrounds one point. From this it follows that all things are related to this one point, no differently from an apple seed which is surrounded and preserved by the fruit … Everything that astronomical theory has profoundly fathomed by studying the planetary aspects and the stars … can also be applied to the firmament of the body. Thinkers in this tradition took the universe to revolve around humanity, and sought to gain knowledge of nature by finding parallels between us and the heavens, seeing reality as a symbolic work of art composed with us in mind (see Figure 3). By the 16th century, the idea that everything revolved around and reflected humanity was in danger, threatened by a number of unsettling discoveries, not least the proposal, advanced by Nicolaus Copernicus (1473–1543), that the earth was not actually at the centre of the universe. The old tradition struggled against the rise of the new. Faced with the news that Galileo’s telescopes had detected moons orbiting Jupiter, the traditionally minded scholar Francesco Sizzi argued that such observations were obviously mistaken. According to Sizzi, there could not possibly be more than seven ‘roving planets’ (or heavenly bodies other than the stars), given that there are seven holes in an animal’s head (two eyes, two ears, two nostrils and a mouth), seven metals, and seven days in a week. Sizzi didn’t win that battle. It’s not just that we agree with Galileo that there are more than seven things moving around in the solar system. More fundamentally, we have a different way of thinking about nature and knowledge. We no longer expect there to be any special human significance to natural facts (‘Why seven planets as opposed to eight or 15?’) and we think knowledge will be gained by systematic and open-minded observations of nature rather than the sorts of analogies and patterns to which Sizzi appeals. However, the transition into the Modern era was not an easy one. The pattern-oriented ways of thinking characteristic of pre-Modern thought naturally appeal to meaning-hungry creatures like us. These ways of thinking are found in a great variety of cultures: in classical Chinese thought, for example, the five traditional elements (wood, water, fire, earth, and metal) are matched up with the five senses in a similar correspondence between the inner and the outer. As a further attraction, pre-Modern views often fit more smoothly with our everyday sense experience: naively, the earth looks to be stable and fixed while the sun moves across the sky, and it takes some serious discipline to convince oneself that the mathematically more simple models (like the sun-centred model of the solar system) are right.

I write again (…) the symbols for the interrelationship between matter and energy as it was understood in my day: E = Mc2 It was a flawed equation, as far as I was concerned. There should have been an A in there somewhere for Awareness - without which the E and the M and the c, which was the mathematical constant, could not exist.

I wrote again (…) the symbols for the interrelationship between matter and energy as it was understood in my day: E=Mc2 it was a flawed equation, as far as I was concerned. There should have been an A in there somewhere for Awareness - without which the E and the M and the c, which was a mathematical constant, could not exist.

We have heard that when it arrived in Europe, zero was treated with suspicion. We don't think of the absence of sound as a type of sound, so why should the absence of numbers be a number, argued its detractors. It took centuries for zero to gain acceptance. It is certainly not like other numbers. To work with it requires some tough intellectual contortions, as mathemati­cian Ian Stewart explains. "Nothing is more interesting than nothing, nothing is more puzzling than nothing, and nothing is more important than nothing. For mathematicians, nothing is one of their favorite topics, a veritable Pandora's box of curiosities and paradoxes. What lies at the heart of mathematics? You guessed it: nothing. "Word games like this are almost irresistible when you talk about nothing, but in the case of math this is cheat­ing slightly. What lies at the heart of math is related to nothing, but isn't quite the same thing. 'Nothing' is ­well, nothing. A void. Total absence of thingness. Zero, however, is definitely a thing. It is a number. It is, in fact, the number you get when you count your oranges and you haven't got any. And zero has caused mathematicians more heartache, and given them more joy, than any other number. "Zero, as a symbol, is part of the wonderful invention of 'place notation.' Early notations for numbers were weird and wonderful, a good example being Roman numerals, in which the number 1,998 comes out as MCMXCVIII ­one thousand (M) plus one hundred less than a thousand (CM) plus ten less than a hundred (XC) plus five (V) plus one plus one plus one (III). Try doing arithmetic with that lot. So the symbols were used to record numbers, while calculations were done using the abacus, piling up stones in rows in the sand or moving beads on wires.

Essential to the order of things is the principle of correspondences. Hidden connections underlie diverse phenomena that impress the mind with similar qualities and associations, such as colour, shape, weight, movement and even names with similar sounds and spellings. The material world, operating as it does according to God's design, can be studied to understand His will (‘as above, so below’). The universe becomes a multilayered tableau of symbols. Chemicals and stars, for alchemists and astrologers, are symbolic and can be aligned with other symbols – mathematical, alphabetical, mythic and cosmic – all considered to be mystical. Humanity's divine spark inspires us to seek reunion with the Divinity. Toward this end, the Hermetist employs alchemy, astrology and magic too. Magical formulae are based on the correspondences already noted: a ritual to induce creativity might be addressed to the Sun and might entail lamps, gold, ‘Apollonian’ music and a sunny mood.

Even the most carefully defined philosophical or mathematical concept, which we are sure does not contain more than we put into it, is nevertheless more than we assume. It is a psychic event and as such partly unknowable. The very numbers you use in counting are more than you take them to be. They are at the same time mythological elements (for the Pythagoreans, they were even divine); but you are certainly unaware of this when you use numbers for practical purpose.

It was a vast, low-ceilinged room in the lower levels of the basement. The ceiling was supported by pillars at regular intervals. The room was almost impossible to navigate, being crammed with sixty years’ worth of electronic flotsam and jetsam. He slowly worked his way backward, deeper into the room and further into the past. Toward the back, he came across a large cabinet that he mistook at first for an antique computer. It contained over a hundred vacuum tubes, each with its own set of inductors and capacitors. Then he uncovered the piano-style keyboard with the name HAMMOND above it. “Oh, that must be the Novachord,” said the Teleplay Director. “It’s like an organ, except not. It was used on various radio dramas for a few years, but when we got the Hammond B3’s it went into storage.” Philo told Viridios about it. “They have a Novachord?” Viridios said in surprise. “I’ve heard of it, but I’ve never played one. It was so far ahead of its time that nobody really knew what to do with it. It’s not an organ at all. It’s more like a polyphonic synthesizer.” “That’s not all,” said Philo. “I found some of your old equipment. It’s marked ‘Valence Sound Laboratory.’ It doesn’t look like musical equipment at all, more like scientific equipment. There’s an eight-foot metal cabinet full of circuitry like nothing I’ve ever seen before. The front panel is full of knobs and jacks labeled with mathematical symbols.” Viridios was astonished. “It still exists!” he exclaimed. “I thought it was dismantled and sold for scrap.” “What is it?” “That’s the instrument we used to create the soundtrack for Prisoners of the Iron Star. It’s called a Magneto-Thermion.

People who don’t read it, and even some of those who write it, like to assume or pretend that the ideas used in science fiction all rise from intimate familiarity with celestial mechanics and quantum theory, and are comprehensible only to readers who work for NASA and know how to program their VCR. This fantasy, while making the writers feel superior, gives the non-readers an excuse. I just don’t understand it, they whimper, taking refuge in the deep, comfortable, anaerobic caves of technophobia. It is of no use to tell them that very few science fiction writers understand “it” either. We, too, generally find we have twenty minutes of I Love Lucy and half a wrestling match on our videocassettes when we meant to record Masterpiece Theater. Most of the scientific ideas in science fiction are totally accessible and indeed familiar to anybody who got through sixth grade, and in any case you aren’t going to be tested on them at the end of the book. The stuff isn’t disguised engineering lectures, after all. It isn’t that invention of a mathematical Satan, “story problems.” It’s stories. It’s fiction that plays with certain subjects for their inherent interest, beauty, relevance to the human condition. Even in its ungainly and inaccurate name, the “science” modifies, is in the service of, the “fiction.” For example, the main “idea” in my book The Left Hand of Darkness isn’t scientific and has nothing to do with technology. It’s a bit of physiological imagination—a body change. For the people of the invented world Gethen, individual gender doesn’t exist. They’re sexually neuter most of the time, coming into heat once a month, sometimes as a male, sometimes as a female. A Getheian can both sire and bear children. Now, whether this invention strikes one as peculiar, or perverse, or fascinating, it certainly doesn’t require a great scientific intellect to grasp it, or to follow its implications as they’re played out in the novel. Another element in the same book is the climate of the planet, which is deep in an ice age. A simple idea: It’s cold; it’s very cold; it’s always cold. Ramifications, complexities, and resonance come with the detail of imagining. The Left Hand of Darkness differs from a realistic novel only in asking the reader to accept, pro tem, certain limited and specific changes in narrative reality. Instead of being on Earth during an interglacial period among two-sexed people, (as in, say, Pride and Prejudice, or any realistic novel you like), we’re on Gethen during a period of glaciation among androgynes. It’s useful to remember that both worlds are imaginary. Science-fictional changes of parameter, though they may be both playful and decorative, are essential to the book’s nature and structure; whether they are pursued and explored chiefly for their own interest, or serve predominantly as metaphor or symbol, they’re worked out and embodied novelistically in terms of the society and the characters’ psychology, in description, action, emotion, implication, and imagery. The description in science fiction is likely to be somewhat “thicker,” to use Clifford Geertz’s term, than in realistic fiction, which calls on an assumed common experience. The description in science fiction is likely to be somewhat “thicker,” to use Clifford Geertz’s term, than in realistic fiction, which calls on an assumed common experience. All fiction offers us a world we can’t otherwise reach, whether because it’s in the past, or in far or imaginary places, or describes experiences we haven’t had, or leads us into minds different from our own. To some people this change of worlds, this unfamiliarity, is an insurmountable barrier; to others, an adventure and a pleasure.

Perhaps the most surprising and powerful aspect of place-value arithmetic is how it reduces any calculation to a set of purely abstract symbolic manipulations. In principle, I suppose, one could even be trained to perform such symbol-jiggling procedures without any comprehension whatever of the underlying meaning. We could even (if we can possible imagine being so cruel) force young children to memorize tables of symbols and meaningless step-by-step procedures, and then reward or punish them for their skill (or lack thereof) in this dreary and soulless activity. This would help protect our future office workers from accidentally gaining a personal relationship to arithmetic as a craft or enjoying the perspective that outlook would provide. We could turn the entire enterprise into a rote mechanical process and then reward those who show the most willingness to be made into reliable and obedient tools. I wonder if you can imagine such a nightmarish, dystopian world? Let's try not to think about it.

Indeed, to generalize, almost all of our experiences in this world do not fall under the domain of science or mathematics. Furthermore, we know (at least we think we do) that from Godel's theorem there are definite limits to what pure logical manipulation of symbols can do, there are limits to the domain of mathematics. It has been an act of faith on the part of scientists that the world can be explained in the simple terms that mathematics handles. When you consider how much science has not answered then you see that our successes are not so impressive as they might otherwise appear.

Leibniz predicted modern computers after the isomorphism between his binary numbers and I Ching became clear to him; it was obvious to his fine mathematical mind that such a symbolism could be mechanically reproduced and we would then have something akin to a "thinking machine." It is amusing that those who think computers think (or will soon think) generally consider themselves materialists, while those who claim I Ching thinks call themselves mystics, but if thought is defined in these terms, then both computers and I Ching must be considered to be thinking. (The fact that the human nervous system operates on a similar binary code may account for our occasional impression that humans also think, at least outside the areas of politics and religion.)

English mathematician Ada Lovelace and scientist Charles Babbage invented a machine called the “Difference Engine” and then later postulated a more advanced “Analytical Engine,” which used a series of predetermined steps to solve mathematical problems. Babbage hadn’t conceived that the machine could do anything beyond calculating numbers. It was Lovelace who, in the footnotes of a scientific paper she was translating, went off on a brilliant tangent speculating that a more powerful version of the Engine could be used in other ways.13 If the machine could manipulate symbols, which themselves could be assigned to different things (such as musical notes), then the Engine could be used to “think” outside of mathematics. While she didn’t believe that a computer would ever be able to create original thought, she did envision a complex system that could follow instructions and thus mimic a lot of what everyday people did. It seemed unremarkable to some at the time, but Ada had written the first complete computer program for a future, powerful machine—decades before the light bulb was invented. A

Physical laws are not amusing. Mathematical symbols do not readily lend themselves to the double entendre. Chemical properties are seldom cause for levity. These facts make it intolerable for a gathering ever to include more than one scientist.

The view on the equations was even more stupid--the extraterrestrials thought that because the equations used scientific notation, the imaginary unit, sigma notation, and alphabetical letters that were magically and ironically used in mathematics, they theorized and overall concluded that these equations were indisputably correct, since the hoo-mans (humans) on Earth used a multitude of calculus symbols and large numbers to learn more about the universe.

The analogy with physics is not a digression since the symbolical schema itself represents the descent into matter and requires the identity of the outside with the inside. Psyche cannot be totally different from matter, for how otherwise could it move matter? And matter cannot be alien to psyche, for how else could matter produce psyche? Psyche and matter exist in one and the same world, and each partakes of the other, otherwise any reciprocal action would be impossible. If research could only advance far enough, therefore, we should arrive at an ultimate agreement between physical and psychological concepts. Our present attempts may be bold, but I believe they are on the right lines. Mathematics, for instance, has more than once proved that its purely logical constructions which transcend all experience subsequently coincided with the behaviour of things. This, like the events I call synchronistic, points to a profound harmony between all forms of existence.

My dying words to you are “Say good-by to mathematical logic if you wish to preserve your relations with concrete realities!” ’ (CWJ 12: 103, 1908). Russell made his reply in a letter to the logician Philip Jourdain: ‘I would much rather, of the two, preserve my relations with symbolic logic.’14

REALITY-TUNNEL: An emic reality established by a system of coding, or a structure of metaphors, and transmitted by language, art, mathematics or other symbolism.

The final, two-way arrow indicates the most subtle and nefarious stage of this neurological programming, the feedback between the incoming energy (plus additions and minus subtractions) and the language system (including symbolic, abstract languages like mathematics) which the brain happens to use habitually. The final precept in humans is always verbal or symbolic and hence coded into the pre-existing structure of whatever languages or systems the brain has been taught. The process is not one of linear reaction but of synergetic transaction. This finished product is thus a neurosemantic construct, a kind of metaphor.

As Keynes explained his own methods in The General Theory: “The object of our analysis is, not to provide a machine, or method of blind manipulation, which will furnish an infallible answer, but to provide ourselves with an organised and orderly method of thinking out particular problems….Too large a proportion of recent ‘mathematical’ economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.”8

If to-day you ask a physicist what he has finally made out the æther or the electron to be, the answer will not be a description in terms of billiard balls or fly-wheels or anything concrete; he will point instead to a number of symbols and a set of mathematical equations which they satisfy. What do the symbols stand for? The mysterious reply is given that physics is indifferent to that; it has no means of probing beneath the symbolism. To understand the phenomena of the physical world it is necessary to know the equations which the symbols obey but not the nature of that which is being symbolised.

Marcus du Sautoy, professor of mathematics at Oxford University, was involved in the study of Bakhshali Manuscript. He realises the importance of the text and says[86], ‘Today we take it for granted that the concept of zero is used across the globe and is a key building block of the digital world. But the creation of zero as a number in its own right, which evolved from the placeholder dot symbol found in the Bakhshali manuscript, was one of the greatest breakthroughs in the history of mathematics.

She took a pen from her purse, pulled the napkin from under her drink, slightly damp, but a workable drawing space, and wrote from memory. “It may look hard, but it’s really not. It’s a mathematical relationship between compression and expansion. On the left side, the Greek symbol tau represents compression. On the right, the letter d is the expansion factor for a quantum dimension. The rest are coefficients that we’ve determined from lab tests.

Square,” she said. She then erased it and drew a circle, and repeated the show and naming. She then drew some simple mathematics symbols and calculations. The opaque eyes of the wolf stared and it remained as still as a statue.

We now come to the decisive step of mathematical abstraction: we forget about what the symbols stand for. ...[The mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for. Hermann Weyl, The Mathematical Way of Thinking

Through the judicious employment of symbols, diagrams, and calculations, mathematics enables us to acquire significant facts about extremely significant things (universal laws, even), not by first forging out into the cosmos with teams of scientists, but rather from the comforts and confines of coffee tables in our living rooms! p. 72

A key ingredient in appreciating what mathematics is about is to realize that it is concerned with ideas, understanding, and communication more than it is with any specific brand of symbols....It is almost as if ideas set in mathematical form melt and become liquid and just as rivers can, from the most humble beginnings, flow for thousands of miles, through the most varied topography bringing nourishment and life with them wherever they go, so too can ideas cast in mathematical form flow far from their original sources, along well-defined paths, electrifying and dramatically affecting much of what they touch. pp. xii - xiii.

Fourier succeeded in proving a theorem concerning sine waves which astonished his, at first, incredulous contemporaries. He showed that any variation of a quantity with time can be accurately represented as the sum of a number of sinusoidal variations of different amplitudes, phases, and frequencies.

progressive enrichment of children’s intuitions, leaning heavily on their precocious understanding of quantitative manipulations and of counting. One should first arouse their curiosity with some amusing numerical puzzles and problems. Then, little by little, one may introduce them to the power of symbolic mathematical notation and the shortcuts it provides — but at this stage, great care should be taken never to divorce such symbolic knowledge from the child’s quantitative intuitions. Eventually, formal axiomatic systems may be introduced. Even then, they should never be imposed on the child, but rather they should always be justified by a demand for greater simplicity and effectiveness. Ideally, each pupil should mentally, in condensed form, retrace the history of mathematics and its motivations.

To talk about information theory without communicating its real mathematical content would be like endlessly telling a man about a wonderful composer yet never letting him hear an example of the composer’s music.

MEN HAVE BEEN at odds concerning the value of history. Some have studied earlier times in order to find a universal system of the world, in whose inevitable unfolding we can see the future as well as the past. Others have sought in the past prescriptions for success in the present. Thus, some believe that by studying scientific discovery in another day we can learn how to make discoveries. On the other hand, one sage observed that we learn nothing from history except that we never learn anything from history, and Henry Ford asserted that history is bunk.

But no organism could survive in the rarefied world that the physicist, up to the present generation, regarded as the real one, the abstract area of mass and motion-any more than man could survive without massive equipment on the life-forsaken moon. The actual world occupied by organisms is one of literally indescribable richness and complexity: a life-furthering accumulation of molecules, organisms, species, each bearing the impress of countless functional adaptations and selective transformations, the residue of billions of years of evolution. Of these vast transformations only an infinitesimal part is visible or can be reduced to any mathematical order. Form, color, odor, tactile sensations, emotions, appetites, feelings, images, dreams, words, symbolic abstractions-that plenitude of life which even the humblest being in some degree exhibits-cannot be resolved in any mathematical equation or converted into a geometric metaphor without eliminating a large part of the relevant experience.

Even discoverers themselves sometimes seem incredibly dense as well as inexplicably wonderful. One might expect of Maxwell’s treatise on electricity and magnetism a bold and simple pronouncement concerning the great step he had taken. Instead, it is cluttered with all sorts of such lesser matters as once seemed important, so that a naïve reader might search long to find the novel step and to restate it in the simple manner familiar to us.

Symbolically, at the entrance to the new pyramid complexes stands the nuclear reactor, which first manifested its powers to the multitude by a typical trick of Bronze Age deities: the instant extermination of all the inhabitants of a populous city. Of this early display of nuclear power, as of all the vastly augmented potentialities for destruction that so rapidly followed, one can say what Melville's mad captain in 'Moby Dick' said of himself: "All my means and methods are sane: my purpose is mad." For the splitting of the atom was the beautiful consummation-and the confirmation-of the experimental and mathematical modes of thinking that since the seventeenth century have inordinately increased the human command of physical forces.

Mathematics main goal is to describe the relationship between the points on the numberline by internationally recognized symbols. The most basic symbol involving the numberline is the equal sign (=).

Between the fifteenth and nineteenth centuries, the New World opened by terrestrial explorers, adventurers, soldiers, and administrators joined forces with the scientific and technical new world that the scientists, the inventors, and the engineers explored and cultivated: they were part and parcel of the same movement. One mode of exploration was concerned with abstract symbols, rational systems, universal laws, repeatable and predictable events, objective mathematical measurements: it sought to understand, utilize, and control the forces that derive ultimately from the cosmos, and the solar system. The other model dwelt on the concrete and the organic, the adventurous, the tangible: to sail uncharted oceans, to conquer new lands, to subdue and overawe strange peoples, to discover new foods and medicines, perhaps to find the fountain of youth, or if not, to seize by shameless force of arms the wealth of the Indies. In both modes of exploration, there was from the beginning a touch of defiant pride and demonic frenzy.

And the rest of us? We should grasp the basics of math and statistics-certainly better than most of us do today-but still follow what we love. The world doesn't need millions of mediocre mathematicians, and there's plenty of opportunity for specialists in other fields. Even in the heart of opportunity for specialists in other fields. Even in the heart of the math economy, at IBM Research, geometers and engineers work on teams with linguists and anthropologists and cognitive psychologists. They detail the behavior of humans to those who are trying to build mathematical models of it. All of these ventures, from Samer Takriti's gang at IBM to the secretive researchers laboring behind the barricades at the National Security Agency, feed from the knowledge and smarts of diverse groups. The key to finding a place on such world-class teams is not necessarily to become a math whiz but to become a whiz at something. And that something should be in an area that sparks the most enthusiasm and creativity within each of us. Somewhere on those teams, of course, whether it's in advertising, publishing, counterterrorism, or medical research, there will be at least a few Numerati. They'll be the ones distilling this knowledge into numbers and symbols and feeding them to their powerful tools.

Because while the approximate number system gives us the early ability to intuitively estimate numbers without relying on words or symbols, the ability to proceed to higher levels of mathematics is absolutely language dependent.

First, let me frame what I'm calling beautiful. It's not simply the equation's neat little string of symbols. Rather, it's the entire nimbus of meaning surrounding the formula, including its funneling of many concepts into a statement of stunning brevity, its arresting combination of apparent simplicity and hidden complexity, the way its derivation bridges disparate topics in mathematics, and the fact that it's rich with implications, some of which weren't apparent until many years after it was proved to be true. I think most mathematicians would agree that the equation's beauty concerns something like this nimbus.

Any sequence of decimal digits may occur, but only certain sequences of English letters ever occur, that is, the words of the English language. Thus, it is more efficient to encode English words as sequences of binary digits rather than to encode the letters of the words individually. This again emphasizes the gain to be made by encoding sequences of characters, rather than encoding each character separately.

Mathematics isn’t just science, it is poetry – our efforts to crystallise the unglimpsed connections between things. Poetry that bridges and magnifies the mysteries of the galaxy. But the signs and symbols and equations sentients employ to express these connections are not discoveries but the teasing out of secrets that have always existed.

This standard Humean/Kantian dichotomy of analytic and synthetic propositions immediately yields a very problematic implication: Logical and mathematical propositions are dis-connected from experiential reality. Propositions about the world of experience such as Beverly’s car is white are never necessarily true, and propositions of logic and mathematics such as Twice two makes four, being necessarily true, must not be about the world of experience. Logical and mathematical propositions, wrote Schlick, “do not deal with any facts, but only with the symbols by means of which the facts are expressed.”[98] Logic and mathematics, accordingly, tell us absolutely nothing about the experiential world of facts. As Wittgenstein put it succinctly in the Tractatus: “All propositions of logic say the same thing. That is, nothing.”[99] Logic and mathematics, then, are on their way to becoming mere games of symbolic manipulation.[100]

The quarks were originally introduced as mathematical symbols-the carriers of certain properties that would facilitate mathematical understanding of elementary particles. This very role was assigned by Plato to his infinitely thin triangles. But there is a difference: Plato's composite "elementary particles," the polyhedra that we discussed, are models of thought, not physical entities. To Plato it made no difference whether their existence was ideal or real. Triangles and the structures built out of them, to him, served only to show that nature could be described in terms of such mathematical entities.

The letter is susceptible of operations which enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms. It is this power of transformation that lifts algebra above the level of a convenient shorthand.

Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines which conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view.

In this book, you will encounter various interesting geometries that have been thought to hold the keys to the universe. Galileo Galilei (1564-1642) suggested that "Nature's great book is written in mathematical symbols." Johannes Kepler (1571-1630) modeled the solar system with Platonic solids such as the dodecahedron. In the 1960s, physicist Eugene Wigner (1902-1995) was impressed with the "unreasonable effectiveness of mathematics in the natural sciences." Large Lie groups, like E8-which is discussed in the entry "The Quest for Lie Group E8 (2007)"- may someday help us create a unified theory of physics. in 2007, Swedish American cosmologist Max Tegmark published both scientific and popular articles on the mathematical universe hypothesis, which states that our physical reality is a mathematical structure-in other words, our universe in not just described by mathematics-it is mathematics.

Jessica has two hourglasses, an 11 minute and a 13 minute hourglass. She wants to time accurately 15 minutes. How can she do that? Give me a clue | Answer 14. Ten More Strawberries You and Margaret have the same amount of strawberries. How many does Margaret need to give you in order to have 10 more strawberries than her? Give me a clue | Answer 15. With Just Two Numbers What is the largest possible number you can write using only 2 numbers - just 2 numbers, no other mathematical symbols?

Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those

Any impatient student of mathematics or science or engineering who is irked by having algebraic symbolism thrust on him should try to get on without it for a week.

This would mean that amount of information should be measured, not by the number of possible messages, but by the logarithm of this number.

Fourier analysis, which makes it possible to represent any signal as a sum of sine waves of various frequencies.

The octal system is very important to people who use computers.

information in terms of the number of binary digits rather than in terms of the number of different messages that the binary digits can form. This would mean that amount of information should be measured, not by the number of possible messages, but by the logarithm of this number.

As we have seen with reference to the experiences of Gauss and Poincare, the mathematicians also discovered the fact that our representations are "ordered" before we become aware of them. B.L. van der Waerden, who cites many examples of essential mathematical insights arising from the unconscious, concludes: "...the unconscious is not only able to associate and combine, but even to judge. The judgment of the unconscious is an intuitive one, but it is under favorable circumstances completely sure.

Recent discussions of how natural selection changes biological populations tend to be expressed in the form of mathematical models. These models are written down, of course. They are formulated using mathematical symbolism, and they have to be supplemented with a commentary telling us (for example) which phenomena in the real world are being represented by the model. But we should not expect an analysis of how mathematical models relate to the world to use the same concepts as an analysis of how hypotheses expressed in ordinary language relate to the world.

The Arabian word for Orion, The Giant, has become the familiar word Algebra, the system of letters and symbols referring to particular quantitative relationships in mathematics.

Two writings of al-Hassār have survived. The first, entitled Kitāb al-bayān wa t-tadhkār [Book of proof and recall] is a handbook of calculation treating numeration, arithmetical operations on whole numbers and on fractions, extraction of the exact or approximate square root of a whole of fractionary number and summation of progressions of whole numbers (natural, even or odd), and of their squares and cubes. Despite its classical content in relation to the Arab mathematical tradition, this book occupies a certain important place in the history of mathematics in North Africa for three reasons: in the first place, and notwithstanding the development of research, this manual remains the most ancient work of calculation representing simultaneously the tradition of the Maghrib and that of Muslim Spain. In the second place, this book is the first wherein one has found a symbolic writing of fractions, which utilises the horizontal bar and the dust ciphers i.e. the ancestors of the digits that we use today (and which are, for certain among them, almost identical to ours) [Woepcke 1858-59: 264-75; Zoubeidi 1996]. It seems as a matter of fact that the utilisation of the fraction bar was very quickly generalised in the mathematical teaching in the Maghrib, which could explain that Fibonacci (d. after 1240) had used in his Liber Abbaci, without making any particular remark about it [Djebbar 1980 : 97-99; Vogel 1970-80]. Thirdly, this handbook is the only Maghribian work of calculation known to have circulated in the scientific foyers of south Europe, as Moses Ibn Tibbon realised, in 1271, a Hebrew translation. [Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa]

Digital computers have either two states, on or off, and so respond only to binary messages, which consist of ones (on) and zeros (off). Every term in a program ultimately must be expressed through these two numbers, ensuring that ordinary mathematical statements quickly grow dizzyingly complex. In the late 1940s, programming a computer was, as one observer put it, “maddeningly difficult.” Before long programmers found ways to produce binary strings more easily. They first devised special typewriters that automatically spit out the desired binary code. Then they shifted to more expansive “assembly” languages, in which letters and symbols stood for ones and zeros. Writing in assembly was an advance, but it still required fidelity to a computer’s rigid instruction set. The programmer had to know the instruction set cold in order to write assembly code effectively. Moreover, the instruction set differed from computer model to computer model, depending on its microprocessor design. This meant that a programmer’s knowledge of an assembly language, so painfully acquired, could be rendered worthless whenever a certain computer fell out of use. By

The Exodus also = salvation; Egypt = sin; Pharoah = Satan; Moses = Christ; the Jews = the Church; the Red Sea = death; the wilderness = Purgatory; the Old Law = the New Law; the gospel; the old Mount (Sinai) = the new mount from which Jesus preached His “sermon on the mount” (Mt 5-7); and the Promised Land = Heaven. The “=” is not mathematical but symbolic.

As one of the central defining aspects of the “other world,” time will be discussed at some length in the chapter that follows; here I will note only that time, commonly taken to be an objective fact of the natural universe (chiefly due to the incontrovertible effects of deterioration and decay), is in fact a construct like any other, bound to language and culture and by no means absolute. We are fooled by time into granting it greater ontological status than it deserves because it may be divided and expressed uniformly through the symbolic system of mathematics. We speak of how long “a day” is on other planets and make adjustments for the rate of rotation and circumference of those planets, and yet in the end we are still playing with the clumsy tool of our arbitrary divisions of time, with hours and minutes, which can be made to divide one year on earth into 365 (almost perfectly) even days.

But when the agricultural villages of the Neolithic expanded into larger towns that grew to more than two thousand inhabitants, the capacity of the human brain to know and recognize all of the members of a single community was stretched beyond its natural limits. Nevertheless, the tribal cultures that had evolved during the Upper Paleolithic with the emergence of symbolic communication enabled people who might have been strangers to feel a collective sense of belonging and solidarity. It was the formation of tribes and ethnicities that enabled the strangers of the large Neolithic towns to trust each other and interact comfortably with each other, even if they were not all personally acquainted. The transformation of human society into urban civilizations, however, involved a great fusion of people and societies into groups so large that there was no possibility of having personal relationships with more than a tiny fraction of them. Yet the human capacity for tribal solidarity meant that there was literally no upper limit on the size that a human group could attain. And if we mark the year 3000 BC as the approximate time when all the elements of urban civilization came together to trigger this new transformation, it has taken only five thousand years for all of humanity to be swallowed up by the immense nation-states that have now taken possession of every square inch of the inhabited world. The new urban civilizations produced the study of mathematics, astronomy, philosophy, history, biology, and medicine. They greatly advanced and refined the technologies of metallurgy, masonry, architecture, carpentry, shipbuilding, and weaponry. They invented the art of writing and the practical science of engineering. They developed the modern forms of drama, poetry, music, painting, and sculpture. They built canals, roads, bridges, aqueducts, pyramids, tombs, temples, shrines, castles, and fortresses by the thousands all over the world. They built ocean-going ships that sailed the high seas and eventually circumnavigated the globe. From their cultures emerged the great universal religions of Christianity, Buddhism, Confucianism, Islam, and Hinduism. And they invented every form of state government and political system we know, from hereditary monarchies to representative democracies. The new urban civilizations turned out to be dynamic engines of innovation, and in the course of just a few thousand years, they freed humanity from the limitations it had inherited from the hunting and gathering cultures of the past.

It was Hypatia’s fault, said the Christians, that the governor was being so stubborn. It was she, they murmured, who was standing between Orestes and Cyril, preventing them from reconciling. Fanned by the parabalani, the rumours started to catch, and flame. Hypatia was not merely a difficult woman, they said. Hadn’t everyone seen her use symbols in her work, and astrolabes? The illiterate parabalani (‘bestial men – truly abominable’ as one philosopher would later call them) knew what these instruments were. They were not the tools of mathematics and philosophy, no: they were the work of the Devil. Hypatia was not a philosopher: she was a creature of Hell. It was she who was turning the entire city against God with her trickery and her spells. She was ‘atheizing’ Alexandria. Naturally, she seemed appealing enough – but that was how the Evil One worked. Hypatia, they said, had ‘beguiled many people through satanic wiles’. Worst of all, she had even beguiled Orestes. Hadn’t he stopped going to church? It was clear: she had ‘beguiled him through her magic’. This could not be allowed to continue. One day in March AD 415, Hypatia set out from her home to go for her daily ride through the city. Suddenly, she found her way blocked by a ‘multitude of believers in God’. They ordered her to get down from her chariot. Knowing what had recently happened to her friend Orestes, she must have realized as she climbed down that her situation was a serious one. She cannot possibly have realized quite how serious. As soon as she stood on the street, the parabalani, under the guidance of a Church magistrate called Peter – ‘a perfect believer in all respects in Jesus Christ’ – surged round and seized ‘the pagan woman’. They then dragged Alexandria’s greatest living mathematician through the streets to a church. Once inside, they ripped the clothes from her body then, using broken pieces of pottery as blades, flayed her skin from her flesh. Some say that, while she still gasped for breath, they gouged out her eyes. Once she was dead, they tore her body into pieces and threw what was left of the ‘luminous child of reason’ onto a pyre and burned her.

Countless gods filled countless gaps,” Langdon said. “And yet, over the centuries, scientific knowledge increased.” A collage of mathematical and technical symbols flooded the sky overhead. “As the gaps in our understanding of the natural world gradually disappeared, our pantheon of gods began to shrink.

I went easy on him, didn't wave, didn't stare. I kept my glasses on. I wrote again on my tabletop, scrawled the symbols for the interrelationship between matter and energy as it was understood in my day: E = Mc2 It was a flawed equation, as far as I was concerned. There should have been an "A" in there somewhere for Awareness - without which the "E" and the "M" and the "c", which was a mathematical constant, could not exist.

It's beautiful that our brains have evolved to recognize patterns. It is one of our greatest strengths. It's what allows me to communicate ideas to you with these squiggly little symbols you're looking at right now. It's what allows us to understand concepts like mathematics and physics, its how we manage to engineer buildings that don't collapse and planes that stay in the air. So in a way, there is a "grand novelist." It's us. We are the ones writing this epic saga, pulling out the plot points from the scenery.

Mental arithmetic poses serious problems for the human brain. Nothing ever prepared it for the task of memorizing dozens of intermingled multiplication facts, or of flawlessly executing the ten or fifteen steps of a two-digit subtraction. An innate sense of approximate numerical quantities may well be embedded in our genes; but when faced with exact symbolic calculation, we lack the proper resources. Our brain has to tinker with alternate circuits in order to make up for the lack of a cerebral organ specifically designed for calculation. This tinkering takes a heavy toll. Loss of speed, increased concentration, and frequent errors illuminate the smallness of the mechanisms that our brain contrives in order to "incorporate" arithmetic.

What Born realized was that the symbols Heisenberg was manipulating in his equations were mathematical objects called matrices, and there was an entire field of mathematics devoted to them, called matrix algebra. For example, Heisenberg had found that there was something strange about his symbols: when entity A was multiplied by entity B, it was not the same as B multiplied by A; the order of multiplication mattered. Real numbers don’t behave this way. But matrices do. A matrix is an array of elements. The array can be a single row, a single column, or a combination of rows and columns. Heisenberg had brilliantly intuited a way of representing the quantum world and asking questions about it using such symbols, while being unaware of matrix algebra.

rangoli on the floor; made of powdered colours, it was an ethereal mix of fractals, mathematics, philosophy, and spiritual symbolism.

Zero-order approximation

conditional probability

A grammar must specify not only rules for putting different types of words together to make grammatical structures; it must divide the actual words of English into classes on the basis of the places in which they can appear in grammatical structures. Linguists make such a division purely on the basis of grammatical function without invoking any idea of meaning. Thus, all we can expect of a grammar is the generation of grammatical sentences, and this includes the example given earlier: “The chartreuse

semiquaver skinned the feelings of the manifold.” Certainly the division of words into grammatical categories such as nouns, adjectives, and verbs is not our sole guide concerning the use of words in producing English text. What does influence the choice among words when the words used in constructing grammatical sentences are chosen, not at random by a machine, but rather by a live human being who, through long training, speaks or writes English according to the rules of the grammar? This question is not to be answered by a vague appeal to the word meaning. Our criteria in producing English sentences can be very complicated indeed. Philosophers and psychologists have speculated about and studied the use of words and language for generations, and it is as hard to say anything entirely new about this as it is to say anything entirely true. In particular, what Bishop Berkeley wrote in the eighteenth century concerning the use of language is so sensible that one can scarcely make a reasonable comment without owing him credit. Let us suppose that a poet of the scanning, rhyming school sets out to write a grammatical poem. Much of his choice will be exercised in selecting words which fit into the chosen rhythmic pattern, which rhyme, and which have alliteration and certain consistent or agreeable sound values. This is particularly notable in Poe’s “The Bells,” “Ulalume,” and “The Raven.

Maxwell’s equations apply to all electrical systems, not merely to a specialized and idealized class of electrical circuits.

Maxwell’s equations are more general than network theory,

Most physical phenomena are not reversible. Irreversible phenomena always involve an increase of entropy.

Morse code had been devised by 1838. In this code, letters of the alphabet are represented by spaces, dots, and dashes. The space is the absence of an electric current, the dot is an electric current of short duration, and the dash is an electric current of longer duration.

The circuits used in the transmission of electrical signals do not change with time, and they behave in what is called a linear fashion.

In a linear electrical circuit or transmission system, signals act as if they were present independently of one another; they do not interact. This is, indeed, the very criterion for a circuit being called a linear circuit.

While linearity is a truly astonishing property of nature, it is by no means a rare one. All circuits made up of the resistors, capacitors, and inductors discussed in Chapter I in connection with network theory are linear, and so are telegraph lines and cables. Indeed, usually electrical circuits are linear, except when they include vacuum tubes, or transistors, or diodes, and sometimes even such circuits are substantially linear.

A sine wave is a rather simple sort of variation with time. It can be characterized, or described, or differentiated completely from any other sine wave by means of just three quantities. One of these is the maximum height above zero, called the amplitude. Another is the time at which the maximum is reached, which is specified as the phase. The third is the time T between maxima, called the period. Usually, we use instead of the period the reciprocal of the period called the frequency, denoted by the letter f. If the period T of a sine wave is 1/100 second, the frequency f is 100 cycles per second, abbreviated cps. A cycle is a complete variation from crest, through trough, and back to crest again. The sine wave is periodic in that one variation from crest through trough to crest again is just like any other.

He showed that any variation of a quantity with time can be accurately represented as the sum of a number of sinusoidal variations of different amplitudes, phases, and frequencies. The quantity concerned might be the displacement of a vibrating string, the height of the surface of a rough ocean, the temperature of an electric iron, or the current or voltage in a telephone or telegraph wire. All are amenable to Fourier’s analysis.

The amounts of the attenuation and delay depend on the frequency of the sine wave.

In fact, the circuit may fail entirely to transmit sine waves of some frequencies. Thus, corresponding to an input signal made up of several sinusoidal components, there will be an output signal having components of the same frequencies but of different relative phases or delays and of different amplitudes.

Thus, in general the shape of the output signal will be different from the shape of the input signal. However, the difference can be thought of as caused by the changes in the relative delays and amplitudes of the various components, differences associated with their different frequencies. If the attenuation and delay of a circuit is the same for all frequencies, the shape of the output wave will be the same as that of the input wave; such a circuit is distortionless.

There was much dispute concerning the efficacy of various signals in ameliorating the limitations imposed by circuit speed, intersymbol interference, noise, and limitations on transmitted power.

He noted that the line speed, and hence also the speed of transmission, was proportional to the width or extent of the range or band (in the sense of strip) of frequencies used in telegraphy; we now call this range of frequencies the band width of a circuit or of a signal.

He noted that the line speed, and hence also the speed of transmission, was proportional to the width or extent of the range or band (in the sense of strip) of frequencies used in telegraphy; we now call this range of frequencies the band width of a circuit or of a signal. Finally, in analyzing one

a steady sinusoidal component of constant amplitude.

it was useless at the receiver,

could have been supplied at the receiver rather than transmitted thence over the circuit.

this useless component of the signal, which, he said, conveyed no intelligence, as redundant,

R. V. L. Hartley, the inventor of the Hartley oscillator, was thinking philosophically about the transmission of information at about this time, and he summarized his reflections in a paper, “Transmission of Information,” which he published in 1928.

H, the information of the message, as the logarithm of the number of possible sequences of symbols which might have been selected and showed that H = n log s Here n is the number of symbols selected, and s is the number of different symbols in the set from which symbols are selected. This is acceptable in the light of our present knowledge of information theory only if successive symbols are chosen independently and if any of the s symbols is equally likely to be selected. In this case, we need merely note, as before, that the logarithm of s, the number of symbols, is the number of independent 0-or-1 choices that can be represented or sent simultaneously, and it is reasonable that the rate of transmission of information should be the rate of sending symbols per second n, times the number of independent 0-or-1 choices that can be conveyed per symbol. Hartley goes on to the problem of encoding

Finally, Hartley stated, in accord with Nyquist, that the amount of information which can be transmitted is proportional to the band width times the time of transmission. But this makes us wonder about the number of allowable current values, which is also important to speed of transmission. How are we to enumerate them?

Wiener is a mathematician whose background ideally fitted him to deal with this sort of problem,

Communication theory tells us how to represent, or encode, messages from a particular message source efficiently for transmission over a particular sort of channel, such as an electrical circuit, and it tells us when we can avoid errors in transmission.

As an example, let us choose base six. To write the quantities from zero to five we would use the symbols 0, 1, 2, 3, 4, 5, as in base ten. The first essential difference comes up when we wish to denote six objects. Since six is to be the base we indicate this larger quantity by the symbols 10, the 1 denoting one times the base, just as in base ten the 1 in 10 denotes one times the base, or the quantity ten. Thus, the symbols 10 can mean different quantities, depending upon the base being employed. To write seven in base six we would write 11, because in base six these symbols mean 1.6+ 1, just as 11 in base ten means 1. 10 + 1. Similarly, to denote twenty in base six we write 32 because these symbols now mean 3 · 6 + 2. To indicate the quantity forty in base six we write 104, because these symbols mean 1 . 62 + 0 . 6 + 4, just as in base ten 104 means 1 · 102 + 0 · 10 + 4 or one hundred and four. It is clear that we can express quantity in base six. Moreover, we can perform the usual arithmetic operations in this base. We would, however, have to learn new addition and multiplication tables. For example, in base ten 4 + 5 = 9, but in base six 9 would be written 13.

It is, in fact, natural to think that man may be a finite-state machine, not only in his function as a message source which produces words, but in all his other behavior as well. We can think if we like of all possible conditions and configurations of the cells of the nervous system as constituting states (states of mind, perhaps). We can think of one state passing to another, sometimes with the production of a letter, word, sound, or a part thereof, and sometimes with the production of some other action or of some part of an action. We can think of sight, hearing, touch, and other senses as supplying inputs which determine or influence what state the machine passes into next. If man is a finite-state- machine, the number of states must be fantastic and beyond any detailed mathematical treatment. But, so are the configurations of the molecules in a gas, and yet we can explain much of the significant behavior of a gas in terms of pressure and temperature merely. Can we someday say valid, simple, and important things about the working of the mind in producing written text and other things as well? As we have seen, we can already predict a good deal concerning the statistical nature of what a man will write down on paper, unless he is deliberately trying to behave eccentrically, and, even then, he cannot help conforming to habits of his own.

Several physicists and mathematicians have been anxious to show that communication theory and its entropy are extremely important in connection with statistical mechanics.

Gottlob Burmann, a German poet who lived from 1737 to 1805, wrote 130 poems, including a total of 20,000 words, without once using the letter R. Further, during the last seventeen years of his life, Burmann even omitted the letter from his daily conversation. In

The entropy of communication theory is measured in bits. We may say that the entropy of a message source is so many bits per letter, or per word, or per message. If the source produces symbols at a constant rate, we can say that the source has an entropy of so many bits per second.

This measure of amount of information is called entropy. If

It is best to illustrate entropy first in a simple case. The mathematical theory of communication treats the message source as an ergodic process, a process which produces a string of symbols that are to a degree unpredictable. We must imagine the message source as selecting a given message by some random, i.e., unpredictable means, which, however, must be ergodic. Perhaps the simplest case we can imagine is that in which there are only two possible symbols, say, X and Y, between which the message source chooses repeatedly, each choice uninfluenced by any previous choices. In this case we can know only that X will be chosen with some probability p0 and Y with some probability p1, as in the outcomes of the toss of a biased coin. The recipient can determine these probabilities by examining a long string of characters (X’s, Y’s) produced by the source. The probabilities p0 and p1 must not change with time if the source is to be ergodic.

To coin a Uexküllian-Heideggerian neologism, Jews were to Uexküll the epitome of Umweltvergessenheit or the “forgetfulness of Umwelt”—an inability to grasp and experience one’s own preordained environment that is both brought about and glossed over by vague appeals to universal liberty and justice. But this was nothing specifically or uniquely Jewish; historical circumstances conspired to make the Jews the avant-garde of modern decline universal, a portent of what was to come if the world succumbed to newfangled notions of absolute time, absolute space, absolute symbolic exchange in the shape of money and mathematics, and the abstractions of modern science. This “regrettable laying-waste of the worlds-as-sensed [that] has arisen from the superstition started by the physicists”38 could be averted if people—or rather, the elites—were to accept his new biology, but while Uexküll could pass on the knowledge of what it means to inhabit and shape one’s own Umwelt next to all the myriads of other human and animal Umwelten, he was not able to impart the experience. That is the business of artists.

ergodic

radio waves, which lies outside of the scope of network theory.

all of the mathematical results of network theory apply to certain specialized and idealized mechanical systems,

Network theory is essentially a mathematical theory.

Euclidean geometry

This is because the phenomena in man’s machines are simplified and ordered in comparison with those occurring naturally, and it is these simplified phenomena that man understands most easily.

The Syrians called the constellation Orion, associated with Osiris, Gabbara, and the Arabians named it Al Jabbar—both meaning “the giant.”[700] The Arabian word for Orion, The Giant, has become the familiar word Algebra, the system of letters and symbols referring to particular quantitative relationships in mathematics.

Having been a Ship’s Captain, a Naval Officer a Mathematics & Science Teacher, most people would believe that my primary interests would be directed towards the sciences. On the other hand, those that know me to be an author interested in history, may believe me to be interested in the arts. University degrees usually fall into the general category of Art or Science. It’s as if we have to pick sides and back one or the other team…. With my degree in Marine Science I am often divided and pigeon holed into this specific discipline or area of interest. One way or the other, this holds true for most of us but is this really true for any of us. As a father I can certainly do other things. Being a navigator doesn’t preclude me from driving a car. Hopefully this article does more than just introduce Cuban Art and in addition gives us all good reason to be accepted as more than a “Johnny One Note.“ My quote that “History is not owned solely by historians. It is a part of everyone’s heritage” hopefully opens doors allowing that we be defined as a sum of all our parts, not just a solitary or prominent one. As it happens, I believe that “Just as science feeds our intellect, art feeds our soul.” For the years that Cuba was under Spanish rule, the island was a direct reflection of Spanish culture. Cuba was thought of as an extension of Spain's empire in the Americas, with Havana and Santiago de Cuba being as Spanish as any city in Spain. Although the early Renaissance concentrated on the arts of Ancient Greece and Rome, it spread to Spain during the 15th and 16th centuries. The new interest in literature and art that Europe experienced quickly spread to Cuba in the years following the colonization of the island. Following their counterparts in Europe, Cuban Professionals, Government Administrators and Merchants demonstrated an interest in supporting the arts. In the 16th century painters and sculptors from Spain painted and decorated the Catholic churches and public buildings in Cuba and by the mid-18th century locally born artists continued this work. During the early part of the 20th century Cuban artists such as Salvador Dali, Joan Miró and Pablo Picasso introduced modern classicism and surrealism to Europe. Cuban artist Wilfred Lam can be credited for bringing this artistic style to Cuba. Another Cuban born painter of that era, Federico Beltran Masses, known to be a master of colorization as well as a painter of seductive images of women, sometimes made obvious artistic references to the tropical settings of his childhood. As Cuban art evolved it encompassed the cultural blend of African, European and American features, thereby producing its own unique character. One of the best known works of Cuban art, of this period, is La Gitana Tropical, painted in 1929, by Víctor Manuel. After the 1959 Cuban Revolution, during the early 1960’s, government agencies such as the Commission of Revolutionary Orientation had posters produced for propaganda purposes. Although many of them showed Soviet design features, some still contained hints of the earlier Cuban style for more colorful designs. Towards the end of the 1960’s, a new Cuban art style came into its own. A generation of artists including Félix Beltran, Raul Martinez, Rene Mederos and Alfredo Rostgaard created vibrantly powerful and intense works which remained distinctively Cuban. Though still commissioned by the State to produce propaganda posters, these artists were accepted on the world stage for their individualistic artistic flair and graphic design. After bringing the various and distinct symbols of the island into their work, present day Cuban artists presented their work at the Volumen Uno Exhibit in Havana. Some of these artists were Jose Bedia, Juan Francisco Elso, Lucy Lippard, Ana Mendieta and Tomas Sanchezare. Their intention was to make a nationalistic statement as to who they were without being concerned over the possibility of government rep

This quest to take a problem and see what happens in different situations is called generalizing, and it is this force that drives mathematics forward. Mathematicians constantly want to find solutions and patterns which apply to as many situations as possible, i.e. are as general as possible. A maths puzzle is not complete when you merely find an answer, a maths puzzle is complete when you've then tried to generalize it to other situations as well-and minds including Leonard Euler and Lord Kelvin have excelled in mathematics by displaying just this kind of curiosity. Because mathematicians like the puzzles which work on the pure number rather than the symbolic digit and the system we happen to be writing our numbers down in, there is a sense that, when a puzzle works only in one given base, there is something rather, well, 'secind class' about it. Mathematicians do not like things which work only in base-10; it is only because we have ten fingers that we find that system interesting at all. Mathematics is the search for universal, not base-specific, truth.

Subconscious thought is biased toward regularity and structure, and it is limited in formal power. It may not be capable of symbolic manipulation, of careful reasoning through a sequence of steps. Conscious thought is quite different. It is slow and labored. Here is where we slowly ponder decisions, think through alternatives, compare different choices. Conscious thought considers first this approach, then that—comparing, rationalizing, finding explanations. Formal logic, mathematics, decision theory: these are the tools of conscious thought. Both conscious and subconscious modes of thought are powerful and essential aspects of human life. Both can provide insightful leaps and creative moments. And both are subject to errors, misconceptions, and failures.

The Cubists are entitled to the serious attention of all who find enjoyment in the colored puzzle pictures of the Sunday newspapers. Of course there is no reason for choosing the cube as a symbol, except that it is probably less fitted than any other mathematical expression for any but the most formal decorative art. There is no reason why people should not call themselves Cubists, or Octagonists, or Parallelopipedonists, or Knights of the Isosceles Triangle, or Brothers of the Cosine, if they so desire; as expressing anything serious and permanent, one term is as fatuous as another.

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

Further, if there is cybernetics, then someone must practice it, and cyberneticist has been anonymously coined to designate such a person.

Her exposition took the form of notes lettered A through G, extending to nearly three times the length of Menabrea’s essay. They offered a vision of the future more general and more prescient than any expressed by Babbage himself. How general? The engine did not just calculate; it performed operations, she said, defining an operation as “any process which alters the mutual relation of two or more things,” and declaring: “This is the most general definition, and would include all subjects in the universe.” The science of operations, as she conceived it, is a science of itself, and has its own abstract truth and value; just as logic has its own peculiar truth and value, independently of the subjects to which we may apply its reasonings and processes.… One main reason why the separate nature of the science of operations has been little felt, and in general little dwelt on, is the shifting meaning of many of the symbols used. Symbols and meaning: she was emphatically not speaking of mathematics alone. The engine “might act upon other things besides number.” Babbage had inscribed numerals on those thousands of dials, but their working could represent symbols more abstractly. The engine might process any meaningful relationships. It might manipulate language. It might create music. “Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.

To begin with, the child of five, six, or seven is in many ways an extremely competent individual. Not only can she use skillfully a raft of symbolic forms, but she has evolved a galaxy of robust theories that prove quite serviceable for most purposes and can even be extended in generative fashion to provide cogent accounts of unfamiliar materials or processes. The child is also capable of intensive and extensive involvement in cognitive activities, ranging from experimenting with fluids in the bathtub to building complex block structures and mastering board games, card games, and sports. While some of these creations are derivative, at least a few of them may exhibit genuine creativity and originality. And quite frequently in at least one area, the young child has achieved the competence expected from much older children. Such precocity is particularly likely when youngsters have pursued a special passion, like dinosaurs, dolls, or guns, or when there is a strain of special talent in areas like mathematics, music, or chess or simply a flexibility, a willingness to try new things.

At the risk of repetitiveness I must once more mention here the Pythagoreans, the chief engineers of that epoch-making change. I have spoken in more detail elsewhere of the inspired methods by which, in their religious order, they transformed the Orphic mystery cult into a religion which considered mathematical and astronomical studies as the main forms of divine worship and prayer. The physical intoxication which had accompanied the Bacchic rites was superseded by the mental intoxication derived from philo-sophia, the love of knowledge. It was one of the many key concepts they coined and which are still basic units in our verbal currency. The cliche' about the 'mysteries of nature' originates in the revolutionary innovation of applying the word referring to the secret rites of the worshippers of Orpheus, to the devotions of stargazing. 'Pure science' is another of their coinages; it signified not merely a contrast to the 'applied' sciences, but also that the contemplation of the new mysteria was regarded as a means of purifying the soul by its immersion in the eternal. Finally, 'theorizing' comes from Theoria, again a word of Orphic origin, meaning a state of fervent contemplation and participation in the sacred rites (thea spectacle, theoris spectator, audience). Contemplation of the 'divine dance of numbers' which held both the secrets of music and of the celestial motions became the link in the mystic union between human thought and the anima mundi. Its perfect symbol was the Harmony of the Spheres-the Pythagorean Scale, whose musical intervals corresponded to the intervals between the planetary orbits; it went on reverberating through 'soft stillness and the night' right into the poetry of the Elizabethans, and into the astronomy of Kepler.

Poincare was equally specific: 'It may be surprising to see emotional sensibility invoked a propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true aesthetic feeling that all real mathematicians know. The useful combinations [of ideas] are precisely the most beautiful, I mean those best able to charm this special sensibility.' Max Planck, the father of quantum theory wrote in his autobiography that the pioneer scientist must have 'a vivid intuitive imagination for new ideas not generated by deduction, but by artistically creative imagination.' The quotations could be continued indefinitely, yet I cannot recall any explicit statement to the contrary by any eminent mathematician or physicist. Here, then, is the apparent paradox. A branch of knowledge which operates predominantly with abstract symbols, whose entire rationale and credo are objectivity, verifiability, logicality, turns out to be dependent on mental processes which are subjective, irrational, and verifiable only after the event.

In the classic symbolism of myth and religion, the number forty (= 4 × 10) marks a passing beyond (see chapter ten) a worldly or fourfold material phase. This symbol of passage lends significance to Noah’s rain of forty days and nights; it is also reflected in the life of Moses, whose 120 years encompassed three forty-year phases and who waited forty days on Mount Sinai to receive the Ten Commandments. The Israelites spent forty years wandering in the desert. Jesus’ forty days in the wilderness, the forty days of Lent, and Ali Baba’s forty thieves each recall the transformation of earth and self, often through physical ordeal. At the fortieth day of human pregnancy, the embryo becomes a fetus.