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.

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

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

The 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

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—

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.

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.

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.

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.

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.

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.

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.

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

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]

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.

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).

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.

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 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.

“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—”

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,

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.

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?

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.

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.

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.

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.

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

pretty sure it’s a mathematic symbol.” He removes a beautiful fountain pen from his pocket, turns over his coaster, and writes. He holds up the coaster. Ethan looks at it and says, “The Kronecker delta symbol?” Leo points at him, delighted. “Exactly! I guess you studied it as part of computer science, right?

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.

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.

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.

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.

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.

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.

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.

... not only did perspective elevate art to a "science"... the subjective visual impression was indeed so far rationalized that this very impression could itself become the foundation for a solidly grounded and yet, in an entirely modern sense, "infinite" experiential world. One could even compare the function of Renaissance perspective with that of critical philosophy... The result was a translation of psychophysiological space into mathematical space; in other words, an objectification of the subjective.

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.

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 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.

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.

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.

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

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.

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

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.

[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.

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.

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

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.

Those who are condemned to see the world in shadows will be forced to formulate a description of reality based on insufficient information and, therefore, contrivance. Human history verifies the axiom, “In the absence of knowledge, superstition prevails.” Impaired perception impedes comprehension and breeds fabrication. We find the gaps in our knowledge irritating and uncomfortable, like a road riddled with potholes, and so we fill them in with fabrications to make the ride smoother. The rutted highway of human knowledge is mended with all sorts of contrivances concerning the nature of the universe, a query that for many centuries was beyond investigation. The tools of modern science have enabled us to repave the road, in a manner of speaking, and to upend, one by one, the falsities of our former ignorance. But the road is long, and the work is slow. We must concede that our current conception of the universe is still infantile. Like a child staring bewilderedly at a blackboard chalked from end to end with the esoteric figures of a complex mathematical formula, we are able to recognize some of the numbers and symbols but cannot hope to comprehend the equation, much less solve it. But rather than accept the irreducible complexity before us, many Christians have endeavored to reduce what they cannot comprehend into facile religious concepts that they can. This “Sunday school reductionism” tends to transform profound truths into coloring book illustrations and connect-the-dot puzzles. Instead of illuminating the problem with the lamp of logic and admitting our ignorance, we tend to obscure the problem beneath a canopy of nebulous abstractions, commending ourselves with the false satisfaction of having “solved” it.

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.

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.

Article 10: Whether symbolic logic is superior to Aristotelian logic for philosophizing? Objection 1 : It seems that it is, for it is a modern development, and would not have become popular if it were not superior. In fact, 99% of all formal logic textbooks in print today use symbolic rather than Aristotelian logic. Objection 2: It is as superior in efficiency to Aristotelian logic as Arabic numerals to Roman numerals, or a computer to an abacus. Objection 3: Aristotelian logic presupposes metaphysical and epistemological realism, which are no longer universally accepted. Symbolic logic is ideologically neutral. It is like mathematics not only in efficiency but also in that it carries less “philosophical baggage.” On the contrary , the authority of common sense is still on the side of Aristotelian rather than symbolic logic. But common sense is the origin, basis, and foundation of all further refinements of reason, including symbolic logic; and a branch should not contradict its trunk, an upper story should not contradict its foundation. All philosophical systems, including symbolic logic, since they are refinements of, begin with, and depend on the validity of common sense, even while they greatly refine and expand this foundation, should not contradict it, as symbolic logic does. (See below.) I answer that at least two essential principles of symbolic logic contradict common sense: (1) the counter-intuitive “paradox of material implication,” according to which a false proposition materially implies any proposition, false as well as true, including contradictories (see Socratic Logic , pp. 266-369); and (2) the assumption that a particular proposition (like “some elves are evil”) claims more, not less, than a universal proposition (like “all elves are evil'’), since it is assumed to have “existential import” while a universal proposition is assumed to lack it, since symbolic logic assumes the metaphysical position (or “metaphysical baggage”) of Nominalism. See Socratic Logic , pp. 179-81, 262-63 and The Two Logics by Henry Veatch. Furthermore, no one ever actually argues in symbolic logic except professional philosophers. Its use coincides with the sudden decline of interest in philosophy among students. If you believe that is a coincidence, I have a nice timeshare in Florida that I would like to sell to you. Reply to Objection 1: Popularity is no index of truth. If it were, truth would change, and contradict itself, as popularity changed — including the truth of that statement. And thus it is self-contradictory. Reply to Objection 2: It is not more efficient in dealing with ordinary language. We never hear people actually argue any of the great philosophical questions in symbolic logic, but we hear a syllogism every few sentences. Reply to Objection 3: Symbolic logic is not philosophically neutral but presupposes Nominalism, as shown by the references in the “/ answer that ” above.

The outlines of birds and beasts and sailing ships could still be seen in the apothecary's attic in Grantham, along with the drawings of men and mathematical symbols. But the boy who who had made them was gone. He had taken his notebook with him: his secret world of star names and tawny lions and golden ink made from quicksilver.

Such attempts lacked all semblance of mathematical rigor because of the lack at that time of satisfactory definitions of either the infinite of the infinitesimal. Arithmetic had not become sufficiently abstract and symbolic to free itself of spatial interpretations, for number was still interpreted metrically as a ratio of geometrical magnitudes.

Newton had considered the calculus as a scientific description of the generation of magnitudes, and Leibniz had viewed it as a metaphysical explanation of such generation. The formalism of the nineteenth century took from the calculus any such preconceptions, leaving only the bare symbolic relationships between abstract mathematical entities.

Materialistic and idealistic philosophies have both failed to appreciate the nature of mathematics, as accepted at the present time. 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.

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 what 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. Nevertheless, in the final rigorous formulation and elaboration of such concepts as have been introduced, mathematics must necessarily be unprejudiced by an irrelevant elements in the experiences from which they have arisen.

Statistical mechanics interprets an increase of entropy as a decrease in order or, if we wish, as a decrease in our knowledge.

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.

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.