Mathematical Expression Quotes

Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.

Do you know what the mathematical expression is for longing? ... The negative numbers. The formalization of the feeling that you are missing something.

I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.

Naturally, we are inclined to be so mathematical and calculating that we look upon uncertainty as a bad thing...Certainty is the mark of the common-sense life. To be certain of God means that we are uncertain in all our ways, we do not know what a day may bring forth. This is generally said with a sigh of sadness; it should rather be an expression of breathless expectation.

Time was simple, is simple. We can divide it into simple parts, measure it, arrange dinner by it, drink whisky to its passage. We can mathematically deploy it, use it to express ideas about the observable universe, and yet if asked to explain it in simple language to a child–in simple language which is not deceit, of course–we are powerless. The most it ever seems we know how to do with time is to waste it.

Only someone who doesn’t understand art tells an artist their art somehow failed. How the fuck can art fail? Art can’t be graded, because it’s going to mean something different to everyone. You can’t apply a mathematical absolute to art because there is no one formula for self-expression.

Every formula which expresses a law of nature is a hymn of praise to God.

Eternal truths are ultimately invisible, and you won't find them in material things or natural phenomenon, or even in human emotions. Mathematics, however, can illuminate them, can give the expression – in fact, nothing can prevent it from doing so.

Her concentration was gone, and last night she had had a nightmare about discovering a formalism that let her translate arbitrary concepts into mathematical expressions: then she had proven that life and death were equivalent.

{Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was 'dull', showing off his spontaneous mathematical genius} No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.

The Couple Overfloweth We sometimes go on as though people can’t express themselves. In fact they’re always expressing themselves. The sorriest couples are those where the woman can’t be preoccupied or tired without the man saying “What’s wrong? Say something…,” or the man, without the woman saying … and so on. Radio and television have spread this spirit everywhere, and we’re riddled with pointless talk, insane quantities of words and images. Stupidity’s never blind or mute. So it’s not a problem of getting people to express themselves but of providing little gaps of solitude and silence in which they might eventually find something to say. Repressive forces don’t stop people expressing themselves but rather force them to express themselves; What a relief to have nothing to say, the right to say nothing, because only then is there a chance of framing the rare, and ever rarer, thing that might be worth saying. What we’re plagued by these days isn’t any blocking of communication, but pointless statements. But what we call the meaning of a statement is its point. That’s the only definition of meaning, and it comes to the same thing as a statement’s novelty. You can listen to people for hours, but what’s the point? . . . That’s why arguments are such a strain, why there’s never any point arguing. You can’t just tell someone what they’re saying is pointless. So you tell them it’s wrong. But what someone says is never wrong, the problem isn’t that some things are wrong, but that they’re stupid or irrelevant. That they’ve already been said a thousand times. The notions of relevance, necessity, the point of something, are a thousand times more significant than the notion of truth. Not as substitutes for truth, but as the measure of the truth of what I’m saying. It’s the same in mathematics: Poincaré used to say that many mathematical theories are completely irrelevant, pointless; He didn’t say they were wrong – that wouldn’t have been so bad. (Negotiations)

There cannot be a language more universal and more simple, more free from errors and obscurities...more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes

No one shall expel us from the paradise which Cantor has created for us. {Expressing the importance of Georg Cantor's set theory in the development of mathematics.}

What drove me? I think most creative people want to express appreciation for being able to take advantage of the work that's been done by others before us. I didn't invent the language or mathematics I use. I make little of my own food, none of my own clothes. Everything I do depends on other members of our species and the shoulders that we stand on. And a lot of us want to contribute something back to our species and to add something to the flow. It's about trying to express something in the only way that most of us know how-because we can't write Bob Dylan songs or Tom Stoppard plays. We try to use the talents we do have to express our deep feelings, to show our appreciation of all the contributions that came before us, and to add something to that flow. That's what has driven me.

Within mathematics, assertions must always be proven mathematically and expressed in a valid and scientifically correct formula. The mathematician must be able to stand on a podium and say the words 'This is so because …

Nothing is less applicable to life than a mathematical argument. A proposition expressed in numbers is definitely false or true. In all other relations, the truth is so mingled with the false that often only instinct can help us to decide among virtuous influences, sometimes equally as strong in one direction as in the other.

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.

When we say that the ancestors of the Blacks, who today live mainly in Black Africa, were the first to invent mathematics, astronomy, the calendar, sciences in general, arts, religion, agriculture, social organization, medicine, writing, technique, architecture; that they were the first to erect buildings out of 6 million tons of stone (the Great Pyramid) as architects and engineers—not simply as unskilled laborers; that they built the immense temple of Karnak, that forest of columns with its famed hypostyle hall large enough to hold Notre-Dame and its towers; that they sculpted the first colossal statues (Colossi of Memnon, etc.)—when we say all that we are merely expressing the plain unvarnished truth that no one today can refute by arguments worthy of the name.

Alfonse invested everything he did with a sense of all-consuming purpose. He knew four languages, had photographic memory, did complex mathematics in his head. He'd once told me that the art of getting ahead in New York was based on learning how to express dissatisfaction in an interesting way. The air was full of rage and complaint. People had no tolerance for your particular hardship unless you knew how to entertain them with it.

I do not know if God is a mathematician, but mathematics is the loom upon which God weaves the fabric of the universe....The fact that reality can be described or approximated by simple mathematical expressions suggests to me that nature has mathematics at its core.

Art, and above all, music has a fundamental function, which is to catalyze the sublimation that it can bring about through all means of expression. It must aim through fixations which are landmarks to draw towards a total exaltation in which the individual mingles, losing his consciousness in a truth immediate, rare, enormous, and perfect. If a work of art succeeds in this undertaking even for a single moment, it attains its goal. This tremendous truth is not made of objects, emotions, or sensations; it is beyond these, as Beethoven's Seventh Symphony is beyond music. This is why art can lead to realms that religion still occupies for some people.

Mathematics has always shown a curious ability to be applicable to nature, and this may express a deep link between our minds and nature. We are the Universe speaking out, a part of nature. So it is not so surprising that our systems of logic and mathematics sing in tune with nature.

These rules, the sign language and grammar of the Game, constitute a kind of highly developed secret language drawing upon several sciences and arts, but especially mathematics and music (and/or musicology), and capable of expressing and establishing interrelationships between the content and conclusions of nearly all scholarly disciplines. The Glass Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colours on his palette.

Language as putative science. - The significance of language for the evolution of culture lies in this, that mankind set up in language a separate world beside the other world, a place it took to be so firmly set that, standing upon it, it could lift the rest of the world off its hinges and make itself master of it. To the extent that man has for long ages believed in the concepts and names of things as in aeternae veritates he has appropriated to himself that pride by which he raised himself above the animal: he really thought that in language he possessed knowledge of the world. The sculptor of language was not so modest as to believe that he was only giving things designations, he conceived rather that with words he was expressing supreame knowledge of things; language is, in fact, the first stage of occupation with science. Here, too, it is the belief that the truth has been found out of which the mightiest sources of energy have flowed. A great deal later - only now - it dawns on men that in their belief in language they have propagated a tremendous error. Happily, it is too late for the evolution of reason, which depends on this belief, to be put back. - Logic too depends on presuppositions with which nothing in the real world corresponds, for example on the presupposition that there are identical things, that the same thing is identical at different points of time: but this science came into existence through the opposite belief (that such conditions do obtain in the real world). It is the same with mathematics, which would certainly not have come into existence if one had known from the beginning that there was in nature no exactly straight line, no real circle, no absolute magnitude.

. . . we come astonishingly close to the mystical beliefs of Pythagoras and his followers who attempted to submit all of life to the sovereignty of numbers. Many of our psychologists, sociologists, economists and other latter-day cabalists will have numbers to tell them the truth or they will have nothing. . . . We must remember that Galileo merely said that the language of nature is written in mathematics. He did not say that everything is. And even the truth about nature need not be expressed in mathematics. For most of human history, the language of nature has been the language of myth and ritual. These forms, one might add, had the virtues of leaving nature unthreatened and of encouraging the belief that human beings are part of it. It hardly befits a people who stand ready to blow up the planet to praise themselves too vigorously for having found the true way to talk about nature.

A painter, who finds no satisfaction in mere representation, however artistic, in his longing to express his inner life, cannot but envy the ease with which music, the most non-material of the arts today, achieves this end. He naturally seeks to apply the methods of music to his own art. And from this results that modern desire for rhythm in painting, for mathematical, abstract construction, for repeated notes of colour, for setting colour in motion.

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

Our experience teaches us that there are indeed laws of nature, regularities in the way things behave, and that these laws are best expressed using the language of mathematics. This raises the interesting possibility that mathematical consistency might be used to guide us, along with experimental observation, to the laws that describe physical reality, and this has proved to be the case time and again throughout the history of science. We will see this happen during the course of this book, and it is truly one of the wonderful mysteries of our universe that it should be so.

It's not for nothing that advanced mathematics tend to be invented in hot countries. It's because of the morphic resonance of all the camels who have that disdainful expression and famous curled lip as a natural result of an ability to do quadratic equations.

But then, the sky! Blue, untainted by a single cloud (the Ancientes had such barbarous tastes given that their poets could have been inspired by such stupid, sloppy, silly-lingering clumps of vapour). I love - and i'm certain that i'm not mistaken if i say we love - skies like this, sterile and flawless! On days like these, the whole world is blown from the same shatterproof, everlasting glass as the glass of the Green Wall and of all our structures. On days like these, you can see to the very blue depths of things, to their unknown surfaces, those marvelous expressions of mathematical equality - which exist in even the most usual and everyday objects.

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.

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.

Where was the star? Take concepts like "distant," "isolate," "faint," and give them precise mathematical expression. They'll vanish under such articulation. But just before they do, that's where it lay. "My star." Lorq swept vanes aside so they could see. "That's my sun. That's my nova, with eight-hundred-year-old-light.

Certainly not! I didn't build a machine to solve ridiculous crossword puzzles! That's hack work, not Great Art! Just give it a topic, any topic, as difficult as you like..." Klapaucius thought, and thought some more. Finally he nodded and said: "Very well. Let's have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit." "Love and tensor algebra?" Have you taken leave of your senses?" Trurl began, but stopped, for his electronic bard was already declaiming: Come, let us hasten to a higher plane, Where dyads tread the fairy fields of Venn, Their indices bedecked from one to n, Commingled in an endless Markov chain! Come, every frustum longs to be a cone, And every vector dreams of matrices. Hark to the gentle gradient of the breeze: It whispers of a more ergodic zone. In Reimann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our asymptotes no longer out of phase, We shall encounter, counting, face to face. I'll grant thee random access to my heart, Thou'lt tell me all the constants of thy love; And so we two shall all love's lemmas prove, And in bound partition never part. For what did Cauchy know, or Christoffel, Or Fourier, or any Boole or Euler, Wielding their compasses, their pens and rulers, Of thy supernal sinusoidal spell? Cancel me not--for what then shall remain? Abscissas, some mantissas, modules, modes, A root or two, a torus and a node: The inverse of my verse, a null domain. Ellipse of bliss, converge, O lips divine! The product of our scalars is defined! Cyberiad draws nigh, and the skew mind Cuts capers like a happy haversine. I see the eigenvalue in thine eye, I hear the tender tensor in thy sigh. Bernoulli would have been content to die, Had he but known such a^2 cos 2 phi!

Accept the abundant life in your own mind. Your mental acceptance and expectancy of wealth has its own mathematics and mechanics of expression.

The best mathematics is serious as well as beautiful—‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better

To know what is real, one must subject one’s ideas to the rigorous, error-correcting mechanism of science, seeking verification that can be expressed mathematically.

Underlying our approach to this subject is our conviction that "computer science" is not a science and that its significance has little to do with computers. The computer revolution is a revolution in the way we think and in the way we express what we think. The essence of this change is the emergence of what might best be called procedural epistemology—the study of the structure of knowledge from an imperative point of view, as opposed to the more declarative point of view taken by classical mathematical subjects. Mathematics provides a framework for dealing precisely with notions of "what is". Computation provides a framework for dealing precisely with notions of "how to".

So a)To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers? Plus and minus, self- evidently; sometimes multiplication, and yes. division. But these signs are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically insoluble?

Alchemy is a science, but a science that acknowledges certain principles of magic. This. . . this is a mathematical expression of quintessence, Archimedes' fifth element, which binds all things together.

Well, regular math, or applied math, is what I suppose you could call practical math," he said. "It's used to solve problems, to provide solutions, whether it's in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn't exist to provide immediate, or necessarily obvious, practical applications. It's purely an expression of form, if you will - the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.

The keel-mounted rail gun pushed the whole ship backward in a solid mathematical relationship to the mass of the two-kilo tungsten round moving at a measurable fraction of c. Newton’s third law expressed as violence. Holden’s

I see, in place of that empty figment of one linear history which can be kept up only by shutting one’s eyes to the overwhelming multitude of facts, the drama of a number of mighty Cultures, each springing with primitive strength from the soil of a mother-region to which it remains firmly bound throughout it’s whole life-cycle; each stamping its material, its mankind, in its own image; each having its own idea, its own passions, its own life, will and feelings, its own death. Here indeed are colours, lights, movements, that no intellectual eye has yet discovered. Here the Cultures, peoples, languages, truths, gods, landscapes bloom and age as the oaks and the pines, the blossoms, twigs and leaves - but there is no ageing “Mankind.” Each Culture has its own new possibilities of self-expression which arise, ripen, decay and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in the deepest essence different from the others, each limited in duration and self-contained, just as each species of plant has its peculiar blossom or fruit, its special type of growth and decline.

Are there Laws of Humanics as there are Laws of Robotics? How many Laws of Humanics might there be and how can they be expressed mathematically? I don’t know. “Perhaps, though, there may come a day when someone will work out the Laws of Humanics and then be able to predict the broad strokes of the future, and know what might be in store for humanity, instead of merely guessing as I do, and know what to do to make things better, instead of merely speculating. I dream sometimes of founding a mathematical science which I think of as ‘psychohistory,’ but I know I can’t and I fear no one ever will.

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.

Indeed, the quality that made Newton's theories truly stand out-the inherent characteristic that turned them into inevitable laws of nature-was precisely the fact that they were all expressed as crystal-clear, self-consistent mathematical relations.

Eternal truths are ultimately invisible, and you won't find them in material things or natural phenomena, or even in human emotions. Mathematics, however, can illuminate them, can give them expression - in fact, nothing can prevent it from doing so.

But man is a frivolous and incongruous creature, and perhaps, like a chess player, loves the process of the game, not the end of it. And who knows (there is no saying with certainty), perhaps the only goal on earth to which mankind is striving lies in this incessant process of attaining, in other words, in life itself, and not in the thing to be attained, which must always be expressed as a formula, as positive as twice two makes four, and such positiveness is not life, gentlemen, but is the beginning of death. Anyway, man has always been afraid of this mathematical certainty, and I am afraid of it now. Granted that man does nothing but seek that mathematical certainty, he traverses oceans, sacrifices his life in the quest, but to succeed, really to find it, dreads, I assure you. He feels that when he has found it there will be nothing for him to look for. When workmen have finished their work they do at least receive their pay, they go to the tavern, then they are taken to the police-station–and there is occupation for a week. But where can man go? Anyway, one can observe a certain awkwardness about him when he has attained such objects. He loves the process of attaining, but does not quite like to have attained, and that, of course, is very absurd. In fact, man is a comical creature; there seems to be a kind of jest in it all. But yet mathematical certainty is after all, something insufferable. Twice two makes four seems to me simply a piece of insolence. Twice two makes four is a pert coxcomb who stands with arms akimbo barring your path and spitting. I admit that twice two makes four is an excellent thing, but if we are to give everything its due, twice two makes five is sometimes a very charming thing too.

Every soul has religion, which is only another word for its existence. All living forms in which it expresses itself—all arts, doctrines, customs, all metaphysical and mathematical form-worlds, all ornament, every column and verse and idea—are ultimately religious, and must be so.

Newton was the first person to discover principles in nature which unify large tracts of experience. He abstracted certain unifying concepts from the endless diversity of nature and gave those concepts mathematical expression. Because of this, more than anything else, Newton’s work

Indeed, except for the very simplest physical systems, virtually everything and everybody in the world is caught up in a vast, nonlinear web of incentives and constraints and connections. The slightest change in one place causes tremors everywhere else. We can't help but disturb the universe, as T.S. Eliot almost said. The whole is almost always equal to a good deal more than the sum of its parts. And the mathematical expression of that property-to the extent that such systems can be described by mathematics at all-is a nonlinear equation: one whose graph is curvy.

(The secret of unification, we will see, lies in expanding Riemann's metric to N-dimensional space and then chopping it up into rectangular pieces. Each rectangular piece corresponds to a different force. In this way, we can describe the various forces of nature by slotting them into the metric tensor like pieces of a puzzle. This is the mathematical expression of the principle that higher-dimensional space unifies the laws of nature, that there is "enough room" to unite them in N-dimensional space. More precisely, there is "enough room" in Riemann's metric to unite the forces of nature.)

Riemann concluded that electricity, magnetism, and gravity are caused by the crumpling of our three-dimensional universe in the unseen fourth dimension. Thus a "force" has no independent life of its own; it is only the apparent effect caused by the distortion of geometry. By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would become one of the dominant themes in modern theoretical physics, that the laws of nature appear simple when expressed in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed.

Do you know what the foundation of mathematics is?" I ask. "The foundation of mathematics is numbers. If anyone asked me what makes me truly happy, I would say: numbers. Snow and ice and numbers. And do you know why?" He splits the claws with a nutcracker and pulls out the meat with curved tweezers. "Because the number system is like human life. First you have the natural numbers. The ones that are whole and positive. The numbers of a small child. But human consciousness expands. The child discovers a sense of longing, and do you know what the mathematical expression is for longing?" He adds cream and several drops of orange juice to the soup. "The negative numbers. The formalization of the feeling that you are missing something. And human consciousness expands and grows even more, and the child discovers the in between spaces. Between stones, between pieces of moss on the stones, between people. And between numbers. And do you know what that leads to? It leads to fractions. Whole numbers plus fractions produce rational numbers. And human consciousness doesn't stop there. It wants to go beyond reason. It adds an operation as absurd as the extraction of roots. And produces irrational numbers." He warms French bread in the oven and fills the pepper mill. "It's a form of madness.' Because the irrational numbers are infinite. They can't be written down. They force human consciousness out beyond the limits. And by adding irrational numbers to rational numbers, you get real numbers.

Complexity and simplicity,” he replied. “Time was simple, is simple. We can divide it into simple parts, measure it, arrange dinner by it, drink whisky to its passage. We can mathematically deploy it, use it to express ideas about the observable universe, and yet if asked to explain it in simple language to a child–in simple language which is not deceit, of course–we are powerless. The most it ever seems we know how to do with time is to waste it.

Ever since his first ecstasy or vision of Christminster and its possibilities, Jude had meditated much and curiously on the probable sort of process that was involved in turning the expressions of one language into those of another. He concluded that a grammar of the required tongue would contain, primarily, a rule, prescription, or clue of the nature of a secret cipher, which, once known, would enable him, by merely applying it, to change at will all words of his own speech into those of the foreign one. His childish idea was, in fact, a pushing to the extremity of mathematical precision what is everywhere known as Grimm's Law—an aggrandizement of rough rules to ideal completeness. Thus he assumed that the words of the required language were always to be found somewhere latent in the words of the given language by those who had the art to uncover them, such art being furnished by the books aforesaid.

Once, probably, I used to think that vagueness was a loftier kind of poetry, truer to the depths of consciousness, and maybe when I started to read mathematics and science back in the mid-70s I found an unexpected lyricism in the necessarily precise language that scientists tend to use My instinct, my superstition is that the closer I see a thing and the more accurately I describe it, the better my chances of arriving at a certain sensuality of expression.

I didn’t invent the language or mathematics I used. I make little of my one food, none of my own clothes. Everything I do depends on other members of our species and the shoulders that we stand on. And a lot of us want to contribute something back to our species and to add something to the flow. It’s about trying to express something in the only way that most of us know how because we can’t write Bob Dylan songs or Tom Stoppard plays. We try to use the talents we do have to express our deep feelings, to show our appreciation of all the contributions that came before us, and to add something to that flow. That’s what has driven me.

I’m trying to justify it somehow, he thought, meaning it not in the moral sense but rather in the mathematical one. Buildings are built by observing certain natural laws; natural laws may be expressed by equations; equations must be justified. Where was the justification in what had happened less than half an hour ago?

There was yet another disadvantage attaching to the whole of Newton’s physical inquiries, ... the want of an appropriate notation for expressing the conditions of a dynamical problem, and the general principles by which its solution must be obtained. By the labours of LaGrange, the motions of a disturbed planet are reduced with all their complication and variety to a purely mathematical question. It then ceases to be a physical problem; the disturbed and disturbing planet are alike vanished: the ideas of time and force are at an end; the very elements of the orbit have disappeared, or only exist as arbitrary characters in a mathematical formula.

That's because, if correct, a mathematical formula expresses an eternal truth about the universe. Hence no one can claim ownership of it; it is ours to share. Rich or poor, black or white, young or old - no one can take these formulas away from us. Nothing in this world is so profound and elegant, and yet so available to all.

And who knows (there is no saying with certainty), perhaps the only goal on earth to which mankind is striving lies in this incessant process of attaining, in other words, in life itself, and not in the thing to be attained, which must always be expressed as a formula, as positive as twice two makes four, and such positiveness is not life, gentlemen, but is the beginning of death. Anyway, man has always been afraid of this mathematical certainty, and I am afraid of it now. Granted that man does nothing but seek that mathematical certainty, he traverses oceans, sacrifices his life in the quest, but to succeed, really to find it, dreads, I assure you.

“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—”

In the West, there was an old debate as to whether mathematical reality was made by mathematicians or, existing independently, was merely discovered by them. Ramanujan was squarely in the latter camp; for him, numbers and their mathematical relationships fairly threw off clues to how the universe fit together. Each new theorem was one more piece of the Infinite unfathomed. So he wasn’t being silly, or sly, or cute when later he told a friend, “An equation for me has no meaning unless it expresses a thought of God.

Previous knowledge is required for all scientific studies or methods of instruction. Examples from Mathematics, Dialectic and Rhetoric. Previous knowledge as variously expressed in theses concerning either the existence of a thing or the meaning of the word denoting it. Learning consists in the conversion of universal into particular knowledge.

Literature, like mathematics, is a language, and a language in itself represents no truth, though it may provide the means for expressing any number of them. But poets and critics alike have always believed in some kind of imaginative truth, and perhaps the justification for the belief is in the containment by the language of what it can express.

The demonstration must be against learning—science. But not every science will do. The attack must have all the shocking senselessness of gratuitous blasphemy. Since bombs are your means of expression, it would be really telling if one could throw a bomb into pure mathematics. But that is impossible… What do you think of having a go at astronomy?

The Pythagoreans were probably the first to recognize the concept that the basic forces in the universe may be expressed through the language of mathematics.

It was in 1742 that Christian Goldbach put forward his famous conjecture that every even number greater than 2 can be expressed as the sum of two primes.

Mathematics is the means by which we deduce the consequences of physical principles. More than that, it is the indispensable language in which the principles of physical science are expressed.

In mathematics, just as in the arts, it is dangerous to depart from the ‘Schaffe Künstler, rede nicht’, since also here the basic principles cannot be expressed, but can only be read between the lines.

The central idea is that you can represent reality using a mathematical function that the algorithm doesn’t know in advance but can guess after having seen some data. You can express reality and all its challenging complexity in terms of unknown mathematical functions that machine learning algorithms find and make advantageous. This concept is the core idea for all kinds of machine learning algorithms.

once i asked myself ," what is time? " , in a second or two , i find the answer - " 't' for tension , 'i' for imaginative character of time , 'm' as it is mathematically expressed , 'e' as it has elegance

The most distinct and beautiful statement of any form must take at last the mathematical form.We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both

The keel-mounted rail gun pushed the whole ship backward in a solid mathematical relationship to the mass of the two-kilo tungsten round moving at a measurable fraction of c. Newton’s third law expressed as violence.

The origins of any productive system seem to be traceable to conditions in which the self-interest driven purposes of individuals are allowed expression. These include the respect for autonomy and inviolability of personal boundaries that define liberty and peace and allow for cooperation for mutual ends. Support for such an environment has led to the flourishing of human activity not only in the production of material well-being, but in the arts, literature, philosophy, entrepreneurship, mathematics, spiritual inquiries, the sciences, medicine, engineering, invention, exploration, and other dimensions that fire the varied imaginations and energies of mankind.

Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation.

The integrals which we have obtained are not only general expressions which satisfy the differential equation, they represent in the most distinct manner the natural effect which is the object of the phenomenon... when this condition is fulfilled, the integral is, properly speaking, the equation of the phenomenon; it expresses clearly the character and progress of it, in the same manner as the finite equation of a line or curved surface makes known all the properties of those forms.

theoretically . . . if there was a computer that could hold all of the world’s facts and if it was perfectly programmed to mathematically express all of the relationships between all of the world’s parts, the future could be perfectly foretold.

Eternal truths are ultimately invisible, and you won’t find them in material things or natural phenomena, or even in human emotions. Mathematics, however, can illuminate them, can give them expression—in fact, nothing can prevent it from doing so.

Eternal truths are ultimately invisible, and you won't find them in material things or natural phenomena, or even in human emotions. Mathematics, however, can illuminate them, can give them expression – in fact, nothing can prevent it from doing so.

The Theorem reste upon the validity of my longstanding argument that the world contains precisely two kinds of people: Dumpers and Dumpees. Everyone is predisposed to being either one or the other, but of course not all people are COMPLETE Dumpers and Dumpees. Hence the bell curve:" The majority of people fall somewhere close to the vertical dividing line with the occasional statisticaly outliner (e.g., me) representing a tiny percentage of overall individuals. The numerical expression of the graph can be something like 5 being extreme Dumper, and 0 being me. Ergo, if the Great One was a 4 and I am a 0, total size of the Dumper/Dumpee differetial = -4 (Assuming negative numbers if the guy is more of a Dumpee; positive if the girl is.)

We know, however, that the mind is capable of understanding these matters in all their complexity and in all their simplicity. A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball's spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball. People who call this "instinct" are merely giving the phenomenon a name, not explaining anything. I think that the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts - it has no meaning or purpose other than to be itself. Every single aspect of a piece of music can be represented by numbers. From the organization of movements in a whole symphony, down through the patterns of pitch and rhythm that make up the melodies and harmonies, the dynamics that shape the performance, all the way down to the timbres of the notes themselves, their harmonics, the way they change over time, in short, all the elements of a noise that distinguish between the sound of one person piping on a piccolo and another one thumping a drum - all of these things can be expressed by patterns and hierarchies of numbers. And in my experience the more internal relationships there are between the patterns of numbers at different levels of the hierarchy, however complex and subtle those relationships may be, the more satisfying and, well, whole, the music will seem to be. In fact the more subtle and complex those relationships, and the further they are beyond the grasp of the conscious mind, the more the instinctive part of your mind - by which I mean that part of your mind that can do differential calculus so astoundingly fast that it will put your hand in the right place to catch a flying ball- the more that part of your brain revels in it. Music of any complexity (and even "Three Blind Mice" is complex in its way by the time someone has actually performed it on an instrument with its own individual timbre and articulation) passes beyond your conscious mind into the arms of your own private mathematical genius who dwells in your unconscious responding to all the inner complexities and relationships and proportions that we think we know nothing about. Some people object to such a view of music, saying that if you reduce music to mathematics, where does the emotion come into it? I would say that it's never been out of it.

I confess that Fermat's Theorem as an isolated proposition has very little interest for me, for a multitude of such theorems can easily be set up, which one could neither prove nor disprove. But I have been stimulated by it to bring our again several old ideas for a great extension of the theory of numbers. Of course, this theory belongs to the things where one cannot predict to what extent one will succeed in reaching obscurely hovering distant goals. A happy star must also rule, and my situation and so manifold distracting affairs of course do not permit me to pursue such meditations as in the happy years 1796-1798 when I created the principal topics of my Disquisitiones arithmeticae. But I am convinced that if good fortune should do more than I expect, and make me successful in some advances in that theory, even the Fermat theorem will appear in it only as one of the least interesting corollaries. {In reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. The hope Gauss expressed for his success was never realised.}

Sigils are the means of guiding and uniting the partially free belief[27] with an organic desire, its carriage and retention till its purpose served in the sub-conscious self, and its means of reincarnation in the Ego. All thought can be expressed by form in true relation. Sigils are monograms of thought, for the government of energy (all heraldry, crests, monograms, are Sigils and the Karmas they govern), relating to Karma; a mathematical means of symbolising desire and giving it form that has the virtue of preventing any thought and association on that particular desire (at the magical time), escaping the detection of the Ego, so that it does not restrain or attach such desire to its own transitory images, memories and worries, but allows it free passage to the sub-consciousness.

Other countries whose educational systems achieve more than ours often do so in part by attempting less. While school children in Japan are learning science, mathematics, and a foreign language, American school children are sitting around in circles, unburdening their psyches and “expressing themselves” on scientific, economic and military issues for which they lack even the rudiments of competence. Worse than what they are not learning is what they are learning—presumptuous superficiality, taught by practitioners of it. The

The golden ratio, as well as the Great Pyramid as an expression of it, is an important key to our universe containing the Earth and the Moon. ... The ratio between the Earth and the Moon is in fact the basis for the mathematical concept of 'squaring the circle' ...

These forays into the real world sharpened his view that scientists needed the widest possible education. He used to say, “How can you design for people if you don’t know history and psychology? You can’t. Because your mathematical formulas may be perfect, but the people will screw it up. And if that happens, it means you screwed it up.” He peppered his lectures with quotations from Plato, Chaka Zulu, Emerson, and Chang-tzu. But as a professor who was popular with his students—and who advocated general education—Thorne found himself swimming against the tide. The academic world was marching toward ever more specialized knowledge, expressed in ever more dense jargon. In this climate, being liked by your students was a sign of shallowness; and interest in real-world problems was proof of intellectual poverty and a distressing indifference to theory.

A 2013 study by the National Center on Education and the Economy found that “the mathematics that most enables students to be successful in college courses is not high school mathematics, but middle school mathematics, especially arithmetic, ratio, proportion, expressions and simple equations.

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

Of course, the set of logically consistent mathematical structures is many times larger than the set of physical principles. Therefore, some mathematical structures, such as number theory (which some mathematicians claim to be the purest branch of mathematics), have never been incorporated into any physical theory. Some argue that this situation may always exist: Perhaps the human mind will always be able to conceive of logically consistent structures that cannot be expressed through any physical principle. However, there are indications that string theory may soon incorporate number theory into its structure as well.

Imagine what would have happened had the logicist endeavor been entirely successful. This would have implied that mathematics stems fully from logic-literally from the laws of thought. But how could such a deductive science so marvelously fit natural phenomena? What is the relation between formal logic (maybe we should even say human formal logic) and the cosmos? The answer did not become any clearer after Hilbert and Godel. Now all that existed was an incomplete formal "game," expressed in mathematical language. How could models based on such an "unreliable" system produce deep insights about the universe and its workings?

If I had to design a mechanism for the express purpose of destroying a child's natural curiosity and love of pattern-making, I couldn't possibly do as good a job as is currently being done--I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute mathematics education.

we think in a series of metaphors. We can explain nothing in terms of itself, but only in terms of other things. Even mathematics can express itself in terms of itself only so long as it deals with an ideal system of pure numbers; the moment it begins to deal with numbers of things it is forced back into the language of analogy.

A study of kindergartens in Germany compared fifty play-based classes with fifty early-learning centers and found that the children who played excelled over the others in reading and mathematics and were better adjusted socially and emotionally in school. They also excelled in creativity and intelligence, oral expression, and industry.8

Ohm found that the results could be summed up in such a simple law that he who runs may read it, and a schoolboy now can predict what a Faraday then could only guess at roughly. By Ohm's discovery a large part of the domain of electricity became annexed by Coulomb's discovery of the law of inverse squares, and completely annexed by Green's investigations. Poisson attacked the difficult problem of induced magnetisation, and his results, though differently expressed, are still the theory, as a most important first approximation. Ampere brought a multitude of phenomena into theory by his investigations of the mechanical forces between conductors supporting currents and magnets. Then there were the remarkable researches of Faraday, the prince of experimentalists, on electrostatics and electrodynamics and the induction of currents. These were rather long in being brought from the crude experimental state to a compact system, expressing the real essence. Unfortunately, in my opinion, Faraday was not a mathematician. It can scarcely be doubted that had he been one, he would have anticipated much later work. He would, for instance, knowing Ampere's theory, by his own results have readily been led to Neumann's theory, and the connected work of Helmholtz and Thomson. But it is perhaps too much to expect a man to be both the prince of experimentalists and a competent mathematician.

Any concepts or words which have been formed in the past through the interplay between the world and ourselves are not really sharply defined with respect to their meaning: that is to say, we do not know exactly how far they will help us in finding our way in the world. We often know that they can be applied to a wide range of inner or outer experience, but we practically never know precisely the limits of their applicability. This is true even of the simplest and most general concepts like "existence" and "space and time". Therefore, it will never be possible by pure reason to arrive at some absolute truth. The concepts may, however, be sharply defined with regard to their connections. This is actually the fact when the concepts become part of a system of axioms and definitions which can be expressed consistently by a mathematical scheme. Such a group of connected concepts may be applicable to a wide field of experience and will help us to find our way in this field. But the limits of the applicability will in general not be known, at least not completely.

The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression,more exact,compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.

5.4 The question of accumulation. If life is a wager, what form does it take? At the racetrack, an accumulator is a bet which rolls on profits from the success of one of the horse to engross the stake on the next one. 5.5 So a) To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers?Plus and minus, self-evidently; sometimes multiplication, and yes, division. But these sings are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total of zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically improbable and mathematically insoluble? 5.6 Thus how might you express an accumulation containing the integers b, b, a (to the first), a (to the second), s, v? B = s - v (*/+) a (to the first) Or a (to the second) + v + a (to the first) x s = b 5.7 Or is that the wrong way to put the question and express the accumulation? Is the application of logic to the human condition in and of itself self-defeating? What becomes of a chain of argument when the links are made of different metals, each with a separate frangibility? 5.8 Or is "link" a false metaphor? 5.9 But allowing that is not, if a link breaks, wherein lies the responsibility for such breaking? On the links immediately on the other side, or on the whole chain? But what do you mean by "the whole chain"? How far do the limits of responsibility extend? 6.0 Or we might try to draw the responsibility more narrowly and apportion it more exactly. And not use equations and integers but instead express matters in the traditional narrative terminology. So, for instance, if...." - Adrian Finn

[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.

Eleven days ago I coined the expression [Solar Normalized Resonant Frequency of the GPG's Height] after I have demonstrated mathematically the valid link between time and space on the Giza Plateau. But now, the revelation is even more glorious after I have shown the physical link between time and space on that area; in addition to confirming the existence of the normalizing factor.

We admit as legitimate mathematics certain reflections on the grammar of a language that concerns the empirical. If one seeks to formalize such a mathematics, then with each formalization there are problems, which one can understand and express in ordinary language, but cannot express in the given formalized language. It follows (Brouwer) that mathematics is inexhaustible: one must always again draw afresh from the “fountain of intuition”. There is, therefore, no characteristica universalis for the whole mathematics, and no decision procedure for the whole mathematics. In each and every closed language there are only countably many expressions. The continuum appears only in “the whole of mathematics” … If we have only one language, and can only make “elucidations” about it, then these elucidations are inexhaustible, they always require some new intuition again.

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

They consider people who don't know Hamlet from Macbeth to be Philistines, yet they might merrily admit that they don't know the difference between a gene and a chromosome, or a transistor and a capacitor, or an integral and differential equation. These concepts might seem difficult. Yes, but so, too, is Hamlet. And like Hamlet, each of these concepts is beautiful. Like an elegant mathematical equation, they are expressions of the glories of the universe.

So, in one slightly technical line, here's the mathematical skinny. There's an equation in string theory that has a contribution of the form (D-10) times (Trouble), where D represents the number of spacetime dimensions and Trouble is a mathematical expression resulting in troublesome physical phenomena, such as the violation of energy conservation mentioned above. As to why the equation takes this precise form, I can't offer any intuitive, nontechnical explanation. But if you do the calculation, that's where the math leads. Now, this simple but key observation is that if the number of spacetime dimensions is ten, not the four we expect, the contribution becomes 0 times Trouble. And since 0 times anything is 0, in a universe with ten spacetime dimensions the trouble gets wiped away. That's how the math plays out. Really. And that's why string theorists argue for a universe with more than four spacetime dimensions.

The entropy of a system is related to the number of indistinguishable rearrangements of its constituents, but properly speaking is not equal to the number itself. The relationship is expressed by a mathematical operation called a logarithm; don't be put off if this brings back bad memories of high school math class. In our coin example, it simply means that you pick out the exponent in the number of rearrangements-that is, the entropy is defined as 1,000 rather than 2^1000.

For if the Absolute has predicates, then there are predicates; but the proposition “there are predicates” is not one which the present theory can admit. We cannot escape by saying that the predicates merely qualify the Absolute; for the Absolute cannot be qualified by nothing, so that the proposition “there are predicates” is logically prior to the proposition “the Absolute has predicates”. Thus the theory itself demands, as its logical prius, a proposition without a subject and a predicate; moreover this proposition involves diversity, for even if there be only one predicate, this must be different from the one subject. Again, since there is a predicate, the predicate is an entity, and its predicability of the Absolute is a relation between it and the Absolute. Thus the very proposition which was to be non-relational turns out to be, after all, relational, and to express a relation which current philosophical language would describe as purely external.

Wars and chaoses and paradoxes ago, two mathematicians between them ended an age d began another for our hosts, our ghosts called Man. One was Einstein, who with his Theory of Relativity defined the limits of man's perception by expressing mathematically just how far the condition of the observer influences the thing he perceives. ... The other was Goedel, a contemporary of Eintstein, who was the first to bring back a mathematically precise statement about the vaster realm beyond the limits Einstein had defined: In any closed mathematical system--you may read 'the real world with its immutable laws of logic'--there are an infinite number of true theorems--you may read 'perceivable, measurable phenomena'--which, though contained in the original system, can not be deduced from it--read 'proven with ordinary or extraordinary logic.' Which is to say, there are more things in heaven and Earth than are dreamed of in your philosophy, Horatio. There are an infinite number of true things in the world with no way of ascertaining their truth. Einstein defined the extent of the rational. Goedel stuck a pin into the irrational and fixed it to the wall of the universe so that it held still long enough for people to know it was there. ... The visible effects of Einstein's theory leaped up on a convex curve, its production huge in the first century after its discovery, then leveling off. The production of Goedel's law crept up on a concave curve, microscopic at first, then leaping to equal the Einsteinian curve, cross it, outstrip it. At the point of intersection, humanity was able to reach the limits of the known universe... ... And when the line of Goedel's law eagled over Einstein's, its shadow fell on a dewerted Earth. The humans had gone somewhere else, to no world in this continuum. We came, took their bodies, their souls--both husks abandoned here for any wanderer's taking. The Cities, once bustling centers of interstellar commerce, were crumbled to the sands you see today.

In my opinion, the black hole is incomparably the most exciting and the most important consequence of general relativity. Black holes are the places in the universe where general relativity is decisive. But Einstein never acknowledged his brainchild. Einstein was not merely skeptical, he was actively hostile to the idea of black holes. He thought that the black hole solution was a blemish to be removed from his theory by a better mathematical formulation, not a consequence to be tested by observation. He never expressed the slightest enthusiasm for black holes, either as a concept or as a physical possibility. Oddly enough, Oppenheimer too in later life was uninterested in black holes, although in retrospect we can say that they were his most important contribution to science. The older Einstein and the older Oppenheimer were blind to the mathematical beauty of black holes, and indifferent to the question whether black boles actually exist. How did this blindness and this indifference come about?

Discreet as you are, Rohan, one can’t help but notice how ardently you are pursued. It seems you hold quite an appeal for the ladies of London. And from all appearances, you’ve taken full advantage of what’s been offered.” Cam stared at him without expression. “Pardon, but are you leading to an actual point, my lord?” Leaning back in his chair, St. Vincent made a temple of his elegant hands and regarded Cam steadily. “Since you’ve had no problem with lack of desire in the past, I can only assume that, as happens with other appetites, yours has been sated with an overabundance of sameness. A bit of novelty may be just the thing.” Considering the statement, which actually made sense, Cam wondered if the notorious former rake had ever been tempted to stray. Having known Evie since childhood, when she had come to visit her widowed father at the club from time to time, Cam felt as protective of her as if she’d been his younger sister. No one would have paired the gentle-natured Evie with such a libertine. And perhaps no one had been as surprised as St. Vincent himself to discover their marriage of convenience had turned into a passionate love match. “What of married life?” Cam asked softly. “Does it eventually become an overabundance of sameness?” St. Vincent’s expression changed, the light blue eyes warming at the thought of his wife. “It has become clear to me that with the right woman, one can never have enough. I would welcome an overabundance of such bliss—but I doubt such a thing is mortally possible.” Closing the account book with a decisive thud, he stood from the desk. “If you’ll excuse me, Rohan, I’ll bid you good night.” “What about finishing the accounting?” “I’ll leave the rest in your capable hands.” At Cam’s scowl, St. Vincent shrugged innocently. “Rohan, one of us is an unmarried man with superior mathematical abilities and no prospects for the evening. The other is a confirmed lecher in an amorous mood, with a willing and nubile young wife waiting at home. Who do you think should do the damned account books?” And, with a nonchalant wave, St. Vincent had left the office.

In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

The PSR gives rise to ontological mathematics, which is just the exploration of all the different ways in which x = 0 can be explored, and x can be any expression at all, provided it can ultimately be reduced to zero. There are infinite mathematical tautologies, all of which are consistent with the PSR and Occam’s razor. Nothing can be simpler in hypothesis than requiring everything to equal zero, and nothing could be richer in phenomena than this strict requirement since there are infinite ways to generate mathematical expressions that equal zero. So, the law of ultimate simplicity leads, inevitably, to endless variety ... all thanks to mathematics and the equals sign. There is no contradiction whatsoever between total simplicity and infinite variety ... that’s exactly why math is so powerful, and can produce the incredibly varied universe we live in ... all of which is simply “nothing” expressed in different ways. Is that not the ultimate miracle? But it’s not a miracle at all. It is the direct consequence of the PSR, hence is the most rational thing of all.

When one thinks of 'matrices' and 'codes' it is sometimes helpful to bear these figures in mind. The matrix is the pattern before you, representing the ensemble of permissible moves. The code which governs the matrix can be put into simple mathematical equations which contain the essence of the pattern in a compressed, 'coded' form; or it can be expressed by the word 'diagonals'. The code is the fixed, invariable factor in a skill or habit; the matrix its variable aspect. The two words do not refer to different entities, they refer to different aspects of the same activity. When you sit in front of the chessboard your code is the rule of the game determining which moves are permitted, your matrix is the total of possible choices before you. Lastly, the choice of the actual move among the variety of permissible moves is a matter of strategy, guided by the lie of the land-the 'environment' of other chessmen on the board. We have seen that comic effects are produced by the sudden clash of incompatible matrices: to the experienced chess player a rook moving bishopwise is decidedly 'funny'.

Why is there something rather than nothing? Only because something can exist as nothing – via the mathematical capacity to express “ nothing ” in non-zero terms, e.g. e iπ + 1 = 0. In other words, wherever you see “ nothing ” (zero), you might in fact be confronting e iπ + 1 (“something” ), without knowing it. Only mathematics has this unique capacity to the ground state of the universe. The compulsory ground state of the universe is zero, the minimum value possible. There is no sufficient reason for any arbitrary non-zero number to serve as the ground state.

A philosopher/mathematician named Bertrand Russell who lived and died in the same century as Gass once wrote: “Language serves not only to express thought but to make possible thoughts which could not exist without it.” Here is the essence of mankind’s creative genius: not the edifices of civilization nor the bang-flash weapons which can end it, but the words which fertilize new concepts like spermatozoa attacking an ovum. It might be argued that the Siamese-twin infants of word/idea are the only contribution the human species can, will, or should make to the raveling cosmos. (Yes, our DNA is unique but so is a salamander’s. Yes, we construct artifacts but so have species ranging from beavers to the architect ants whose crenellated towers are visible right now off the port bow. Yes, we weave real-fabric things from the dreamstuff of mathematics, but the universe is hardwired with arithmetic. Scratch a circle and π peeps out. Enter a new solar system and Tycho Brahe’s formulae lie waiting under the black velvet cloak of space/time. But where has the universe hidden a word under its outer layer of biology, geometry, or insensate rock?)

Regular expressions are widely used for string matching. Although regular-expression systems are derived from a perfectly good mathematical formalism, the particular choices made by implementers to expand the formalism into useful software systems are often disastrous: the quotation conventions adopted are highly irregular; the egregious misuse of parentheses, both for grouping and for backward reference, is a miracle to behold. In addition, attempts to increase the expressive power and address shortcomings of earlier designs have led to a proliferation of incompatible derivative languages.

was once asked to give a talk to a group of science journalists who were meeting in my hometown. I decided to talk about the design of bridges, explaining how their form does not derive from a set of equations expressing the laws of physics but rather from the creative mind of the engineer. The first step in designing a bridge is for the engineer to conceive of a form in his mind’s eye. This is then translated into words and pictures so that it can be communicated to other engineers on the team and to the client who is commissioning the work. It is only when there is a form to analyze that science can be applied in a mathematical and methodical way. This is not to say that scientific principles might not inform the engineer’s conception of a bridge, but more likely they are embedded in the engineer’s experience with other, existing bridges upon which the newly conceived bridge is based. The journalists to whom I was speaking were skeptical. Surely science is essential to design, they insisted. No, it is not. And it is not a chicken-and-egg paradox. The design of engineering structures is a creative process in the same way that paintings and novels are the products of creative minds.

Pure math,” he replied. “How is that different from”—she laughed—“regular math?” Gillian asked. “Well, regular math, or applied math, is what I suppose you could call practical math,” he said. “It’s used to solve problems, to provide solutions, whether it’s in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn’t exist to provide immediate, or necessarily obvious, practical applications. It’s purely an expression of form, if you will—the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.

Perhaps I don't know enough yet to find the right words for it, but I think I can describe it. It happened again just a moment ago. I don't know how to put it except by saying that I see things in two different ways-everything, ideas included. If I make an effort to find any difference in them, each of them is the same today as it was yesterday, but as soon as I shut my eyes they're suddenly transformed, in a different light. Perhaps I went wrong about the imaginary numbers. If I get to them by going straight along inside mathematics, so to speak, they seem quite natural. It's only if I look at them directly, in all their strangeness, that they seem impossible. But of course I may be all wrong about this, I know too little about it. But I wasn't wrong about Basini. I wasn't wrong when I couldn't turn my ear away from the faint trickling sound in the high wall or my eye from the silent, swirling dust going up in the beam of light from a lamp. No, I wasn't wrong when I talked about things having a second, secret life that nobody takes any notice of! I-I don't mean it literally-it's not that things are alive, it's not that Basini seemed to have two faces-it was more as if I had a sort of second sight and saw all this not with the eyes of reason. Just as I can feel an idea coming to life in my mind, in the same way I feel something alive in me when I look at things and stop thinking. There's something dark in me, deep under all my thoughts, something I can't measure out with thoughts, a sort of life that can't be expressed in words and which is my life, all the same. “That silent life oppressed me, harassed me. Something kept on making me stare at it. I was tormented by the fear that our whole life might be like that and that I was only finding it out here and there, in bits and pieces. . . . Oh, I was dreadfully afraid! I was out of my mind.. .” These words and these figures of speech, which were far beyond what was appropriate to Törless's age, flowed easily and naturally from his lips in this state of vast excitement he was in, in this moment of almost poetic inspiration. Then he lowered his voice and, as though moved by his own suffering, he added: “Now it's all over. I know now I was wrong after all. I'm not afraid of anything any more. I know that things are just things and will probably always be so. And I shall probably go on for ever seeing them sometimes this way and sometimes that, sometimes with the eyes of reason, and sometimes with those other eyes. . . . And I shan't ever try again to compare one with the other. .

What Homer could never have foreseen is the double idiocy into which we now educate our children. We have what look like our equivalent to the Greek “assemblies”; we can watch them on cable television, as long as one can endure them. For they are charades of political action. They concern themselves constantly, insufferably, about every tiniest feature of human existence, but without slow deliberation, without balance, without any commitment to the difficult virtues. We do not have men locked in intellectual battle with other men, worthy opponents both, as Thomas Paine battled with John Dickinson, or Daniel Webster with Robert Hayne. We have men strutting and mugging for women nagging and bickering. We have the sputters of what used to be language, “tweets,” expressions of something less than opinion. It is the urge to join—something, anything—while remaining aloof from the people who live next door, whose names we do not know. Aristotle once wrote that youths should not study politics, because they had not the wealth of human experience to allow for it; all would become for them abstract and theoretical, like mathematics, which the philosopher said was more suitable for them. He concluded that men should begin to study politics at around the age of forty. Whether that wisdom would help us now, I don’t know.

A.N. Kolmogorov and Yasha Sinai had worked out some illuminating mathematics for the way a system's "entropy per unit time" applies to the geometric pictures of surfaces stretching and folding in phase space. The conceptual core of the technique was a matter of drawing some arbitrarily small box around some set of initial conditions, as one might draw a small square on the side of a balloon, then calculating the effect of various expressions or twists on the box. It might stretch in one direction, for example, while remaining narrow in the other. The change in area corresponded to an introduction of uncertainty about the system's past, a gain or loss of information.

I struggle with words. Never could express myself the way I wanted. My mind fights my mouth, and thoughts get stuck in my throat. Sometimes they stay stuck for seconds or even minutes. Some thoughts stay for years; some have stayed hidden all my life. As a child, I stuttered. What was inside couldn't get out. I'm still not real fluent. I don't know a lot of good words. If I were wrongfully accused of a crime, I'd have a tough time explaining my innocence. I'd stammer and stumble and choke up until the judge would throw me in jail. Words aren't my friends. Music is. Sounds, notes, rhythms. I talk through music. Maybe that's why I became a loner, someone who loves privacy and doesn't reveal himself too easily. My friendliness might fool you. Come into my dressing room and I'll shake your hand, pose for a picture, make polite small talk. I'll be as nice as I can, hoping you'll be nice to me. I'm genuinely happy to meet you and exchange a little warmth. I have pleasant acquaintances with thousands of people the world over. But few, if any, really know me. And that includes my own family. It's not that they don't want to; it's because I keep my feelings to myself. If you hurt me, chances are I won't tell you. I'll just move on. Moving on is my method of healing my hurt and, man, I've been moving on all my life. Now it's time to stop. This book is a place for me to pause and look back at who I was and what I became. As I write, I'm seventy hears old, and all the joy and hurts, small and large, that I've stored up inside me...well, I want to pull 'em out and put 'em on the page. When I've been described on other people's pages, I don't recognize myself. In my mind, no one has painted the real me. Writers have done their best, but writers have missed the nitty-gritty. Maybe because I've hidden myself, maybe because I'm not an easy guy to understand. Either way, I want to open up and leave a true account of who I am. When it comes to my own life, others may know the cold facts better than me. Scholars have told me to my face that I'm mixed up. I smile but don't argue. Truth is, cold facts don't tell the whole story. Reading this, some may accuse me of remembering wrong. That's okay, because I'm not writing a cold-blooded history. I'm writing a memory of my heart. That's the truth I'm after - following my feelings, no matter where they lead. I want to try to understand myself, hoping that you - my family, my friends, my fans - will understand me as well. This is a blues story. The blues are a simple music, and I'm a simple man. But the blues aren't a science; the blues can't be broken down like mathematics. The blues are a mystery, and mysteries are never as simple as they look.

Anatol Rapoport, a mathematical psychologist who was famous for his insights into social interactions: You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, ‘Thanks, I wish I’d thought of putting it that way.’ You should list any points of agreement (especially if they are not matters of widespread agreement). You should mention anything that you have learned from your target. Only then are you permitted to say so much as a word of rebuttal or criticism.1 How many times have you heard or participated in a conversation that obeys these rules? Such guidelines have gone out of fashion recently, if they were ever followed.

The two hit it off well, because de Broglie was trying, like Einstein, to see if there were ways that the causality and certainty of classical physics could be saved. He had been working on what he called “the theory of the double solution,” which he hoped would provide a classical basis for wave mechanics. “The indeterminist school, whose adherents were mainly young and intransigent, met my theory with cold disapproval,” de Broglie recalled. Einstein, on the other hand, appreciated de Broglie’s efforts, and he rode the train with him to Paris on his way back to Berlin. At the Gare du Nord they had a farewell talk on the platform. Einstein told de Broglie that all scientific theories, leaving aside their mathematical expressions, ought to lend themselves to so simple a description “that even a child could understand them.” And what could be less simple, Einstein continued, than the purely statistical interpretation of wave mechanics! “Carry on,” he told de Broglie as they parted at the station. “You are on the right track!” But he wasn’t. By 1928, a consensus had formed that quantum mechanics was correct, and de Broglie relented and adopted that view. “Einstein, however, stuck to his guns and continued to insist that the purely statistical interpretation of wave mechanics could not possibly be complete,” de Broglie recalled, with some reverence, years later.

What drove me? I think most creative people want to express appreciation for being able to take advantage of the work that’s been done by others before us. I didn’t invent the language or mathematics I use. I make little of my own food, none of my own clothes. Everything I do depends on other members of our species and the shoulders that we stand on. And a lot of us want to contribute something back to our species and to add something to the flow. It’s about trying to express something in the only way that most of us know how—because we can’t write Bob Dylan songs or Tom Stoppard plays. We try to use the talents we do have to express our deep feelings, to show our appreciation of all the contributions that came before us, and to add something to that flow. That’s what has driven me.

Common sense requires modern man’s recognition of the scientific method as a spectacularly useful instrumentality for transforming our environment. Respect and gratitude are indeed due the scientist for many comforts and conveniences furnished to modern living, often as the fruit of painstakingly sacrificial research and experimentation, although in recent times not often without financial reward. This practical success of science inclines many persons to a tacit acceptance of the scientific world-picture of external reality as a realm merely of impersonal processes and mathematically connectible sequences. Charles H. Malik observes rightly that all too often the highly merited prestige of scientists in their own fields of competence is transferred to areas of publicly expressed opinion in which they are novices.

Nevertheless, Leibniz remains a great man, and his greatness is more apparent now than it was at any earlier time. Apart from his eminence as a mathematician and as the inventor of the infinitesimal calculus, he was a pioneer in mathematical logic, of which he perceived the importance when no one else did so. And his philosophical hypotheses, though fantastic, are very clear, and capable of precise expression. Even his monads can still be useful as suggesting possible ways of viewing perception, though they cannot be regarded as windowless. What I, for my part, think best in his theory of monads is his two kinds of space, one subjective, in the perceptions of each monad, and one objective, consisting of the assemblage of points of view of the various monads. This, I believe, is still useful in relating perception to physics.

Einstein’s “mathematical strategy,” on the other hand, focused on using generic mathematical knowledge about the metric tensor to find a gravitational field equation that was generally (or at least broadly) covariant. The process worked both ways: Einstein would examine equations that were abstracted from his physical requirements to check their covariance properties, and he would examine equations that sprang from elegant mathematical formulations to see if they met the requirements of his physics. “On page after page of the notebook, he approached the problem from either side, here writing expressions suggested by the physical requirements of the Newtonian limit and energy-momentum conservation, there writing expressions naturally suggested by the generally covariant quantities supplied by the mathematics of Ricci and Levi-Civita,” says John Norton.

My current best model of how a market works is fractional Brownian motion of multifractal time. It has been called the Multifractal Model of Asset Returns. The basic ideas are similar to the cartoon versions above-though far more intricate, mathematically. The cartoon of Brownian motion gets replaced by an equation that a computer can calculate. The trading-time process is expressed by another mathematical function, called f(\propto), that can be tuned to fit a wide range of market behavior. My model redistributes time. It compresses it in some places, stretches it out in others. The result appears very wild, very random. The two functions, of time and Brownian motion, work together in what mathematicians call a compound manner: Price is a function of trading time, which in turn is a function of clock time. Again, the two steps in the model combine to produce a "baby" far different from either parent.

Frege ridiculed the formalist conception of mathematics by saying that the formalists confused the unimportant thing, the sign, with the important, the meaning. Surely, one wishes to say, mathematics does not treat of dashes on a bit of paper. Frege’s idea could be expressed thus: the propositions of mathematics, if they were just complexes of dashes, would be dead and utterly uninteresting, whereas they obviously have a kind of life. And the same, of course, could be said of any proposition: Without a sense, or without the thought, a proposition would be an utterly dead and trivial thing. And further it seems clear that no adding of inorganic signs can make the proposition live. And the conclusion which one draws from this is that what must be added to the dead signs in order to make a live proposition is something immaterial, with properties different from all mere signs. But if we had to name anything which is the life of the sign, we should have to say that it was its use.

Because the number system is like human life. (emphasis added) First you have natural numbers. The ones that are whole and positive. The numbers of a small child. But human consciousness expands. The child discovers a sense of long, and do you know what the mathematical expression is for longing?’ He adds cream and several drops of orange juice to the soup. ‘The negative numbers. The formalization of the feeling that you are missing something. And human consciousness expands and grows even more, and the child discovers the in between spaces. Between stones, between pieces of moss on the stones, between people. And between numbers. And do you know what that leads to? It leads to fractions. Whole numbers plus fractions prouce rational numbers. And human consciousness doesn’t stop there. It wants to go beyond reason. It adds an operation as absurd as the extraction of roots. And produces irrational numbers.’ He warms French bread in the over and fills the pepper mill. ‘It’s a form of madness. Because the irrational numbers are infinite. They can’t be written down. They force human consciousness out beyond the limits. And by adding irrational numbers to rational numbers, you get real numbers.’ I’ve stepped into the middle of the room to have more space. It’s rare that you have a chance to explain yourself to a fellow human being. Usually you have to fight for the floor. And this is important to me. ‘It doesn’t stop. It never stops. Because now, on the spot, we expand the real numbers with imaginary square roots of negative numbers. These are numbers we can’t picture, numbers that normal human consciousness cannot comprehend. And when we add the imaginary numbers to the real numbers, we have the complex number system. The first number system in which it’s possible to explain satisfactorily the crystal formation of ice. It’s like a vast, open landscape. The horizons. You head toward them, and they keep receding. That is Greenland, and that’s what I can’t be without! That’s why I don’t want to be locked up

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.

These rules, the sign language and grammar of the Game, constitute a kind of highly developed secret language drawing upon several sciences and arts, but especially mathematics and music (and/or musicology), and capable of expressing and establishing interrelationships between the content and conclusions of nearly all scholarly disciplines. The Glass Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colors on his palette. All the insights, noble thoughts, and works of art that the human race has produced in its creative eras, all that subsequent periods of scholarly study have reduced to concepts and converted into intellectual property on all this immense body of intellectual values the Glass Bead Game player plays like the organist on an organ. And this organ has attained an almost unimaginable perfection; its manuals and pedals range over the entire intellectual cosmos; its stops are almost beyond number. Theoretically this instrument is capable of reproducing in the Game the entire intellectual content of the universe.

I taught mathematics here at the university for many years. But before that time, I used to create beautiful music with my beloved wife, who passed away. Her name was Sarah.” He paused for a moment to gather himself after uttering her name, but then continued. “I didn’t play for many years after she died, but one thing that loss has taught me is that keeping a gift locked away only brings harm. It is essential to express ourselves. Walls and doors keep out not just the bad but also the good. It is our job to own the keys of our freedom, and to be able to open those doors. I spent many years behind a closed door without knowing where to find the key to my life, but thanks to my music, I am now finding my way out. To quote an extraordinary but lesser-known philosopher, my late wife Sarah Held, ‘What is life without the beauty of art or music or poetry to help us interpret it, encourage us to know how to feel, how to love and how to live?’ In music I see the darkness, the light, the messiness, the beauty, and the complexities of life that simply can’t be summed up like an equation. Music, for me, helps bring down the walls, to open those locked doors. And I hope it has been that way for you too, tonight.

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.

The first is the belief that the universe is essentially harmonious, and that the source of that concord lies in mathematical proportions which can be directly related to musical harmonies. Pythagoras, in an oft-repeated legend, was said to have meditated on the sound of smiths beating hammers upon anvils, and to have argued that a hammer half as heavy produced a note an octave above its full-sized fellow.3 More important were the experiments with a single string, or monochord, attributed to him by his successors. If a stretched string is divided exactly into two it produces a sound an octave higher than the fundamental pitch (the ratio 2:1), the intervals of the fourth and fifth can similarly be expressed as the ratios 4:3 and 3:2 respectively, and all other intervals can be described in mathematical terms.4 These numerical proportions were then extended to describe the relationships of the planetary spheres, both in their relative distance one from another, and in the speed of their movement. The ideas were given influential (if obscure) expression in Plato’s Timaeus, and endlessly elaborated in succeeding centuries up to the Renaissance. One of the final manifestations of this understanding is provided in the illustration of cosmic harmony from Robert Fludd’s Utriusque cosmi … historia 1

For almost all astronomical objects, gravitation dominates, and they have the same unexpected behavior. Gravitation reverses the usual relation between energy and temperature. In the domain of astronomy, when heat flows from hotter to cooler objects, the hot objects get hotter and the cool objects get cooler. As a result, temperature differences in the astronomical universe tend to increase rather than decrease as time goes on. There is no final state of uniform temperature, and there is no heat death. Gravitation gives us a universe hospitable to life. Information and order can continue to grow for billions of years in the future, as they have evidently grown in the past. The vision of the future as an infinite playground, with an unending sequence of mysteries to be understood by an unending sequence of players exploring an unending supply of information, is a glorious vision for scientists. Scientists find the vision attractive, since it gives them a purpose for their existence and an unending supply of jobs. The vision is less attractive to artists and writers and ordinary people. Ordinary people are more interested in friends and family than in science. Ordinary people may not welcome a future spent swimming in an unending flood of information. A darker view of the information-dominated universe was described in the famous story “The Library of Babel,” written by Jorge Luis Borges in 1941.§ Borges imagined his library, with an infinite array of books and shelves and mirrors, as a metaphor for the universe. Gleick’s book has an epilogue entitled “The Return of Meaning,” expressing the concerns of people who feel alienated from the prevailing scientific culture. The enormous success of information theory came from Shannon’s decision to separate information from meaning. His central dogma, “Meaning is irrelevant,” declared that information could be handled with greater freedom if it was treated as a mathematical abstraction independent of meaning. The consequence of this freedom is the flood of information in which we are drowning. The immense size of modern databases gives us a feeling of meaninglessness. Information in such quantities reminds us of Borges’s library extending infinitely in all directions. It is our task as humans to bring meaning back into this wasteland. As finite creatures who think and feel, we can create islands of meaning in the sea of information. Gleick ends his book with Borges’s image of the human condition: We walk the corridors, searching the shelves and rearranging them, looking for lines of meaning amid leagues of cacophony and incoherence, reading the history of the past and of the future, collecting our thoughts and collecting the thoughts of others, and every so often glimpsing mirrors, in which we may recognize creatures of the information.

The concept of absolute time—meaning a time that exists in “reality” and tick-tocks along independent of any observations of it—had been a mainstay of physics ever since Newton had made it a premise of his Principia 216 years earlier. The same was true for absolute space and distance. “Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external,” he famously wrote in Book 1 of the Principia. “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.” But even Newton seemed discomforted by the fact that these concepts could not be directly observed. “Absolute time is not an object of perception,” he admitted. He resorted to relying on the presence of God to get him out of the dilemma. “The Deity endures forever and is everywhere present, and by existing always and everywhere, He constitutes duration and space.”45 Ernst Mach, whose books had influenced Einstein and his fellow members of the Olympia Academy, lambasted Newton’s notion of absolute time as a “useless metaphysical concept” that “cannot be produced in experience.” Newton, he charged, “acted contrary to his expressed intention only to investigate actual facts.”46 Henri Poincaré also pointed out the weakness of Newton’s concept of absolute time in his book Science and Hypothesis, another favorite of the Olympia Academy. “Not only do we have no direct intuition of the equality of two times, we do not even have one of the simultaneity of two events occurring in different places,” he wrote.

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.

A more complex way to understand this is the method used by Hermann Minkowski, Einstein’s former math teacher at the Zurich Polytechnic. Reflecting on Einstein’s work, Minkowski uttered the expression of amazement that every beleaguered student wants to elicit someday from condescending professors. “It came as a tremendous surprise, for in his student days Einstein had been a lazy dog,” Minkowski told physicist Max Born. “He never bothered about mathematics at all.”63 Minkowski decided to give a formal mathematical structure to the theory. His approach was the same one suggested by the time traveler on the first page of H. G. Wells’s great novel The Time Machine, published in 1895: “There are really four dimensions, three which we call the three planes of Space, and a fourth, Time.” Minkowski turned all events into mathematical coordinates in four dimensions, with time as the fourth dimension. This permitted transformations to occur, but the mathematical relationships between the events remained invariant. Minkowski dramatically announced his new mathematical approach in a lecture in 1908. “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength,” he said. “They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”64 Einstein, who was still not yet enamored of math, at one point described Minkowski’s work as “superfluous learnedness” and joked, “Since the mathematicians have grabbed hold of the theory of relativity, I myself no longer understand it.” But he in fact came to admire Minkowski’s handiwork and wrote a section about it in his popular 1916 book on relativity.

Absolute continuity of motion is not comprehensible to the human mind. Laws of motion of any kind become comprehensible to man only when he examines arbitrarily selected elements of that motion; but at the same time, a large proportion of human error comes from the arbitrary division of continuous motion into discontinuous elements. There is a well known, so-called sophism of the ancients consisting in this, that Achilles could never catch up with a tortoise he was following, in spite of the fact that he traveled ten times as fast as the tortoise. By the time Achilles has covered the distance that separated him from the tortoise, the tortoise has covered one tenth of that distance ahead of him: when Achilles has covered that tenth, the tortoise has covered another one hundredth, and so on forever. This problem seemed to the ancients insoluble. The absurd answer (that Achilles could never overtake the tortoise) resulted from this: that motion was arbitrarily divided into discontinuous elements, whereas the motion both of Achilles and of the tortoise was continuous. By adopting smaller and smaller elements of motion we only approach a solution of the problem, but never reach it. Only when we have admitted the conception of the infinitely small, and the resulting geometrical progression with a common ratio of one tenth, and have found the sum of this progression to infinity, do we reach a solution of the problem. A modern branch of mathematics having achieved the art of dealing with the infinitely small can now yield solutions in other more complex problems of motion which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when it deals with separate elements of motion instead of examining continuous motion. In seeking the laws of historical movement just the same thing happens. The movement of humanity, arising as it does from innumerable arbitrary human wills, is continuous. To understand the laws of this continuous movement is the aim of history. But to arrive at these laws, resulting from the sum of all those human wills, man's mind postulates arbitrary and disconnected units. The first method of history is to take an arbitrarily selected series of continuous events and examine it apart from others, though there is and can be no beginning to any event, for one event always flows uninterruptedly from another. The second method is to consider the actions of some one man—a king or a commander—as equivalent to the sum of many individual wills; whereas the sum of individual wills is never expressed by the activity of a single historic personage. Historical science in its endeavor to draw nearer to truth continually takes smaller and smaller units for examination. But however small the units it takes, we feel that to take any unit disconnected from others, or to assume a beginning of any phenomenon, or to say that the will of many men is expressed by the actions of any one historic personage, is in itself false. It needs no critical exertion to reduce utterly to dust any deductions drawn from history. It is merely necessary to select some larger or smaller unit as the subject of observation—as criticism has every right to do, seeing that whatever unit history observes must always be arbitrarily selected. Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.

Mathematical analysis and computer modelling are revealing to us that the shapes and processes we encounter in nature -the way that plants grow, the way that mountains erode or rivers flow, the way that snowflakes or islands achieve their shapes, the way that light plays on a surface, the way the milk folds and spins into your coffee as you stir it, the way that laughter sweeps through a crowd of people — all these things in their seemingly magical complexity can be described by the interaction of mathematical processes that are, if anything, even more magical in their simplicity. Shapes that we think of as random are in fact the products of complex shifting webs of numbers obeying simple rules. The very word “natural” that we have often taken to mean ”unstructured” in fact describes shapes and processes that appear so unfathomably complex that we cannot consciously perceive the simple natural laws at work.They can all be described by numbers. We know, however, that the mind is capable of understanding these matters in all their complexity and in all their simplicity. A ball flying through the air is responding to the force and direction with which it was thrown, the action of gravity, the friction of the air which it must expend its energy on overcoming, the turbulence of the air around its surface, and the rate and direction of the ball's spin. And yet, someone who might have difficulty consciously trying to work out what 3 x 4 x 5 comes to would have no trouble in doing differential calculus and a whole host of related calculations so astoundingly fast that they can actually catch a flying ball. People who call this "instinct" are merely giving the phenomenon a name, not explaining anything. I think that the closest that human beings come to expressing our understanding of these natural complexities is in music. It is the most abstract of the arts - it has no meaning or purpose other than to be itself.

My own observations had by now convinced me that the mind of the average Westerner held an utterly distorted image of Islam. What I saw in the pages of the Koran was not a ‘crudely materialistic’ world-view but, on the contrary, an intense God-consciousness that expressed itself in a rational acceptance of all God-created nature: a harmonious side-by-side of intellect and sensual urge, spiritual need and social demand. It was obvious to me that the decline of the Muslims was not due to any shortcomings in Islam but rather to their own failure to live up to it. For, indeed, it was Islam that had carried the early Muslims to tremendous cultural heights by directing all their energies toward conscious thought as the only means to understanding the nature of God’s creation and, thus, of His will. No demand had been made of them to believe in dogmas difficult or even impossible of intellectual comprehension; in fact, no dogma whatsoever was to be found in the Prophet’s message: and, thus, the thirst after knowledge which distinguished early Muslim history had not been forced, as elsewhere in the world, to assert itself in a painful struggle against the traditional faith. On the contrary, it had stemmed exclusively from that faith. The Arabian Prophet had declared that ‘Striving after knowledge is a most sacred duty for every Muslim man and woman’: and his followers were led to understand that only by acquiring knowledge could they fully worship the Lord. When they pondered the Prophet’s saying, ‘God creates no disease without creating a cure for it as well’, they realised that by searching for unknown cures they would contribute to a fulfilment of God’s will on earth: and so medical research became invested with the holiness of a religious duty. They read the Koran verse, ‘We create every living thing out of water’ - and in their endeavour to penetrate to the meaning of these words, they began to study living organisms and the laws of their development: and thus they established the science of biology. The Koran pointed to the harmony of the stars and their movements as witnesses of their Creator’s glory: and thereupon the sciences of astronomy and mathematics were taken up by the Muslims with a fervour which in other religions was reserved for prayer alone. The Copernican system, which established the earth’s rotation around its axis and the revolution of the planet’s around the sun, was evolved in Europe at the beginning of the sixteenth century (only to be met by the fury of the ecclesiastics, who read in it a contradiction of the literal teachings of the Bible): but the foundations of this system had actually been laid six hundred years earlier, in Muslim countries - for already in the ninth and tenth centuries Muslim astronomers had reached the conclusion that the earth was globular and that it rotated around its axis, and had made accurate calculations of latitudes and longitudes; and many of them maintained - without ever being accused of hearsay - that the earth rotated around the sun. And in the same way they took to chemistry and physics and physiology, and to all the other sciences in which the Muslim genius was to find its most lasting monument. In building that monument they did no more than follow the admonition of their Prophet that ‘If anybody proceeds on his way in search of knowledge, God will make easy for him the way to Paradise’; that ‘The scientist walks in the path of God’; that ‘The superiority of the learned man over the mere pious is like the superiority of the moon when it is full over all other stars’; and that ‘The ink of the scholars is more precious that the blood of martyrs’. Throughout the whole creative period of Muslim history - that is to say, during the first five centuries after the Prophet’s time - science and learning had no greater champion than Muslim civilisation and no home more secure than the lands in which Islam was supreme.

As physicist Edward Witten once said, “String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it’s obligatory in string theory.” Ten Dimensions But as the theory began to evolve, more and more fantastic, totally unexpected features began to be revealed. For example, it was found that the theory can only exist in ten dimensions! This shocked physicists, because no one had ever seen anything like it. Usually, any theory can be expressed in any dimension you like. We simply discard these other theories because we obviously live in a three-dimensional world. (We can only move forward, sideways, and up and down. If we add time, then it takes four dimensions to locate any event in the universe. If we want to meet someone in Manhattan, for example, we might say, Let’s meet at the corner of 5th Avenue and 42nd Street, on the tenth floor, at noon. However, moving in dimensions beyond four is impossible for us, no matter how we try. In fact, our brains cannot even visualize how to move in higher dimensions. Therefore all the research done in higher-dimensional string theory is done using pure mathematics.) But in string theory, the dimensionality of space-time is fixed at ten dimensions. The theory breaks down mathematically in other dimensions. I still remember the shock that physicists felt when string theory posited that we live in a universe of ten dimensions. Most physicists saw this as proof that the theory was wrong. When John Schwarz, one of the leading architects of string theory, was in the elevator at Caltech, Richard Feynman would prod him, asking, “Well, John, and how many dimensions are you in today?” Yet over the years, physicists gradually began to show that all rival theories suffered from fatal flaws. For example, many could be ruled out because their quantum corrections were infinite or anomalous (that is, mathematically inconsistent). So over time, physicists began to warm up to the idea that perhaps our universe might be ten-dimensional after all. Finally, in 1984, John Schwarz and Michael Green showed that string theory was free of all the problems that had doomed previous candidates for a unified field theory. If string theory is correct, then the universe might have originally been ten-dimensional. But the universe was unstable and six of these dimensions somehow curled up and became too small to be observed. Hence, our universe might actually be ten-dimensional, but our atoms are too big to enter these tiny higher dimensions.

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

The textbooks of history prepared for the public schools are marked by a rather naive parochialism and chauvinism. There is no need to dwell on such futilities. But it must be admitted that even for the most conscientious historian abstention from judgments of value may offer certain difficulties. As a man and as a citizen the historian takes sides in many feuds and controversies of his age. It is not easy to combine scientific aloofness in historical studies with partisanship in mundane interests. But that can and has been achieved by outstanding historians. The historian's world view may color his work. His representation of events may be interlarded with remarks that betray his feelings and wishes and divulge his party affiliation. However, the postulate of scientific history's abstention from value judgments is not infringed by occasional remarks expressing the preferences of the historian if the general purport of the study is not affected. If the writer, speaking of an inept commander of the forces of his own nation or party, says "unfortunately" the general was not equal to his task, he has not failed in his duty as a historian. The historian is free to lament the destruction of the masterpieces of Greek art provided his regret does not influence his report of the events that brought about this destruction. The problem of Wertfreíheit must also be clearly distinguished from that of the choice of theories resorted to for the interpretation of facts. In dealing with the data available, the historian needs ali the knowledge provided by the other disciplines, by logic, mathematics, praxeology, and the natural sciences. If what these disciplines teach is insufficient or if the historian chooses an erroneous theory out of several conflicting theories held by the specialists, his effort is misled and his performance is abortive. It may be that he chose an untenable theory because he was biased and this theory best suited his party spirit. But the acceptance of a faulty doctrine may often be merely the outcome of ignorance or of the fact that it enjoys greater popularity than more correct doctrines. The main source of dissent among historians is divergence in regard to the teachings of ali the other branches of knowledge upon which they base their presentation. To a historian of earlier days who believed in witchcraft, magic, and the devil's interference with human affairs, things hàd a different aspect than they have for an agnostic historian. The neomercantilist doctrines of the balance of payments and of the dollar shortage give an image of presentday world conditions very different from that provided by an examination of the situation from the point of view of modern subjectivist economics.

Interestingly enough, creative geniuses seem to think a lot more like horses do. These people also spend a rather large amount of time engaging in that favorite equine pastime: doing nothing. In his book Fire in the Crucible: The Alchemy of Creative Genius, John Briggs gathers numerous studies illustrating how artists and inventors keep their thoughts pulsating in a field of nuance associated with the limbic system. In order to accomplish this feat against the influence of cultural conditioning, they tend to be outsiders who have trouble fitting into polite society. Many creative geniuses don’t do well in school and don’t speak until they’re older, thus increasing their awareness of nonverbal feelings, sensations, and body language cues. Einstein is a classic example. Like Kathleen Barry Ingram, he also failed his college entrance exams. As expected, these sensitive, often highly empathic people feel extremely uncomfortable around incongruent members of their own species, and tend to distance themselves from the cultural mainstream. Through their refusal to fit into a system focusing on outside authority, suppressed emotion, and secondhand thought, creative geniuses retain and enhance their ability to activate the entire brain. Information flows freely, strengthening pathways between the various brain functions. The tendency to separate thought from emotion, memory, and sensation is lessened. This gives birth to a powerful nonlinear process, a flood of sensations and images interacting with high-level thought functions and aspects of memory too complex and multifaceted to distill into words. These elements continue to influence and build on each other with increasing ferocity. Researchers emphasize that the entire process is so rapid the conscious mind barely registers that it is happening, let alone what is happening. Now a person — or a horse for that matter — can theoretically operate at this level his entire life and never receive recognition for the rich and innovative insights resulting from this process. Those called creative geniuses continuously struggle with the task of communicating their revelations to the world through the most amenable form of expression — music, visual art, poetry, mathematics. Their talent for innovation, however, stems from an ability to continually engage and process a complex, interconnected, nonlinear series of insights. Briggs also found that creative geniuses spend a large of amount of time “doing nothing,” alternating episodes of intense concentration on a project with periods of what he calls “creative indolence.” Albert Einstein once remarked that some of his greatest ideas came to him so suddenly while shaving that he was prone to cut himself with surprise.

At this point, the cautious reader might wish to read over the whole argument again, as presented above, just to make sure that I have not indulged in any 'sleight of hand'! Admittedly there is an air of the conjuring trick about the argument, but it is perfectly legitimate, and it only gains in strength the more minutely it is examined. We have found a computation Ck(k) that we know does not stop; yet the given computational procedure A is not powerful enough to ascertain that facet. This is the Godel(-Turing) theorem in the form that I require. It applies to any computational procedure A whatever for ascertaining that computations do not stop, so long as we know it to be sound. We deduce that no knowably sound set of computational rules (such as A) can ever suffice for ascertaining that computations do not stop, since there are some non-stopping computations (such as Ck(k)) that must elude these rules. Moreover, since from the knowledge of A and of its soundness, we can actually construct a computation Ck(k) that we can see does not ever stop, we deduce that A cannot be a formalization of the procedures available to mathematicians for ascertaining that computations do not stop, no matter what A is. Hence: (G) Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth. It seems to me that this conclusion is inescapable. However, many people have tried to argue against it-bringing in objections like those summarized in the queries Q1-Q20 of 2.6 and 2.10 below-and certainly many would argue against the stronger deduction that there must be something fundamentally non-computational in our thought processes. The reader may indeed wonder what on earth mathematical reasoning like this, concerning the abstract nature of computations, can have to say about the workings of the human mind. What, after all, does any of this have to do with the issue of conscious awareness? The answer is that the argument indeed says something very significant about the mental quality of understanding-in relation to the general issue of computation-and, as was argued in 1.12, the quality of understanding is something dependent upon conscious awareness. It is true that, for the most part, the foregoing reasoning has been presented as just a piece of mathematics, but there is the essential point that the algorithm A enters the argument at two quite different levels. At the one level, it is being treated as just some algorithm that has certain properties, but at the other, we attempt to regard A as being actually 'the algorithm that we ourselves use' in coming to believe that a computation will not stop. The argument is not simply about computations. It is also about how we use our conscious understanding in order to infer the validity of some mathematical claim-here the non-stopping character of Ck(k). It is the interplay between the two different levels at which the algorithm A is being considered-as a putative instance of conscious activity and as a computation itself-that allows us to arrive at a conclusion expressing a fundamental conflict between such conscious activity and mere computation.

And who knows (there is no saying with certainty), perhaps the only goal on earth to which mankind is striving Free eBooks at Planet eBook.com lies in this incessant process of attaining, in other words, in life itself, and not in the thing to be attained, which must always be expressed as a formula, as positive as twice two makes four, and such positiveness is not life, gentlemen, but is the beginning of death. Anyway, man has always been afraid of this mathematical certainty, and I am afraid of it now. Granted that man does nothing but seek that math- ematical certainty, he traverses oceans, sacri ces his life in the quest, but to succeed, really to nd it, dreads, I assure you. He feels that when he has found it there will be noth- ing for him to look for. When workmen have nished their work they do at least receive their pay, they go to the tavern, then they are taken to the police-station—and there is oc- cupation for a week. But where can man go? Anyway, one can observe a certain awkwardness about him when he has attained such objects. He loves the process of attaining, but does not quite like to have attained, and that, of course, is very absurd. In fact, man is a comical creature; there seems to be a kind of jest in it all. But yet mathematical certainty is a er all, something insu erable. Twice two makes four seems to me simply a piece of insolence. Twice two makes four is a pert coxcomb who stands with arms akimbo bar- ring your path and spitting. I admit that twice two makes four is an excellent thing, but if we are to give everything its due, twice two makes ve is sometimes a very charming thing too. And why are you so rmly, so triumphantly, convinced that only the normal and the positive—in other words, only what is conducive to welfare—is for the advantage of man? Notes from the Underground Is not reason in error as regards advantage? Does not man, perhaps, love something besides well-being? Perhaps he is just as fond of su ering? Perhaps su ering is just as great a bene t to him as well-being? Man is sometimes extraor- dinarily, passionately, in love with su ering, and that is a fact. ere is no need to appeal to universal history to prove that; only ask yourself, if you are a man and have lived at all. As far as my personal opinion is concerned, to care only for well-being seems to me positively ill-bred. Whether it’s good or bad, it is sometimes very pleasant, too, to smash things. I hold no brief for su ering nor for well-being either. I am standing for ... my caprice, and for its being guaran- teed to me when necessary. Su ering would be out of place in vaudevilles, for instance; I know that. In the ‘Palace of Crystal’ it is unthinkable; su ering means doubt, negation, and what would be the good of a ‘palace of crystal’ if there could be any doubt about it? And yet I think man will never renounce real su ering, that is, destruction and chaos. Why, su ering is the sole origin of consciousness. ough I did lay it down at the beginning that consciousness is the great- est misfortune for man, yet I know man prizes it and would not give it up for any satisfaction. Consciousness, for in- stance, is in nitely superior to twice two makes four. Once you have mathematical certainty there is nothing le to do or to understand. ere will be nothing le but to bottle up your ve senses and plunge into contemplation. While if you stick to consciousness, even though the same result is attained, you can at least og yourself at times, and that will, at any rate, liven you up. Reactionary as it is, corporal punishment is better than nothing.

Self-government" is a marriage of two terms, expressible mathematically as a ratio (self-government). The first thing we should observe about this relationship is that the first term is always static while the second is potentially infinite. The smaller the second term, which is to say, the fewer are the "others" that go to make up the apparatus of government, which within democracy is theoretically everyone, the more tolerable we find the arrangement. But as the second term approaches infinity, the more we feel our isolated "self' dissolving into insignificance. The wider the circumference of the "self-government," the smaller the share of each self in the governing of the selves which comprise it. We begin to understand that what was flattering in theory can become terrifying in practice.

Every mathematical idea presents itself to us with the character of a construction after the fact, a reconquest. Cultural constructions never have the solidity of natural objects. They are never there in the same way. Each morning, after night has intervened, we must make contact with them again. They remain impalpable; they float in the air of the village but the countryside does contain them. If, nevertheless, in the fullness of thought, the truths of culture seem to us the measure of being, and if so many philosophies posit the world upon them, it is because knowledge continues upon the thrust of perception. It is because knowledge uses the world-thesis which is its fundamental sound. We believe truth is eternal because truth expresses the perceived world and perception implies a world which was functioning before it and according to principles which it discovers and does not posit. In one and the same movement knowledge roots itself in perception and distinguishes itself from perception. Knowledge is an effort to recapture, to internalize, truly to possess a meaning that escapes perception at the very moment that it takes shape there, because it is interested only in the echo that being draws from itself, not in this resonator, its own other which makes the echo possible. Perception opens us to a world already constituted and can only reconstitute it.

So, which is it? Does matter create mind? Or does mind create matter? Scientific materialists have never made any progress whatsoever in explaining how matter produces mind. Mind can easily explain the production of matter – via autonomous Fourier immaterial frequency functions finding collective expression in a Fourier material spacetime world. It’s the collective rather than individual nature of this mental activity that makes it seem “physical”, and which gives rise to the delusion of the existence of matter. It’s all in the math.

Gödel (and indeed the whole mathematical community) failed to realise that all valid mathematical axioms must be tautological, i.e. must be shown to have a common root, of which they are equivalent expressions. Any mathematical axioms that are not tautologous automatically fall foul of Cartesian substance dualism, i.e. they imply different ontologies and epistemologies – different and incompatible versions of mathematics – hence cannot be complete and consistent with regard to each other. In other words, Gödel simply came up with an ingenious way of showing that existence must be predicated on monism, and not on dualism or pluralism.

rumors—had prepared these men for actually seeing a woman in their ranks. They almost looked silly with their eyes bulging and their jaws dropping, but I knew better than to laugh. I willed myself to pay their expressions no heed, to ignore the doughy faces of my fellow students, who were desperately trying to appear older than their eighteen years with their heavily waxed mustaches. A determination to master physics and mathematics brought me to the Polytechnic, not a desire to make friends or please others. I reminded myself of this simple fact as I steeled myself to face my instructor.

Consider Euler’s identity: eiπ + 1 = 0. Here we have the exact situation where a something – the expression on the left – is exactly equal to zero (nothing). The expression on the left is not nonexistence. It has properties, capacities, potentialities, the ability to interact with others of its kind, indeed, to contribute to the entire infinite system of mathematics, which, in the end, reduces to nothing but a set of infinite tautologies expressing 0 = 0. Since we know that eiπ + 1 = 0, we can substitute 0 for eiπ + 1, leaving 0 = 0. So, here we have something whose essence is to exist and yet be nothing. This is true of the whole of ontological mathematics, and it’s the only system of which this is true, hence it’s the only true system.

Consider Euler’s identity: eiπ + 1 = 0. Here we have the exact situation where a something – the expression on the left – is exactly equal to zero (nothing). The expression on the left is not nonexistence. It has properties, capacities, potentialities, the ability to interact with others of its kind, indeed, to contribute to the entire infinite system of mathematics, which, in the end, reduces to nothing but a set of infinite tautologies expressing 0 = 0. Since we know that eiπ + 1 = 0, we can substitute 0 for eiπ + 1, leaving 0 = 0. So, here we have something whose essence is to exist and yet be nothing. This is true of the whole of ontological mathematics, and it’s the only system of which this is true, hence it’s the only true system.

I have long exercised an honest introspection, the exquisitely painful approach to wisdom. Self−scrutiny, relentless observance of one's thoughts, is a stark and shattering experience. It pulverizes the stoutest ego. But true self−analysis mathematically operates to produce seers. The way of 'self−expression,' individual acknowledgements, results in egotists, sure of the right to their private interpretations of God and the universe. Truth humbly retires, no doubt, before such arrogant originality.

Reason is indeed all about identity, or, rather, tautology. Mathematics is the eternal, necessary system of rational, analytic tautology. Tautology is not “empty”, as it is so often characterized by philosophers. It is in fact the fullest thing there, the analytic ground of existence, and the basis of everything. Mathematical tautology has infinite masks to wear, hence delivers infinite variety. Mathematical tautology provides Leibniz’s world that is “simplest in hypothesis and the richest in phenomena.” No hypothesis cold be simpler than the one revolving around tautologies concerning “nothing.” There is something – existence – because nothing is tautologous, and “something” is how that tautology is expressed. If we write x = 0, where x is any expression that has zero as its net result, then we have a world of infinite possibilities where something (“x”) equals nothing (0).

Language teleologically seeks to express its most general form, to attain the universal. The universal language is unique. It is mathematics.

Much of Buddhist cosmology seems fantastical. The physical layout of the cosmos has a logic mostly unrelated to what we know of the universe, and the mathematical calculations consist of apparently arbitrary numbers which, however, are strangely precise. Naturally, ideas such as the centrality of the Indian experience to the cosmology and the existence of hells inside the earth do not hold true today. All the same, we do not necessarily gain by interpreting Buddhist cosmology purely in terms of geography and revealing all its deficiencies, for to a certain extent it was constructed as a symbolic representation. For example, we may infer that the authors of the cosmology depicted it symbolically from the first, given the overly schematic description of the universe and the too-artificial numbers of the worlds' dimensions. Of course those authors did not come out and say that their cosmology was symbolic. Because of the boldness of expression though, both speakers and hearers must have understood it as being so. If we comprehend it in this way, we can appreciate the sophistication of ancient Buddhist cosmology. It explained graphically, in a way that is easy to memorize, the entire picture of the world and the universe. In this sense it is of little import the the individual numbers and configurations do not match reality.

It’s not for nothing that advanced mathematics tends to be invented in hot countries. It’s because of the morphic resonance of all the camels, who have that disdainful expression and famous curled lip as a natural result of an ability to do quadratic equations.

Those who view mathematical science, not merely as a vast body of abstract and immutable truths, whose intrinsic beauty, symmetry and logical completeness, when regarded in their connexion together as a whole, entitle them to a prominent place in the interest of all profound and logical minds, but as possessing a yet deeper interest for the human race, when it is remembered that this science constitutes the language through which alone we can adequately express the great facts of the natural world, and those unceasing changes of mutual relationship which, visibly or invisibly, consciously or unconsciously to our immediate physical perceptions, are interminably going on in the agencies of the creation we live amidst...

Self-scrutiny, relentless observance of one’s thoughts, is a stark and shattering experience. It pulverises the stoutest ego. But true self-analysis mathematically operates to produce seers. The way of ‘self-expression,’ individual acknowledgments, results in egotists, sure of the right to their private interpretations of God and the universe.

Discoveries far outpaced my expectations. I learned that waiting on God is not just doing nothing. Waiting on God is faith expressed in persevering obedience while trusting God to work all things according to His perfect plan in His perfect time; or put mathematically, waiting on God is trust multiplied by time.

The Friendship Formula consists of the four basic building blocks: proximity, frequency, duration, and intensity. These four elements can be expressed using the following mathematical formula: Friendship =Proximity + Frequency + Duration + Intensity

Some people have thought I might be subtly decrying reason or exalting emotion. But I should remind you that both hemispheres are involved in reasoning and in emotion. The left hemisphere is especially good at voluntary and social expressions of emotion and one of the most clearly lateralised emotional registers is that of anger, which lateralises to the left hemisphere. Deeper and more complex expressions of emotion, and the reading of faces, are best dealt with, however, by the right hemisphere. As far as reason goes, the left hemisphere is better at carrying out certain procedures that involve manipulating numbers, but has less of a grasp than the right hemisphere of what those numbers mean. Much of mathematics is dependent on the right hemisphere: most of its great discoveries were perceived as complex patterns of relationships, and only later, often much later, translated painstakingly into linear sets of propositions. Deductive logic, it turns out, depends on the right hemisphere.

...we use images to express the boundlessness of human hope and desire--for eternal life, unconditional love, unlimited communion.... None of these articles of faith, including those that recall the story of Jesus, are provable in the way of mathematics or science. Or even of history. It is central to my own belief that Jesus was a real historical person; historical study can support my belief but it cannot prove it. The truths of faith are precisely, according to traditional Roman Catholic teaching, those that cannot be known by reason alone; that is why they are 'revealed.

your positivity ratio is your frequency of positivity over any given time span, divided by your frequency of negativity over that same time span. In mathematical terms, the ratio is captured by the simple expression P/N.

Mankind,’ however, has no aim, no idea, no plan, any more than the family of butterflies or orchids. ‘Mankind’ is a zoological expression, or an empty word. But conjure away the phantom, break the magic circle, and at once there emerges an astonishing wealth of actual forms the Living with all its immense fullness, depth and movement hitherto veiled by a catchword, a dryasdust scheme, and a set of personal ‘ideals.’ I see, in place of that empty figment of one linear history which can only be kept up by shutting one’s eyes to the overwhelming multitude of the facts, the drama of a number of mighty Cultures, each springing with primitive strength from the soil of a mother region to which it remains firmly bound throughout its whole life-cycle, each stamping its material, its mankind, in its own image; each having its own idea, its own passions, its own life, will, and feeling, its own death Here indeed are colours, lights, movements, that no intellectual eye has yet discovered. Here the Cultures, peoples, languages, truths, gods, landscapes bloom and age as the oaks and the stone-pines, the blossoms, twigs and leaves but there is no ageing ‘Mankind.’ Each Culture has its own new possibilities of self-expression which arise, ripen, decay, and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in its deepest essence different from the others, each limited in duration and self-contained, just as each species of plant has its peculiar blossom or fruit, its special type of growth and decline. These cultures, sublimated life-essences, grow with the same superb aimlessness as the flowers of the field. They belong, like the plants and the animals, to the living Nature of Goethe, and not to the dead Nature of Newton. I see world-history as a picture of endless formations and transformations, of the marvelous waxing and waning of organic forms. The professional historian, on the contrary, sees it as a sort of tapeworm industriously adding on to itself one epoch after another.

From the epistemological side, Hilbert's point of view reduces to a strict limitation to the finite; all mathematical sentences in which the infinity enters one way or the other are declared devoid of any meaning. True, with a brilliant skill, Hilbert recovers rejected mathematical theories in the form of a formal consistent game of symbols. Yet, this way out, giving no explanation of what sustained mathematics to date, of why, while expressing judgments about infinity that have no meaning, mathematicians understood each other, is dictated only by inability to find a more satisfactory way out.

the idea that mathematics expresses the essential reality of nature was first put explicitly, in modern times, by scientists, such as Sir James Jeans and Werner Heisenberg, but within a few decades, these ideas were being transmitted almost subliminally. As a result, after passing through graduate school, most physicists have come to regard this attitude toward mathematics as being perfectly natural. However, in earlier generations such views would have been regarded as strange and perhaps even a little crazy—at all events irrelevant to a proper scientific view of reality.

What drove me? I think most creative people want to express appreciation for being able to take advantage of the work that's been done by others before us. I didn't invent the language or mathematics I use. I make little of my own food, none of my own clothes. Everything I do depends on other members of our species and the shoulders that we stand on. And a lot of us want to contribute something back to our species and to add something to the flow. It's about trying to express something in the only way that most of us know how—because we can't write Bob Dylan songs or Tom Stoppard plays. We try to use the talents and we do have to express our deep feelings, to show our appreciation of all the contributions that came before us, and to add something to that flow. That's what has driven me.

Mathiness: British economic journalist John Kay defines mathiness as a “use of algebraic symbols and quantitative data to give an appearance of scientific content to ideological preconceptions.” Expressing an idea in mathematical symbols instead of straightforward literary terms helps legitimize it in the minds of many people, thanks to a seeming similarity with natural science. In this respect math is basically a form of numerical rhetoric. “The American economist Paul Romer has recently written of ‘mathiness,’ by analogy with ‘truthiness,’ a term coined by American talk show host Stephen Colbert. Truthiness presents narratives which are not actually true, but consistent with the world view of the person who spins the story. It is exemplified in rightwing fabrications about European health systems – their death panels and forced euthanasia.” Paul Samuelson, for instance, trivialized economics in terms that give the outward appearance of science by being expressed mathematically, even when its assumptions are purely hypothetical (and not all realistic)and there are no quantitative statistics to illustrate its categories.

[She Stoops to Conquer] seems to be a comedy of pure plot; yet there is also psychological acuity under the mathematical ingenuity. This is seen most clearly in the character of Marlow, who is painted as a palpable victim of the English class system. His dilemma is that he is a tongue-tied wreck amongst women of his own class but brimming with sexual bravura with a barmaid or college bedmaker. He himself expresses his dilemma with painful clarity: MARLOW: My life has been chiefly spent in a college or an inn, in seclusion from that lovely part of the creation that chiefly teach men confidence. I don't know that I was ever familiarly acquainted with a single modest woman except my mother. But among females of another class, you know - HASTINGS: Ay, among them you are impudent enough of all conscience. Marlow himself rightly calls this 'the English malady': a paralysing fear, resulting from a monastic education, of women of his own class and an ability to be at ease only with social inferiors whom he can bully, dominate or treat as purchasable commodities. It took an observant Irishman to pin down the damage done to the English male psyche by a punitive educational system. [...] And there is further evidence of Marlow's split personality when Kate accosts him in the guise of a household drudge. Marlow the psychological wreck turns into a brazen lech who, within seconds, is asking to taste the nectar of Kate's lips. Not only that. He is soon bragging of his sexual exploits at a louche London club attended by the likes of Mrs Mantrap, Lady Betty Blackleg, the Countess of Sligo, Mrs Langhorns and old Miss Biddy Buckskin.

A mathematical proposition expresses a certain expectation. For example, the proposition, “Euler constant C is rational” expresses the expectation that we could find two integers a and b such that C = a/b. Perhaps, the word “intention”, coined by the phenomenologists, expresses even better what is meant here.

According to Kolmogorov, Hilbert was seriously worried by what would happen to the Mathematische Annalen cover in 500 years: he thought that the names of the former Editors would fill up all the space. Kolmogorov in return expressed to Hilbert his own worries that our culture would probably not survive for such a long period: the united bureaucrats of all countries would soon be able to stop all kind of creativity, making further mathematical discoveries impossible, as are geographical discoveries today. In our time, we can imagine that some of the most appealing domains of mathematics will be transformed into wilderness preserves, where rich people will be able to buy for an expensive price the pleasure to hunt one or two theorems, guided by scientific jaegermeisters.

The true geometrician makes this selection judiciously, because he is guided by a sure instinct, or by some vague consciousness of I know not what profounder and more hidden geometry, which alone gives a value to the constructed edifice. To seek the origin of this instinct, and to study the laws of this profound geometry which can be felt but not expressed, would be a noble task for the philosophers who will not allow that logic is all. But this is not the point of view I wish to take, and this is not the way I wish to state the question. This instinct I have been speaking of is necessary to the discoverer, but it seems at first as if we could do without it for the study of the science once created. Well, what I want to find out is, whether it is true that once the principles of logic are admitted we can, I will not say discover, but demonstrate all mathematical truths without making a fresh appeal to intuition.

It has been contended, by Rudolf Carnap and others, that since we are unable to find in application an absolute standard by which the validity of a formal system may be tested we are free to choose what formalisation of mathematics we please, technical considerations alone leading us to prefer one system to another. If we accept this standpoint then the distinction between constructive and non- constructive systems is a distinction without a difference and the constructive system becomes little more than a poor relation of the non-constructive. I consider this view to be wholly mistaken. Even if we leave out of account the question of demonstrable freedom from contradiction, the Principia [Mathematica of Whitehead and Russell] and the Grundlagen[der Mathematik of Hilbert and Bernays] must be rejected as formalisations of mathematics for their failure to express adequately the concepts of universality and existence. Even though we do not discover a contradiction in a formal system by showing that the existential quantifier fails to express the notion of existence, for we have no right to pre-judge the meaning of the signs of the system—and to this extent Carnap is right—none-the-less when a mathematician seeks to establish the existence of a number with a certain property he will not, and should not, be satisfied to find that all he has proved is a formula in some formal system, which whatever it may affirm assuredly does not say that a number exists with the desired property.

It has been contended, by Rudolf Carnap and others, that since we are unable to find in application an absolute standard by which the validity of a formal system may be tested we are free to choose what formalisation of mathematics we please, technical considerations alone leading us to prefer one system to another. If we accept this standpoint then the distinction between constructive and non- constructive systems is a distinction without a difference and the constructive system becomes little more than a poor relation of the non-constructive. I consider this view to be wholly mistaken. Even if we leave out of account the question of demonstrable freedom from contradiction, the Principia [Mathematica of Whitehead and Russell] and the Grundlagen [der Mathematik of Hilbert and Bernays] must be rejected as formalisations of mathematics for their failure to express adequately the concepts of universality and existence. Even though we do not discover a contradiction in a formal system by showing that the existential quantifier fails to express the notion of existence, for we have no right to pre-judge the meaning of the signs of the system—and to this extent Carnap is right—nonetheless when a mathematician seeks to establish the existence of a number with a certain property he will not, and should not, be satisfied to find that all he has proved is a formula in some formal system, which whatever it may affirm assuredly does not say that a number exists with the desired property.

I see, in place of that empty figment of one linear history which can only be kept up by shutting one’s eyes to the overwhelming multitude of the facts, the drama of a number of mighty Cultures, each springing with primitive strength from the soil of a mother-region to which it remains firmly bound throughout its whole life-cycle; each stamping its material, its mankind, in its own image; each having its own idea, its own passions, its own life, will and feeling, its own death. Here indeed are colours, lights, movements, that no intellectual eye has yet discovered. Here the Cultures, peoples, languages, truths, gods, landscapes bloom and age as the oaks and the stone-pines, the blossoms, twigs and leaves—but there is no ageing “Mankind.” Each Culture has its own new possibilities of self-expression which arise, ripen, decay, and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in its deepest essence different from the others, each limited in duration and self-contained, just as each species of plant has its peculiar blossom or fruit, its special type of growth and decline.

For a long time I took a purely theological standpoint on the issue, which is actually so fundamental that it can be used as a springboard for any debate – if environment is the operative factor, for example, if man at the outset is both equal and shapeable and the good man can be shaped by engineering his surroundings, hence my parents’ generation’s belief in the state, the education system and politics, hence their desire to reject everything that had been and hence their new truth, which is not found within man’s inner being, in his detached uniqueness, but on the contrary in areas external to his intrinsic self, in the universal and collective, perhaps expressed in its clearest form by Dag Solstad, who has always been the chronicler of his age, in a text from 1969 containing his famous statement “We won’t give the coffee pot wings”: out with spirituality, out with feeling, in with the new materialism, but it never struck them that the same attitude could lie behind the demolition of old parts of town to make way for roads and parking lots, which naturally the intellectual Left opposed, and perhaps it has not been possible to be aware of this until now when the link between the idea of equality and capitalism, the welfare state and liberalism, Marxist materialism and the consumer society is obvious because the biggest equality creator of all is money, it levels all differences, and if your character and your fate are entities that can be shaped, money is the most natural shaper, and this gives rise to the fascinating phenomena whereby crowds of people assert their individuality and originality by shopping in an identical way while those who ushered all this in with their affirmation of equality, their emphasis on material values and belief in change, are now inveighing against their own handiwork, which they believed the enemy created, but like all simple reasoning this is not wholly true either, life is not a mathematical quantity, it has no theory, only practice, and though it is tempting to understand a generation’s radical rethink of society as being based on its view of the relationship between heredity and environment, this temptation is literary and consists more in the pleasure of speculating, that is, of weaving one’s thoughts through the most disparate areas of human activity, than in the pleasure of proclaiming the truth.

The theorem: ‘if a triangle is isosceles it is an acute triangle’ is expressed as a logical theorem: the predicate ‘isosceles’ in the case of triangles is considered to imply the predicate ‘acute’, i.e. one imagines all the triangles of a given plane represented by the points of an R6 and one then sees that the domain of R6 representing isosceles triangles is contained in the domain representing all acute triangles. This is in fact true, and logical formulation and logical language can therefore safely be applied. But the thoughts of the mathematician, who because of the poverty of his language formulated this theorem as a logical theorem, proceed in a way quite different from the above interpretation. He imagines that he is going to construct an isosceles triangle, and then finds that either at the end of the construction all angles appear to be acute or that on the postulation of a right or obtuse angle the construction cannot be executed. In other words, he thinks the construction mathematically, not in its logical interpretation.

Most cliches have a respectable past: if you only listen with a fresh ear, they express an imaginative thought or a clever analogy.

There is a fundamental conflict between intellectual and political life... The essence of the intellectual life is the impulse to analyze everything indiscriminately, to understand the explanations for everything. The essence of political success is the contrary habit. Whatever one's internal understanding of the world, one must train oneself never to express those conclusions needlessly, for they may offend. Truth must be offered up by teaspoons, always with a purpose.

throughout the whole of the Middle Ages Islamic culture ranked far above Christian culture. Of the three great areas of culture which inherited the Roman-Hellenic culture, the Roman-German, the Greek-Slav, and the Egyptian-Syrian, Arab culture, the latter took over the whole of the knowledge of antiquity in the fields of mathematics, astronomy, chemistry, mechanics and medicine; it was not Rome and not Constantinople but Alexandria which was the centre of science in the Roman Empire. Now the religious expression of the Germanic-Roman sphere of culture was the Roman church, and that of the Greek-Slav sphere was the Greek Church, but that of the Arab Egyptian-Syrian sphere was Islam.

Each culture has its own new possibilities of self expression which arise, ripen, decay and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in its deepest essence different from the other, each limited in duration and self contained…Western European physics let no-one deceive himself has reached the limit of its possibilities. This is the origin of the sudden and annihilating doubt that has arisen about things that even yesterday were the unchallenged foundation of physical theory, about the meaning of the energy principle, the concepts of mass, space, absolute time, and causal laws generally.

Mathematical answers are not always expressed numerically. How does one calculate the worth of humanity, or of a single human life?

In terms of mathematical information theory, each great painting is important because it contains information. In this theory, information is, roughly, that which you haven't encountered before. A painting you have encountered often contains new information again if you suddenly see it in a new way. Norbert Wiener, one of the inventors of information theory, expressed this thought by saying great poetry contains more information than political speeches. In a great poem, as in a great painting, you encounter a new and different emic reality, a new way of perceiving/experiencing humanity-in-universe. A political speech, typically, merely regurgitates old reality-tunnels. Great art, in terms of this metaphor, is merely the opposite of cliché — it takes us to a new window, instead of gazing through a habitual window. That is why the greatest art, notoriously, is always denounced as "bizarre" and "barbarous" when it first appears. The best books are called "unreadable" because people at first do not know how to read them.

In order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world — to show, that is, in how far culture in the "become" state can express or portray an idea of human existence — I have chosen number, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind.

In order to exemplify the way in which a soul seeks to actualize itself in the picture of its outer world — to show, that is, in how far culture in the "become" state can express or portray an idea of human existence — I have chosen number, the primary element on which all mathematics rests. I have done so because mathematics, accessible in its full depth only to the very few, holds a quite peculiar position amongst the creations of the mind. [...] Every philosophy has hitherto grown up in conjunction with a mathematic belonging to it. Number is the symbol of causal necessity. Like the conception of God, it contains the ultimate meaning of the world-as-nature. [...] But the actual number with which the mathematician works, the figure, formula, sign, diagram, in short the number-sign which he thinks, speaks or writes exactly, is (like the exactly-used word) from the first a symbol of these depths, something imaginable, communicable, comprehensible to the inner and the outer eye, which can be accepted as representing the demarcation. The origin of numbers resembles that of the myth. Primitive man elevates indefinable nature-impressions (the "alien," in our terminology) into deities, numina, at the same time capturing and impounding them by a name which limits them. [...] Nature is the numerable, while History, on the other hand, is the aggregate of that which has no relation to mathematics hence the mathematical certainty of the laws of Nature, the astounding Tightness of Galileo's saying that Nature is "written in mathematical language," and the fact, emphasized by Kant, that exact natural science reaches just as far as the possibilities of applied mathematics allow it to reach.

The primary concern of mathematics is number, and this means the positive integers. ... In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

When Poincaré said that there are no solved problems, there are only problems which are more or less solved, he was implying that any question formulated in a yes/no fashion is an expression of narrow-mindedness.

The Brouwerian believed that this conception was wholly wrong from the beginning. They accused it of misunderstanding the nature of mathematics and of unjustifiedly transferring to the realm of infinity methods of reasoning that are valid only in the realm of the finite. By regaining the right perspective, mathematics could be constructed on a basis whose intuitive soundness could not be doubted. The antinomies were only the symptoms of a disease by which mathematics was infected. Once this disease was cured, one need worry no longer about the symptoms. All Russellians thought that our naiveness consisted in taking for granted that every grammatically correct indicative sentence expresses something which either is or is not the case, and some — among them Russell himself — believed, in addition, that through some carelessness a certain type of viciously circular concept formation had been allowed to enter logico-mathematical thinking. By restricting the language — and proscribing the dangerous types of concept formation— the known antinomies could be made to disappear. Their faith in the consistency of the resulting, somewhat mutilated, systems was less strong than that of the Brouwerians, since certain intuitively not too well founded devices had to be used in order to restore at least part of the lost strength and maneuverability. Zermelians, finally, thought that our blunder consisted in naively assuming that to every condition there must correspond a certain entity, namely the set of all those objects that satisfy this condition. By suitable restriction of the axiom of comprehension, in which this assumption is formulated, they tried to construct systems which were free of the known antinomies yet strong enough to allow for the reconstruction of a sufficient part of classical mathematics.

Statements like ‘a quantity x has a completely definite value’ (expressed by a real number and represented by a point in the mathematical continuum) seem to me to have no physical meaning.

Google’s semantic search is the first step toward a search engine that is like Star Trek’s onboard computer and the brains of that semantic search is what Google calls the Knowledge Graph. The word “Graph” here has been taken from mathematics but in this context it was coined by Facebook’s founder and CEO, Mark Zuckerberg, who used it to express the social network of relationships within Facebook’s digital boundaries. He called it the Social Graph.

Most of us were required to take three or four years of coursework in high school, starting with algebra and working up the chain: geometry, algebra 2, trigonometry, precalculus, calculus. Lockhart writes, “If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done—I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

We saw in the last chapter that for notes played on conventional instruments, where partials occur at integer multiples of the fundamental frequency, intervals corresponding to frequency ratios expressable as a ratio of small integers are favoured as consonant.

Accept the abundant life in your own mind. Your mental acceptance and expectancy of wealth has its own mathematics and mechanics of expression. As you enter into the mood of opulence, all things necessary for the abundant life will come to pass.

Your mental acceptance and expectancy of wealth has its own mathematics and mechanics of expression. As you enter into the mood of opulence, all things necessary for the abundant life will come to pass.

We call a set [i.e., species] denumerably unfinished if it has the following properties: we can never construct in a well-defined way more than a denumerable subset of it, but when we have constructed such a subset, we can immediately deduce from it, following some previously defined mathematical process, new elements which are counted to the original set. But from a strictly mathematical point of view this set does not exist as a whole, nor does its power exist; however we can introduce these words here as an expression for a known intention.