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.

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.

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.}

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.

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)

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.

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

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.

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 …

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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!

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?

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.

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.

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

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".

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.

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.

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.

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.

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.

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.

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.

(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.

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?

“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.

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.

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.

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.

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.

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.

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.)

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

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?

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.

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

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?

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.

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.}

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 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.

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.

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.

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.

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?)

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.

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.

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.

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.

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.

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.

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.

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).