Mathematics Related Quotes

Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.

Since the mathematicians have invaded the theory of relativity I do not understand it myself any more.

Whereas I think: I’m lying here in a haystack... The tiny space I occupy is so infinitesimal in comparison with the rest of space, which I don’t occupy and which has no relation to me. And the period of time in which I’m fated to live is so insignificant beside the eternity in which I haven’t existed and won’t exist... And yet in this atom, this mathematical point, blood is circulating, a brain is working, desiring something... What chaos! What a farce!

A relativist is an individual who doesn't know the difference between an adjective and an adverb.

What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant.

Einstein's relativity work is a magnificent mathematical garb which fascinates, dazzles and makes people blind to the underlying errors. The theory is like a beggar clothed in purple whom ignorant people take for a king... its exponents are brilliant men but they are metaphysicists rather than scientists.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain esthetic criteria - that is, in relation to how much it describes, it must be rather simple.

What I'm thinking is: here I am, lying under a haystack ... The tiny little place I occupy is so small in relation to the rest of space where I am not and where it's none of my business; and the amount of time which I'll succeed in living is so insignificant by comparison with the eternity where I haven't been and never will be ... And yet in this atom, in this mathematical point, the blood circulates, the brain works and even desires something as well .. What sheer ugliness! What sheer nonsense!

A mathematician says that an electromagnetic wave travels from Andromeda to your eye and that it also extends from Andromeda to your eye.

Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so.

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

In the early universe—when the universe was small enough to be governed by both general relativity and quantum theory—there were effectively four dimensions of space and none of time. That means that when we speak of the “beginning” of the universe, we are skirting the subtle issue that as we look backward toward the very early universe, time as we know it does not exist! We must accept that our usual ideas of space and time do not apply to the very early universe. That is beyond our experience, but not beyond our imagination, or our mathematics.

Each pleasure we feel is a pleasure less; each day a stroke on a calendar. What we will not accept is that the joy in the day and the passing of the day are inseparable. What makes our existence worthwhile is precisely that its worth and its while - its quality and duration - are as impossible to unravel as time and space in mathematics of relativity.

If you cannot solve the proposed problem...try to solve first some related problem.

Whatever— the soup is getting cold. [Last sentence of a mathematical theorem in Leonardo da Vinci’s notebook, 1518]

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 generally recognized that women are better than men at languages, personal relations and multitasking, but less good at map-reading and spatial awareness. It is therefore not unreasonable to suppose that women might be less good at mathematics and physics. It is not politically correct to say such things....But it cannot be denied that there are differences between men and women. Of course, these are differences between the averages only. There are wide variations about the mean.

It is the definition of the word 'object' which destroys all religions.

A mathematician tells you that the wall of warped space prevents the Moon from flying out of its orbit yet can't tell you why an astronaut can go back and forth across that same space.

Saint Bartleby's School for Young Gentlemen Annual Report Student: Artemis Fowl II Year: First Fees: Paid Tutor: Dr Po Language Arts As far as I can tell, Artemis has made absolutely no progress since the beginning of the year. This is because his abilities are beyond the scope of my experience. He memorizes and understands Shakespeare after a single reading. He finds mistakes in every exercise I administer, and has taken to chuckling gently when I attempt to explain some of the more complex texts. Next year I intend to grant his request and give him a library pass during my class. Mathematics Artemis is an infuriating boy. One day he answers all my questions correctly, and the next every answer is wrong. He calls this an example of the chaos theory, and says that he is only trying to prepare me for the real world. He says the notion of infinity is ridiculous. Frankly, I am not trained to deal with a boy like Artemis. Most of my pupils have trouble counting without the aid of their fingers. I am sorry to say, there is nothing I can teach Artemis about mathematics, but someone should teach him some manners. Social Studies Artemis distrusts all history texts, because he says history was written by the victors. He prefers living history, where survivors of certain events can actually be interviewed. Obviously this makes studying the Middle Ages somewhat difficult. Artemis has asked for permission to build a time machine next year during double periods so that the entire class may view Medieval Ireland for ourselves. I have granted his wish and would not be at all surprised if he succeeded in his goal. Science Artemis does not see himself as a student, rather as a foil for the theories of science. He insists that the periodic table is a few elements short and that the theory of relativity is all very well on paper but would not hold up in the real world, because space will disintegrate before lime. I made the mistake of arguing once, and young Artemis reduced me to near tears in seconds. Artemis has asked for permission to conduct failure analysis tests on the school next term. I must grant his request, as I fear there is nothing he can learn from me. Social & Personal Development Artemis is quite perceptive and extremely intellectual. He can answer the questions on any psychological profile perfectly, but this is only because he knows the perfect answer. I fear that Artemis feels that the other boys are too childish. He refuses to socialize, preferring to work on his various projects during free periods. The more he works alone, the more isolated he becomes, and if he does not change his habits soon, he may isolate himself completely from anyone wishing to be his friend, and, ultimately, his family. Must try harder.

The importance of C.F. Gauss for the development of modern physical theory and especially for the mathematical fundament of the theory of relativity is overwhelming indeed; also his achievement of the system of absolute measurement in the field of electromagnetism. In my opinion it is impossible to achieve a coherent objective picture of the world on the basis of concepts which are taken more or less from inner psychological experience.

At the federal level, this problem could be greatly alleviated by abolishing the Electoral College system. It's the winner-take-all mathematics from state to state that delivers so much power to a relative handful of voters. It's as if in politics, as in economics, we have a privileged 1 percent. And the money from the financial 1 percent underwrites the microtargeting to secure the votes of the political 1 percent. Without the Electoral College, by contrast, every vote would be worth exactly the same. That would be a step toward democracy.

Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things. In my opinion the answer to this question is, briefly, this:--As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

Fundamental ideas play the most essential role in forming a physical theory. Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulae, are the beginning of every physical theory.

In a civilization devoted to the strictly abstract and mathematical ideal of making the most money in the least time, the only sure method of success is to cheat the customer, to sell various kinds of nothingness in pretentious packages.

A mathematician is an individual who proves his beliefs with equations.

For example, knowing that it takes only about eleven and a half days for a million seconds to tick away, whereas almost thirty-two years are required for a billion seconds to pass, gives one a better grasp of the relative magnitudes of these two common numbers.

I liked numbers because they were solid, invariant; they stood unmoved in a chaotic world. There was in numbers and their relation something absolute, certain, not to be questioned, beyond doubt.

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

The philosophers make still another objection: "What you gain in rigour," they say, "you lose in objectivity. You can rise toward your logical ideal only by cutting the bonds which attach you to reality. Your science is infallible, but it can only remain so by imprisoning itself in an ivory tower and renouncing all relation with the external world. From this seclusion it must go out when it would attempt the slightest application.

Science has nothing in common with religion. Facts and miracles never did, and never will agree. They are not in the least related. They are deadly foes. What has religion to do with facts? Nothing. Can there be Methodist mathematics, Catholic astronomy, Presbyterian geology, Baptist biology, or Episcopal botany?

Mathematical science shows what is. It is the language of unseen relations between things. —Ada Lovelace

All of our ideas in physics require a certain amount of common sense in their application; they are not purely mathematical or abstract ideas.

When the noise died down, she could lose herself in mathematics—in attempts to combine quantum mechanics with the theory of relativity—and forget the world around her.

I think a strong claim can be made that the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well constructed theory is in some respects undoubtedly an artistic production. A fine example is the famous Kinetic Theory of Maxwell. ... The theory of relativity by Einstein, quite apart from any question of its validity, cannot but be regarded as a magnificent work of art.

The assumption that numbers and mathematical or logical laws are mental is due to the even more widespread notion that only particular sensible entities exist in nature, and that relations abstractions, or universals cannot have any such objective existence - hence they are given a shadowy existence in the mind.

The habit of looking at life as a social relation — an affair of society — did no good. It cultivated a weakness which needed no cultivation. If it had helped to make men of the world, or give the manners and instincts of any profession — such as temper, patience, courtesy, or a faculty of profiting by the social defects of opponents — it would have been education better worth having than mathematics or languages; but so far as it helped to make anything, it helped only to make the college standard permanent through life.

If memory within nature sounds mysterious, we should bear in mind that mathematical laws transcending nature are more rather than less so; they are metaphysical rather than physical. The way mathematical laws can exist independently of the evolving universe and at the same time act upon it remains a profound mystery. For those who accept God, this mystery is an aspect of God's relation to the realm of nature; for those who deny God, the mystery is even more obscure: A quasi-mental realm of mathematical laws somehow exists independently of nature, yet not in God, and governs the evolving physical world without itself being physical.

If Einstein had upended our everyday notions about the physical world with his theory of relativity, the younger man, Kurt Gödel, had had a similarly subversive effect on our understanding of the abstract world of mathematics.

The forms of mathematics, the harmonies of music, the motions of the planets, and the gods of the mysteries were all essentially related for Pythagoreans, and the meaning of that relation was revealed in an education that culminated in the human soul’s assimilation to the world soul, and thence to the divine creative mind of the universe.

When Charles Darwin was trying to decide whether he should propose to his cousin Emma Wedgwood, he got out a pencil and paper and weighed every possible consequence. In favor of marriage he listed children, companionship, and the 'charms of music and female chit-chat.' Against marriage he listed the 'terrible loss of time,' lack of freedom to go where he wished, the burden of visiting relatives, the expense and anxiety provoked by children, the concern that 'perhaps my wife won't like London,' and having less money to spend on books. Weighing one column against the other produced a narrow margin of victory, and at the bottom Darwin scrawled, 'Marry—Marry—Marry Q.E.D.' Quod erat demonstrandum, the mathematical sign-off that Darwin himself restated in English: 'It being proved necessary to Marry.

Really neat that human beings conquered the Earth invented poetry and mathematics and the combustion engine, discovered that time and space are relative, built machines big and small to ferry us to the moon for some rocks or carry us to McDonald's for a strawberry-banana smoothie. Very cool we split the atom and bestowed upon the Earth the Internet and smartphones and, of course, the selfie stick. But the most wonderful thing of all, our highest achievement and the one thing for which I pray we will always be remembered, is stuffing wads of polyester into an anatomically incorrect, cartoonish ideal of one of nature's most fearsome predators for no other reason than to soothe a child.

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 don’t know,” he says. “I guess so. In space, everything has its relative position. Space is an entity, right, but also limitless. It’s less dense the farther out you go, but you can always keep going. There’s no definitive border between the start and the end. We’ll never fully understand or know it. We can’t.” “You don’t think?” “Dark matter makes up the majority of all matter, and it’s still a mystery.” “Dark matter?” “It’s invisible. It’s all the extra mass we can’t see that makes the formation of galaxies and the rotational velocities of stars around galaxies mathematically possible.” “I’m glad we don’t know everything.

There is no reason, therefore, so far as I am able to perceive, to deny the ultimate and absolute philosophical validity of a theory of geometry which regards space as composed of points, and not as a mere assemblage of relations between non-spatial terms.

The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholely ‘useless’ (and this is true of ‘applied’ as of ‘pure’ mathematics). It is not possible to justify the life of any genuine professional mathematician on the ground of the ‘utility’ of his work.… The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. It

Well, suppose we use our brains. We see things solid. Solidities are important to us in nature. In solidities, there are measures that greatly affect us. There are rhythms in the ins and outs of form. Music, the forest and to many the most impressive of arts deals in measures which seem to go in every direction. They combine, they move together, they deflect and they oppose. Music is a structure of highly mathematical measures. According to the selection and relative value of these measures the music is great or small in its effect on us.

Another time somebody gave a talk about poetry. He talked about the structure of the poem and the emotions that come with it; he divided everything up into certain kinds of classes. In the discussion that came afterwards, he said, “Isn’t that the same as in mathematics, Dr. Eisenhart?” Dr. Eisenhart was the dean of the graduate school and a great professor of mathematics. He was also very clever. He said, “I’d like to know what Dick Feynman thinks about it in reference to theoretical physics.” He was always putting me on in this kind of situation. I got up and said, “Yes, it’s very closely related. In theoretical physics, the analog of the word is the mathematical formula, the analog of the structure of the poem is the interrelationship of the theoretical bling-bling with the so-andso”–and I went through the whole thing, making a perfect analogy. The speaker’s eyes were _beaming_ with happiness. Then I said, “It seems to me that no matter _what_ you say about poetry, I could find a way of making up an analog with _any_ subject, just as I did for theoretical physics. I don’t consider such analogs meaningful.

The oldest problem in economic education is how to exclude the incompetent. A certain glib mastery of verbiage-the ability to speak portentously and sententiously about the relation of money supply to the price level-is easy for the unlearned and may even be aided by a mildly enfeebled intellect. The requirement that there be ability to master difficult models, including ones for which mathematical competence is required, is a highly useful screening device.

Investors look at economic fundamentals; traders look at each other; ‘quants’ look at the data. Dealing on the basis of historic price series was once described as technical analysis, or chartism (and there are chartists still). These savants identify visual patterns in charts of price data, often favouring them with arresting names such as ‘head and shoulders’ or ‘double bottoms’. This is pseudo-scientific bunk, the financial equivalent of astrology. But more sophisticated quantitative methods have since proved profitable for some since the 1970s’ creation of derivative markets and the related mathematics. Profitable

t is generally recognized that women are better than men at languages, personal relations and multitasking, but less good at map-reading and spatial awareness. It is therefore not unreasonable to suppose that women might be less good at mathematics and physics. It is not politically correct to say such things....But it cannot be denied that there are differences between men and women. Of course, these are differences between the averages only. There are wide variations about the mean.

Einstein, twenty-six years old, only three years away from crude privation, still a patent examiner, published in the Annalen der Physik in 1905 five papers on entirely different subjects. Three of them were among the greatest in the history of physics. One, very simple, gave the quantum explanation of the photoelectric effect—it was this work for which, sixteen years later, he was awarded the Nobel prize. Another dealt with the phenomenon of Brownian motion, the apparently erratic movement of tiny particles suspended in a liquid: Einstein showed that these movements satisfied a clear statistical law. This was like a conjuring trick, easy when explained: before it, decent scientists could still doubt the concrete existence of atoms and molecules: this paper was as near to a direct proof of their concreteness as a theoretician could give. The third paper was the special theory of relativity, which quietly amalgamated space, time, and matter into one fundamental unity. This last paper contains no references and quotes to authority. All of them are written in a style unlike any other theoretical physicist's. They contain very little mathematics. There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.

It was a hundred years later that Einstein gave a theory (general relativity) which said that the geometry of the universe is determined by its content of matter, so that no one geometry is intrinsic to space itself. Thus to the question, "Which geometry is true?" nature gives an ambiguous answer not only in mathematics, but also in physics

Under another aspect the same is true for music. If any art is devoid of lessons, it is certainly music. It is too closely related to mathematics not to have borrowed their gratuitousness. That game the mind plays with itself according to set and measured laws takes place in the sonorous compass that belongs to us and beyond which the vibrations nevertheless meet in an inhuman universe. There is no purer sensation. These examples are too easy. The absurd man recognizes as his own these harmonies and these forms.

The peculiarity of the evidence of mathematical truths is, that all the argument is on one side. There are no objections, and no answers to objections. But on every subject on which difference of opinion is possible, the truth depends on a balance to be struck between two sets of confliting reasons. Even in natural philosophy, there is always some other explanation possible of the same facts; some geocentric theory instead of heliocentric, some phlogiston instead of oxygen; and it has to be shown why that other theory cannot be the true on: and until this is shown, and until we know how it is shown, we do not understand the grounds of our opinion. But when we turn to subjects infinitely more complicated, to morals, religion, politics, social relations, and the business of life, three-fourths of the arguments for every disputed opinion consist in dispelling the appearances which favour some opinion different from it.

Always preoccupied with his profound researches, the great Newton showed in the ordinary-affairs of life an absence of mind which has become proverbial. It is related that one day, wishing to find the number of seconds necessary for the boiling of an egg, he perceived, after waiting a minute, that he held the egg in his hand, and had placed his seconds watch (an instrument of great value on account of its mathematical precision) to boil! This absence of mind reminds one of the mathematician Ampere, who one day, as he was going to his course of lectures, noticed a little pebble on the road; he picked it up, and examined with admiration the mottled veins. All at once the lecture which he ought to be attending to returned to his mind; he drew out his watch; perceiving that the hour approached, he hastily doubled his pace, carefully placed the pebble in his pocket, and threw his watch over the parapet of the Pont des Arts.

The Golden Proportion, sometimes called the Divine Proportion, has come down to us from the beginning of creation. The harmony of this ancient proportion, built into the very structure of creation, can be unlocked with the 'key' ... 528, opening to us its marvelous beauty. Plato called it the most binding of all mathematical relations, and the key to the physics of the cosmos.

The difference between Lorentz's Transformation in Lorentz's theory and Lorentz's Transformation in Einstein's Special Relativity is not mathematical but ontological and epistemological and, being so, it was to be expected the emergence of historians, scientists, and philosophers that, not having understood in depth the philosophical content and transcendence of the theory, would minimize Einstein's contribution.

Evidence in support of general relativity came quickly. Astronomers had long known that Mercury’s orbital motion around the sun deviated slightly from what Newton’s mathematics predicted. In 1915, Einstein used his new equations to recalculate Mercury’s trajectory and was able to explain the discrepancy, a realization he later described to his colleague Adrian Fokker as so thrilling that for some hours it gave him heart palpitations.

The Night Dances       A smile fell in the grass. Irretrievable!   And how will your night dances Lose themselves. In mathematics?   Such pure leaps and spirals—— Surely they travel   The world forever, I shall not entirely Sit emptied of beauties, the gift   Of your small breath, the drenched grass Smell of your sleeps, lilies, lilies.   Their flesh bears no relation. Cold folds of ego, the calla,   And the tiger, embellishing itself——

the relation of mathematics to the world of temporal change and of phenomenal particularity is direct: less by induction than by what Pierce called abduction – an imaginative jumping off from an open-ended series of particulars.

For an idea to have survived so long across so many cycles is indicative of its relative fitness. Noise, at least some noise, was filtered out. Mathematically, progress means that some new information is better than past information, not that the average of new information will supplant past information, which means that it is optimal for someone, when in doubt, to systematically reject the new idea, information, or method. Clearly and shockingly, always. Why?

The aborted research project wasn’t important in and of itself. What mattered was the instruction that Ye Wenjie had given him, so that’s where Luo Ji’s mind was stuck. Over and over again he recalled her words: Suppose a vast number of civilizations are distributed throughout the universe, on the order of the number of detectable stars. Lots and lots of them. The mathematical structure of cosmic sociology is far clearer than that of human sociology. The factors of chaos and randomness in the complex makeups of every civilized society in the universe get filtered out by the immense distance, so those civilizations can act as reference points that are relatively easy to manipulate mathematically. First: Survival is the primary need of civilization. Second: Civilization continuously grows and expands, but the total matter in the universe remains constant. One more thing: To derive a basic picture of cosmic sociology from these two axioms, you need two other important concepts: chains of suspicion and the technological explosion. I’m afraid there won’t be that opportunity.… Well, you might as well just forget I said anything. Either way, I’ve fulfilled my duty. He

The notion that a term can be modified arises from neglect to observe the eternal self-identity of all terms and all logical concepts, which alone form the constituents of propositions.* What is called modification consists merely in having at one time, but not at another, some specific relation to some other specific term; but the term which sometimes has and sometimes has not the relation in question must be unchanged, otherwise it would not be that term which had ceased to have the relation.

This book is an essay in what is derogatorily called "literary economics," as opposed to mathematical economics, econometrics, or (embracing them both) the "new economic history." A man does what he can, and in the more elegant - one is tempted to say "fancier" - techniques I am, as one who received his formation in the 1930s, untutored. A colleague has offered to provide a mathematical model to decorate the work. It might be useful to some readers, but not to me. Catastrophe mathematics, dealing with such events as falling off a height, is a new branch of the discipline, I am told, which has yet to demonstrate its rigor or usefulness. I had better wait. Econometricians among my friends tell me that rare events such as panics cannot be dealt with by the normal techniques of regression, but have to be introduced exogenously as "dummy variables." The real choice open to me was whether to follow relatively simple statistical procedures, with an abundance of charts and tables, or not. In the event, I decided against it. For those who yearn for numbers, standard series on bank reserves, foreign trade, commodity prices, money supply, security prices, rate of interest, and the like are fairly readily available in the historical statistics.

Thinking in art and morals and even mathematics is neither the reflection in consciousness of a mechanical order in the brain nor the tracing with the mind’s eye of some empirical order in its object, but an endeavour to realize in thought an ideal order which would satisfy an inner demand. The nearer thought comes to its goal, the more it finds itself under constraint by that goal, and dominated in its creative effort by aesthetic or moral or logical relevance. These relations of relevance are not physical or psychological relations. They are normative relations that can enter into the mental current because that current is . . . teleological. Their operation marks the presence of a different type of law, which supervenes upon physical and psychological laws when purpose takes control.

I always felt that someone, a long time ago, organized the affairs of the world into areas that made sense-catagories of stuff that is perfectible, things that fit neatly in perfect bundles. The world of business, for example, is this way-line items, spreadsheets, things that add up, that can be perfected. The legal system-not always perfect, but nonetheless a mind-numbing effort to actually write down all kinds of laws and instructions that cover all aspects of being human, a kind of umbrella code of conduct we should all follow. Perfection is crucial in building an aircraft, a bridge, or a high-speed train. The code and mathematics residing just below the surface of the Internet is also this way. Things are either perfectly right or they will not work. So much of the world we work and live in is based upon being correct, being perfect. But after this someone got through organizing everything just perfectly, he (or probably a she) was left with a bunch of stuff that didn't fit anywhere-things in a shoe box that had to go somewhere. So in desperation this person threw up her arms and said, 'OK! Fine. All the rest of this stuff that isn't perfectible, that doesn't seem to fit anywhere else, will just have to be piled into this last, rather large, tattered box that we can sort of push behind the couch. Maybe later we can come back and figure where it all is supposed to fit in. Let's label the box ART.' The problem was thankfully never fixed, and in time the box overflowed as more and more art piled up. I think the dilemma exists because art, among all the other tidy categories, most closely resembles what it is like to be human. To be alive. It is our nature to be imperfect. The have uncategorized feelings and emotions. To make or do things that don't sometimes necessarily make sense. Art is all just perfectly imperfect. Once the word ART enters the description of what you're up to , it is almost getting a hall pass from perfection. It thankfully releases us from any expectation of perfection. In relation to my own work not being perfect, I just always point to the tattered box behind the couch and mention the word ART, and people seem to understand and let you off the hook about being perfect a go back to their business.

In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just thoroughly existent as the finite positive integers.

Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical conceptsãthe four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable.

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?

Nothing universal can be rationally affirmed on any moral, or any political subject. Pure metaphysical abstraction does not belong to these matters. The lines of morality are not like the ideal lines of mathematics. They are broad and deep as well as long. They admit of exceptions; they demand modifications. These exceptions and modifications are not made by the process of logic, but by the rules of prudence. Prudence is not only the first in rank of the virtues political and moral, but she is the director, the regulator, the standard of them all. ­Metaphysics cannot live without ­definition; but prudence is cautious how she defines.

No one is alone in this world. No act is without consequences for others. It is a tenet of chaos theory that, in dynamical systems, the outcome of any process is sensitive to its starting point-or, in the famous cliche, the flap of a butterfly's wings in the Amazon can cause a tornado in Texas. I do not assert markets are chaotic, though my fractal geometry is one of the primary mathematical tools of "chaology." But clearly, the global economy is an unfathomably complicated machine. To all the complexity of the physical world of weather, crops, ores, and factories, you add the psychological complexity of men acting on their fleeting expectations of what may or may not happen-sheer phantasms. Companies and stock prices, trade flows and currency rates, crop yields and commodity futures-all are inter-related to one degree or another, in ways we have barely begun to understand. In such a world, it is common sense that events in the distant past continue to echo in the present.

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

Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic orbits of flying rocks (explaining our penchant for baseball). A cavewoman thinking too hard about what matter is ultimately made of might fail to notice the tiger sneaking up behind and get cleaned right out of the gene pool. Darwin’s theory thus makes the testable prediction that whenever we use technology to glimpse reality beyond the human scale, our evolved intuition should break down. We’ve repeatedly tested this prediction, and the results overwhelmingly support Darwin. At high speeds, Einstein realized that time slows down, and curmudgeons on the Swedish Nobel committee found this so weird that they refused to give him the Nobel Prize for his relativity theory. At low temperatures, liquid helium can flow upward. At high temperatures, colliding particles change identity; to me, an electron colliding with a positron and turning into a Z-boson feels about as intuitive as two colliding cars turning into a cruise ship. On microscopic scales, particles schizophrenically appear in two places at once, leading to the quantum conundrums mentioned above. On astronomically large scales… weirdness strikes again: if you intuitively understand all aspects of black holes [then you] should immediately put down this book and publish your findings before someone scoops you on the Nobel Prize for quantum gravity… [also,] the leading theory for what happened [in the early universe] suggests that space isn’t merely really really big, but actually infinite, containing infinitely many exact copies of you, and even more near-copies living out every possible variant of your life in two different types of parallel universes.

Highly complex numbers like the Comma of Pythagoras, Pi and Phi (sometimes called the Golden Proportion), are known as irrational numbers. They lie deep in the structure of the physical universe, and were seen by the Egyptians as the principles controlling creation, the principles by which matter is precipitated from the cosmic mind. Today scientists recognize the Comma of Pythagoras, Pi and the Golden Proportion as well as the closely related Fibonacci sequence are universal constants that describe complex patterns in astronomy, music and physics. ... To the Egyptians these numbers were also the secret harmonies of the cosmos and they incorporated them as rhythms and proportions in the construction of their pyramids and temples.

This “Hawking temperature” of a black hole and its “Hawking radiation” (as they came to be called) were truly radical—perhaps the most radical theoretical physics discovery in the second half of the twentieth century. They opened our eyes to profound connections between general relativity (black holes), thermodynamics (the physics of heat) and quantum physics (the creation of particles where before there were none). For example, they led Stephen to prove that a black hole has entropy, which means that somewhere inside or around the black hole there is enormous randomness. He deduced that the amount of entropy (the logarithm of the hole’s amount of randomness) is proportional to the hole’s surface area. His formula for the entropy is engraved on Stephen’s memorial stone at Gonville and Caius College in Cambridge, where he worked. For the past forty-five years, Stephen and hundreds of other physicists have struggled to understand the precise nature of a black hole’s randomness. It is a question that keeps on generating new insights about the marriage of quantum theory with general relativity—that is, about the ill-understood laws of quantum gravity.

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.

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.

Suppose a vast number of civilizations are distributed throughout the universe, on the order of the number of detectable stars. Lots and lots of them. The mathematical structure of cosmic sociology is far clearer than that of human sociology. The factors of chaos and randomness in the complex makeups of every civilized society in the universe get filtered out by the immense distance, so those civilizations can act as reference points that are relatively easy to manipulate mathematically. First: Survival is the primary need of civilization. Second: Civilization continuously grows and expands, but the total matter in the universe remains constant. One more thing: To derive a basic picture of cosmic sociology from these two axioms, you need two other important concepts: chains of suspicion and the technological explosion.

I want economists to quit concerning themselves with allocation problems, per se, with the problem, as it has been traditionally defined. The vocabulary of science is important here, and as T. D. Weldon once suggested, the very word "problem" in and of itself implies the presence of "solution." Once the format has been established in allocation terms, some solution is more or less automatically suggested. Our whole study becomes one of applied maximization of a relatively simple computational sort. Once the ends to be maximized are provided by the social welfare function, everything becomes computational, as my colleague, Rutledge Vining, has properly noted. If there is really nothing more to economics than this, we had as well turn it all over to the applied mathematicians. This does, in fact, seem to be the direction in which we are moving, professionally, and developments of note, or notoriety, during the past two decades consist largely in improvements in what are essentially computing techniques, in the mathematics of social engineering. What I am saying is that we should keep these contributions in perspective; I am urging that they be recognized for what they are, contributions to applied mathematics, to managerial science if you will, but not to our chosen subject field which we, for better or for worse, call "economics.

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.

Nash’s lifelong quest for meaning, control, and recognition in the context of a continuing struggle, not just in society, but in the warring impulses of his paradoxical self, was now reduced to a caricature. Just as the overconcreteness of a dream is related to the intangible themes of waking life, Nash’s search for a piece of paper, a carte d’identité, mirrored his former pursuit of mathematical insights. Yet the gulf between the two recognizably related Nashes was as great as that between Kafka, the controlling creative genius, struggling between the demands of his self-chosen vocation and ordinary life, and K, a caricature of Kafka, the helpless seeker of a piece of paper that will validate his existence, rights, and duties. Delusion is not just fantasy but compulsion. Survival, both of the self and the world, appears to be at stake. Where once he had ordered his thoughts and modulated them, he was now subject to their peremptory and insistent commands.

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?

Most of us didn’t feel too enthusiastic about making a collapsar jump, either. We’d been assured that we wouldn’t even feel it happen, just free fall all the way. I wasn’t convinced. As a physics student, I’d had the usual courses in general relativity and theories of gravitation. We only had a little direct data at that time — Stargate was discovered when I was in grade school — but the mathematical model seemed clear enough. The collapsar Stargate was a perfect sphere about three kilometers in radius. It was suspended forever in a state of gravitational collapse that should have meant its surface was dropping toward its center at nearly the speed of light. Relativity propped it up, at least gave it the illusion of being there … the way all reality becomes illusory and observer-oriented when you study general relativity. Or Buddhism. Or get drafted. At any rate, there would be a theoretical point in space-time when one end of our ship was just above the surface of the collapsar, and the other end was a kilometer away (in our frame of reference). In any sane universe, this would set up tidal stresses and tear the ship apart, and we would be just another million kilograms of degenerate matter on the theoretical surface, rushing headlong to nowhere for the rest of eternity or dropping to the center in the next trillionth of a second. You pays your money and you takes your frame of reference. But they were right. We blasted away from Stargate 1, made a few course corrections and then just dropped, for about an hour.

All knowledge of reality starts from experience and ends in it.” But he immediately proceeded to emphasize the role that “pure reason” and logical deductions play. He conceded, without apology, that his success using tensor calculus to come up with the equations of general relativity had converted him to a faith in a mathematical approach, one that emphasized the simplicity and elegance of equations more than the role of experience. The fact that this method paid off in general relativity, he said, “justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas.”45

Germany had been united in empire for only eight years when Einstein was born in Ulm on March 14, 1879. He grew up in Munich. He was slow to speak, but he was not, as legend has it, slow in his studies; he consistently earned the highest or next-highest marks in mathematics and Latin in school and Gymnasium. At four or five the “miracle” of a compass his father showed him excited him so much, he remembered, that he “trembled and grew cold.” It seemed to him then that “there had to be something behind objects that lay deeply hidden.”624 He would look for the something which objects hid, though his particular genius was to discover that there was nothing behind them to hide; that objects, as matter and as energy, were all; that even space and time were not the invisible matrices of the material world but its attributes. “If you will not take the answer too seriously,” he told a clamorous crowd of reporters in New York in 1921 who asked him for a short explanation of relativity, “and consider it only as a kind of joke, then I can explain it as follows. It was formerly believed that if all material things disappeared out of the universe, time and space would be left. According to the relativity theory, however, time and space disappear together with the things.

Hume had shown that we naively read causality into this world and think that we grasp necessary succession in intuition. The same is true of everything that makes the body of the everyday surrounding world into an identical thing with identical properties, relations, etc. (and Hume had in fact worked this out in detail in the Treatise, which was unknown to Kant). Data and complexes of data come and go, but the thing, presumed to be simply experienced sensibly, is not something sensible which persists through this alteration. The sensationalist thus declares it to be a fiction. He is substituting, we shall say, mere sense-data for perception, which after all places things (everyday things) before our eyes. In other words, he overlooks the fact that mere sensibility, related to mere data of sense, cannot account for objects of experience. Thus he overlooks the fact that these objects of experience point to a hidden mental accomplishment and to the problem of what kind of an accomplishment this can be. From the very start, after all, it must be a kind which enables the objects of pre-scientific experience, through logic, mathematics, mathematical natural science, to be knowable with objective validity, i.e., with a necessity which can be accepted by and is binding for everyone.

(William) Hamilton recast the central ideas (of the evolutionary theory of aging) in mathematical form. Though this work tells us a good deal about why human lives take the course they do, Hamilton was a biologist whose great love was insects and their relatives, especially insects which make both our lives and an octopus’s life seem rather humdrum. Hamilton found mites in which the females hang suspended in the air with their swollen bodies packed with newly hatched young, and the males in the brood search out and copulate with their sisters there inside the mother. He found tiny beetles in which the males produce “and manhandle sperm cells longer than their whole bodies. Hamilton died in 2000, after catching malaria on a trip to Africa to investigate the origins of HIV. About a decade before his death, he wrote about how he would like his own burial to go. He wanted his body carried to the forests of Brazil and laid out to be eaten from the inside by an enormous winged Coprophanaeus beetle using his body to nurture its young, who would emerge from him and fly off. 'No worm for me nor sordid fly, I will buzz in the dusk like a huge bumble bee. I will be many, buzz even as a swarm of motorbikes, be borne, body by flying body out into the Brazilian wilderness beneath the stars, lofted under those beautiful and un-fused elytra [wing covers] which we will all hold over our “backs. So finally I too will shine like a violet ground beetle under a stone.

In using the present in order to reveal the past, we assume that the forces in the world are essentially the same through all time; for these forces are based on the very nature of matter, and could not have changed. The ocean has always had its waves, and those waves have always acted in the same manner. Running water on the land has ever had the same power of wear and transportation and mathematical value to its force. The laws of chemistry, heat, electricity, and mechanics have been the same through time. The plan of living structures has been fundamentally one, for the whole series belongs to one system, as much almost as the parts of an animal to the one body; and the relations of life to light and heat, and to the atmosphere, have ever been the same as now.

But that there is a simple relation between literary and other fictions seems, if one attends to it, more obvious than has appeared. If we think first of modern fictions, it can hardly be an accident that ever since Nietzsche generalized and developed the Kantian insights, literature has increasingly asserted its right to an arbitrary and private choice of fictional norms, just as historiography has become a discipline more devious and dubious because of our recognition that its methods depend to an unsuspected degree on myths and fictions. After Nietzsche it was possible to say, as Stevens did, that 'the final belief must be in a fiction.' This poet, to whom the whole question was of perpetual interest, saw that to think in this way was to postpone the End--when the fiction might be said to coincide with reality--for ever; to make of it a fiction, an imaginary moment when 'at last' the world of fact and the mundo of fiction shall be one. Such a fiction--the last section of Notes toward a Supreme Fiction is, appropriately, the place where Stevens gives it his fullest attention--such a fiction of the end is like infinity plus one and imaginary numbers in mathematics, something we know does not exist, but which helps us to make sense of and to move in the world. Mundo is itself such a fiction. I think Stevens, who certainly thought we have to make our sense out of whatever materials we find to hand, borrowed it from Ortega. His general doctrine of fictions he took from Vaihinger, from Nietzsche, perhaps also from American pragmatism.

I took care to replace the Compendium in its correct pamphlet, and in doing so dislodged a slim pamphlet by Grastrom, one of the most eccentric authors in Solarist literature. I had read the pamphlet, which was dictated by the urge to understand what lies beyond the individual, man, and the human species. It was the abstract, acidulous work of an autodidact who had previously made a series of unusual contributions to various marginal and rarefied branches of quantum physics. In this fifteen-page booklet (his magnum opus!), Grastrom set out to demonstrate that the most abstract achievements of science, the most advanced theories and victories of mathematics represented nothing more than a stumbling, one or two-step progression from our rude, prehistoric, anthropomorphic understanding of the universe around us. He pointed out correspondences with the human body-the projections of our sense, the structure of our physical organization, and the physiological limitations of man-in the equations of the theory of relativity, the theorem of magnetic fields and the various unified field theories. Grastrom’s conclusion was that there neither was, nor could be any question of ‘contact’ between mankind and any nonhuman civilization. This broadside against humanity made no specific mention of the living ocean, but its constant presence and scornful, victorious silence could be felt between every line, at any rate such had been my own impression. It was Gibarian who drew it to my attention, and it must have been Giarian who had added it to the Station’s collection, on his own authority, since Grastrom’s pamphlet was regarded more as a curiosity than a true contribution to Solarist literature

The European and the African have an entirely different concept of time. In the European worldview, time exists outside man, exists objectively, and has measurable and linear characteristics. According to Newton, time is absolute: “Absolute, true, mathematical time of itself and from its own nature, it flows equitably and without relation to anything external.” The European feels himself to be time’s slave, dependent on it, subject to it. To exist and function, he must observe its ironclad, inviolate laws, its inflexible principles and rules. He must heed deadlines, dates, days, and hours. He moves within the rigors of time and cannot exist outside them. They impose upon him their requirements and quotas. An unresolvable conflict exists between man and time, one that always ends with man’s defeat—time annihilates him. Africans apprehend time differently. For them, it is a much looser concept, more open, elastic, subjective. It is man who influences time, its shape, course, and rhythm (man acting, of course, with the consent of gods and ancestors ). Time is even something that man can create outright, for time is made manifest through events, and whether an event takes place or not depends, after all, on man alone. If two armies do not engage in a battle, then that battle will not occur (in other words, time will not have revealed its presence, will not have come into being). Time appears as a result of our actions, and vanishes when we neglect or ignore it. It is something that springs to life under our influence, but falls into a state of hibernation, even nonexistence, if we do not direct our energy toward it. It is a subservient, passive essence, and, most importantly, one dependent on man.

But the whole theory rests, if I am not mistaken, upon neglect of the fundamental distinction between an idea and its object. Misled by neglect of being, people have supposed that what does not exist is nothing. Seeing that numbers, relations, and many other objects of thought, do not exist outside the mind, they have supposed that the thoughts in which we think of these entities actually create their own objects. Every one except a philosopher can see the difference between a post and my idea of a post, but few see the difference between the number 2 and my idea of the number 2. Yet the distinction is as necessary in one case as in the other. The argument that 2 is mental requires that 2 should be essentially an existent. But in that case it would be particular, and it would be impossible for 2 to be in two minds, or in one mind at two times. Thus 2 must be in any case an entity, which will have being even if it is in no mind.* But further, there are reasons for denying that 2 is created by the thought which thinks it. For, in this case, there could never be two thoughts until some one thought so; hence what the person so thinking supposed to be two thoughts would not have been two, and the opinion, when it did arise, would be erroneous. And applying the same doctrine to 1; there cannot be one thought until some one thinks so. Hence Adam’s first thought must have been concerned with the number 1; for not a single thought could precede this thought. In short, all knowledge must be recognition, on pain of being mere delusion; Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. The number 2 is not purely mental, but is an entity which may be thought of. Whatever can be thought of has being, and its being is a precondition, not a result, of its being thought of.

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

Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas-mixtures” and “systems of solutions of linear equations”. It is left to metaphysics to make this “comprehensible” or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

If the law of gravitation be regarded as universal, the point may be stated as follows. The laws of motion require to be stated by reference to what have been called kinetic axes: these are in reality axes having no absolute acceleration and no absolute rotation. It is asserted, for example, when the third law is combined with the notion of mass, that, if m, m' be the masses of two particles between which there is a force, the component accelerations of the two particles due to this force are in the ratio m2 : m1. But this will only be true if the accelerations are measured relative to axes which themselves have no acceleration. We cannot here introduce the centre of mass, for, according to the principle that dynamical facts must be, or be derived from, observable data, the masses, and therefore the centre of mass, must be obtained from the acceleration, and not vice versâ. Hence any dynamical motion, if it is to obey the laws of motion, must be referred to axes which are not subject to any forces. But, if the law of gravitation be accepted, no material axes will satisfy this condition. Hence we shall have to take spatial axes, and motions relative to these are of course absolute motions. 465. In order to avoid this conclusion, C. Neumann* assumes as an essential part of the laws of motion the existence, somewhere, of an absolutely rigid “Body Alpha”, by reference to which all motions are to be estimated. This suggestion misses the essence of the discussion, which is (or should be) as to the logical meaning of dynamical propositions, not as to the way in which they are discovered. It seems sufficiently evident that, if it is necessary to invent a fixed body, purely hypothetical and serving no purpose except to be fixed, the reason is that what is really relevant is a fixed place, and that the body occupying it is irrelevant. It is true that Neumann does not incur the vicious circle which would be involved in saying that the Body Alpha is fixed, while all motions are relative to it; he asserts that it is rigid, but rightly avoids any statement as to its rest or motion, which, in his theory, would be wholly unmeaning. Nevertheless, it seems evident that the question whether one body is at rest or in motion must have as good a meaning as the same question concerning any other body; and this seems sufficient to condemn Neumann’s suggested escape from absolute motion.

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 Sumerian pantheon was headed by an "Olympian Circle" of twelve, for each of these supreme gods had to have a celestial counterpart, one of the twelve members of the Solar System. Indeed, the names of the gods and their planets were one and the same (except when a variety of epithets were used to describe the planet or the god's attributes). Heading the pantheon was the ruler of Nibiru, ANU whose name was synonymous with "Heaven," for he resided on Nibiru. His spouse, also a member of the Twelve, was called ANTU. Included in this group were the two principal sons of ANU: E.A ("Whose House Is Water"), Anu's Firstborn but not by Antu; and EN.LIL ("Lord of the Command") who was the Heir Apparent because his mother was Antu, a half sister of Anu. Ea was also called in Sumerian texts EN.KI ("Lord Earth"), for he had led the first mission of the Anunnaki from Nibiru to Earth and established on Earth their first colonies in the E.DIN ("Home of the Righteous Ones")—the biblical Eden. His mission was to obtain gold, for which Earth was a unique source. Not for ornamentation or because of vanity, but as away to save the atmosphere of Nibiru by suspending gold dust in that planet's stratosphere. As recorded in the Sumerian texts (and related by us in The 12th Planet and subsequent books of The Earth Chronicles), Enlil was sent to Earth to take over the command when the initial extraction methods used by Enki proved unsatisfactory. This laid the groundwork for an ongoing feud between the two half brothers and their descendants, a feud that led to Wars of the Gods; it ended with a peace treaty worked out by their sister Ninti (thereafter renamed Ninharsag). The inhabited Earth was divided between the warring clans. The three sons of Enlil—Ninurta, Sin, Adad—together with Sin's twin children, Shamash (the Sun) and Ishtar (Venus), were given the lands of Shem and Japhet, the lands of the Semites and Indo-Europeans: Sin (the Moon) lowland Mesopotamia; Ninurta, ("Enlil's Warrior," Mars) the highlands of Elam and Assyria; Adad ("The Thunderer," Mercury) Asia Minor (the land of the Hittites) and Lebanon. Ishtar was granted dominion as the goddess of the Indus Valley civilization; Shamash was given command of the spaceport in the Sinai peninsula. This division, which did not go uncontested, gave Enki and his sons the lands of Ham—the brown/black people—of Africa: the civilization of the Nile Valley and the gold mines of southern and western Africa—a vital and cherished prize. A great scientist and metallurgist, Enki's Egyptian name was Ptah ("The Developer"; a title that translated into Hephaestus by the Greeks and Vulcan by the Romans). He shared the continent with his sons; among them was the firstborn MAR.DUK ("Son of the Bright Mound") whom the Egyptians called Ra, and NIN.GISH.ZI.DA ("Lord of the Tree of Life") whom the Egyptians called Thoth (Hermes to the Greeks)—a god of secret knowledge including astronomy, mathematics, and the building of pyramids. It was the knowledge imparted by this pantheon, the needs of the gods who had come to Earth, and the leadership of Thoth, that directed the African Olmecs and the bearded Near Easterners to the other side of the world. And having arrived in Mesoamerica on the Gulf coast—just as the Spaniards, aided by the same sea currents, did millennia later—they cut across the Mesoamerican isthmus at its narrowest neck and—just like the Spaniards due to the same geography—sailed down from the Pacific coast of Mesoamerica southward, to the lands of Central America and beyond. For that is where the gold was, in Spanish times and before.

That such a surprisingly powerful philosophical method was taken seriously can be only partially explained by the backwardness of German natural science in those days. For the truth is, I think, that it was not at first taken really seriously by serious men (such as Schopenhauer, or J. F. Fries), not at any rate by those scientists who, like Democritus2, ‘would rather find a single causal law than be the king of Persia’. Hegel’s fame was made by those who prefer a quick initiation into the deeper secrets of this world to the laborious technicalities of a science which, after all, may only disappoint them by its lack of power to unveil all mysteries. For they soon found out that nothing could be applied with such ease to any problem whatsoever, and at the same time with such impressive (though only apparent) difficulty, and with such quick and sure but imposing success, nothing could be used as cheaply and with so little scientific training and knowledge, and nothing would give such a spectacular scientific air, as did Hegelian dialectics, the mystery method that replaced ‘barren formal logic’. Hegel’s success was the beginning of the ‘age of dishonesty’ (as Schopenhauer3 described the period of German Idealism) and of the ‘age of irresponsibility’ (as K. Heiden characterizes the age of modern totalitarianism); first of intellectual, and later, as one of its consequences, of moral irresponsibility; of a new age controlled by the magic of high-sounding words, and by the power of jargon. In order to discourage the reader beforehand from taking Hegel’s bombastic and mystifying cant too seriously, I shall quote some of the amazing details which he discovered about sound, and especially about the relations between sound and heat. I have tried hard to translate this gibberish from Hegel’s Philosophy of Nature4 as faithfully as possible; he writes: ‘§302. Sound is the change in the specific condition of segregation of the material parts, and in the negation of this condition;—merely an abstract or an ideal ideality, as it were, of that specification. But this change, accordingly, is itself immediately the negation of the material specific subsistence; which is, therefore, real ideality of specific gravity and cohesion, i.e.—heat. The heating up of sounding bodies, just as of beaten or rubbed ones, is the appearance of heat, originating conceptually together with sound.’ There are some who still believe in Hegel’s sincerity, or who still doubt whether his secret might not be profundity, fullness of thought, rather than emptiness. I should like them to read carefully the last sentence—the only intelligible one—of this quotation, because in this sentence, Hegel gives himself away. For clearly it means nothing but: ‘The heating up of sounding bodies … is heat … together with sound.’ The question arises whether Hegel deceived himself, hypnotized by his own inspiring jargon, or whether he boldly set out to deceive and bewitch others. I am satisfied that the latter was the case, especially in view of what Hegel wrote in one of his letters. In this letter, dated a few years before the publication of his Philosophy of Nature, Hegel referred to another Philosophy of Nature, written by his former friend Schelling: ‘I have had too much to do … with mathematics … differential calculus, chemistry’, Hegel boasts in this letter (but this is just bluff), ‘to let myself be taken in by the humbug of the Philosophy of Nature, by this philosophizing without knowledge of fact … and by the treatment of mere fancies, even imbecile fancies, as ideas.’ This is a very fair characterization of Schelling’s method, that is to say, of that audacious way of bluffing which Hegel himself copied, or rather aggravated, as soon as he realized that, if it reached its proper audience, it meant success.