Mathematical Modelling Quotes

Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?

The core of science is not a mathematical modeling--it is intellectual honesty. It is a willingness to have our certainties about the world constrained by good evidence and good argument.

Essentially, all models are wrong, but some are useful

I would say, if you like, that the party is like an out-moded mathematics...that is to say, the mathematics of Euclid. We need to invent a non-Euclidian mathematics with respect to political discipline.

We pose only those questions whose answers are the pre-given conditions of the questions themselves.

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.

Here we see that models, despite their reputation for impartiality, reflect goals and ideology. When I removed the possibility of eating Pop-Tarts at every meal, I was imposing my ideology on the meals model. It’s something we do without a second thought. Our own values and desires influence our choices, from the data we choose to collect to the questions we ask. Models are opinions embedded in mathematics.

Models are just opinions disguised as mathematics!

Our own values and desires influence our choices, from the data we choose to collect to the questions we ask. Models are opinions embedded in mathematics.

Fieldwork is probably always more likely to be holistic than lab work or mathematical modeling because in the field you can’t get away from the whole when a research project starts.

Lulled in the countless chambers of the brain, Our thoughts are linked by many a hidden chain. Awake but one, and lo, what myriads rise! Each stamps its image as the other flies!

How is that perceiving beings can arise from out of the physical world, and how is that mentality is able seemingly to 'create' mathematical concepts out of some kind of mental model.

... toxic derivatives were underpinned by toxic economics, which, in turn, were no more than motivated delusions in search of theoretical justification; fundamentalist tracts that acknowledged facts only when they could be accommodated to the demands of the lucrative faith. Despite their highly impressive labels and technical appearance, economic models were merely mathematized versions of the touching superstition that markets know best, both at times of tranquility and in periods of tumult.

if you take a positivist position, as I do, questions about reality don’t have any meaning. All one can ask is whether imaginary time is useful in formulating mathematical models that describe what we observe.

The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?

The basic principle of the new education is to be that dunces and idlers must not be made to feel inferior to intelligent and industrious pupils. That would be ‘undemocratic’. These differences between the pupils—for they are obviously and nakedly individual differences—must be disguised. This can be done on various levels. At universities, examinations must be framed so that nearly all the students get good marks. Entrance examinations must be framed so that all, or nearly all, citizens can go to universities, whether they have any power (or wish) to profit by higher education or not. At schools, the children who are too stupid or lazy to learn languages and mathematics and elementary science can be set to doing the things that children used to do in their spare time. Let them, for example, make mud-pies and call it modelling. But all the time there must be no faintest hint that they are inferior to the children who are at work. Whatever nonsense they are engaged in must have—I believe the English already use the phrase—‘parity of esteem’. An even more drastic scheme is not impossible. Children who are fit to proceed to a higher class may be artificially kept back, because the others would get a trauma—Beelzebub, what a useful word!—by being left behind. The bright pupil thus remains democratically fettered to his own age-group throughout his school career, and a boy who would be capable of tackling Aeschylus or Dante sits listening to his coaeval’s attempts to spell out A CAT SAT ON THE MAT.

Those of us who teach math should try to turn this bug into a feature. We should be up front about the fact that word problems force us to make simplifying assumptions. That’s a valuable skill—it’s called mathematical modeling.

Do we use models to help us find the truth? Or do we know the truth first, and then develop the mathematics to explain it?

Models are the mothers of invention.

In mathematical modeling, as in all of science, we always have to make choices about what to stress and what to ignore. The art of abstraction lies in knowing what is essential and what is minutia, what is signal and what is noise, what is trend and what is wiggle. It's an art because such choices always involve an element of danger; they come close to wishful thinking and intellectual dishonesty.

Economics [...] has the advantage of joining an extremely simple model of human nature with extremely complicated mathematical formulae that non-specialists can rarely understand, much less criticize.

In our lives we often act like we can reach an equilibrium: once we get into a relationship, we’ll be happy; once we move, we’ll be productive; once X thing happens, we’ll be in Y state. But things are always in flux. We don’t reach a certain steady state and then stay there forever. The endless adjustments are our lives.

Moreover, we look in vain to philosophy for the answer to the great riddle. Despite its noble purpose and history, pure philosophy long ago abandoned the foundational questions about human existence. The question itself is a reputation killer. It has become a Gorgon for philosophers, upon whose visage even the best thinkers fear to gaze. They have good reason for their aversion. Most of the history of philosophy consists of failed models of the mind. The field of discourse is strewn with the wreckage of theories of consciousness. After the decline of logical positivism in the middle of the twentieth century, and the attempt of this movement to blend science and logic into a closed system, professional philosophers dispersed in an intellectual diaspora. They emigrated into the more tractable disciplines not yet colonized by science – intellectual history, semantics, logic, foundational mathematics, ethics, theology, and, most lucratively, problems of personal life adjustment. Philosophers flourish in these various endeavors, but for the time being, at least, and by a process of elimination, the solution of the riddle has been left to science. What science promises, and has already supplied in part, is the following. There is a real creation story of humanity, and one only, and it is not a myth. It is being worked out and tested, and enriched and strengthened, step by step. (9-10)

The credit crunch was based on a climate (the post-Cold War victory party of free-market capitalism), a problem (the sub-prime mortgages), a mistake (the mathematical models of risk) and a failure, that of the regulators.

At schools, the children who are too stupid or lazy to learn languages, mathematics and elementary science can be set to doing the things that children used to do in their spare time. Let them, for example, make mud pies and call it modelling. But all the time there must be no faintest hint that they are inferior to the children who are at work. Whatever nonsense they are engaged in must have—I believe the English already use the phrase—"parity of esteem." An even more drastic scheme is not impossible. Children who are fit to proceed to a higher class may be artificially kept back, because the others would get a trauma—Beelzebub, what a useful word!—by being left behind. The bright pupil thus remains democratically fettered to his own age group throughout his school career, and a boy who would be capable of tackling Aeschylus or Dante sits listening to his coeval's attempts to spell out 'A Cat Sat On A Mat'.

Neither parents nor schools are very effective at teaching the young to find pleasure in the right things. Adults, themselves often deluded by infatuation with fatuous models, conspire in the deception. They make serious tasks seem dull and hard, and frivolous ones exciting and easy. Schools generally fial to teach how exciting, how mesmerizingly beautiful science or mathematics can be; they teach the routine of literature or history rather than the adventure.

In many schools today, the phrase "computer-aided instruction" means making the computer teach the child. One might say the computer is being used to program the child. In my vision, the child programs the computer and, in doing so, both acquires a sense of mastery over a piece of the most modern and powerful technology and establishes an intimate contact with some of the deepest ideas from science, from mathematics, and from the art of intellectual model building.

This is a point I’ll be returning to in future chapters: we’ve seen time and again that mathematical models can sift through data to locate people who are likely to face great challenges, whether from crime, poverty, or education. It’s up to society whether to use that intelligence to reject and punish them—or to reach out to them with the resources they need.

Yes," I continued, "I discovered this model recently and her style never fails to be mathematically perfect. She seems to come by it naturally. As if she were born resonant. I notice Japanese models tend to do this. Like I said, they seem to have resonance somewhere deep in their culture. But Yuri Nakagawa, she's the best I've ever seen. The best model, with the most powerful resonance. I need her to probe deeper into this profound mathematical instinct, which I call resonance.

. . .a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, "real" or "imaginary" time? It is simply a matter of which is the more useful description.

The math-powered applications powering the data economy were based on choices made by fallible human beings. Some of these choices were no doubt made with the best intentions. Nevertheless, many of these models encoded human prejudice, misunderstanding, and bias into the software systems that increasingly managed our lives. Like gods, these mathematical models were opaque, their workings invisible to all but the highest priests in their domain: mathematicians and computer scientists. Their verdicts, even when wrong or harmful, were beyond dispute or appeal. And they tended to punish the poor and the oppressed in our society, while making the rich richer.

We’ve seen time and again that mathematical models can sift through data to locate people who are likely to face great challenges, whether from crime, poverty, or educations. It’s up to society whether to use that intelligence to reject and punish them—or to reach out to them with the resources they need. We can use the scale and efficiency that make WMDs so pernicious in order to help people. It all depends on the objective we choose.

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.

The reader will no doubt understand that this was no arbitrary fancy, but just such another ‘tool’ as the hypothesis of Copernicus; an intellectual construction devised to accommodate the phenomena observed. We have recently been reminded13 how much mathematics, and how good, went to the building of the Model.

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.

The principle, now known as Hebbian learning, is succinctly described by the phrase ‘neurons that fire together wire together’.

I’m not really an epidemiologist, if that’s what you mean. It’s outside my area. I build mathematical models based on biological systems.

Marx did not use mathematical models, and his prose was not always limpid,

As the ancient Greeks replaced myth-based explanations with mechanistic models of the Solar System, their emphasis shifted from asking why to asking how.

irrational behavior is as unacceptable to a certain species of economist as the irrational magnitude of the hypotenuse was to the Pythagoreans. It doesn’t fit their model of what can be; and yet it is.

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.

Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates.

Magic is like a lot of other disciplines that people have recently begun developing, in historic terms. Working with magic is a way of understanding the universe and how it functions. You can approach it from a lot of different angles, applying a lot of different theories and mental models to it. You can get to the same place through a lot of different lines of theory and reasoning, kind of like really advanced mathematics. There's no truly right or wrong way to get there, either--there are just different ways, some more or less useful than others for a given application. And new vistas of thought, theory, and application open up on a pretty regular basis, as the Art develops and expands through the participation of multiple brilliant minds. But that said, once you have a good grounding in it,you get a pretty solid idea of what's possible and what isn't. No matter how much circumlocution you do with your formulae, two plus two doesn't equal five. (Except maybe very, very rarely, sometimes, in extremely specific and highly unlikely circumstances.)

Be wary, though, of the way news media use the word “significant,” because to statisticians it doesn’t mean “noteworthy.” In statistics, the word “significant” means that the results passed mathematical tests such as t-tests, chi-square tests, regression, and principal components analysis (there are hundreds). Statistical significance tests quantify how easily pure chance can explain the results. With a very large number of observations, even small differences that are trivial in magnitude can be beyond what our models of change and randomness can explain. These tests don’t know what’s noteworthy and what’s not—that’s a human judgment.

What if one were to want to hunt for these hidden presences? You can’t just rummage around like you’re at a yard sale. You have to listen. You have to pay attention. There are certain things you can’t look at directly. You need to trick them into revealing themselves. That’s what we’re doing with Walter, Jaz. We’re juxtaposing things, listening for echoes. It’s not some silly cybernetic dream of command and control, modeling the whole world so you can predict the outcome. It’s certainly not a theory of everything. I don’t have a theory of any kind. What I have is far more profound.’ ‘What’s that?’ ‘A sense of humor.’ Jaz looked at him, trying to find a clue in his gaunt face, in the clear gray eyes watching him with such - what? Amusement? Condescension? There was something about the man which brought on a sort of hermeneutic despair. He was a forest of signs. ‘We’re hunting for jokes.’ Bachman spoke slowly, as if to a child. ‘Parapraxes. Cosmic slips of the tongue. They’re the key to the locked door. They’ll help us discover it.’ ‘Discover what?’ ‘The face of God. What else would we be looking for?

Of course, I’ve only brought up two examples. Other universal laws of physics have been used as weapons as well, though we don’t know all of them. It’s very possible that every law of physics has been weaponized. It’s possible that in some parts of the universe, even … Forget it, I don’t even believe that.” “What were you going to say?” “The foundation of mathematics.” Cheng Xin tried to imagine it, but it was simply impossible. “That’s … madness.” Then she asked, “Will the universe turn into a war ruin? Or, maybe it’s more accurate to ask: Will the laws of physics turn into war ruins?” “Maybe they already are.… The physicists and cosmologists of the new world are focused on trying to recover the original appearance of the universe before the wars more than ten billion years ago. They’ve already constructed a fairly clear theoretical model describing the pre-war universe. That was a really lovely time, when the universe itself was a Garden of Eden. Of course, the beauty could only be described mathematically. We can’t picture it: Our brains don’t have enough dimensions.” Cheng Xin thought back to the conversation with the Ring again. Did you build this four-dimensional fragment? You told me that you came from the sea. Did you build the sea? “You are saying that the universe of the Edenic Age was four-dimensional, and that the speed of light was much higher?” “No, not at all. The universe of the Edenic Age was ten-dimensional. The speed of light back then wasn’t only much higher—rather, it was close to infinity. Light back then was capable of action at a distance, and could go from one end of the cosmos to the other within a Planck time.… If you had been to four-dimensional space, you would have some vague hint of how beautiful that ten-dimensional Garden must have been.” “You’re saying—” “I’m not saying anything.” Yifan seemed to have awakened from a dream. “We’ve only seen small hints; everything else is just guessing. You should treat it as a guess, just a dark myth we’ve made up.” But Cheng Xin continued to follow the course of the discussion taken so far. “—that during the wars after the Edenic Age, one dimension after another was imprisoned from the macroscopic into the microscopic, and the speed of light was reduced again and again.…” “As I said, I’m not saying anything, just guessing.” Yifan’s voice grew softer. “But no one knows if the truth is even darker than our guesses.… We are certain of only one thing: The universe is dying.” The

The difference between the Platonic theory and the morphic-resonance hypothesis can be illustrated by analogy with a television set. The pictures on the screen depend on the material components of the set and the energy that powers it, and also on the invisible transmissions it receives through the electromagnetic field. A sceptic who rejected the idea of invisible influences might try to explain everything about the pictures and sounds in terms of the components of the set – the wires, transistors, and so on – and the electrical interactions between them. Through careful research he would find that damaging or removing some of these components affected the pictures or sounds the set produced, and did so in a repeatable, predictable way. This discovery would reinforce his materialist belief. He would be unable to explain exactly how the set produced the pictures and sounds, but he would hope that a more detailed analysis of the components and more complex mathematical models of their interactions would eventually provide the answer. Some mutations in the components – for example, by a defect in some of the transistors – affect the pictures by changing their colours or distorting their shapes; while mutations of components in the tuning circuit cause the set to jump from one channel to another, leading to a completely different set of sounds and pictures. But this does not prove that the evening news report is produced by interactions among the TV set’s components. Likewise, genetic mutations may affect an animal’s form and behaviour, but this does not prove that form and behaviour are programmed in the genes. They are inherited by morphic resonance, an invisible influence on the organism coming from outside it, just as TV sets are resonantly tuned to transmissions that originate elsewhere.

The transmission of SARS, Dwyer said, seems to depend much on super spreaders—and their behavior, not to mention the behavior of people around them, can be various. The mathematical ecologist’s term for variousness of behavior is “heterogeneity,” and Dwyer’s models have shown that heterogeneity of behavior, even among forest insects, let alone among humans, can be very important in damping the spread of infectious disease. “If you hold mean transmission rate constant,” he told me, “just adding heterogeneity by itself will tend to reduce the overall infection rate.” That sounds dry. What it means is that individual effort, individual discernment, individual choice can have huge effects in averting the catastrophes that might otherwise sweep through a herd. An individual gypsy moth may inherit a slightly superior ability to avoid smears of NPV as it grazes on a leaf. An individual human may choose not to drink the palm sap, not to eat the chimpanzee, not to pen the pig beneath mango trees, not to clear the horse’s windpipe with his bare hand, not to have unprotected sex with the prostitute, not to share the needle in a shooting gallery, not to cough without covering her mouth, not to board a plane while feeling ill, or not to coop his chickens along with his ducks. “Any tiny little thing that people do,” Dwyer said, if it makes them different from one another, from the idealized standard of herd behavior, “is going to reduce infection rates.

Every now and then, I'm lucky enough to teach a kindergarten or first-grade class. Many of these children are natural-born scientists - although heavy on the wonder side, and light on skepticism. They're curious, intellectually vigorous. Provocative and insightful questions bubble out of them. They exhibit enormous enthusiasm. I'm asked follow-up questions. They've never heard of the notion of a 'dumb question'. But when I talk to high school seniors, I find something different. They memorize 'facts'. By and large, though, the joy of discovery, the life behind those facts has gone out of them. They've lost much of the wonder and gained very little skepticism. They're worried about asking 'dumb' questions; they are willing to accept inadequate answers, they don't pose follow-up questions, the room is awash with sidelong glances to judge, second-by-second, the approval of their peers. They come to class with their questions written out on pieces of paper, which they surreptitiously examine, waiting their turn and oblivious of whatever discussion their peers are at this moment engaged in. Something has happened between first and twelfth grade. And it's not just puberty. I'd guess that it's partly peer pressure not to excel - except in sports, partly that the society teaches short-term gratification, partly the impression that science or mathematics won't buy you a sports car, partly that so little is expected of students, and partly that there are few rewards or role-models for intelligent discussion of science and technology - or even for learning for it's own sake. Those few who remain interested are vilified as nerds or geeks or grinds. But there's something else. I find many adults are put off when young children pose scientific questions. 'Why is the Moon round?', the children ask. 'Why is grass green?', 'What is a dream?', 'How deep can you dig a hole?', 'When is the world's birthday?', 'Why do we have toes?'. Too many teachers and parents answer with irritation, or ridicule, or quickly move on to something else. 'What did you expect the Moon to be? Square?' Children soon recognize that somehow this kind of question annoys the grown-ups. A few more experiences like it, and another child has been lost to science.

For far too long, economists have neglected the distribution of wealth, partly because of Kuznets’s optimistic conclusions and partly because of the profession’s undue enthusiasm for simplistic mathematical models based on so-called representative agents.

Among this bewildering multiplicity of ideals which shall we choose? The answer is that we shall choose none. For it is clear that each one of these contradictory ideals is the fruit of particular social circumstances. To some extent, of course, this is true of every thought and aspiration that has ever been formulated. Some thoughts and aspirations, however, are manifestly less dependent on particular social circumstances than others. And here a significant fact emerges: all the ideals of human behaviour formulated by those who have been most successful in freeing themselves from the prejudices of their time and place are singularly alike. Liberation from prevailing conventions of thought, feeling and behaviour is accomplished most effectively by the practice of disinterested virtues and through direct insight into the real nature of ultimate reality. (Such insight is a gift, inherent in the individual; but, though inherent, it cannot manifest itself completely except where certain conditions are fulfilled. The principal pre-condition of insight is, precisely, the practice of disinterested virtues.) To some extent critical intellect is also a liberating force. But the way in which intellect is used depends upon the will. Where the will is not disinterested, the intellect tends to be used (outside the non-human fields of technology, science or pure mathematics) merely as an instrument for the rationalization of passion and prejudice, the justification of self-interest. That is why so few even of die acutest philosophers have succeeded in liberating themselves completely from the narrow prison of their age and country. It is seldom indeed that they achieve as much freedom as the mystics and the founders of religion. The most nearly free men have always been those who combined virtue with insight. Now, among these freest of human beings there has been, for the last eighty or ninety generations, substantial agreement in regard to the ideal individual. The enslaved have held up for admiration now this model of a man, now that; but at all times and in all places, the free have spoken with only one voice. It is difficult to find a single word that will adequately describe the ideal man of the free philosophers, the mystics, the founders of religions. 'Non-attached* is perhaps the best. The ideal man is the non-attached man. Non-attached to his bodily sensations and lusts. Non-attached to his craving for power and possessions. Non-attached to the objects of these various desires. Non-attached to his anger and hatred; non-attached to his exclusive loves. Non-attached to wealth, fame, social position. Non-attached even to science, art, speculation, philanthropy. Yes, non-attached even to these. For, like patriotism, in Nurse Cavel's phrase, 'they are not enough, Non-attachment to self and to what are called 'the things of this world' has always been associated in the teachings of the philosophers and the founders of religions with attachment to an ultimate reality greater and more significant than the self. Greater and more significant than even the best things that this world has to offer. Of the nature of this ultimate reality I shall speak in the last chapters of this book. All that I need do in this place is to point out that the ethic of non-attachment has always been correlated with cosmologies that affirm the existence of a spiritual reality underlying the phenomenal world and imparting to it whatever value or significance it possesses.

Through the works of Weinberg, Glashow, and Salam on the electroweak theory and the elegant framework developed by the physicists David Gross, David Politzer, and Frank Wilczek for quantum chromodynamics, the characteristic group of the standard model has been identified with a product of three Lie groups denoted by U(1), SU(2), and SU(3). In some sense, therefore, the road toward the ultimate unification of the forces of nature has to go through the discovery of the most suitable Lie group that contains the product U(1) X SU(2) x SU(3).

To build his cathedral dome—a self-supporting structure of close to four million bricks that is still the largest masonry dome in the world—Brunelleschi had to develop sophisticated mathematical modeling techniques and invent an array of hoists and other engineering tools. In

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?

certain professionals, while believing they are experts, are in fact not. Based on their empirical record, they do not know more about their subject matter than the general population, but they are much better at narrating—or, worse, at smoking you with complicated mathematical models. They are also more likely to wear a tie.

I advise you to look for a chance to break away, to find a subject you can make your own. That is where the quickest advances are likely to occur, as measured by discoveries per investigator per year. Therein you have the best chance to become a leader and, as time passes, to gain growing freedom to set your own course. If a subject is already receiving a great deal of attention, if it has a glamorous aura, if its practitioners are prizewinners who receive large grants, stay away from that subject. Listen to the news coming from the hubbub, learn how and why the subject became prominent, but in making your own long-term plans be aware it is already crowded with talented people. You would be a newcomer, a private amid bemedaled first sergeants and generals. Take a subject instead that interests you and looks promising, and where established experts are not yet conspicuously competing with one another, where few if any prizes and academy memberships have been given, and where the annals of research are not yet layered with superfluous data and mathematical models.

Professor Simon Newcomb was expounding this to the New York Mathematical Society only a month or so ago. You know how on a flat surface, which has only two dimensions, we can represent a figure of a three-dimensional solid, and similarly they think that by models of three dimensions they could represent one of four—if they could master the perspective of the thing. See?

The fallacy that dynamic processes must be modeled as if the system is in continuous equilibrium is probably the most important reason for the intellectual failure of neoclassical economics. Mathematics, science and engineering developed tools long ago to model outside of equilibrium processes. This dynamic approach to thinking about the economy should become second nature to economists.

Nevertheless, many of these models encoded human prejudice, misunderstanding, and bias into the software systems that increasingly managed our lives. Like gods, these mathematical models were opaque, their workings invisible to all but the highest priests in their domain: mathematicians and computer scientists. Their verdicts, even when wrong or harmful, were beyond dispute or appeal. And

Szilard began explaining. “Five or ten minutes” later, he says, Einstein understood. After only a year of university physics, Szilard had worked out a rigorous mathematical proof that the random motion of thermal equilibrium could be fitted within the framework of the phenomenological theory in its original, classical form, without reference to a limiting atomic model—“and [Einstein] liked this very much.

What can you prove about space? How do you know where you are? Can space be curved? How many dimensions are there? How does geometry explain the natural order and unity of the cosmos? These are the questions behind the five geometric revolutions of world history. It started with a little scheme hatched by Pythagoras: to employ mathematics as the abstract system of rules that can model the physical universe.

Every now and then, I’m lucky enough to teach a kindergarten or first-grade class. Many of these children are natural-born scientists—although heavy on the wonder side and light on skepticism. They’re curious, intellectually vigorous. Provocative and insightful questions bubble out of them. They exhibit enormous enthusiasm. I’m asked follow-up questions. They’ve never heard of the notion of a “dumb question.” But when I talk to high school seniors, I find something different. They memorize “facts.” By and large, though, the joy of discovery, the life behind those facts, has gone out of them. They’ve lost much of the wonder, and gained very little skepticism. They’re worried about asking “dumb” questions; they’re willing to accept inadequate answers; they don’t pose follow-up questions; the room is awash with sidelong glances to judge, second-by-second, the approval of their peers. They come to class with their questions written out on pieces of paper, which they surreptitiously examine, waiting their turn and oblivious of whatever discussion their peers are at this moment engaged in. Something has happened between first and twelfth grade, and it’s not just puberty. I’d guess that it’s partly peer pressure not to excel (except in sports); partly that the society teaches short-term gratification; partly the impression that science or mathematics won’t buy you a sports car; partly that so little is expected of students; and partly that there are few rewards or role models for intelligent discussion of science and technology—or even for learning for its own sake. Those few who remain interested are vilified as “nerds” or “geeks” or “grinds.

Science begins with the world we have to live in, accepting its data and trying to explain its laws. From there, it moves toward the imagination: it becomes a mental construct, a model of a possible way of interpreting experience. The further it goes in this direction, the more it tends to speak the languages of mathematics, which is really one of the languages of the imagination, along with literature and music.

This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a figment of our imaginations. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just an idea that we invent to help us describe what we think the universe is like. But according to the approach I described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter of which is the more useful description.

we’ve seen time and again that mathematical models can sift through data to locate people who are likely to face great challenges, whether from crime, poverty, or education. It’s up to society whether to use that intelligence to reject and punish them—or to reach out to them with the resources they need. We can use the scale and efficiency that make WMDs so pernicious in order to help people. It all depends on the objective we choose.

In other words, even though an observer moment objectively occupies less than a liter of volume and a second of time, it subjectively feels as if it occupies all the space you're aware of and all the time you remember. You feel as if you're observing this space and time form here and now, but all that space and time are just part of the reality model that you're experiencing. This is why you subjectively feel that time flows even though it doesn't.

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

In sum, the fruition of 50 years of research, and several hundred million dollars in government funds, has given us the following picture of sub-atomic matter. All matter consists of quarks and leptons, which interact by exchanging different types of quanta, described by the Maxwell and Yang-Mills fields. In one sentence, we have captured the essence of the past century of frustrating investigation into the subatomic realm, From this simple picture one can derive, from pure mathematics alone, all the myriad and baffling properties of matter. (Although it all seems so easy now, Nobel laureate Steven Weinberg, one of the creators of the Standard Model, once reflected on how tortuous the 50-year journey to discover the model had been. He wrote, "There's a long tradition of theoretical physics, which by no means affected everyone but certainly affected me, that said the strong interactions [were] too complicated for the human mind.")

Supporters of the "modified Platonic view" of mathematics like to point out that, over the centuries, mathematicians have produced (or "discovered") numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unexpectedly feeding into physics, but there are many more.

Most decisions, and nearly all human interaction, can be incorporated into a contingencies model. For example, a President may start a war, a man may sell his business, or divorce his wife. Such an action will produce a reaction; the number of reactions is infinite but the number of probable reactions is manageably small. Before making a decision, an individual can predict various reactions, and he can assess his original, or primary-mode, decision more effectively. But there is also a category which cannot be analyzed by contingencies. This category involves events and situations which are absolutely unpredictable, not merely disasters of all sorts, but those also including rare moments of discovery and insight, such as those which produced the laser, or penicillin. Because these moments are unpredictable, they cannot be planned for in any logical manner. The mathematics are wholly unsatisfactory. We may only take comfort in the fact that such situations, for ill or for good, are exceedingly rare.

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.

To understand what rationality is, why it seems scarce, and why it matters, we must begin with the ground truths of rationality itself: the ways an intelligent agent ought to reason, given its goals and the world in which it lives. These “normative” models come from logic, philosophy, mathematics, and artificial intelligence, and they are our best understanding of the “correct” solution to a problem and how to find it. They serve as an aspiration for those who want to be rational, which should mean everyone.

We have already seen that gauge symmetry that characterizes the electroweak force-the freedom to interchange electrons and neturinos-dictates the existence of the messenger electroweak fields (photon, W, and Z). Similarly, the gauge color symmetry requires the presence of eight gluon fields. The gluons are the messengers of the strong force that binds quarks together to form composite particles such as the proton. Incidentally, the color "charges" of the three quarks that make up a proton or a neutron are all different (red, blue, green), and they add up to give zero color charge or "white" (equivalent to being electrically neutral in electromagnetism). Since color symmetry is at the base of the gluon-mediated force between quarks, the theory of these forces has become known as quantum chromodynamics. The marriage of the electroweak theory (which describes the electromagnetic and weak forces) with quantum chromodynamics (which describes the strong force) produced the standard model-the basic theory of elementary particles and the physical laws that govern them.

From a mathematical point of view, however, trust is hard to quantify. That's a challenge for people building models. Sadly, it's far easier to keep counting arrests, to build models that assume we're birds of a feather and treat us as such. Innocent people surrounded by criminals get treated badly, and criminals surrounded by law-abiding public get a pass. And because of the strong correlation between poverty and reported crime, the poor continue to get caught up in the digital dragnets. The rest of us barely have to think about them.

Every culture has its own creation myth, its own cosmology. And in some respects every cosmology is true, even if I might flatter myself in assuming mine is somehow truer because it is scientific. But it seems to me that no culture, including scientific culture, has cornered the market on definitive answers when it comes to the ultimate questions. Science may couch its models in the language of mathematics and observational astronomy, while other cultures use poetry and sacrificial propitiations to defend theirs. But in the end, no one knows, at least not yet. The current flux in the state of scientific cosmology attests to this, as we watch physicists and astronomers argue over string theory and multiverses and the cosmic inflation hypothesis. Many of the postulates of modern cosmology lie beyond, or at least at the outer fringes, of what can be verified through observation. As a result, aesthetics—as reflected by the “elegance” of the mathematical models—has become as important as observation in assessing the validity of a cosmological theory. There is the assumption, sometimes explicit and sometimes not, that the universe is rationally constructed, that it has an inherent quality of beauty, and that any mathematical model that does not exemplify an underlying, unifying simplicity is to be considered dubious if not invalid on such criteria alone. This is really nothing more than an article of faith; and it is one of the few instances where science is faith-based, at least in its insistence that the universe can be understood, that it “makes sense.” It is not entirely a faith-based position, in that we can invoke the history of science to support the proposition that, so far, science has been able to make sense, in a limited way, of much of what it has scrutinized. (The psychedelic experience may prove to be an exception.)

He wrote up the mathematics and everything fitted together. James had shown how the electrical and magnetic forces which we experience could have their seat not in physical objects like magnets and wires but in energy stored in the space between and around the bodies. Electrostatic energy was potential energy, like that of a spring; magnetic energy was rotational, like that in a flywheel, and both could exist in empty space. And these two forms of energy were immutably linked: a change in one was always accompanied by a change in the other. The model demonstrated how they acted together to produce all known electromagnetic phenomena.

There is a definite set of biomorphs, each permanently sitting in its own unique place in a mathematical space. It is permanently sitting there in the sense that, if only you knew its genetic formula, you could instantly find it; moreover, its neighbours in this special kind of space are the biomorphs that differ from it by only one gene. [W]hen you first evolve a new creature by artificial selection in [a] computer model, it feels like a creative process. So it is, indeed. But what you are really doing is finding the creature, for it is, in a mathematical sense, already sitting in its own place in the genetic space of Biomorph Land.

In a representative statement from 1963, he claimed, “Man does not know most of the rules on which he acts; and even what we call his intelligence is largely a system of rules which operate on him but which he does not know.”60 This deference to the precognitive or the pre-rational is what separated him from the rational choice and rational expectations models of Chicago School economists, who professed much more faith in the possibility of both formal mathematical modeling and forecasting. As he explained in his Nobel speech, Hayek saw such efforts as not only presumptuous but misleading. The best one could hope for was pattern prediction.

Wolfram, one of the most innovative thinkers in scientific computing and in the theory of complex systems, has been best known for the development of Mathematica, a computer program/system that allows a range of calculations not accessible before. After ten years of virtual silence, Wolfram is about to emerge with a provocative book that makes the bold claim that he can replace the basic infrastructure of science. In a world used to more than three hundred years of science being dominated by mathematical equations as the basic building blocks of models for nature, Wolfram proposes simple computer programs instead. He suggests that nature's main secret is the use of simple programs to generate complexity.

It was this situation that led mathematician Chris Hauert and his colleagues to consider another possibility in an important evolutionary model published in Science in 2002. In Axelrod's study and in most previous theoretical models, individuals were forced to interact with each other. But what if they could choose not to interact? Rather than attempting to cooperate and risking being taken advantage of, a person could fend for herself. In other words, she could sever her connections to others in the network. Hauert called the people who adopt this strategy "loners." Using some beautiful mathematics, Hauert and his colleagues showed that in a world full of loners it is easy for cooperation to evolve because there are no people to take advantage of the cooperators that appear. The loners fend for themselves, and the cooperators form networks with other cooperators. Soon, the cooperators take over the population because they always do better together than the loners. But once the world is full of cooperators, it is very easy for free riders to evolve and enjoy the fruits of cooperation without contributing (like parasites). As the free riders become the dominant type in the population, there is no one left for them to take advantage of; then, the loners once again take over -- because they want nothing to do, as it were, with those bastards. In short, cooperating can emerge because we can do more together than we can apart. But because of the free-rider problem, cooperation is not guaranteed to succeed.

Taking least squares is no longer optimal, and the very idea of ‘accuracy’ has to be rethought. This simple fact is as important as it is neglected. This problem is easily illustrated in the Logistic Map: given the correct mathematical formula and all the details of the noise model – random numbers with a bell-shaped distribution – using least squares to estimate α leads to systematic errors. This is not a question of too few data or insufficient computer power, it is the method that fails. We can compute the optimal least squares solution: its value for α is too small at all noise levels. This principled approach just does not apply to nonlinear models because the theorems behind the principle of least squares repeatedly assume bell-shaped distributions.

Computational models of the mind would make sense if what a computer actually does could be characterized as an elementary version of what the mind does, or at least as something remotely like thinking. In fact, though, there is not even a useful analogy to be drawn here. A computer does not even really compute. We compute, using it as a tool. We can set a program in motion to calculate the square root of pi, but the stream of digits that will appear on the screen will have mathematical content only because of our intentions, and because we—not the computer—are running algorithms. The computer, in itself, as an object or a series of physical events, does not contain or produce any symbols at all; its operations are not determined by any semantic content but only by binary sequences that mean nothing in themselves. The visible figures that appear on the computer’s screen are only the electronic traces of sets of binary correlates, and they serve as symbols only when we represent them as such, and assign them intelligible significances. The computer could just as well be programmed so that it would respond to the request for the square root of pi with the result “Rupert Bear”; nor would it be wrong to do so, because an ensemble of merely material components and purely physical events can be neither wrong nor right about anything—in fact, it cannot be about anything at all. Software no more “thinks” than a minute hand knows the time or the printed word “pelican” knows what a pelican is. We might just as well liken the mind to an abacus, a typewriter, or a library. No computer has ever used language, or responded to a question, or assigned a meaning to anything. No computer has ever so much as added two numbers together, let alone entertained a thought, and none ever will. The only intelligence or consciousness or even illusion of consciousness in the whole computational process is situated, quite incommutably, in us; everything seemingly analogous to our minds in our machines is reducible, when analyzed correctly, only back to our own minds once again, and we end where we began, immersed in the same mystery as ever. We believe otherwise only when, like Narcissus bent above the waters, we look down at our creations and, captivated by what we see reflected in them, imagine that another gaze has met our own.

... might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a figment of our imaginations. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just an idea that we invent to help us describe what we think that universe is like. (....) a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, 'real' or 'imaginary' time? It is simply a matter of which is the more useful description.

In a seminal 1981 paper, the economist Sherwin Rosen worked out the mathematics behind these “winner-take-all” markets. One of his key insights was to explicitly model talent—labeled, innocuously, with the variable q in his formulas—as a factor with “imperfect substitution,” which Rosen explains as follows: “Hearing a succession of mediocre singers does not add up to a single outstanding performance.” In other words, talent is not a commodity you can buy in bulk and combine to reach the needed levels: There’s a premium to being the best. Therefore, if you’re in a marketplace where the consumer has access to all performers, and everyone’s q value is clear, the consumer will choose the very best. Even if the talent advantage of the best is small compared to the next rung down on the skill ladder, the superstars still win the bulk of the market.

We shall never know what Faraday would have achieved had he mastered mathematics, but, paradoxically, his ignorance may have been an advantage. It led him to derive his theories entirely from experimental observation rather than to deduce them from mathematical models. Over time, this approach gave him a deep-seated intuition into electromagnetic phenomena. It enabled him to ask questions that had not occurred to others, to devise experiments that no one else had thought of, and to see possibilities that others had missed. He thought boldly but would never commit himself to an opinion until it had withstood the most rigorous experimental testing. As he explained in a letter to Ampère: I am unfortunate in a want to mathematical knowledge and the power of entering with facility any abstract reasoning. I am obliged to feel my way by facts placed closely together.

If we shuffle three colored quarks and the equations remain the same, then we say that the equations possess something called SU(3) symmetry. The 3 represents the fact that we have three types of colors, and the SU stands for a specific mathematical property of the symmetry. We say that there are three quarks in a multiplet. The quarks in a multiplet can be shuffled among one another without changing the physics of the theory. Similarly, the weak force governs the properties of two particles, the electron and the neutrino. The symmetry that interchanges these particles, yet leaves the equation the same, is called SU(2). This means that a multiplet of the weak force contains an electron and a neutrino, which can be rotated into each other. Finally, the electromagnetic force has U(1) symmetry, which rotates the components of the Maxwell field into itself. Each of these symmetries is simple and elegant. However, the most controversial aspect of the Standard Model is that it "unifies" the three fundamental forces by simply splicing all three theories into one large symmetry. SU(3) X SU(2) X U(1), which is just the product of the symmetries of the individual forces. (This can be compared to assembling a jigsaw puzzle. If we have three jigsaw pieces that don't quite fit, we can always take Scotch tape and splice them together by hand. This is how the Standard Model is formed, by taping three distinct multiplets together. This may not be aesthetically pleasing, but at least the three jigsaw puzzles now hang together by tape.) Ideally, one might have expected that "the ultimate theory" would have all the particles inside just a single multiplet. Unfortunately, the Standard Model has three distinct multiplets, which cannot be rotated among one another.

What can we conclude from all of these insights in terms of the role of symmetry in the cosmic tapestry? My humble personal summary is that we don't know yet whether symmetry will turn out to be the most fundamental concept in the workings of the universe. Some of the symmetries physicists have discovered or discussed over the years have later been recognized as being accidental or only approximate. Other symmetries, such as general covariance in general relativity and the gauge symmetries of the standard model, became the buds from which forces and new particles bloomed. All in all, there is absolutely no doubt in my mind that symmetry principles almost always tells us something important, and they may provide the most valuable clues and insights toward unveiling and deciphering the underlying principles of the universe, whatever those may be. Symmetry, in this sense, is indeed fruitful.

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

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 beauty of the principle idea of string theory is that all the known elementary particles are supposed to represent merely different vibration modes of the same basic string. Just as a violin or a guitar string can be plucked to produce different harmonics, different vibrational patterns of a basic string correspond to distinct matter particles, such as electrons and quarks. The same applies to the force carriers as well. Messenger particles such as gluons or the W and Z owe their existence to yet other harmonics. Put simply, all the matter and force particles of the standard model are part of the repertoire that strings can play. Most impressively, however, a particular configuration of vibrating string was found to have properties that match precisely the graviton-the anticipated messenger of the gravitational force. This was the first time that the four basic forces of nature have been housed, if tentatively, under one roof.

Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, if so, does he have any other effect on the universe? And who created him? Up to now, most scientists have been too occupied with the development of new theories that describe what the universe is to ask the question why. On the other hand, the people whose business it is to ask why, the philosophers, have not been able to keep up with the advance of scientific theories. In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, “The sole remaining task for philosophy is the analysis of language.” What a comedown from the great tradition of philosophy from Aristotle to Kant! However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason – for then we would know the mind of God.

In 1963, the chaos theorist Edward Lorenz presented an often-referenced lecture entitled “Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” Lorenz’s main point was that chaotic mathematical functions are very sensitive to initial conditions. Slight differences in initial conditions can lead to dramatically different results after many iterations. Lorenz believed that this sensitivity to slight differences in the beginning made it impossible to determine an answer to his question. Underlying Lorenz’s lecture was the assumption of determinism, that each initial condition can theoretically be traced as a cause of a final effect. This idea, called the “Butterfly Effect,” has been taken by the popularizers of chaos theory as a deep and wise truth. However, there is no scientific proof that such a cause and effect exists. There are no well-established mathematical models of reality that suggest such an effect. It is a statement of faith. It has as much scientific validity as statements about demons or God.

There were also many cases of feedback between physics and mathematics, where a physical phenomenon inspired a mathematical model that later proved to be the explanation of an entirely different physical phenomenon. An excellent example is provided by the phenomenon known as Brownian motion. In 1827, British botanist Robert Brown (1773-1858) observed that wen pollen particles are suspended in water, they get into a state of agitated motion. This effect was explained by Einstein in 1905 as resulting from the collisions that the colloidal particles experience with the molecules of the surrounding fluid. Each single collision has a negligible effect, because the pollen grains are millions of times more massive than the water molecules, but the persistent bombardment has a cumulative effect. Amazingly, the same model was found to apply to the motions of stars in star clusters. There the Brownian motion is produced by the cumulative effect of many stars passing by any given star, with each passage altering the motion (through gravitational interaction) by a tiny amount.

So they rolled up their sleeves and sat down to experiment -- by simulation, that is mathematically and all on paper. And the mathematical models of King Krool and the beast did such fierce battle across the equation-covered table, that the constructors' pencils kept snapping. Furious, the beast writhed and wriggled its iterated integrals beneath the King's polynomial blows, collapsed into an infinite series of indeterminate terms, then got back up by raising itself to the nth power, but the King so belabored it with differentials and partial derivatives that its Fourier coefficients all canceled out (see Riemann's Lemma), and in the ensuing confusion the constructors completely lost sight of both King and beast. So they took a break, stretched their legs, had a swig from the Leyden jug to bolster their strength, then went back to work and tried it again from the beginning, this time unleashing their entire arsenal of tensor matrices and grand canonical ensembles, attacking the problem with such fervor that the very paper began to smoke. The King rushed forward with all his cruel coordinates and mean values, stumbled into a dark forest of roots and logarithms, had to backtrack, then encountered the beast on a field of irrational numbers (F_1) and smote it so grievously that it fell two decimal places and lost an epsilon, but the beast slid around an asymptote and hid in an n-dimensional orthogonal phase space, underwent expansion and came out fuming factorially, and fell upon the King and hurt him passing sore. But the King, nothing daunted, put on his Markov chain mail and all his impervious parameters, took his increment Δk to infinity and dealt the beast a truly Boolean blow, sent it reeling through an x-axis and several brackets—but the beast, prepared for this, lowered its horns and—wham!!—the pencils flew like mad through transcendental functions and double eigentransformations, and when at last the beast closed in and the King was down and out for the count, the constructors jumped up, danced a jig, laughed and sang as they tore all their papers to shreds, much to the amazement of the spies perched in the chandelier—perched in vain, for they were uninitiated into the niceties of higher mathematics and consequently had no idea why Trurl and Klapaucius were now shouting, over and over, "Hurrah! Victory!!

To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The definition of the Golden Ratio emerged originally from the axioms of Euclidean geometry; the definition of the Fibonacci sequence from the axioms of the theory of numbers. Yet the fact that the ratio of successive Fibonacci numbers converges to the Golden Ratio was imposed on us-humans had not choice in the matter. Therefore, mathematical objects, albeit imaginary, do have real properties. Second, the explanation of the unreasonable power of mathematics cannot be based entirely on evolution in the restricted sense. For example, when Newton proposed his theory of gravitation, the data that he was trying to explain were at best accurate to three significant figures. Yet his mathematical model for the force between any two masses in the universe achieved the incredible precision of better than one part in a million. Hence, that particular model was not forced on Newton by existing measurements of the motions of planets, nor did Newton force a natural phenomenon into a preexisting mathematical pattern. Furthermore, natural selection in the common interpretation of that concept does not quite apply either, because it was not the case that five competing theories were proposed, of which one eventually won. Rather, Newton's was the only game in town!

When I first started coming to the seminar, Gelfand had a young physicist, Vladimir Kazakov, present a series of talks about his work on so-called matrix models. Kazakov used methods of quantum physics in a novel way to obtain deep mathematical results that mathematicians could not obtain by more conventional methods. Gelfand had always been interested in quantum physics, and this topic had traditionally played a big role at his seminar. He was particularly impressed with Kazakov’s work and was actively promoting it among mathematicians. Like many of his foresights, this proved to be golden: a few years later this work became famous and fashionable, and it led to many important advances in both physics and math. In his lectures at the seminar, Kazakov was making an admirable effort to explain his ideas to mathematicians. Gelfand was more deferential to him than usual, allowing him to speak without interruptions longer than other speakers. While these lectures were going on, a new paper arrived, by John Harer and Don Zagier, in which they gave a beautiful solution to a very difficult combinatorial problem.6 Zagier has a reputation for solving seemingly intractable problems; he is also very quick. The word was that the solution of this problem took him six months, and he was very proud of that. At the next seminar, as Kazakov was continuing his presentation, Gelfand asked him to solve the Harer–Zagier problem using his work on the matrix models. Gelfand had sensed that Kazakov’s methods could be useful for solving this kind of problem, and he was right. Kazakov was unaware of the Harer–Zagier paper, and this was the first time he heard this question. Standing at the blackboard, he thought about it for a couple of minutes and immediately wrote down the Lagrangian of a quantum field theory that would lead to the answer using his methods. Everyone in the audience was stunned.

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

Unlike classically spinning bodies, such as tops, however, where the spin rate can assume any value fast or slow, electrons always have only one fixed spin. In the units in which this spin is measured quantum mechanically (called Planck's constant) the electrons have half a unit, or they are "spin-1/2" particles. In fact, all the matter particles in the standard model-electrons, quarks, neutrinos, and two other types called muons and taus-all have "spin 1/2." Particles with half-integer spin are known collectively as fermions (after the Italian physicist Enrico Fermi). On the other hand, the force carriers-the photon, W, Z, and gluons-all have one unit of spin, or they are "spin-1" particles in the physics lingo. The carrier of gravity-the graviton-has "spin 2," and this was precisely the identifying property that one of the vibrating strings was found to possess. All the particles with integer units of spin are called bosons (after the Indian physicist Satyendra Bose). Just as ordinary spacetime is associated with a supersymmetry that is based on spin. The predictions of supersymmetry, if it is truly obeyed, are far-reaching. In a universe based on supersymmetry, every known particle in the universe must have an as-yet undiscovered partner (or "superparrtner"). The matter particles with spin 1/2, such as electrons and quarks, should have spin 0 superpartners. the photon and gluons (that are spin 1) should have spin-1/2 superpartners called photinos and gluinos respectively. Most importantly, however, already in the 1970s physicists realized that the only way for string theory to include fermionic patterns of vibration at all (and therefore to be able to explain the constituents of matter) is for the theory to be supersymmetric. In the supersymmetric version of the theory, the bosonic and fermionic vibrational patters come inevitably in pairs. Moreover, supersymmetric string theory managed to avoid another major headache that had been associated with the original (nonsupersymmetric) formulation-particles with imaginary mass. Recall that the square roots of negative numbers are called imaginary numbers. Before supersymmetry, string theory produced a strange vibration pattern (called a tachyon) whose mass was imaginary. Physicists heaved a sigh of relief when supersymmetry eliminated these undesirable beasts.

To a theoretician, all these criticisms are troublesome but not fatal. But what does cause problems for a theoretician is that the model seems to predict a multiverse of parallel universes, many of which are crazier than those in the imagination of a Hollywood scriptwriter. String theory has an infinite number of solutions, each describing a perfectly well-behaved finite theory of gravity, which do not resemble our universe at all. In many of these parallel universes, the proton is not stable, so it would decay into a vast cloud of electrons and neutrinos. In these universes, complex matter as we know it (atoms and molecules) cannot exist. They only consist of a gas of subatomic particles. (Some might argue that these alternate universes are only mathematical possibilities and are not real. But the problem is that the theory lacks predictive power, since it cannot tell you which of these alternate universes is the real one.) This problem is actually not unique to string theory. For example, how many solutions are there to Newton’s or Maxwell’s equations? There are an infinite number, depending on what you are studying. If you start with a light bulb or a laser and you solve Maxwell’s equations, you find a unique solution for each instrument. So Maxwell’s or Newton’s theories also have an infinite number of solutions, depending on the initial conditions—that is, the situation you start with. This problem is likely to exist for any theory of everything. Any theory of everything will have an infinite number of solutions depending on the initial conditions. But how do you determine the initial conditions of the entire universe? This means you have to input the conditions of the Big Bang from the outside, by hand. To many physicists this seems like cheating. Ideally, you want the theory itself to tell you the conditions that gave rise to the Big Bang. You want the theory to tell you everything, including the temperature, density, and composition of the original Big Bang. A theory of everything should somehow contain its own initial conditions, all by itself. In other words, you want a unique prediction for the beginning of the universe. So string theory has an embarrassment of riches. Can it predict our universe? Yes. That is a sensational claim, the goal of physicists for almost a century. But can it predict just one universe? Probably not. This is called the landscape problem. There are several possible solutions to this problem, none of them widely accepted. The first is the anthropic principle, which says that our universe is special because we, as conscious beings, are here to discuss this question in the first place. In other words, there might be an infinite number of universes, but our universe is the one that has the conditions that make intelligent life possible. The initial conditions of the Big Bang are fixed at the beginning of time so that intelligent life can exist today. The other universes might have no conscious life in them.

Many models are constructed to account for regularly observed phenomena. By design, their direct implications are consistent with reality. But others are built up from first principles, using the profession’s preferred building blocks. They may be mathematically elegant and match up well with the prevailing modeling conventions of the day. However, this does not make them necessarily more useful, especially when their conclusions have a tenuous relationship with reality. Macroeconomists have been particularly prone to this problem. In recent decades they have put considerable effort into developing macro models that require sophisticated mathematical tools, populated by fully rational, infinitely lived individuals solving complicated dynamic optimization problems under uncertainty. These are models that are “microfounded,” in the profession’s parlance: The macro-level implications are derived from the behavior of individuals, rather than simply postulated. This is a good thing, in principle. For example, aggregate saving behavior derives from the optimization problem in which a representative consumer maximizes his consumption while adhering to a lifetime (intertemporal) budget constraint.† Keynesian models, by contrast, take a shortcut, assuming a fixed relationship between saving and national income. However, these models shed limited light on the classical questions of macroeconomics: Why are there economic booms and recessions? What generates unemployment? What roles can fiscal and monetary policy play in stabilizing the economy? In trying to render their models tractable, economists neglected many important aspects of the real world. In particular, they assumed away imperfections and frictions in markets for labor, capital, and goods. The ups and downs of the economy were ascribed to exogenous and vague “shocks” to technology and consumer preferences. The unemployed weren’t looking for jobs they couldn’t find; they represented a worker’s optimal trade-off between leisure and labor. Perhaps unsurprisingly, these models were poor forecasters of major macroeconomic variables such as inflation and growth.8 As long as the economy hummed along at a steady clip and unemployment was low, these shortcomings were not particularly evident. But their failures become more apparent and costly in the aftermath of the financial crisis of 2008–9. These newfangled models simply could not explain the magnitude and duration of the recession that followed. They needed, at the very least, to incorporate more realism about financial-market imperfections. Traditional Keynesian models, despite their lack of microfoundations, could explain how economies can get stuck with high unemployment and seemed more relevant than ever. Yet the advocates of the new models were reluctant to give up on them—not because these models did a better job of tracking reality, but because they were what models were supposed to look like. Their modeling strategy trumped the realism of conclusions. Economists’ attachment to particular modeling conventions—rational, forward-looking individuals, well-functioning markets, and so on—often leads them to overlook obvious conflicts with the world around them.