Mathematics Inspirational Quotes

A mathematical formula for happiness:Reality divided by Expectations.There were two ways to be happy:improve your reality or lower your expectations.

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.

The ‘Muse’ is not an artistic mystery, but a mathematical equation. The gift are those ideas you think of as you drift to sleep. The giver is that one you think of when you first awake.

The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for all time, that its very elementalness can never be altered. But it is impossible to prove. Always, absolutes, nevers: these are the words, as much as numbers, that make up the world of mathematics. Not everyone liked the axiom of equality––Dr. Li had once called it coy and twee, a fan dance of an axiom––but he had always appreciated how elusive it was, how the beauty of the equation itself would always be frustrated by the attempts to prove it. It was the kind of axiom that could drive you mad, that could consume you, that could easily become an entire life. But now he knows for certain how true the axiom is, because he himself––his very life––has proven it. The person I was will always be the person I am, he realizes. The context may have changed: he may be in this apartment, and he may have a job that he enjoys and that pays him well, and he may have parents and friends he loves. He may be respected; in court, he may even be feared. But fundamentally, he is the same person, a person who inspires disgust, a person meant to be hated.

Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite.

It is not knowledge, but the act of learning, not the possession of but the act of getting there, which grants the greatest enjoyment.

We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.

At times there's something so precise and mathematically chilling about nationalism. Build a dam to take away water AWAY from 40 million people. Build a dam to pretend to BRING water to 40 million people. Who are these gods that govern us? Is there no limit to their powers?

We are mathematical equations where your life is the sum of all choices you've made until now. The good news is you can change the equation so that you start making a difference in your life.

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

Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion—not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.

The thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was "out there" all along I just couldn't see it. And now I can! This is really what keeps me in the math game-- the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods.

(existential mathematics...) the degree of slowness is directly proportional to the intensity of memory; the degree of speed is directly proportional to the intensity of forgetting.” –p. 39

Both for practical reasons and for mathematically verifiable moral reasons, authority and responsibility must be equal - else a balancing takes place as surely as current flows between points of unequal potential. To permit irresponsible authority is to sow disaster; to hold a man responsible for anything he does not control is to behave with blind idiocy. The unlimited democracies were unstable because their citizens were not responsible for the fashion in which they exerted their sovereign authority... other than through the tragic logic of history... No attempt was made to determine whether a voter was socially responsible to the extent of his literally unlimited authority. If he voted the impossible, the disastrous possible happened instead - and responsibility was then forced on him willy-nilly and destroyed both him and his foundationless temple.

As he soars, he thinks, suddenly, of Dr. Kashen. Or not of Dr. Kashen, necessarily, but the question he had asked him when he was applying to be his advisee: What's your favorite axiom? (The nerd pickup line, CM had once called it.) "The axiom of equality," he'd said, and Kashen had nodded, approvingly. "That's a good one," he'd said. The axiom of equality states that x always equals x: it assumes that if you have a conceptual thing named x, that it must always be equivalent to itself, that it has a uniqueness about it, that it is in possession of something so irreducible that we must assume it is absolutely, unchangeably equivalent to itself for all time, that its very elementalness can never be altered. But it is impossible to prove. Always, absolutes, nevers: these are the words, as much as numbers, that make up the world of mathematics. Not everyone liked the axiom of equality––Dr. Li had once called it coy and twee, a fan dance of an axiom––but he had always appreciated how elusive it was, how the beauty of the equation itself would always be frustrated by the attempts to prove it. I was the kind of axiom that could drive you mad, that could consume you, that could easily become an entire life. But now he knows for certain how true the axiom is, because he himself––his very life––has proven it. The person I was will always be the person I am, he realizes. The context may have changed: he may be in this apartment, and he may have a job that he enjoys and that pays him well, and he may have parents and friends he loves. He may be respected; in court, he may even be feared. But fundamentally, he is the same person, a person who inspires disgust, a person meant to be hated. And in that microsecond that he finds himself suspended in the air, between ecstasy of being aloft and the anticipation of his landing, which he knows will be terrible, he knows that x will always equal x, no matter what he does, or how many years he moves away from the monastery, from Brother Luke, no matter how much he earns or how hard he tries to forget. It is the last thing he thinks as his shoulder cracks down upon the concrete, and the world, for an instant, jerks blessedly away from beneath him: x = x, he thinks. x = x, x = x.

I was advised to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.

Going back to the basis, the phrase ‘Fight Like A Girl’, and we’ve all heard that growing up. And by that they mean that you’re some kind of weakling and have no skills as a male. It’s said to little boys when they can’t fight yet, and it ridicules us. By the time we were born, the most of us hear things which program you to accept and know that you are less than your male counter part. It comes apparent in the way you’re paid for your job, it comes apparent when yóu are not allowed to go outside after a certain hour because you stand a good chance of getting raped while no one says that to your boyfriend. While women, anywhere, live in some kind of fear, there is no equality and that is mathematically impossible. We cannot see that change or solved in our lifetimes, but we have to do everything that we can. We should remind ourselves that we are fifty-one percent. Everyone should know that fighting like a girl is a positive thing and that there is not inherently anything wrong with us by the fact that we are born like ladies. That is a beautiful thing that we should never be put down because of. Being compared to a woman should only make a man feel stronger. It should be a compliment. In this world we’re creating it actually is. I remember this one guy who came to our show in Texas or something and he had painted his shirt “real men fight like a girl”, and I cried, because he was going away in the army next day. He bought my book because he wanted something he could read over there. I just hoped that this men, fully straight and fully male can maintain and retain all of those things that make him understand us, and what makes him so beautiful. A lot of military training is step one: you take all those guys and put them in front of bunch of hardcore videogames where you kill a bunch of people and become desensitised. But that is NOT power! I will not do that. I will not become less of a human being and I refuse to give up my femininity because that’s bullshit. I’m not going to have to shave my head and become all buff and all that to be able to say “now I’m powerful” because that’s bullshit. All of this, all of us, we are power. You don’t have to change anything to be strong.

[...] confusing time with its mathematical progression, as the old do, to whom all the past is not a diminishing road but, instead, a huge meadow which no winter ever touches.

Faraday was asked: "What is the use of this discovery?" He answered: "What is the use of a child - it grows to be a man.

A wall is happy when it is well designed, when it rests firmly on its foundation, when its symmetry balances its part and produces no unpleasant stresses. Good design can be worked out on the mathematical principles of mechanics.

David Hilbert, the towering mathematical intellect of the previous thirty years, had put it thus:9 ‘Mathematics knows no races … for mathematics, the whole cultural world is a single country’,

A lot of scientific evidence suggests that the difference between those who succeed and those who don't is not the brains they were born with, but their approach to life, the messages they receive about their potential, and the opportunities they have to learn.

I noticed that the [drawing] teacher didn't tell people much... Instead, he tried to inspire us to experiment with new approaches. I thought of how we teach physics: We have so many techniques - so many mathematical methods - that we never stop telling the students how to do things. On the other hand, the drawing teacher is afraid to tell you anything. If your lines are very heavy, the teacher can't say, "Your lines are too heavy." because *some* artist has figured out a way of making great pictures using heavy lines. The teacher doesn't want to push you in some particular direction. So the drawing teacher has this problem of communicating how to draw by osmosis and not by instruction, while the physics teacher has the problem of always teaching techniques, rather than the spirit, of how to go about solving physical problems.

It is by a mathematical point only that we are wise, as the sailor or fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.

By simple mathematics giving is key to the world you seek to live in. If I take I alone gain. If I give or share then two at least are enriched.

Every time a student makes a mistake in math, they grow a synapse.” There

Education makes your maths better, not necessarily your manners.

To be a scholar study math, to be a smart study magic.

Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics.

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

Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity-- to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs-- you deny them mathematics itself.

Teaching mathematics, like teaching any art, requires the ability to inspire the student. Inspiration requires marketing, and marketing requires stirring communication.

If all the tough situations in our life are problems, mathematics thought us to solve any problem. Just need a formula which you need to derive on your own.

The distance between your Dreams and Reality is inversely proportional to your Efforts.

Is it possible that the Pentateuch could not have been written by uninspired men? that the assistance of God was necessary to produce these books? Is it possible that Galilei ascertained the mechanical principles of 'Virtual Velocity,' the laws of falling bodies and of all motion; that Copernicus ascertained the true position of the earth and accounted for all celestial phenomena; that Kepler discovered his three laws—discoveries of such importance that the 8th of May, 1618, may be called the birth-day of modern science; that Newton gave to the world the Method of Fluxions, the Theory of Universal Gravitation, and the Decomposition of Light; that Euclid, Cavalieri, Descartes, and Leibniz, almost completed the science of mathematics; that all the discoveries in optics, hydrostatics, pneumatics and chemistry, the experiments, discoveries, and inventions of Galvani, Volta, Franklin and Morse, of Trevithick, Watt and Fulton and of all the pioneers of progress—that all this was accomplished by uninspired men, while the writer of the Pentateuch was directed and inspired by an infinite God? Is it possible that the codes of China, India, Egypt, Greece and Rome were made by man, and that the laws recorded in the Pentateuch were alone given by God? Is it possible that Æschylus and Shakespeare, Burns, and Beranger, Goethe and Schiller, and all the poets of the world, and all their wondrous tragedies and songs are but the work of men, while no intelligence except the infinite God could be the author of the Pentateuch? Is it possible that of all the books that crowd the libraries of the world, the books of science, fiction, history and song, that all save only one, have been produced by man? Is it possible that of all these, the bible only is the work of God?

Worth as I use it here is immeasurable, not as in mathematics towards infinity. But that it can not be measured. There are no measurable parameters for it! Certainly not a material-communal measurable parameter for it! Such, it is what the being holds that cannot and should never be traded. When it is there, every essence of your being knows it, and takes commands from it that will be able to override any personal or imposed sense of value.

History, Geology, Psychology, Philosophy, Chemistry, Physics, Theology, Mathematics, Technology, Sociology, Biology, and the list goes on and on. If all this body of knowledge exist for human consumption, why would I specialize in only one field?

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

the global impact of pure science rises above all national boundaries, and the sheer timelessness of pure mathematics transcends the limitations of his twentieth-century span. When Turing returned to the prime numbers in 1950 they were unchanged from when he left them in 1939, wars

This is rather an awe-inspiring statement to get out of a straightforward uniqueness theorem in mathematics.

(which has inspired at least one novel, Apostolos Doxiadis's Uncle Petros and Goldbach's Conjecture29).

Always give help when needed, always ask for help when you need it

The spirit of Edison, not Einstein, still governed their image of the scientist. Perspiration, not inspiration. Mathematics was unfathomable and unreliable.

Life is not mathematics and the human being is not made for the sake of politics. I want a change in the present social system and do not believe in mere party politics.

Diagnostic, comment-based feedback is now known to promote learning, and it should be the standard way in which students’ progress is reported.

Some people are awe-inspired by majestic mountains, some by poetry and others by abstruse mathematics. But whatever the source, we all need a little awe in our lives.

A bird is like an instrument working according to mathematical law, and it is in the capacity of man to reproduce such an instrument

That this blind and aging man forged ahead with such gusto is a remarkable lesson, a tale for the ages. Euler's courage, determination, and utter unwillingness to be beaten serves, in the truest sense of the word, as an inspiration for mathematician and non-mathematician alike. The long history of mathematics provides no finer example of the triumph of the human spirit.

had explicitly been concerned to treat mathematics as if it were a chess game, without asking for a connection with the world. That question was, as it were, always left for someone else to tackle.

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

The popular image of the lone (and possibly slight mad) genius-who ignores the literature and other conventional wisdom and manages by some inexplicable inspiration (enhanced, perhaps, with a liberal dash of suffering) to come up with a breathtakingly original solution to a problem that confounded all the experts-is a charming and romantic image, but also a wildly inaccurate one, at least in the world of modern mathematics. We do have spectacular, deep and remarkable results and insights in this subject, of course, but they are the hard-won and cumulative achievement of years, decades, or even centuries of steady work and progress of many good and great mathematicians; the advance from one stage of understanding to the next can be highly non-trivial, and sometimes rather unexpected, but still builds upon the foundation of earlier work rather than starting totally anew....Actually, I find the reality of mathematical research today-in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck-to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of "geniuses.

They were aware that the symbols of mythology and the the symbols of mathematical science were different aspects of the same, indivisible Reality. They did not live in a 'divided house of faith and reason'; the two were interlocking, like ground-plan and elevation on an architect's drawing. It is a state of mind very difficult for twentieth-century man to imagine- or even to believe that it could ever have existed. It may help to remember though, that some of the greatest pre-Socratic sages formulated their philosophies in verse; the unitary source of inspiration of prophet, poet, and philosopher was still taken for granted.

five suggestions that can work to open mathematics tasks and increase their potential for learning: Open up the task so that there are multiple methods, pathways, and representations. Include inquiry opportunities. Ask the problem before teaching the method. Add a visual component and ask students how they see the mathematics. Extend the task to make it lower floor and higher ceiling. Ask students to convince and reason; be skeptical.

there was no shame in admitting you didn't have the answer, it was a necessary step toward the truth. It was as important to teach us about the unknown or the unknowable as it was to teach us what had already been safely proven.

I was not born happy. As a child, my favourite hymn was :'Weary of earth and laden with my sin.' At the age of five, I reflected that, if I should live to be seventy, I had only endured, so far, a fourteenth part of my whole life, and I felt the long-spread-out boredom ahead of me to be almost unedurable. In adolescense, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics.

I've been thinking about all the things I might have done differently. All the choices I didn't make. All the decisions that made and unmade me, all the actions and inactions I did or didn't take. With the shades drawn and the garbage overflowing, I've been thinking about all the bold steps I never took, all the gut instincts I didn't listen to, all the people I let down. I've been thinking about the cruel mathematics of my life, looking at my sums and wishing I'd shown my work.

It turns out that even believing you are smart—one of the fixed mindset messages—is damaging, as students with this fixed mindset are less willing to try more challenging work or subjects because they are afraid of slipping up and no longer being seen as smart.

We don't know that we don't own or owe Anything (In mathematical terms we are sifar), so keep walking as long as your legs are moving towards the infinite. Your own Ego is the only obstacle in this journey, earlier you get rid of it sooner will you find the infinite peace.

Another misconception about mathematics that is pervasive and damaging—and wrong—is the idea that people who can do math are the smartest or cleverest people. This makes math failure particularly crushing for students, as they interpret it as meaning that they are not smart.

Mathematics is at the center of thinking about how to spend the day, how many events and jobs can fit into the day, what size of space can be used to fit equipment or turn a car around, how likely events are to happen, knowing how tweets are amplified and how many people they reach.

In adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics. Now, on the contrary, I enjoy life; I might almost say that with every year that passes I enjoy it more…very largely it is due to a diminishing preoccupation with myself. Like others who had a Puritan education, I had the habit of meditating on my sins, follies, and shortcomings. I seemed to myself - no doubt justly - a miserable specimen. Gradually I learned to be indifferent to myself and my deficiencies; I came to center my attention increasingly upon external objects: the state of the world, various branches of knowledge, individuals for whom I felt affection…And every external interest inspires some activity which, so long as the interest remains alive, is a complete preventive of ennui. Interest in oneself, on the contrary, leads to no activity of a progressive kind. It may lead to the keeping of a diary, to getting psychoanalyzed, or perhaps to becoming a monk. But the monk will not be happy until the routine of the monastery has made him forget his own soul. The happiness which he attributes to religion he could have obtained from becoming a crossing-sweeper, provided he were compelled to remain one. External discipline is the only road to happiness for those unfortunates whose self-absorption is too profound to be cured in any other way.

The human mind is an incredible thing. It can conceive of the magnificence of the heavens and the intricacies of the basic components of matter. Yet for each mind to achieve its full potential, it needs a spark. The spark of enquiry and wonder. Often that spark comes from a teacher. Allow me to explain. I wasn’t the easiest person to teach, I was slow to learn to read and my handwriting was untidy. But when I was fourteen my teacher at my school in St Albans, Dikran Tahta, showed me how to harness my energy and encouraged me to think creatively about mathematics. He opened my eyes to maths as the blueprint of the universe itself. If you look behind every exceptional person there is an exceptional teacher. When each of us thinks about what we can do in life, chances are we can do it because of a teacher. [...] The basis for the future of education must lie in schools and inspiring teachers. But schools can only offer an elementary framework where sometimes rote-learning, equations and examinations can alienate children from science. Most people respond to a qualitative, rather than a quantitative, understanding, without the need for complicated equations. Popular science books and articles can also put across ideas about the way we live. However, only a small percentage of the population read even the most successful books. Science documentaries and films reach a mass audience, but it is only one-way communication.

The paradox of quantum physics in the 21st Century, awakens us to the realisation that "nothing matters" in and of itself. That nothing can be stated with certainty; but everything is just a mathematical probability occurring in an instance of space-time convergence, which forms our objective reality in the present moment.

How do we learn? Is there a better way? What can we predict? Can we trust what we’ve learned? Rival schools of thought within machine learning have very different answers to these questions. The main ones are five in number, and we’ll devote a chapter to each. Symbolists view learning as the inverse of deduction and take ideas from philosophy, psychology, and logic. Connectionists reverse engineer the brain and are inspired by neuroscience and physics. Evolutionaries simulate evolution on the computer and draw on genetics and evolutionary biology. Bayesians believe learning is a form of probabilistic inference and have their roots in statistics. Analogizers learn by extrapolating from similarity judgments and are influenced by psychology and mathematical optimization.

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better if for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Grothendieck transformed modern mathematics. However, much of the credit for this transformation should go to a lesser-known forerunner of his, Emmy Noether. It was Noether, born in Bavaria in 1882, who largely created the abstract approach that inspired category theory.1 Yet as a woman in a male academic world, she was barred from holding a professorship in Göttingen, and the classicists and historians on the faculty even tried to block her from giving unpaid lectures—leading David Hilbert, the dean of German mathematics, to comment, “I see no reason why her sex should be an impediment to her appointment. After all, we are a university, not a bathhouse.” Noether, who was Jewish, fled to the United States when the Nazis took power, teaching at Bryn Mawr until her death from a sudden infection in 1935.

More often than not, at the end of the day (or a month, or a year), you realize that your initial idea was wrong, and you have to try something else. These are the moments of frustration and despair. You feel that you have wasted an enormous amount of time, with nothing to show for it. This is hard to stomach. But you can never give up. You go back to the drawing board, you analyze more data, you learn from your previous mistakes, you try to come up with a better idea. And every once in a while, suddenly, your idea starts to work. It's as if you had spent a fruitless day surfing, when you finally catch a wave: you try to hold on to it and ride it for as long as possible. At moments like this, you have to free your imagination and let the wave take you as far as it can. Even if the idea sounds totally crazy at first.

In confronting the theoretical falsity and absurdity of that [Social Democrat] doctrine with the reality of the phenomenon, I gradually acquired a clear picture of its aims... At such times, I was overcome by dark forebodings and fear of something evil. I saw before me a teaching inspired by egoism and hatred, mathematically calculated to win a victory-but the triumph of which would be a mortal blow to humanity.

In my opinion, defining intelligence is much like defining beauty, and I don’t mean that it’s in the eye of the beholder. To illustrate, let’s say that you are the only beholder, and your word is final. Would you be able to choose the 1000 most beautiful women in the country? And if that sounds impossible, consider this: Say you’re now looking at your picks. Could you compare them to each other and say which one is more beautiful? For example, who is more beautiful— Katie Holmes or Angelina Jolie? How about Angelina Jolie or Catherine Zeta-Jones? I think intelligence is like this. So many factors are involved that attempts to measure it are useless. Not that IQ tests are useless. Far from it. Good tests work: They measure a variety of mental abilities, and the best tests do it well. But they don’t measure intelligence itself.

now I want to speak about the word ‘theory’. This was originally an Orphic word, which Cornford interprets as ‘passionate sympathetic contemplation’. In this state, he says, ‘The spectator is identified with the suffering God, dies in his death, and rises again in his new birth.’ for Pythagoras, the ‘passionate sympathetic contemplation’ was intellectual, and issued in mathematical knowledge. In this way, through Pythagoreanism, ‘theory’ gradually acquired its modern meaning; but for all who were inspired by Pythagoras it retained an element of ecstatic revelation. To those who have reluctantly learnt a little mathematics in school this may seem strange; but to those who have experienced the intoxicating delight of sudden understanding that mathematics gives, from time to time, to those who love it, the Pythagorean view will seem completely natural even if untrue.

Sometimes all it takes is a simple word, a mere nothing, a well-intentioned but over-protective gesture, like the gesture made, quite unwittingly, by the Mathematics teacher, for the pacific, docile, submissive person suddenly to vanish and be replaced, to the dismay and incomprehension of those who thought they knew all there was to know about the human soul, by the blind, devastating wrath of the meek. It doesn't usually last very long, but while it does, it inspires real fear.

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

theory'. This was originally an Orphic word, which Cornford interprets as 'passionate sympathetic contemplation'. In this state, he says, 'The spectator is identified with the suffering God, dies in his death, and rises again in his new birth.' For Pythagoras, the 'passionate sympathetic contemplation' was intellectual, and issued in mathematical knowledge. In this way, through Pythagoreanism, 'theory' gradually acquired its modern meaning; but for all who were inspired by Pythagoras it retained an element of ecstatic revelation.

How to describe the excitement I felt when I saw this beautiful work and realized its potential? I guess it's like when, after a long journey, suddenly a mountain peak comes in full view. You catch your breath, take in its majestic beauty, and all you can say is "Wow!" It's the moment of revelation. You have not yet reached the summit, you don't even know yet what obstacles lie ahead, but its allure is irresistible, and you already imagine yourself at the top. It's yours to conquer now. But do you have the strength and stamina to do it?

The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them. The researchers published their results with the title “A Time for Telling,” and they argued that the question is not “Should we tell or explain methods?” but “When is the best time do this?

Glad that you thus continue your resolve To suck the sweets of sweet philosophy Only, good master, while we do admire This virtue and this moral discipline, Let's be no Stoics nor no stocks, I pray, Or so devote to Aristotle's checks As Ovid be an outcast quite abjured. Balk logic with acquaintance that you have, And practise rhetoric in your common talk, Music and poesy use to quicken you, The mathematics and the metaphysics Fall to them as you find your stomach serves you. No profit grows where is no pleasure ta'en. In brief, sir, study what you most affect.

The study of invisible writings was a new discipline made available by the discovery of the bi-directional nature of Library-Space. The thaumic mathematics are complex, but boil down to the fact that all books, everywhere, affect all other books. This is obvious: books inspire other books written in the future, and cite books written in the past. But the General Theory** of L-Space suggests that, in that case, the contents of books as yet unwritten can be deduced from books now in existence. **There’s a Special Theory as well, but no one bothers much it much because it’s self-evidently a load of marsh gas.

It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true.

I grow little of the food I eat, and of the little I do grow I did not breed or perfect the seeds. I do not make any of my own clothing. I speak a language I did not invent or refine. I did not discover the mathematics I use. I am protected by freedoms and laws I did not conceive of or legislate, and do not enforce or adjudicate. I am moved by music I did not create myself. When I needed medical attention, I was helpless to help myself survive. I did not invent the transistor, the microprocessor, object oriented programming, or most of the technology I work with. I love and admire my species, living and dead, and am totally dependent on them for my life and well being.

A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult. What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

The predominant thoughts and feelings of a pregnant woman are lodged in some of the major chakras of the unborn baby. They will therefore affect the character of the unborn baby. To produce better babies, it is very important for a pregnant woman to see and hear things that are beautiful, inspiring, and strong. The feelings and thoughts should be harmonious and progressive or positive. Anger, pessimism, hopelessness, injurious words, negative feelings and thoughts should be avoided. It is advisable for a pregnant mother to read books that are inspirational like the biographies of great yogis or great people, books on spiritual teachings, mathematics, sciences, business and languages. All of these will have beneficial effects on the unborn baby and will tend to make the baby not only spiritual, but also sharp-minded and practical.

G. Stanley Hall, a creature of his times, believed strongly that adolescence was determined – a fixed feature of human development that could be explained and accounted for in scientific fashion. To make his case, he relied on Haeckel's faulty recapitulation idea, Lombroso's faulty phrenology-inspired theories of crime, a plethora of anecdotes and one-sided interpretations of data. Given the issues, theories, standards and data-handling methods of his day, he did a superb job. But when you take away the shoddy theories, put the anecdotes in their place, and look for alternate explanations of the data, the bronze statue tumbles hard. I have no doubt that many of the street teens of Hall's time were suffering or insufferable, but it's a serious mistake to develop a timeless, universal theory of human nature around the peculiarities of the people of one's own time and place.

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.

The legendary inscription above the Academy's door speaks loudly about Plato's attitude toward mathematics. In fact, most of the significant mathematical research of the fourth century BC was carried out by people associated in one way or another with the Academy. Yet Plato himself was not a mathematician of great technical dexterity, and his direct contributions to mathematical knowledge were probably minimal. Rather, he was an enthusiastic spectator, a motivating source of challenge, an intelligent critic, an an inspiring guide. The first century philosopher and historian Philodemus paints a clear picture: "At that time great progress was seen in mathematics, with Plato serving as the general architect setting out problems, and the mathematicians investigating them earnestly." To which the Neoplatonic philosopher and mathematician Proclus adds: "Plato...greatly advanced mathematics in general and geometry in particular because of his zeal for these studies. It is well known that his writings are thickly sprinkled with mathematical terms and that he everywhere tries to arouse admiration for mathematics among students of philosophy." In other words, Plato, whose mathematical knowledge was broadly up to date, could converse with the mathematicians as an equal and as a problem presenter, even though his personal mathematical achievements were not significant.

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.

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

The goal was ambitious. Public interest was high. Experts were eager to contribute. Money was readily available. Armed with every ingredient for success, Samuel Pierpont Langley set out in the early 1900s to be the first man to pilot an airplane. Highly regarded, he was a senior officer at the Smithsonian Institution, a mathematics professor who had also worked at Harvard. His friends included some of the most powerful men in government and business, including Andrew Carnegie and Alexander Graham Bell. Langley was given a $50,000 grant from the War Department to fund his project, a tremendous amount of money for the time. He pulled together the best minds of the day, a veritable dream team of talent and know-how. Langley and his team used the finest materials, and the press followed him everywhere. People all over the country were riveted to the story, waiting to read that he had achieved his goal. With the team he had gathered and ample resources, his success was guaranteed. Or was it? A few hundred miles away, Wilbur and Orville Wright were working on their own flying machine. Their passion to fly was so intense that it inspired the enthusiasm and commitment of a dedicated group in their hometown of Dayton, Ohio. There was no funding for their venture. No government grants. No high-level connections. Not a single person on the team had an advanced degree or even a college education, not even Wilbur or Orville. But the team banded together in a humble bicycle shop and made their vision real. On December 17, 1903, a small group witnessed a man take flight for the first time in history. How did the Wright brothers succeed where a better-equipped, better-funded and better-educated team could not? It wasn’t luck. Both the Wright brothers and Langley were highly motivated. Both had a strong work ethic. Both had keen scientific minds. They were pursuing exactly the same goal, but only the Wright brothers were able to inspire those around them and truly lead their team to develop a technology that would change the world. Only the Wright brothers started with Why. 2.

However, mathematics-inspired books such as Agatha Christie’s Ten Little Indians (with its valuable, realworld lessons on subtraction) are fair game.

Since the development of mathematics is often inspired and guided by aesthetic considerations, mathematics can be described as “amphibious”: It is both a science and one of the humanities.

The issue, then, is not, What is the best way to teach? but, What is mathematics really all about?... Controversies about…teaching cannot be resolved without confronting problems about the nature of mathematics.

It was not just joy. There was something more, something deeper: After all, what could be more mysterious, what could be more awe-inspiring, than to find that the structure of the physical world is intimately tied to the deep mathematical concepts, concepts which were developed out of considerations rooted only in logic and the beauty of form?

mathematical problems solved; feature problems solved...

Pythagoras (550 BCE), with his theory of numbers, had been a source of inspiration for those who sought harmony in the Universe. His aim was to show in his philosophy that there was a high, structural, divine order to the Universe. This was a natural habitat for the souls. Mathematics was the tool to investigate this order.

But he needed to be certain before committing to something so—the word, certain, arrested his thoughts. A person can’t be absolutely certain about anything, not certainty in the sense of a mathematical proof. He wasn’t certain about Kate. He saw her, observed her, wanted to be with her. Somehow, he just knew. For reasons already set in his heart, the way he was wired, Josh knew Kate was a person he wanted in his life. She was the proof. Would it be the same way with God?

…Just walking one short path could make you feel hopeful, frustrated, bored, excited, or even nothing at all. And that this could change from one step to the next. You’re aware that you want to reach the center, and also aware that the labyrinth keeps taking you away from it. Just as you seem to be getting close, you turn and end up walking almost around its outer limits.... You love the labyrinth and you hate it at different moments, but you never feel like you’ve conquered it, because that would be ridiculous.... This is a path that is determined for you in advance, but no one can tell you what to think while you’re walking it. It’s not like a maze, you can’t get lost. No one’s playing any tricks on you. There aren’t any monsters lurking around any corners. You can see the end and yet, you walk calmly towards it, following perhaps the least logical route (in mathematical terms at least). Perhaps the labyrinth tells us why we don’t simply read the last pages of books. Why we don’t hurry through life looking for outcomes all the time, however many times we’re told that we should, and that we should be overtaking people and overcoming things as we go. The labyrinth doesn’t tell us how to live. It shows us how we do live.

I walk around in a near-constant state of inspiration with a great hunger of knowledge, and I read everything I can about math and physics, often developing my own theories along the way.

I'm still just as slow… At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn't have a precise relation to intelligence. 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. (Schwartz, 2001)

Such results should prompt educators to abandon the traditional fixed ideas of the brain and learning that currently fill schools—ideas that children are smart or dumb, quick or slow. If brains can change in three weeks, imagine what can happen in a year of math class if students are given the right math materials and they receive positive messages about their potential and ability.

In this chapter we will look at the entire edifice of QFT. We will see that it is based on three simple principles. We will also list some of its achievements, including some new insights and understandings not previously mentioned. THE FOUNDATION QFT is an axiomatic theory that rests on a few basic assumptions. Everything you have learned so far, from the force of gravity to the spectrum of hydrogen, follows almost inevitably from these three basic principles. (To my knowledge, Julian Schwinger is the only person who has presented QFT in this axiomatic way, at least in the amazing courses he taught at Harvard University in the 1950's.) 1. The field principle. The first pillar is the assumption that nature is made of fields. These fields are embedded in what physicists call flat or Euclidean three-dimensional space-the kind of space that you intuitively believe in. Each field consists of a set of physical properties at every point of space, with equations that describe how these particles or field intensities influence each other and change with time. In QFT there are no particles, no round balls, no sharp edges. You should remember, however, that the idea of fields that permeate space is not intuitive. It eluded Newton, who could not accept action-at-a-distance. It wasn't until 1845 that Faraday, inspired by patterns of iron filings, first conceived of fields. The use of colors is my attempt to make the field picture more palatable. 2. The quantum principle (discetization). The quantum principle is the second pillar, following from Planck's 1900 proposal that EM fields are made up of discrete pieces. In QFT, all physical properties are treated as having discrete values. Even field strengths, whose values are continues, are regarded as the limit of increasingly finer discrete values. The principle of discretization was discovered experimentally in 1922 by Otto Stern and Walther Gerlach. Their experiment (Fig. 7-1) showed that the angular momentum (or spin) of the electron in a given direction can have only two values: +1/2 or -1/2 (Fig. 7-1). The principle of discretization leads to another important difference between quantum and classical fields: the principle of superposition. Because the angular momentum along a certain axis can only have discrete values (Fig. 7-1), this means that atoms whose angular momentum has been determined along a different axis are in a superposition of states defined by the axis of the magnet. This same superposition principle applies to quantum fields: the field intensity at a point can be a superposition of values. And just as interaction of the atom with a magnet "selects" one of the values with corresponding probabilities, so "measurement" of field intensity at a point will select one of the possible values with corresponding probability (see "Field Collapse" in Chapter 8). It is discretization and superposition that lead to Hilbert space as the mathematical language of QFT. 3. The relativity principle. There is one more fundamental assumption-that the field equations must be the same for all uniformly-moving observers. This is known as the Principle of Relativity, famously enunciated by Einstein in 1905 (see Appendix A). Relativistic invariance is built into QFT as the third pillar. QFT is the only theory that combines the relativity and quantum principles.