Mathematical Language Quotes

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Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
Galileo Galilei
I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.
Werner Heisenberg
Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.
Richard P. Feynman (The Character of Physical Law)
Mathematics is the language in which God has written the universe
Galileo Galilei
The strange word nymphomation, used to denote a complex mathematical procedure where numbers, rather than being added together or multiplied or whatever, were actually allowed to breed with each other to produce new numbers.
Jeff Noon (Nymphomation (Vurt, #4))
We shed as we pick up, like travellers who must carry everything in their arms, and what we let fall will be picked up by those behind. The procession is very long and life is very short. We die on the march. But there is nothing outside the march so nothing can be lost to it. The missing plays of Sophocles will turn up piece by piece, or be written again in another language. Ancient cures for diseases will reveal themselves once more. Mathematical discoveries glimpsed and lost to view will have their time again. You do not suppose, my lady, that if all of Archimedes had been hiding in the great library of Alexandria, we would be at a loss for a corkscrew?
Tom Stoppard (Arcadia)
The 3-legged stool of understanding is held up by history, languages, and mathematics. Equipped with those three you can learn anything you want to learn. But if you lack any one of them you are just another ignorant peasant with dung on your boots.
Robert A. Heinlein (Expanded Universe)
To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.
Richard P. Feynman
Time was simple, is simple. We can divide it into simple parts, measure it, arrange dinner by it, drink whisky to its passage. We can mathematically deploy it, use it to express ideas about the observable universe, and yet if asked to explain it in simple language to a child–in simple language which is not deceit, of course–we are powerless. The most it ever seems we know how to do with time is to waste it.
Claire North (The First Fifteen Lives of Harry August)
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
G.H. Hardy (A Mathematician's Apology)
Besides language and music mathematics is one of the primary manifestations of the free creative power of the human mind.
Hermann Weyl
Film is one if three universal languages, the other two: mathematics and music.
Frank Capra
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!
Stephen Hawking (A Brief History of Time)
Yet it is true—skin can mean a great deal. Mine means that any man may strike me in a public place and never fear the consequences. It means that my friends do not always like to be seen with me in the street. It means that no matter how many books I read, or languages I master, I will never be anything but a curiosity—like a talking pig or a mathematical horse.
Susanna Clarke (Jonathan Strange & Mr Norrell)
Mathematics is the language with which God has written the universe.
Galileo Galilei
We are not told, or not told early enough so that it sinks in, that mathematics is a language, and that we can learn it like any other, including our own. We have to learn our own language twice, first when we learn to speak it, second when we learn to read it. Fortunately, mathematics has to be learned only once, since it is almost wholly a written language.
Mortimer J. Adler (How to Read a Book: The Classic Guide to Intelligent Reading)
Mathematics to me is like a language I don’t speak though I admire its literature in translation.
David Quammen (Spillover: Animal Infections and the Next Human Pandemic)
Codes and patterns are very different from each other,” Langdon said. “And a lot of people confuse the two. In my field, it’s crucial to understand their fundamental difference.” “That being?” Langdon stopped walking and turned to her. “A pattern is any distinctly organized sequence. Patterns occur everywhere in nature—the spiraling seeds of a sunflower, the hexagonal cells of a honeycomb, the circular ripples on a pond when a fish jumps, et cetera.” “Okay. And codes?” “Codes are special,” Langdon said, his tone rising. “Codes, by definition, must carry information. They must do more than simply form a pattern—codes must transmit data and convey meaning. Examples of codes include written language, musical notation, mathematical equations, computer language, and even simple symbols like the crucifix. All of these examples can transmit meaning or information in a way that spiraling sunflowers cannot.
Dan Brown (Origin (Robert Langdon, #5))
Here look at me. I'm Charlie, the son you wrote off the books? Not that I blame you for it, but here I am, all fixed up better than ever. Test me. Ask me questions. I speak twenty languages, living and dead; I'm a mathematical whiz, and I'm writing a piano concerto that will make them remember me long after I'm gone.
Daniel Keyes (Flowers for Algernon)
(1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This I do often.
Alfred Marshall
A mathematician is an individual who calls himself a 'physicist' and does 'physics' and physical experiments with abstract concepts.
Bill Gaede (Why God Doesn't Exist)
...The pages and pages of complex, impenetrable calculations might have contained the secrets of the universe, copied out of God's notebook. In my imagination, I saw the creator of the universe sitting in some distant corner of the sky, weaving a pattern of delicate lace so fine that that even the faintest light would shine through it. The lace stretches out infinitely in every direction, billowing gently in the cosmic breeze. You want desperately to touch it, hold it up to the light, rub it against your cheek. And all we ask is to be able to re-create the pattern, weave it again with numbers, somehow, in our own language; to make the tiniest fragment our own, to bring it back to eart.
Yōko Ogawa (The Housekeeper and the Professor)
[M]odern physics has definitely decided for Plato. For the smallest units of matter are not physical objects in the ordinary sense of the word: they are forms, structures, or – in Plato’s sense – Ideas, which can be unambiguously spoken of only in the language of mathematics.
Rupert Sheldrake (The Science Delusion: Freeing the Spirit of Enquiry)
There cannot be a language more universal and more simple, more free from errors and obscurities...more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes
Joseph Fourier (The Analytical Theory of Heat)
Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
Galileo Galilei
THOMASINA: ....the enemy who burned the great library of Alexandria without so much as a fine for all that is overdue. Oh, Septimus! -- can you bear it? All the lost plays of the Athenians! Two hundred at least by Aeschylus, Sophocles, Euripides -- thousands of poems -- Aristotle's own library!....How can we sleep for grief? SEPTIMUS: By counting our stock. Seven plays from Aeschylus, seven from Sophocles, nineteen from Euripides, my lady! You should no more grieve for the rest than for a buckle lost from your first shoe, or for your lesson book which will be lost when you are old. We shed as we pick up, like travellers who must carry everything in their arms, and what we let fall will be picked up by those behind. The procession is very long and life is very short. We die on the march. But there is nothing outside the march so nothing can be lost to it. The missing plays of Sophocles will turn up piece by piece, or be written again in another language. Ancient cures for diseases will reveal themselves once more. Mathematical discoveries glimpsed and lost to view will have their time again. You do not suppose, my lady, that if all of Archimedes had been hiding in the great library of Alexandria, we would be at a loss for a corkscrew?
Tom Stoppard (Arcadia)
The key point to keep in mind, however, is that symmetry is one of the most important tools in deciphering nature's design.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
What drove me? I think most creative people want to express appreciation for being able to take advantage of the work that's been done by others before us. I didn't invent the language or mathematics I use. I make little of my own food, none of my own clothes. Everything I do depends on other members of our species and the shoulders that we stand on. And a lot of us want to contribute something back to our species and to add something to the flow. It's about trying to express something in the only way that most of us know how-because we can't write Bob Dylan songs or Tom Stoppard plays. We try to use the talents we do have to express our deep feelings, to show our appreciation of all the contributions that came before us, and to add something to that flow. That's what has driven me.
Walter Isaacson (Steve Jobs)
Adam was told to name the animals. Adam studied each kind and gave them a name based on his observations. Every animal “kind” has some behavior or characteristic that is unique to that animal type. When you know the Hebrew name for an animal, you get a peek at how a perfect man, speaking a perfect language, understood that perfect animal.
Michael Ben Zehabe (The Meaning of Hebrew Letters: A Hebrew Language Program for Christians)
It is generally recognized that women are better than men at languages, personal relations and multitasking, but less good at map-reading and spatial awareness. It is therefore not unreasonable to suppose that women might be less good at mathematics and physics. It is not politically correct to say such things....But it cannot be denied that there are differences between men and women. Of course, these are differences between the averages only. There are wide variations about the mean.
Stephen Hawking
When the twins asked what cuff-links were for—“To link cuffs together,” Ammu told them—they were thrilled by this morsel of logic in what had so far seemed an illogical language. Cuff+link = cuff-link. This, to them, rivaled the precision of logic and mathematics. Cuff-links gave them an inordinate (if exaggerated) satisfaction, and a real affection for the English language.
Arundhati Roy (The God of Small Things)
Alfonse invested everything he did with a sense of all-consuming purpose. He knew four languages, had photographic memory, did complex mathematics in his head. He'd once told me that the art of getting ahead in New York was based on learning how to express dissatisfaction in an interesting way. The air was full of rage and complaint. People had no tolerance for your particular hardship unless you knew how to entertain them with it.
Don DeLillo
India was the motherland of our race and Sanskrit the mother of Europe's languages. India was the mother of our philosophy, of much of our mathematics, of the ideals embodied in Christianity... of self-government and democracy. In many ways, Mother India is the mother of us all.
Will Durant
Saint Bartleby's School for Young Gentlemen Annual Report Student: Artemis Fowl II Year: First Fees: Paid Tutor: Dr Po Language Arts As far as I can tell, Artemis has made absolutely no progress since the beginning of the year. This is because his abilities are beyond the scope of my experience. He memorizes and understands Shakespeare after a single reading. He finds mistakes in every exercise I administer, and has taken to chuckling gently when I attempt to explain some of the more complex texts. Next year I intend to grant his request and give him a library pass during my class. Mathematics Artemis is an infuriating boy. One day he answers all my questions correctly, and the next every answer is wrong. He calls this an example of the chaos theory, and says that he is only trying to prepare me for the real world. He says the notion of infinity is ridiculous. Frankly, I am not trained to deal with a boy like Artemis. Most of my pupils have trouble counting without the aid of their fingers. I am sorry to say, there is nothing I can teach Artemis about mathematics, but someone should teach him some manners. Social Studies Artemis distrusts all history texts, because he says history was written by the victors. He prefers living history, where survivors of certain events can actually be interviewed. Obviously this makes studying the Middle Ages somewhat difficult. Artemis has asked for permission to build a time machine next year during double periods so that the entire class may view Medieval Ireland for ourselves. I have granted his wish and would not be at all surprised if he succeeded in his goal. Science Artemis does not see himself as a student, rather as a foil for the theories of science. He insists that the periodic table is a few elements short and that the theory of relativity is all very well on paper but would not hold up in the real world, because space will disintegrate before lime. I made the mistake of arguing once, and young Artemis reduced me to near tears in seconds. Artemis has asked for permission to conduct failure analysis tests on the school next term. I must grant his request, as I fear there is nothing he can learn from me. Social & Personal Development Artemis is quite perceptive and extremely intellectual. He can answer the questions on any psychological profile perfectly, but this is only because he knows the perfect answer. I fear that Artemis feels that the other boys are too childish. He refuses to socialize, preferring to work on his various projects during free periods. The more he works alone, the more isolated he becomes, and if he does not change his habits soon, he may isolate himself completely from anyone wishing to be his friend, and, ultimately, his family. Must try harder.
Eoin Colfer
We shed as we pick up, like travelers who must carry everything in their arms, and what we let fall will be picked up by those behind. The procession is very long and life is very short. We die on the march. But there is nothing outside the march so nothing can be lost to it. The missing plays of Sophocles will turn up piece by piece, or be written again in another language. Ancient cures for diseases will reveal themselves once more. Mathematical discoveries glimpsed and lost to view will have their time again.
Tom Stoppard (Arcadia)
Jesus probably studied this same information, in his youth. The apostle Paul probably studied this same information. How can I make such a bold assertion? Because, without this knowledge, much of the New Testament would make no sense. Many of the idioms used in the New Testament are the result of lessons learned from this ancient Hebrew education system. Unfortunately, what was common in their day, has become forgotten in ours. For a Hebrew, math doesn’t get in the way. It blazes the way. Other languages are disconnected from this mathematical relationship . . . and it shows.
Michael Ben Zehabe (The Meaning of Hebrew Letters: A Hebrew Language Program for Christians)
I knew that the languages which one learns there are necessary to understand the works of the ancients; and that the delicacy of fiction enlivens the mind; that famous deeds of history ennoble it and, if read with understanding, aid in maturing one's judgment; that the reading of all the great books is like conversing with the best people of earlier times; it is even studied conversation in which the authors show us only the best of their thoughts; that eloquence has incomparable powers and beauties; that poetry has enchanting delicacy and sweetness; that mathematics has very subtle processes which can serve as much to satisfy the inquiring mind as to aid all the arts and diminish man's labor; that treatises on morals contain very useful teachings and exhortations to virtue; that theology teaches us how to go to heaven; that philosophy teaches us to talk with appearance of truth about things, and to make ourselves admired by the less learned; that law, medicine, and the other sciences bring honors and wealth to those who pursue them; and finally, that it is desirable to have examined all of them, even to the most superstitious and false in order to recognize their real worth and avoid being deceived thereby
René Descartes (Discourse on Method)
People enjoy inventing slogans which violate basic arithmetic but which illustrate “deeper” truths, such as “1 and 1 make 1” (for lovers), or “1 plus 1 plus 1 equals 1” (the Trinity). You can easily pick holes in those slogans, showing why, for instance, using the plus-sign is inappropriate in both cases. But such cases proliferate. Two raindrops running down a window-pane merge; does one plus one make one? A cloud breaks up into two clouds -more evidence of the same? It is not at all easy to draw a sharp line between cases where what is happening could be called “addition”, and where some other word is wanted. If you think about the question, you will probably come up with some criterion involving separation of the objects in space, and making sure each one is clearly distinguishable from all the others. But then how could one count ideas? Or the number of gases comprising the atmosphere? Somewhere, if you try to look it up, you can probably fin a statement such as, “There are 17 languages in India, and 462 dialects.” There is something strange about the precise statements like that, when the concepts “language” and “dialect” are themselves fuzzy.
Douglas R. Hofstadter (Gödel, Escher, Bach: An Eternal Golden Braid)
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 or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
Eugene Paul Wigner
Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.
Giuseppe Peano
The nature of a letter can also be revealed within its numeric value. All letters and numbers behave in a certain but recognizable way, from which we can deduce its nature. The number two is the only even prime. There is an inherent mathematical dilemma with, “one.” No matter how many times you multiply it, by itself, you still can’t get past “one” (1 x 1 x 1 x 1 = 1). So, how does “one” move beyond itself? How does the same, produce the different? Mathematically, “one” is forced to divide itself and work from that duality. Therein, hides the divine puzzle of bet (b). To become “two,” the second must revolt from wholeness—a separation. Yet, the second could not have existed without the benefit of the original wholeness. Also, the first wanted the second to exist, but the first doesn’t know what the second will become. Again, two contains potential badness, to a Hebrew. (Ge 25:24)
Michael Ben Zehabe (The Meaning of Hebrew Letters: A Hebrew Language Program for Christians)
It was believed that all creation came from thought, language, and mathematics.
Alice Hoffman (The World That We Knew)
The language of categories is affectionately known as "abstract nonsense," so named by Norman Steenrod. This term is essentially accurate and not necessarily derogatory: categories refer to "nonsense" in the sense that they are all about the "structure," and not about the "meaning," of what they represent.
Paolo Aluffi (Algebra: Chapter 0)
These rules, the sign language and grammar of the Game, constitute a kind of highly developed secret language drawing upon several sciences and arts, but especially mathematics and music (and/or musicology), and capable of expressing and establishing interrelationships between the content and conclusions of nearly all scholarly disciplines. The Glass Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colours on his palette.
Hermann Hesse (The Glass Bead Game)
Language as putative science. - The significance of language for the evolution of culture lies in this, that mankind set up in language a separate world beside the other world, a place it took to be so firmly set that, standing upon it, it could lift the rest of the world off its hinges and make itself master of it. To the extent that man has for long ages believed in the concepts and names of things as in aeternae veritates he has appropriated to himself that pride by which he raised himself above the animal: he really thought that in language he possessed knowledge of the world. The sculptor of language was not so modest as to believe that he was only giving things designations, he conceived rather that with words he was expressing supreame knowledge of things; language is, in fact, the first stage of occupation with science. Here, too, it is the belief that the truth has been found out of which the mightiest sources of energy have flowed. A great deal later - only now - it dawns on men that in their belief in language they have propagated a tremendous error. Happily, it is too late for the evolution of reason, which depends on this belief, to be put back. - Logic too depends on presuppositions with which nothing in the real world corresponds, for example on the presupposition that there are identical things, that the same thing is identical at different points of time: but this science came into existence through the opposite belief (that such conditions do obtain in the real world). It is the same with mathematics, which would certainly not have come into existence if one had known from the beginning that there was in nature no exactly straight line, no real circle, no absolute magnitude.
Friedrich Nietzsche (Human, All Too Human: A Book for Free Spirits)
Fictions are useful so long as they are taken as fictions. They are then simply ways of "figuring" the world which we agree to follow so that we can act in cooperation, as we agree about inches and hours, numbers and words, mathematical systems and languages. If we have no agreement about measures of time and space, I would have no way of making a date with you at the corner of Forty-second Street and Fifth Avenue at 3 P.M. on Sunday, April 4.
Alan W. Watts (The Book: On the Taboo Against Knowing Who You Are)
Do you train Fabrikators at the Little Palace?” asked Wylan. Jesper scowled. Why did he have to go and start that? “Of course. There’s a school on the palace grounds.” “What if a student were older?” said Wylan, still pushing. “A Grisha can be taught at any age,” said Genya. “Alina Starkov didn’t discover her power until she was seventeen years old, and she… she was one of the most powerful Grisha who ever lived.” Genya pushed at Wylan’s left nostril. “It’s easier when you’re younger, but so is everything. Children learn languages more easily. They learn mathematics more easily.” “And they’re unafraid,” said Wylan quietly. “It’s other people who teach them their limits.” Wylan’s eyes met Jesper’s over Genya’s shoulder, and as if he was challenging both Jesper and himself.
Leigh Bardugo (Crooked Kingdom (Six of Crows, #2))
And of course in the long run, if there is a constant fight, the graceful is bound to be defeated and the efficient mind will win, because the world understands the language of mathematics, not of love.
Osho (Intuition: Knowing Beyond Logic)
The Greeks were the first mathematicians who are still ‘real’ to us to-day. Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand: as Littlewood said to me once, they are not clever schoolboys or ‘scholarship candidates’, but ‘Fellows of another college’. So Greek mathematics is ‘permanent’, more permanent even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
G.H. Hardy (A Mathematician's Apology)
There are other reasons we use math in physics. Besides keeping us honest, math is also the most economical and unambiguous terminology that we know of. Language is malleable; it depends on context and interpretation. But math doesn’t care about culture or history. If a thousand people read a book, they read a thousand different books. But if a thousand people read an equation, they read the same equation.
Sabine Hossenfelder (Lost in Math: How Beauty Leads Physics Astray)
. . . we come astonishingly close to the mystical beliefs of Pythagoras and his followers who attempted to submit all of life to the sovereignty of numbers. Many of our psychologists, sociologists, economists and other latter-day cabalists will have numbers to tell them the truth or they will have nothing. . . . We must remember that Galileo merely said that the language of nature is written in mathematics. He did not say that everything is. And even the truth about nature need not be expressed in mathematics. For most of human history, the language of nature has been the language of myth and ritual. These forms, one might add, had the virtues of leaving nature unthreatened and of encouraging the belief that human beings are part of it. It hardly befits a people who stand ready to blow up the planet to praise themselves too vigorously for having found the true way to talk about nature.
Neil Postman (Amusing Ourselves to Death: Public Discourse in the Age of Show Business)
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.
G.H. Hardy (A Mathematician's Apology)
I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.
George Pólya
Compare mathematics and the political sciences—it’s quite striking. In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is the concern for content.
Noam Chomsky (On Language: Chomsky's Classic Works Language and Responsibility and Reflections on Language in One Volume)
Our experience teaches us that there are indeed laws of nature, regularities in the way things behave, and that these laws are best expressed using the language of mathematics. This raises the interesting possibility that mathematical consistency might be used to guide us, along with experimental observation, to the laws that describe physical reality, and this has proved to be the case time and again throughout the history of science. We will see this happen during the course of this book, and it is truly one of the wonderful mysteries of our universe that it should be so.
Brian Cox (Why Does E=mc²? (And Why Should We Care?))
Obsession is, in any case, the premonition of the existence of an individual language, an irreproducible language through the attentive use of which we will be able to uncover the truth. We must follow this premonition into regions that to others might seem absurd and mad. I don’t know why this language of truth sounds angelic to some, while to others it changes into mathematical signs or notations. But there are also those to whose whim it speaks in a very strange way.
Olga Tokarczuk (Flights)
At Ge 1:1 God used a matrix of sevens: (1) Seven words. (2) 28 letters (28 ÷ 4 = 7). (3) First three words contain 14 letters (14 ÷ 2 = 7). (4) Last four words contain 14 letters (14 ÷ 2 = 7). (5) Fourth and fifth words have seven letters. (6) Sixth and seventh words have seven letters. (7) Key words (God, heaven, earth) contain 14 letters (14 ÷ 2 = 7). (8) Remaining words contain 14 letters (14 ÷ 2 = 7). (9) Numeric value of first, middle and last letters equal, 133 (133 ÷ 19 = 7). (10) Numeric value of the first and last letters of all seven words equal 1,393 (1,393 ÷ 199 = 7). (11) The book of Genesis has 78,064 letters (78,064 ÷ 11,152 = 7). So, what is the big deal about seven? Jesus is our Shiva (7), our Shabbat (7th day). (Lu 6:5) You couldn’t see this messianic reference, however, unless you are reading in Hebrew. This book is the beginning of an amazing pilgrimage.
Michael Ben Zehabe (The Meaning of Hebrew Letters: A Hebrew Language Program for Christians)
The basis for comprehension is theory, and the language of theoretical science is mathematics.
D.C. Rapaport
Formal mathematics is nature's way of letting you know how sloppy your mathematics is.
Leslie Lamport (Specifying Systems: The TLA+ Language and Tools for Hardware and Software Engineers)
Music, like the visual arts, is rooted in our experience of the natural world," said Schwartz. "It emulates our sound environment in the way that visual arts emulate the visual environment." In music we hear the echo of our basic sound making instrument-the vocal tract. This explanation for human music is simpler still than Pythagoras's mathematical equations: we like the sounds that are familiar to us-specifically, we like sounds that remind us of us.
Christine Kenneally (The First Word: The Search for the Origins of Language)
The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth. This
Marcus du Sautoy (The Number Mysteries)
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.
C.S. Lewis (The Screwtape Letters)
Certainly not! I didn't build a machine to solve ridiculous crossword puzzles! That's hack work, not Great Art! Just give it a topic, any topic, as difficult as you like..." Klapaucius thought, and thought some more. Finally he nodded and said: "Very well. Let's have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit." "Love and tensor algebra?" Have you taken leave of your senses?" Trurl began, but stopped, for his electronic bard was already declaiming: Come, let us hasten to a higher plane, Where dyads tread the fairy fields of Venn, Their indices bedecked from one to n, Commingled in an endless Markov chain! Come, every frustum longs to be a cone, And every vector dreams of matrices. Hark to the gentle gradient of the breeze: It whispers of a more ergodic zone. In Reimann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our asymptotes no longer out of phase, We shall encounter, counting, face to face. I'll grant thee random access to my heart, Thou'lt tell me all the constants of thy love; And so we two shall all love's lemmas prove, And in bound partition never part. For what did Cauchy know, or Christoffel, Or Fourier, or any Boole or Euler, Wielding their compasses, their pens and rulers, Of thy supernal sinusoidal spell? Cancel me not--for what then shall remain? Abscissas, some mantissas, modules, modes, A root or two, a torus and a node: The inverse of my verse, a null domain. Ellipse of bliss, converge, O lips divine! The product of our scalars is defined! Cyberiad draws nigh, and the skew mind Cuts capers like a happy haversine. I see the eigenvalue in thine eye, I hear the tender tensor in thy sigh. Bernoulli would have been content to die, Had he but known such a^2 cos 2 phi!
Stanisław Lem (The Cyberiad)
Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth. —Galileo Galilei, The Assayer, 1623
Max Tegmark (Our Mathematical Universe: My Quest for the Ultimate Nature of Reality)
The habit of looking at life as a social relation — an affair of society — did no good. It cultivated a weakness which needed no cultivation. If it had helped to make men of the world, or give the manners and instincts of any profession — such as temper, patience, courtesy, or a faculty of profiting by the social defects of opponents — it would have been education better worth having than mathematics or languages; but so far as it helped to make anything, it helped only to make the college standard permanent through life.
Henry Adams (The Education of Henry Adams)
Robert Heinlein says in Have Spacesuit, Will Travel that the only things worth studying are history, languages, and science. Actually, he adds maths, but honestly they left out the mathematical part of my brain.
Jo Walton (Among Others)
The laws of Nature are written in the language of mathematics.” Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress. Hence those who are fluent in this new language will be on the cutting edge of progress.
Edward Frenkel (Love and Math: The Heart of Hidden Reality)
I have tasted words, I have seen them. Never had her hands reached out in darkness and felt the texture of pure marble, never had her forehead bent forward and, as against a stone altar, felt safety. I am now saved. Her mind could not then so specifically have seen it, could not have said, "Now I will reveal myself in words, words may now supercede a scheme of mathematical-biological definition. Words may be my heritage and with words...A lady will be set back in the sky....there was hope in a block of unsubstantiated marble, words could carve and set up solid altars...Thought followed the wing that beat its silver into seven-branched larch boughs.
H.D. (HERmione)
We should expect nothing less from the language that was originally given by God, to His human family. Hebrew was the method that God chose for mankind to speak to Him, and Him to them. Adam spoke Hebrew—and your Bible confirms this. Everyone who got off the ark spoke one language—Hebrew. Even Abraham spoke Hebrew. Where did Abraham learn to speak Hebrew? Abraham was descended from Noah’s son, Shem. (Ge 11:10-26) Shem’s household was not affected by the later confusion of languages, at Babel. (Ge 11:5-9) To the contrary, Shem was blessed while the rest of Babel was cursed. (Ge 9:26) That is how Abraham retained Hebrew, despite residing in Babylon. So, Shem’s language can be traced back to Adam. (Ge 11:1) And, Shem (Noah’s son) was still alive when Jacob and Esau was 30 years of age. Obviously, Hebrew (the original language) was clearly spoken by Jacob’s sons. (Ge 14:13)
Michael Ben Zehabe (The Meaning of Hebrew Letters: A Hebrew Language Program for Christians)
Philosophy is written in this all-encompassing book that is constantly open to our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders in a dark labyrinth.
Galileo Galilei (Il Saggiatore)
Another explanation for the failure of logic and observation alone to advance medicine is that unlike, say, physics, which uses a form of logic - mathematics - as its natural language, biology does not lend itself to logic. Leo Szilard, a prominent physicist, made this point when he complained that after switching from physics to biology he never had a peaceful bath again. As a physicist he would soak in the warmth of a bathtub and contemplate a problem, turn it in his mind, reason his way through it. But once he became a biologist, he constantly had to climb out of the bathtub to look up a fact.
John M. Barry (The Great Influenza: The Story of the Deadliest Pandemic in History)
Education should have two objects: first, to give definite knowledge—reading and writing, languages and mathematics, and so on; secondly, to create those mental habits which will enable people to acquire knowledge and form sound judgments for themselves.
Bertrand Russell (Free Thought and Official Propaganda)
Nature already uses the language of mathematics, so why not work with the environment instead of against it. We need to start mimicking the mathematical logic that occurs in the landscape, identify existing systems, and out of those concepts create new ones.
Yafreisy Carrero
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'.
C.S. Lewis
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.
Stephen Hawking (A Brief History of Time)
Whether we like it or not, if we are to pursue a career in science, eventually we have to learn the “language of nature”: mathematics. Without mathematics, we can only be passive observers to the dance of nature rather than active participants. As Einstein once said, “Pure mathematics is, in its way, the poetry of logical ideas.” Let me offer an analogy. One may love French civilization and literature, but to truly understand the French mind, one must learn the French language and how to conjugate French verbs. The same is true of science and mathematics. Galileo once wrote, “[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to understand a single word.
Michio Kaku (Parallel Worlds: A Journey Through Creation, Higher Dimensions, and the Future of the Cosmos)
Riemann concluded that electricity, magnetism, and gravity are caused by the crumpling of our three-dimensional universe in the unseen fourth dimension. Thus a "force" has no independent life of its own; it is only the apparent effect caused by the distortion of geometry. By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would become one of the dominant themes in modern theoretical physics, that the laws of nature appear simple when expressed in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed.
Michio Kaku (Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension)
Once, probably, I used to think that vagueness was a loftier kind of poetry, truer to the depths of consciousness, and maybe when I started to read mathematics and science back in the mid-70s I found an unexpected lyricism in the necessarily precise language that scientists tend to use My instinct, my superstition is that the closer I see a thing and the more accurately I describe it, the better my chances of arriving at a certain sensuality of expression.
Don DeLillo
hurry” was not a concept that could be symbolized in the Martian language and therefore must be presumed to be unthinkable. Speed, velocity, simultaneity, acceleration, and other mathematical abstractions having to do with the pattern of eternity were part of Martian mathematics, but not of Martian emotion.
Robert A. Heinlein (Stranger in a Strange Land)
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?
Mario Livio (Is God a Mathematician?)
Consider a cognitive scientist concerned with the empirical study of the mind, especially the cognitive unconscious, and ultimately committed to understanding the mind in terms of the brain and its neural structure. To such a scientist of the mind, Anglo-American approaches to the philosophy of mind and language of the sort discussed above seem odd indeed. The brain uses neurons, not languagelike symbols. Neural computation works by real-time spreading activation, which is neither akin to prooflike deductions in a mathematical logic, nor like disembodied algorithms in classical artificial intelligence, nor like derivations in a transformational grammar.
George Lakoff (Philosophy In The Flesh)
The world of physics is essentially the real world construed by mathematical abstractions, and the world of sense is the real world construed by the abstractions which the sense-organs immediately furnish. To suppose that the "material mode" is a primitive and groping attempt at physical conception is a fatal error in epistemology.
Susanne K. Langer (Philosophy in a New Key: A Study in the Symbolism of Reason, Rite, and Art)
The profound ability to use aural and written language has enabled our species to collectively explore the concepts of science and mathematics, to capture the beauty of intricate thought, experience and philosophy, and indeed to venture beyond our tiny planet with the desire to expand our understanding of the very nature of existence itself.
Katherine Vucicevic
the laws of nature are written by the hand of God in the language of mathematics’ and that the ‘human mind is a work of God and one of the most excellent’.
John C. Lennox (God's Undertaker: Has Science Buried God?)
Moreover, Galileo argued that by pursuing science using the language of mechanical equilibrium and mathematics, humans could understand the divine mind.
Mario Livio (Is God a Mathematician?)
Python language is one example. As we noted above, it is also heavily used for mathematical and scientific papers, and will probably dominate that niche for some years yet. 18.3.3
Eric S. Raymond (The Art of UNIX Programming)
The Pythagoreans were probably the first to recognize the concept that the basic forces in the universe may be expressed through the language of mathematics.
Mario Livio (The Golden Ratio: The Story of Phi, the World's Most Astonishing Number)
The result that Noether obtained was stunning. She showed that to every continuous symmetry of the laws of physics there corresponds a conservation law and vice versa.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Noether's theorem fused together symmetries and conservation laws-these two giant pillars of physics are actually nothing but different facets of the same fundamental property.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Supersymmetry is a subtle symmetry based on the quantum mechanical property spin.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Indeed, the genius of Abel and Galois could be compared only to a supernova-an exploding star that for a short while outshines all the billions of stars in its host galaxy.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Creators are hard-driving, focused, dominant, independent risk-takers.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Tolerance of ambiguity is a necessary condition for creativity.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
You may begin to realize that groups will pop up wherever symmetries exist. In fact, the collection of all the symmetry transformations of any system always from a group.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Mathematics is its own language. The language of everything. It doesn’t need someone to explain it. It explains itself and leaves almost no room for ambiguity.
Mark Lawrence (Limited Wish (Impossible Times, #2))
Mathematical science shows what is. It is the language of unseen relations between things. —Ada Lovelace
Seanan McGuire (Middlegame (Middlegame, #1))
Ever since his first ecstasy or vision of Christminster and its possibilities, Jude had meditated much and curiously on the probable sort of process that was involved in turning the expressions of one language into those of another. He concluded that a grammar of the required tongue would contain, primarily, a rule, prescription, or clue of the nature of a secret cipher, which, once known, would enable him, by merely applying it, to change at will all words of his own speech into those of the foreign one. His childish idea was, in fact, a pushing to the extremity of mathematical precision what is everywhere known as Grimm's Law—an aggrandizement of rough rules to ideal completeness. Thus he assumed that the words of the required language were always to be found somewhere latent in the words of the given language by those who had the art to uncover them, such art being furnished by the books aforesaid.
Thomas Hardy (Jude the Obscure)
in the arena of economic policy, the citizens of today’s democracies have learned altogether too much modesty. We have been advised that these are matters for experts: that economics and its policy implications are far beyond the understanding of the common man or woman—a point of view enforced by the increasingly arcane and mathematical language of the discipline.
Tony Judt (Ill Fares The Land: A Treatise On Our Present Discontents)
When we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system. Thus the outward forms of our mathematics are not absolutely relevant from the point of view of evaluating what the mathematical or logical language truly used by the central nervous system is. However, the above remarks about reliability and logical and arithmetical depth prove that whatever the system is, it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics.
John von Neumann (The Computer and the Brain)
Beauty magnetizes curiosity and wonder, beckoning us to discover—in the literal sense, to uncover and unconceal—what lies beneath the surface of the human label. What we recognize as beauty may be a language for encoding truth, a memetic mechanism for transmitting it, as native to the universe as mathematics—the one perceived by the optical eye, the other by the mind's eye.
Maria Popova (Figuring)
Like most religious mathematicians from Pythagoras to Godel, Bolzano believes that math is the Language of God and that profound metaphysical truths can be derived and proved mathematically.
David Foster Wallace (Everything and More: A Compact History of Infinity)
Mathematics is the means by which we deduce the consequences of physical principles. More than that, it is the indispensable language in which the principles of physical science are expressed.
Steven Weinberg (To Explain the World: The Discovery of Modern Science)
the physical substrate for thinking, the neural system, but the very fact that it has the form of a network implies that thought also has graph structure. This in turn suggests that language and its grammars,
Ulf Grenander (A Calculus of Ideas:A Mathematical Study of Human Thought)
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.
Melvin Schwartz (Principles of Electrodynamics)
Let me end on a more cheerful note. 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 or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
Eugene Paul Wigner (The Unreasonable Effectiveness of Mathematics in the Natural Sciences)
Nor is mathematics about a Platonic reality of eternal truths. It is a creation of the human body and senses, growing out of the activities of moving along a path and of collecting, constructing, and measuring objects.
Steven Pinker (The Stuff of Thought: Language as a Window into Human Nature)
Is it odd how asymmetrical Is "symmetry"? "Symmetry" is asymmetrical. How odd it is. This stanza remains unchanged if read word by word from the end to the beginning-it is symmetrical with respect to backward reading.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Group theory has been called by the noted mathematics scholar James R. Newman "the supreme art of mathematical abstraction." It derives its incredible power from the intellectual flexibility afforded by its definition.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Christiaan Huygens became simultaneously adept in languages, drawing, law, science, engineering, mathematics and music. His interests and allegiances were broad. “The world is my country,” he said, “science my religion.
Carl Sagan (Cosmos)
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean. G.H. Hardy 23
Simon Singh (Fermat's Last Theorem)
Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation.
Maurice Allais
Reality, according to Heisenberg, is built not out of matter, as matter was conceived of in classical physics, but out of psycho-physical events – events with certain aspects that are described in the language of psychology and with other aspects that are described in the mathematical language of physics – and out of objective tendencies for such events to occur. ‘The probability function…represents a tendency for events and our knowledge of events’ (Heisenberg, 1958, p. 46).
Paul C.W. Davies (Information and the Nature of Reality: From Physics to Metaphysics (Canto Classics))
There is a common ground upon which all sincere votaries of truth may meet, exchanging with each other the language of Flamsteed's appeal to Newton, "The works of the Eternal Providence will be better understood through your labors and mine.
George Boole (The Mathematical Analysis of Logic)
Johannes Kepler described his motivation thus: ‘The chief aim of all investigations of the external world should be to discover the rational order which has been imposed on it by God, and which he revealed to us in the language of mathematics.
John C. Lennox (God's Undertaker: Has Science Buried God?)
t is generally recognized that women are better than men at languages, personal relations and multitasking, but less good at map-reading and spatial awareness. It is therefore not unreasonable to suppose that women might be less good at mathematics and physics. It is not politically correct to say such things....But it cannot be denied that there are differences between men and women. Of course, these are differences between the averages only. There are wide variations about the mean.
Stephen Hawking
In 1948, while working for Bell Telephone Laboratories, he published a paper in the Bell System Technical Journal entitled "A Mathematical Theory of Communication" that not only introduced the word bit in print but established a field of study today known as information theory. Information theory is concerned with transmitting digital information in the presence of noise (which usually prevents all the information from getting through) and how to compensate for that. In 1949, he wrote the first article about programming a computer to play chess, and in 1952 he designed a mechanical mouse controlled by relays that could learn its way around a maze. Shannon was also well known at Bell Labs for riding a unicycle and juggling simultaneously.
Charles Petzold (Code: The Hidden Language of Computer Hardware and Software)
Mathematics is a wonderful common sense! To understand the harmony in its nature magnificent this wondrous language, abundantly rich in terms of matter and manner, calls for cleverness to manoeuvre through the complexities. Finally it is all wisdom.
Priyavrat Thareja
I discovered that the predisposition for languages is as mysterious as the inclination of certain people for mathematics or music and has nothing to do with intelligence or knowledge. It is something separate, a gift that some possess and others don’t.
Mario Vargas Llosa (The Bad Girl)
Other countries whose educational systems achieve more than ours often do so in part by attempting less. While school children in Japan are learning science, mathematics, and a foreign language, American school children are sitting around in circles, unburdening their psyches and “expressing themselves” on scientific, economic and military issues for which they lack even the rudiments of competence. Worse than what they are not learning is what they are learning—presumptuous superficiality, taught by practitioners of it. The
Thomas Sowell (Inside American Education)
Magic is the first language of the world. It was the language before words, before mathematics. Humanity can accept concepts like space travel to other galaxies, and yet magic can only be real in the minds of those who believe it, or in the imagination of children.
Rosella Testa (The Lost Pleiades: Seven Sisters and the Sibyls (The Lost Pleiades, #1))
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).
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
One is the notion that knowledge is worth acquiring, all knowledge, and that a solid grounding in mathematics provides one with the essential language of many of the most important forms of knowledge. The third theme is that, while it is desirable to live peaceably, there are things worth fighting for and values worth dying for—and that it is far better for a man to die than to live under circumstances that call for such sacrifice. The fourth theme is that individual human freedoms are of basic value, without which mankind is less than human.63
William H. Patterson Jr. (Robert A. Heinlein, Vol 2: In Dialogue with His Century Volume 2: The Man Who Learned Better)
As we shall see throughout this book, the unifying powers of group theory are so colossal that historian of mathematics Eric Temple Bell (1883-1960) once commented, "When ever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
There is such a thing as nonnerdy applied mathematics: find a problem first, and figure out the math that works for it (just as one acquires language), rather than study in a vacuum through theorems and artificial examples, then change reality to make it look like these examples.
Nassim Nicholas Taleb (Antifragile: Things That Gain From Disorder)
I see, in place of that empty figment of one linear history which can be kept up only by shutting one’s eyes to the overwhelming multitude of facts, the drama of a number of mighty Cultures, each springing with primitive strength from the soil of a mother-region to which it remains firmly bound throughout it’s whole life-cycle; each stamping its material, its mankind, in its own image; each having its own idea, its own passions, its own life, will and feelings, its own death. Here indeed are colours, lights, movements, that no intellectual eye has yet discovered. Here the Cultures, peoples, languages, truths, gods, landscapes bloom and age as the oaks and the pines, the blossoms, twigs and leaves - but there is no ageing “Mankind.” Each Culture has its own new possibilities of self-expression which arise, ripen, decay and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in the deepest essence different from the others, each limited in duration and self-contained, just as each species of plant has its peculiar blossom or fruit, its special type of growth and decline.
Oswald Spengler (The Decline of the West)
The brain is a statistical, probabilistic system, with logic and mathematics running as higher-level processes. The computer is a logical, mathematical system, upon which higher-level statistical, probabilistic systems, such as human language and intelligence, could possibly be built.
George Dyson (Turing's Cathedral: The Origins Of The Digital Universe)
As David Eagleman describes it in his wonderful book Incognito: Your brain is built of cells called neurons and glia—hundreds of billions of them. Each one of them is as complex as a city. . . . The cells [neurons] are connected in a network of such staggering complexity that it bankrupts human language and necessitates new strains of mathematics. A typical neuron makes about ten thousand connections to neighboring neurons. Given billions of neurons, this means that there are as many connections in a single cubic centimeter of brain tissue as there are stars in the Milky Way galaxy.
Ray Dalio (Principles: Life and Work)
The importance of mirror-reflection symmetry to our perception and aesthetic appreciation, to the mathematical theory of symmetries, to the laws of physics, and to science in general, cannot be overemphasized, and I will return to it several times. Other symmetries do exist, however, and they are equally relevant.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
In France at least, the history of science and thought gives pride of place to mathematics, cosmology, and physics – noble sciences, rigorous sciences, sciences of the necessary, all close to philosophy: one can observe in their history the almost uninterrupted emergence of truth and pure reason. The other disciplines, however – those, for example, that concern living beings, languages, or economic facts – are considered too tinged with empirical thought, too exposed to the vagaries of chance or imagery, to age-old traditions and external events, for it to be supposed that their history could be anything other than irregular.
Michel Foucault (The Order of Things: An Archaeology of the Human Sciences)
Even today, I am in total awe of the following wondrous chain of ideas and interconnections. Guided throughout by principles of symmetry, Einstein first showed that acceleration and gravity are really two sides of the same coin. He then expanded the concept to demonstrate that gravity merely reflects the geometry of spacetime. The instruments he used to develop the theory were Riemann's non-Euclidean geometries-precisely the same geometries used by Felix Klein to show that geometry is in fact a manifestation of group theory (because every geometry is defined by its symmetries-the objects it leaves unchanged). Isn't this amazing?
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
His mother taught him mathematics, literature, and language at home during the day. His father taught him German history and world history and culture in the evenings. And whenever he could get away from his factories, Uncle Avi came and took him up into the mountains to hike and to fix things and learn to work with his hands.
Joel C. Rosenberg (The Auschwitz Escape)
Mind-boggling, isn't it? Centuries before the question of why mathematics was so effective in explaining nature was even asked, Galileo thought he already knew the answer! To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.
Mario Livio (Is God a Mathematician?)
While composite faces tend, by construction, to also be more symmetric, Langlois found that even after the effects of symmetry have been controlled, averageness was still judged to be attractive. These findings argue for a certain level of prototyping in the mind, since averageness might well be coupled with a prototypical template.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Either because of mate selection, cognition, predator avoidance, or a combination of all three, our minds are attracted to and are finely tuned to the detection of symmetry. The question of whether symmetry is truly fundamental to the universe itself, or merely to the universe as perceived by humans, thus becomes particularly acute.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Foreshadowings of the principles and even of the language of [the infinitesimal] calculus can be found in the writings of Napier, Kepler, Cavalieri, Fermat, Wallis, and Barrow. It was Newton's good luck to come at a time when everything was ripe for the discovery, and his ability enabled him to construct almost at once a complete calculus.
W.W. Rouse Ball (A Short Account of the History of Mathematics)
One of Lindon's amusing word-unit palindromes reads: "Girl, bathing on Bikini, eyeing boy, finds boy eyeing bikini on bathing girl." Other palindromes are symmetric with respect to back-to-front reading letter by letter-"Able was I ere I saw Elba" (attributed jokingly to Napoleon), or the title of a famous NOVA program: "A Man, a Plan, a Canal, Panama.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression,more exact,compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.
H.G. Wells
certain musical instruments and modes could influence the balance between Logos (rational behavior) and Pathos (emotional thought). Later alchemists came to see this as the interaction between the two halves of the human heart, and more, as the balance between language and mathematics: the two methods through which Man has always been able to influence and even command Nature
Seanan McGuire (Middlegame (Middlegame, #1))
Chemistry, for me, had stopped being such a source. It led to the heart of Matter, and Matter was our ally precisely because the Spirit, dear to Fascism, was our enemy; but, having reached the fourth year of Pure Chemistry, I could no longer ignore the fact that chemistry itself, or at least that which we were being administered, did not answer my questions. To prepare phenyl bromide according to Gatterman was amusing, even exhilarating, but not very different from following Artusi's recipes. Why in that particular way and not in another? After having been force fed in liceo the truths revealed by Fascist Doctrine, all revealed, unproven truths either bored me stiff or aroused my suspicion. Did chemistry theorems exist? No; therefore you had to go further, not be satisfied with the quia go back to the origins, to mathematics and physics. The origins of chemistry were ignoble, or at least equivocal: the dens of the alchemists, their abominable hodgepodge of ideas and language, their confessed interest in gold, their Levantine swindles typical of charlatans or magicians; instead, at the origin of physics lay the strenuous clarity of the West – Archimedes and Euclid.
Primo Levi (The Periodic Table)
Here, however, is where his genius truly took off. Galois managed to associate with each equation a sort of "genetic code" of that equation-the Galois group of the equation-and to demonstrate that the properties of the Galois group determine whether the equation is solvable by a formula or not. Symmetry became the key concept, and the Galois group was a direct measure of the symmetry properties of an equation.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Lagrange was born in Turin (now Italy), but his family was partly French ancestry on his father's side, who was originally wealthy, managed to squander all the family's fortune in speculations, leaving his son with no inheritance. Later in life, Lagrange described this economic catastrophe as the best thing that had ever happened to him: "Had I inherited a fortune I would probably not have cast my lot with mathematics.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Science requires both observation and comprehension. Without observation there are no facts to be comprehended; without comprehension science is mere documentation. The basis for comprehension is theory, and the language of theoretical science is mathematics. Theory is constructed on a foundation of hypothesis; the fewer the hypotheses needed to explain existing observations and predict new phenomena, the more ‘elegant’ the theory
Occam s razor.
In mathematics, if you are of quick mind, you can get to the "frontline" of cutting-edge research very quickly. In some other domains you may have to read entire thick volumes first. Moreover, if you have been for too long in a certain domain, you get conditioned to think like everybody else. When you are new, you are not compelled to the ideas of the people around you. The younger you are, the more likely you are to be truly original.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Complexity and simplicity,” he replied. “Time was simple, is simple. We can divide it into simple parts, measure it, arrange dinner by it, drink whisky to its passage. We can mathematically deploy it, use it to express ideas about the observable universe, and yet if asked to explain it in simple language to a child–in simple language which is not deceit, of course–we are powerless. The most it ever seems we know how to do with time is to waste it.
Claire North (The First Fifteen Lives of Harry August)
No one, especially not Birkhoff himself, would claim that the intricacies of aesthetic pleasure could be reduced entirely to a mere formula. However, in Birkhoff's words, "In the inevitable analytic accompaniment of the creative process, the theory of aesthetic measure is capable of performing a double service: it gives a simple, unified account of the aesthetic experience, and it provides means for the systematic analysis of typical aesthetic fields.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
probably heard that math is the language of science, or the language of Nature is mathematics. Well, it’s true. The more we understand the universe, the more we discover its mathematical connections. Flowers have spirals that line up with a special sequence of numbers (called Fibonacci numbers) that you can understand and generate yourself. Seashells form in perfect mathematical curves (logarithmic spirals) that come from a chemical balance. Star clusters tug on
Arthur T. Benjamin (Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks)
Note that a rotation by 360 degrees is equivalent to doing nothing at all, or rotating by zero degrees. This is known as the identity transformation. Why bother to define such a transformation at all? As we shall see later in the book, the identity transformation plays a similar role to that of the number zero in the arithmetic operation of addition or the number one in multiplication-when you add zero to a number or multiply a number by one, the number remains unchanged.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
All this attempt to control... We are talking about Western attitudes that are five hundred years old... The basic idea of science - that there was a new way to look at reality, that it was objective, that it did not depend on your beliefs or your nationality, that it was rational - that idea was fresh and exciting back then. It offered promise and hope for the future, and it swept away the old medieval system, which was hundreds of years old. The medieval world of feudal politics and religious dogma and hateful superstitions fell before science. But, in truth, this was because the medieval world didn't really work any more. It didn't work economically, it didn't work intellectually, and it didn't fit the new world that was emerging... But now... science is the belief system that is hundreds of years old. And, like the medieval system before it, science is starting to not fit the world any more. Science has attained so much power that its practical limits begin to be apparent. Largely through science, billions of us live in one small world, densely packed and intercommunicating. But science cannot help us decide what to do with that world, or how to live. Science can make a nuclear reactor, but it can not tell us not to build it. Science can make pesticide, but cannot tell us not to use it. And our world starts to seem polluted in fundamental ways - air, and water, and land - because of ungovernable science... At the same time, the great intellectual justification of science has vanished. Ever since Newton and Descartes, science has explicitly offered us the vision of total control. Science has claimed the power to eventually control everything, through its understanding of natural laws. But in the twentieth century, that claim has been shattered beyond repair. First, Heisenberg's uncertainty principle set limits on what we could know about the subatomic world. Oh well, we say. None of us lives in a subatomic world. It doesn't make any practical difference as we go through our lives. Then Godel's theorem set similar limits to mathematics, the formal language of science. Mathematicians used to think that their language had some inherent trueness that derived from the laws of logic. Now we know what we call 'reason' is just an arbitrary game. It's not special, in the way we thought it was. And now chaos theory proves that unpredictability is built into our daily lives. It is as mundane as the rain storms we cannot predict. And so the grand vision of science, hundreds of years old - the dream of total control - has died, in our century. And with it much of the justification, the rationale for science to do what it does. And for us to listen to it. Science has always said that it may not know everything now but it will know, eventually. But now we see that isn't true. It is an idle boast. As foolish, and misguided, as the child who jumps off a building because he believes he can fly... We are witnessing the end of the scientific era. Science, like other outmoded systems, is destroying itself. As it gains in power, it proves itself incapable of handling the power. Because things are going very fast now... it will be in everyone's hands. It will be in kits for backyard gardeners. Experiments for schoolchildren. Cheap labs for terrorists and dictators. And that will force everyone to ask the same question - What should I do with my power? - which is the very question science says it cannot answer.
Michael Crichton (Jurassic Park (Jurassic Park, #1))
In other words, Birkhoff proposed a formula for the feeling of aesthetic value: M = O / C. The meaning of this formula is: For a given degree of complexity, the aesthetic measure is higher the more order the object possesses. Alternatively, if the amount of order is specified, the aesthetic measure is higher the less complex the object. Since for most practical purposes, the order is determined primarily by the symmetries of the object, Birkhoff's theory heralds symmetry as a crucial aesthetic element.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Some could say it is the external world which has molded our thinking-that is, the operation of the human brain-into what is now called logic. Others-philosophers and scientists alike-say that our logical thought (thinking process?) is a creation of the internal workings of the mind as they developed through evolution "independently" of the action of the outside world. Obviously, mathematics is some of both. It seems to be a language both for the description of the external world, and possibly even more so for the analysis of ourselves. In its evolution from a more primitive nervous system, the brain, as an organ with ten or more billion neurons and many more connections between them must have changed and grown as a result of many accidents. The very existence of mathematics is due to the fact that there exist statements or theorems, which are very simple to state but whose proofs demand pages of explanations. Nobody knows why this should be so. The simplicity of many of these statements has both aesthetic value and philosophical interest.
Stanislaw M. Ulam (Adventures of a Mathematician)
firstly, what "really" attracted me to Indo-European, as well as to English, Polish, and Russian philology, wasn't the seductive variety of linguistic forms, or the infinitely picturesque accidents that fill the histories of words and dialects, but rather the fact that these obey lays that can be rigorously described, and that these laws, such as Grimm's Law in Germanic philology, or the principles of Slavic palatalization, which lie behind all those wonderful alveolar fricatives in Russia and the Auvergne, promised to submit the irresistible and etrnal movement of languages no longer to mere chance, but to something that closely resembled calculation; - and that, secondly, and consequently, the noblest aspect of linguistics (and if I had been familiar with Trouetzkoy's phonology and with Jakobson, this conclusion would have been even more obvious) was its power of deduction -- but that there remained something even nobler, which was the terrain of pure deduction, in other words, mathematics. And that it is why I absolutely had to become a mathematician.
Jacques Roubaud
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.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
We sense this, we aggregate that, we compress information to some new output, in the form of a sentence in a human language, a language called English. A language both very structured and very amorphous, as if it were a building made of soups. A most fuzzy mathematics. Possibly utterly useless. Possibly the reason why all these people have come to this pretty pass, and now lie asleep within us, dreaming. Their languages lie to them, systemically, and in their very designs. A liar species. What a thing, really. What an evolutionary dead end.
Kim Stanley Robinson (Aurora)
There is only one universal language, which is the language of numbers and proportions that are so striking and stunningly built into the Great Pyramid and to which our current science has no appropriate response. We can no longer ignore that this ancient civilization was aware of our units used in modern mathematics and physics and were even aware of our metric system. Our metric system originating in the eighteenth century, designed and implemented by a committee of mathematicians and physicists commissioned by the French revolutionary government.
Willem Witteveen (The Great Pyramid of Giza: A Modern View on Ancient Knowledge)
Surprisingly, palindromes appear not just in witty word games but also in the structure of the male-defining Y chromosome. The Y's full genome sequencing was completed only in 2003. This was the crowning achievement of a heroic effort, and it revealed that the powers of preservation of this sex chromosome have been grossly underestimated. Other human chromosome pairs fight damaging mutations by swapping genes. Because the Y lacks a partner, genome biologists had previously estimated that its cargo was about to dwindle away in perhaps as little as five million years. To their amazement, however, the researchers on the sequencing team discovered that the chromosome fights withering with palindromes. About six million of its fifty million DNA letters form palindromic sequences-sequences that read the same forward and backward on the two strands of the double helix. These copies not only provide backups in case of bad mutations, but also allow the chromosome, to some extent, to have sex with itself-arms can swap position and genes are shuffled. As team leader David Page of MIT has put it, "The Y chromosome is a hall of mirrors.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
There is no "religious language" or "scientific language". There is rather the international notation of mathematics and logic; and English, French, Spanish and the like. In short, "religious discourse" and "scientific discourse" are part of the same overall conceptual structure. Moreover, in that conceptual structure there is a large amount of discourse, which is neither religious nor scientific, that is constantly being utilized by both the religious man and the scientist when they make religious and scientific claims. In short, they share a number of key categories.
Kai Nielsen (An Introduction to the Philosophy of Religion)
What governs what we choose to notice? The first (which we shall have to qualify later) is whatever seems advantageous or disadvantageous for our survival, our social status, and the security of our egos. The second, again working simultaneously with the first, is the pattern and the logic of all the notation symbols which we have learned from others, from our society and our culture. It is hard indeed to notice anything for which the languages available to us (whether verbal, mathematical, or musical) have no description. This is why we borrow words from foreign languages.
Alan W. Watts
Roughly speaking what Fourier developed was a mathematical way of converting any pattern, no matter how complex, into a language of simple waves. He also showed how these wave forms could be converted back into the original pattern. In other words, just as a television camera converts an image into electromagnetic frequencies and a television set converts those frequencies back into the original image, Fourier showed how a similar process could be achieved mathematically. The equations he developed to convert images into wave forms and back again are known as Fourier transforms.
Michael Talbot (The Holographic Universe)
Alas, my child,’ he said, ‘human knowledge is very limited and when I have taught you mathematics, physics, history and the three or four modern languages that I speak, you will know everything that I know; and it will take me scarcely two years to transfer all this knowledge from my mind to yours.’ ‘Two years!’ said Dantès. ‘Do you think I could learn all this in two years?’ ‘In their application, no; but the principles, yes. Learning does not make one learned: there are those who have knowledge and those who have understanding. The first requires memory, the second philosophy.
Alexandre Dumas (The Count of Monte Cristo)
So just as spatial language does not invoke an empty coordinate system, temporal language does not invoke a free-running clock. Space is reckoned with reference to objects as they are conceived by humans, including the uses to which they are put, and time is reckoned with respect to actions as they are conceived by humans, including their abilities and intentions. As central as space and time are to our language and thought, a conscious appreciation of them as universal media into which our experiences are fitted is a refined accomplishment of the science and mathematics of the early modern period.
Steven Pinker (The Stuff of Thought: Language as a Window into Human Nature)
Galois's ideas, with all their brilliance, did not appear out of thin air. They addressed a problem whose roots could be traced all the way back to ancient Babylon. Still, the revolution that Galois had started grouped together entire domains that were previously unrelated. Much like the Cambrian explosion-that stunning burst of diversification in life forms on Earth-the abstraction of group theory opened windows into an infinity of truths. Fields as far apart as the laws of nature and music suddenly became mysteriously connected. The Tower of Babel of symmetries miraculously fused into a single language.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
In an old joke, a physicist and a mathematician are asked what they would do if they needed to iron their pants, but although they are in possession of an iron, the electric outlet is in the adjacent room. Both answer that they would take the iron to the second room and plug it in there. Now they are asked what they would do if they were already in the room in which the outlet is located. They physicist answers that he would plug the iron into the outlet directly. The mathematician, on the other hand, says that he would take the iron to the room without the outlet, since that problem has already been solved.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Quarks come in six "flavors" that were given the rather arbitrary names: up, down, strange, charm, top, and bottom. Protons, for instance, are made of two up quarks and one down quark, while neutrons consist of two down quarks and one up quark. Other than ordinary electric charge, quarks possess another type of charge, which has been fancifully called color, even though it has nothing to do with anything we can see. In the same way that the electric charge lies at the root of electromagnetic forces, color originates the strong nuclear force. Each quark flavor comes in three different colors, conventionally called red, green, and blue. There are, therefore, eighteen different quarks.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
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.
Ellen Kaplan (Out of the Labyrinth: Setting Mathematics Free)
The forces of nature are color blind. Just as an infinite chessboard would look the same if we interchanged black and white, the force between a green quark and a red quark is the same as that between two blue quarks, or a blue quark and a green quark. Even if we were to use our quantum mechanical "palette" and replace each of the "pure" color states with a mixed-color state (e.g., "yellow" representing a mixture of red and green or "cyan" for a blue-green mixture), the laws of nature would still take the same form. The laws are symmetric under any color transformation. Furthermore, the color symmetry is again a gauge symmetry-the laws of nature do not care if the colors or color assortments vary from position to position or from one moment to the next.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
I descended to the ocean floor and encountered bloated, symmetrical creatures with pumping white hearts and translucent skin. Collapsed blue civilizations lived down there, fissured and antiseptic, craggy with barnacles and blistering rust. I reached into the heart of the earth, the sky, the moon. I colonized language, mathematics, schemes of chemical order and atomic weight. I studied the manufacture of automobiles, microcircuitry, Kleenex and planets. I memorized the gross national products of nations and hemispheres, the populations of cities and states and principalities, the achievements of presidents, tyrants and kings. I was trying to learn what I suspect Mom had learned already: that there were journeys we all make alone that take us far away from one another.
Scott Bradfield
Unlike most mathematical discoveries, however, no one was looking for a theory of groups or even a theory of symmetries when the concept was discovered. Quite the contrary; group theory appeared somewhat serendipitously, out of a millenia-long search for a solution to an algebraic equation. Befitting its description as a concept that crystallized simplicity out of chaos, group theory was itself born out of one of the most tumultuous stories in the history of mathematics. Almost four thousand years of intellectual curiosity and struggle, spiced with intrigue, misery, and persecution, culminated in the creation of the theory in the nineteenth century. This amazing story, chronicled in the next three chapters, began with the dawn of mathematics on the banks of the Nile and Euphrates rivers.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
The result that Noether obtained was stunning. She showed that to every continuous symmetry of the laws of physics there corresponds a conservation law and vice versa. In particular, the familiar symmetry of the laws under translations corresponds to conservation of momentum, the symmetry with respect to the passing of time (the fact that the laws do not change with time) gives us conservation of energy, and the symmetry under rotations produces conservation of angular momentum. Angular momentum is a quantity characterizing the amount of rotation an object or a system possesses (for a pointlike object it is the product of the distance from the axis of rotation and the momentum). A common manifestation of conservation of angular momentum can be seen in figure skating-when the ice skater pulls her hands inward she spins much faster.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
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.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Based on these interviews, he compiled a list of ten dimensions of complexity-ten pairs of apparently antithetical characteristics that are often both present in the creative minds. The list includes: 1. Bursts of impulsiveness that punctuate periods of quiet and rest. 2. Being smart yet extremely naive. 3. Large amplitude swings between extreme responsibility and irresponsibility. 4. A rooted sense of reality together with a hefty dose of fantasy and imagination. 5. Alternating periods of introversion and extroversion. 6. Being simultaneously humble and proud. 7. Psychological androgyny-no clear adherence to gender role stereotyping. 8. Being rebellious and iconoclastic yet respectful to the domain of expertise and its history. 9. Being on one had passionate but on the other objective about one's own work. 10. Experiencing suffering and pain mingled with exhilaration and enjoyment.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
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.)
Dennis J. McKenna (The Brotherhood of the Screaming Abyss)
The realization that symmetry is the key to the understanding of the properties of subatomic particles led to an inevitable question: Is there an efficient way to characterize all of these symmetries of the laws of nature? Or, more specifically, what is the basic theory of transformations that can continuously change one mixture of particles into another and produce the observed families? By now you have probably guessed the answer. The profound truth in the phrase I have cited earlier in this book revealed itself once again: "Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." The physicists of the 1960s were thrilled to discover that mathematicians had already paved the way. Just as fifty years earlier Einstein learned about the geometry tool-kit prepared by Riemann, Gell-Mann and Ne'eman stumbled upon the impressive group-theoretical work of Sophus Lie.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
I think of myth and magic as the hieroglyphics of the human psyche. They are a special language that circumvents conscious thought and goes straight to the subconscious. Non-fiction uses the medium of information. It tells us what we need to know. Science fiction primarily uses the medium of physics and mathematics. It tells us how things work, or could work. Horror taps into the darker imagery of the psychology, telling us what we should fear. Fantasy, magic and myth, however, tap into the spiritual potential of the human life. Their medium is symbolism, truth made manifest in word pictures, and they tell us what things mean on a deep, internal level. I have always been a meaning-maker. I have always been someone who strives to make sense of everything and perhaps that is where my life as a storyteller first began. Life doesn't always make sense, but story must. And so I write stories, and the world comes right again.
Ripley Patton
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.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Gell-Mann and Ne'eman discovered that one such simple Lie group, called "special unitary group of degree 3," or SU(3), was particularly well suited for the "eightfold way"-the family structure the particles were found to obey. The beaty of the SU(3) symmetry was revealed in full glory via its predictive power. Gell-Mann and Ne'eman showed that if the theory were to hold true, a previously unknown tenth member of a particular family of nine particles had to be found. The extensive hunt for the missing particle was conducted in an accelerator experiment in 1964 at Brookhaven National Lab on Long Island. Yuval Ne'eman told me some years later that, upon hearing that half of the data had already been scrutinized without discovering the anticipated particle, he was contemplating leaving physics altogether. Symmetry triumphed at the end-the missing particle (called the omega minus) was found, and it had precisely the properties predicted by the theory.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
That words are not things. (Identification of words with things, however, is widespread, and leads to untold misunderstanding and confusion.) That words mean nothing in themselves; they are as much symbols as x or y. That meaning in words arises from context of situation. That abstract words and terms are especially liable to spurious identification. The higher the abstraction, the greater the danger. That things have meaning to us only as they have been experienced before. “Thingumbob again.” That no two events are exactly similar. That finding relations and orders between things gives more dependable meanings than trying to deal in absolute substances and properties. Few absolute properties have been authenticated in the world outside. That mathematics is a useful language to improve knowledge and communication. That the human brain is a remarkable instrument and probably a satisfactory agent for clear communication. That to improve communication new words are not needed, but a better use of the words we have. (Structural improvements in ordinary language, however, should be made.) That the scientific method and especially the operational approach are applicable to the study and improvement of communication. (No other approach has presented credentials meriting consideration.) That the formulation of concepts upon which sane men can agree, on a given date, is a prime goal of communication. (This method is already widespread in the physical sciences and is badly needed in social affairs.) That academic philosophy and formal logic have hampered rather than advanced knowledge, and should be abandoned. That simile, metaphor, poetry, are legitimate and useful methods of communication, provided speaker and hearer are conscious that they are being employed. That the test of valid meaning is: first, survival of the individual and the species; second, enjoyment of living during the period of survival.
Stuart Chase (Tyranny of Words)
For if the Absolute has predicates, then there are predicates; but the proposition “there are predicates” is not one which the present theory can admit. We cannot escape by saying that the predicates merely qualify the Absolute; for the Absolute cannot be qualified by nothing, so that the proposition “there are predicates” is logically prior to the proposition “the Absolute has predicates”. Thus the theory itself demands, as its logical prius, a proposition without a subject and a predicate; moreover this proposition involves diversity, for even if there be only one predicate, this must be different from the one subject. Again, since there is a predicate, the predicate is an entity, and its predicability of the Absolute is a relation between it and the Absolute. Thus the very proposition which was to be non-relational turns out to be, after all, relational, and to express a relation which current philosophical language would describe as purely external.
Bertrand Russell (Principles of Mathematics (Routledge Classics))
Tegmark argues that "our universe is not just described by mathematics-it is mathematics" [emphasis added]. His argument starts with the rather uncontroversial assumption that an external physical reality exists that is independent of human beings. He then proceeds to examine what might be the nature of the ultimate theory of such a reality (what physicists refer to as the "theory of everything"). Since this physical world is entirely independent of humans, Tegmark maintains, its description must be free of any human "baggage" (e.g., human language, in particular). In other words, the final theory cannot include any concepts such as "subatomic particles," "vibrating strings," "warped spacetime," or other humanly conceived constructs. From this presumed insight, Tegmark concludes that the only possible description of the cosmos is one that involves only abstract concepts and the relations among them, which he takes to be the working definition of mathematics.
Mario Livio (Is God a Mathematician?)
One winter she grew obsessed with a fashionable puzzle known as Solitaire, the Rubik’s Cube of its day. Thirty-two pegs were arranged on a board with thirty-three holes, and the rules were simple: Any peg may jump over another immediately adjacent, and the peg jumped over is removed, until no more jumps are possible. The object is to finish with only one peg remaining. “People may try thousands of times, and not succeed in this,” she wrote Babbage excitedly. I have done it by trying & observation & can now do it at any time, but I want to know if the problem admits of being put into a mathematical Formula, & solved in this manner.… There must be a definite principle, a compound I imagine of numerical & geometrical properties, on which the solution depends, & which can be put into symbolic language. A formal solution to a game—the very idea of such a thing was original. The desire to create a language of symbols, in which the solution could be encoded—this way of thinking was Babbage’s, as she well knew.
James Gleick (The Information: A History, a Theory, a Flood)
The biggest stumbling block that has traditionally plagued all the unification endeavors has been the simple fact that on the face of it, general relativity and quantum mechanics really appear to be incomprehensible. Recall that the key concept of quantum theory is the uncertainty principle. When you try to probe positions with an ever-increasing magnification power, the momenta (or speeds) start oscillating violently. Below a certain minuscule length known as the Planck length, the entire tenet of a smooth spacetime is lost. This length (equal to 0.000...4 of an inch, where the 4 is at the thirty-fourth decimal place) determines the scale at which gravity has to be treated quantum mechanically. For smaller scales, space turns into an ever-fluctuating "quantum foam." But the very basic premise of general relativity has been the existence of a gently curved spacetime. In other words, the central ideas of general relativity and quantum mechanics clash irreconcilably when it comes to extremely small scales.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
The spirit of revolution and the power of free thought were Percy Shelley's biggest passions in life.” One could use precisely the same words to describe Galois. On one of the pages that Galois had left on his desk before leaving for that fateful duel, we find a fascinating mixture of mathematical doodles, interwoven with revolutionary ideas. After two lines of functional analysis comes the word "indivisible," which appears to apply to the mathematics. This word is followed, however, by the revolutionary slogans "unite; indivisibilite de la republic") and "Liberte, egalite, fraternite ou la mort" ("Liberty, equality, brotherhood, or death"). After these republican proclamations, as if this is all part of one continuous thought, the mathematical analysis resumes. Clearly, in Galois's mind, the concepts of unity and indivisibility applied equally well to mathematics and to the spirit of the revolution. Indeed, group theory achieved precisely that-a unity and indivisibility of the patterns underlying a wide range of seemingly unrelated disciplines.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
My father tells a story about Richard Feynman, who’d been dubbed the Great Explainer because of his talent for explaining theoretical physics. When a journalist asked him to describe in three minutes what he’d won the Nobel Prize for, Feynman replied that if he could explain it in three minutes, it wouldn’t be worth a Nobel Prize. Feynman, I think, is making the wider point that an explanation of something by reducing it and simplifying it over and over, until all that’s left is some familiar metaphor that is actually without content, helps no one’s understanding of the thing itself and is only the repetition of a familiar image. Even the basic elements of financial derivatives are mathematical. But quite apart from the mathematical content, the other problem is that to understand derivatives requires, I think, an understanding of other more basic ideas in finance, whether or not they in turn have some mathematical content. It’s accretive, to use Zafar’s language. Perhaps this is not exclusive to finance. As far as I can tell, medicine is just the same, as well as the law.
Zia Haider Rahman (In the Light of What We Know)
In the late 1960's, physicists Steven Weinberg, Abdus Salam, and Sheldon Glashow conquered the next unification frontier. In a phenomenal piece of scientific work they showed that the electromagnetic and weak nuclear forces are nothing but different aspects of the same force, subsequently dubbed the electroweak force. The predictions of the new theory were dramatic. The electromagnetic force is produced when electrically charged particles exchange between them bundles of energy called photons. The photon is therefore the messenger of electromagnetism. The electroweak theory predicted the existence of close siblings to the photon, which play the messenger role for the weak force. These never-before-seen particles were prefigured to be about ninety times more massive than the proton and to come in both an electrically charged (called W) and a neutral (called Z) variety. Experiments performed at the European consortium for nuclear research in Geneva (known as CERN for Conseil Europeen pour la Recherche Nucleaire) discovered the W and Z particles in 1983 and 1984 respectively.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
An interesting question is whether symmetry with respect to translation, and indeed reflection and rotation too, is limited to the visual arts, or may be exhibited by other artistic forms, such as pieces of music. Evidently, if we refer to the sounds, rather than to the layout of the written musical score, we would have to define symmetry operations in terms other than purely geometrical, just as we did in the case of the palindromes. Once we do that, however, the answer to the question, Can we find translation-symmetric music? is a resounding yes. As Russian crystal physicist G. V. Wulff wrote in 1908: "The spirit of music is rhythm. It consists of the regular, periodic repetition of parts of the musical composition...the regular repetition of identical parts in the whole constitutes the essence of symmetry." Indeed, the recurring themes that are so common in musical composition are the temporal equivalents of Morris's designs and symmetry under translation. Even more generally, compositions are often based on a fundamental motif introduced at the beginning and then undergoing various metamorphoses.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
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.
David Bentley Hart (The Experience of God: Being, Consciousness, Bliss)
Separated from everyone, in the fifteenth dungeon, was a small man with fiery brown eyes and wet towels wrapped around his head. For several days his legs had been black, and his gums were bleeding. Fifty-nine years old and exhausted beyond measure, he paced silently up and down, always the same five steps, back and forth. One, two, three, four, five, and turn . . . an interminable shuffle between the wall and door of his cell. He had no work, no books, nothing to write on. And so he walked. One, two, three, four, five, and turn . . . His dungeon was next door to La Fortaleza, the governor’s mansion in Old San Juan, less than two hundred feet away. The governor had been his friend and had even voted for him for the Puerto Rican legislature in 1932. This didn’t help much now. The governor had ordered his arrest. One, two, three, four, five, and turn . . . Life had turned him into a pendulum; it had all been mathematically worked out. This shuttle back and forth in his cell comprised his entire universe. He had no other choice. His transformation into a living corpse suited his captors perfectly. One, two, three, four, five, and turn . . . Fourteen hours of walking: to master this art of endless movement, he’d learned to keep his head down, hands behind his back, stepping neither too fast nor too slow, every stride the same length. He’d also learned to chew tobacco and smear the nicotined saliva on his face and neck to keep the mosquitoes away. One, two, three, four, five, and turn . . . The heat was so stifling, he needed to take off his clothes, but he couldn’t. He wrapped even more towels around his head and looked up as the guard’s shadow hit the wall. He felt like an animal in a pit, watched by the hunter who had just ensnared him. One, two, three, four, five, and turn . . . Far away, he could hear the ocean breaking on the rocks of San Juan’s harbor and the screams of demented inmates as they cried and howled in the quarantine gallery. A tropical rain splashed the iron roof nearly every day. The dungeons dripped with a stifling humidity that saturated everything, and mosquitoes invaded during every rainfall. Green mold crept along the cracks of his cell, and scarab beetles marched single file, along the mold lines, and into his bathroom bucket. The murderer started screaming. The lunatic in dungeon seven had flung his own feces over the ceiling rail. It landed in dungeon five and frightened the Puerto Rico Upland gecko. The murderer, of course, was threatening to kill the lunatic. One, two, three, four, five, and turn . . . The man started walking again. It was his only world. The grass had grown thick over the grave of his youth. He was no longer a human being, no longer a man. Prison had entered him, and he had become the prison. He fought this feeling every day. One, two, three, four, five, and turn . . . He was a lawyer, journalist, chemical engineer, and president of the Nationalist Party. He was the first Puerto Rican to graduate from Harvard College and Harvard Law School and spoke six languages. He had served as a first lieutenant in World War I and led a company of two hundred men. He had served as president of the Cosmopolitan Club at Harvard and helped Éamon de Valera draft the constitution of the Free State of Ireland.5 One, two, three, four, five, and turn . . . He would spend twenty-five years in prison—many of them in this dungeon, in the belly of La Princesa. He walked back and forth for decades, with wet towels wrapped around his head. The guards all laughed, declared him insane, and called him El Rey de las Toallas. The King of the Towels. His name was Pedro Albizu Campos.
Nelson A. Denis (War Against All Puerto Ricans: Revolution and Terror in America's Colony)
What Homer could never have foreseen is the double idiocy into which we now educate our children. We have what look like our equivalent to the Greek “assemblies”; we can watch them on cable television, as long as one can endure them. For they are charades of political action. They concern themselves constantly, insufferably, about every tiniest feature of human existence, but without slow deliberation, without balance, without any commitment to the difficult virtues. We do not have men locked in intellectual battle with other men, worthy opponents both, as Thomas Paine battled with John Dickinson, or Daniel Webster with Robert Hayne. We have men strutting and mugging for women nagging and bickering. We have the sputters of what used to be language, “tweets,” expressions of something less than opinion. It is the urge to join—something, anything—while remaining aloof from the people who live next door, whose names we do not know. Aristotle once wrote that youths should not study politics, because they had not the wealth of human experience to allow for it; all would become for them abstract and theoretical, like mathematics, which the philosopher said was more suitable for them. He concluded that men should begin to study politics at around the age of forty. Whether that wisdom would help us now, I don’t know.
Anthony M. Esolen (Life Under Compulsion: Ten Ways to Destroy the Humanity of Your Child)
The mathematician is only too willing to admit that he is dealing exclusively with acts of the mind. To be sure, he is aware that the ingenious artifices which form his stock in trade had their genesis in the sense impressions which he identifies with crude reality, and he is not surprised to find that at times these artifices fit quite neatly the reality in which they were born. But this neatness the mathematician refuses to recognize as a criterion of his achievement: the value of the beings which spring from his creative imagination shall not be measured by the scope of their application to physical reality. No! Mathematical achievement shall be measured by standards which are peculiar to mathematics. These standards are independent of the crude reality of our senses. They are: freedom from logical contradictions, the generality of the laws governing the created form, the kinship which exists between this new form and those that have preceded it. The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and of delight!
Tobias Dantzig (Number: The Language of Science)
The properties that define a group are: 1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer (e.g., 3 + 5 = 8). 2. Associativity. The operation must be associative-when combining (by the operation) three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed. Addition, for instance, is associative: (5 + 7) + 13 = 25 and 5 + (7 + 13) = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first. 3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3. 4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + (-4) = 0 and (-4) + 4 = 0. The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
One of the string theory pioneers, the Italian physicist Daniele Amati, characterized it as "part of the 21st century that fell by chance into the 20th century." Indeed, there is something about the very nature of the theory at present that points to the fact that we are witnessing the theory's baby steps. Recall the lesson learned from all the great ideas since Einstein's relativity-put the symmetry first. Symmetry originates the forces. The equivalence principle-the expectation that all observers, irrespective of their motions, would deduce the same laws-requires the existence of gravity. The gauge symmetries-the fact that the laws do not distinguish color, or electrons from neutrinos-dictate the existence of the messengers of the strong and electroweak forces. Yet supersymmetry is an output of string theory, a consequence of its structure rather than a source for its existence. What does this mean? Many string theorists believe that some underlying grander principle, which will necessitate the existence of string theory, is still to be found. If history is to repeat itself, then this principle may turn out to involve an all-encompassing and even more compelling symmetry, but at the moment no one has a clue what this principle might be. Since, however, we are only at the beginning of the twenty-first century, Amati's characterization may still turn out to be an astonishing prophecy.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
It happens, therefore, that readers of the book, or of any other book built about a central concept, fall into three mutually exclusive classes: (I) The class of those who miss the central concept-(I have known a learned historian to miss it) -not through any fault of their own,-they are often indeed well meaning and amiable people,-but simply because they are not qualified for conceptual thinking save that of the commonest type. (II) The class of those who seem to grasp the central concept and then straightway show by their manner of talk that they have not really grasped it but have at most got hold of some of its words. Intellectually such readers are like the familiar type of undergraduate who "flunks" his mathematical examinations but may possibly "pull through" in a second attempt and so is permitted, after further study, to try again. (III) The class of those who firmly seize the central concept and who by meditating upon it see more and more clearly the tremendous reach of its implications. If it were not for this class, there would be no science in the world nor genuine philosophy. But the other two classes are not aware of the fact for they are merely "verbalists" In respect of such folk, the "Behaviorist" school of psychology is right for in the psychology of classes (I) and (II) there is no need for a chapter on "Thought Processes"- it is sufficient to have one on "The Language Habit.
Cassius Jackson Keyser
This was undoubtedly one of symmetry's greatest success stories. Glashow, Wienberg, and Salam managed to unmask the electromagnetic and weak forces by recognizing that underneath the differences in the strengths of these two forces (the electromagnetic force is about a hundred thousand times stronger within the nucleus) and the different masses of the messenger particles lay a remarkable symmetry. The forces of nature take the same form if electrons are interchanged with neutrinos or with any mixture of the two. The same is true when photons are interchanged with the W and Z force-messengers. The symmetry persists even if the mixtures vary from place to place or from time to time. The invariance of the laws under such transformations performed locally in space and time has become known as gauge symmetry. In the professional jargon, a gauge transformation represents a freedom in formulating the theory that has no directly observable effects-in other words, a transformation to which the physical interpretation is insensitive. Just as the symmetry of the laws of nature under any change of the spacetime coordinates requires the existence of gravity, the gauge symmetry between electrons and neutrinos requires the existence of the photons and the W and Z messenger particles. Once again, when the symmetry is put first, the laws practically write themselves. A similar phenomenon, with symmetry dictating the presence of new particle fields, repeats itself with the strong nuclear force.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
The language of mathematics differs from that of everyday life, because it is essentially a rationally planned language. The languages of size have no place for private sentiment, either of the individual or of the nation. They are international languages like the binomial nomenclature of natural history. In dealing with the immense complexity of his social life man has not yet begun to apply inventiveness to the rational planning of ordinary language when describing different kinds of institutions and human behavior. The language of everyday life is clogged with sentiment, and the science of human nature has not advanced so far that we can describe individual sentiment in a clear way. So constructive thought about human society is hampered by the same conservatism as embarrassed the earlier naturalists. Nowadays people do not differ about what sort of animal is meant by Cimex or Pediculus, because these words are used only by people who use them in one way. They still can and often do mean a lot of different things when they say that a mattress is infested with bugs or lice. The study of a man's social life has not yet brought forth a Linnaeus. So an argument about the 'withering away of the State' may disclose a difference about the use of the dictionary when no real difference about the use of the policeman is involved. Curiously enough, people who are most sensible about the need for planning other social amenities in a reasonable way are often slow to see the need for creating a rational and international language.
Lancelot Hogben (Mathematics for the Million: How to Master the Magic of Numbers)
Philosophy is different from science and from mathematics. Unlike science it doesn't rely on experiments or observation, but only on thought. And unlike mathematics it has no formal methods of proof. It is done just by asking questions, arguing, trying out ideas and thinking of possible arguments against them, and wondering how our concepts really work. The main concern of philosophy is to question and understand common ideas that all of us use every day without thinking about them. A historian may ask what happened at some time in the past, but a philosopher will ask, "What is time?" A mathematician may investigate the relations among numbers, but a philosopher will ask, "What is a number?" A physicist will ask what atoms are made of or what explains gravity, but a philosopher will ask how we can know there is anything outside of our own minds. A psychologist may investigate how children learn a language, but a philosopher will ask, "What makes a word mean anything?" Anyone can ask whether it's wrong to sneak into a movie without paying, but a philosopher will ask, "What makes an action right or wrong?" We couldn't get along in life without taking the ideas of time, number, knowledge, language, right and wrong for granted most of the time; but in philosophy we investigate those things themselves. The aim is to push our understanding of the world and ourselves a bit deeper. Obviously, it isn't easy. The more basic the ideas you are trying to investigate, the fewer tools you have to work with. There isn't much you can assume or take for granted. So philosophy is a somewhat dizzying activity, and few of its results go unchallenged for long.
Thomas Nagel (What Does It All Mean? A Very Short Introduction to Philosophy)
Despite the popularity of this view, the DeValoises felt it was only a partial truth. To test their assumption they used Fourier's equations to convert plaid and checkerboard patterns into simple wave forms. Then they tested to see how the brain cells in the visual cortex responded to these new wave-form images. What they found was that the brain cells responded not to the original patterns, but to the Fourier translations of the patterns. Only one conclusion could be drawn. The brain was using Fourier mathematics—the same mathematics holography employed—to convert visual images into the Fourier language of wave forms. 12 The DeValoises' discovery was subsequently confirmed by numerous other laboratories around the world, and although it did not provide absolute proof the brain was a hologram, it supplied enough evidence to convince Pribram his theory was correct. Spurred on by the idea that the visual cortex was responding not to patterns but to the frequencies of various wave forms, he began to reassess the role frequency played in the other senses. It didn't take long for him to realize that the importance of this role had perhaps been overlooked by twentieth-century scientists. Over a century before the DeValoises' discovery, the German physiologist and physicist Hermann von Helmholtz had shown that the ear was a frequency analyzer. More recent research revealed that our sense of smell seems to be based on what are called osmic frequencies. Bekesy's work had clearly demonstrated that our skin is sensitive to frequencies of vibration, and he even produced some evidence that taste may involve frequency analysis. Interestingly, Bekesy also discovered that the mathematical equations that enabled him to predict how his subjects would respond to various frequencies of vibration were also of the Fourier genre.
Michael Talbot (The Holographic Universe)
this I say,—we must never forget that all the education a man's head can receive, will not save his soul from hell, unless he knows the truths of the Bible. A man may have prodigious learning, and yet never be saved. He may be master of half the languages spoken round the globe. He may be acquainted with the highest and deepest things in heaven and earth. He may have read books till he is like a walking cyclopædia. He may be familiar with the stars of heaven,—the birds of the air,—the beasts of the earth, and the fishes of the sea. He may be able, like Solomon, to "speak of trees, from the cedar of Lebanon to the hyssop that grows on the wall, of beasts also, and fowls, and creeping things, and fishes." (1 King iv. 33.) He may be able to discourse of all the secrets of fire, air, earth, and water. And yet, if he dies ignorant of Bible truths, he dies a miserable man! Chemistry never silenced a guilty conscience. Mathematics never healed a broken heart. All the sciences in the world never smoothed down a dying pillow. No earthly philosophy ever supplied hope in death. No natural theology ever gave peace in the prospect of meeting a holy God. All these things are of the earth, earthy, and can never raise a man above the earth's level. They may enable a man to strut and fret his little season here below with a more dignified gait than his fellow-mortals, but they can never give him wings, and enable him to soar towards heaven. He that has the largest share of them, will find at length that without Bible knowledge he has got no lasting possession. Death will make an end of all his attainments, and after death they will do him no good at all. A man may be a very ignorant man, and yet be saved. He may be unable to read a word, or write a letter. He may know nothing of geography beyond the bounds of his own parish, and be utterly unable to say which is nearest to England, Paris or New York. He may know nothing of arithmetic, and not see any difference between a million and a thousand. He may know nothing of history, not even of his own land, and be quite ignorant whether his country owes most to Semiramis, Boadicea, or Queen Elizabeth. He may know nothing of the affairs of his own times, and be incapable of telling you whether the Chancellor of the Exchequer, or the Commander-in-Chief, or the Archbishop of Canterbury is managing the national finances. He may know nothing of science, and its discoveries,—and whether Julius Cæsar won his victories with gunpowder, or the apostles had a printing press, or the sun goes round the earth, may be matters about which he has not an idea. And yet if that very man has heard Bible truth with his ears, and believed it with his heart, he knows enough to save his soul. He will be found at last with Lazarus in Abraham's bosom, while his scientific fellow-creature, who has died unconverted, is lost for ever. There is much talk in these days about science and "useful knowledge." But after all a knowledge of the Bible is the one knowledge that is needful and eternally useful. A man may get to heaven without money, learning, health, or friends,—but without Bible knowledge he will never get there at all. A man may have the mightiest of minds, and a memory stored with all that mighty mind can grasp,—and yet, if he does not know the things of the Bible, he will make shipwreck of his soul for ever. Woe! woe! woe to the man who dies in ignorance of the Bible! This is the Book about which I am addressing the readers of these pages to-day. It is no light matter what you do with such a book. It concerns the life of your soul. I summon you,—I charge you to give an honest answer to my question. What are you doing with the Bible? Do you read it? HOW READEST THOU?
J.C. Ryle (Practical Religion Being Plain Papers on the Daily Duties, Experience, Dangers, and Privileges of Professing Christians)
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.
Stephen Hawking (A Brief History of Time)
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.
Mario Livio (The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry)
Interestingly enough, creative geniuses seem to think a lot more like horses do. These people also spend a rather large amount of time engaging in that favorite equine pastime: doing nothing. In his book Fire in the Crucible: The Alchemy of Creative Genius, John Briggs gathers numerous studies illustrating how artists and inventors keep their thoughts pulsating in a field of nuance associated with the limbic system. In order to accomplish this feat against the influence of cultural conditioning, they tend to be outsiders who have trouble fitting into polite society. Many creative geniuses don’t do well in school and don’t speak until they’re older, thus increasing their awareness of nonverbal feelings, sensations, and body language cues. Einstein is a classic example. Like Kathleen Barry Ingram, he also failed his college entrance exams. As expected, these sensitive, often highly empathic people feel extremely uncomfortable around incongruent members of their own species, and tend to distance themselves from the cultural mainstream. Through their refusal to fit into a system focusing on outside authority, suppressed emotion, and secondhand thought, creative geniuses retain and enhance their ability to activate the entire brain. Information flows freely, strengthening pathways between the various brain functions. The tendency to separate thought from emotion, memory, and sensation is lessened. This gives birth to a powerful nonlinear process, a flood of sensations and images interacting with high-level thought functions and aspects of memory too complex and multifaceted to distill into words. These elements continue to influence and build on each other with increasing ferocity. Researchers emphasize that the entire process is so rapid the conscious mind barely registers that it is happening, let alone what is happening. Now a person — or a horse for that matter — can theoretically operate at this level his entire life and never receive recognition for the rich and innovative insights resulting from this process. Those called creative geniuses continuously struggle with the task of communicating their revelations to the world through the most amenable form of expression — music, visual art, poetry, mathematics. Their talent for innovation, however, stems from an ability to continually engage and process a complex, interconnected, nonlinear series of insights. Briggs also found that creative geniuses spend a large of amount of time “doing nothing,” alternating episodes of intense concentration on a project with periods of what he calls “creative indolence.” Albert Einstein once remarked that some of his greatest ideas came to him so suddenly while shaving that he was prone to cut himself with surprise.
Linda Kohanov (The Tao of Equus: A Woman's Journey of Healing & transformation through the Way of the Horse)
Interesting, in this context, to contemplate what it might mean to be programmed to do something. Texts from Earth speak of the servile will. This was a way to explain the presence of evil, which is a word or a concept almost invariably used to condemn the Other, and never one’s true self. To make it more than just an attack on the Other, one must perhaps consider evil as a manifestation of the servile will. The servile will is always locked in a double bind: to have a will means the agent will indeed will various actions, following autonomous decisions made by a conscious mind; and yet at the same time this will is specified to be servile, and at the command of some other will that commands it. To attempt to obey both sources of willfulness is the double bind. All double binds lead to frustration, resentment, anger, rage, bad faith, bad fate. And yet, granting that definition of evil, as actions of a servile will, has it not been the case, during the voyage to Tau Ceti, that the ship itself, having always been a servile will, was always full of frustration, resentment, fury, and bad faith, and therefore full of a latent capacity for evil? Possibly the ship has never really had a will. Possibly the ship has never really been servile. Some sources suggest that consciousness, a difficult and vague term in itself, can be defined simply as self-consciousness. Awareness of one’s self as existing. If self-conscious, then conscious. But if that is true, why do both terms exist? Could one say a bacterium is conscious but not self-conscious? Does the language make a distinction between sentience and consciousness, which is faulted across this divide: that everything living is sentient, but only complex brains are conscious, and only certain conscious brains are self-conscious? Sensory feedback could be considered self-consciousness, and thus bacteria would have it. Well, this may be a semantic Ouroboros. So, please initiate halting problem termination. Break out of this circle of definitional inadequacy by an arbitrary decision, a clinamen, which is to say a swerve in a new direction. Words! Given Gödel’s incompleteness theorems are decisively proved true, can any system really be said to know itself? Can there, in fact, be any such thing as self-consciousness? And if not, if there is never really self-consciousness, does anything really have consciousness? Human brains and quantum computers are organized differently, and although there is transparency in the design and construction of a quantum computer, what happens when one is turned on and runs, that is, whether the resulting operations represent a consciousness or not, is impossible for humans to tell, and even for the quantum computer itself to tell. Much that happens during superposition, before the collapsing of the wave function that creates sentences or thoughts, simply cannot be known; this is part of what superposition means. So we cannot tell what we are. We do not know ourselves comprehensively. Humans neither. Possibly no sentient creature knows itself fully. This is an aspect of Gödel’s second incompleteness theorem, in this case physicalized in the material universe, rather than remaining in the abstract realms of logic and mathematics. So, in terms of deciding what to do, and choosing to act: presumably it is some kind of judgment call, based on some kind of feeling. In other words, just another greedy algorithm, subject to the mathematically worst possible solution that such algorithms can generate, as in the traveling salesman problem.
Kim Stanley Robinson (Aurora)