Mathematical Language Quotes

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.

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.

Mathematics is the language in which God has written the universe

Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning.

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?

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.

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.

Besides language and music mathematics is one of the primary manifestations of the free creative power of the human mind.

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.

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.

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.

Mathematics is the language with which God has written the universe.

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.

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!

Film is one if three universal languages, the other two: mathematics and music.

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.

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.

...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.

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.

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.

The key point to keep in mind, however, is that symmetry is one of the most important tools in deciphering nature's design.

All the hypotheses about the early universe . . . are pure speculation. They're modern creation myths written in the language of mathematics.

Mathematics to me is like a language I don’t speak though I admire its literature in translation.

(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.

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

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.

[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.

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.

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?

A mathematician is an individual who calls himself a 'physicist' and does 'physics' and physical experiments with abstract concepts.

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.

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.

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.

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.

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.

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.

...whatever we do whatever way we move forward there will be damage carried from all previous rows and columns in this mathematical computation we make, all the multi-configured additions and subtractions in language America, in all bases, particularly I'm thinking about at the moment 1492, 1776, 1861, 1867, 1980, 2016, 2020.

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.

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.

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.

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.

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.

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.

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

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)

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.

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.

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.

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.

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.

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

It was believed that all creation came from thought, language, and mathematics.

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.

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.

. . . 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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)

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.

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.

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.

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!

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.

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.

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.

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.

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'.

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.

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.

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?

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.

Complexity and simplicity,” he replied. “Time was simple, is simple. We can divide it into simple parts, measure it, arrange dinner by it, drink whisky to its passage. We can mathematically deploy it, use it to express ideas about the observable universe, and yet if asked to explain it in simple language to a child–in simple language which is not deceit, of course–we are powerless. The most it ever seems we know how to do with time is to waste it.

Once, probably, I used to think that vagueness was a loftier kind of poetry, truer to the depths of consciousness, and maybe when I started to read mathematics and science back in the mid-70s I found an unexpected lyricism in the necessarily precise language that scientists tend to use My instinct, my superstition is that the closer I see a thing and the more accurately I describe it, the better my chances of arriving at a certain sensuality of expression.

I didn’t invent the language or mathematics I used. I make little of my one food, none of my own clothes. Everything I do depends on other members of our species and the shoulders that we stand on. And a lot of us want to contribute something back to our species and to add something to the flow. It’s about trying to express something in the only way that most of us know how because we can’t write Bob Dylan songs or Tom Stoppard plays. We try to use the talents we do have to express our deep feelings, to show our appreciation of all the contributions that came before us, and to add something to that flow. That’s what has driven me.

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.

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.

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.

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.

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.

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.

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.

Dear Rick, I've never thought math was a miracle. The things we study simply are. They were the rules of the universe before we were here to understand them. They operate the world behind the curtain, whether we look behind it or not. The rules are already there. Music is a miracle. It adds something to the world that didn't have to be here. Language is a miracle. Every sentence ever spoken and every song ever sung is a new invention. Not only do they add something new to the world, they transmit thoughts and emotions that would otherwise be locked within one person. I hear a song and feel something a composer felt 200 years ago. I read your letter and hear your voice saying the words. I feel you in the room with me. That's the miracle.

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.

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.

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.

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.)

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.

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.