Geometry Mathematics 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.

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.

Reading list (1972 edition)[edit] 1. Homer – Iliad, Odyssey 2. The Old Testament 3. Aeschylus – Tragedies 4. Sophocles – Tragedies 5. Herodotus – Histories 6. Euripides – Tragedies 7. Thucydides – History of the Peloponnesian War 8. Hippocrates – Medical Writings 9. Aristophanes – Comedies 10. Plato – Dialogues 11. Aristotle – Works 12. Epicurus – Letter to Herodotus; Letter to Menoecus 13. Euclid – Elements 14. Archimedes – Works 15. Apollonius of Perga – Conic Sections 16. Cicero – Works 17. Lucretius – On the Nature of Things 18. Virgil – Works 19. Horace – Works 20. Livy – History of Rome 21. Ovid – Works 22. Plutarch – Parallel Lives; Moralia 23. Tacitus – Histories; Annals; Agricola Germania 24. Nicomachus of Gerasa – Introduction to Arithmetic 25. Epictetus – Discourses; Encheiridion 26. Ptolemy – Almagest 27. Lucian – Works 28. Marcus Aurelius – Meditations 29. Galen – On the Natural Faculties 30. The New Testament 31. Plotinus – The Enneads 32. St. Augustine – On the Teacher; Confessions; City of God; On Christian Doctrine 33. The Song of Roland 34. The Nibelungenlied 35. The Saga of Burnt Njál 36. St. Thomas Aquinas – Summa Theologica 37. Dante Alighieri – The Divine Comedy;The New Life; On Monarchy 38. Geoffrey Chaucer – Troilus and Criseyde; The Canterbury Tales 39. Leonardo da Vinci – Notebooks 40. Niccolò Machiavelli – The Prince; Discourses on the First Ten Books of Livy 41. Desiderius Erasmus – The Praise of Folly 42. Nicolaus Copernicus – On the Revolutions of the Heavenly Spheres 43. Thomas More – Utopia 44. Martin Luther – Table Talk; Three Treatises 45. François Rabelais – Gargantua and Pantagruel 46. John Calvin – Institutes of the Christian Religion 47. Michel de Montaigne – Essays 48. William Gilbert – On the Loadstone and Magnetic Bodies 49. Miguel de Cervantes – Don Quixote 50. Edmund Spenser – Prothalamion; The Faerie Queene 51. Francis Bacon – Essays; Advancement of Learning; Novum Organum, New Atlantis 52. William Shakespeare – Poetry and Plays 53. Galileo Galilei – Starry Messenger; Dialogues Concerning Two New Sciences 54. Johannes Kepler – Epitome of Copernican Astronomy; Concerning the Harmonies of the World 55. William Harvey – On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals 56. Thomas Hobbes – Leviathan 57. René Descartes – Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy 58. John Milton – Works 59. Molière – Comedies 60. Blaise Pascal – The Provincial Letters; Pensees; Scientific Treatises 61. Christiaan Huygens – Treatise on Light 62. Benedict de Spinoza – Ethics 63. John Locke – Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education 64. Jean Baptiste Racine – Tragedies 65. Isaac Newton – Mathematical Principles of Natural Philosophy; Optics 66. Gottfried Wilhelm Leibniz – Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology 67. Daniel Defoe – Robinson Crusoe 68. Jonathan Swift – A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal 69. William Congreve – The Way of the World 70. George Berkeley – Principles of Human Knowledge 71. Alexander Pope – Essay on Criticism; Rape of the Lock; Essay on Man 72. Charles de Secondat, baron de Montesquieu – Persian Letters; Spirit of Laws 73. Voltaire – Letters on the English; Candide; Philosophical Dictionary 74. Henry Fielding – Joseph Andrews; Tom Jones 75. Samuel Johnson – The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

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

Me, and thousands of others in this country like me, are half-baked, because we were never allowed to complete our schooling. Open our skulls, look in with a penlight, and you'll find an odd museum of ideas: sentences of history or mathematics remembered from school textbooks (no boy remembers his schooling like the one who was taken out of school, let me assure you), sentences about politics read in a newspaper while waiting for someone to come to an office, triangles and pyramids seen on the torn pages of the old geometry textbooks which every tea shop in this country uses to wrap its snacks in, bits of All India Radio news bulletins, things that drop into your mind, like lizards from the ceiling, in the half hour before falling asleep--all these ideas, half formed and half digested and half correct, mix up with other half-cooked ideas in your head, and I guess these half-formed ideas bugger one another, and make more half-formed ideas, and this is what you act on and live with.

I guess a sock is also a geometric shape—technically—but I don't know what you'd call it. A socktagon?

What music is to the heart, mathematics is to the mind.

[The golden proportion] is a scale of proportions which makes the bad difficult [to produce] and the good easy.

Be honest: did you actually read [the above geometric proof]? Of course not. Who would want to? The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious by insisting that it be 'rigorously proved' ... is to say to a student 'Your feelings and ideas are suspect. You need to think and speak our way.

The Golden Number is a mathematical definition of a proportional function which all of nature obeys, whether it be a mollusk shell, the leaves of plants, the proportions of the animal body, the human skeleton, or the ages of growth in man.

When the ancients discovered ‘Phi’, they were certain they had stumbled across God’s building block for the world.

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

He could not believe that any of them might actually hit somebody. If one did, what a nowhere way to go: killed by accident; slain not as an individual but by sheer statistical probability, by the calculated chance of searching fire, even as he himself might be at any moment. Mathematics! Mathematics! Algebra! Geometry! When 1st and 3d Squads came diving and tumbling back over the tiny crest, Bell was content to throw himself prone, press his cheek to the earth, shut his eyes, and lie there. God, oh, God! Why am I here? Why am I here? After a moment's thought, he decided he better change it to: why are we here. That way, no agency of retribution could exact payment from him for being selfish.

We see that music, like the world, is formed from unchanging mathematical principles deployed in time, creating complexity, variety and beauty.

[As a young teenager] Galois read Legendre]'s geometry from cover to cover as easily as other boys read a pirate yarn.

One of the best examples of a polymath is Leonardo da Vinci. Born in Italy in 1452, Leonardo was a sculptor, painter, architect, mathematician, musician, engineer, inventor, anatomist, botanist, geologist, cartographer and writer. Although he received an informal education that included geometry, Latin and mathematics, he was essentially an autodidact, or a self-taught individual.

As for methods I have sought to give them all the rigour that one requires in geometry, so as never to have recourse to the reasons drawn from the generality of algebra.

Maths is at only one remove from magic.

Life is geometry and chemistry, not biology.

The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.

Is everyone with one face called a Milo?" "Oh no," Milo replied; "some are called Henry or George or Robert or John or lots of other things." "How terribly confusing," he cried. "Everything here is called exactly what it is. The triangles are called triangles, the circles are called circles, and even the same numbers have the same name. Why, can you imagine what would happen if we named all the twos Henry or George or Robert or John or lots of other things? You'd have to say Robert plus John equals four, and if the four's name were Albert, things would be hopeless." "I never thought of it that way," Milo admitted. "Then I suggest you begin at once," admonished the Dodecahedron from his admonishing face, "for here in Digitopolis everything is quite precise.

The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.

People joke, in our field, about Pythagoras and his religious cult based on perfect geometry and other abstract mathematical forms, but if we are going to have religion at all then a religion of mathematics seems ideal, because if God exists then what is He but a mathematician?

Even there, something inside me (and, I suspect, inside many other computer scientists!) is suspicious of those parts of mathematics that bear the obvious imprint of physics, such as partial differential equations, differential geometry, Lie groups, or anything else that's “too continuous.

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

From all this we concluded that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of the matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry (tr. Toomer, p. 6).

The human mind is only capable of absorbing a few things at a time. We see what is taking place in front of us in the here and now, and cannot envisage simultaneously a succession of processes, no matter how integrated and complementary. Our faculties of perception are consequently limited even as regards fairly simple phenomena. The fate of a single man can be rich with significance, that of a few hundred less so, but the history of thousands and millions of men does not mean anything at all, in any adequate sense of the word. The symmetriad is a million—a billion, rather—raised to the power of N: it is incomprehensible. We pass through vast halls, each with a capacity of ten Kronecker units, and creep like so many ants clinging to the folds of breathing vaults and craning to watch the flight of soaring girders, opalescent in the glare of searchlights, and elastic domes which criss-cross and balance each other unerringly, the perfection of a moment, since everything here passes and fades. The essence of this architecture is movement synchronized towards a precise objective. We observe a fraction of the process, like hearing the vibration of a single string in an orchestra of supergiants. We know, but cannot grasp, that above and below, beyond the limits of perception or imagination, thousands and millions of simultaneous transformations are at work, interlinked like a musical score by mathematical counterpoint. It has been described as a symphony in geometry, but we lack the ears to hear it.

The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics.

The Ark of the Covenant is a Golden Rectangle because its rectangular shape is in the proportions of the Golden Ratio.

In other words, if geometry was top-down mathematics, the method of indivisibles was bottom-up mathematics.

The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It's that the world is full of non-proofs, and grown-ups need to know the difference. It's hard to settle for a non-proof once you've really familiarized yourself with the genuine article.

In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]

We teach our children the mathematics of certainty—geometry and trigonometry—but not the mathematics of uncertainty, statistical thinking. And we teach our children biology but not the psychology that shapes their fears and desires. Even experts, shockingly, are not trained how to communicate risks to the public in an understandable way. And there can be positive interest in scaring people: to get an article on the front page, to persuade people to relinquish civil rights, or to sell a product.

The description of this proportion as Golden or Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for one number to play, but then again this one number has played an incredible role in human history and the universe at large.

I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry.

Therefor I doubt not but, if it had been a thing contrary to any man’s right of dominion, or to the interest of men that have dominion, ‘that the three angles of a triangle should be equal to two angles of a square,’ that doctrine should have been, if not disputed, yet by the burning of all books of geometry suppressed, as far as he whom it concerned was able.

Faith in God, then, is not at all the same as the kind of logical certainty that we attain in Euclidean geometry. God is not the conclusion to a process of reasoning, the solution to a mathematical problem. To believe in God is not to accept the possibility of his existence because it has been “proved” to us by some theoretical argument, but it is to put our trust in One whom we know and love. Faith is not the supposition that something might be true, but the assurance that someone is there.

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?

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

Geometry exist everywhere.It is necessary, however, to have eyes to see it, intelligence to understand it , and spirit to wonder at it.The wild Bedouin sees geometric forms but doesn't understand them ; the Sunni understands them but does not admire them; the artist, finally, perceives the perfection of figures, understands beauty, and admires order and harmony.God was the Great Geometer.He geometrized heaven and earth.

Here is the essence of mankind's creative genius: not the edifices of civilization nor the bang-flash weapons which can end it, but the words which fertilize new concepts like spermatoza attacking an ovum. It might be argued that the Siamese-twin infants of word/idea are the only contribution the human species can, will, or should make to the reveling cosmos. (Yes, our DNA is unique, but so is a salamander's. Yes, we construct artifacts, but so have species ranging from beavers to the architecture ants... Yes, we weave real fabric things from the dreamstuff of mathematics, but the universe is hardwired with arithmetic. Scratch a circle and pi peeps out. Enter a new solar system and Tycho Brahe's formulae lie waiting under the black velvet cloak of space/time. But where has the universe hidden a word under its outer layer of biology, geometry, or insensate rock?)

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer...

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.

In physics, theories are made of math. We don’t use math because we want to scare away those not familiar with differential geometry and graded Lie algebras; we use it because we are fools. Math keeps us honest—it prevents us from lying to ourselves and to each other. You can be wrong with math, but you can’t lie.

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

Some people gain their understanding of the world by symbols and mathematics. Others gain their understanding by pure geometry and space. There are some others that find an acceleration in the muscular effort that is brought to them in understanding, in feeling the force of objects moving through the world. What they want are words of power that stir their souls like the memory of childhood. For the sake of persons of these different types, whether they want the paleness and tenuity of mathematical symbolism, or they want the robust aspects of this muscular engagement, we should present all of these ways. It’s the combination of them that give us our best access to truth

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

To a scholar, mathematics is music.

The golden ratio is a reminder of the relatedness of the created world to the perfection of its source and of its potential future evolution.

The play is independent of the pages on which it is printed, and ‘pure geometries’ are independent of lecture rooms, or of any other detail of the physical world.

Geometry . . . is the science that it hath pleased God hitherto to bestow on mankind. —THOMAS HOBBES

The impulse to all movement and all form is given by [the golden ratio], since it is the proportion that summarizes in itself the additive and the geometric, or logarithmic, series.

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

Kepler’s discovery would not have been possible without the doctrine of conics. Now contemporaries of Kepler—such penetrating minds as Descartes and Pascal—were abandoning the study of geometry ... because they said it was so UTTERLY USELESS. There was the future of the human race almost trembling in the balance; for had not the geometry of conic sections already been worked out in large measure, and had their opinion that only sciences apparently useful ought to be pursued, the nineteenth century would have had none of those characters which distinguish it from the ancien régime.

Sandor Boatly had never guessed that, properly played, baseball consisted of mathematics, geometry, art, philosophy, ballet, and carnival, all intertwined like the mystical ribbons of color in a rainbow.

The mathematical order is beautiful precisely because it has no effect on the real world. Life isn't going to be easier, nor is anyone going to make a fortune, just because they know something about prime numbers. Of course, lots of mathematical discoveries have practical applications, no matter how esoteric they may seem. Research on ellipses made it possible to determine the orbits of the planets, and Einstein used non-Euclidean geometry to describe the form of the universe. Even prime numbers were used during the war to create codes—to cite a regrettable example. But those things aren't the goal of mathematics. The only goal is to discover the truth.

We therefore find that the triangles and rectangles herein described, enclose a large majority of the temples and cathedrals of the Greek and Gothic masters, for we have seen that the rectangle of the Egyptian triangle is a perfect generative medium, its ratio of five in width to eight in length 'encouraging impressions of contrast between horizontal and vertical lines' or spaces; and the same practically may be said of the Pythagorean triangle

In the 1950s, John Nash disrupted the balance between geometry and analysis when he discovered that the abstract geometric problem of isometric embedding could be solved by the fine “peeling” of partial differential equations.

but it should be noted even so that a well-thought out, well laid-out fantasy follows a complex and impossible geometry. The mathematics of the imagination adheres more rigorously to terms and theorems than the mathematics of reality.

Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. —Johannes Kepler

But Miss Ferguson preferred science over penmanship. Philosophy over etiquette. And, dear heavens preserve them all, mathematics over everything. Not simply numbering that could see a wife through her household accounts. Algebra. Geometry. Indecipherable equations made up of unrecognizable symbols that meant nothing to anyone but the chit herself. It was enough to give Miss Chase hives. The girl wasn’t even saved by having any proper feminine skills. She could not tat or sing or draw. Her needlework was execrable, and her Italian worse. In fact, her only skills were completely unacceptable, as no one wanted a wife who could speak German, discuss physics, or bring down more pheasant than her husband.

In medieval Europe, logic, grammar and rhetoric formed the educational core, while the teaching of mathematics seldom went beyond simple arithmetic and geometry. Nobody studied statistics. The undisputed monarch of all sciences was theology.

The teacher manages to get along still with the cumbersome algebraic analysis, in spite of its difficulties and imperfections, and avoids the smooth infinitesimal calculus, although the eighteenth century shyness toward it had long lost all point.

The only logical meaning of necessity seems to be derived from implication. A proposition is more or less necessary according as the class of propositions for which it is a premiss is greater or smaller.* In this sense the propositions of logic have the greatest necessity, and those of geometry have a high degree of necessity. But this sense of necessity yields no valid argument from our inability to imagine holes in space to the conclusion that there cannot really be any space at all except in our imaginations.

Euler’s Formula encapsulates the whole of existence. It contains 0, the number of the monad (ontological zero); the number e that determines exponentiation; the number i that determines the imaginary domain (time); the number 1 that determines the domain of counting numbers (and with 0 creates the binary system of computing), and real numbers (space); -1, the number of the negative domain (antimatter); and the number π that determines the world of the circle and geometry. Euler’s Formula is the unquestionable God Equation.

The Great Pyramid was a fractal resonator for the entire Earth. It is designed according to the proportions of the cosmic temple, the natural pattern that blends the two fundamental principles of creation. The pyramid has golden ratio, pi, the base of natural logarithms, the precise length of the year and the dimensions of the Earth built into its geometry. It demonstrates.... As John Michell has pointed out in his wonderful little book, City of Revelation, 'Above all, the Great Pyramid is a monument to the art of 'squaring the circle''.

It is to geometry that we owe in some sort the source of this discovery [of beryllium]; it is that [science] that furnished the first idea of it, and we may say that without it the knowledge of this new earth would not have been acquired for a long time, since according to the analysis of the emerald by M. Klaproth and that of the beryl by M. Bindheim one would not have thought it possible to recommence this work without the strong analogies or even almost perfect identity that Citizen Haüy found for the geometrical properties between these two stony fossils.

In the heaven of the great god Indra is said to be a vast and shimmering net, finer than a spider’s web, stretching to the outermost reaches of space. Strung at the each intersection of its diaphanous threads is a reflecting pearl. Since the net is infinite in extent, the pearls are infinite in number. In the glistening surface of each pearl are reflected all the other pearls, even those in the furthest corners of the heavens. In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end.

Often people think of developments in computation as arising when we make our computers more blazingly fast, so they can compute more stuff, bigger data. It's actually just as important to prune away big parts of the data that aren't relevant to the problem at hand! The fastest computation is the one you don't do.

What he discovered was that these intervals were harmonious because the relationship between them was a precise and simple mathematical ratio. This series, which we now know as the harmonic series, confirmed for him that the elegance of the mathematics he had found in abstract geometry also existed in the natural world.

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

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

There is a mathematical underpinning that you must first acquire, mastery of each mathematical subdiscipline leading you to the threshold of the next. In turn you must learn arithmetic, Euclidian geometry, high school algebra, differential and integral calculus, ordinary and partial differential equations, vector calculus, certain special functions of mathematical physics, matrix algebra, and group theory. For most physics students, this might occupy them from, say, third grade to early graduate school—roughly 15 years. Such a course of study does not actually involve learning any quantum mechanics, but merely establishing the mathematical framework required to approach it deeply.

... I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for convenience sake, I verified the result at my leisure.

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

And John Nash, my mathematical hero, revolutionized analysis and geometry with the proof of three theorems in scarcely more than five years before succumbing to paranoid schizophrenia. There is a fine line, it is often said, between genius and madness. Neither of these concepts is well defined, however. And in the case not only of Grothendieck but also of Gödel and Nash, periods of mental derangement, so far from promoting mathematical productivity, actually precluded it. Innate versus acquired, a classic debate. Fischer, Grothendieck, Erdős, and Perelman were all Jewish. Of these, Fischer and Erdős were Hungarian. No one who is familiar with the world of science can have failed to notice how many of the most gifted mathematicians and physicists of the twentieth century were Jews, or how many of the greatest geniuses were Hungarian (many

Mathematics enjoys the greatest reputation as a diversion from sexuality. This had been the very advice to which Jean-Jacques Rousseau was obliged to listen from a lady who was dissatisfied with him: 'Lascia le donne e studia la matematica!' So too our fugitive threw himself with special eagerness into the mathematics and geometry which he was taught at school, till suddenly one day his powers of comprehension were paralysed in the face of some apparently innocent problems. It was possible to establish two of these problems; 'Two bodies come together, one with a speed of ... etc' and 'On a cylinder, the diameter of whose surface is m, describe a cone ... etc' Other people would certainly not have regarded these as very striking allusions to sexual events; but he felt that he had been betrayed by mathematics as well, and took flight from it too.

In mathematical physics, quantum field theory and statistical mechanics are characterized by the probability distribution of exp(−βH(x)) where H(x) is a Hamiltonian function. It is well known in [12] that physical problems are determined by the algebraic structure of H(x). Statistical learning theory can be understood as mathematical physics where the Hamiltonian is a random process defined by the log likelihood ratio function.

Euclid's Elements has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written—shortly after the foundation of the Alexandrian Museum—Mathematics was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning for the rest to follow her.

This picture of matter curving space and curvaceous space dictating how matter and light will move has several striking features. It brings the non-Euclidean geometries that we talked about in the last chapter out from the library of pure mathematics into the arena of science. The vast collection of geometries describing spaces that are not simply the flat space of Euclid are the ones that Einstein used to capture the possible structures of space distorted by the presence of mass and energy.

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

A philosopher/mathematician named Bertrand Russell who lived and died in the same century as Gass once wrote: “Language serves not only to express thought but to make possible thoughts which could not exist without it.” Here is the essence of mankind’s creative genius: not the edifices of civilization nor the bang-flash weapons which can end it, but the words which fertilize new concepts like spermatozoa attacking an ovum. It might be argued that the Siamese-twin infants of word/idea are the only contribution the human species can, will, or should make to the raveling cosmos. (Yes, our DNA is unique but so is a salamander’s. Yes, we construct artifacts but so have species ranging from beavers to the architect ants whose crenellated towers are visible right now off the port bow. Yes, we weave real-fabric things from the dreamstuff of mathematics, but the universe is hardwired with arithmetic. Scratch a circle and π peeps out. Enter a new solar system and Tycho Brahe’s formulae lie waiting under the black velvet cloak of space/time. But where has the universe hidden a word under its outer layer of biology, geometry, or insensate rock?)

The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music-all of which, the Pythagorean Archytas tells us, fell under the general definition of "mathematics." According to legend, when Alexander the Great asked his teacher Menaechmus (who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola) for a shortcut to geometry, he got the reply: "O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry there is one road for all.

To do so he used something called a tensor. In Euclidean geometry, a vector is a quantity (such as of velocity or force) that has both a magnitude and a direction and thus needs more than a single simple number to describe it. In non-Euclidean geometry, where space is curved, we need something more generalized—sort of a vector on steroids—in order to incorporate, in a mathematically orderly way, more components. These are called tensors. A metric tensor is a mathematical tool that tells us how to calculate the distance between points in a given space. For two-dimensional maps, a metric tensor has three components. For three-dimensional space, it has six independent components. And once you get to that glorious four-dimensional entity known as spacetime, the metric tensor needs ten independent components.

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.

You too can make the golden cut, relating the two poles of your being in perfect golden proportion, thus enabling the lower to resonate in tune with the higher, and the inner with the outer. In doing so, you will bring yourself to a point of total integration of all the separate parts of your being, and at the same time, you will bring yourself into resonance with the entire universe.... Nonetheless the universe is divided on exactly these principles as proven by literally thousands of points of circumstantial evidence, including the size, orbital distances, orbital frequencies and other characteristics of planets in our solar system, many characteristics of the sub-atomic dimension such as the fine structure constant, the forms of many plants and the golden mean proportions of the human body, to mention just a few well known examples. However the circumstantial evidence is not that on which we rely, for we have the proof in front of us in the pure mathematical principles of the golden mean.

This development had dramatic philosophical consequences. As in the case of the non-Euclidean geometries in the nineteenth century, there wasn't just one definitive set theory, but rather at least four! One could make different assumptions about infinite sets and end up with mutually exclusive set theories. For instance, once could assume that both the axiom of choice and the continuum hypothesis hold true and obtain one version, or that both do not hold, and obtain an entirely different theory. Similarly, assuming the validity of one of the two axioms and the negation of the other would have led to yet two other set theories. This was the non-Euclidean crisis revisited, only worse. The fundamental role of set theory as the potential basis for the whole of mathematics made the problem for the Platonists much more acute. If indeed one could formulate many set theories simply by choosing a different collection of axioms, didn't this argue for mathematics being nothing but a human invention? The formalists' victory looked virtually assured.

Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

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

The cosmic sculptor had felt compelled to dot pupils onto the universe, yet had a tremendous terror of granting it sight. This balance of fear and desire resulted in the tininess of the stars against the hugeness of space, a declaration of caution above all. “See how the stars are points? The factors of chaos and randomness in the complex makeups of every civilized society in the universe get filtered out by the distance, so those civilizations can act as reference points that are relatively easy to manipulate mathematically.” “But there’s nothing concrete to study in your cosmic sociology, Dr. Ye. Surveys and experiments aren’t really possible.” “That means your ultimate result will be purely theoretical. Like Euclidean geometry, you’ll set up a few simple axioms at first, then derive an overall theoretic system using those axioms as a foundation.” “It’s all fascinating, but what would the axioms of cosmic sociology be?” “First: Survival is the primary need of civilization. Second: Civilization continuously grows and expands, but the total matter in the universe remains constant.” The

The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems that are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers. When the Declaration of Independence says 'we hold these truths to be self-evident', it is modelling itself on Euclid. The eighteenth-century doctrine of natural rights is a search for Euclidean axioms in politics.8 The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source. Personal religion is derived from ecstasy, theology from mathematics; and both are to be found in Pythagoras.

One method that Einstein employed to help people visualize this notion was to begin by imagining two-dimensional explorers on a two-dimensional universe, like a flat surface. These “flatlanders” can wander in any direction on this flat surface, but the concept of going up or down has no meaning to them. Now, imagine this variation: What if these flatlanders’ two dimensions were still on a surface, but this surface was (in a way very subtle to them) gently curved? What if they and their world were still confined to two dimensions, but their flat surface was like the surface of a globe? As Einstein put it, “Let us consider now a two-dimensional existence, but this time on a spherical surface instead of on a plane.” An arrow shot by these flatlanders would still seem to travel in a straight line, but eventually it would curve around and come back—just as a sailor on the surface of our planet heading straight off over the seas would eventually return from the other horizon. The curvature of the flatlanders’ two-dimensional space makes their surface finite, and yet they can find no boundaries. No matter what direction they travel, they reach no end or edge of their universe, but they eventually get back to the same place. As Einstein put it, “The great charm resulting from this consideration lies in the recognition that the universe of these beings is finite and yet has no limits.” And if the flatlanders’ surface was like that of an inflating balloon, their whole universe could be expanding, yet there would still be no boundaries to it.10 By extension, we can try to imagine, as Einstein has us do, how three-dimensional space can be similarly curved to create a closed and finite system that has no edge. It’s not easy for us three-dimensional creatures to visualize, but it is easily described mathematically by the non-Euclidean geometries pioneered by Gauss and Riemann. It can work for four dimensions of spacetime as well. In such a curved universe, a beam of light starting out in any direction could travel what seems to be a straight line and yet still curve back on itself. “This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived,” the physicist Max Born has declared.

Even male children of affluent white families think that history as taught in high school is “too neat and rosy.” 6 African American, Native American, and Latino students view history with a special dislike. They also learn history especially poorly. Students of color do only slightly worse than white students in mathematics. If you’ll pardon my grammar, nonwhite students do more worse in English and most worse in history.7 Something intriguing is going on here: surely history is not more difficult for minorities than trigonometry or Faulkner. Students don’t even know they are alienated, only that they “don’t like social studies” or “aren’t any good at history.” In college, most students of color give history departments a wide berth. Many history teachers perceive the low morale in their classrooms. If they have a lot of time, light domestic responsibilities, sufficient resources, and a flexible principal, some teachers respond by abandoning the overstuffed textbooks and reinventing their American history courses. All too many teachers grow disheartened and settle for less. At least dimly aware that their students are not requiting their own love of history, these teachers withdraw some of their energy from their courses. Gradually they end up going through the motions, staying ahead of their students in the textbooks, covering only material that will appear on the next test. College teachers in most disciplines are happy when their students have had significant exposure to the subject before college. Not teachers in history. History professors in college routinely put down high school history courses. A colleague of mine calls his survey of American history “Iconoclasm I and II,” because he sees his job as disabusing his charges of what they learned in high school to make room for more accurate information. In no other field does this happen. Mathematics professors, for instance, know that non-Euclidean geometry is rarely taught in high school, but they don’t assume that Euclidean geometry was mistaught. Professors of English literature don’t presume that Romeo and Juliet was misunderstood in high school. Indeed, history is the only field in which the more courses students take, the stupider they become. Perhaps I do not need to convince you that American history is important. More than any other topic, it is about us. Whether one deems our present society wondrous or awful or both, history reveals how we arrived at this point. Understanding our past is central to our ability to understand ourselves and the world around us. We need to know our history, and according to sociologist C. Wright Mills, we know we do.8

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

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

Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof. Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did! Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.

In learning general relativity, and then in teaching it to classes at Berkeley and MIT, I became dissatisfied with what seemed to be the usual approach to the subject. I found that in most textbooks geometric ideas were given a starring role, so that a student...would come away with an impression that this had something to do with the fact that space-time is a Riemannian [curved] manifold. Of course, this was Einstein's point of view, and his preeminent genius necessarily shapes our understanding of the theory he created. However, I believe that the geometrical approach has driven a wedge between general relativity and [Quantum Field Theory]. As long as it could be hoped, as Einstein did hope, that matter would eventually be understood in geometrical terms, it made sense to give Riemannian geometry a primary role in describing the theory of gravitation. But now the passage of time has taught us not to expect that the strong, weak, and electromagnetic interactions can be understood in geometrical terms, and too great an emphasis on geometry can only obscuret he deep connections between gravitation and the rest of physics...[My] book sets out the theory of gravitation according to what I think is its inner logic as a branch of physics, and not according to its historical development. It is certainly a historical fact that when Albert Einstein was working out general relativity, there was at hand a preexisting mathematical formalism, that of Riemannian geometry, that he could and did take over whole. However, this historical fact does not mean that the essence of general relativity necessarily consists in the application of Riemannian geometry to physical space and time. In my view, it is much more useful to regard general relativity above all as a theory of gravitation, whose connection with geometry arises from the peculiar empirical properties of gravitation.

When I threw the stick at Jamie, I hadn't intended to hit him with it. But the moment it left my hand, I knew that's what was going to happen. I didn't yet know any calculus or geometry, but I was able to plot, with some degree of certainty, the trajectory of that stick. The initial velocity, the acceleration, the impact. The mathematical likelihood of Jamie's bloody cheek. It had good weight and heft, that stick. It felt nice to throw. And it looked damn fine in the overcast sky, too, flying end over end, spinning like a heavy, two-pronged pinwheel and (finally, indifferently, like math) connecting with Jamie's face. Jamie's older sister took me by the arm and she shook me. Why did you do that? What were you thinking? The anger I saw in her eyes. Heard in her voice. The kid I became to her then, who was not the kid I thought I was. The burdensome regret. I knew the word "accident" was wrong, but I used it anyway. If you throw a baseball at a wall and it goes through a window, that is an accident. If you throw a stick directly at your friend and it hits your friend in the face, that is something else. My throw had been something of a lob and there had been a good distance between us. There had been ample time for Jamie to move, but he hadn't moved. There had been time for him to lift a hand and protect his face from the stick, but he hadn't done that either. He just stood impotent and watched it hit him. And it made me angry: That he hadn't tried harder at a defense. That he hadn't made any effort to protect himself from me. What was I thinking? What was he thinking? I am not a kid who throws sticks at his friends. But sometimes, that's who I've been. And when I've been that kid, it's like I'm watching myself act in a movie, reciting somebody else's damaging lines. Like this morning, over breakfast. Your eyes asking mine to forget last night's exchange. You were holding your favorite tea mug. I don't remember what we were fighting about. It doesn't seem to matter any more. The words that came out of my mouth then, deliberate and measured, temporarily satisfying to throw at the bored space between us. The slow, beautiful arc. The spin and the calculated impact. The downward turn of your face. The heavy drop in my chest. The word "accident" was wrong. I used it anyway.