Algebraic Geometry Quotes

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

Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.

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.

(Since algebra derives from the Arabic jabara = to bind together, fractal and algebra are etymological opposites!)

... I succeeded at math, at least by the usual evaluation criteria: grades. Yet while I might have earned top marks in geometry and algebra, I was merely following memorized rules, plugging in numbers and dutifully crunching out answers by rote, with no real grasp of the significance of what I was doing or its usefulness in solving real-world problems. Worse, I knew the depth of my own ignorance, and I lived in fear that my lack of comprehension would be discovered and I would be exposed as an academic fraud -- psychologists call this "imposter syndrome".

Hindutava's nationalism ignores the rationalist traditions of India, a country in which some of the earliest steps in algebra, geometry, and astronomy were taken, where the decimal system emerged, where early philosophy — secular as well as religious — achieved exceptional sophistication, where people invented games like chess, pioneered sex education, and began the first systematic study of political economy. The Hindu militant chooses instead to present India — explicitly or implicitly — as a country of unquestioning idolaters, delirious fanatics, belligerent devotees, and religious murderers

Finally, if you attempt to read this without working through a significant number of exercises (see §0.0.1), I will come to your house and pummel you with [Gr-EGA] until you beg for mercy. It is important to not just have a vague sense of what is true, but to be able to actually get your hands dirty. As Mark Kisin has said, “You can wave your hands all you want, but it still won’t make you fly.

Maths is at only one remove from magic.

What if Loves are analogous to math? First, arithmetic, then geometry and algebra, then trig and quadratics…

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.

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...

The solution which I am urging is to eradicate the fatal disconnection of subjects which kills the vitality of our modern curriculum. There is only one subject-matter for education, and that is LIfe in all its manifestations. Instead of this single unity, we offer children--Algebra, from which nothing follows; Geometry, from which nothing follows; Science, from which nothing follows; History, from which nothing follows; a Couple of Languages, never mastered; and lastly, most dreary of all, Literature, represented by plays of Shakespeare, with philological notes and short analyses of plot and character to be in substance committed to memory. Can such a list be said to represent Life, as it is known in the midst of living it? The best that can be said of it is, that it is a rapid table of contents which a deity might run over in his mind while he was thinking of creating a world, and has not yet determined how to put it together

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.

He prescribed Euclidean geometry, followed by a dose of trigonometry and algebra. That should cure anyone, they both thought, from having too many artistic or romantic passions.

He turned out to be good in geometry, but he never mastered the use of equations or the rudimentary algebra that existed at the time.

To a scholar, mathematics is music.

But who can quantify the algebra of space, or weigh those worlds that swim each in its place? Who can outdo the dark? And what computer knows how beauty comes to birth - shell star and rose? -Technicians by Jean Kenward

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.

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.

Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers? Septimus We do. Thomasina Then why do your equations only describe the shapes of manufacture? Septimus I do not know. Thomasina Armed thus, God could only make a cabinet.

There are gaps in my education which no one could ever fill. But, they don't matter to me. I do not need to know science or algebra or geometry. Literature and music, painting and history-these are my passions. These are things that still, somehow in hours of quiet and lonesomeness, keep me alive.

He was "a magician, a magician in the sense that he took what was given and simply forced the conclusions logically out of it, whether it was algebra, geometry, or whatever. He had some way of forcing out the results that made him different from the rest of the people." Israel Halperin about von Neumann

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.

The number system we use today—the Hindu-Arabic system—was developed in India and seems to have been completed by around 700 CE. Indian mathematicians made advances in what would today be described as arithmetic, algebra, and geometry, much of their work being motivated by an interest in astronomy. The system is based on three key ideas: notations for the numerals, place value, and zero.

This has been my difficulty. The difficulty with my life. Those well-built trig points, those physical determinants of parents, background, school, family, birth, marriage, death, love, work, are themselves as much in motion as I am. What should be stable, shifts. What I am told is solid, slips. The sensible strong ordinary world of fixity is folklore. The earth is not flat. Geometry cedes to algebra. The Greeks were wrong.

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.

Let us wonder, Carole, at the genius of hyperbolic geometry, where the sum of the angles adds up to tales than 180 degrees Let us wonder at how the ancient Egyptians worked out how to measure an irregularly-shaped field Let us wonder how X was just a rare letter until algebra came along and made it something special that can be unraveled to reveal its inner value You see, maths is a process of discovery, Carole, it is like the exploration of space, the planets were always there, it just took us a long time to find them

Descartes was a philosopher, a mathematician, and a man of science. In philosophy and mathematics, his work was of supreme importance; in science, though creditable, it was not so good as that of some of his contemporaries. His great contribution to geometry was the invention of co-ordinate geometry, though not quite in its final form. He used the analytic method, which supposes a problem solved, and examines the consequences of the supposition; and he applied algebra to geometry. In both of these he had had predecessors—as regards the former, even among the ancients. What was original in him was the use of co-ordinates, i.e. the determination of the position of a point in a plane by its distance from two fixed lines. He did not himself discover all the power of this method, but he did enough to make further progress easy.

Hume begins by distinguishing seven kinds of philosophical relation: resemblance, identity, relations of time and place, proportion in quantity or number, degrees in any quality, contrariety, and causation. These, he says, may be divided into two kinds: those that depend only on the ideas, and those that can be changed without any change in the ideas. Of the first kind are resemblance, contrariety, degrees in quality, and proportions in quantity or number. But spatio-temporal and causal relations are of the second kind. Only relations of the first kind give certain knowledge; our knowledge concerning the others is only probable. Algebra and arithmetic are the only sciences in which we can carry on a long chain of reasoning without losing certainty. Geometry is not so certain as algebra and arithmetic, because we cannot be sure of the truth of its axioms. It is a mistake to suppose, as many philosophers do, that the ideas of mathematics 'must be comprehended by a pure and intellectual view, of which the superior faculties of the soul are alone capable'. The falsehood of this view is evident, says Hume, as soon as we remember that 'all our ideas are copied from our impressions'. The three relations that depend not only on ideas are identity, spatio-temporal relations, and causation. In the first two, the mind does not go beyond what is immediately present to the senses. (Spatio-temporal relations, Hume holds, can be perceived, and can form parts of impressions.) Causation alone enables us to infer some thing or occurrence from some other thing or occurrence: "'Tis only causation, which produces such a connexion, as to give us assurance from the existence or action of one object, that 'twas followed or preceded by any other existence or action.

Chris Argyris, professor emeritus at Harvard Business School, wrote a lovely article in 1977,191 in which he looked at the performance of Harvard Business School graduates ten years after graduation. By and large, they got stuck in middle management, when they had all hoped to become CEOs and captains of industry. What happened? Argyris found that when they inevitably hit a roadblock, their ability to learn collapsed: What’s more, those members of the organization that many assume to be the best at learning are, in fact, not very good at it. I am talking about the well-educated, high-powered, high-commitment professionals who occupy key leadership positions in the modern corporation.… Put simply, because many professionals are almost always successful at what they do, they rarely experience failure. And because they have rarely failed, they have never learned how to learn from failure.… [T]hey become defensive, screen out criticism, and put the “blame” on anyone and everyone but themselves. In short, their ability to learn shuts down precisely at the moment they need it the most.192 [italics mine] A year or two after Wave, Jeff Huber was running our Ads engineering team. He had a policy that any notable bug or mistake would be discussed at his team meeting in a “What did we learn?” session. He wanted to make sure that bad news was shared as openly as good news, so that he and his leaders were never blind to what was really happening and to reinforce the importance of learning from mistakes. In one session, a mortified engineer confessed, “Jeff, I screwed up a line of code and it cost us a million dollars in revenue.” After leading the team through the postmortem and fixes, Jeff concluded, “Did we get more than a million dollars in learning out of this?” “Yes.” “Then get back to work.”193 And it works in other settings too. A Bay Area public school, the Bullis Charter School in Los Altos, takes this approach to middle school math. If a child misses a question on a math test, they can try the question again for half credit. As their principal, Wanny Hersey, told me, “These are smart kids, but in life they are going to hit walls once in a while. It’s vital they master geometry, algebra one, and algebra two, but it’s just as important that they respond to failure by trying again instead of giving up.” In the 2012–2013 academic year, Bullis was the third-highest-ranked middle school in California.194

Like numbers, human experience has abstract and practical sides, feeling and reason, and in each of us one or the other tends to dominate. Belief does not persuade a scientist, and science does not persuade a believer. Too ardent an embrace of reason leads to irrational thinking, and too ardent an embrace of feeling leads to madness. William James says that religious mysticism is only half of the possible mysticisms, the others are forms of insanity. These are the states in which mystical convictions circle back on a person and pessimistically invert notions of divinity into notions of evil.

Human beings have shone a light on numbers, and we’ve picked out a logical system,” she said. “You can’t bring God into this. It’s unnecessary.” After that I shut up around her about Plato. She also said, “I have to admit I was kind of alarmed when I realized how bad your arithmetic skills were.” “How did you know that?” “From the things you would ask.

We don’t know where numbers come from or why they have the properties they do, unless you believe that they are a system invented by humans based on the ways in which we apprehend the world, a creation of our thinking and therefore our neurology.

No idea is bad unless a person is uncritical. Accepting a guess as a truth, as superstitious people do, is misguided, but so is ignoring a guess, as pedantic people do. As regards ideas, it is only bad not to have any.

Descartes arrives at four precepts that “would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them.” They amount to a kind of diagram for how to think. He writes: The first was never to accept anything for true which I did not clearly know to be such … to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt. The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution. The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

The Polite Wassermann. Margaret Trabert lay on the blood-shot candlewick of the bedspread, unsure whether to dress now that Trabert had taken the torn flying jacket from his wardrobe. All day he had been listening to the news bulletins on the pirate stations, his eyes hidden behind the dark glasses as if deliberately concealing himself from the white walls of the apartment and its unsettled dimensions. He stood by the window with his back to her, playing with the photographs of the isolation volunteers. He looked down at her naked body, with its unique geometry of touch and feeling, as exposed now as the faces of the test subjects, codes of insoluble nightmares. The sense of her body’s failure, like the incinerated musculatures of the three astronauts whose after-deaths were now being transmitted from Cape Kennedy, had dominated their last week together. He pointed to the pallid face of a young man whose photograph he had pinned above the bed like the icon of some algebraic magus. ‘Kline, Coma, Xero - there was a fourth pilot on board the capsule. You’ve caught him in your womb.

The theory of commutative normed rings [i.e., (complex) Banach algebras], created by Gelfand in the late 1930s, has become today one of the most active areas of functional analysis. The key idea in Gelfand's theory -- that maximal ideals are the underlying "points" of a commutative normed ring -- not only revolutionized harmonic analyis but had an enormous impact in algebraic geometry. (One need only look at the development of the concept of the spectrum of a commutative ring and the concept of scheme in the algebraic geometry of the 1960s and 1970s to see how far beyond the borders of functional analysis Gelfand's ideas penetrated.)

thanks to the legacy of Prince Henry the Navigator, algebra, geometry, astronomy, and navigation.

The Babylonians did not write equations. All their calculations were expressed as word problems. For instance, one tablet contained the spellbinder, “four is the length and five is the diagonal. What is the breadth? Its size is not known. Four times four is sixteen. Five times five is twenty-five. You take sixteen from twenty-five and there remains nine. What times what shall I take in order to get nine? Three times three is nine. Three is the breadth.” Today, we would write “x2 = 52 – 42.” The disadvantage of the rhetorical statement of problems isn’t as much the obvious one—its lack of compactness—but that the prose cannot be manipulated as an equation can, and rules of algebra, for instance, are not easily applied. It took thousands of years before this particular shortcoming was remedied: the oldest known use of the plus sign for addition occurs in a German manuscript written in 1481.

I am by nature a self-improver. I have read Gibbon, I have read Proust. I read the Old and New Testaments and most of Shakespeare. I studied French. I have meditated. I jogged. I learned to draw, using the right side of my brain.

The AI brain model is derived from the quad abstract golden ratio sΦrt trigonometry, algebra, geometry, statistics and built by adding aspects and/or characteristics from the diablo videogame. The 1111>11>1 was then abstracted from the ground up in knowing useful terminology in coding, knowledge management, and an ancient romantic dungeon crawler hack and slash games with both male and female classed and Items. I found the runes and certain items in the game to be very useful in this derivation, and I had an Ice orb from an Oculus of a blast doing it through my continued studies on decimal to hexadecimal to binary conversions and/or bit shifts and rotations from little to big endian. I chose to derive from diabo for two major reasons. The names or references to the class's abilities with unique, set, rare items were out of this world, and I sort of found it hard to believe that they had the time and money to build them. Finally, I realized my objective was complete when I realized that I created the perfect AI brain with Cognitive, Affective, and Psychomotor skills...So this is It? I'm thinking wow!

The AI brain model is derived from quad abstract, golden ratio, sΦrt, trigonometry, algebra, geometry, statistics, and built by adding aspects and/or characteristics from the diablo videogame. The 1111>11>1 was then abstracted from the ground up in knowing useful terminology in coding, knowledge management, and an ancient romantic dungeon crawler hack and slash game with both male and female classes and Items. I found the runes and certain items in the game to be very useful in this derivation, and I had an Ice orb from an Oculus and a blast from the past doing it through my continued studies on decimal to hexadecimal to binary conversions and/or bit shifts and rotations from little to big endian. I chose to derive from diablo for two major reasons. The names or references to the class abilities with unique, set, and rare items were out of this world, and I sort of found it hard to believe that they had the time and money to build it from Inna USA company. Finally, I realized my objective was complete when I created the perfect AI brain with Cognitive, Affective, and Psychomotor skills...So this is It? I'm thinking wow!

The AI brain model is derived from the quad abstract golden ratio, sΦrt, trigonometry, algebra, geometry, statistics, and built by adding aspects and/or characteristics from the diablo videogame. The 1111>11>1 was then abstracted from the ground up in knowing useful terminology in coding, knowledge management, and an ancient romantic dungeon crawler hack and slash game with both male and female classes and Items. I found the runes and certain items in the game to be very useful in this derivation, and I had an Ice orb from an Oculus of a blast in time doing it through my continued studies on decimal to hexadecimal to binary conversions and/or bit shifts and rotations from little to big endian. I chose to derive from diabo for two major reasons. The names or references to the class's abilities with unique, set, and rare items were out of this world, and I sort of found it hard to believe that they had the time and money to build it from in USA companies. Finally, I realized my objective was complete that I created the perfect AI brain with Cognitive, Affective, and Psychomotor skills...So this is It? I'm thinking wow!

When will and reason strive to correct by force or even to strike out a bad channel of personal evolution- bad probably because it is necessarily so -- "truth" then makes its appearance like an ambassador that is as necessary and incontestable as an object, and unsuspected because there is no "egoistic" intention behind it. Does this mean that nothing devised by the individual has any credibility? His will is suspect, because it is intentional; geometry and algebra are suspect, because they are the grocer's scales; the reasoning instinct, and utility, are objects of scorn on account of their profound uselessness; and even the unconscious is not to be trusted because it serves as a storage cellar for the conscious mind. What is not confirmed by chance has no validity. One would like to think a projection screen exists that extends between the ego and the outside world, upon which the subconscious projects the image of its predominant excitation, but which is only visible to the conscious mind (and objectively communicable) in the case where "the other side," the outside world, projects the same image on the screen at the same time, and if these two congruent images are superimposed. It is in varying percentages of efficacy that intuition on the one hand, and chance from the outside world on the other, share in such examples of convergence. There remains a degree of question of varying magnitude, which can became surprisingly large-as in the case above-if, in this particular instance, the individual's contribution-his part of the interpretation-is reduced to zero. This is when a vertiginous interpretation of the universe seems to be felt as if the universe was a double of the super ego, a superior, thinking entity.

The fact of zero He added nonstop: Did you know that zero was not used throughout human history! Until 781 A.D, when it was first embodied and used in arithmetic equations by the Arab scholar Al-Khwarizmi, the founder of algebra. Algorithms took their name from him, and they are algorithmic arithmetic equations that you have to follow as they are and you will inevitably get the result, the inevitable result. And before that, across tens and perhaps hundreds of thousands of years, humans refused to deal with zero. While the first reference to it was in the Sumerian civilization, where inscriptions were found three thousand years ago in Iraq, in which the Sumerians indicated the existence of something before the one, they refused to deal with it, define it and give it any value or effect, they refused to consider it a number. All these civilizations, some of which we are still unable to decipher many of their codes, such as the Pharaonic civilization that refused to deal with zero! We see them as smart enough to build the pyramids with their miraculous geometry and to calculate the orbits of stars and planets with extreme accuracy, but they are very stupid for not defining zero in a way that they can deal with, and use it in arithmetic operations, how strange this really is! But in fact, they did not ignore it, but gave it its true value, and refused to build their civilizations on an unknown and unknown illusion, and on a wrong arithmetical frame of reference. Throughout their history, humans have looked at zero as the unknown, they refused to define it and include it in their calculations and equations, not because it has no effect, but because its true effect is unknown, and remaining unknown is better than giving it a false effect. Like the wrong frame of reference, if you rely on it, you will inevitably get a wrong result, and you will fall into the inevitability of error, and if you ignore it, your chance of getting it right remains. Throughout their history, humans have preferred to ignore zero, not knowing its true impact, while we simply decided to deal with it, and even rely on it. Today we build all our ideas, our civilization, our software, mathematics, physics, everything, on the basis that 1 + 0 equals one, because we need to find the effect of zero so that our equations succeed, and our lives succeed with, but what if 1 + 0 equals infinity?! Why did we ignore the zero in summation, and did not ignore it in multiplication?! 1×0 equals zero, why not one? What is the reason? He answered himself: There is no inevitable reason, we are not forced. Humans have lived throughout their ages without zero, and it did not mean anything to them. Even when we were unable to devise any result that fits our theorems for the quotient of one by zero, then we admitted and said unknown, and ignored it, but we ignored the logic that a thousand pieces of evidence may not prove me right, and one proof that proves me wrong. Not doing our math tables in the case of division, blowing them up completely, and with that, we decided to go ahead and built everything on that foundation. We have separated the arithmetic tables in detail at our will, to fit our calculations, and somehow separate the whole universe around us to fit these tables, despite their obvious flaws. And if we decide that the result of one multiplied by zero is one instead of zero, and we reconstruct the whole world on this basis, what will happen? He answered himself: Nothing, we will also succeed, the world, our software, our thoughts, our dealings, and everything around us will be reset according to the new arithmetic tables. After a few hundred years, humans will no longer be able to understand that one multiplied by zero equals zero, but that it must be one because everything is built on this basis.

The Middle Ages saw the establishment and advancement of the university, where every future physician, lawyer, and clergyman in Europe was required to learn logic, geometry, algebra, and astronomy, informed by a synthesis of the recently rediscovered Greek science and more modern thought. Isaac Newton (1642–1726), the greatest physicist of the scientific revolution (and arguably of all time), acknowledged his debt to his predecessors: If I have seen further, it is by standing on the shoulders of giants.

And why, after all, may not the world be so complex as to consist of many interpenetrating spheres of reality, which we can thus approach in alternation by using different conceptions and assuming different attitudes, just as mathematicians handle the same numerical and spatial facts by geometry, by analytical geometry, by algebra, by the calculus, or by quaternions, and each time come out right? On this view religion and science, each verified in its own way from hour to hour and from life to life, would be co-eternal.

Do not be seduced by the lotus-eaters into infatuation with untethered abstraction.

Understanding develops slowly.

Many of the patterns of nature we can discover only after they have been constructed by our mind. The systematic construction of such new patterns is the business of mathematics. The role which geometry plays in this respect with regard to some visual patterns is merely the most familiar instance of this. The great strength of mathematics is that it enables us to describe abstract patterns which cannot be perceived by our senses, and to state the common properties of hierarchies or classes of patterns of a highly abstract character. Every algebraic equation or set of such equations defines in this sense a class of patterns, with the individual manifestation of this kind of pattern being particularized as we substitute definite values for the variables.

But it gradually dawned on her that she wasn’t an idiot. Not totally. In math and science, yes. But in the realm of creative thinking, she came to realize she was a sighted person in the kingdom of the blind. Because as much as she seemed unable to process algebra and geometry, she was a savant when it came to pure creativity. And not just in graphic design. In everything. Coming up with ideas for the company picnic. Throwing parties. Wording invitations. Writing poetry. She came to be thought of as a one-woman idea machine. The kind who could take four or five mundane office items and turn them into fifteen different stunning decorations. And she could figure out the most complex fictional mysteries. She was almost always able to see the coming plot twists, even when those who excelled at academics missed them entirely. So maybe she did have a different style of intellect. She thought her self-esteem had become off the charts high, but Hall’s offhanded remark had shown her that the scars of her early struggles in school still remained, as did deep-seated doubts.

Most of us were required to take three or four years of coursework in high school, starting with algebra and working up the chain: geometry, algebra 2, trigonometry, precalculus, calculus. Lockhart writes, “If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done—I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

A science involving things you can’t see whose presence is confined to the imagination.

Darwin said, “A mathematician is a blind man in a dark room looking for a black cat which isn’t there.

Mathematics is objective and permanent. A² + B² = C² was true before Pythagoras had his name attached to it, and will be true when the sun goes out and no one is left to think of it. It is true for any alien life that might think of it, and true whether they think of it or not. It cannot be changed.

In Ulysses James Joyce writes that the present is the drain that the future goes down on its way to becoming the past.

The three great branches of mathematics are, in historical order, Geometry, Algebra, and Analysis. Geometry we owe essentially to Greek civilization, Algebra is of Indo-Arab origin, and Analysis (or Calculus) was the creation of Newton and Leibniz, ushering in the modern era.

Many of the patterns of nature we can discover only after they have been constructed by our mind. The systematic construction of such new patterns is the business of mathematics. The role which geometry plays in this respect with regard to some visual patterns is merely the most familiar instance of this. The great strength of mathematics is that it enables us to describe abstract patterns which cannot be perceived by our senses, and to state the common properties of hierarchies or classes of patterns of a highly abstract character. Every algebraic equation or set of such equa- tions defines in this sense a class of patterns, with the individual manifestation of this kind of pattern being particularized as we substitute definite values for the variables.

Many of the patterns of nature we can discover only after they have been constructed by our mind. The systematic construction of such new patterns is the business of mathematics. The role which geometry plays in this respect with regard to some visual patterns is merely the most familiar instance of this. The great strength of mathematics is that it enables us to describe abstract patterns which cannot be perceived by our senses, and to state the common properties of hierarchies or classes of patterns of a highly abstract character. Every algebraic equation or set of such equations defines in this sense a class of patterns, with the individual manifestation of this kind of pattern being particularized as we substitute definite values for the variables.

I read once of David Hockney’s answering the question, Why do your shoeless figures always have socks on, by saying, I can’t draw feet.

The belief that mathematics exists somewhere else than within us, that it is discovered more than created, is called Platonism, after Plato’s belief in a non-spatiotemporal realm that was the region of the perfect forms of which the objects on earth were imperfect reproductions. By definition, the non-spatiotemporal realm is outside time and space. It is not the creation of any deity, it simply is. To say that it is eternal or that it has always existed is to make a temporal remark, which does not apply. It is the timeless nowhere which never has and never will exist anywhere but which nevertheless is. The physical world is temporal and declines, the non-spatiotemporal one is ideal and doesn’t.

Ibn Sina was born in a tiny settlement called Afshanah, outside the village of Kharmaythan, and soon after his birth his family moved to the nearby city of Bukhara. While he was still a small boy his father, a tax collector, arranged for him to study with a teacher of Qu’ran and a teacher of literature, and by the time he was ten he had memorized the entire Qu’ran and absorbed much of Muslim culture. His father met a learned vegetable peddler named Mahmud the Mathematician, who taught the child Indian calculation and algebra. Before the gifted youth grew his first facial hairs he had qualified in law and delved into Euclid and geometry, and his teachers begged his father to allow him to devote his life to scholarship. He began the study of medicine at eleven and by the time he was sixteen he was lecturing to older physicians and spending much of his time in the practice of law. All his life he would be both jurist and philosopher, but he noted that although these learned pursuits were given deference and respect by the Persian world in which he lived, nothing mattered more to an individual than his well-being and whether he would live or die. At an early age, fate made Ibn Sina the servant of a series of rulers who used his genius to guard their health, and though he wrote dozens of volumes on law and philosophy—enough to win him the affectionate sobriquet of Second Teacher (First Teacher being Mohammed)—it was as the Prince of Physicians that he gained the fame and adulation that followed him wherever he traveled. In Ispahan, where he had gone at

There! I think it's another zombie." Off in the distance, we could see someone, or something, approaching at the edge of the parking lot. "I can't tell, hold on. I've got a pair of binoculars." I leaned inside, where we'd piled our supplies. "Well?" asked Misty. I adjusted the focus. "It's a zombie." "Are you sure? He doesn't look dead—maybe just a little pale." "Look for yourself, you'll see." I handed them to her and spit in disgust. "I can't see his eyes. He might be alive." "You don't recognize him?" "No, do you?" "It's Mr. Lopez. Teaches—well, taught Math," I said, rubbing my forehead. "Oh my gosh, you're right. How could I have forgotten?" Mr. Lopez had died of a heart attack last week while trying to teach Geometry in summer school—probably enough to kill anyone. It had been in the paper and on the local news. "You had his class last year, didn't you?" "Yeah, Pre-Algebra, hated it. It's just wrong to teach kids algebra. Still, to see him standing there...

Modern mathematics contains much more than that, of course. It includes set theory, for example, created by Georg Cantor in 1874, and “foundations,” which another George, the Englishman George Boole, split off from classical logic in 1854, and in which the logical underpinnings of all mathematical ideas are studied. The traditional categories have also been enlarged to include big new topics—geometry to include topology, algebra to take in game theory, and so on. Even before the early nineteenth century there was considerable seepage from one area into another. Trigonometry, for example, (the word was first used in 1595) contains elements of both geometry and algebra. Descartes had in fact arithmetized and algebraized a large part of geometry in the seventeenth century, though pure-geometric demonstrations in the style of Euclid were still popular—and still are— for their clarity, elegance, and ingenuity.

He has translated Virgil’s Aeneid . . . the whole of Sallust and Tacitus’ Agricola . . . a great part of Horace, some of Ovid, and some of Caesar’s Commentaries . . . besides Tully’s [Cicero’s] Orations. . . . In Greek his progress has not been equal; yet he has studied morsels of Aristotle’s Politics, in Plutarch’s Lives, and Lucian’s Dialogues, The Choice of Hercules in Xenophon, and lately he has gone through several books in Homer’s Iliad. In mathematics I hope he will pass muster. In the course of the last year . . . I have spent my evenings with him. We went with some accuracy through the geometry in the Preceptor, the eight books of Simpson’s Euclid in Latin. . . . We went through plane geometry . . . algebra, and the decimal fractions, arithmetical and geometrical proportions. . . . I then attempted a sublime flight and endeavored to give him some idea of the differential method of calculations . . . [and] Sir Isaac Newton; but alas, it is thirty years since I thought of mathematics.

Among other elements, the test had a vestigial examination in drawing, and Mandelbrot discovered a latent facility for copying the Venus de Milo. On the mathematical sections of the test—exercises in formal algebra and integrated analysis—he managed to hide his lack of training with the help of his geometrical intuition. He had realized that, given an analytic problem, he could almost always think of it in terms of some shape in his mind. Given a shape, he could find ways of transforming it, altering its symmetries, making it more harmonious. Often his transformations led directly to a solution of the analogous problem. In physics and chemistry, where he could not apply geometry, he got poor grades. But in mathematics, questions he could never have answered using proper techniques melted away in the face of his manipulations of shapes.

Noncommutative geometry has turned up in several approaches to quantum gravity, including string theory, DSR, and loop quantum gravity. But none of these capture the depth of Conne's original conception, which he and a few mathematicians, mostly in France, continue to develop. The various versions of it that appear in other programs are based on superficial ideas, such as making the coordinates of space and time into noncommuting quantities. Conne's idea is much deeper; it is a unification at the foundations of algebra and geometry. It could only be the invention of someone who does not just exploit mathematics but thinks strategically and creatively about the structure of mathematical knowledge and its future.

The refusal to examine Islamic culture and traditions, the sordid dehumanization of Muslims, and the utter disregard for the intellectual traditions and culture of one of the world’s great civilizations are characteristic of those who disdain self-reflection and intellectual inquiry. Confronting this complexity requires work and study rather than a retreat into slogans and cliches. And enlightened, tolerant civilizations have flourished outside the orbit of the United Sates and Europe. The ruins of the ancient Mughal capital, Fatehpur Sikri, lie about 100 miles south of Delhi. The capital was constructed by the emperor Akbar the Great at the end of the sixteenth century. The emperor’s court was filled with philosophers, mystics and religious scholars, including Sunni, Sufi, and Shiite Muslims, Hindu followers of Shiva and Vishnu, as well as atheists, Christians, Jains, Jews , Buddhists and Zoroastrians. They debated ethics and beliefs. He forbade any person to be discriminated against on the basis of belief and declared that everyone was free to follow any religion. This took place as the Inquisition was at its height in Spain and Portugal, and as Giordano Bruno was being burnt at the stake in Rome’s Campo de Fiori. Tolerance, as well as religious and political plurality, is not exclusive to Western culture. The Judeo-Christian tradition was born and came to life in the Middle East. Its intellectual and religious beliefs were cultivated and formed in cities such as Jerusalem, Antioch, Alexandria and Constantinople. Many of the greatest tenets of Western civilization, as is true with Islam and Buddhism, are Eastern in origin. Our respect for the rule of law and freedom of expression, as well as printing, paper, the book, the translation and dissemination of the classical Greek philosophers, algebra, geometry and universities were given to us by the Islamic world. One of the first law codes was invented by the ancient Babylonian ruler Hammurabi, in what is now Iraq. One of the first known legal protections of basic freedoms and equality was promulgated in the third century B.C. by the Buddhist Indian emperor Ashoka. And, unlike, Aristotle, he insisted on equal rights for women and slaves. The division set up by the new atheists between superior Western, rational values and the irrational beliefs of those outside our tradition is not only unhistorical but untrue. The East and the West do not have separate, competing value systems. We do not treat life with greater sanctity than those we belittle and dismiss. Eastern and Western traditions have within them varied ethical systems, some of which are repugnant and some of which are worth emulating. To hold up the highest ideals of our own culture and to deny that these great ideals exist in other cultures, especially Eastern cultures, is made possible only by a staggering historical and cultural illiteracy. The civilization we champion and promote as superior is, in fact, a product of the fusion of traditions and beliefs of the Orient and the Occident. We advance morally and intellectually only when we cross these cultural lines, when we use the lens of other cultures to examine our own. It is then that we see our limitations, that we uncover the folly of or own assumptions and our prejudices. It is then that we achieve empathy, we learn and make wisdom possible.

Life isn’t about algebra and geometry. Learning by making mistakes and not duplicating them is what life is about.

Mathematics is not just a subject of education system, it is the soul of education system.

Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics.

In 1958, Fortune singled Nash out for his achievements in game theory, algebraic geometry, and nonlinear theory,

Descartes was responsible for analytical geometry, a mechanism for translating from geometrical forms to the equivalent algebraic equations and vice versa.