Abstract Algebra Quotes

I deliberately and consciously give preference to a dramatic, mythological way of thinking and speaking, because this is not only more expressive but also more exact than an abstract scientific terminology, which is wont to toy with the notion that its theoretic formulations may one fine day be resolved into algebraic equations.

The neurons that do expire are the ones that made imitation possible. When you are capable of skillful imitation, the sweep of choices before you is too large; but when your brain loses its spare capacity, and along with it some agility, some joy in winging it, and the ambition to do things that don't suit it, then you finally have to settle down to do well the few things that your brain really can do well--the rest no longer seems pressing and distracting, because it is now permanently out of reach. The feeling that you are stupider than you were is what finally interests you in the really complex subjects of life: in change, in experience, in the ways other people have adjusted to disappointment and narrowed ability. You realize that you are no prodigy, your shoulders relax, and you begin to look around you, seeing local color unrivaled by blue glows of algebra and abstraction.

I do not deny the power and the beauty of reductionist science, as exemplified in the axioms and theorems of abstract algebra....But I assert the equal power and beauty of constructive science as exemplified in Godel's construction of an undecidable proposition....

There had always been the rumour that one of the old heptarchs had squirreled away a collection of heretical calendrical erotica. Just how you made abstract algebra erotic was going to have to remain a mystery.

The language of categories is affectionately known as "abstract nonsense," so named by Norman Steenrod. This term is essentially accurate and not necessarily derogatory: categories refer to "nonsense" in the sense that they are all about the "structure," and not about the "meaning," of what they represent.

The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations.

The feeling that you are stupider than you were is what finally interests you in the really complex subjects of life: in change, in experience, in the ways other people have adjusted to disappointment and narrowed ability. You realize that you are no prodigy, your shoulders relax, and you begin to look around you, seeing local color unrivaled by blue glows of algebra and abstraction.

Visual thinkers, on the other hand, see images in their mind’s eye that allow them to make rapid-fire associations. Generally, visual thinkers like maps, art, and mazes, and often don’t need directions at all. Some visual thinkers can easily locate a place they’ve been to only once, their internal GPS having logged the visual landmarks. Visual thinkers tend to be late talkers who struggle with school and traditional teaching methods. Algebra is often their undoing, because the concepts are too abstract, with little or nothing concrete to visualize. Visual thinkers tend to be good at arithmetic that is directly related to practical

The Minds did not assume such distinctions; to them, there was no cutoff between the two. Tactics cohered into strategy, strategy disintegrated into tactics, in the sliding scale of their dialectical moral algebra. It was all more than they ever expected the mammal brain to cope with. He recalled what Sma had said to him, long long ago back in that new beginning (itself the product of so much guilt and pain); that they dealt in the intrinsically untoward, where rules were forged as you went along and were never the same twice anyway, where just by the nature of things nothing could be known or predicted or even judged with any real certainty. It all sounded very sophisticated and abstract and challenging to work with, but in the end it came down to people and problems.

Tactics cohered into strategy, strategy disintegrated into tactics, in the sliding scale of their dialectical moral algebra. It was all more than they ever expected the mammal brain to cope with. He recalled what Sma had said to him, long long ago back in that new beginning (itself the product of so much guilt and pain); that they dealt in the intrinsically untoward, where rules were forged as you went along and were never the same twice anyway, where just by the nature of things nothing could be known or predicted or even judged with any real certainty. It all sounded very sophisticated and abstract and challenging to work with, but in the end it came down to people and problems.

Entirely my own opinion,” said Ivanov. “I am glad that we have reached the heart of the matter soon. In other words: you are convinced that “we” – that is to say, the Party, the State and the masses behind it – no longer represent the interests of the Revolution.” “I should leave the masses out of it,” said Rubashov. […] “Leave the masses out of it, “ he repeated. “You understand nothing about them. Nor, probably, do I any more. Once, when the great “we” still existed, we understood them as no one had ever understood them before. We had penetrated into their depths, we worked in the amorphous raw material of history itself…” […] “At that time,” Rubashov went on, “we were called the Party of the Plebs. What did the others know of history? Passing ripples, little eddies and breaking waves. They wondered at the changing forms of the surface and could not explain them. But we had descended into the depths, into the formless, anonymous masses, which at all times constituted the substance of history; and we were the first to discover her laws of motion. We had discovered the laws of her inertia, of the slow changing of her molecular structure, and of her sudden eruptions. That was the greatness of our doctrine. The Jacobins were moralists; we were empirics. We dug in the primeval mud of history and there we found her laws. We knew more than ever men have known about mankind; that is why our revolution succeeded. And now you have buried it all again….” […] “Well,” said Rubashov, “one more makes no difference. Everything is buried: the men, their wisdom and their hopes. You killed the “We”; you destroyed it. Do you really maintain that the masses are still behind you? Other usurpers in Europe pretend the same thing with as much right as you….” […] “Forgive my pompousness,” he went on, “but do you really believe the people are still behind you? It bears you, dumb and resigned, as it bears others in other countries, but there is no response in their depths. The masses have become deaf and dumb again, the great silent x of history, indifferent as the sea carrying the ships. Every passing light is reflected on its surface, but underneath is darkness and silence. A long time ago we stirred up the depths, but that is over. In other words” – he paused and put on his pince-nez – “in those days we made history; now you make politics. That’s the whole difference.” […] "A mathematician once said that algebra was the science for lazy people - one does not work out x, but operates with it as if one knew it. In our case, x stands for the anonymous masses, the people. Politics mean operating with this x without worrying about its actual nature. Making history is to recognize x for what it stands for in the equation." "Pretty," said Ivanov. "But unfortunately rather abstract. To return to more tangible things: you mean, therefore, that "We" - namely, Party and State - no longer represent the interests of the Revolution, of the masses or, if you like, the progress of humanity." "This time you have grasped it," said Rubashov smiling. Ivanov did not answer his smile.

In the history of science it happens not infrequently that a reductionist approach leads to a spectacular success. Frequently the understanding of a complicated system as a whole is impossible without an understanding of its component parts. And sometimes the understanding of a whole field of science is suddenly advanced by the discover of a single basic equation. Thus it happened that the Schrodinger equation in 1926 and the Dirac equation in 1927 brought a miraculous order into the previously mysterious processes of atomic physics. The equations of Erwin Schrodinger and Paul Dirac were triumphs of reductionism. Bewildering complexities of chemistry and physics were reduced to two lines of algebraic symbols. These triumphs were in Oppenheimer's mind when he belittled his own discovery of black holes. Compared with the abstract beauty and simplicity of the Dirac equation, the black hole solution seemed to him ugly, complicated, and lacking in fundamental significance.

What this means is that the (Infinity) of points involved in continuity is greater than the (Infinity) of points comprised by any kind of discrete sequence, even an infinitely dense one. (2) Via his Diagonal Proof that c is greater than Aleph0, Cantor has succeeded in characterizing arithmetical continuity entirely in terms of order, sets, denumerability, etc. That is, he has characterized it 100% abstractly, without reference to time, motion, streets, noses, pies, or any other feature of the physical world-which is why Russell credits him with 'definitively solving' the deep problems behind the dichotomy. (3) The D.P. also explains, with respect to Dr. G.'s demonstration back in Section 2e, why there will always be more real numbers than red hankies. And it helps us understand why rational numbers ultimately take up 0 space on the Real Line, since it's obviously the irrational numbers that make the set of all reals nondenumerable. (4) An extension of Cantor's proof helps confirm J. Liouville's 1851 proof that there are an infinite number of transcendental irrationals in any interval on the Real Line. (This is pretty interesting. You'll recall from Section 3a FN 15 that of the two types of irrationals, transcendentals are the ones like pi and e that can't be the roots of integer-coefficient polynomials. Cantor's proof that the reals' (Infinity) outweighs the rationals' (Infinity) can be modified to show that it's actually the transcendental irrationals that are nondenumerable and that the set of all algebraic irrationals has the same cardinality as the rationals, which establishes that it's ultimately the transcendetnal-irrational-reals that account for the R.L.'s continuity.)

To wit, researchers recruited a large group of college students for a seven-day study. The participants were assigned to one of three experimental conditions. On day 1, all the participants learned a novel, artificial grammar, rather like learning a new computer coding language or a new form of algebra. It was just the type of memory task that REM sleep is known to promote. Everyone learned the new material to a high degree of proficiency on that first day—around 90 percent accuracy. Then, a week later, the participants were tested to see how much of that information had been solidified by the six nights of intervening sleep. What distinguished the three groups was the type of sleep they had. In the first group—the control condition—participants were allowed to sleep naturally and fully for all intervening nights. In the second group, the experimenters got the students a little drunk just before bed on the first night after daytime learning. They loaded up the participants with two to three shots of vodka mixed with orange juice, standardizing the specific blood alcohol amount on the basis of gender and body weight. In the third group, they allowed the participants to sleep naturally on the first and even the second night after learning, and then got them similarly drunk before bed on night 3. Note that all three groups learned the material on day 1 while sober, and were tested while sober on day 7. This way, any difference in memory among the three groups could not be explained by the direct effects of alcohol on memory formation or later recall, but must be due to the disruption of the memory facilitation that occurred in between. On day 7, participants in the control condition remembered everything they had originally learned, even showing an enhancement of abstraction and retention of knowledge relative to initial levels of learning, just as we’d expect from good sleep. In contrast, those who had their sleep laced with alcohol on the first night after learning suffered what can conservatively be described as partial amnesia seven days later, forgetting more than 50 percent of all that original knowledge. This fits well with evidence we discussed earlier: that of the brain’s non-negotiable requirement for sleep the first night after learning for the purposes of memory processing. The real surprise came in the results of the third group of participants. Despite getting two full nights of natural sleep after initial learning, having their sleep doused with alcohol on the third night still resulted in almost the same degree of amnesia—40 percent of the knowledge they had worked so hard to establish on day 1 was forgotten.

seemed almost certain to the mathematicians that since the general first, second, third, and fourth degree equations can be solved by means of the usual algebraic operations such as addition, subtraction, and roots, then the general fifth degree equation and still higher degree equations could also be solved. For three hundred years this problem was a classic one. Hundreds of mature and expert mathematicians sought the solution, but a little boy found the full answer. The Frenchman Évariste Galois (1811— 1832), who refused to conform to school examinations but worked brilliantly and furiously on his own, showed that general equations of degree higher than the fourth cannot be solved by algebraic operations. To establish this result Galois created the theory of groups, a subject that is now at the base of modern abstract algebra and that transformed algebra from a a series of elementary techniques to a broad, abstract, and basic branch of mathematics.

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.

Walking on these streets, until the night falls, my life feels to me like the life they have. By day they’re full of meaningless activity; by night they’re full of a meaningless lack of it. By day I am nothing, and by night I am I. There is no difference between me and these streets, save they being streets and I a soul, which perhaps is irrelevant when we consider the essence of things. There is an equal, abstract destiny for men and for things; both have an equally indifferent designation in the algebra of the world’s mystery.

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!

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

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.

By convention, I have been taught to isolate the variables on the left side and the constants on the right But you still have that characteristic freewill of wallowing in the abstraction of algebra You could also isolate the variables on the right side and the constants on the left.

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.

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.

Men and objects share a common abstract destiny: to be of equally insignificant value in the algebra of life’s mystery.

In our own time, algebra has become the most rarefied and demanding of all mental disciplines, whose objects are abstractions of abstractions of abstractions, yet whose results have a power and beauty that are all too little known outside the world of professional mathematicians. Most amazing, most mysterious of all, these ethereal mental objects seem to contain, within their nested abstractions, the deepest, most fundamental secrets of the physical world.

Young people need looking after,” she said. “Think of that beautiful boy Galois. People felt there was something secret in his character. They were right. The secret was mathematics. His father a suicide. His own death a horrible farce. Dawn in the fields. Caped and whiskered seconds. Sinister marksman poised to fire.” I need all my courage to die at twenty. “Then there was Abel, not much older, desperately poor, Abel in delirium, hemorrhaging. So often mathematical experience consists of time segments too massive to be contained in the usual frame. Lives overstated. Themes pursued to extreme points. Adventure, romance and tragedy.” I will fight for my life. “Look at Pascal, who rid himself of physical pain by dwelling on mathematics. He was just a bit older than you when he constructed his mystic hexagram. The loveliest aspect of the mystic hexagram is that it is mystic. That’s what’s so lovely about it. It’s able to become its own shadow.” Keep believing it. “The tricky thing about mathematical genius,” she said, “is that its sources are so often buried. Galois for one. Ramanujan for another. No indication anywhere in their backgrounds that these boys would one day display such natural powers. Figures jumping out of sequence. Or completely misplaced.” (...) “Numbers have supernatural harmonies, according to Hermite. They exist beyond human thought. Divine order through number. Number as absolute reality. Someone said of Hermite: ‘The most abstract entities are for him like living creatures.’ That’s what someone said.” “People invented numbers,” he said. “You don’t have numbers without people.” “Good, let’s argue.” “I don’t want to argue.” “Secret lives,” she said. “Dedekind listed as dead twelve years before the fact. Poncelet scratching calculations on the walls of his cell. Lobachevski mopping the floors of an old museum. Sophie Germain using a man’s name. Do I have the order right? Sometimes I get it mixed up or completely backwards. (...) “Tell me about your mathematical dreams.” “Never had one.” “Cardano did, born half dead, his inner life a neon web of treachery and magic. Gambler, astrologer, heretic, court physician. Schemed his way through the algebra wars.” “Can I see the baby?” “Ramanujan had algebraic dreams. Wrote down the results after getting out of bed. Vast intuitive powers but poor education. Taken to Cambridge like a jungle boy. Sonja Kowalewski wasn’t allowed to attend university lectures. We both know why. When her husband died she spent days and days without food, coming out of her room only after she’d restored herself by working on her mathematics. Tell me, was it Kronecker who thought mathematics similar to poetry? I know Hamilton and many others tried their hands at verse. Our superduper Sonja preferred the novel.

Instead of the ordinary numbers used in conventional mathematics, the quantum nature of the fields requires the use of Hilbert algebra, in which the physical properties are described by "vectors" in an abstract Hilbert space and by "operators" that act on those vectors.

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.

What is important though, is that these catastrophes can be described inde- pendently of the physical substrate that may instantiate them: “One of the basic postulates of my model is that there are coherent systems of catastrophes (chre- ods) organized in archetypes and that these structures exist as abstract algebraic entities independent of any substrate.”⁵³ This allows Thom to make some surprising claims for his studies in morpho- genesis. He insists that we must “accept the idea that a sequence of stable trans- formations of our space-time could be directed or programmed by an organizing center consisting of an algebraic structure outside space-time itself.”⁵⁴

According to Thom “coherent systems of catastrophes” (i.e., chreods) con- stellate into structures that become “abstract algebraic entities independent of any substrate.” He furthermore posits that the transformations of our space-time may be directed or “programed” by another “algebraic structure” that is itself outside our space-time entirely (refer to quotation above).