algebra

"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

"But mathematicians know that the number system we study in school is but one of many possibilities. And indeed, other kinds of numbers are important for understanding geometry and physics. Among the strangest alternatives is the octonions. Largely neglected since their discovery in 1843, in the last few decades they have assumed a curious importance in string theory. And indeed, if string theory is a correct representation of the universe, they may explain why the universe has the number of dimensions it does."

"Despite its counter-culture status, the octonions have long drawn the curiosity of generations of physicists. The algebra is known to appear without warning in apparently disparate areas of mathematics, within algebra, geometry, and topology. However, despite its ubiquity, its practical uses in physics have remained elusive, due to the algebra's non-associativity, which must be handled with care."

"Besides their possible role in physics, the octonions are important because they tie together some algebraic structures that otherwise appear as isolated and inexplicable exceptions."

"It is possible to form an analogous theory with seven imaginary roots of (-1)."

"There is still something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties. ... If with your alchemy you can make three pounds of gold, why should you stop there?"

"It is nearly irresistible to ask if the octonions, \mathbb O, the last of the set of four normed division algebras over \mathbb R, have a calling in nature. Certainly several have thought so, but for the most part, the octonions have remained as a well kept secret from mainstream physics. More often than not, the octonions are passed by in haste because they are non-associative, and hence at times temperamental. As we will show, this property is in fact a gift, which will offer a way to streamline some of the standard model’s complex structure."

"But perhaps most important, it wasn’t clear in Hamilton’s time just what the octonions would be good for. They are closely related to the geometry of 7 and 8 dimensions, and we can describe rotations in those dimensions using the multiplication of octonions. But for over a century that was a purely intellectual exercise. It would take the development of modern particle physics—and string theory in particular—to see how the octonions might be useful in the real world."

"Octonions are to physics what the Sirens were to ."

"The octonions are much stranger. Not only are they noncommutative, they are also break another familiar law of arithmetic: the associative law (xy)z = x(yz)."

"In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."

"The historical associations of the word algebra almost substantiate the sordid character of the subject. The word comes from the title of a book written by... Al Khawarizmi. In this title, al-jebr w' almuqabala, the word al-jebr meant transposing a quantity from one side of an equation to another and muqabala meant simplification of the resulting expressions. Figuratively, al-jebr meant restoring the balance of an equation... When the Moors reached Spain... algebrista... came to mean a bonesetter... and signs reading Algebrista y Sangrador (bonesetter and bloodletter) were found over Spanish barber shops. Thus it might be said that there is a good historical basis for the fact that the word algebra stirs up disagreeable thoughts."

"The Study of Algebra may be pursued in three very different schools, the Practical, the Philological, or the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contemplation; according as ease of operation, or symmetry of expression, or clearness of thought, (the agere, the fari, or the sapere,) is eminently prized and sought for."

"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 marvellous machine.'"

"[T]he sciences that are expressed by numbers or by other small signs, are easily learned; and... this facility rather than its demonstrability is what has made the fortune of algebra."

"The motives which give rise to the use of alphabetic letters as symbols of number in preference to any other system of symbols, arbitrarily selected for the same purpose, are principally the following. First, As they have no numerical signification in themselves, they are subject to no ambiguity, having in reference to numbers no other signification than they are defined to have in the outset of each problem… Secondly, Being familiar to the eye, the tongue, the hand, and the mind, that is, having a well-known form and name, they are easily read, written, spoken, remembered, and discriminated from one another, which could not be the case were they mere arbitrary marks, formed according to the caprice of each individual who used them. Thirdly, The order in which the letters are arranged in the alphabet, facilitates the classification of them into groups much more easy to survey and comprehend in the expressions which arise…, and thereby renders the investigator much less likely to omit any of them by an imperfect enumeration."

"It has seemed to me for a long time that commutative algebra is best practiced with knowledge of the geometric ideas that played a great role in its formation: in short, with a view toward algebraic geometry."

"It has been said that Poincaré did not invent topology, but that he gave it wings. This is surely true, and verges on understatement. His six great topological papers created, almost out of nothing, the field of algebraic topology."

"At the end of the 1930s algebraic topology had amassed a stock of problems which its then available tools were unable to attack. Sammy was prominent among a small group of mathematicians—among them, for example, J. H. C. Whitehead, Hassler Whitney, Saunders Mac Lane, and Norman Steenrod—who dedicated themselves to building a more adequate armamentarium. Their success in doing this was attested to by the fact that by the end of the 1960s most of those problems had been solved (inordinately many of them by J. F. Adams)."

"Group theory is the mathematical expression of symmetry, providing appropriate mathematical tools for the implementation of symmetry considerations in theoretical physics."

"The introduction of the digit 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps...-Alexander Grothendieck"

"Arithmetic automorphic representation theory is one of the most active areas in current mathematical research, centering around what is known as the Langlands program."

"Abel did not deny that we might solve quintics using techniques other than algebraic ones of adding, subtracting, multiplying, dividing, and extracting roots. ...the general quintic can be solved by introducing... "elliptic functions," but these require operations considerably more complicated than those of elementary algebra. In addition, Abel's result did not preclude our approximating solutions... as accurately as we... wish. What Abel did do was prove that there exists no algebraic formula... The analogue of the quadratic formula for second-degree equations and Cardano's formula for cubics simply does not exist... This situation is reminiscent of that encountered when trying to square the circle, for in both cases mathematicians are limited by the tools they can employ. ...the restriction to "solution by radicals"... hampers mathematicians... what Abel actually demonstrated was that algebra does have... limits, and for no obvious reason, these limits appear precisely as we move from the fourth to the fifth degree."

"The following Treatise... has been endeavoured to make the theory of limits, or ultimate ratios... the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit..."

"The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition."

"Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions."

"By the help of God and with His precious assistance, I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to "things" which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of Algebra as stated above. The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns."

"I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him."

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of Elements."

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making."

"The geometrical algebra of the Greeks has been in evidence all through our history from Pythagoras downwards, and no more need be said of it here except that its arithmetical application was no new thing to Diophantus. It is probable, for example, that the solution of the quadratic equation, discovered first by geometry, was applied for the purpose of finding numerical values for the unknown as early as Euclid, if not earlier still. In Heron the numerical solution of equations is well established, so that Diophantus was not the first to treat equations algebraically. What he did was to take a step forward towards an algebraic notation."

"My specific... object has been to contain, within the prescribed limits, the whole of the student's course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathematical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers. or of any one of our English mathematicians, under the idea that it is too hard for him."

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"With the help of books only he Wilhelm Xylander] studied the subject of Algebra, as far as was possible from what men like Cardan had written and by his own reflection, with such success that not only did he fall into what Herakleitos called... the conceit of "being somebody" in the field of Arithmetic and "Logistic," but others too who were themselves learned men thought him an arithmetician of exceptional merit. But when he first became acquainted with the problems of Diophantos his pride had a fall so sudden and so humiliating that he might reasonably doubt whether he ought previously to have bewailed, or laughed at himself. He considers it therefore worth while to confess publicly in how disgraceful a condition of ignorance he had previously been content to live, and to do something to make known the work of Diophantos, which had so opened his eyes."

"al-Khwārizmī “not having taken algebra from the Greeks,. . . must have either invented it himself, or taken it from the Indians. Of the two, the second appears to me the most probable”"

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions."

"When the relation between all points of a curve and all points of a straight line are known, in the way I have already explained, it is easy to find the relation between the points of the curve and all other given points and lines; and from these relations to find its diameters, axes, center, and other lines or points which have especial significance for this curve, and to choose the easiest. By this method alone it is then possible to find out all that can be determined about the magnitude of their areas, and there is no need for further explanation from me."

"Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since."

"The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention... And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning... It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimensional: extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might (no doubt) be represented by points upon a line, yet I thought that their simple successiveness was better conceived by comparing them with moments of time, divested, however, of all reference to cause and effect; so that the "time" here considered might be said to be abstract, ideal, or pure, like that "space" which is the object of geometry. In this manner I was led, many years ago, to regard Algebra as the Science of Pure Time: and an Essay, containing my views respecting it as such, was published in 1835. ...[I]f the letters A and B were employed as dates, to denote any two moments of time, which might or might not be distinct, the case of the coincidence or identity of these two moments, or of equivalence of these two dates, was denoted by the equation,B = Awhich symbolic assertion was thus interpreted as not involving any original reference to quantity, nor as expressing the result of any comparison between two durations as measured. It corresponded to the conception of simultaneity or synchronism; or, in simpler words, it represented the thought of the present in time. Of all possible answers to the general question, "When," the simplest is the answer, "Now:" and it was the attitude of mind, assumed in the making of this answer, which (in the system here described) might be said to be originally symbolized by the equation above written."

"Admitting the Hindu and Alexandrian authors [such as Diophantus], to be nearly equally ancient, it must be conceded in favor of the Indian algebraist, that he was more advanced in the science […] In the whole science [of algebra], he [Diophantus] is very far behind the Hindu writers […] he is hardly to be considered as the inventor, since he seems to treat the art as already known."

"This principle, which is thus made the foundation of the operations and results of Symbolical Algebra, has been called "The principle of the permanence of equivalent forms", and may be stated as follows: "Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form.""

"In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a - b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and \frac{a}{b} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science."

"All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations."

"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."

"You can ask the question about these ancient topics, such as s and ... and ask, are these good problems... I'd like to give a small amount of evidence... that they are... [S]tudying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra.."

"No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hid, the ars rei et census [the art of the evaluation of wealth or tax] which to-day they call by the Arabic name of Algebra."

"That he [Al-Khwarizmi] should have borrowed from Diophantus is not at all probable; … It is far more probable that the Arabs received their first knowledge of algebra from the Hindus, who furnished them with the decimal notation of numerals, and with various important points of mathematical and astronomical information."

"Ninety per cent of all the mathematics we know has been discovered (or invented), in the last hundred years... the advances made in each of some dozen directions are converging into one single discipline uniting algebra, topology and analysis."

"Although I wish the present work to be regarded principally as a history, yet there are two other aspects... It may claim the title of a comprehensive treatise on the Theory of Probability, for it assumes in the reader only so much know much knowledge as can be gained from an elementary book on Algebra, and introduces him to almost every process and every species of problem which the literature of the subject can furnish; or the work can be considered more specially as a commentary on the celebrated treatise of Laplace,—and perhaps no mathematical treatise ever more required or more deserved such an accomplishment."

"In … a series of lectures at the University of Padua in 1464, he [Regiomontanus] introduced the idea that Arabic algebra descended from Diophantus’s Arithmetica. This heralded the initiation of a myth cultivated by humanists for centuries. Diophantus … became the alleged origin of European algebra. … By overrating the importance of Diophantus … humanist writers created a new mythical identity of European mathematics."